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1NE0T4oBgHgl3EQf_gL8/content/tmp_files/2301.02829v1.pdf.txt

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+Dark Matter and MOND: Two sides of the same coin?
+D. F. Roscoe (The Open University; [email protected])
+ORCID: 0000-0003-3561-7425
+1
+arXiv:2301.02829v1  [astro-ph.GA]  7 Jan 2023
+
+Abstract
+It has recently been reported that the application of convolutional neural-network tech-
+niques to infer the dark-matter distribution in the local cosmos has revealed how it follows
+the D ≈ 2 hierarchical distribution of galaxies in the locality, rather than exhibiting the
+expected homogeneity throughout the IGM. Taken at face value, this implies that the Hub-
+ble Law, observed to be followed on scales which are deep inside the observed hierarchical
+structures, can no longer be assumed to arise from universal expansion. So, if not universal
+expansion, then what?
+As a possibility, it has been recognized for a considerable time that if the lower cut-off
+scales of a D ≈ 2 hierarchical cosmos are identified with the scales of a typical galaxy, then
+gravitational redshift automatically follows the Hubble Law with Hg ≈ 70 km/sec/Mpc.
+Inter alia, this suggests a model of galaxy formation in a D ≈ 2 hierarchical IGM in which
+all of the material M0 within a sphere R0 coalesces about a unique center so that hierachical
+symmetry is broken on the scale (M0, R0).
+Putting these things together leads unambiguously to the conclusion that, in an hierachical
+cosmos, the Dark Matter hypothesis and Milgrom’s MOND hypothesis are two sides of the
+same coin.
+2
+
+1
+Introduction:
+It is now widely accepted that on scales up to about 200 Mpc galaxies are distributed in a quasi-
+fractal D ≈ 2 fashion. For fairly recent work see Tekhanovich & Baryshev (2016), but many
+others have contributed over the years.
+It then becomes a point of considerable significance that the Hubble Law is well established
+on scales that are deep inside the accepted fractal structure of the general galaxy distribution.
+This very interesting circumstance is the primary evidence supporting the idea that the IGM is
+largely populated by an homogeneous distribution of dark matter on the small scales required
+- for, without homogeneity, the linear nature of Hubble’s Law cannot be understood within the
+context of universal expansion.
+It is for this reason that the paper of Hong et al (2021) caused so much consternation: specifically,
+the authors used state-of-the-art convolutional neural-network techniques combined with modern
+positional and peculiar velocity data to compute and map the local dark matter distribution.
+Against expectation, this distribution is found to trace the hierarchical distribution of galaxies
+very closely - there is no indication of homogeneity, and hence no indication that the Hubble
+Law can be understood in terms of universal expansion.
+The only immediately plausible alternative is some form of gravitational redshift: Baryshev
+et al (1998) point out that inside a D = 2 hierarchical galaxy distribution (with an assumed
+homogeneous distribution of dark matter) the gravitational part of redshift is also purely linear
+with distance and cannot be distinguished from the expansion component. But if the results of
+Hong et al (2021) are to be taken at face value, then any contribution to redshift from expansion
+must manifest itself as a departure from linearity. Since such a departure is not observed then,
+according to the results of Hong et al, there can be no expansion effect at all.
+This line of argument is reinforced by the further observation of Baryshev et al that if the
+lower cut-off mass and length scales of the hierarchy are identified with the mass and length
+scales of the typical galaxy, then a gravitational redshift of Hg ≈ 70 km/sec/Mpc is to be ex-
+pected.
+Inter alia, the foregoing considerations suggest a process of galaxy formation according to which
+an isolated galactic object can be modelled as a finite bounded spherically symmetric peturbation
+of the hierarchical IGM (assumed in the first instance to be a mix of baryonic and non-baryonic
+mass) - this automatically entails that all of the mass M0 within the sphere R0 has coalesced
+around a unique centre so that fractal symmetry is broken on the scales of (M0, R0).
+2
+Consequences on the lower cut-off scales:
+From these general considerations we may conclude:
+1. The lower cut-off radial and mass scales (M0, R0) must behave according to
+M0 = 4πR2
+0ΣF
+(1)
+3
+
+where ΣF is the mass surface density of the D = 2 hierarchical mass distribution in the
+local cosmos;
+2. Since galaxies in general appear to be stable structures, there must be an equilibrium
+constraint at the lower cut-off scales of the hierarchy. Using simple Newtonian arguments,
+we show in appendix §A that equilibrium at these lower cut-off scales requires:
+V 2
+0
+R0
+= aF ≡ 4πGΣF
+(2)
+where aF is the characteristic acceleration scale associated with ΣF;
+3. The relationship
+V 4
+0 = aF GM0 ,
+(3)
+which is formally identical to the Baryonic Tully-Fisher Relationship (BTFR), is now de-
+rived directly by eliminating R0 between (1) and (2).
+It is to be noted that whilst (3) is formally identical to the BTFR of Milgrom’s MOND, it differs
+fundamentally in the assumption (expressed in the last paragraph of §1) that M0 is an unknown
+mix of baryonic and non-baryonic mass whereas, by definition, the BTFR asserts that this mass
+is purely baryonic.
+3
+Empirical support for the BTFR hypothesis
+3.1
+The analysis of Lelli, McGaugh & Schombert (2016B)
+It has only recently been possible to explore the BTFR hypothesis in a statistically rigorous
+fashion. Specifically, the SPARC sample of Lelli, McGaugh & Schombert (2016A) contains high
+quality rotation curves and high quality modern surface photometry at 3.6 µm for a sample of 175
+nearby disk galaxies. The high quality of the surface photometry over this sample allowed Lelli,
+McGaugh & Schombert (2016B) to construct photometric models of baryonic mass distributions
+in that particular subsample of 118 disks which also had rotation curves extending to flatness,
+making it ideal for a statistically rigorous testing of the BTFR hypothesis.
+Subsequently, the authors used regression analysis techniques to demonstrate how the subsample
+really does fit the BTFR with very small scatter. In this way, they argued that the observed
+scatter is sufficiently below the instrinsic-scatter expectations of ΛCDM cosmology to present a
+fundamental difficulty for that cosmology and for the associated idea of dynamically significant
+quantities of non-baryonic matter in the generality of galaxy disks.
+Since (3) is derived from the hypothesis that galaxies form by coalescing in a stable way out
+of the D = 2 hierarchical IGM, this result implies that the IGM itself consists primarily of
+undetected baryonic matter and so, in effect, (3) itself represents a derivation of the hitherto
+empirical BTFR from a fundamental theoretical position.
+4
+
+3.2
+The estimation of (aF, ΣF)
+The data used by Lelli, McGaugh & Schombert (2016B) is available as an on-line data-sheet giv-
+ing estimates for the photometrically modelled baryonic masses M0 and flat rotation velocities
+V0 for the 118 disk galaxies. Given this data, an alternative demonstration supporting the BTFR
+hypothesis is provided by showing how the hypothesis, applied differently to the data, yields a
+very sharp estimate of the characteristic acceleration parameter aF, thereby demonstrating how,
+for all practical purposes, its value is identical to that of Milgrom’s critical acceleration parame-
+ter, a0.
+In order to estimate aF ≡ 4πGΣF from this data, we rearrange (3) as
+V 4
+0
+GM0
+= 4πGΣF ≡ constant
+and hence form the empirical sample distribution
+J ≡
+� V 4
+0i
+GM0i
+, i = 1...118
+�
+.
+Then, from J, we generate N = 10000 bootstrapped distributions, ˆJi, i = 1..N in the usual way.
+For each ˆJi we then compute its geometric mean, ˆaFi say, to obtain, finally, the distribution
+AF ≡ (log ˆaFi, i = 1..N) .
+The density distribution of AF is given in figure 1 from which it is clear that the estimate for
+aF is very tightly constrained around the modal value of 1.3 × 10−10 mtrs/sec2 which, for all
+practical purposes, is identical to Milgrom’s value of MOND’s critical acceleration parameter
+a0. This estimate of aF corresponds to ΣF ≈ 0.15 kg/mtr2 for the mass surface density of the
+hierarchical cosmos.
+The grey curve in figure 1 arises from the same analysis but applied to shuffled velocity and
+mass data. It is clear that the signal so powerfully present in the unshuffled data is destroyed
+by shuffling. We conclude that, for all practical purposes the signal for the mass surface density
+ΣF ≈ 0.15 kg/mtr2 in the hierarchical cosmos is real.
+5
+
+Dotted = 1.1 × 10−10 mtrs/sec2
+Mode = 1.3 × 10−10 mtrs/sec2
+Dashed = 1.6 × 10−10 mtrs/sec2
+0
+5
+10
+15
+log(aF)
+Density
+Distribution of bootstrapped geometric means
+Figure 1: Solid black curve = density distribution of log ˆaF. Solid grey curve arises when velocity
+and mass data are shuffled with respect to each other. The signal represented by the black curve
+is destroyed on the shuffled data.
+4
+Full circle to MOND
+Taking the results of Hong et al (2021) at face value, together with the results of Lelli, McGaugh
+& Schombert (2016B) and observations of Baryshev et al (1998), it has been shown that the Dark
+Matter of modern astrophysics, rather than being an homogeneous distribution of non-baryonic
+matter, can reasonably be identified as a D = 2 hierarchical distribution of undetected baryonic
+matter which, as the BTFR (3) shows, provides exactly the dynamical support for galaxies that
+the original non-baryonic Dark Matter hypothesis was formulated for in the first place.
+6
+
+However, hitherto the prominence of the originally empirically derived BTFR has rested en-
+tirely upon the fact that it is central to the architecture of MOND Milgrom (1983a,b,c). To see
+this, (3) gives directly
+V 2
+0
+R0
+=
+√aF GM0
+R0
+which, in effect, MOND extrapolates by hypothesis to give g = √aF gN, where gN ≡ GM0/R2, as
+the effective gravitational force for all R ≥ R0 . Consequently, MOND automatically receives the
+direct interpretation as a first-order descriptor of gravitational dynamics in a D = 2 hierarchical
+IGM.
+In short, in an hierachical cosmos, the Dark Matter hypothesis and Milgrom’s MOND hypothesis
+are two sides of the same coin.
+5
+The nature of baryonic Dark Matter?
+We have argued that the Dark Matter of the IGM is distributed in a D = 2 hierarchy and consists
+of undetected baryonic material. So, the immediate question is: how can such a distribution of
+baryonic material remain undetected? There are three potential strands to the answer, easily
+conceived to be acting in concert.
+5.1
+Conventional possibilities
+Given that the IGM forms a D = 2 hierarchy, then its volume density in a spherical volume of
+radius R tends to zero as R → ∞, making its detection at large radii intrinsically difficult.
+Furthermore, it can reasonably be assumed that the IGM is at least close to being in thermal
+equilibrium with the general background, again making its detection against the background also
+intrinsically difficult.
+5.2
+Unconventional possibilities
+Until recently, it has always been assumed that there is no such thing in nature as a perfect, or
+near perfect, blackbody absorber - the reason being that no such thing had ever been observed.
+However, Mizuno et al (2009) showed how to fabricate, from agglomerations of single-walled
+carbon nanotubes (SWCNTs), material distributions having specific bulk statistics which act
+as near-perfect blackbody absorbers (emissivity > 0.98) across a very wide range of incident
+wavelengths from UV at 200nm to the far IR at 200µm.
+This behaviour has been shown to be independent of the specific properties of the individual
+SWCNTs, but is rather a consequence of the bulk statistical characteristics of the fabricated
+SWCNT distributions. We know that many allotropes of carbon exist in interstellar space and
+these must to some extent be blown into the IGM from the generality of galactic interiors. It is a
+short step to visualizing the existence of clouds of SWCNTs dispersed throughout the hierachical
+7
+
+IGM containing sub-populations which, when viewed in projection along any given line of sight,
+possess the bulk statistical characteristics required to mimic the properties of the fabricated
+SWCNT distributions of Mizuno et al (2009).
+In this way, it is possible to conceive how SWCNT clouds within the IGM have the potential to
+act as ‘dispersed near-perfect blackbody objects’ making them virtually undetectable.
+6
+Summary and conclusions
+There is general agreement that the distribution of galaxies in particular is quasi-fractal D ≈ 2
+out to about 200 Mpc and Baryshev et al (1998) has pointed out that gravitational redshift in
+such an hierarchical cosmos will follow the Hubble Law. Furthermore, these authors point out
+that if the lower cut-off scales of the hierarchy are identified with the mass and radial scales of
+the typical galaxy, then Hg ≈ 70 km/sec/Mpc is to be expected.
+Notwithstanding the hierarchical distribution of galaxies in the local cosmos, the conventional
+view holds that the IGM itself consists of non-baryonic Dark Matter, the assumed homogeneous
+distribution of which makes the Hubble Law a consequence of universal expansion. It is for this
+reason that the paper of Hong et al (2021), which computes and maps the distribution of Dark
+Matter in the local cosmos, caused so much consternation: specifically, they report that the
+distribution of local Dark Matter shows no indication of homogeneity, but instead closely follows
+the fractal structures of the galaxy distribution.
+These results, taken together, suggest a model of galaxy formation in a D = 2 hierarchical
+IGM according to which all of the matter M0 in a sphere R0 coalesces about a unique center
+so that hierachical symmetry is broken, with (M0, R0) then representing the lower cut-off scales
+of the hierarchy. Given the results of Lelli, McGaugh & Schombert (2016B) to the effect that,
+within any given galaxy, M0 primarily consists of baryonic matter then, given that the resulting
+galactic objects are in equilibrium with the general environment, this model of galaxy formation
+gives a direct derivation of the Baryonic Tully-Fisher Relationship and, consequently, provides
+a natural interpretation of MOND as a first-order descriptor of gravitational dynamics in an
+hierachical cosmos.
+In conclusion, in an hierchical cosmos, the Dark Matter hypothesis and Milgrom’s MOND hy-
+pothesis are seen to be two sides of the same coin.
+References
+Baryshev, Yu V., Sylos Labini, F., Montuori, M., Pietronero, L., astro-ph/9803142
+Hong, S.E., Jeong, D., Hwang, H.S., Kim, J., 2021. Ap. J.; 913, 76
+Lelli, F., McGaugh, SS, Schombert, JM., ApJ., 152, 6, 2016A
+Lelli, F., McGaugh, SS, Schombert, JM., ApJL., 816, L14, 2016B
+8
+
+Milgrom, M., 1983a, Ap. J. 270: 365.
+Milgrom, M., 1983b, Ap. J. 270: 371
+Milgrom, M. 1983c. Ap. J. 270: 384-389
+Mizuno, K., Ishii, J., Kishida, H., Hayamizu, Y., Yasuda, S., Futaba, D., Yumura, M., Hata, K.,
+2009. PNAS, 106, 15, 6044-6047
+Tekhanovich D.I.I and Baryshev Yu.V., Astro.ph/1610.05206
+A
+Equilibrium at the lower cut-off scales of the hierarchy
+In the cosmos of our experience, galaxies in general appear to be stable and long-lasting struc-
+tures. Since the matter distribution in the D = 2 fractal hierarchy is isotropic (by definition)
+about any arbitrarily chosen centre, then the notional gravitational acceleration imparted to
+a particle at radius R from the centre, and generated by the material contained within R, is
+directed towards the chosen centre and has magnitude given by
+M(R) G
+R2
+= 4πG ΣF ≡ aF, R < ∞.
+(4)
+On this basis, it is clear that the net actual gravitational acceleration imparted to a material
+particle immersed anywhere in the global hierarchy is zero, from which it can be concluded that
+a D = 2 fractal distribution of material is in a state of dynamical equilibrium.
+It follows that:
+• if a finite spherical volume, radius R0, is imagined emptied of all material, then the net
+actual gravitational acceleration of any material particle placed on R0 will be aF directed
+radially outwards from the centre of the empty volume;
+• the empty spherical volume is unstable since all accelerations on R0 are outward. It follows
+that stability requires the volume to be occupied by a stablizing mass, a galaxy say, creating
+a state of zero net radial acceleration on R0. In other words, the equilibrium condition
+g0 ≡ V 2
+0
+R0
+= aF
+(5)
+must be satisfied.
+9
+

1NE0T4oBgHgl3EQf_gL8/content/tmp_files/load_file.txt

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+page_content='Dark Matter and MOND: Two sides of the same coin?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
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+page_content=' Roscoe (The Open University;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='Roscoe@open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='uk)  ORCID: 0000-0003-3561-7425  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='02829v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='GA]  7 Jan 2023  Abstract  It has recently been reported that the application of convolutional neural-network tech-  niques to infer the dark-matter distribution in the local cosmos has revealed how it follows  the D ≈ 2 hierarchical distribution of galaxies in the locality, rather than exhibiting the  expected homogeneity throughout the IGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' Taken at face value, this implies that the Hub-  ble Law, observed to be followed on scales which are deep inside the observed hierarchical  structures, can no longer be assumed to arise from universal expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' So, if not universal  expansion, then what?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  As a possibility, it has been recognized for a considerable time that if the lower cut-off  scales of a D ≈ 2 hierarchical cosmos are identified with the scales of a typical galaxy, then  gravitational redshift automatically follows the Hubble Law with Hg ≈ 70 km/sec/Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  Inter alia, this suggests a model of galaxy formation in a D ≈ 2 hierarchical IGM in which  all of the material M0 within a sphere R0 coalesces about a unique center so that hierachical  symmetry is broken on the scale (M0, R0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  Putting these things together leads unambiguously to the conclusion that, in an hierachical  cosmos, the Dark Matter hypothesis and Milgrom’s MOND hypothesis are two sides of the  same coin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  2  1  Introduction:  It is now widely accepted that on scales up to about 200 Mpc galaxies are distributed in a quasi-  fractal D ≈ 2 fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' For fairly recent work see Tekhanovich & Baryshev (2016), but many  others have contributed over the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  It then becomes a point of considerable significance that the Hubble Law is well established  on scales that are deep inside the accepted fractal structure of the general galaxy distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  This very interesting circumstance is the primary evidence supporting the idea that the IGM is  largely populated by an homogeneous distribution of dark matter on the small scales required  for, without homogeneity, the linear nature of Hubble’s Law cannot be understood within the  context of universal expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  It is for this reason that the paper of Hong et al (2021) caused so much consternation: specifically,  the authors used state-of-the-art convolutional neural-network techniques combined with modern  positional and peculiar velocity data to compute and map the local dark matter distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  Against expectation, this distribution is found to trace the hierarchical distribution of galaxies  very closely - there is no indication of homogeneity, and hence no indication that the Hubble  Law can be understood in terms of universal expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  The only immediately plausible alternative is some form of gravitational redshift: Baryshev  et al (1998) point out that inside a D = 2 hierarchical galaxy distribution (with an assumed  homogeneous distribution of dark matter) the gravitational part of redshift is also purely linear  with distance and cannot be distinguished from the expansion component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' But if the results of  Hong et al (2021) are to be taken at face value, then any contribution to redshift from expansion  must manifest itself as a departure from linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' Since such a departure is not observed then,  according to the results of Hong et al, there can be no expansion effect at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  This line of argument is reinforced by the further observation of Baryshev et al that if the  lower cut-off mass and length scales of the hierarchy are identified with the mass and length  scales of the typical galaxy, then a gravitational redshift of Hg ≈ 70 km/sec/Mpc is to be ex-  pected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  Inter alia, the foregoing considerations suggest a process of galaxy formation according to which  an isolated galactic object can be modelled as a finite bounded spherically symmetric peturbation  of the hierarchical IGM (assumed in the first instance to be a mix of baryonic and non-baryonic  mass) - this automatically entails that all of the mass M0 within the sphere R0 has coalesced  around a unique centre so that fractal symmetry is broken on the scales of (M0, R0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  2  Consequences on the lower cut-off scales:  From these general considerations we may conclude:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' The lower cut-off radial and mass scales (M0, R0) must behave according to  M0 = 4πR2  0ΣF  (1)  3  where ΣF is the mass surface density of the D = 2 hierarchical mass distribution in the  local cosmos;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' Since galaxies in general appear to be stable structures, there must be an equilibrium  constraint at the lower cut-off scales of the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' Using simple Newtonian arguments,  we show in appendix §A that equilibrium at these lower cut-off scales requires:  V 2  0  R0  = aF ≡ 4πGΣF  (2)  where aF is the characteristic acceleration scale associated with ΣF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' The relationship  V 4  0 = aF GM0 ,  (3)  which is formally identical to the Baryonic Tully-Fisher Relationship (BTFR), is now de-  rived directly by eliminating R0 between (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  It is to be noted that whilst (3) is formally identical to the BTFR of Milgrom’s MOND, it differs  fundamentally in the assumption (expressed in the last paragraph of §1) that M0 is an unknown  mix of baryonic and non-baryonic mass whereas, by definition, the BTFR asserts that this mass  is purely baryonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  3  Empirical support for the BTFR hypothesis  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='1  The analysis of Lelli, McGaugh & Schombert (2016B)  It has only recently been possible to explore the BTFR hypothesis in a statistically rigorous  fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' Specifically, the SPARC sample of Lelli, McGaugh & Schombert (2016A) contains high  quality rotation curves and high quality modern surface photometry at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='6 µm for a sample of 175  nearby disk galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' The high quality of the surface photometry over this sample allowed Lelli,  McGaugh & Schombert (2016B) to construct photometric models of baryonic mass distributions  in that particular subsample of 118 disks which also had rotation curves extending to flatness,  making it ideal for a statistically rigorous testing of the BTFR hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  Subsequently, the authors used regression analysis techniques to demonstrate how the subsample  really does fit the BTFR with very small scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' In this way, they argued that the observed  scatter is sufficiently below the instrinsic-scatter expectations of ΛCDM cosmology to present a  fundamental difficulty for that cosmology and for the associated idea of dynamically significant  quantities of non-baryonic matter in the generality of galaxy disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  Since (3) is derived from the hypothesis that galaxies form by coalescing in a stable way out  of the D = 2 hierarchical IGM, this result implies that the IGM itself consists primarily of  undetected baryonic matter and so, in effect, (3) itself represents a derivation of the hitherto  empirical BTFR from a fundamental theoretical position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='2  The estimation of (aF, ΣF)  The data used by Lelli, McGaugh & Schombert (2016B) is available as an on-line data-sheet giv-  ing estimates for the photometrically modelled baryonic masses M0 and flat rotation velocities  V0 for the 118 disk galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' Given this data, an alternative demonstration supporting the BTFR  hypothesis is provided by showing how the hypothesis, applied differently to the data, yields a  very sharp estimate of the characteristic acceleration parameter aF, thereby demonstrating how,  for all practical purposes, its value is identical to that of Milgrom’s critical acceleration parame-  ter, a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  In order to estimate aF ≡ 4πGΣF from this data, we rearrange (3) as  V 4  0  GM0  = 4πGΣF ≡ constant  and hence form the empirical sample distribution  J ≡  � V 4  0i  GM0i  , i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='118  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  Then, from J, we generate N = 10000 bootstrapped distributions, ˆJi, i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='.N in the usual way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  For each ˆJi we then compute its geometric mean, ˆaFi say, to obtain, finally, the distribution  AF ≡ (log ˆaFi, i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='.N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  The density distribution of AF is given in figure 1 from which it is clear that the estimate for  aF is very tightly constrained around the modal value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='3 × 10−10 mtrs/sec2 which, for all  practical purposes, is identical to Milgrom’s value of MOND’s critical acceleration parameter  a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' This estimate of aF corresponds to ΣF ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='15 kg/mtr2 for the mass surface density of the  hierarchical cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  The grey curve in figure 1 arises from the same analysis but applied to shuffled velocity and  mass data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' It is clear that the signal so powerfully present in the unshuffled data is destroyed  by shuffling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' We conclude that, for all practical purposes the signal for the mass surface density  ΣF ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='15 kg/mtr2 in the hierarchical cosmos is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  5  Dotted = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='1 × 10−10 mtrs/sec2  Mode = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='3 × 10−10 mtrs/sec2  Dashed = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='6 × 10−10 mtrs/sec2  0  5  10  15  log(aF)  Density  Distribution of bootstrapped geometric means  Figure 1: Solid black curve = density distribution of log ˆaF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' Solid grey curve arises when velocity  and mass data are shuffled with respect to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' The signal represented by the black curve  is destroyed on the shuffled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  4  Full circle to MOND  Taking the results of Hong et al (2021) at face value,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' together with the results of Lelli,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' McGaugh  & Schombert (2016B) and observations of Baryshev et al (1998),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' it has been shown that the Dark  Matter of modern astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' rather than being an homogeneous distribution of non-baryonic  matter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' can reasonably be identified as a D = 2 hierarchical distribution of undetected baryonic  matter which,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' as the BTFR (3) shows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' provides exactly the dynamical support for galaxies that  the original non-baryonic Dark Matter hypothesis was formulated for in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  6  However, hitherto the prominence of the originally empirically derived BTFR has rested en-  tirely upon the fact that it is central to the architecture of MOND Milgrom (1983a,b,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' To see  this, (3) gives directly  V 2  0  R0  =  √aF GM0  R0  which, in effect, MOND extrapolates by hypothesis to give g = √aF gN, where gN ≡ GM0/R2, as  the effective gravitational force for all R ≥ R0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' Consequently, MOND automatically receives the  direct interpretation as a first-order descriptor of gravitational dynamics in a D = 2 hierarchical  IGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  In short, in an hierachical cosmos, the Dark Matter hypothesis and Milgrom’s MOND hypothesis  are two sides of the same coin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  5  The nature of baryonic Dark Matter?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  We have argued that the Dark Matter of the IGM is distributed in a D = 2 hierarchy and consists  of undetected baryonic material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' So, the immediate question is: how can such a distribution of  baryonic material remain undetected?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' There are three potential strands to the answer, easily  conceived to be acting in concert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='1  Conventional possibilities  Given that the IGM forms a D = 2 hierarchy, then its volume density in a spherical volume of  radius R tends to zero as R → ∞, making its detection at large radii intrinsically difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  Furthermore, it can reasonably be assumed that the IGM is at least close to being in thermal  equilibrium with the general background, again making its detection against the background also  intrinsically difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='2  Unconventional possibilities  Until recently, it has always been assumed that there is no such thing in nature as a perfect, or  near perfect, blackbody absorber - the reason being that no such thing had ever been observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  However, Mizuno et al (2009) showed how to fabricate, from agglomerations of single-walled  carbon nanotubes (SWCNTs), material distributions having specific bulk statistics which act  as near-perfect blackbody absorbers (emissivity > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='98) across a very wide range of incident  wavelengths from UV at 200nm to the far IR at 200µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  This behaviour has been shown to be independent of the specific properties of the individual  SWCNTs, but is rather a consequence of the bulk statistical characteristics of the fabricated  SWCNT distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' We know that many allotropes of carbon exist in interstellar space and  these must to some extent be blown into the IGM from the generality of galactic interiors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' It is a  short step to visualizing the existence of clouds of SWCNTs dispersed throughout the hierachical  7  IGM containing sub-populations which, when viewed in projection along any given line of sight,  possess the bulk statistical characteristics required to mimic the properties of the fabricated  SWCNT distributions of Mizuno et al (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  In this way, it is possible to conceive how SWCNT clouds within the IGM have the potential to  act as ‘dispersed near-perfect blackbody objects’ making them virtually undetectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  6  Summary and conclusions  There is general agreement that the distribution of galaxies in particular is quasi-fractal D ≈ 2  out to about 200 Mpc and Baryshev et al (1998) has pointed out that gravitational redshift in  such an hierarchical cosmos will follow the Hubble Law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' Furthermore, these authors point out  that if the lower cut-off scales of the hierarchy are identified with the mass and radial scales of  the typical galaxy, then Hg ≈ 70 km/sec/Mpc is to be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  Notwithstanding the hierarchical distribution of galaxies in the local cosmos, the conventional  view holds that the IGM itself consists of non-baryonic Dark Matter, the assumed homogeneous  distribution of which makes the Hubble Law a consequence of universal expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' It is for this  reason that the paper of Hong et al (2021), which computes and maps the distribution of Dark  Matter in the local cosmos, caused so much consternation: specifically, they report that the  distribution of local Dark Matter shows no indication of homogeneity, but instead closely follows  the fractal structures of the galaxy distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  These results, taken together, suggest a model of galaxy formation in a D = 2 hierarchical  IGM according to which all of the matter M0 in a sphere R0 coalesces about a unique center  so that hierachical symmetry is broken, with (M0, R0) then representing the lower cut-off scales  of the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' Given the results of Lelli, McGaugh & Schombert (2016B) to the effect that,  within any given galaxy, M0 primarily consists of baryonic matter then, given that the resulting  galactic objects are in equilibrium with the general environment, this model of galaxy formation  gives a direct derivation of the Baryonic Tully-Fisher Relationship and, consequently, provides  a natural interpretation of MOND as a first-order descriptor of gravitational dynamics in an  hierachical cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  In conclusion, in an hierchical cosmos, the Dark Matter hypothesis and Milgrom’s MOND hy-  pothesis are seen to be two sides of the same coin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  References  Baryshev, Yu V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
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+page_content=', McGaugh, SS, Schombert, JM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
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+page_content=', McGaugh, SS, Schombert, JM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=', ApJL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=', 816, L14, 2016B  8  Milgrom, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=', 1983a, Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' 270: 365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  Milgrom, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=', 1983b, Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' 270: 371  Milgrom, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' 1983c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' 270: 384-389  Mizuno, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=', Ishii, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=', Kishida, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=', Hayamizu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=', Yasuda, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=', Futaba, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=', Yumura, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=', Hata, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=',  2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' PNAS, 106, 15, 6044-6047  Tekhanovich D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='I and Baryshev Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=', Astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='ph/1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='05206  A  Equilibrium at the lower cut-off scales of the hierarchy  In the cosmos of our experience, galaxies in general appear to be stable and long-lasting struc-  tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' Since the matter distribution in the D = 2 fractal hierarchy is isotropic (by definition)  about any arbitrarily chosen centre, then the notional gravitational acceleration imparted to  a particle at radius R from the centre, and generated by the material contained within R, is  directed towards the chosen centre and has magnitude given by  M(R) G  R2  = 4πG ΣF ≡ aF, R < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  (4)  On this basis, it is clear that the net actual gravitational acceleration imparted to a material  particle immersed anywhere in the global hierarchy is zero, from which it can be concluded that  a D = 2 fractal distribution of material is in a state of dynamical equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  It follows that:  if a finite spherical volume, radius R0, is imagined emptied of all material, then the net  actual gravitational acceleration of any material particle placed on R0 will be aF directed  radially outwards from the centre of the empty volume;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  the empty spherical volume is unstable since all accelerations on R0 are outward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' It follows  that stability requires the volume to be occupied by a stablizing mass, a galaxy say, creating  a state of zero net radial acceleration on R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content=' In other words, the equilibrium condition  g0 ≡ V 2  0  R0  = aF  (5)  must be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}
+page_content='  9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'}

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@@ -0,0 +1,781 @@
+ 
+ 
+ 
+ 
+ 
+ 
+ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII 
+WEB: https://www.ankarakongresi.org 
+E-MAIL: [email protected] 
+1657 
+PREDICTING THE STUDENTS INVOLVEMENTS AND IT’S IMPACTS ON 
+LEARNING OUTCOMES THROUGH ONLINE EDUCATION DURING COVID-19 
+ 
+Muhammad Nadeem  
+Computer Science Department, University of the Punjab 
+              
+  
+Faisal Bukhari 
+Data Science Department, University of the Punjab 
+           
+Ali Hussain 
+Computer Science Department, University of the Punjab 
+ 
+Abstract 
+Everybody knows very well about the COVID-19 pandemic, lockdown, and its impacts and 
+effects on every field of life, from childhood to senior citizens, from local to global. The 
+underlying research study focuses on students' involvement in online classes. This paper 
+assesses the effect of the COVID-19 pandemic on the students' participation and involvement 
+during online classes compared to the physical classes, cheating behavior, health effects, and 
+study styles of the students of diverse degrees and age groups. This research study contributes 
+to the real problems and challenges that students faced during online classes during the 
+COVID-19 pandemic. The percentages of the students' responses with different color schemes 
+shown in Fig. 1, Fig. 2, Fig.3(a), Fig.3(b) and Fig.4 are conveying powerful and meaningful 
+insight. These figures and the results given in Table I and Table II indicate that most students 
+are not fully involved during online classes due to technical issues, remote distance, etc. We 
+applied the Test here because we do not have exact population means. We used ttest_1samp 
+with default value 0 to compute the variables' statistics and p-value. These values are minimal 
+in favor of rejecting the null or H0 (hypothesis) and accepting the alternate or H1 
+(hypothesis). It further means that students' involvement during online classes is severely 
+affected.  
+Keywords: COVID-19, e-Learning, Students Involvements, Cheating Concerns of Students, 
+Class Participation. 
+ 
+I. INTRODUCTION  
+The primary motivation for selecting this topic is that the quality of education is directly 
+proportional to the involvement of the students during the lecture. Firstly, I found it as a 
+teacher that many students have left the online lecture physically, but logically they showed 
+their status as a present. This problem has multiple issues. The respected teacher cannot be 
+confident about the presence of the students physically during online lectures. Secondly, the 
+students are facing different issues during online lectures. The impact of these issues is that 
+they lose interest in learning during online lectures. This research work is a new study focused 
+mainly on the level of student involvement during online lectures. All the countries attacked 
+by the villainous COVID-19 virus that has upset each area of life as per economy, from 
+producers to consumers [1]. During the Covid19 pandemic, the Education sector was also 
+severely impacted. The forceful impact of this virus sent the students and teachers to study 
+and teach remotely from face to face system of education. Resultantly, Educational 
+institutions are searching for another way to teach and evaluate the students [2]. So to keep 
+every student and teacher safe, all the Educational Institutions closed because of the citywide, 
+districtwide, and countrywide lockdowns. In such lockup situations, the students and teachers 
+cannot interact face-to-face [3].  
+ 
+
+ 
+ 
+ 
+ 
+ 
+ 
+ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII 
+WEB: https://www.ankarakongresi.org 
+E-MAIL: [email protected] 
+1658 
+To keep the chain of teaching in COVID-19 virus, the World Bank has been actively trying to 
+give financial assistance to the underdeveloped or more affected countries. The ultimate goal 
+of [4] is to provide basic education rights to every student during this viral disease. As far as 
+online learning is concerned, there is much use of technology. This technology-dependent 
+way of education becomes a barrier for learners who did not train to use technology [5]. 
+Similarly, in Pakistan, in 2021, all the educational institutions have closed as the previous 
+year due to the severity of COVID-19. Pakistani Ministry of Education and Higher Education 
+Commission (HEC) also provides online and distance learning ways to teach the students. [6]. 
+The HEC provided the design for online policy guidance notes and guidelines for the 
+Universities. However, It’s a reality that practical work is not being taught during online 
+education. This also demotivated the students, and it made an impact on their involvement in 
+online lectures [7]. In addition to the problems mentioned above and issues of students and 
+teachers, there are also the problems of admin staff [8]. 
+Therefore, the teachers are not satisfied with the student’s involvement in online classes 
+compared to physical classes. 
+In this connection, to find the answers, this study would work on the following research 
+objectives: 
+• To predict why the students are involved is not as much as physical class. 
+• To find why the students are not interested in attending the full online lecture.  
+• To discover the issue faced by the students during online lecture. 
+• To find the impact of taking lectures in class room with the lecture taking online on the 
+students' learning outcomes. 
+• To find the family members' realization about their children's online study. 
+The outcomes of the research would be necessary for the following concerning levels: 
+• Student 
+• Teacher 
+• Parents 
+• Educational Institution 
+• Education Ministries 
+The most crucial stakeholder in the learning process are teachers, and students are aware of 
+the issues and the factors involved as per the student involvement during an online class. The 
+parents would also notice the difference in attitude and aptitude to study in the classroom and 
+at home via online education. The Educational Institution may send reports to the Ministry of 
+Education and HEC based on the outcomes of the student's involvement during an online 
+class. In this way, the Ministries can inform the Government to look after the policies to plan 
+a different mature online education system or to open the educational institution as soon as 
+possible. 
+ 
+II. LITERATURE REVIEW 
+The impacts of COVID-19 on health, society, and education are highlighted in [9]. The 
+researchers divided their research into four different groups: general demographics, 
+information about daily online routine, assessment of the learning of online experience and 
+level of satisfaction of the students, and evaluation of health due to change in lifestyle. 
+Cheating during the exam is one of the main problems. The research work done by [10] on 
+cheating shows that an individual's strengths vary according to the achievement settings. 
+Their findings also concluded that the cheating rate was higher in educational settings than in 
+work areas and in work sites than in sports venues. Study 1 further suggests that the strengths 
+of individuals' cheating intentions differ across achievement settings.  
+ 
+
+ 
+ 
+ 
+ 
+ 
+ 
+ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII 
+WEB: https://www.ankarakongresi.org 
+E-MAIL: [email protected] 
+1659 
+Intentions to cheat were higher in educational settings than in work settings and higher in 
+work settings than in sports settings. The outcomes of this research [11] concluded that the 
+online examination during COVID-19 increased the cheating ratio, which is unrelated to 
+achievement goals. The studies provided different guidelines to the teachers for setting the 
+questions and time duration for online exams. The researchers of [12] highlighted the levels of 
+students' stress, depressive symptoms, loneliness, effects of missing social life, and specific 
+worries for their undergraduate studies. They also showed extreme crises of the students on 
+health and research during lockdown due to COVID-19. The authors discussed that they got 
+212 responses out of 266 from students for the crises suffered. They also recommended 
+different plans for teachers and academic institution administrators to develop online events 
+so that they can prepare newcomers very well. The research efforts of [13] discover the 
+critical problems faced by the students in the present e-learning system. They have also found 
+the factors influencing online learning during COVID-19.  
+The authors also discussed the impacts of students' willingness to study alone in an e-learning 
+environment. In addition, they interviewed 30 students from six Universities and conducted 
+meetings with 31 e-learning system experts to find the main problems. They also suggested 
+applicable plans for policymakers, developers, designers, and researchers, enabling them to be 
+better acquainted with the critical aspects of the e-learning system during the COVID-19 
+pandemic. The researchers of [14] have found too much dissatisfaction during the online 
+study on the COVID-19 situation. The outcomes of this research concluded that the students 
+of the dental study were dissatisfied with the online teaching during COVID-19. The results 
+of this research crying that online study is disturbing the student's level of involvement in the 
+study very severely. The efforts of the analyses highlighted different aspects of students 
+during the online study in the COVID-19 pandemic worldwide. They discussed and evaluated 
+severe issues such as technical and economic issues, psychological problems, and students' 
+fears about the future. It badly affects the study taste of the students and their pace in the 
+learning process. They also offered different plans and suggestions for the policymakers and 
+higher authorities to overcome the issues faced by the students and the teachers. The research 
+study by the authors of [15] observed and evaluated the impact of the perception of e-learning 
+crashes. They discovered its impact on psychological upset in the students during the COVID-
+19 pandemic. They concluded that fear of academic loss had become the main reason for 
+mental upset during the issues of online study in corona disease. They also suggested 
+remedies for the policymakers and educational institutions to manage the student's stress 
+during the online study. The researchers analyzed different types of challenges faced by the 
+students in Pakistani Universities [16]. The main obstacles highlighted are economic, 
+technical, lack of skills, family support, etc. They also recommended that the Govt. take a 
+severe step to overcome the challenges faced by the students. The outcomes of this research 
+work [17] show that the students do not want to study online. The students expressed their 
+problems during the survey that they were not prepared and trained for such a learning shift. 
+They do not have a non-stop electricity facility and well-equipped information technology-
+based infrastructure at their homes. 
+ 
+III. PROBLEM STATEMENT 
+To find the effect of the COVID-19 pandemic on the involvement of the students during 
+online classes as compared to the physical classes, cheating behavior, health effects, and study 
+styles from the students of diverse degrees and age groups. 
+Hypothesis: 
+H0 = Student’s involvement during online classes is the same as in physical classes. 
+H1 = Student’s involvement during online classes is not the same as in physical classes. 
+
+ 
+ 
+ 
+ 
+ 
+ 
+ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII 
+WEB: https://www.ankarakongresi.org 
+E-MAIL: [email protected] 
+1660 
+Methodology and Data Collection 
+The survey methodology used to accomplish this research. Survey is a method for the 
+collection of the information for the sample of individuals [18].  The findings of the survey 
+analyzed through statistical analysis. 
+ 
+• OBJECTIVES OF THE SURVEY  
+To analyze the levels of the student’s involvement and its impacts on learning outcomes 
+during online lectures during COVID-19. 
+ 
+•TARGET POPULATION 
+Graduate, Undergraduate and Intermediate students of the Universities and Colleges 
+ 
+•DATA TO BE COLLECTED  
+A questionnaire developed based on the literature review. Then this questionnaire circulated 
+online as much as possible to find the maximum responses from the target population due to 
+the COVID-19 situation. 
+ 
+•MEASUREMENT `INSTRUMENT'  
+The measurement instrument of the required survey is a questionnaire. The questions of this 
+questionnaire were   
+closed-ended with a Likert scale. The definition of the Likert scale is given below: 
+1. SA (Strongly Agreed) 
+2. A (Agreed) 
+3. U (Undecided),  
+4. D (Disagreed) 
+5. SD (Strongly Disagreed) 
+This questionnaire would be distributed through Google docs to make it available to the 
+targeted population and to get a maximum number of responses. 
+ 
+IV. DESIGN OF RESEARCH STUDY 
+An online survey performed using Google online forms. However, the questionnaire of this 
+survey consists of the following subsections: 
+A. Respondents will be requested to answer their following usual demographics: 
+• Age  
+• Gender 
+• Area of residence 
+B. Getting information routine wise online learning during the shift from face to face study to 
+online study in colleges/Universities in Pakistan. These information consists of the following: 
+• Average time given for online study in hours per day 
+• Quality and the problems of the communication medium 
+• Actual involvement in virtual lecture same as face to face lecture in physical class 
+• Level of interruption by the family members during online study period  
+• Attention and focus level from joining to the end of online class. 
+• Effects of online learning on Cheating behavior and students involvement to  
+C. Evaluation of the experience of the student’s level of involvement in virtual class to find 
+the overall students involvement in online lecture. 
+D. Evaluation of health during change in learning style from physical class environment 
+provided by the College/University to the virtual class environment provided by your parents 
+at home and the effects of virtual class on your involvement of class. 
+ 
+
+ 
+ 
+ 
+ 
+ 
+ 
+ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII 
+WEB: https://www.ankarakongresi.org 
+E-MAIL: [email protected] 
+1661 
+The pictorial survey responses are given below: 
+ 
+ 
+Fig. 1. Getting General Info 
+ 
+ 
+Fig. 2. Getting General Info 
+ 
+ 
+Fig. 3(a). Getting Specific Info 
+ 
+ 
+
+Section A: Demographics Info
+Chart
+100%
+80%
+60%
+40%
+20%
+0%
+6
+1
+4
+4
+5
+8
+5
+9
+3
+2
+3
+3
+3
+4
+4
+4
+5
+5
+5
+Gender
+Age
+Degree_ Level
+Area of ResidenceSection B: Getting General Info Chart
+100%
+80%
+60%
+40%
+20%
+0%
+3
+5
+9
+2
+8
+5
+8
+385
+417
+1.49
+3
+4
+8
+3
+6
+1
+1
+T
+2
+3
+3
+5
+5
+5
+6
+Time_Spent_SociaMediaLaptopComputerAvail
+ISmartPhonesAvail
+Class Participation Level
+CheatingConcern
+StudyLevelifdontExam
+I Lack of IT Skills
+BetterOnlineLearnSectionC:GettingSpecificInfo(11-17
+Questions)
+100%
+80%
+60%
+40%
+20%
+0%
+8
+6
+OnlineAndOffineEqual
+Technicallssuelmpact
+Economiclssuelmpact
+TeacherVoicelssue
+Lessrsinteraction
+AcademicLossfearinClassParticipation
+LackDeficiencyforNonITSt 
+ 
+ 
+ 
+ 
+ 
+ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII 
+WEB: https://www.ankarakongresi.org 
+E-MAIL: [email protected] 
+1662 
+We have created questionnaire. Its soft copy is available at the following link:  
+https://docs.google.com/forms/d/1zqnXC9EXRXjmNL7VX2FP4hh6OL0NVu-
+C_w8QZOFxRsc/edit ) 
+We have collected 623 responses from different level of degree students.  
+ 
+ 
+Fig. 3(b). Getting Specific Info 
+ 
+ 
+Fig. 4. Getting Health Issues info 
+ 
+V. EXPEIRMENTAL RESULTS 
+The means and standard deviation of all the variables as per questionnaire are given below: 
+ 
+ 
+
+SectionC:GettingSpecificInfo(1-10Questions)
+100%
+80%
+60%
+40%
+20%
+0%
+8
+5
+2
+5
+BetterTimeUtilization
+CheatingBehavior
+Umwilingness_of_ResponsibilityStudentsHesitancylmpact
+TechDificultylmpact
+HaveNetAccess
+HaveElectricSuppy
+InteractionWihTeacher
+ClassParticipationChance
+AttensionAndFocusDisturbSection D: Getting Health Issues Info
+Chart
+100%
+80%
+60%
+40%
+20%
+0%
+4
+8
+2
+3
+80
+365
+393
+2
+4
+3
+89
+9
+3
+3
+4
+5
+5
+5
+5
+6
+FitnessOfAttensionLonelinessEffectAnxietyLevelPsycholmpacts 
+ 
+ 
+ 
+ 
+ 
+ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII 
+WEB: https://www.ankarakongresi.org 
+E-MAIL: [email protected] 
+1663 
+TABLE I. 
+S.# 
+Variable 
+Value 
+Mean of all the variables 
+SECTION A:  DEMOGRAPHICS INFO 
+1 
+Gender 
+1.400000 
+2 
+Age 
+2.028571 
+3 
+Degree_Level 
+2.257143 
+4 
+Area_of_Residence 
+1.628571 
+SECTION B: GETTING GENERAL INFO 
+1 
+Time_Spent_ SociaMedia 
+2.457143 
+2 
+LaptopComputerAvail 
+1.771429 
+3 
+SmartPhonesAvail 
+1.571429 
+4 
+Class_Participation_Level 
+2.885714 
+5 
+CheatingConcern 
+1.942857 
+6 
+StudyLevelIfdontExam 
+3.028571 
+7 
+Lack_of_IT_Skills 
+2.485714 
+8 
+BetterOnlineLearn 
+3.600000 
+SECTION C: GETTING SPECIFIC INFO 
+1 
+BetterTimeUtilization 
+3.342857 
+2 
+CheatingBehavior 
+2.285714 
+3 
+Unwilingness_of_Responsibility 
+2.114286 
+4 
+StudentsHesitancyImpact 
+2.371429 
+5 
+TechDifficultyImpact 
+2.200000 
+6 
+HaveNetAccess 
+2.514286 
+7 
+HaveElectricSupply 
+3.000000 
+8 
+InteractionWihTeacher 
+3.085714 
+9 
+ClassParticipationChance 
+3.057143 
+10 
+AttensionAndFocusDisturb 
+2.342857 
+11 
+OnlineAndOfflineEqual 
+3.800000 
+12 
+TechnicalIssueImpact 
+1.857143 
+13 
+EconomicIssueImpact 
+2.000000 
+14 
+TeacherVoiceIssue 
+1.971429 
+15 
+LessTSInteraction 
+1.857143 
+16 
+AcademicLossFearinClassParticipation 
+2.028571 
+16 
+AcademicLossFearinClassParticipation 
+0.970588 
+17 
+LackDeficiencyforNonITSt 
+0.747240 
+Standard Deviation of all the variables 
+SECTION A:  DEMOGRAPHICS INFO 
+1 
+Gender 
+0.489898 
+2 
+Age 
+0.376883 
+3 
+Degree_Level 
+0.552545 
+4 
+Area_of_Residence 
+0.483187 
+ 
+ 
+
+ 
+ 
+ 
+ 
+ 
+ 
+ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII 
+WEB: https://www.ankarakongresi.org 
+E-MAIL: [email protected] 
+1664 
+SECTION B: GETTING GENERAL INFO 
+1 
+Time_Spent_ SociaMedia 
+1.078169 
+2 
+LaptopComputerAvail 
+0.897161 
+3 
+SmartPhonesAvail 
+0.766652 
+4 
+Class_Participation_Level 
+1.259738 
+5 
+CheatingConcern 
+1.093954 
+6 
+StudyLevelIfdontExam 
+1.502107 
+7 
+Lack_of_IT_Skills 
+1.105090 
+8 
+BetterOnlineLearn 
+1.515633 
+SECTION C: GETTING SPECIFIC INFO 
+1 
+BetterTimeUtilization 
+1.413059 
+2 
+CheatingBehavior 
+1.110249 
+3 
+Unwilingness_of_Responsibility 
+1.259738 
+4 
+StudentsHesitancyImpact 
+1.332789 
+5 
+TechDifficultyImpact 
+0.979796 
+6 
+HaveNetAccess 
+1.273273 
+7 
+HaveElectricSupply 
+1.309307 
+8 
+InteractionWihTeacher 
+1.295518 
+9 
+ClassParticipationChance 
+1.286032 
+10 
+AttensionAndFocusDisturb 
+1.392692 
+11 
+OnlineAndOfflineEqual 
+1.214202 
+12 
+TechnicalIssueImpact 
+0.797957 
+13 
+EconomicIssueImpact 
+0.956183 
+3 
+TeacherVoiceIssue 
+1.027777 
+4 
+LessTSInteraction 
+1.045886 
+ 
+TABLE II: 
+TTest Outcomes 
+Ttest_1sampResult(statistic=array([16.663333  , 31.38507589, 
+23.81939622, 19.65311057, 13.28871279, 
+11.51311097, 11.95187108, 13.35711613, 10.35574591, 11.7564528 , 
+13.11574349, 13.84994208, 13.79421828, 12.0044142 ,  9.78640192, 
+10.375     , 13.09261879, 11.51416659, 13.36038922, 13.88838218, 
+13.86128572,  9.80912102, 18.24871239, 13.57080199, 12.19631092, 
+11.18462458, 10.35381536, 12.18694645, 13.15416906, 11.9272551 , 
+11.34226868, 10.64348064, 11.2720409 ]), pvalue=array([6.29551067e-
+18, 1.08636299e-26, 8.60469002e-23, 3.85366635e-20, 
+5.07753770e-15, 2.80770597e-13, 1.00580609e-13, 4.38220401e-15, 
+4.72979440e-12, 1.58421613e-13, 7.38588068e-15, 1.53985258e-15, 
+1.73086219e-15, 8.90861219e-14, 2.02190243e-11, 4.50637008e-12, 
+7.76732461e-15, 2.80069994e-13, 4.35148773e-15, 1.42079500e-15, 
+1.50369222e-15, 1.90647656e-11, 3.87615125e-19, 2.77545554e-15, 
+5.73530395e-14, 6.15085107e-13, 4.75281129e-12, 5.85928814e-14, 
+6.79392859e-15, 1.06477004e-13, 4.21455437e-13, 2.30654437e-12, 
+4.98568395e-13])) 
+ 
+ 
+ 
+
+ 
+ 
+ 
+ 
+ 
+ 
+ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII 
+WEB: https://www.ankarakongresi.org 
+E-MAIL: [email protected] 
+1665 
+VIII. CONCLUSTION 
+To evaluate and find the correctness and applicability of the hypothesis as per the problem 
+statement, we used an online survey approach using Google docs. According to the 
+percentages of survey responses given in Fig.1  Fig.2, Fig.3(a), Fig.3(b) and Fig. 4, 
+availability of Laptop/Computer at student homes was 85.3% and smart phones was 87.5%. 
+Time spent on social media during the online lecture was 71%. The level of Class 
+participation was 49.6%. The concern of students cheating during the online exam was 
+70.8%—level of cheating behavior to ignore interest in online due to online exams 
+encouraged by 60.8% of students. The student's unwillingness was found at 73.2%. Impacts 
+of technical issues during online classes were 84.4% . The pace of the teacher's voice due to 
+the Net problem was discovered at 79.5% and Impacts of less interaction of teacher-student 
+found to be 78%. As per Fig. 4, psychological impacts on learning participation during online 
+classes were discovered at 73.5%  , the stress of loneliness affects students' level of 
+involvement was 68.5% and the anxiety levels disturb students' level of motivation by 77.9%. 
+As per the above experiments, the means and standard deviations are given in Table I and 
+Table II above. Most of the high values of  means shows that much percentage of the students 
+are not fully involved during online lecture. Similarly, most of the values of standard 
+deviations are far from zero. It shows that data points are far from the mean. We applied the 
+test here because we don't have actual population means. We used ttest_1samp (Dataset 
+[:35],0) with a default value of 0 to compute the variables' statistics and p-value. The results 
+of this test are provided in Table II. These values are minimal, which is in favor of rejecting 
+the null or H0 hypothesis and accepting the alternate or H1 hypothesis. It further means that 
+students' involvement during online classes is severely affected.  
+ 
+ACKNOWLEDGMENT  
+The authors are very grateful to the management of server room of Faculty of Computing and 
+Information Technology (FCIT), University of the Punjab to forward our questionnaire to the 
+students for the responses. We are also thankful to the Students of Undergraduates and 
+Gradates students of FCIT for the warm participation and sincere responses during the survey 
+of this research study. 
+ 
+REFERENCES 
+[1].  
+Fernando, R., “The COVID-19 Pandemic: A call for a reality check.”, 
+published in Galle Medical Journal, 25 (1). 2020. 
+[2].  
+Myers, A., “After COVID-19: Recalibrating the American educational 
+system”, Retrieved from https://hub.jhu.edu/2020/04/07/bob-balfanz-education-reform-covid-
+19/, accessed 2021. 
+[3].  
+Tam, G., & El-Azar, D. (2020), “3 ways the coronavirus pandemic could 
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+ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII 
+WEB: https://www.ankarakongresi.org 
+E-MAIL: [email protected] 
+1666 
+[6].  
+Noor Ul Ain, “Is online education the new future of Pakistan?”  Retrieved from 
+https://dailytimes.com.pk/579663/blessing-in-disguise-is-online-education-the-new-future-of-
+pakistan/ , published in 2020 but accessed 2021. 
+[7].  
+Noor Ul Ain Ali., “Students disappointed with online teaching system amid 
+COVID-19”. Retrieved from https://dailytimes.com.pk/587446/students-disappointed-with-
+online-teachingsystem-amid-covid-19/, published in 2020 but accessed 2021. 
+[8].  
+Kebritchi, M., Lipschuetz, A., & Santiague, L., “Issues and challenges for 
+teaching successful online courses in higher education: A literature review”, Journal of 
+Educational Technology Systems, 46 (1), 4–29, 2017. 
+[9].  
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+and its impact on education, social life and mental health of students: A survey”, published in 
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+[10].  Nico W. Van Yperen, Melvyn R.W. Hamstra and Marloes van der Klauw, “ To 
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+published in British Journal of Management, Vol. 22, S5–S15 (2011) 
+[11].  Lia M. Daniels, Lauren D, Goeganx Patti C. Parker, “ The impact of COVID 
+19 triggered changes to instruction and assessment on university students’ self reported 
+motivation, engagement and perceptions”, published in Social Psychology of Education 
+(2021) 24:299–318, Springer. 
+[12].  Timon ElmerID, Kieran Mepham, Christoph Stadtfeld, “Students under 
+lockdown: Comparisons of students’ social networks and mental health before and during the 
+COVID-19 crisis in Switzerland”, published in PLOS ONE, pp: 1-22, 2020. 
+[13].  Mohammed Amin Almaiah,  Ahmad Al-Khasawneh and  Ahmad Althunibat, “ 
+Exploring the critical challenges and factors influencing the E-learning system usage during 
+COVID-19 pandemic”, published in Journal of Education and Information Technologies,  
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+[14].  Huma Sarwar, Hira Akhtar, Meshal Muhammad Naeem, Javeria Ali Khan, 
+Khadija Waraich, Sumaiya Shabbir,  Arshad Hasan and Zohaib Khurshid, “ Self-Reported 
+Effectiveness of e-Learning Classes during COVID-19 Pandemic: A Nation-Wide Survey of 
+Pakistani Undergraduate Dentistry Students”, published in European Journal of Dentistry, 
+2020;14(suppl S1):S34–S43. 
+[15].  Aqsa Arshad, Madiha Afzal, * Dr. Muhammad Sabboor Hussain, “ Sudden 
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+Critical Analysis of Discourse of Fear”, published in Research Journal of Social Sciences & 
+Economics Review, Vol. 1, Issue 3, 2020, PP: 188-199. 
+[16].  Najmul Hasan, Yukun Bao, “ Impact of “e-Learning crack-up” perception on 
+psychological distress among college students during COVID-19 pandemic: A mediating role 
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+PP: 1-10. 
+[17].  Muhammad Anwar, Anwar Khan, Khalid Sultan, “The barriers and challenges 
+faced by students in online education during covid-19 pandemic in Pakistan”, published in 
+Gomal University Journal of Research, Volume 36, Issue 1, JUNE, 2020. PP. 52-62. 
+[18].  Brochure, what is a survey? Bill Kalsbeek, 1995 publications officer, ASA 
+section on survey research methods, 1995. 
+

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+page_content=' University of the Punjab  Abstract Everybody knows very well about the COVID-19 pandemic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' lockdown,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' and its impacts and effects on every field of life,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' from childhood to senior citizens,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' from local to global.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=" The underlying research study focuses on students' involvement in online classes." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=" This paper assesses the effect of the COVID-19 pandemic on the students' participation and involvement during online classes compared to the physical classes, cheating behavior, health effects, and study styles of the students of diverse degrees and age groups." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' This research study contributes to the real problems and challenges that students faced during online classes during the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=" The percentages of the students' responses with different color schemes shown in Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' 1, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' 2, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='3(a), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='3(b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='4 are conveying powerful and meaningful insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' These figures and the results given in Table I and Table II indicate that most students are not fully involved during online classes due to technical issues, remote distance, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' We applied the Test here because we do not have exact population means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=" We used ttest_1samp with default value 0 to compute the variables' statistics and p-value." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' These values are minimal in favor of rejecting the null or H0 (hypothesis) and accepting the alternate or H1 (hypothesis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=" It further means that students' involvement during online classes is severely affected." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' Keywords: COVID-19, e-Learning, Students Involvements, Cheating Concerns of Students, Class Participation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' INTRODUCTION The primary motivation for selecting this topic is that the quality of education is directly proportional to the involvement of the students during the lecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' Firstly, I found it as a teacher that many students have left the online lecture physically, but logically they showed their status as a present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' This problem has multiple issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' The respected teacher cannot be confident about the presence of the students physically during online lectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' Secondly, the students are facing different issues during online lectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' The impact of these issues is that they lose interest in learning during online lectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' This research work is a new study focused mainly on the level of student involvement during online lectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' All the countries attacked by the villainous COVID-19 virus that has upset each area of life as per economy, from producers to consumers [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' During the Covid19 pandemic, the Education sector was also severely impacted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' The forceful impact of this virus sent the students and teachers to study and teach remotely from face to face system of education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' Resultantly, Educational institutions are searching for another way to teach and evaluate the students [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' So to keep every student and teacher safe, all the Educational Institutions closed because of the citywide, districtwide, and countrywide lockdowns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' In such lockup situations, the students and teachers cannot interact face-to-face [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='  ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII WEB: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='ankarakongresi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='org E-MAIL: bilgi@ankarakongresi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='org 1658 To keep the chain of teaching in COVID-19 virus, the World Bank has been actively trying to give financial assistance to the underdeveloped or more affected countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' The ultimate goal of [4] is to provide basic education rights to every student during this viral disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' As far as online learning is concerned, there is much use of technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' This technology-dependent way of education becomes a barrier for learners who did not train to use technology [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' Similarly, in Pakistan, in 2021, all the educational institutions have closed as the previous year due to the severity of COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' Pakistani Ministry of Education and Higher Education Commission (HEC) also provides online and distance learning ways to teach the students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' The HEC provided the design for online policy guidance notes and guidelines for the Universities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' However, It’s a reality that practical work is not being taught during online education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' This also demotivated the students, and it made an impact on their involvement in online lectures [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' In addition to the problems mentioned above and issues of students and teachers, there are also the problems of admin staff [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' Therefore, the teachers are not satisfied with the student’s involvement in online classes compared to physical classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='  In this connection, to find the answers, this study would work on the following research objectives: • To predict why the students are involved is not as much as physical class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' • To find why the students are not interested in attending the full online lecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' • To discover the issue faced by the students during online lecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=" • To find the impact of taking lectures in class room with the lecture taking online on the students' learning outcomes." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=" • To find the family members' realization about their children's online study." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' The outcomes of the research would be necessary for the following concerning levels: • Student • Teacher • Parents • Educational Institution • Education Ministries The most crucial stakeholder in the learning process are teachers, and students are aware of the issues and the factors involved as per the student involvement during an online class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' The parents would also notice the difference in attitude and aptitude to study in the classroom and at home via online education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=" The Educational Institution may send reports to the Ministry of Education and HEC based on the outcomes of the student's involvement during an online class." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' In this way, the Ministries can inform the Government to look after the policies to plan a different mature online education system or to open the educational institution as soon as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' LITERATURE REVIEW The impacts of COVID-19 on health, society, and education are highlighted in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' The researchers divided their research into four different groups: general demographics, information about daily online routine, assessment of the learning of online experience and level of satisfaction of the students, and evaluation of health due to change in lifestyle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' Cheating during the exam is one of the main problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=" The research work done by [10] on cheating shows that an individual's strengths vary according to the achievement settings." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' Their findings also concluded that the cheating rate was higher in educational settings than in work areas and in work sites than in sports venues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=" Study 1 further suggests that the strengths of individuals' cheating intentions differ across achievement settings." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='  ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII WEB: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='ankarakongresi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='org E-MAIL: bilgi@ankarakongresi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='org 1659 Intentions to cheat were higher in educational settings than in work settings and higher in work settings than in sports settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' The outcomes of this research [11] concluded that the online examination during COVID-19 increased the cheating ratio, which is unrelated to achievement goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' The studies provided different guidelines to the teachers for setting the questions and time duration for online exams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=" The researchers of [12] highlighted the levels of students' stress, depressive symptoms, loneliness, effects of missing social life, and specific worries for their undergraduate studies." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' They also showed extreme crises of the students on health and research during lockdown due to COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' The authors discussed that they got 212 responses out of 266 from students for the crises suffered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' They also recommended different plans for teachers and academic institution administrators to develop online events so that they can prepare newcomers very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' The research efforts of [13] discover the critical problems faced by the students in the present e-learning system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' They have also found the factors influencing online learning during COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=" The authors also discussed the impacts of students' willingness to study alone in an e-learning environment." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='  In addition, they interviewed 30 students from six Universities and conducted meetings with 31 e-learning system experts to find the main problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' They also suggested applicable plans for policymakers, developers, designers, and researchers, enabling them to be better acquainted with the critical aspects of the e-learning system during the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' The researchers of [14] have found too much dissatisfaction during the online study on the COVID-19 situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' The outcomes of this research concluded that the students of the dental study were dissatisfied with the online teaching during COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=" The results of this research crying that online study is disturbing the student's level of involvement in the study very severely." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' The efforts of the analyses highlighted different aspects of students during the online study in the COVID-19 pandemic worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=" They discussed and evaluated severe issues such as technical and economic issues, psychological problems, and students' fears about the future." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' It badly affects the study taste of the students and their pace in the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' They also offered different plans and suggestions for the policymakers and higher authorities to overcome the issues faced by the students and the teachers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' The research study by the authors of [15] observed and evaluated the impact of the perception of e-learning crashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' They discovered its impact on psychological upset in the students during the COVID- 19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='  They concluded that fear of academic loss had become the main reason for mental upset during the issues of online study in corona disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=" They also suggested remedies for the policymakers and educational institutions to manage the student's stress during the online study." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' The researchers analyzed different types of challenges faced by the students in Pakistani Universities [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' The main obstacles highlighted are economic, technical, lack of skills, family support, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' They also recommended that the Govt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' take a severe step to overcome the challenges faced by the students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' The outcomes of this research work [17] show that the students do not want to study online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' The students expressed their problems during the survey that they were not prepared and trained for such a learning shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' They do not have a non-stop electricity facility and well-equipped information technology- based infrastructure at their homes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' PROBLEM STATEMENT To find the effect of the COVID-19 pandemic on the involvement of the students during online classes as compared to the physical classes, cheating behavior, health effects, and study styles from the students of diverse degrees and age groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' Hypothesis: H0 = Student’s involvement during online classes is the same as in physical classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' H1 = Student’s involvement during online classes is not the same as in physical classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='  ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII WEB: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='ankarakongresi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='org E-MAIL: bilgi@ankarakongresi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='org 1660 Methodology and Data Collection The survey methodology used to accomplish this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' Survey is a method for the collection of the information for the sample of individuals [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' The findings of the survey analyzed through statistical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='  OBJECTIVES OF THE SURVEY To analyze the levels of the student’s involvement and its impacts on learning outcomes during online lectures during COVID  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='  TARGET POPULATION Graduate, Undergraduate and Intermediate students of the Universities and Colleges  DATA TO BE COLLECTED A questionnaire developed based on the literature review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' Then this questionnaire circulated online as much as possible to find the maximum responses from the target population due to the COVID  19 situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content="  MEASUREMENT `INSTRUMENT' The measurement instrument of the required survey is a questionnaire." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' The questions of this questionnaire were closed  ended with a Likert scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' The definition of the Likert scale is given below: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' SA (Strongly Agreed) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' A (Agreed) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' U (Undecided), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' D (Disagreed) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' SD (Strongly Disagreed) This questionnaire would be distributed through Google docs to make it available to the targeted population and to get a maximum number of responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' DESIGN OF RESEARCH STUDY An online survey performed using Google online forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' However, the questionnaire of this survey consists of the following subsections: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' Respondents will be requested to answer their following usual demographics: • Age • Gender • Area of residence B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' Getting information routine wise online learning during the shift from face to face study to online study in colleges/Universities in Pakistan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' These information consists of the following: • Average time given for online study in hours per day • Quality and the problems of the communication medium • Actual involvement in virtual lecture same as face to face lecture in physical class • Level of interruption by the family members during online study period • Attention and focus level from joining to the end of online class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' • Effects of online learning on Cheating behavior and students involvement to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' Evaluation of the experience of the student’s level of involvement in virtual class to find the overall students involvement in online lecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' Evaluation of health during change in learning style from physical class environment provided by the College/University to the virtual class environment provided by your parents at home and the effects of virtual class on your involvement of class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='  ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII WEB: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='ankarakongresi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='org E-MAIL: bilgi@ankarakongresi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='org 1661 The pictorial survey responses are given below:  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' Getting General Info  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' Getting General Info  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' Getting Specific Info  Section A: Demographics Info Chart 100% 80% 60% 40% 20% 0% 6 1 4 4 5 8 5 9 3 2 3 3 3 4 4 4 5 5 5 Gender Age Degree_ Level Area of ResidenceSection B: Getting General Info Chart 100% 80% 60% 40% 20% 0% 3 5 9 2 8 5 8 385 417 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='49 3 4 8 3 6 1 1 T 2 3 3 5 5 5 6 Time_Spent_SociaMediaLaptopComputerAvail ISmartPhonesAvail Class Participation Level CheatingConcern StudyLevelifdontExam I Lack of IT Skills BetterOnlineLearnSectionC:GettingSpecificInfo(11-17 Questions) 100% 80% 60% 40% 20% 0% 8 6 OnlineAndOffineEqual Technicallssuelmpact Economiclssuelmpact TeacherVoicelssue Lessrsinteraction AcademicLossfearinClassParticipation LackDeficiencyforNonITSt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII WEB: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='ankarakongresi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='org E-MAIL: bilgi@ankarakongresi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='org 1662 We have created questionnaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' Its soft copy is available at the following link: https://docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='com/forms/d/1zqnXC9EXRXjmNL7VX2FP4hh6OL0NVu- C_w8QZOFxRsc/edit ) We have collected 623 responses from different level of degree students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' Getting Specific Info  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' Getting Health Issues info  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' EXPEIRMENTAL RESULTS The means and standard deviation of all the variables as per questionnaire are given below:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
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+page_content='98568395e-13]))  ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII WEB: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='ankarakongresi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='org E-MAIL: bilgi@ankarakongresi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='org 1665 VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' CONCLUSTION To evaluate and find the correctness and applicability of the hypothesis as per the problem statement, we used an online survey approach using Google docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' According to the percentages of survey responses given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='1  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='2, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='3(a), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='3(b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' 4, availability of Laptop/Computer at student homes was 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='3% and smart phones was 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' Time spent on social media during the online lecture was 71%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' The level of Class participation was 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' The concern of students cheating during the online exam was 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='8%—level of cheating behavior to ignore interest in online due to online exams encouraged by 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='8% of students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=" The student's unwillingness was found at 73." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' Impacts of technical issues during online classes were 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='4% .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=" The pace of the teacher's voice due to the Net problem was discovered at 79." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='5% and Impacts of less interaction of teacher-student found to be 78%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' As per Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' 4, psychological impacts on learning participation during online classes were discovered at 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content="5%  , the stress of loneliness affects students' level of involvement was 68." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content="5% and the anxiety levels disturb students' level of motivation by 77." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='9%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' As per the above experiments, the means and standard deviations are given in Table I and Table II above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='  Most of the high values of  means shows that much percentage of the students are not fully involved during online lecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' Similarly, most of the values of standard deviations are far from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' It shows that data points are far from the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=" We applied the test here because we don't have actual population means." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=" We used ttest_1samp (Dataset [:35],0) with a default value of 0 to compute the variables' statistics and p-value." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' The results of this test are provided in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' These values are minimal, which is in favor of rejecting the null or H0 hypothesis and accepting the alternate or H1 hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=" It further means that students' involvement during online classes is severely affected." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content='  ACKNOWLEDGMENT The authors are very grateful to the management of server room of Faculty of Computing and Information Technology (FCIT), University of the Punjab to forward our questionnaire to the students for the responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' We are also thankful to the Students of Undergraduates and Gradates students of FCIT for the warm participation and sincere responses during the survey of this research study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
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+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' Hamstra and Marloes van der Klauw, “ To Win, or Not to Lose, At Any Cost: The Impact of Achievement Goals on Cheating”, published in British Journal of Management, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
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+page_content=' [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
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+page_content=' Mohammed Amin Almaiah,  Ahmad Al-Khasawneh and  Ahmad Althunibat, “ Exploring the critical challenges and factors influencing the E-learning system usage during COVID-19 pandemic”, published in Journal of Education and Information Technologies, Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
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+page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' Aqsa Arshad, Madiha Afzal, * Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
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+page_content=' 1, Issue 3, 2020, PP: 188-199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
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+page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' Muhammad Anwar, Anwar Khan, Khalid Sultan, “The barriers and challenges faced by students in online education during covid-19 pandemic in Pakistan”, published in Gomal University Journal of Research, Volume 36, Issue 1, JUNE, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' PP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' 52-62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
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+page_content=' Brochure, what is a survey?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}
+page_content=' Bill Kalsbeek, 1995 publications officer, ASA section on survey research methods, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'}

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+arXiv:2301.04406v1  [cs.DS]  11 Jan 2023
+A Note on Property Testing of the Binary Rank
+Nader H. Bshouty
+Dept. of Computer Science
+Technion, Haifa, Israel.
+January 12, 2023
+Abstract
+Let M be a n × m (0, 1)-matrix. We define the s-binary rank, brs(M), of M to be the
+minimal integer d such that there are d monochromatic rectangles that cover all the 1-entries in
+the matrix, and each 1-entry is covered by at most s rectangles. When s = 1, this is the binary
+rank, br(M), known from the literature.
+Let R(M) and C(M) be the set of rows and columns of M, respectively. We use the result of
+Sgall [8] to prove that if M has s-binary rank at most d, then |R(M)| · |C(M)| ≤
+� d
+≤s
+�
+2d where
+� d
+≤s
+�
+= �s
+i=0
+�d
+i
+�
+. This bound is tight; that is, there exists a matrix M ′ of s-binary rank d such
+that |R(M ′)| · |C(M ′)| =
+� d
+≤s
+�
+2d.
+Using this result, we give a new one-sided adaptive and non-adaptive testers for (0, 1)-
+matrices of s-binary rank at most d (and exactly d) that makes ˜O
+�� d
+≤s
+�
+2d/ǫ
+�
+and ˜O
+�� d
+≤s
+�
+2d/ǫ2�
+queries, respectively.
+For a fixed s, this improves the query complexity of the tester of Parnas et al. in [7] by a
+factor of ˜Θ(2d).
+1
+Introduction
+Let M be a n × m (0, 1)-matrix. We define the s-binary rank, brs(M), of M to be the minimal
+integer d such that there are d sets (rectangles) Ik×Jk where Ik ⊆ [n] := {1, . . . , n}, Jk ⊂ [m], k ∈ [d]
+such that1 M[i, j] = 1 for all (i, j) ∈ Ik × Jk, k ∈ [d] (monochromatic rectangles), and for every
+(i, j) ∈ [n] × [m] where M[i, j] = 1, there are at least one and at most s integers t ∈ [d] such
+that (i, j) ∈ It × Jt (each 1-entry in M is covered by at least one and at most s monochromatic
+rectangles). When s = 1, br1(M), is the binary rank, br(M), and when s = ∞, br∞(M) is the
+Boolean rank. Both are known from the literature. See, for example, [4].
+The binary rank can also be defined as follows. The binary rank of a n × m (0, 1)-matrix M
+is equal to the minimal d, where there are n × d (0, 1)-matrix N and d × m (0, 1)-matrix L such
+that M = NL. It is also equal to the minimal number of bipartite cliques needed to partition all
+the edges of a bipartite graph whose adjacent matrix is M. The s-binary rank of M is the minimal
+number of bipartite cliques needed to cover all edges of a bipartite graph whose adjacent matrix
+is M, where each edge is covered by at most s bipartite cliques. In [2], it was shown that it is
+NP-hard to approximating the binary rank to within a factor of n1−δ for any given δ.
+1For M, the (i, j) entry of the matrix is denoted by M[i, j].
+1
+
+A property-testing algorithm (tester) of the s-binary rank [7] is given as input 0 < ǫ < 1, integers
+d, n, m, and query access to the entries of a n×m (0, 1)-matrix M. If M has s-binary rank at most
+d (resp. equal d), then the tester accepts with probability at least 2/3. If M is ǫ-far from having
+s-binary rank at most d (resp. equal d), i.e., more than ǫ-fraction of the entries of M should be
+modified to get a matrix with s-binary rank at most d (resp. equal to d), then the tester rejects
+with probability at least 2/3. If the tester accepts matrices having s-binary rank at most d (resp.
+equal to d) with probability 1, then we call it a one-sided error tester. In adaptive testing, the
+queries can depend on the answers to the previous queries, whereas in non-adaptive testing, all the
+queries are fixed in advance by the tester. The goal is to construct a tester that makes a minimal
+number of queries.
+The testability of s-binary rank at most d of (0, 1)-matrices was studied in [6, 7]. In [6], Nakar
+and Ron gave a non-adaptive one-sided error tester for s = 1, that makes ˜O(24d/ǫ4). In [7], Parnas
+et al. gave a non-adaptive and adaptive one-sided error tester for s = 1 that makes O(22d/ǫ2) and
+O(22d/ǫ) queries, respectively. The results in [7] also hold for s-binary rank at most d. In this
+paper, for s-binary at most d and equal to d, we prove
+Theorem 1. There exists an adaptive one-sided error tester for s-binary rank of n × m (0, 1)-
+matrices that makes ˜O
+�� d
+≤s
+�
+2d/ǫ
+�
+queries.
+Theorem 2. There exists a non-adaptive one-sided error tester for s-binary rank of n × m (0, 1)-
+matrices that makes ˜O
+�� d
+≤s
+�
+2d/ǫ2�
+queries.
+For fixed s, this improves the query complexity of Parnas et al. in [7] by a factor of ˜O(2d).
+1.1
+Our Approach
+The tester of Parnas et al. [7] uses the fact that if M′ is a k × k sub-matrix of M and M′ is of
+s-binary rank at most d, then
+1. M′ has at most 2d distinct rows and at most 2d distinct columns.
+2. If M is ǫ-far from having s-binary rank at most d, then extending M′ by one more uniformly
+at random row and column of M, gives a (k + 1) × (k + 1) sub-matrix M′′ of M that, with
+probability at least Ω(ǫ), satisfies: the number of distinct rows in M′′ is greater by one than
+the number of distinct rows in M′, or, the number of distinct columns in M′′ is greater by
+one than the number of distinct columns in M′.
+So, their adaptive tester runs O(2d/ǫ) iterations. At each iteration, it extends M′ by uniformly at
+random one row and one column. Let M′′ be the resulting sub-matrix. If the s-binary rank of M′′
+is greater than d, the tester rejects. If the number of distinct rows or columns in M′′ is greater
+than the number in M′, then it continues to the next iteration with M′ ← M′′. Otherwise, it
+continues to the next iteration with M′. If, after O(2d/ǫ) iterations, M′ has s-binary rank d, the
+tester accepts.
+If the s-binary rank of M is d, then every sub-matrix has a s-binary rank d, and the tester
+accepts. If M is ǫ-far from having s-binary rank at most d, then: since, at each iteration, with
+probability at least Ω(ǫ), the number of distinct rows or columns of M′ is increased by one, and
+since matrices of s-binary rank d has at most 2d distinct rows and at most 2d distinct columns,
+with high probability, we get M′ with s-binary rank greater than d and the tester rejects. The
+2
+
+query complexity of the tester is O(22d/ǫ), which is the number of entries of the matrix M′, O(22d),
+times the number of trials O(1/ǫ) for extending M′ by one row and one column.
+We now give our approach. Call a sub-matrix M′ of M perfect if it has distinct rows and distinct
+columns. Our adaptive tester uses the fact that if M′ is a perfect k×k′ sub-matrix of M of s-binary
+rank d, then
+1. kk′ ≤
+� d
+≤s
+�
+2d.
+2. If M is ǫ-far from having s-binary rank at most d, then at least one of the following occurs
+(a) With probability at least Ω(ǫ), extending M′ by one uniformly at random column of M,
+gives a perfect k × (k′ + 1) sub-matrix M′′ of M.
+(b) With probability at least Ω(ǫ), extending M′ by one uniformly at random row of M,
+gives a perfect (k + 1) × k′ sub-matrix M′′ of M.
+(c) With probability at least Ω(ǫ), extending M′ by one uniformly at random column and
+one uniformly at random row of M, gives a perfect2 (k + 1) × (k′ + 1) sub-matrix M′′ of
+M.
+Item 1 follows from Sgall result in [8] (See Section 3), and item 2 is Claim 10 in [7]. Now, the
+tester strategy is as follows. If k ≤ k′, the tester first tries to extend M′ with a new column. If
+it succeeds, it moves to the next iteration. Otherwise, it tries to extend M′ with a new row. If it
+succeeds, it moves to the next iteration. Otherwise, it tries to extend M′ with a new row and a
+new column. If it succeeds, it moves to the next iteration. If it fails, it accepts. If k′ < k, it starts
+with the row, then the column, and then both.
+Using this strategy, we show that the query complexity will be, at most, the order of the size
+kk′ ≤
+� d
+≤s
+�
+2d of M′ times the number of trials, ˜O(1/ǫ), to find the new row, column, or both. This
+achieves the query complexity in Theorem 1.
+For the non-adaptive tester, the tester, uniformly at random, chooses t = ˜O
+�� d
+≤s
+�
+2d/ǫ2�
+rows
+r1, . . . , rt ∈ [n] and t columns c1, . . . , ct ∈ [m] and queries all M[ri, cj] for all i · j ≤ t and puts them
+in a table. Then it runs the above non-adaptive tester. When the non-adaptive tester asks for
+uniformly at random row or column, it provides the next element ri or cj, respectively. The queries
+are then answered from the table. We show that the adaptive algorithm does not need to make
+queries that are not in the table before it halts. This achieves the query complexity in Theorem 2.
+1.2
+Other Rank Problems
+The real rank of a n × m-matrix M over any field F is the minimal d, such that there is a n × d
+matrix N over F and a d × m matrix L over F such that M = NL. The testability of the real
+rank was studied in [1, ?, 5]. In [1], Balcan et al. gave a non-adaptive tester for the real rank that
+makes ˜O(d2/ǫ) queries. They also show that this query complexity is optimal.
+The Boolean rank (∞-binary rank) was studied in [6, 7]. Parnas et al. in [7] gave a non-adaptive
+tester for the Boolean rank that makes ˜O(d4/ǫ4) queries3.
+2It may happen that events (a) and (b) do not occur and (c) does
+3The query complexity in [7] is ˜O(d4/ǫ6).
+We’ve noticed that Lemma 3 in [7] is also true when we replace
+(ǫ2/64)n2 with (ǫ/4)n2. To prove that, in the proof of Lemma 3, replace Modification rules 1 and 2 with the following
+modification: Modify to 0 all beneficial entries. This gives the result stated here,[3].
+3
+
+2
+Definitions and Preliminary Results
+Let M be a n × m (0, 1)-matrix. We denote by R(M) and C(M) the set of rows and columns
+of M, respectively. The number of distinct rows and columns of M are denoted by r(M) = |R(M)|
+and, c(M) = |C(M)|, respectively. The binary rank of a n × m-matrix M, br(M), is equal to the
+minimal d, where there is a n × d (0, 1)-matrix N and a d × m (0, 1)-matrix L such that M = NL.
+We define the s-binary rank, brs(M), of M to be the minimal integer d such that there are d
+sets (rectangles) Ik × Jk where Ik ⊆ [n] := {1, . . . , n}, Jk ⊂ [m], k ∈ [d] such that M[i, j] = 1 for all
+(i, j) ∈ Ik ×Jk, k ∈ [d] (monochromatic rectangles) and for every (i, j) ∈ [n]×[m] where M[i, j] = 1
+there are at least one and at most s integers t ∈ [d] such that (i, j) ∈ It × Jt (each 1-entry in M is
+covered by at least one and at most s monochromatic rectangles).
+We now prove.
+Lemma 1. Let M be a n × m (0, 1)-matrix. The s-binary rank of M, brs(M), is equal to the
+minimal integer d, where there is a n × d (0, 1)-matrix N and a d × m (0, 1)-matrix L such that:
+For P = NL,
+1. For every (i, j) ∈ [n] × [m], M[i, j] = 0 if and only if P[i, j] = 0.
+2. For every (i, j) ∈ [n] × [m], P[i, j] ≤ s.
+Proof. If M is of s-binary rank d, then there are rectangles {Ik ×Jk}k∈[d], Ik ⊆ [n], Jk ⊂ [m], k ∈ [d]
+such that M[i, j] = 1 for all (i, j) ∈ Ik ×Jk, k ∈ [d] and for every (i, j) ∈ [n]×[m] where M[i, j] = 1
+there are at least one and at most s integers t ∈ [d] such that (i, j) ∈ It × Jt. Define row vectors
+a(k) ∈ {0, 1}n and b(k) ∈ {0, 1}m where a(k)
+i
+= 1 iff (if and only if) i ∈ Ik, and b(k)
+j
+= 1 iff j ∈ Jk.
+Then define4 P = a(1)′b(1)+· · ·+a(d)′b(d). It is easy to see that (a(k)′b(k))[i, j] = 1 iff (i, j) ∈ Ik ×Jk.
+Therefore, P[i, j] = 0 iff M[i, j] = 0 and P[i, j] ≤ s for all (i, j) ∈ [n]×[m]. Define the n×d matrix
+N =
+�
+a(1)′| · · · |a(d)′�
+and the d × m matrix L =
+�
+b(1)′| · · · |b(d)′�′
+.
+It is again easy to see that
+P = NL.
+The other direction can be easily seen by tracing backward in the above proof.
+We now prove the following,
+Lemma 2. Let M be a n × m matrix. Let N and L be n × d (0, 1)-matrix and d × m (0, 1)-matrix,
+respectively, such that P = NL. Then r(P) ≤ r(N) and c(P) ≤ c(L).
+Proof. We prove the result for r.
+The proof for c is similar.
+Let r1, . . . , rn be the rows of N
+and p1, . . . , pn be the rows of P.
+Then pi = riL.
+Therefore, if ri = rj, then pi = pj.
+Thus,
+r(P) ≤ r(N).
+Let M be a n × m matrix. For x ∈ X ⊆ [n], y ∈ Y ⊆ [m], we denote by M[X, Y ] the |X| × |Y |
+sub-matrix of M, (M[x′, y′])x′∈X,y′∈Y . Denote by M[X, y] the column vector (M[x′, y])x′∈X and by
+M[x, Y ] the row vector (M(x, y′))y′∈Y .
+For x ∈ [n] (resp. y ∈ [m]) we say that M[X, y] is a new column (resp. M[x, Y ] is a new row)
+to M[X, Y ] if it is not equal to any of the columns (resp. rows) of M[X, Y ].
+4Here x′ is the transpose of x.
+4
+
+Lemma 3. Let M be a n × m matrix, x ∈ [n], X ⊆ [n], y ∈ [m], and Y ⊆ [m]. Suppose M[x, Y ] is
+not a new row to M[X, Y ], and M[X, y] is not a new column to M[X, Y ]. Then M[x, Y ∪ {y}] is
+not a new row to M[X, Y ∪{y}] if and only if M[X ∪{x}, y] is not a new column to M[X ∪{x}, Y ].
+Proof. If M[x, Y ∪ {y}] is not a new row to M[X, Y ∪ {y}], then there is x′ ∈ X such that
+M[x, Y ∪ {y}] = M[x′, Y ∪ {y}]. Since M[X, y] is not a new column to M[X, Y ], there is y′ ∈ Y
+such that M[X, y] = M[X, y′]. Since M[x, Y ∪ {y}] = M[x′, Y ∪ {y}], we have M[x′, y′] = M[x, y′]
+and M[x, y] = M[x′, y].
+Since M[X, y] = M[X, y′], we have M[x′, y] = M[x′, y′].
+Therefore,
+M[x, y] = M[x, y′] and M[X ∪ {x}, y] = M[X ∪ {x}, y′]. Thus, M[X ∪ {x}, y] is not a new column
+to M[X ∪ {x}, Y ].
+Similarly, the other direction follows.
+3
+Matrices of s-Binary Rank d
+In this section, we prove the following two Lemmas.
+Lemma 4. For any n × m (0, 1)-matrix M of s-binary rank at most d, we have
+r(M) · c(M) ≤
+� d
+≤ s
+�
+2d.
+Lemma 5. There is a (0, 1)-matrix M′ of s-binary rank d that satisfies r(M′) · c(M′) =
+� d
+≤s
+�
+2d.
+To prove Lemma 4, we use the following Sgall’s lemma.
+Lemma 6. [8]. Let A, B ⊆ 2[d] be such that for every A ∈ A and B ∈ B, |A ∩ B| ≤ s. Then
+|A| · |B| ≤
+� d
+≤s
+�
+2d.
+We now prove Lemma 4.
+Proof. Since the s-binary rank of M is at most d, by Lemma 1, there is a n × d (0, 1)-matrix N
+and a d × m (0, 1)-matrix L such that, for P = NL
+1. For every (i, j) ∈ [n] × [m], M[i, j] = 0 if and only if P[i, j] = 0.
+2. For every (i, j) ∈ [n] × [m], P[i, j] ≤ s.
+Obviously, r(M) ≤ r(P) and c(M) ≤ c(P).
+Consider A = {A1, . . . , An} ⊆ 2[d] and B =
+{B1, . . . , Bm} ⊆ 2[d], where Ai = {j|Ni,j = 1} and Bk = {j|Lj,k = 1}.
+Since the entries of
+P = NL are at most s, for every i ∈ [n] and k ∈ [m], |Ai ∩ Bk| ≤ s.
+By Lemma 2 and 6,
+r(M) · c(M) ≤ r(P) · c(P) ≤ r(N) · c(L) = |A| · |B| ≤
+� d
+≤ s
+�
+2d.
+We now prove Lemma 5
+5
+
+Proof. Let N be a 2d × d (0, 1)-matrix where its rows contain all the vectors in {0, 1}d. Let L be a
+d ×
+� d
+≤s
+�
+matrix where its columns contain all the vectors in {0, 1}d of weight at most s. Obviously,
+P = NL is 2d ×
+� d
+≤s
+�
+with entries that are less than or equal to s. Define a 2d ×
+� d
+≤s
+�
+(0, 1)-matrix
+M′ where M′[i, j] = 0 if and only if P[i, j] = 0. Then, by Lemma 1, M′ is of s-binary rank at
+most d. We now show that r(M′) · c(M′) =
+� d
+≤s
+�
+2d.
+Since the identity d × d matrix Id is a sub-matrix of L, we have that NId = N is (0, 1)-matrix
+and a sub-matrix of P and therefore of M′. Therefore, r(M′) ≥ r(N) = 2d. Since Id is a sub-
+matrix of N, by the same argument, c(M′) ≥ c(L) =
+� d
+≤s
+�
+. Therefore r(M′) · c(M′) ≥
+� d
+≤s
+�
+2d.
+Thus, r(M′) · c(M′) =
+� d
+≤s
+�
+2d.
+We now show that M′ has s-binary rank d. Suppose the contrary, i.e., M′ has binary rank
+d′ < d. Then there are 2d × d′ (0, 1)-matrix N and d′ ×
+� d
+≤s
+�
+(0, 1)-matrix L such that P = NL
+and M′[i, j] = 0 iff P[i, j] = 0. Now by Lemma 2, r(M′) ≤ r(P) ≤ r(N) ≤ 2d′ < 2d, which gives a
+contradiction.
+4
+Testing The s-Binary Rank
+In this section, we present the adaptive and non-adaptive testing algorithms for s-binary rank at
+most d. We first give the adaptive algorithm and prove Theorem 1.
+4.1
+The Adaptive Tester
+In this section, we prove Theorem 1.
+Consider the tester Adaptive-Test-Rank in Figure 1. The tester, at every iteration of the
+main While-loop (step 2) has a set X of rows of M and a set Y of columns of M. If |X| ≥ |Y |
+(step 5), the tester first tries to extend M[X, Y ] with a new column (steps 6-8). If it succeeds, it
+moves to the next iteration. Otherwise, it tries to extend M[X, Y ] with a new row (steps 9-12). If
+it succeeds, it moves to the next iteration. Otherwise, it tries to extend M[X, Y ] with a new row
+and a new column (steps 21-26). If it succeeds, it moves to the next iteration. If it fails, it accepts
+(step 27). If |X| < |Y | (step 13), it starts with the row of M[X, Y ] (steps 14-16), then the column
+(steps 18-20), and then both (steps 21-26). If it fails, it accepts (step 27).
+If |X| · |Y | >
+� d
+≤s
+�
+2d (step 2 and then step 28) or the s-binary rank of M[X, Y ] is greater than
+d (step 3), then it rejects.
+We first prove
+Lemma 7. Let t = 9d/ǫ. Tester Adaptive-Test-Rank makes at most 2
+� d
+≤s
+�
+2dt = ˜O
+�� d
+≤s
+�
+2d�
+/ǫ
+queries.
+Proof. We prove by induction that at every iteration of the main While-loop (step 2), the tester
+knows the entries of M[X, Y ], and the total number of queries, qX,Y , is at most 2|X||Y |t. Since
+the While-loop condition is |X||Y | ≤
+� d
+≤s
+�
+2d, the result follows.
+At the beginning of the algorithm, no queries are made, and |X| = |Y | = 1. Then 2|X||Y |t =
+2t > 0 = qX,Y .
+Suppose, at the kth iteration, the tester knows the entries of M[X, Y ] and
+qX,Y ≤ 2|X||Y |t. We prove the result for the (k + 1)th iteration.
+We have the following cases (at the (k + 1)th iteration)
+Case I. |X| ≥ |Y | (step 5) and, for some x, M[x, Y ] is a new row to M[X, Y ] (step 8).
+6
+
+Adaptive-Test-Rank(d, s, M, n, m, ǫ)
+Input: Oracle that accesses the entries of n × m (0, 1)-matrix M.
+Output: Either “Accept” or “Reject”
+1. X ← {1}; Y ← {1}; t = 9d/ǫ.
+2. While |X| · |Y | ≤
+� d
+≤s
+�
+2d do
+3.
+If the s-binary rank of M[X, Y ] is greater than d, then Reject.
+4.
+Finish ← False; X′ ← Ø; Y ′ ← Ø. /∗ X′ and Y ′ are multi-sets.
+5.
+If |X| ≥ |Y | then
+6.
+While (NOT Finish) AND |X′| < t
+7.
+Draw uniformly at random x ∈ [n]\X; X′ ← X′ ∪ {x};
+8.
+If M[x, Y ] is a new row to M[X, Y ] then X ← X ∪ {x}; Finish ← True.
+9.
+If (NOT Finish) then
+10.
+While (NOT Finish) AND |Y ′| < t
+11.
+Draw uniformly at random y ∈ [m]\Y ; Y ′ ← Y ′ ∪ {y}.
+12.
+If M[X, y] is new column to M[X, Y ] then Y ← Y ∪ {y}; Finish ← True.
+13.
+Else (|X| < |Y |)
+14.
+While (NOT Finish) AND |Y ′| < t
+15.
+Draw uniformly at random y ∈ [m]\Y ; Y ′ ← Y ′ ∪ {y};
+16.
+If M[X, y] is a new column to M[X, Y ] then Y ← Y ∪ {y}; Finish ← True.
+17.
+If (NOT Finish) then
+18.
+While (NOT Finish) AND |X′| < t
+19.
+Draw uniformly at random x ∈ [n]\X; X′ ← X′ ∪ {x}
+20.
+If M[x, Y ] is a new row to M[X, Y ] then X ← X ∪ {x}; Finish ← True.
+21.
+While (NOT Finish) AND X′ ̸= Ø do
+22.
+Draw uniformly at random x ∈ X′ and y ∈ Y ′
+23.
+If M[x, Y ∪ {y}] is a new row to M[X, Y ∪ {y}] OR, equivalently,
+24.
+M[X ∪ {x}, y] is a new column to M[X ∪ {x}, Y ]
+25.
+then X ← X ∪ {x}; Y ← Y ∪ {y}; Finish ← True.
+26.
+else X′ ← X′\{x}; Y ′ ← Y ′\{y}.
+27.
+If (NOT Finish) then Accept
+28.Reject
+Figure 1: An adaptive tester for s-binary rank at most d.
+In that case, Finish becomes true, and no other sub-while-loop is executed. Therefore, the
+number of queries made at this iteration is at most |Y |t (to find all M[x, Y ]), and one element x is
+added to X. Then, the tester knows all the entries of M[X ∪ {x}, Y ] and
+qX∪{x},Y = qX,Y + |Y |t ≤ 2|X||Y |t + |Y |t ≤ 2|X ∪ {x}| · |Y |t,
+and the result follows.
+Case II. |X| ≥ |Y | (step 5), for all x′ ∈ X′, M[x′, Y ] is not a new row to M[X, Y ] (step 8), and
+for some y, M[X, y] is a new column to M[X, Y ] (step 12).
+7
+
+In that case, Finish becomes true, and no other sub-while-loop is executed after the second
+sub-while-loop (step 10).
+Therefore, in this case, the number of queries made at this iteration is at most |Y |t + |X|t.
+|X|t queries in the first sub-while-loop (to find M[x, Y ] for all x ∈ X′), and at most |Y |t queries
+in the second sub-while-loop (to find M[X, y′] for all y′ ∈ Y ′). Then one element y is added to Y .
+Therefore, the tester knows the entries of M[X, Y ∪ {y}] and, since |Y | ≤ |X|,
+qX,Y ∪{y} = qX,Y + |X|t + |Y |t ≤ 2|X||Y |t + 2|X|t = 2|X| · |Y ∪ {y}|t,
+and the result follows.
+Case III. |X| ≥ |Y |, for all x′ ∈ X′, M[x′, Y ] is not a new row to M[X, Y ], for all y′ ∈ Y ′, M[X, y′]
+is not a new column to M[X, Y ], and for some x ∈ X′, y ∈ Y ′, M[x, Y ∪ {y}] is a new row to
+M[X, Y ∪ {y}] (step 23).
+In this case, |X′| = |Y ′| = t, the number of queries is |X|t + |Y |t + t. Exactly |X|t queries in
+the first sub-while-loop, |Y |t queries in the second sub-while-loop, and at most5 t queries in the
+sub-while-loop in step 21. Then one element x is added to X, and one element y is added to Y .
+Then the tester knows the entries of M[X ∪ {x}, Y ∪ {y}] and
+qX∪{x},Y ∪{y} = qX,Y + |X|t + |Y |t + t ≤ 2|X| · |Y |t + |X|t + |Y |t + t ≤ 2|X ∪ {x}| · |Y ∪ {y}|t.
+Case IV. |X| ≥ |Y |, for all x′ ∈ X′, M[x′, Y ] is not a new row to M[X, Y ], for all y′ ∈ Y ′, M[X, y′]
+is not a new column to M[X, Y ], and for all the drawn pairs x ∈ X′, y ∈ Y ′, M[x, Y ∪ {y}] is not
+a new row to M[X, Y ∪ {y}] (step 23).
+In this case, Finish will have value False, and the tester accepts in step 27.
+The analysis of the case when |X| < |Y | is similar to the above analysis.
+We now prove the completeness of the tester.
+Lemma 8. If M is a n × m (0, 1)-matrix of s-binary rank at most d, then the tester Adaptive-
+Test-Rank accepts with probability 1.
+Proof. The tester rejects if and only if one of the following occurs,
+1. M[X, Y ] has s-binary rank greater than d.
+2. |X| · |Y | >
+� d
+≤s
+�
+2d.
+If M[X, Y ] has s-binary rank greater than d, then M has s-binary rank greater than d. This is
+because, if M = NL, then M[X, Y ] = N[X, [d]] · L[[d], Y ]. So item 1 cannot occur.
+Before we show that item 2 cannot occur, we prove the following:
+Claim 1. The rows (resp. columns) of M[X, Y ] are distinct.
+Proof. The steps in the tester where we add rows or columns are steps 8, 12 16, 20, and 23. In
+steps 8, 12 16, 20 it is clear that a row (resp. column) is added only if it is a new row (resp.
+column) to M[X, Y ]. Consider step 23 and suppose, w.l.o.g |X| ≥ |Y |. This step is executed only
+when Finish = False. This happens when |X′| = |Y ′| = t, for every x ∈ X′, M[x, Y ] is not a
+new row to M[X, Y ], and for every y ∈ Y ′, M[X, y] is not a new column to M[X, Y ]. Then x
+5This is because, for x ∈ X′, y ∈ Y ′, the tester already knows M[x, Y ] and M[X, y] from the first and second
+sub-while-loop and only needs to query M[x, y].
+8
+
+and y are added to X and Y , respectively, if M[x, Y ∪ {y}] is a new row to M[X, Y ∪ {y}]. Then,
+by Lemma 3, M[X ∪ {x}, y] is a new column to M[X ∪ {x}, Y ]. So, the rows (and columns) in
+M[X ∪ {x}, Y ∪ {y}] are distinct. This implies the result.
+Suppose, to the contrary, |X| · |Y | >
+� d
+≤s
+�
+2d. Since M′ = M[X, Y ] satisfies r(M′)c(M′) =
+|X| · |Y | >
+� d
+≤s
+�
+2d, by Lemma 4, the s-binary rank of M′, and therefore of M, is greater than d. A
+contradiction.
+We now prove the soundness of the tester.
+We first prove the following.
+Claim 2. Let M be a n×m (0, 1)-matrix, X ⊆ [n], and Y ⊆ [m]. Suppose there are two functions,
+′ : [n] → X and ′′ : [m] → Y , such that
+1. For every x ∈ [n], M[x, Y ] = M[x′, Y ].
+2. For every y ∈ [m], M[X, y] = M[X, y′′].
+3. For every x ∈ [n] and y ∈ [m], M[x, y] = M[x′, y′′].
+Then M has at most |X| distinct rows and |Y | distinct columns, and its s-binary rank is the s-binary
+rank of M[X, Y ].
+Proof. Let x ∈ [n]\X. For every y, M[x, y] = M[x′, y′′] = M[x′, y]. Therefore, row x in M is equal
+to row x′. Similarly, column y in M is equal to column y′′.
+Since adding equal columns and rows to a matrix does not change the s-binary rank6, we have
+brs(M[X, Y ]) = brs(M[X, [m]]) = brs(M).
+The following Claim is proved in [7] (Claim 10). Here, we give the proof for completeness.
+Claim 3. Let M be a (0, 1)-matrix that is ǫ-far from having s-binary rank at most d. Let X ⊆ [n]
+and Y ⊆ [m], such that brs(M[X, Y ]) ≤ d, the columns of M[X, Y ] are distinct, and the rows of
+M[X, Y ] are distinct. Then one of the following must hold:
+1. The number of rows x ∈ [n] where M[x, Y ] is a new row to M[X, Y ] is at least nǫ/3.
+2. The number of columns y ∈ [m] where M[X, y] is a new column to M[X, Y ] is at least mǫ/3.
+3. The number pairs (x, y), x ̸∈ X, y ̸∈ Y , where, M[x, Y ] = M[x′, Y ] for some x′ ∈ X,
+M[X, y] = M[X, y′′] for some y′′ ∈ Y , and M[x, y] ̸= M[x′, y′′], is at least mnǫ/3.
+Proof. Assume, to the contrary, that none of the above statements holds. Change every row x in
+M where M[x, Y ] is a new row to M[X, Y ] to a zero row. Let X′ be the set of such rows. Change
+every column y in M where M[X, y] is a new row to M[X, Y ] to a zero column. Let Y ′ be the
+set of such columns. For every other entry (x, y), x ̸∈ X, y ̸∈ Y that is not changed to zero and
+M[x, y] ̸= M[x′, y′′], change M[x, y] to M[x′, y′′]. Let M′ be the matrix obtained from the above
+changes.
+The number of entries (x, y) where M[x, y] ̸= M′[x, y] is less than (nǫ/3)m + (mǫ/3)n +
+mnǫ/3 = ǫmn. Therefore, M′ is ǫ-close to M. By claim 3, brs(M′) = brs(M[[n]\X′, [m]\Y ′]) =
+brs(M[X, Y ]) ≤ d. A contradiction.
+6If we add a column to a matrix that is equal to column y, then the rectangles that cover column y can be extended
+to cover the added column.
+9
+
+We now prove the completeness of the tester.
+Lemma 9. If M is ǫ-far from having s-binary rank d, then with probability at least 2/3, Adaptive-
+Test-Rank rejects.
+Proof. Consider the while-loop in step 2 at some iteration i. If brs(M[X, Y ]) > d, then the tester
+rejects in step 3. We will now show that if brs(M[X, Y ]) ≤ d, then, with probability at most 3e−2d,
+the tester accepts at iteration i.
+To this end, let brs(M[X, Y ]) ≤ d. Then, by Claim 3, one of the following holds.
+1. The number of rows x ∈ [n] where M[x, Y ] is a new row to M[X, Y ] is at least nǫ/3.
+2. The number of columns y ∈ [m] where M[X, y] is a new column to M[X, Y ] is at least mǫ/3.
+3. The number pairs (x, y), x ̸∈ X, y ̸∈ Y , where, M[x, Y ] = M[x′, Y ] for some x′ ∈ X,
+M[X, y] = M[X, y′′] for some y′′ ∈ Y , and M[x, y] ̸= M[x′, y′′], is at least mnǫ/3.
+Now at the ith iteration, suppose w.l.o.g, |X| ≥ |Y | (the other case |Y | < |X| is similar). If item 1
+occurs, then with probability at least p = 1 − (1 − ǫ/3)t ≥ 1 − e−2d, the tester finds a new row
+to M[X, Y ] and does not accept at iteration i. If item 2 occurs, then if it does not find a new
+row to M[X, Y ], with probability at least p, the tester finds a new column to M[X, Y ] and does
+not accept. If item 3 occurs, and it does not find a new row or column to M[X, Y ], then with
+probability at least p, it finds such a pair and does not accept. Therefore, with probability at most
+3(1 − p) ≤ 3e−2d, the tester accepts at iteration i.
+Since the while-loop runs at most |X| + |Y | ≤ 2|X||Y | ≤ 2
+� d
+≤s
+�
+2d ≤ 22d+1 iterations, with
+probability at most 3e−2d22d+1 ≤ 1/3, the tester accepts in while-loop. Therefore, with proba-
+bility at least 2/3, the tester does not accept in the while-loop. Thus, it either rejects because
+brs(M[X, Y ]) > d or rejects in step 28.
+4.2
+The Non-Adaptive Tester
+In this section, we prove Theorem 2.
+First, consider Adaptive-Test-Rank in Figure 1. Consider steps 7,11,15, and 19, where it
+draws a new column or row. We prove.
+Lemma 10. Let t = 9d/ǫ. At each iteration of Adaptive-Test-Rank, the total number of uni-
+formly at random rows x ∈ [n] drawn is at most (|X| + min(|X|, |Y | − 1))t, and the number of
+uniformly at random rows y ∈ [m] drawn is at most (|Y | + min(|X|, |Y |))t.
+Proof. We prove by induction that at every iteration of the main While-loop (step 2), the total
+number of random rows drawn by the tester, nX,Y , is at most (|X| + min(|X|, |Y | − 1))t, and the
+total number of random columns drawn, mX,Y , is at most (|Y | + min(|X|, |Y |))t.
+At the beginning, |X| = |Y | = 1, and the number of columns and rows is 1. In that case,7,
+nX,Y = 1 ≤ t and mX,Y = 1 ≤ 2t. Suppose, at the kth iteration, the induction statement is true.
+We prove the result for the (k + 1)th iteration.
+At the (k + 1)th iteration, we have the following cases.
+Case I. |X| ≥ |Y | (step 5) and, for some x, M[x, Y ] is a new row to M[X, Y ] (step 8).
+7We assume that the first column/row drawn is column/row one
+10
+
+Non-Adaptive-Test-Rank(d, s, M, n, m, ǫ)
+Input: Oracle that accesses the entries of (0, 1)-matrix M.
+Output: Either “Accept” or “Reject”.
+1.
+T ←
+324·d2( d
+≤s)2d
+ǫ2
+.
+2.
+Dray uniformly at random x(1), . . . , x(T) ∈ [n].
+3.
+Dray uniformly at random y(1), . . . , y(T) ∈ [m].
+4.
+For every i ∈ [T] and j ∈ [T] such that i · j ≤ T
+5.
+D[i, j] ← Query M[x(i), y(j)]
+6.
+u = 1; w = 1.
+7.
+Run Adaptive-Test-Rank(d, s, M, n, m, ǫ)
+When the tester asks for a uniform at random x - return x(u); u ← u + 1
+When the tester asks for a uniform at random y - return y(w); w ← w + 1
+When the tester makes the Query M[x(i), y(j)] - return D[i, j]
+Figure 2: A non-adaptive tester for s-binary rank at most d.
+In that case, Finish becomes true, and no other sub-while-loop is executed. Therefore, the
+number of rows drawn at this iteration is at most t, and one element x is added to X. No columns
+are drawn. Then,
+nX∪{x},Y ≤ nX,Y + t ≤ (|X| + min(|X|, |Y | − 1) + 1)t ≤ (|X ��� {x}| + min(|X ∪ {x}|, |Y | − 1))t,
+and
+mX∪{x},Y = mX,Y ≤ (|Y | + min(|X|, |Y |))t ≤ (|Y | + min(|X ∪ {x}|, |Y |))t.
+Thus, the result follows for this case.
+Case II. |X| ≥ |Y | (step 5), for all x′ ∈ X′, M[x′, Y ] is not a new row to M[X, Y ] (step 8), and
+for some y, M[X, y] is a new column to M[X, Y ] (step 12).
+In that case, Finish becomes true, and no other sub-while-loop is executed after the second
+sub-while-loop (step 10).
+Therefore, in this case, the number of rows drawn at this iteration is t, one element y is added
+to Y , and the number of columns drawn is at most t. Then
+nX,Y ∪{y} = nX,Y + t
+≤
+(|X| + min(|X|, |Y | − 1) + 1)t
+=
+(|X| + |Y |)t = (|X| + min(|X|, |Y ∪ {y}| − 1))t,
+and
+mX,Y ∪{y} ≤ mX,Y + t ≤ (|Y | + min(|X|, |Y |) + 1)t ≤ (|Y ∪ {y}| + min(|X|, |Y ∪ {y}|))t.
+Thus, the result follows for this case.
+Case III. |X| < |Y | (step 13), and for some y, M[X, y] is a new column to M[X, Y ] (step 16).
+11
+
+In that case, Finish becomes true, and no other sub-while-loop is executed. Therefore, the
+number of columns drawn at this iteration is at most t, and one element y is added to Y . No rows
+are drawn. Then,
+nX,Y ∪{y} = nX,Y ≤ (|X| + min(|X|, |Y | − 1))t ≤ (|X| + min(|X|, |Y ∪ {y}| − 1))t,
+and
+mX,Y ∪{y} ≤ mX,Y + t ≤ (|Y | + min(|X|, |Y |) + 1)t = (|Y ∪ {y}| + min(|X|, |Y ∪ {y}|))t.
+Thus, the result follows for this case.
+Case IV. |X| < |Y | (step 13), for all y′ ∈ Y ′, M[X, y′] is not a new row to M[X, Y ], and for some
+x, M[x, Y ] is a new column to M[X, Y ] (step 20). In that case, Finish becomes true, and no other
+sub-while-loop is executed after the fourth sub-while-loop (step 18).
+In this case, the number of rows drawn at this iteration is t, one element x is added to X, and
+the number of columns drawn is at most t. Then
+nX∪{x},Y = nX,Y + t
+≤
+(|X| + min(|X|, |Y | − 1) + 1)t
+≤
+(|X ∪ {x}| + min(|X ∪ {x}|, |Y | − 1))t
+mX∪{x},Y ≤ mX,Y + t ≤ (|Y | + min(|X|, |Y |) + 1)t = (|Y | + min(|X ∪ {x}|, |Y |))t.
+Thus, the result follows for this case.
+Case V. For all x′ ∈ X′, M[x′, Y ] is not a new row to M[X, Y ], for all y′ ∈ Y ′, M[X, y′] is not a
+new column to M[X, Y ], and for some x ∈ X′, y ∈ Y ′, M[x, Y ∪{y}] is a new row to M[X, Y ∪{y}]
+(step 23).
+In this case, the number of rows drawn at this iteration is t, the number of columns drawn is t,
+one element x is added to X, and one element y is added to Y . Then
+nX∪{x},Y ∪{y} = nX,Y + t
+≤
+(|X| + min(|X|, |Y | − 1) + 1)t
+≤
+(|X ∪ {x}| + min(|X ∪ {x}|, |Y ∪ {y}| − 1))t.
+mX∪{x},Y ∪{y} = mX,Y + t
+≤
+(|Y | + min(|X|, |Y |) + 1)t
+≤
+(|Y ∪ {y}| + min(|X ∪ {x}|, |Y ∪ {y}|))t.
+We are now ready to prove Theorem 2.
+Proof. By Lemma 10, the total number of rows and columns drawn in Adaptive-Test-Rank
+up to iteration t is at most n′ := 9(|X| + min(|X|, |Y | − 1))d/ǫ ≤ 18|X|d/ǫ and m′ := 9(|Y | +
+min(|X|, |Y |)d/ǫ ≤ 18|Y |d/ǫ, respectively. We also have |X| · |Y | ≤
+� d
+≤s
+�
+2d. So
+n′ · m′ ≤ 324|X||Y |d2/ǫ2 ≤ T :=
+324 · d2� d
+≤s
+�
+2d
+ǫ2
+.
+Consider the tester Non-Adaptive-Test-Rank in Figure 2. The tester draws T rows x(1), . . . ,
+x(T) ∈ [n], and columns y(1), . . . , y(T) ∈ [m] and queries all M[x(i), y(j)] where ij ≤ T and puts the
+12
+
+result in the table D. Then it runs Adaptive-Test-Random using the above-drawn rows and
+columns. We now show that all the queries that Adaptive-Test-Random makes can be fetched
+from the table D.
+At any iteration, the number of rows drawn is at most n′, and the number of rows drawn is at
+most m′. Therefore, the tester needs to know (in the worst case) all the entries M[x(i), y(j)] where
+i ≤ n′ and j ≤ m′. Since ij ≤ n′m′ ≤ T, the result follows.
+The number of queries that the tester makes is
+T
+�
+i=1
+T
+i = O(T ln T) = ˜O
+�� d
+≤s
+�
+2d
+ǫ2
+�
+.
+5
+Testing the Exact s-Binary Rank
+We first prove the following.
+Lemma 11. Let M and M′ be n × m (0, 1)-matrices that differ in one row (or column). Then
+|brs(M) − brs(M′)| ≤ 1.
+Proof. Suppose brs(M) = d and M′ differ from M in row k. Let N and L be n × d (0, 1)-matrix
+and d × m (0, 1)-matrix, respectively, such that P = NL, for every (i, j) ∈ [n] × [m], P[i, j] ≤ s,
+and P[i, j] = 0 if and only if M[i, j] = 0. Add to N a column (as a (d + 1)th column) that all its
+entries are zero except the k-th entry, which equals 1. Then change N[k, j] to zero for all j ∈ [d].
+Let N ′ be the resulting matrix. Add to L another row (as a (d + 1)th row) equal to the k-th row
+of M′. Let L′ be the resulting matrix. Let P ′ = N ′L′. It is easy to see that P ′[i, j] = P[i, j] for all
+i ̸= k and j, and the kth row of P ′ is equal to the kth row of M′. Then, for every (i, j) ∈ [n] × [m],
+P ′[i, j] ≤ s, and P ′[i, j] = 0 if and only if M′[i, j] = 0. Therefore, brs(M′) ≤ d + 1 = brs(M) + 1.
+In the same way, brs(M) ≤ brs(M′) + 1.
+Lemma 12. Let η = d2/(nm). Let M be n × m (0, 1)-matrix. If M is ǫ-close to having s-binary
+rank at most d, then M is (ǫ + η)-close to having s-binary rank d.
+Proof. We will show that for every n × m (0, 1)-matrix H of s-binary rank at most d − 1, there is a
+n × m (0, 1)-matrix G of s-binary rank d that is η-close to H. Therefore, if M is ǫ-close to having
+s-binary rank at most d, then it is (ǫ + η)-close to having s-binary rank d.
+Define the n×m (0, 1)-matrices Gk, k ∈ [d]∪{0}, where G0 = H and for k ≥ 1, Gk[i, j] = H[i, j]
+if j > k or i > d, and Gk[[d], [k]] = Id[[d], [k]] where Id is the d × d identity matrix.
+Since
+Gd[[d], [d]] = Id, we have brs(Gd) ≥ d. It is clear that for every k ∈ [d] ∪ {0}, Gk is (d2/nm)-close
+to H. If brs(Gd) = d, then take G = Gd, and we are done. Otherwise, suppose brs(Gd) > d.
+Now consider a sequence H = G0, G1, G2, . . . , Gd. By Lemma 11, we have brs(Gi−1) − 1 ≤
+brs(Gi) ≤ brs(Gi−1) + 1. Now since brs(G0) = brs(H) ≤ d − 1 and brs(Gd) > d, by the discrete
+intermediate value theorem, there must be k ∈ [d] such that brs(Gk) = d. Then take G = Gk, and
+we are done.
+Now, the tester for testing the s-binary rank d runs as follows.
+If mn < 2d2/ǫ, then find
+all the entries of M with mn < 2d2/ǫ queries. If brs(M) = d, then accept. Otherwise, reject.
+13
+
+If mn ≥ 2d2/ǫ, then run Adaptive-Test-Rank(d, s, M, n, m, ǫ/2) (for the non-adaptive, we run
+Non-Adaptive-Test-Rank(d, s, M, n, m, ǫ/2)) and output its answer.
+We now show the correctness of this algorithm. If M is of s-binary rank d, then it is of s-binary
+rank at most d, and the tester accepts.
+Now, suppose f is ǫ-far from having s-binary rank d. If mn < 2d2/ǫ, the tester rejects. If
+mn ≥ 2d2/ǫ, then, by Lemma 12, f is (ǫ − η)-far from having s-binary rank at most d, where
+η = d2/(nm). Since η = d2/(nm) ≤ ǫ/2, the function f is (ǫ/2)-far from having s-binary rank at
+most d, and therefore the tester, with probability at least 2/3, rejects.
+References
+[1] Maria-Florina Balcan, Yi Li, David P. Woodruff, and Hongyang Zhang. Testing matrix rank,
+optimally. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algo-
+rithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 727–746, 2019.
+doi:10.1137/1.9781611975482.46.
+[2] Parinya Chalermsook, Sandy Heydrich, Eugenia Holm, and Andreas Karrenbauer. Nearly tight
+approximability results for minimum biclique cover and partition. In Andreas S. Schulz and
+Dorothea Wagner, editors, Algorithms - ESA 2014 - 22th Annual European Symposium, Wro-
+claw, Poland, September 8-10, 2014. Proceedings, volume 8737 of Lecture Notes in Computer
+Science, pages 235–246. Springer, 2014. doi:10.1007/978-3-662-44777-2\_20.
+[3] Dana Ron. Private Communication.
+[4] David A. Gregory, Norman J. Pullman, Kathryn F. Jones, and J. Richard Lundgren. Biclique
+coverings of regular bigraphs and minimum semiring ranks of regular matrices. J. Comb. Theory,
+Ser. B, 51(1):73–89, 1991. doi:10.1016/0095-8956(91)90006-6.
+[5] Yi Li, Zhengyu Wang, and David P. Woodruff.
+Improved testing of low rank matrices.
+In The 20th ACM SIGKDD International Conference on Knowledge Discovery and Data
+Mining, KDD ’14, New York, NY, USA - August 24 - 27, 2014, pages 691–700, 2014.
+doi:10.1145/2623330.2623736.
+[6] Yonatan Nakar and Dana Ron.
+On the testability of graph partition properties.
+In
+Eric Blais, Klaus Jansen, Jos´e D. P. Rolim, and David Steurer, editors, Approxima-
+tion, Randomization, and Combinatorial Optimization. Algorithms and Techniques, AP-
+PROX/RANDOM 2018,
+August 20-22,
+2018
+- Princeton,
+NJ, USA, volume
+116
+of
+LIPIcs,
+pages
+53:1–53:13.
+Schloss
+Dagstuhl
+-
+Leibniz-Zentrum
+f¨ur
+Informatik,
+2018.
+doi:10.4230/LIPIcs.APPROX-RANDOM.2018.53.
+[7] Michal Parnas, Dana Ron, and Adi Shraibman. Property testing of the boolean and binary
+rank. Theory Comput. Syst., 65(8):1193–1210, 2021. doi:10.1007/s00224-021-10047-8.
+[8] Jir´ı Sgall. Bounds on pairs of families with restricted intersections. Comb., 19(4):555–566, 1999.
+doi:10.1007/s004939970007.
+14
+

49AyT4oBgHgl3EQfcPf_/content/tmp_files/2301.00281v1.pdf.txt

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+arXiv:2301.00281v1  [cs.LG]  31 Dec 2022
+arXiv® 2022 (cs.LG) 1-7
+Submitted 12/22; Published 12/22
+Lightmorphic Signatures Analysis Toolkit
+Dumitru Damian
+[email protected]
+Information and Communication Engineering
+Research and development consultant
+Timis,oara, RO
+Abstract
+In this paper we discuss the theory used in the design of an open source lightmorphic sig-
+natures analysis toolkit (LSAT). In addition to providing a core functionality, the software
+package enables specific optimizations with its modular and customizable design.
+To promote its usage and inspire future contributions, LSAT is publicly available. By
+using a self-supervised neural network and augmented machine learning algorithms, LSAT
+provides an easy-to-use interface with ample documentation.
+The experiments demonstrate that LSAT improves the otherwise tedious and error-
+prone tasks of translating lightmorphic associated data into usable spectrograms, enhanced
+with parameter tuning and performance analysis.
+With the provided mathematical functions, LSAT validates the nonlinearity encoun-
+tered in the data conversion process while ensuring suitability of the forecasting algorithms.
+Keywords:
+lightmorphic, machine learning, spectrogram, graph chord, neural network
+1. Introduction
+It is common knowledge, in the machine learning domain, to use differential values, since
+they provide a simple way to model the data. However, such algorithms may not fit the
+lightmorphic signature properly, leading to a reduced quality of the obtained results. Train-
+ing a neural network to predict the lightmorphic signature can significantly increase the data
+quality. This is the task that LSAT tries to accomplish.
+As such we define the lightmorphic metric learning (LML) as a branch of machine
+learning algorithms, set out with the purpose of learning lightmorphic signatures from
+multiple datasets trough usage of vibrating graph chords.
+In the pursuing sections we describe the main features of the toolkit, explain the general
+mathematical concepts and finally detail the plans regarding future functionalities.
+2. General mathematical concepts
+In this section we expand the mathematical concepts and link them with the reasoning
+encountered in the implemented code.
+We define the lightmorphic signature as a function of: light intensity (I) that varies
+according to seasons and local weather conditions, trajectory distribution characteristics
+©2022 Dumitru Damian.
+License: CC-BY 4.0, see https://creativecommons.org/licenses/by/4.0/.
+Typeset in LaTex using the JMLR LaTeX style file https://jmlr.org/author-info.html.
+
+Damian
+(D), and specific adjustments (T):
+fL⊙ =
+I
+�
+1
+D
+�
+1
+T
+�
+1
+Γtζtdt
+(1)
+where:
+• Γt – trajectory tensor
+• ζt – point in time specificity
+Storage of these trajectory specific lightmorphic signatures is done in a database (Θ).
+The segments containing isochronous surfaces with similarities are stored in another database
+(Φ) that serves as a baseline for training the neural network implementation.
+The isochronous surfaces that constitute the lightmorphic signature are interlinked
+trough the definition and usage of graph chords (δ(t)). Observing their vibrational am-
+plitude allows the prediction of alternative lightmorphic signatures and, at the same time,
+correction of the already known values.
+Since the primary light source considered is the Earth’s Sun, specific spacetime metrics
+(ex. gµν, ηµν, h+, h×, Gµν) have to be used in order to describe the encountered anisotropies.
+These are implemented as a function of distant astrophysical forces that stretch and com-
+press the fabric of spacetime.
+According to special relativity, spacetime is seen as a four dimensional manifold de-
+scribed by a flat Minkowski metric defined in Cartesian coordinates (t, x, y, z, c = 1)
+as:
+ηµν =
+
+
+
+
+−1
+0
+0
+0
+0
+1
+0
+0
+0
+0
+1
+0
+0
+0
+0
+1
+
+
+
+ ,
+(2)
+When considering the geometry of curved space, we have made use of the metric gµν,
+that replaces the flat Minkowski metric ηµν. This substitution was done considering that
+the geometry of curved space will eventually reduce to the flat spacetime of special relativity
+at a sufficiently small scale.
+The interaction between curvature of spacetime and the mass distribution was modeled
+following (Blackburn (2010)) work, as:
+Gµν = kTµν
+(3)
+where Gµν is defined as the Einstein curvature tensor, Tµν is the stress-energy tensor and
+represents the mass-energy distribution, while k describes the Einstein constant of gravita-
+tion defined as:
+k = 8πG
+c4
+(4)
+where c is the speed of light in a vacuum.
+2
+
+Lightmorphic Signatures Analysis Toolkit
+At the same time, in order to improve the results quality, the Einstein tensor was also
+considered under the form:
+Gµν = Rµν − 1
+2gµνR,
+(5)
+where Rµν is the Riemann tensor for the local spacetime, and R is the Ricci scalar.
+Since there is not one general solution for the complex Einstein equations, but a large
+variety of possible solutions that apply to particular circumstances, we’ve considered a weak-
+field approximation, where the nonlinear Einstein equations where approximated towards
+linearity.
+For example, a very small perturbation specific to a gravitational wave, will impact the
+flat spacetime and it is defined as hµν(x) and it’s value will be |hµν| << 1.
+Thus, the Einstein equation becomes:
+gµν(x) = ηµν + hµν(x).
+(6)
+or by simply considering the induced strain variations:
+□hµν(x) = 0,
+(7)
+By further pursuing such linearization, we can represent in the TT gauge, a propagating
+wave, under the following form:
+hTT
+µν =
+
+
+
+
+0
+0
+0
+0
+0
+h+
+h×
+0
+0
+h×
+−h+
+0
+0
+0
+0
+0
+
+
+
+ ,
+(8)
+where the constant amplitudes (h+, h×) represent the two gravitational wave polariza-
+tions, the plus- and cross-polarization.
+We represent the distance between two neighboring points as defined by (Berit (2013))
+for a flat spacetime, trough the following expression:
+ds2 = −c2dt2 + dx2 + dy2 + dz2 = −c2dt2 + [1 + h+(t)]dx2 + [1 − h+(t)]dy2
+That allows us to model in the TT gauge, the gravitational wave stretching along the x
+axis and compression along the y axis with the specific factor of:
+�
+1 ± h+(t) ≃ 1 + 1
+2h+(t)
+Having modeled the photon’s traveling path in outer space, in order to simplify the
+inherent path inhomogeneities, we separated the domains into outer space domain, atmo-
+spheric domain and Earth specific domains (lithosphere, hydrosphere, biosphere, noises,
+etc).
+We further define the phase of an electromagnetic wave of frequency ω0 as φ. Following
+Driggers (2015)’s work, we consider that the starting light phase is at 0 and it travels at
+the speed of light c. After a distance L it will have a phase δφspace that can be expressed as
+a distance integral over the spacetime metric,
+δφspace = ω0
+c
+� L
+0
+gdx,
+(9)
+3
+
+Damian
+with g(t) = η+h(t), where η is the Minkowski metric and h(t) is the dimensionless spacetime
+strain.
+Summing the light phase shift δφatm and the δφEarth which is derived from the noise
+sources like seismic or electromagnetic interferences, leads to the dataset of trajectory spe-
+cific lightmorphic signatures:
+ΦΓIDT =
+N
+�
+j=1
+Γj
+IDT
+(10)
+The signature parameter estimation is performed considering a prior distribution p(Φ|L⊙)
+that is updated upon receiving the new data d to give a posterior distribution p(Φ|d, L⊙)
+p(Φ|d, L⊙) = p(Φ|L⊙)p(d|Φ, L⊙)
+p(d|L⊙)
+(11)
+While observing the distribution of multiple light segments within the dataset ΦΓIDT ,
+it will be possible to estimate the probability for trajectory specific lightmorphic evolution:
+pΦ = f(ρk · pΦk)
+(12)
+where pΦk is the database’s k-th segment specific probability, ρk is the prediction weight
+for the k-th segment.
+3. Software package design
+The distribution matrices specific to the isochronous segmentation surfaces, which define
+the lightmorphic signature model, form the LSAT core.
+As such we’ve used a design principle that ensures simplicity for the whole package,
+while making the source codes easy to read and maintain. As the toolkit is written in a
+modular way, new functionalities can be easily plugged in. This makes the LSAT not only
+a lightmorphic signature machine learning tool but also an experimental platform.
+LSAT comes with plenty of documentation for all the interface functionalities and related
+data structures. The README file describes the installation process and interface usage.
+For developers who use the toolkit in their applications, the API documentation can provide
+additional information related to functionality calls.
+4. Practical Usage
+In the examples, we provide sample values for the lightmorphic signature updates, as a
+function of δφatm derived by the neural network from the values of a large dataset of at-
+mospheric meteorological data for 317 cities in Romania, with hundreds of thousands data
+points.
+Automatic learning is supported trough API calls to the domain specific data
+providers.
+Beyond this simple way of running the lightmorphic signatures analysis toolkit, there are
+several enhancement options for advanced usage. As example, one may activate additional
+functionalities that consider input parameters like complex space weather forecasting, dif-
+ferent electromagnetic wave disturbances or lithosphere, hydrosphere and biosphere specific
+localized data.
+4
+
+Lightmorphic Signatures Analysis Toolkit
+5. Conclusion and Future Work
+With the lightmorphic signatures analysis toolkit we provided an open source SW package
+that is simple and easy-to-use.
+Experiments and analysis conclude that the modular design and customization support
+are performing excellent in practice and can provide the base for additional research on
+lightmorphic signatures.
+The toolkit is constantly being improved by new research results and user feed-back
+with the ultimate goal of having an automated toolkit to use in maintaining and updating
+a large database of high-quality light signatures.
+Future work will focus on probability estimates, additional functionalities that mitigate
+the large uncertainties in the available observational input data which arise from the complex
+interaction processes. In addition, the inclusion of artificial intelligence (AI) options will
+be considered while building a national/international network for lightmorphic signature
+analysis.
+5
+
+Damian
+6. References
+References
+Rana Adhikari. Sensitivity and noise analysis of 4 km laser interferometric gravitational
+wave antennae. PhD thesis, Massachusetts Institute of Technology, 2004.
+Behnke Berit. A Directed Search for Continuous Gravitational Waves from Unknown Iso-
+lated Neutron Stars at the Galactic Center. PhD thesis, Leibniz University Hannover,
+2013.
+Sylvia Biscoveanu, Maximiliano Isi, Salvatore Vitale, and Vijay Varma.
+New Spin
+on LIGO-Virgo Binary Black Holes.
+Phys. Rev. Lett., 126(17):171103, 2021.
+doi:
+10.1103/PhysRevLett.126.171103.
+Lindy Blackburn. Open Issues in the Search for Gravitational Wave Transients. PhD thesis,
+Massachusetts Institute of Technology, 2010.
+Daniel E. Clark. Control of Differential Motion between Adjacent Advanced LIGO Seismic
+Isolation Platforms. PhD thesis, Stanford University, 2013.
+Katherine Laird Dooley. Design and performance of high laser power interferometers for
+gravitational-wave detection. PhD thesis, University of Florida, 2011.
+Jennifer Clair Driggers. Noise Cancellation for Gravitational Wave Detectors. PhD thesis,
+California Institute of Technology, 2015.
+Tobin Thomas Fricke. Homodyne detection for laser-interferometric gravitational wave de-
+tectors. PhD thesis, Louisiana State University and Agricultural and Mechanical College,
+2011.
+Paul D. Lasky et al. Gravitational-wave cosmology across 29 decades in frequency. Phys.
+Rev. X, 6(1):011035, 2016. doi: 10.1103/PhysRevX.6.011035.
+Michele Maggiore. Gravitational wave experiments and early universe cosmology. Phys.
+Rept., 331:283–367, 2000. doi: 10.1016/S0370-1573(99)00102-7.
+Denis Martynov. Lock Acquisition and Sensitivity Analysis of Advanced LIGO Interferom-
+eters. PhD thesis, California Institute of Technology, 2015.
+Ryan Quitzow-James. Search for Long-Duration Transient Gravitational Waves Associated
+with Magnetar Bursts during LIGO’s Sixth Science Run. PhD thesis, Oregon U., 2016.
+Joseph D. Romano and Neil J. Cornish. Detection methods for stochastic gravitational-
+wave backgrounds: a unified treatment. Living Rev. Rel., 20(1):2, 2017. doi: 10.1007/
+s41114-017-0004-1.
+Michael P. Ross. Precision Mechanical Rotation Sensors for Terrestrial Gravitational Wave
+Observatories. PhD thesis, University of Washington, 2020.
+6
+
+Lightmorphic Signatures Analysis Toolkit
+Darkhan Tuyenbayev. Extending the scientific reach of Advanced LIGO by compensating
+for temporal variations in the calibration of the detectors. PhD thesis, The University of
+Texas at San Antonio, 2017.
+Madeline Wade. Gravitational-Wave Science with the Laser Interferometer Gravitational-
+Wave Observatory. PhD thesis, University of Wisconsin–Milwaukee, 2015.
+7
+

49AyT4oBgHgl3EQfcPf_/content/tmp_files/load_file.txt

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='00281v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='LG]  31 Dec 2022  arXiv® 2022 (cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='LG) 1-7  Submitted 12/22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Published 12/22  Lightmorphic Signatures Analysis Toolkit  Dumitru Damian  dumitrudamian@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='com  Information and Communication Engineering  Research and development consultant  Timis,oara, RO  Abstract  In this paper we discuss the theory used in the design of an open source lightmorphic sig-  natures analysis toolkit (LSAT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' In addition to providing a core functionality, the software  package enables specific optimizations with its modular and customizable design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  To promote its usage and inspire future contributions, LSAT is publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' By  using a self-supervised neural network and augmented machine learning algorithms, LSAT  provides an easy-to-use interface with ample documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  The experiments demonstrate that LSAT improves the otherwise tedious and error-  prone tasks of translating lightmorphic associated data into usable spectrograms, enhanced  with parameter tuning and performance analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  With the provided mathematical functions, LSAT validates the nonlinearity encoun-  tered in the data conversion process while ensuring suitability of the forecasting algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  Keywords:  lightmorphic, machine learning, spectrogram, graph chord, neural network  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Introduction  It is common knowledge, in the machine learning domain, to use differential values, since  they provide a simple way to model the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' However, such algorithms may not fit the  lightmorphic signature properly, leading to a reduced quality of the obtained results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Train-  ing a neural network to predict the lightmorphic signature can significantly increase the data  quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' This is the task that LSAT tries to accomplish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  As such we define the lightmorphic metric learning (LML) as a branch of machine  learning algorithms, set out with the purpose of learning lightmorphic signatures from  multiple datasets trough usage of vibrating graph chords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  In the pursuing sections we describe the main features of the toolkit, explain the general  mathematical concepts and finally detail the plans regarding future functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' General mathematical concepts  In this section we expand the mathematical concepts and link them with the reasoning  encountered in the implemented code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  We define the lightmorphic signature as a function of: light intensity (I) that varies  according to seasons and local weather conditions, trajectory distribution characteristics  ©2022 Dumitru Damian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  License: CC-BY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='0, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='0/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  Typeset in LaTex using the JMLR LaTeX style file https://jmlr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='org/author-info.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  Damian  (D), and specific adjustments (T):  fL⊙ =  I  �  1  D  �  1  T  �  1  Γtζtdt  (1)  where:  Γt – trajectory tensor  ζt – point in time specificity  Storage of these trajectory specific lightmorphic signatures is done in a database (Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  The segments containing isochronous surfaces with similarities are stored in another database  (Φ) that serves as a baseline for training the neural network implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  The isochronous surfaces that constitute the lightmorphic signature are interlinked  trough the definition and usage of graph chords (δ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Observing their vibrational am-  plitude allows the prediction of alternative lightmorphic signatures and, at the same time,  correction of the already known values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  Since the primary light source considered is the Earth’s Sun, specific spacetime metrics  (ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' gµν, ηµν, h+, h×, Gµν) have to be used in order to describe the encountered anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  These are implemented as a function of distant astrophysical forces that stretch and com-  press the fabric of spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  According to special relativity, spacetime is seen as a four dimensional manifold de-  scribed by a flat Minkowski metric defined in Cartesian coordinates (t, x, y, z, c = 1)  as:  ηµν =  \uf8eb  \uf8ec  \uf8ec  \uf8ed  −1  0  0  0  0  1  0  0  0  0  1  0  0  0  0  1  \uf8f6  \uf8f7  \uf8f7  \uf8f8 ,  (2)  When considering the geometry of curved space, we have made use of the metric gµν,  that replaces the flat Minkowski metric ηµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' This substitution was done considering that  the geometry of curved space will eventually reduce to the flat spacetime of special relativity  at a sufficiently small scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  The interaction between curvature of spacetime and the mass distribution was modeled  following (Blackburn (2010)) work, as:  Gµν = kTµν  (3)  where Gµν is defined as the Einstein curvature tensor, Tµν is the stress-energy tensor and  represents the mass-energy distribution, while k describes the Einstein constant of gravita-  tion defined as:  k = 8πG  c4  (4)  where c is the speed of light in a vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  2  Lightmorphic Signatures Analysis Toolkit  At the same time, in order to improve the results quality, the Einstein tensor was also  considered under the form:  Gµν = Rµν − 1  2gµνR,  (5)  where Rµν is the Riemann tensor for the local spacetime, and R is the Ricci scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  Since there is not one general solution for the complex Einstein equations, but a large  variety of possible solutions that apply to particular circumstances, we’ve considered a weak-  field approximation, where the nonlinear Einstein equations where approximated towards  linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  For example, a very small perturbation specific to a gravitational wave, will impact the  flat spacetime and it is defined as hµν(x) and it’s value will be |hµν| << 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  Thus, the Einstein equation becomes:  gµν(x) = ηµν + hµν(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  (6)  or by simply considering the induced strain variations:  □hµν(x) = 0,  (7)  By further pursuing such linearization, we can represent in the TT gauge, a propagating  wave, under the following form:  hTT  µν =  \uf8eb  \uf8ec  \uf8ec  \uf8ed  0  0  0  0  0  h+  h×  0  0  h×  −h+  0  0  0  0  0  \uf8f6  \uf8f7  \uf8f7  \uf8f8 ,  (8)  where the constant amplitudes (h+, h×) represent the two gravitational wave polariza-  tions, the plus- and cross-polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  We represent the distance between two neighboring points as defined by (Berit (2013))  for a flat spacetime,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' trough the following expression:  ds2 = −c2dt2 + dx2 + dy2 + dz2 = −c2dt2 + [1 + h+(t)]dx2 + [1 − h+(t)]dy2  That allows us to model in the TT gauge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' the gravitational wave stretching along the x  axis and compression along the y axis with the specific factor of:  �  1 ± h+(t) ≃ 1 + 1  2h+(t)  Having modeled the photon’s traveling path in outer space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' in order to simplify the  inherent path inhomogeneities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' we separated the domains into outer space domain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' atmo-  spheric domain and Earth specific domains (lithosphere,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' hydrosphere,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' biosphere,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' noises,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  We further define the phase of an electromagnetic wave of frequency ω0 as φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Following  Driggers (2015)’s work, we consider that the starting light phase is at 0 and it travels at  the speed of light c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' After a distance L it will have a phase δφspace that can be expressed as  a distance integral over the spacetime metric,  δφspace = ω0  c  � L  0  gdx,  (9)  3  Damian  with g(t) = η+h(t), where η is the Minkowski metric and h(t) is the dimensionless spacetime  strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  Summing the light phase shift δφatm and the δφEarth which is derived from the noise  sources like seismic or electromagnetic interferences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' leads to the dataset of trajectory spe-  cific lightmorphic signatures:  ΦΓIDT =  N  �  j=1  Γj  IDT  (10)  The signature parameter estimation is performed considering a prior distribution p(Φ|L⊙)  that is updated upon receiving the new data d to give a posterior distribution p(Φ|d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' L⊙)  p(Φ|d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' L⊙) = p(Φ|L⊙)p(d|Φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' L⊙)  p(d|L⊙)  (11)  While observing the distribution of multiple light segments within the dataset ΦΓIDT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  it will be possible to estimate the probability for trajectory specific lightmorphic evolution:  pΦ = f(ρk · pΦk)  (12)  where pΦk is the database’s k-th segment specific probability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' ρk is the prediction weight  for the k-th segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Software package design  The distribution matrices specific to the isochronous segmentation surfaces, which define  the lightmorphic signature model, form the LSAT core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  As such we’ve used a design principle that ensures simplicity for the whole package,  while making the source codes easy to read and maintain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' As the toolkit is written in a  modular way, new functionalities can be easily plugged in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' This makes the LSAT not only  a lightmorphic signature machine learning tool but also an experimental platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  LSAT comes with plenty of documentation for all the interface functionalities and related  data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' The README file describes the installation process and interface usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  For developers who use the toolkit in their applications, the API documentation can provide  additional information related to functionality calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Practical Usage  In the examples, we provide sample values for the lightmorphic signature updates, as a  function of δφatm derived by the neural network from the values of a large dataset of at-  mospheric meteorological data for 317 cities in Romania, with hundreds of thousands data  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  Automatic learning is supported trough API calls to the domain specific data  providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  Beyond this simple way of running the lightmorphic signatures analysis toolkit, there are  several enhancement options for advanced usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' As example, one may activate additional  functionalities that consider input parameters like complex space weather forecasting, dif-  ferent electromagnetic wave disturbances or lithosphere, hydrosphere and biosphere specific  localized data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  4  Lightmorphic Signatures Analysis Toolkit  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Conclusion and Future Work  With the lightmorphic signatures analysis toolkit we provided an open source SW package  that is simple and easy-to-use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  Experiments and analysis conclude that the modular design and customization support  are performing excellent in practice and can provide the base for additional research on  lightmorphic signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  The toolkit is constantly being improved by new research results and user feed-back  with the ultimate goal of having an automated toolkit to use in maintaining and updating  a large database of high-quality light signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  Future work will focus on probability estimates, additional functionalities that mitigate  the large uncertainties in the available observational input data which arise from the complex  interaction processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' In addition, the inclusion of artificial intelligence (AI) options will  be considered while building a national/international network for lightmorphic signature  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  5  Damian  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' References  References  Rana Adhikari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Sensitivity and noise analysis of 4 km laser interferometric gravitational  wave antennae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' PhD thesis, Massachusetts Institute of Technology, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  Behnke Berit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' A Directed Search for Continuous Gravitational Waves from Unknown Iso-  lated Neutron Stars at the Galactic Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' PhD thesis, Leibniz University Hannover,  2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  Sylvia Biscoveanu, Maximiliano Isi, Salvatore Vitale, and Vijay Varma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  New Spin  on LIGO-Virgo Binary Black Holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=', 126(17):171103, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='171103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  Lindy Blackburn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Open Issues in the Search for Gravitational Wave Transients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' PhD thesis,  Massachusetts Institute of Technology, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  Daniel E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Clark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Control of Differential Motion between Adjacent Advanced LIGO Seismic  Isolation Platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' PhD thesis, Stanford University, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  Katherine Laird Dooley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Design and performance of high laser power interferometers for  gravitational-wave detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' PhD thesis, University of Florida, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  Jennifer Clair Driggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Noise Cancellation for Gravitational Wave Detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' PhD thesis,  California Institute of Technology, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  Tobin Thomas Fricke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Homodyne detection for laser-interferometric gravitational wave de-  tectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' PhD thesis, Louisiana State University and Agricultural and Mechanical College,  2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  Paul D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Lasky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Gravitational-wave cosmology across 29 decades in frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' X, 6(1):011035, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='1103/PhysRevX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='011035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  Michele Maggiore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Gravitational wave experiments and early universe cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=', 331:283–367, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='1016/S0370-1573(99)00102-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  Denis Martynov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Lock Acquisition and Sensitivity Analysis of Advanced LIGO Interferom-  eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' PhD thesis, California Institute of Technology, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  Ryan Quitzow-James.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Search for Long-Duration Transient Gravitational Waves Associated  with Magnetar Bursts during LIGO’s Sixth Science Run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' PhD thesis, Oregon U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=', 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  Joseph D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Romano and Neil J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Cornish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Detection methods for stochastic gravitational-  wave backgrounds: a unified treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Living Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=', 20(1):2, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='1007/  s41114-017-0004-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  Michael P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Ross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Precision Mechanical Rotation Sensors for Terrestrial Gravitational Wave  Observatories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' PhD thesis, University of Washington, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  6  Lightmorphic Signatures Analysis Toolkit  Darkhan Tuyenbayev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Extending the scientific reach of Advanced LIGO by compensating  for temporal variations in the calibration of the detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' PhD thesis, The University of  Texas at San Antonio, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  Madeline Wade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' Gravitational-Wave Science with the Laser Interferometer Gravitational-  Wave Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content=' PhD thesis, University of Wisconsin–Milwaukee, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}
+page_content='  7' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'}

59E3T4oBgHgl3EQfQwmQ/content/tmp_files/2301.04416v1.pdf.txt

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+pyssam – a Python library for statistical modelling of
+biomedical shape and appearance
+Josh Williams1, Ali Ozel1, and Uwe Wolfram1
+1 School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, UK
+DOI: TBD
+Software
+• Repository
+• Archive
+Editor: Pending Editor
+Reviewers:
+• Pending Reviewers
+Submitted: N/A
+Published: N/A
+License
+Authors of papers retain
+copyright and release the work
+under a Creative Commons
+Attribution 4.0 International
+License (CC BY 4.0).
+Summary
+pyssam is a Python library for creating statistical shape and appearance models (SSAMs)
+for biological (and other) shapes such as bones, lungs or other organs. A point cloud best
+describing the anatomical ‘landmarks’ of the organ are required from each sample in a small
+population as an input. Additional information such as landmark gray-value can be included to
+incorporate joint correlations of shape and ‘appearance’ into the model. Our library performs
+alignment and scaling of the input data and creates a SSAM based on covariance across the
+population. The output SSAM can be used to parameterise and quantify shape change across
+a population. pyssam is a small and low dependency codebase with examples included as
+Jupyter notebooks for several common SSAM computations. The given examples can easily be
+extended to alternative datasets, and also alternative tasks such as medical image segmentation
+by incorporating a SSAM as a constraint for segmented organs.
+Statement of need
+Statistical shape (and appearance) models (SSAMs) have drawn significant interest in biomed-
+ical engineering and computer vision research due to their ability to automatically deduce a
+linear parameterisation of shape covariances across a small population of training data (Baka
+et al., 2011; Cootes et al., 1995; Heimann & Meinzer, 2009; Väänänen et al., 2015). The
+classic statistical shape model (SSM) approach uses a point cloud of landmarks which are
+in correspondence across several instances of a shape. The covariances of how the shape
+changes across the training population are computed, and principal component analysis (PCA)
+is used to parameterise the different modes of shape variation (Cootes et al., 1995). This
+approach paved the way for automatic algorithms which could significantly aid medical image
+segmentation (similar to an atlas) (Irving et al., 2011), characterise how the organ shape varies
+over a population as a diagnostic tool (Osanlouy et al., 2020), or even reconstruct a full 3D
+structure from a sparser imaging modality such as planar X-ray images (Baka et al., 2011;
+Väänänen et al., 2015).
+We have found that available open-source toolkits such as Statismo and Scalismo (Lüthi et
+al., 2012) suffer from an exhaustive number of dependencies and are difficult to adapt to
+new tasks, datasets and I/O datatypes. ShapeWorks (Cates et al., 2017) is another strongly
+developed library for statistical shape modelling, but it uses an alternative method of extracting
+landmarks (a so-called particle-based method) which is less broadly used and more complex
+than a landmark-based system (where landmarks can be defined in any desired way for different
+anatomical shapes). Additionally, as the machine learning ecosystem has strong foundations in
+Python, building statistical models in C++, Scala or other languages reduces compatibility
+with the majority of modern machine learning developments (Bhalodia et al., 2018). We
+therefore implemented a lightweight Python framework for SSAMs which is easily adaptable
+with few dependencies, making it suitable for integrating as part of a broader codebase, as
+well as installing and running on high-performance computing clusters where users do not have
+root access to install many dependencies. We provide Jupyter notebooks on readthedocs and
+Williams et al. (2023). pyssam – a Python library for statistical modelling of biomedical shape and appearance. Journal of Open Source Software,
+TBD, TBD. https://doi.org/TBD
+1
+arXiv:2301.04416v1  [q-bio.QM]  11 Jan 2023
+
+two example datasets that allow users new to coding or SSAMs to learn how these models
+work in an interactive way to ease access when learning a new research topic and library.
+Overview
+The main modelling classes are built on the abstract base class StatisticalModelBase,
+which has several methods for pre-processing data and performing PCA (Figure 1). There
+are also several global variables that are inherited which are related to principal components,
+component variances and model parameters. The classes for SSM and SAM pre-process the data
+(align to zero mean and standard deviation of one) and can compute the population mean
+shape/appearance. Finally, the SSAM class for shape and appearance modelling inherits all
+of these, but also imports the SSM and SAM methods to pre-process shape and appearance
+features separately, before they are merged into one dataset for modelling.
+StatisticalModelBase
+SSM
+StatisticalModelBase
+SAM
+SSAM
+StatisticalModelBase
+StatisticalModelBase
+SSM
+SAM
+Figure 1: Schematic overview of the codebase. Each modelling class is abstracted from the Statis
+ticalModelBase class and contains several inherited variables such as model weights and principal
+components. The SSAM class inherits from StatisticalModelBase, but also uses pre-processing
+pipelines from SSM and SAM.
+Examples
+Here we present two example applications of pyssam. The first example examines shape
+variations in a toy dataset created for this study, which has a tree structure. Tree structures
+appear often in biology, including the lung airways and vascular system. Toy datasets such as
+these are a simple means to visualise and interpret the modelling and code framework. We then
+provide a more complex example which considers the left lower lobe of human lungs obtained
+from CT data (Tang et al., 2019). This example considers shape and appearance, where the
+appearance is the gray-value at the landmark location on an X-ray projection (obtained with
+the AppearanceFromXray helper class).
+Statistical shape modelling toy dataset
+To understand the shape modelling process, we have provided a dataset class called Tree
+which creates a number of tree shapes which are randomly computed based on global minimum
+and maximum values for angle and branch length ratio (between parent and child). Tree
+parameters are shown in Figure 2a. Tree nodes are converted to a numpy array and used to
+initialise pyssam.SSM. At initialisation of the SSM class, the landmarks are aligned, scaled to
+unit standard deviation and stacked into a matrix of shape (Nf, 3NL) where Nf is the number
+of features (samples in our training dataset) and NL is the number of landmarks (each with
+a x, y, z coordinates). All y coordinates in this case are zero, meaning the data is actually
+2D but we preserve a 3D coordinate system for simplicity in generalising the code to more
+Williams et al. (2023). pyssam – a Python library for statistical modelling of biomedical shape and appearance. Journal of Open Source Software,
+TBD, TBD. https://doi.org/TBD
+2
+
+common 3D applications. The code below shows how we can simply obtain a SSM from a set
+of landmarks.
+from glob import glob
+import numpy as np
+import pyssam
+tree_class = pyssam.datasets.Tree(num_extra_ends=1)
+landmark_coordinates = np.array(
+[tree_class.make_tree_landmarks() for i in range(0, num_samples)]
+)
+ssm_obj = pyssam.SSM(landmark_coordinates)
+ssm_obj.create_pca_model(ssm_obj.landmarks_scale)
+mean_shape_columnvector = ssm_obj.compute_dataset_mean()
+L1
+L2
+θ
+0
+10
+20
+30
+40
+Number of components
+50
+60
+70
+80
+90
+100
+Variance [%]
+(a)
+(b)
+Figure 2: Overview of tree dataset population. Panels show (a) a visualisation of 100 tree samples,
+and (b) cumulative variance versus the number of PCA components constructed by the statistical
+shape model. Inset of (a) shows a legend describing the morphological parameters varied to create
+the tree dataset. These parameters include the initial branch length, L1, the branch length ratio
+LR = L2/L1, and branching angle θ.
+Shape and appearance modelling of lung shape and chest X-ray images
+In the following example, we show a real application where 3D landmark for the left lower
+lung lobe are projected onto digitally reconstructed X-rays (Väänänen et al., 2015) and the
+gray-value is used to obtain appearance. Example landmark data was obtained using an
+automatic algorithm (Ferrarini et al., 2007). Appearance information is extracted from the
+X-ray images using AppearanceFromXray (part of pyssam.utils). We use landmarks,
+X-ray images as well as origin and pixel spacing information for the X-ray images to extract
+appearance as follows
+appearance_xr = pyssam.AppearanceFromXray(
+IMAGE_DATASET, IMAGE_ORIGIN, IMAGE_SPACING
+)
+appearance_values = appearance_xr.all_landmark_density(
+landmarks_coordinates
+)
+Williams et al. (2023). pyssam – a Python library for statistical modelling of biomedical shape and appearance. Journal of Open Source Software,
+TBD, TBD. https://doi.org/TBD
+3
+
+The SSAM can then be trained in a similar way as the SSM in subsection with the following
+code snippet:
+ssam_obj = pyssam.SSAM(landmark_coordinates, appearance_values)
+ssam_obj.create_pca_model(ssam_obj.shape_appearance_columns)
+mean_shape_appearance_columnvector = ssam_obj.compute_dataset_mean()
+The shape and appearance modes can then be computed based on the model parameters
+(ssam.model_parameters). The computed model parameters (eigenvectors and eigenvalues
+of the covariance matrix) can be used to morph the shape and appearance using ssam.morph
+_model (part of StatisticalModelBase in Figure 1) by
+x ≈ ¯x + Φ · b
+(1)
+where x is a new array containing shape and appearance, ¯x is the training dataset mean
+shape and appearance, Φ is the model principal components (eigenvectors of the training data
+covariance matrix), b is the model parameters, which is an array of weights unique to each
+data sample. The model parameter a mode m should be within [−3
+�
+σ2
+m, 3
+�
+σ2
+m], where
+σ2
+m is the explained variance of m (mth largest eigenvalue of the covariance matrix) (Cootes
+et al., 1995).
+Each mode of shape and appearance variation is visualised, as shown for a representative mode
+in Figure 3. This shows how lung shape influences the gray-value of lung pixels on the X-ray
+image. In this case, the change in shape and appearance are mainly due to how the lung
+interacts with adjacent structures such as the heart, rib cage and diaphragm.
+Figure 3: First mode of SSAM variation for lung lobe dataset. Panels show shape and appearance
+morphed using ssam.morph_model method and varying the model parameters (ssam.model_parame
+ters), from -2, 0 (mean shape) and 2.
+Acknowledgement
+JW was funded by a 2019 PhD Scholarship from the Carnegie-Trust for the Universities of
+Scotland.
+Williams et al. (2023). pyssam – a Python library for statistical modelling of biomedical shape and appearance. Journal of Open Source Software,
+TBD, TBD. https://doi.org/TBD
+4
+
+References
+Baka, N., Kaptein, B. L., Bruijne, M. de, Walsum, T. van, Giphart, J., Niessen, W. J., &
+Lelieveldt, B. P. (2011). 2D–3D shape reconstruction of the distal femur from stereo X-ray
+imaging using statistical shape models. Medical Image Analysis, 15(6), 840–850.
+Bhalodia, R., Elhabian, S. Y., Kavan, L., & Whitaker, R. T. (2018). DeepSSM: A deep
+learning framework for statistical shape modeling from raw images. International Workshop
+on Shape in Medical Imaging, 244–257.
+Cates, J., Elhabian, S., & Whitaker, R. (2017). Shapeworks: Particle-based shape corre-
+spondence and visualization software. In Statistical shape and deformation analysis (pp.
+257–298). Elsevier.
+Cootes, T. F., Taylor, C. J., Cooper, D. H., & Graham, J. (1995). Active shape models-their
+training and application. Computer Vision and Image Understanding, 61(1), 38–59.
+Ferrarini, L., Olofsen, H., Palm, W. M., Van Buchem, M. A., Reiber, J. H., & Admiraal-Behloul,
+F. (2007). GAMEs: Growing and adaptive meshes for fully automatic shape modeling and
+analysis. Medical Image Analysis, 11(3), 302–314.
+Heimann, T., & Meinzer, H.-P. (2009).
+Statistical shape models for 3D medical image
+segmentation: A review. Medical Image Analysis, 13(4), 543–563.
+Irving, B., Goussard, P., Gie, R., Todd-Pokropek, A., & Taylor, P. (2011). Segmentation of
+obstructed airway branches in CT using airway topology and statistical shape analysis. 2011
+IEEE International Symposium on Biomedical Imaging: From Nano to Macro, 447–451.
+Lüthi, M., Blanc, R., Albrecht, T., Gass, T., Goksel, O., Büchler, P., Kistler, M., Bousleiman,
+H., Reyes, M., Cattin, P., & others. (2012). Statismo-a framework for PCA based statistical
+models. The Insight Journal, 2012, 1–18.
+Osanlouy, M., Clark, A. R., Kumar, H., King, C., Wilsher, M. L., Milne, D. G., Whyte, K.,
+Hoffman, E. A., & Tawhai, M. H. (2020). Lung and fissure shape is associated with age in
+healthy never-smoking adults aged 20–90 years. Scientific Reports, 10(1), 1–13.
+Tang, H., Zhang, C., & Xie, X. (2019). Automatic pulmonary lobe segmentation using deep
+learning. arXiv Preprint arXiv:1903.09879.
+Väänänen, S. P., Grassi, L., Flivik, G., Jurvelin, J. S., & Isaksson, H. (2015). Generation of
+3D shape, density, cortical thickness and finite element mesh of proximal femur from a
+DXA image. Medical Image Analysis, 24(1), 125–134.
+Williams et al. (2023). pyssam – a Python library for statistical modelling of biomedical shape and appearance. Journal of Open Source Software,
+TBD, TBD. https://doi.org/TBD
+5
+

59E3T4oBgHgl3EQfQwmQ/content/tmp_files/load_file.txt

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+page_content='pyssam – a Python library for statistical modelling of  biomedical shape and appearance  Josh Williams1, Ali Ozel1, and Uwe Wolfram1  1 School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, UK  DOI: TBD  Software  Repository  Archive  Editor: Pending Editor  Reviewers:  Pending Reviewers  Submitted: N/A  Published: N/A  License  Authors of papers retain  copyright and release the work  under a Creative Commons  Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
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+page_content='  Summary  pyssam is a Python library for creating statistical shape and appearance models (SSAMs)  for biological (and other) shapes such as bones, lungs or other organs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' A point cloud best  describing the anatomical ‘landmarks’ of the organ are required from each sample in a small  population as an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' Additional information such as landmark gray-value can be included to  incorporate joint correlations of shape and ‘appearance’ into the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' Our library performs  alignment and scaling of the input data and creates a SSAM based on covariance across the  population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' The output SSAM can be used to parameterise and quantify shape change across  a population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' pyssam is a small and low dependency codebase with examples included as  Jupyter notebooks for several common SSAM computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' The given examples can easily be  extended to alternative datasets, and also alternative tasks such as medical image segmentation  by incorporating a SSAM as a constraint for segmented organs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='  Statement of need  Statistical shape (and appearance) models (SSAMs) have drawn significant interest in biomed-  ical engineering and computer vision research due to their ability to automatically deduce a  linear parameterisation of shape covariances across a small population of training data (Baka  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' Cootes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=', 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' Heimann & Meinzer, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' Väänänen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' The  classic statistical shape model (SSM) approach uses a point cloud of landmarks which are  in correspondence across several instances of a shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' The covariances of how the shape  changes across the training population are computed, and principal component analysis (PCA)  is used to parameterise the different modes of shape variation (Cootes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=', 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' This  approach paved the way for automatic algorithms which could significantly aid medical image  segmentation (similar to an atlas) (Irving et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=', 2011), characterise how the organ shape varies  over a population as a diagnostic tool (Osanlouy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=', 2020), or even reconstruct a full 3D  structure from a sparser imaging modality such as planar X-ray images (Baka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='  Väänänen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='  We have found that available open-source toolkits such as Statismo and Scalismo (Lüthi et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=', 2012) suffer from an exhaustive number of dependencies and are difficult to adapt to  new tasks, datasets and I/O datatypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' ShapeWorks (Cates et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=', 2017) is another strongly  developed library for statistical shape modelling, but it uses an alternative method of extracting  landmarks (a so-called particle-based method) which is less broadly used and more complex  than a landmark-based system (where landmarks can be defined in any desired way for different  anatomical shapes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' Additionally, as the machine learning ecosystem has strong foundations in  Python, building statistical models in C++, Scala or other languages reduces compatibility  with the majority of modern machine learning developments (Bhalodia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' We  therefore implemented a lightweight Python framework for SSAMs which is easily adaptable  with few dependencies, making it suitable for integrating as part of a broader codebase, as  well as installing and running on high-performance computing clusters where users do not have  root access to install many dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' We provide Jupyter notebooks on readthedocs and  Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' pyssam – a Python library for statistical modelling of biomedical shape and appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' Journal of Open Source Software,  TBD, TBD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='org/TBD  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='04416v1  [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='QM]  11 Jan 2023  two example datasets that allow users new to coding or SSAMs to learn how these models  work in an interactive way to ease access when learning a new research topic and library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='  Overview  The main modelling classes are built on the abstract base class StatisticalModelBase,  which has several methods for pre-processing data and performing PCA (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' There  are also several global variables that are inherited which are related to principal components,  component variances and model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' The classes for SSM and SAM pre-process the data  (align to zero mean and standard deviation of one) and can compute the population mean  shape/appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' Finally, the SSAM class for shape and appearance modelling inherits all  of these, but also imports the SSM and SAM methods to pre-process shape and appearance  features separately, before they are merged into one dataset for modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='  StatisticalModelBase  SSM  StatisticalModelBase  SAM  SSAM  StatisticalModelBase  StatisticalModelBase  SSM  SAM  Figure 1: Schematic overview of the codebase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' Each modelling class is abstracted from the Statis  ticalModelBase class and contains several inherited variables such as model weights and principal  components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' The SSAM class inherits from StatisticalModelBase, but also uses pre-processing  pipelines from SSM and SAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='  Examples  Here we present two example applications of pyssam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' The first example examines shape  variations in a toy dataset created for this study, which has a tree structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' Tree structures  appear often in biology, including the lung airways and vascular system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' Toy datasets such as  these are a simple means to visualise and interpret the modelling and code framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' We then  provide a more complex example which considers the left lower lobe of human lungs obtained  from CT data (Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' This example considers shape and appearance, where the  appearance is the gray-value at the landmark location on an X-ray projection (obtained with  the AppearanceFromXray helper class).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='  Statistical shape modelling toy dataset  To understand the shape modelling process, we have provided a dataset class called Tree  which creates a number of tree shapes which are randomly computed based on global minimum  and maximum values for angle and branch length ratio (between parent and child).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' Tree  parameters are shown in Figure 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' Tree nodes are converted to a numpy array and used to  initialise pyssam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='SSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' At initialisation of the SSM class, the landmarks are aligned, scaled to  unit standard deviation and stacked into a matrix of shape (Nf, 3NL) where Nf is the number  of features (samples in our training dataset) and NL is the number of landmarks (each with  a x, y, z coordinates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' All y coordinates in this case are zero, meaning the data is actually  2D but we preserve a 3D coordinate system for simplicity in generalising the code to more  Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' pyssam – a Python library for statistical modelling of biomedical shape and appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' Journal of Open Source Software,  TBD, TBD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='org/TBD  2  common 3D applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' The code below shows how we can simply obtain a SSM from a set  of landmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='  from glob import glob  import numpy as np  import pyssam  tree_class = pyssam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='Tree(num_extra_ends=1)  landmark_coordinates = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='array(  [tree_class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='make_tree_landmarks() for i in range(0, num_samples)]  )  ssm_obj = pyssam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='SSM(landmark_coordinates)  ssm_obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='create_pca_model(ssm_obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='landmarks_scale)  mean_shape_columnvector = ssm_obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='compute_dataset_mean()  L1  L2  θ  0  10  20  30  40  Number of components  50  60  70  80  90  100  Variance [%]  (a)  (b)  Figure 2: Overview of tree dataset population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' Panels show (a) a visualisation of 100 tree samples,  and (b) cumulative variance versus the number of PCA components constructed by the statistical  shape model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' Inset of (a) shows a legend describing the morphological parameters varied to create  the tree dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' These parameters include the initial branch length, L1, the branch length ratio  LR = L2/L1, and branching angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='  Shape and appearance modelling of lung shape and chest X-ray images  In the following example, we show a real application where 3D landmark for the left lower  lung lobe are projected onto digitally reconstructed X-rays (Väänänen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=', 2015) and the  gray-value is used to obtain appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' Example landmark data was obtained using an  automatic algorithm (Ferrarini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' Appearance information is extracted from the  X-ray images using AppearanceFromXray (part of pyssam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='utils).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' We use landmarks,  X-ray images as well as origin and pixel spacing information for the X-ray images to extract  appearance as follows  appearance_xr = pyssam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='AppearanceFromXray(  IMAGE_DATASET, IMAGE_ORIGIN, IMAGE_SPACING  )  appearance_values = appearance_xr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='all_landmark_density(  landmarks_coordinates  )  Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' pyssam – a Python library for statistical modelling of biomedical shape and appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' Journal of Open Source Software,  TBD, TBD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='org/TBD  3  The SSAM can then be trained in a similar way as the SSM in subsection with the following  code snippet:  ssam_obj = pyssam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='SSAM(landmark_coordinates, appearance_values)  ssam_obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='create_pca_model(ssam_obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='shape_appearance_columns)  mean_shape_appearance_columnvector = ssam_obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='compute_dataset_mean()  The shape and appearance modes can then be computed based on the model parameters  (ssam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='model_parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' The computed model parameters (eigenvectors and eigenvalues  of the covariance matrix) can be used to morph the shape and appearance using ssam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='morph  _model (part of StatisticalModelBase in Figure 1) by  x ≈ ¯x + Φ · b  (1)  where x is a new array containing shape and appearance, ¯x is the training dataset mean  shape and appearance, Φ is the model principal components (eigenvectors of the training data  covariance matrix), b is the model parameters, which is an array of weights unique to each  data sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' The model parameter a mode m should be within [−3  �  σ2  m, 3  �  σ2  m], where  σ2  m is the explained variance of m (mth largest eigenvalue of the covariance matrix) (Cootes  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=', 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='  Each mode of shape and appearance variation is visualised, as shown for a representative mode  in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' This shows how lung shape influences the gray-value of lung pixels on the X-ray  image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' In this case, the change in shape and appearance are mainly due to how the lung  interacts with adjacent structures such as the heart, rib cage and diaphragm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='  Figure 3: First mode of SSAM variation for lung lobe dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' Panels show shape and appearance  morphed using ssam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='morph_model method and varying the model parameters (ssam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='model_parame  ters), from -2, 0 (mean shape) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='  Acknowledgement  JW was funded by a 2019 PhD Scholarship from the Carnegie-Trust for the Universities of  Scotland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='  Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
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+page_content=' Medical Image Analysis, 24(1), 125–134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='  Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' pyssam – a Python library for statistical modelling of biomedical shape and appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' Journal of Open Source Software,  TBD, TBD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}
+page_content='org/TBD  5' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'}

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+Wrapping pathways of anisotropic dumbbell
+particles by giant unilamellar vesicles
+Ali Azadbakht,†,∥ Billie Meadowcroft,‡,¶,∥ Thijs Varkevisser,†,§,∥ Anđela Šarić,‡ and
+Daniela J. Kraft∗,†
+†Soft Matter Physics, Huygens-Kamerlingh Onnes Laboratory, Leiden University, PO Box
+9504, 2300 RA Leiden, the Netherlands
+‡Institute of Science and Technology Austria, 3400 Klosterneuburg, Austria
+¶Department of Physics and Astronomy, Institute for the Physics of Living Systems,
+University College London, London WC1E 6BT, United Kingdom
+§Van der Waals-Zeeman Institute, Institute of Physics, University of Amsterdam, Science
+Park 904, 1098 XH Amsterdam, Netherlands
+∥These authors contributed equally to this work.
+E-mail: [email protected]
+Abstract
+Endocytosis is a key cellular process involved in the uptake of nutrients, pathogens
+or the diagnosis and therapy of diseases. Most studies have focused on spherical ob-
+jects, whereas biologically relevant shapes can be highly anisotropic. In this letter, we
+use an experimental model system based on Giant Unilamellar Vesicles (GUVs) and
+dumbbell-shaped colloidal particles to mimic and investigate the first stage of the pas-
+sive endocytic process: engulfment of an anisotropic object by the membrane. Our
+model has specific ligand-receptor interactions realized by mobile receptors on the vesi-
+cles and immobile ligands on the particles. Through a series of experiments, theory
+1
+arXiv:2301.12150v1  [cond-mat.soft]  28 Jan 2023
+
+and molecular dynamics simulations, we quantify the wrapping process of anisotropic
+dumbbells by GUVs and identify distinct stages of the wrapping pathway. We find that
+the strong curvature variation in the neck of the dumbbell as well as membrane tension
+are crucial in determining both the speed of wrapping and the final states.
+The engulfment of objects through the cell membrane is critical for endocytic processes
+such as phagocytosis1–3 and receptor-mediated endocytosis. The latter is often exploited by
+viruses for cell entry and proliferation4 and key to nanomedical applications such as drug
+delivery and imaging.5 To single out receptor-mediated effects from active mechanisms in-
+volved in the engulfment,6 simplified passive model systems can be employed, which recently
+led to a conclusive understanding of the wrapping of spherical objects.7,8 However, biological
+objects such as bacteria and viruses4,9,10 as well as nanoparticles relevant for applications
+in nanomedicine but also nanotoxicology11 often posses non-spherical shapes. Moreover, in
+vitro experiments with nanoparticles and simulations have shown that the size and shape
+influence their likelihood to be taken up by endocytosis.6,12–17
+The wrapping pathways of spheres at sufficiently low membrane tensions have been shown
+to be a continuous transition from attached to fully wrapped, occurring either spontaneously
+or after activation.7,8,18 In contrast, anisotropic particles such as ellipsoids and rods, are
+expected to reorient during the wrapping process or become trapped in metastable states
+due to their varying curvature.19–27 The aspect ratio of these particles as well as the degree of
+rounding of their tip were the key parameters affecting the wrapping orientation with respect
+to the membrane and their metastable and stable states.24,27 Despite the extensive work in
+theory and simulations and exciting observations on shape-dependence in phagocytosis,28 no
+experimental work has investigated the passive wrapping process of anisotropic particles by
+lipid membranes and tested these predictions yet.
+In this letter, we employ an experimental model system based on Giant Unilamellar
+Vesicles (GUVs) and colloidal dumbbell particles to investigate the wrapping of micrometre-
+sized anisotropic objects by lipid membranes. Our model system is designed to have mobile
+2
+
+ligands on the vesicles and immobile receptors on the particles mimicking receptor-mediated
+endocytotic systems.18,29,30 We quantify the wrapping pathways of anisotropic dumbbells
+by lipid membranes and test if their initial orientation affects the final states. Molecular
+dynamics simulations of the same system corroborate our experimental data, allowing us to
+inspect the dynamics of the process that was inaccessible to experiment. We find that the
+strong curvature variation in the neck of the dumbbell as well as membrane tension and not
+their initial orientation are crucial in both determining the speed of wrapping and the final
+states.
+We investigate the wrapping process of anisotropic objects by a lipid membrane using a
+model system consisting of GUVs and colloidal particles, (see Fig. 1a). We chose the simplest
+object that features anisotropy: a dumbbell shaped colloidal particle that consists of two
+equal sized spheres.
+The colloid dumbbells were obtained from aggregating polystyrene
+spheres with diameter ds=0.98± 0.03 µm31 by briefly lowering the pH to 5.3 and then
+quenching the process by increasing the pH to 8.6.32 This process yielded 5-10% dimers with
+a long axis of 1.96 ± 0.06 µm and a short axis of 0.98 ± 0.03 µm. GUVs were prepared by
+electroswelling from 97.5% w/w 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC).
+To realize strong ligand-receptor mediated binding we doped the GUVs with 2% w/w 1,2-
+dioleoyl-sn-glycero-3-phosphoethanolamine-N-[biotin-2000] (DOPE-PEG2000-Biotin) and the
+dumbbells with 2.2×103/µm2 NeutrAvidin following,31 see Fig. 1b and c and see particle
+functionalization and quantification of binding affinity in Supporting Information. We sup-
+press electrostatic interactions by working in 50 mM Phosphate Buffered Saline, and achieve
+colloidal stability by coating the dumbbells with polyethyleneglycol (PEG5000). Imaging of
+the position and orientation of the dumbbells and membranes in three dimensions was made
+possible by dying the colloids with BODIPY, represented by a green color throughout the
+manuscript, as well as including 0.5% w/w 1,2-dioleoyl-sn-glycero-3-phosphoethanolamine-
+N-(lissamine rhodamine B sulfonyl) (DOPE-Rhodamine) into the GUVs, represented by a
+magenta color. See Fig. 1c. Confocal stacks and image sequences were acquired with an
+3
+
+inverted Nikon TI-e microscope, equipped with a 60x (NA 1.2) objective and A1-R scan
+head. 2D image sequences were taken at 59 fps, which enables tracking of the dumbbells in
+real time. Experimental details are described in the Supporting Information.
+To initiate the wrapping process, we used optical tweezers to bring dumbbell particles in
+contact with the GUV. They subsequently diffused on the GUV surface before suddenly and
+quickly becoming wrapped, a process that took between a few seconds and a few minutes
+depending on membrane tension, see Figure 1e and Movie S1. To capture the wrapping pro-
+cess with high speed, we adjusted the focal height during acquisition of the image sequence.
+After wrapping, the dumbbell continued to diffuse on the inside of the vesicle.
+We quantify the wrapping process of a dumbbell by measuring the angle θ between the
+major axis of the dumbbell and surface normal of the GUV and distance d of the dumbbell
+with respect to the undistorted surface of the GUV, see Figure 1d. We inferred the 3D
+position of the dumbbell from the position of its lobes with respect to the GUV. To improve
+the accuracy of tracking, particles were tracked only when their center of mass was between
+-0.8R<z<0.8R, and when both lobes were in focus. Details are described in the Supporting
+Information.
+We show confocal microscopy snapshots of a typical wrapping pathway in Figure 1 e,
+and quantitative data of θ and d for exemplary pathways in Figure 2a and b. Surprisingly,
+we find that the dumbbells end up in one of two states even though they start from different
+initial orientations: (1) both lobes are either being fully wrapped (Fig. 2a), or (2) a single
+lobe is being wrapped, such that the dumbbell is engulfed up to its waist by the membrane
+(Fig. 2b). The green-blue points in Fig. 2a and b represent dumbbells attached almost
+parallel to the membrane at the beginning of the process, whereas the yellow-red points
+represent dumbbells attached roughly perpendicular with respect to the membrane initially.
+Other starting orientations also lead to either a fully wrapped or a half wrapped dumbbell,
+but the probability for reaching either state was influenced by the initial position as we will
+discuss below.
+4
+
+θ
+d
+I
+II
+III
+IV
+V
+VI
+VII
+r
+z
+R
+b)
+d)
+t=10 s
+t=30 s
+t=50 s
+t=70 s
+t=90 s
+t=110 s
+a)
+c)
+e)
+{
+y [µm]
+Figure 1: Experimental setup to quantitatively measure the wrapping process of
+a dumbbell colloid by a GUV a) 3D confocal reconstruction of a GUV in magenta and
+a dumbbell particle in green with an indication of the relative height z from the equator
+of the GUV, radius of GUV R, and cross section radius of the vesicle at the location of
+the dumbbell, r. b) Detailed schematic of ligand-receptor based binding scheme between the
+dumbbell and GUV. I-DOPC lipid II-DOPE lipid III -Biotin IV -NeutrAvidin V-Rhodamine,
+VI -Polyethylene glycol (PEG) VII -Polystyrene particle. (Not to scale) c) Representative
+confocal images reconstructed from two channels, (1) dumbbell excited by 488 nm laser
+light and emission collected between 500-550 nm (2) GUV excited by 561 nm laser light
+and emission collected in 580-630 nm (scale bar 1µm). d) Schematic representation of the
+parameters d and θ used for the quantitative description of the wrapping process. e) Time
+series of snapshots of confocal images of a dumbbell being wrapped by a vesicle (scale bar
+4µm).
+If the dumbbell is oriented parallel to the membrane initially (θ ≈ 90◦ and proceeds to
+a fully wrapped state, then it tilts in the first part of the engulfment process to about 60◦.
+Subsequently, its CoM moves inward to almost d ≈ 1.5ds from the undisturbed membrane
+contour, before returning to a more parallel orientation and an insertion depth about d ≈
+0.7ds. This overshooting and recoil is similar to that observed for spheres previously.8,33
+If the dumbbell initially is roughly perpendicular the membrane, it first becomes oriented
+more precisely perpendicular until it is covered halfway (d = 0 and θ ≈ 10◦) before being
+wrapped further and finally ending in a more parallel orientation at a similar distance from
+5
+
+25
+20
+wr] 
+15
+10
+5
+X [um]
+0
+0
+20
+5
+10
+15
+20
+25
+302 μm
+0:00:09.8112 μm
+0:00:29.9322 μm
+0:00:49.4992 μm
+0:01:09.9972 μm
+0:01:30.0882 μm
+0:01:49.789the undisturbed membrane as the initially parallel dumbbells.
+For final states where one lobe is being wrapped only, an initially perpendicular dumbbell
+first reorients more parallel before becoming engulfed until its waist while becoming perpen-
+dicular again. An initially parallel dumbbell proceeds to reorient perpendicular while being
+engulfed, see Fig. 2b. The gap in the yellow-red trace at θ ≈ 55◦ and d=0.5 µm was caused
+by the dumbbell going through an orientation that was filtered out for accuracy as described
+above.
+To obtain more quantitative results for the dynamics of the system we carried out coarse-
+grained (CG) molecular dynamics (MD) simulations of anisotropic dumbbell particles being
+wrapped by a membrane. Besides the advantage of easily measuring dynamic properties,
+in these simulations we are also able to control the size of the vesicle and dumbbell, the
+membrane tension and the interaction strength between dumbbell and membrane and thus
+probe a wider parameter space than is available to experiments.
+The membrane is modelled using a one particle thick fluid surface developed by Yuan
+et al34 which reproduces the mechanical properties associated with biological membranes.35
+Using this model, we simulate spherical membrane vesicles and change the membrane tension
+by the addition of small solute particles on the inside and outside of the vesicle.36 The solute
+particles only interact via volume exclusion and produce a pressure force when the inside and
+outside concentrations are different. The dumbbell colloid is then placed on the membrane in
+either a vertical or horizontal initial condition and due to the attractive interaction between
+the membrane beads and the dumbbell, the dumbbell is slowly wrapped and engulfed by the
+vesicle. Details can be found in the Supporting Information.
+The results obtained from simulations show qualitatively similar behavior as in the exper-
+iments, see Figure 2. Again, both final states, i.e. i) one lobe attached and ii) fully engulfed,
+could be reached from any initial position, and the pathway they took was influenced by the
+initial orientation. Interestingly, our simulations suggest that the initial position strongly
+influences the first part of the wrapping process and to a lesser degree the second half,
+6
+
+which is observed to be similar for both extreme initial orientations. The observation that
+the wrapping pathways from different initial positions can result in the same final position
+shows that there is an energy minimum for the GUV-dumbbell system independent of the
+initial position of the dumbbell. In all observed pathways towards the fully wrapped state,
+the dumbbell particle tilts during the engulfment suggesting that this requires less bending
+energy.
+A similar reorientation upon wrapping was observed for linear aggregate of particles37 and
+elongated ellipsoids.21,25–27 Ellipsoids have been found to become first adhered by the side,
+before rotating to the tip upon being wrapped by the membrane.25 For sphero-cylindrical
+particles that were initially touching with their tip, a rotation-mediated wrapping was also
+seen,17,23 which can rotate the particle from a standing to a lying position at high aspect
+ratios. The first point of contact has been predicted to be crucial for the ultimate fate of
+a non-spherical particle.26,27 In contrast, for the dumbbell particles used here rotation is
+not driven by a variation of particle curvature, but primarily by thermal fluctuations and
+possibly inhomogeneities in the ligand coating density, because of the constant curvature
+of the constituent spheres of the dumbbells. The only region of curvature variation is the
+dumbbell neck, which we will show to play a crucial role in the wrapping.
+From the many wrapping processes we observed in experiments and simulations, we iden-
+tified a number of key intermediate states during the engulfment that ultimately determined
+the final state. A decisive event during the wrapping of the first lobe is whether the second
+lobe gets bound to the membrane. This is always the case if the particle starts out being
+perfectly parallel and thus with both lobes attached (Figure 3A3). If the particle initially
+is attached with a single lobe (3A1 and A2), however, tilting during the engulfment may
+attach the second lobe (3B). In principle, since one lobe is spherical one may expect engulf-
+ment to proceed uniformly, not inducing or requiring any tilt. However, any inhomogeneity
+in the coating density of the ligands on the dumbbells, as well as thermal fluctuations will
+tilt the particle and may induce contact of the second lobe to the membrane. Since biotin-
+7
+
+a)
+b)
+c)
+d)
+Experiment
+Simulations
+Figure 2: Quantitative wrapping pathway of dumbbell particles by for GUVs.
+Tilt angle θ and distance d of the dumbbell from the vesicle surface obtained from a,b)
+experiments and c,d) simulations as a function of time. In all panels, green-blue pathways
+indicate dumbbells starting from a vertical position with respect to the vesicle surface, and
+yellow-red pathways indicate dumbbells that initially start almost horizontally with respect
+to the membrane.
+Time is indicated by color, specified by colorbars for each panel.
+a)
+Experimentally obtained pathways for a dumbbell initially oriented parallel or perpendicular
+to the membrane surface to a fully wrapped end state. Each data point represents an average
+over 1s. b) Experimentally obtained pathways taken by a dumbbell initially oriented parallel
+or perpendicular to the membrane surface to the half-wrapped end state. Each data point
+represents an average over 5s. c) Simulations of pathways for a dumbbell initially oriented
+parallel and perpendicular to the membrane surface to the fully-wrapped end state. This
+was the most common stable state with ∼ 90% of dumbbells reaching this end state. d)
+Simulation of pathways for a dumbbell initially oriented parallel and perpendicular to the
+membrane surface to the half-wrapped end state. a-d) Circle size indicates the number of
+images used for the average. Simulation time is in expressed in ∆T = 0.01τ0, τ0 being MD
+unit of time.
+Neutravidin interactions are essentially irreversible at room temperature, attachment of the
+second lobe always precludes achieving a final state where only one lobe is wrapped. If
+the second lobe does not attach, the single-wrapped lobe state is reached (3D). Otherwise,
+the dumbbell will wrap both lobes consecutively, either in a symmetric fashion (3E2) or in
+an asymmetric way (3E1), leading to the fully wrapped state. The symmetric wrapping is
+unstable, and eventually leads to Fig. 3F in which both lobes are covered. The angle the
+dumbbell makes with the membrane after wrapping completed can vary. In this end state, a
+small neck connected the fully wrapped dumbbell at one lobe with the vesicle, see Fig. 3F.
+To quantify the time evolution, we measured the transition times between the different
+8
+
+wrapping states. Membrane tension was found to be crucial for the overall wrapping time, see
+below, and therefore simulations were used for quantitative measurements of the transition
+times and experiments for qualitative comparison. While the initial wrapping of the first
+lobe in the simulations is almost equally fast for the different initial states (see Fig. 3G and
+H), the wrapping slowed down significantly when the membrane was crossing the waist (Fig.
+3G and H). This signifies an energy barrier stemming from the high bending energy required
+to adapt to the strong variation in curvature of the particle surface. For dumbbells with both
+lobes attached, we observed slowing down at the waist (Fig. 3G). For dumbbells attached
+with a single lobe only, the wrapping process stopped for a longer time at the waist (Fig. 3H).
+We observed the same qualitative behavior in experiments, both for tense and floppy GUVs,
+indicating that the bending energy required to continue wrapping largely exceeded the energy
+gained from adhesion. In experiments, in less than 10% of the cases, we observed dumbbells
+wrapped with one lobe (3D) to suddenly transition to the fully engulfed state within about
+10 minutes, but never observed this within the timescales used in simulations in line with
+ref.13 The high bending energy costs at the waist and the significantly faster wrapping for
+tilted dumbbells observed in both simulations and experiments suggest that wrapping a
+tilted dumbbell is less energetically costly than one that is oriented perpendicular to the
+membrane.25 The strong trapping at the waist also causes single-lobe wrapped dumbbells to
+attain their stable insertion depth d without overshooting and recoil.
+The probability of following a specific pathway and reaching one of the two final states
+as qualitatively observed in experiments, depended on two factors: the membrane tension of
+the GUV and the dumbbell’s angle θ0 with respect to the membrane’s surface normal during
+the initial wrapping. The higher the surface tension of the GUV, the more likely it was for
+the dumbbell to end up in situation 3D. Large fluctuations of the vesicle’s surface enabled
+the dumbbell to attach to the non-wrapped lobe. The larger the angle θ in situation 3A2,
+and thus the closer to the membrane it started out at the more likely it was for the dumbbell
+to end up in situation 3B and hence E1.
+9
+
+F
+E1
+E2
+C
+D
+A3
+A1
+B
+A2
+A3
+C
+A1
+F
+E1
+B
+G
+H
+Experiments
+Simulations
+Simulations
+Simulations
+Figure 3: Overview of the observed wrapping pathways. A1-F) Confocal images of
+the possible orientation of a dumbbell (All scale bars denote 1µm). Arrows indicate the
+directions of the possible wrapping pathways, and dashed arrows illustrate transitions that
+were rarely observed. G) Measurements of the time between the states for the horizontal
+dumbbell starting position, given in simulation timesteps. H) Measurements of the time
+between the states for the vertical dumbbell starting position.
+The overall time as well as the transition between different stages in the wrapping strongly
+depended on the membrane tension - both the initial tension as well as the tension at later
+times which will increase because of the wrapping, see Figure 4. We experimentally measured
+the membrane tension from the fluctuation spectrum of the lipid vesicle following ref.38 and
+plot the time taken to complete wrapping as a function of membrane tension in Figure
+4a,b. We observed an increase in overall wrapping time with increasing initial membrane
+tension in experiments (e.g. Figure 4a,b) and simulations (Figure 4c). However, the range
+of tensions we could replicate in experiments and simulations was quite limited. To be able
+to fully explore this effect, we extended a previously developed analytical theory describing
+the time to wrap colloids,39,40 which was recently experimentally confirmed,8 and adapted
+it to the shape of a dumbbell (Details of the theory can be found in the SI). In doing so we
+10
+
+could explore the effect of tension on time to wrap the dumbbell for a range of theoretical
+parameters. All the parameters used in the theory were taken directly from the experiment,
+apart from the binding energy per area (W) and the microviscosity of the membrane (ηeff)
+which are both discussed below.
+For a given adhesion energy, we find that the time taken to fully wrap the dumbbell in-
+creases non-linearly with the tension. With increasing adhesion energy, the wrapping process
+becomes faster at the same tension, see Figure 4a. The adhesion energies in experiments
+vary due to the distribution of binding sites between dumbbells18,31 which is also reflected
+in that the experimental data points fall within a range of adhesion energies identified by
+the theory. We note that only a small percentage of the NeutrAvidin sites that have been
+added during synthesis contribute to the effective adhesion energy, as was found previously
+in ref.18 Although fixed in the experiments, varying membrane microviscosity in the theory
+also changes the time taken to wrap. Membrane microviscosity is a measure of how easily
+the lipids slide past each other during rearrangement, and a higher microviscosity is linked
+to a higher frictional force during colloid-membrane wrapping. The comparison between
+the theoretical and experimental results allows us to estimate the membrane microviscosity,
+which is experimentally inaccessible. We find that our experimental measurements best fit
+the theoretical curves for a membrane microviscosity of ηeff ≈ 0.8 Pa·s, Figure 4b, about 10
+times larger than the lower bound estimated in.8 However, the theory in ref8 consistently
+over-estimated the wrapping speed as compared with experiments on spheres, so it could be
+that the experiment microviscosity was larger than their theoretically predicted value.
+Here we have developed the first model system to quantitatively study ligand-receptor
+mediated endocytosis of an anisotropic object by making use of GUVs and colloidal dumbbell
+particles. We followed and quantified their orientation θ and distance d with respect to the
+membrane during wrapping using experiments and molecular dynamics simulations.
+We
+found that there are two final states: 1) only one lobe or 2) both lobes of the dumbbell are
+fully wrapped by the membrane. The two states can be reached from any initial position
+11
+
+a)
+b)
+c)
+Figure 4: Measurement of the time required to fully wrap a dumbbell-shaped
+particle as a function of membrane tension (a) Experimental data (points) and the-
+oretical predictions (lines) for different membrane viscosity in the range of 0.4-1.6 Pa·s at
+a fixed adhesion energy per unit of area of 0.76 µJ/m2. (b) Experimental data (points)
+and theoretical predictions (lines) for different adhesion energy per unit area int he range
+of 0.69-0.79 µJ/m2 at a fixed membrane viscosity of 0.8 Pa·s. c) Time to fully wrap the
+dumbbell-shaped particle in simulations for a range of tensions <10 nN/m.
+except when both lobes were attached initially which necessarily leads to full wrapping of
+both lobes. However, the initial position influenced the pathway towards the final state. We
+identified a number of key intermediate states during the wrapping that determine the final
+state. Wrapping of one lobe was only found for high membrane tensions and if the other lobe
+did not touch the membrane during engulfment. Using molecular dynamics simulations we
+quantified the time required between key intermediate steps, with the slowest step being the
+crossing of the highly curved neck region of the dumbbell. With simulations we confirmed
+the experimentally-observed trend of time to wrap increasing for increasing tension, and
+using analytical theory we estimated the membrane microviscosity.
+Our results contribute to a better understanding of how shape affects endocytosis, nutri-
+tion uptake, and bacterial evasion. Our choice of a simple anisotropic object, a dumbbell,
+enabled a key insight: highly negatively curved regions may dominate the wrapping and
+possibly even prevent full engulfment unless active processes are present. This suggests that
+objects, such as certain viruses such as pox virus4that rely on endocytosis, may profit from
+having a convex shape. Incorporation of active processes, such as those driven by actin or
+ESCRT-III polymers, could provide further insights into how the competition between the
+12
+
+passive and active processes affects wrapping.
+Supporting Information
+• "Supporting Information: Details of the experiments and simulations; experimental
+materials used; methods employed for membrane preparation, particle functionaliza-
+tion, experimental imaging, and quantification of ligands on dumbbells; details of quan-
+tification of dumbbell wrapping and filters applied for the analysis; detail of theory used
+for time taken to wrap a dumbbell"
+• Movie S1: An example of an experimental video of the wrapping process of a dumbbell
+colloid attached to a GUV, recorded with a confocal microscope at 59 frames per second
+and 3× accelerated.
+Movie S1: An example of an experimental video of the wrapping process of a dumbbell
+colloid attached to a GUV, recorded with a confocal microscope at 59 frames per second and
+3× accelerated.
+Acknowledgments
+We sincerely thank Casper van der Wel for providing open-source packages for tracking, as
+well as Yogesh Shelke for his assistance with PAA coverslip preparation and Rachel Doherty
+for her assistance with particle functionalization. We are grateful to Felix Frey for useful
+discussions on the theory of membrane wrapping. B.M. and A.Š. acknowledge funding by the
+European Union’s Horizon 2020 research and innovation programme (ERC Starting Grant
+No. 802960).
+13
+
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+18
+
+Graphical TOC Entry
+time
+Experiment
+Simulation
+19
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@@ -0,0 +1,1585 @@
+ 
+1 
+Critical resolved shear stresses for slip and twinning in Mg-Y-Ca alloys and their 
+effect on the ductility 
+Mingdi Yua, Yuchi Cuib, Jingya Wanga,*, Yiwen Chena, Zhigang Dingc, Tao Yinga, 
+Javier Llorcad,e,*, Xiaoqin Zenga,* 
+a National Engineering Research Center of Light Alloy Net Forming and State Key 
+Laboratory of Metal Matrix Composites, Shanghai Jiao Tong University, Shanghai 
+200240, PR China. 
+ 
+b School of Materials Science and Engineering, Shanghai Jiao Tong University, 
+Shanghai, 200240, PR China. 
+ 
+c Nano and Heterogeneous Materials Center, School of Materials Science and 
+Engineering, Nanjing University of Science and Technology, Nanjing, Jiangsu 
+210094, China. 
+ 
+d IMDEA Materials Institute, 28906 Getafe, Madrid, Spain. 
+ 
+e Department of Materials Science, Polytechnic University of Madrid, E. T. S. de 
+Ingenieros de Caminos, 28040 Madrid, Spain. 
+ 
+* Corresponding author. E-mail address: [email protected]. 
+* Corresponding author. E-mail address: [email protected]. 
+* Corresponding author. E-mail address: [email protected]. 
+ 
+Abstract: 
+The deformation mechanisms of an extruded Mg-5Y-0.08Ca (wt. %) alloy were 
+analyzed by means of micropillar compression tests on single crystals along different 
+orientations -selected to activate specific deformation modes- as well as slip trace 
+analysis, transmission electron microscopy and transmission Kikuchi diffraction. The 
+polycrystalline alloy presented a remarkable ductility in tension (~32%) and negligible 
+
+ 
+2 
+differences in the yield strength between tension and compression. It was found that the 
+presence of Y and Ca in solid solution led to a huge increase in the CRSS for <a> basal 
+slip (29 ± 5 MPa), <c+a> pyramidal slip (203 ± 7 MPa) and tensile twin nucleation 
+(above 148 MPa), while the CRSS for <a> prismatic slip only increases up to 105 ± 4 
+MPa. The changes in the CRSS for slip and tensile twinning in Mg-Y-Ca alloys 
+expectedly modify the dominant deformation mechanisms in polycrystals. In particular, 
+tensile twinning is replaced by <a> prismatic slip during compressive deformation 
+along the a-axis. The reduction of twinning (which generally induces strong anisotropy 
+in the plastic deformation in textured alloys), and the activation of <a> prismatic slip 
+(which provides an additional plastic deformation mechanism with limited hardening) 
+were responsible for the large tensile ductility of the alloy.  
+ 
+Keywords: Mg-Y-Ca alloys; micropillar compression; critical resolved shear stress; 
+plastic anisotropy; tension-compression asymmetry; tensile ductility.  
+ 
+1. Introduction 
+Pure Mg and Mg alloys generally present poor ductility and formability, especially 
+at room temperature (Huang et al., 2022; Sun et al., 2019; Tang et al., 2022; Yaghoobi 
+et al., 2022). As a result, forming of rolled sheets and extruded bars becomes difficult 
+and limits the application of wrought Mg alloys in different industrial sectors (Li and 
+Fang, 2022). Thus, understanding the origin of the lack of ductility and formability is 
+of paramount importance to develop new Mg alloys that overcome these limitations.  
+The poor ductility of Mg alloys is primarily traced to its low-symmetry hexagonal 
+closed packed (HCP) lattice structure, which results in very large differences in the 
+critical resolved shear stress (CRSS) between basal and non-basal slip systems as well 
+as in the easy activation of tensile twinning (Lee et al., 2018). Plastic deformation in 
+pure Mg is initially accommodated by <a> basal slip, which only provides two 
+independent slip systems (Partridge, 1967). This process leads to the development of a 
+strong basal texture during rolling and extrusion. Moreover, plastic deformation along 
+the c-axis (which is necessary to activate five independent slip systems to fulfil the von-
+
+ 
+3 
+Mises criterion for homogeneous plastic deformation) is absorbed by tensile twinning, 
+which is triggered at much lower CRSS than that necessary to produce <c+a> pyramidal 
+slip (Graff et al., 2007; Sukedai and Yokoyama, 2010). However, the plastic strain 
+associated with tensile twinning is very limited (at most 7%), moreover, tensile twining 
+is a polar mechanism that only occurs when the stress along the c-axis of the crystal is 
+tensile (Mayama et al., 2011). This leads to a large buildup of stresses to activate <c+a> 
+pyramidal slip in grains that are not suitably oriented for twinning and/or that cannot 
+accommodate more plastic deformation by twinning (Obara et al., 1973; Reed-Hill and 
+Robertson, 1957). The stress concentrations in these grains facilitate the nucleation of 
+cracks and limit the ductility (Zhang et al., 2022). Moreover, huge differences in the 
+flow stress and the strain hardening rate between tension and compression appear in 
+textured microstructures, which also lead to fracture during bending and forming 
+operations (Agnew and Duygulu, 2005; Basu et al., 2021).  
+The strategies to improve ductility and formability of Mg alloys have been directed 
+towards promoting the activation of multiple slip, including non-basal <a> and non-
+basal <c+a> slip, and to suppress deformation twinning. Multiple slip leads to more 
+homogeneous plastic deformation and limits texture development during rolling and 
+extrusion while twinning promotes plastic anisotropy in textured microstructures 
+(Ahmad et al., 2019; G. Liu et al., 2017; Zhang et al., 2016a). For instance, precipitation 
+hardening in Mg-Zn alloys leads to large enhancements in the CRSS for basal (Alizadeh 
+and LLorca, 2020; Chun and Byrne, 1969; Wang and Stanford, 2015) and pyramidal 
+slip (Alizadeh et al., 2021) and, thus, to an important reduction in the pyramidal-to-
+basal CRSS ratio. Nevertheless, the large increase in flow stress inherently decreases 
+the ductility due to the strong accumulation of geometrically necessary dislocations 
+around the precipitates (Rosalie et al., 2012). In addition, precipitates also increase the 
+CRSS for twin growth but do not affect the CRSS for twin nucleation (Wang et al., 
+2019b). As the latter is normally higher than the former, the presence of precipitates do 
+not contribute to hinder the development of twinning. The only difference induced by 
+the precipitates is a larger number of smaller twins, as compared to the precipitate-free 
+condition (Stanford et al., 2012). Thus, precipitate is not very efficient to enhance the 
+
+ 
+4 
+ductility of Mg alloys (Fu et al., 2019; Jain et al., 2010). 
+Strategies based on solid solution hardening have been more successful to improve 
+the ductility of Mg alloys if the alloying elements are properly chosen. For instance, 
+Sandlöbes et al. (Sandlöbes et al., 2013, 2012, 2011) reported that the addition of 3 wt. % 
+Y led to Mg alloys with a tensile ductility > 25 %, which was associated with the 
+presence of a large density of <c+a> pyramidal dislocations in the deformed sample. 
+This behavior was mainly attributed to a reduction in the ratio between the CRSS of the 
+< c+a > pyramidal slip and < a > basal slip, which was ~3.2 according to in situ high 
+energy X-ray diffraction tests (Huang et al., 2018; Wang et al., 2018) and ~2.8-4.8 from 
+micropillar compression tests (Wu et al., 2020). Large ductility and formability are not 
+achieved, however, by the addition of other elements in solid solution (such as Al or Zn) 
+because the pyramidal-to-basal CRSS ratio in these alloys are > 10 (Li et al., 2021a; 
+Wang et al., 2020). Zhu et al., (2019) found that the addition of 0.47 wt. % of Ca in 
+solid solution enhanced the activity of <a> prismatic and pyramidal I dislocations as 
+well as the cross-slip between basal and non-basal slip planes, improving the tensile 
+ductility to ~18 % in a Mg-0.47 Ca (wt. %) alloy. And several authors reported a large 
+improvement in the ductility of binary Mg-Zn and Mg-Al alloys through the addition 
+of small amount of Ca (Hofstetter et al., 2015; Sandlöbes et al., 2017; Wang et al., 
+2021b). This behavior was supported by our recent micropillar compression tests that 
+showed that the addition of Ca to Mg-Zn alloys reduced the pyramidal-to-basal CRSS 
+ratio values, that were similar to those found in Mg-Y alloys (Wang et al., 2021a). 
+Finally, Wu et al., (2018) showed that the presence of Y and Ca reduces the energy for 
+cross-slip/double cross-slip of <c+a> pyramidal dislocations, leading to new dislocation 
+loops which accommodate plastic deformation. In contrast, the cross-slip is inhibited in 
+pure Mg (or in Mg-Al and Mg-Zn alloys) (Wu et al., 2018), by the favorable 
+dissociation of edge pyramidal <c+a> dislocation segments into sessile segments in the 
+basal plane.  
+Regarding the effect of solid solution on tensile twinning, several investigations 
+reported an increase in the CRSS for twin nucleation and growth with the addition of 
+Al (Wang et al., 2020), Zn (Li et al., 2021a), Y (Li et al., 2021b) as well as Ca to Mg-
+
+ 
+5 
+Zn alloys (Wang et al., 2021a). However, the CRSS for twin nucleation and growth 
+were lower than that for <c+a> pyramidal slip in the corresponding alloy, thus, tensile 
+twinning was still preferred over pyramidal slip to accommodate plastic deformation in 
+grains suitable oriented for twinning. In addition, the addition of 4Y (wt. %) could 
+significantly suppress the tensile twinning (with CRSS larger than 113 MPa) and 
+promote the <c+a> dislocations (with CRSS around 106 MPa) (Wu et al., 2020). The 
+results summarized above point to the beneficial effects of Y and Ca in solid solution 
+to reduce the plastic anisotropy of Mg. Thus, the co-addition of Ca and Y is expected 
+to promote the homogeneous deformation and improve the plastic deformability of Mg 
+alloys, taking advantages of the significant suppression effect of Y on the tensile 
+twinning, the promotion effect of Ca on the non-basal <a> slips, simultaneously the 
+positive effect of Ca and Y on the activation of the <c+a> slips. Ca enhances the 
+activation of <a> prismatic and <a> pyramidal slip while Y has similar effects on <c+a> 
+slip. Moreover, experimental results on the tensile behavior of an extruded Mg – 2.4 
+wt. % Y – 0.3 wt. % Ca (Zhou et al., 2013) showed a very large tensile elongation 
+(~37 %) but there is not information available in the literature -to the authors’ 
+knowledge- on the concurrent effects of Y and Ca in solid solution on the dominant 
+deformation mechanisms and this is the main objective of this investigation. Thus, the 
+CRSS for different slip systems and twinning was determined in a Mg-Y-Ca alloy from 
+micropillar compression tests in single crystals with different orientations. The 
+deformation mechanisms were ascertained from slip trace analysis in the scanning 
+electron microscope (SEM), transmission electron microscopy (TEM) observations of 
+the dislocations as well as transmission Kikuchi diffraction (TKD). This information 
+was used to rationalize the excellent ductility of Mg-Y-Ca and to provide guidelines to 
+design novel Mg alloys with improved ductility and formability. 
+ 
+2. Materials and experimental techniques 
+2.1 Materials  
+The Mg-Y-Ca alloy was prepared from pure Mg (99.99 wt. %), Mg-30 Ca (wt. %) 
+and Mg-30 Y (wt. %) master alloys in a resistance furnace under a protective 
+
+ 
+6 
+atmosphere of CO2 and SF6. The actual chemical composition of the ingot, obtained by 
+inductively coupled plasma atomic emission spectroscopy, was Mg-5Y-0.08Ca (wt. %). 
+The cast alloy was solution treated at 400 ℃ for 12 h, followed by extrusion at 300 ℃ 
+with an extrusion ratio of ~ 18:1. Afterwards, parallelepipedal samples of 10×10×5 mm3 
+were cut from the extruded specimens and homogenized at 550 ℃ for 20 days within 
+quartz capsules filled with Ar to induce grain growth.  
+2.2 Experimental techniques 
+Tensile and compressive tests were carried out along the extrusion direction in 
+polycrystalline specimens at crosshead speed of 0.5 mm/min, using a universal testing 
+machine (Z100-TEW) at room temperature. The dimensions of the gage section of the 
+dog-bone tensile specimens were 18×3.4×1.4 mm3 (length × width × thickness), while 
+cylindrical specimens of 8 mm in diameter and 12 mm in length were used in the 
+compression tests. Deformation was measured with an extensometer and 3 specimens 
+were tested in each condition.  
+The crystallographic orientation of the grains in the sample was characterized via 
+electron back-scattered diffraction (EBSD) in a Tescan Mira-3 SEM with an Oxford 
+Instruments Nordlys EBSD detector at an accelerating voltage of 20 kV. The surface of 
+the sample was mechanically ground using abrasive SiC papers with a grit size of 1200, 
+2000, 3000, 5000 and 7000. Subsequently, the sample surface was electropolished in 
+an ethanol solution with 10 (vol. %) perchloric acid at -30 ℃ and 30 V for 90 s to 
+remove the surface damage induced by grinding and reveal the grain boundaries. The 
+EBSD data were analyzed using the Channel 5 software and the Oxford Instruments 
+AZtec Nanoanalysis software package v6.0 along with AZtec Crystal. Several grains 
+whose orientations were appropriate to active different deformation modes were 
+selected to mill the micropillars.  
+Micropillars of 5 × 5 μm2 square cross and an aspect ratio 2:1 were milled from 
+the selected grains using a FEI Helios G4 UX Focused Ion Beam (FIB)/SEM dual beam 
+microscope operated at 30 kV. These dimensions are known to minimize size effects 
+during mechanical deformation while the time and effort to mill each micropillar are 
+reasonable (Wang et al., 2021a). An initial ion current of 9.3 nA was used to remove the 
+
+ 
+7 
+surrounding material and it was reduced to 2.5 nA when the beam was getting closer to 
+the actual dimensions of the micropillar. A final ion current of 80 pA was used in the 
+final polishing step to minimize the surface damage due to Ga+ ion-implantation. The 
+final taper of the micropillars was < 1.5°.  
+Micropillar compression tests were performed in ex situ using a Hysitron 
+Triboindenter TI950 system though a diamond flat punch of 10 μm in diameter. All the 
+tests were conducted under displacement control up to a maximum strain of 10 % at a 
+nominal strain rate of 10-3 s-1. The experimental displacement was corrected to account 
+for the elastic deflection of the matrix material beneath the micropillars following the 
+Sneddon correction (Sneddon, 1965). To this end, the elastic modulus of each grain was 
+determined via the nanoindentation method with a Berkovich tip in the same grain 
+where the micropillar was milled. More details about micropillar manufacturing and the 
+compression set-up can be found in (Sneddon, 1965; Wang et al., 2021a).  
+The engineering stress-strain curves were obtained from the load and the corrected 
+elastic deflection of the micropillar using the initial cross-sectional area and the height 
+of the micropillars measured in the SEM. The yield stress, σy , was determined from 
+the loss of linearity in the stress-strain curve following the methodology described in 
+(Alizadeh and LLorca, 2020; Maaß et al., 2009). From this information, the CRSS of 
+the active slip system was determined as  
+CRSS = SF × σy                     (1) 
+where SF is the Schmid factor of the corresponding slip system, computed from the 
+crystallographic orientation of each crystal (Table 1). 
+The slip traces on the top and lateral surfaces of the deformed micropillars were 
+characterized in a Tescan Mira-3 SEM to ascertain the active slip planes. The active slip 
+plane and direction were identified from the micropillar orientation using VESTA 
+software (Momma and Izumi, 2008). Moreover, TEM and TKD were used to determine 
+the dislocation activity and the orientation of the micropillar after deformation. To this 
+end, a thin lamella was lifted-out along the loading direction from the deformed pillars 
+and thinned to < 100 nm in thickness using FIB. The TKD maps were collected in a 
+
+ 
+8 
+Tescan Mira-3 SEM at 30 kV with a step size of 20 nm. The TEM observations were 
+carried out using a Talos F200X G2 microscope operated at 200 kV. The two-beam 
+condition was applied to obtain dislocation contrast. Moreover, the “g·b” visibility 
+criterion was used to identify the types of dislocation, i.e., the dislocation is in contrast 
+when g!⃗ · b!⃗ ≠ 0, where g!⃗ is the diffraction vector and b!⃗ the Burgers vector. 
+2.3 First-principles calculations 
+In order to study the influence of Y and/or Ca atoms on the deformation 
+mechanisms in Mg alloys, the generalized stacking fault energy (GSFE) curves of 
+different slip systems were calculated via the first-principles calculations using the 
+Vienna Ab initio Simulation Package (VASP) (Kresse and Furthmüller, 1996). The 
+exchange-correlation function was described using the generalized gradient 
+approximation (GGA) with the Perdew-Burke-Ernzerholf functional (PBE), based on 
+the projector augmented wave (PAW) (Blöchl, 1994) method. 
+A supercell with 12-layers containing 48 atoms was defined for different slip 
+systems, as indicated in Fig. 1. Each supercell was separated by 15 Å vacuum to 
+eliminate the influence of the periodic boundary conditions. The formation energy was 
+initially calculated for different positions of the solute atoms and the configurations 
+with lower formation energy was selected as the most stable ones (Yuasa et al., 2014). 
+In the binary Mg47N1 (N = Y, Ca) alloys, the most stable configuration was found when 
+one Mg atom at the center site of the stacking fault plane was substituted by a solute 
+atom X. In the ternary Mg46N1X1 (N = Y, and X = Ca) alloy, the most stable 
+configuration was found when one Mg atom at the center site of the stacking fault plane 
+was substituted by a Ca atom. Then, one of the eleven nearest Mg atoms from the Ca 
+atom was substituted by one Y atom, as shown in Fig. S1 in the supplementary material. 
+The exact position of the Y atom was determined from the formation energy (Ding et 
+al., 2019; Dong et al., 2018). The formation energies for every occupancy of the Y atom 
+are listed in Table S1 in the supplementary material. 
+The conventional direct crystal slip methods were employed to obtain the GSFE 
+curves of different slip systems The perfect supercell was cut into two free parts and 
+
+ 
+9 
+one part was displaced with respect to the other one along the slip direction. The atomic 
+positions were relaxed only along the direction perpendicular to the stacking fault plane 
+(Wang et al., 2020). A residual force threshold of 0.01 eV/Å was performed in all 
+geometric relaxations until the electronic energy converged to less than 10-5 eV/cell. 
+The Brillouin zone for the GSFE of the basal slip system, the prismatic slip system, and 
+the pyramidal slip system was set as 8×8×1, 10×6×1, and 6×10×1, respectively, with 
+an energy cutoff of 480 eV (Dong et al., 2018; Wang et al., 2013). 
+ 
+ 
+Fig. 1. Schematic illustration of the models to calculate the GSFE for (a) basal slip (b) 
+prismatic slip, and (c) pyramidal Ⅰ slip. The most stable positions of Y and Ca atoms 
+determined by the lowest formation energy are marked by blue and purple atoms, 
+respectively. Stacking fault planes are noted by the dotted lines. 
+ 
+3. Results 
+3.1 Mechanical behavior of polycrystals 
+The inverse pole figure (IPF) map of the as-extruded Mg-Y-Ca alloy along the 
+(a) basal slip
+(b) prismatic slip
+(c) pyramidalⅠslip
+[11!00]
+[0001]
+[112!0]
+[101!1]
+[112!0]
+[112!3]
+105°
+[112!0]
+[11!00]
+[0001]
+Mg
+Ca
+Y
+
+:
+O
+O
+O
+O 
+10 
+extrusion direction is plotted in Fig. 2a. The {0001} pole figure shows that the Mg-Y-
+Ca alloy possesses a weak texture with a strength of ~ 8.21 mrd, as displayed in Fig. 
+2b, compared to pure wrought Mg with a strong basal texture of >15 mrd (Yin et al., 
+2021). The engineering stress-strain curves of the extruded Mg-Y-Ca alloy from the 
+tensile and compressive tests parallel to the extrusion direction are plotted in Fig. 2c. 
+The scatter was very limited and the average tensile elongation was very large (≈ 32%). 
+Moreover, the tensile yield stress was 104 MPa, very close to the yield strength in the 
+compression tests (122 MPa). Thus, the Mg-Y-Ca alloy presented very low 
+tension/compression asymmetry in the yield strength in contrast with the marked 
+asymmetry in extruded Mg and Mg alloys (Sukedai and Yokoyama, 2010; Yin et al., 
+2021; Zhang et al., 2016b).1 It should also be noted that volume fraction of the twinned 
+material after tensile deformation was very low (≈ 1.8%), indicating that twining was 
+not a dominant deformation mechanism in the Mg alloy. 
+ 
+Fig. 2. (a) IPF map of the Mg-Y-Ca along the extrusion direction. (b) {0001} Pole figure 
+of the Mg-Y-Ca alloy illustrating the texture characteristics before the deformation in 
+the TD-ED plane. (c) Engineering stress-strain curves in tension and compression 
+parallel to the extrusion direction of the Mg-Y-Ca alloy.  
+ 
+3.2 Deformation mechanisms  
+ 
+1 The comparison between both curves shows the limited tension-compression anisotropy in the yield 
+strength but the differences in the elastic and fully plastic regions are due to the limitations of the 
+compression tests. Compression tests always underestimate the elastic modulus because it is very difficult 
+to ensure that the specimen surface and the loading plate surface are perfectly parallel. Thus, partial 
+contact between both surface leads to localized plastic deformation and to an apparent elastic modulus 
+that is lower than the real one. Moreover, barreling of the cylindrical specimen during compression leads 
+to non-homogeneous plastic deformation and overestimates the strain hardening for large plastic strains. 
+50μm
+TD
+ED
+Max=8.21
+ED∥ Tensile direction
+8.21
+0.00
+(a)
+(b)
+(c)
+(c)
+
+400
+Tension
+Compression
+300
+200
+0
+10
+20
+30
+40
+Engineering strain (%)Tscedan-(0001) -Magnesium
+8.21
+则量计数：100008
+Subset1
+半宽：10.0*
+样品对称性：三料
+使用样本对疗性：数量
+投射类型：等围积
+透射平面：XY
+率球：上
+00'0 
+11 
+The IPF map with the crystallographic orientation of the grains in the Mg-Y-Ca 
+alloy is depicted in Fig. 3. The grains were larger than 150 μm, and the micropillars 
+were milled from the center of the grains to ensure that they were single crystals. Four 
+grains with appropriate orientations (Fig. 3) were selected to activate different 
+deformation mechanisms. The loading directions in the four grains are listed in Table 1, 
+as well as the maximum Schmid Factor (SF) for the corresponding slip systems (<a> 
+basal slip, <a> prismatic slip, <a> pyramidal Ⅰ slip, <c+a> pyramidal Ⅰ slip and <c+a> 
+pyramidal Ⅱ slip) as well as {101$2} tensile twinning. The inclination angle in Table 1 
+indicates the angle between the c-axis of each grain and the compression direction, as 
+presented. The compression direction is nearly parallel to [112$0], [101$0], and [0001] in 
+grains B, C and D, respectively, and forms an angle of ~ 48.5° with respect to [0001] 
+axis in grain A. Herein, grain A presents the highest SF for <a> basal slip, which is 
+prone to be the dominant deformation mechanism during compression. Plastic 
+deformation along the <c+a> pyramidal I and II systems is favored in Grain D. <a> 
+prismatic and pyramidal as well as <c+a> pyramidal slip systems have similar SFs in 
+grain B, while grain C is suitably oriented to promote tensile twinning and <a> 
+prismatic slip. 
+ 
+Table 1. The loading direction, inclination angle, elastic modulus, and maximum 
+Schmid factor for each slip system and tensile twinning in the selected grains. 
+Grain 
+Loading 
+direction 
+Inclination 
+angle (°) 
+Elastic 
+modulus 
+(GPa) 
+Maximum Schmid factor 
+Basal 
+<a> 
+Prismatic 
+<a> 
+Pyramidal 
+Ⅰ <a> 
+Pyramidal 
+Ⅰ <c+a> 
+Pyramidal 
+Ⅱ <c+a> 
+Tensile 
+twin 
+A 
+[112!3] 
+48.5 
+46.04 
+0.44 
+0.25 
+0.42 
+0.36 
+0.20 
+0.17 
+B 
+[112!0] 
+83.4 
+48.14 
+0.11 
+0.48 
+0.46 
+0.47 
+0.47 
+0.43 
+C 
+[101!0] 
+87.8 
+46.48 
+0.03 
+0.46 
+0.42 
+0.43 
+0.37 
+0.49 
+D 
+[0001] 
+4.5 
+47.47 
+0.06 
+0.00 
+0.03 
+0.44 
+0.47 
+-* 
+*Tensile twinning cannot be activated during compression along the c-axis. 
+
+ 
+12 
+ 
+Fig. 3. Inverse pole figure (IPF) map showing the crystallographic orientation of the 
+grains in the Mg-Y-Ca alloy. (a) The loading direction in micropillars from grains A and 
+B form an angle of ~ 48.5° with the [0001] crystal orientation and are parallel to [112$0], 
+respectively. (b) The loading direction in micropillars from grains C is parallel to [101$0]. 
+(c) The loading direction in micropillars from grain D is parallel to [0001]. The 
+compression loading direction is perpendicular to the paper. 
+ 
+3.2.1 Deformation mechanisms in micropillar of grain A  
+The engineering stress-strain curves obtained from the compression micropillars 
+carved from grain A along [112$3] orientation are plotted in Fig. 4a. For the sake of 
+clarity, the horizontal axis of the green curve in Fig. 4a is shifted by 0.5%. After the 
+initial elastic region, the curves show gradual yielding and reach a plateau in the flow 
+stress at an applied strain of ~ 5%, without significant work hardening afterwards. This 
+behavior is consistent with a plastic deformation dominated by basal slip in pure Mg 
+and Mg alloys (Kiener et al., 2021; Y. Liu et al., 2017; Luo et al., 2022; Wang et al., 
+2020, 2019a; Wu et al., 2020). Small strain bursts (noticed by sudden drops in the stress) 
+are present in the stress-strain curves and they are associated with the activation of 
+dislocation sources in particular basal slip planes. However, the magnitude of the strain 
+400μm
+200μm
+011!0
+0001
+1!21!0
+Loading direction
+200μm
+Grain A
+Grain B
+Grain C
+Grain D
+(a)
+(b)
+(c)
+
+ 
+13 
+bursts is much smaller than that reported in other Mg alloys. In fact, large strain bursts 
+are associated with the localization of deformation in a few slip planes along the 
+micropillar (Wang et al., 2019). However, the lateral and top views of the micropillar 
+after deformation (Figs. 4c and 4d, respectively) show evidence of uniform slip traces 
+along the length and width of the micropillar, indicating that plastic deformation was 
+homogeneous. A yield stress of 65 ± 11 MPa (indicated by the black stars in the inset 
+of Fig. 4a) was determined from the critical points in the engineering stress-strain 
+curves when the curves deviated from linearity, following the procedure detailed in 
+(Alizadeh and LLorca, 2020; Wang et al., 2019a). 
+Secondary electron images of lateral and top views of the deformed micropillars 
+were obtained in the SEM to ascertain the actual deformation mechanisms and are 
+shown in Figs. 4c and 4d, respectively. Many parallel slip traces appear on the top and 
+lateral surfaces, which were not present before deformation (Fig. 4b). The orientation 
+of the slip traces on the micropillar surfaces is indicated by the green dashed lines in 
+Figs. 4c and 4d. The slip steps were obviously observed on the top view from the top 
+right corner to the lower left corner, and the corresponding slip direction is determined 
+as marked with a white arrow in Fig. 4d. They are indicated by blue planes and red 
+arrows, respectively, in Figs. 4e and 4f within the crystallographic lattice. It is evident 
+that the slip traces in the micropillar are parallel to the basal planes and the shear 
+deformation takes place along the [21$1$0] direction, as shown from the top and lateral 
+views of the deformed micropillar. In fact, the (0001) <21$1$0> basal slip system has 
+highest SF (listed in Table 1) and plastic deformation along this slip system is dominant 
+in this micropillar. Therefore, the CRSS for <a> basal slip (based on the yield stress and 
+the corresponding SF) can be estimated as 29 ± 5 MPa. 
+
+ 
+14 
+ 
+Fig. 4. (a) Engineering stress-strain curves obtained from micropillar compression tests 
+in grain A. The yield stress is marked with a black star. SEM images of the micropillar 
+(b) before compression and after compression from the (c) right lateral view and (d) top 
+view. The slip plane trace and slip direction are indicated by the green dashed lines and 
+the white arrow, respectively. The schematic crystallographic lattice of the 
+corresponding slip plane is presented (e) for right side and (f) for top side. The blue 
+planes indicate the theoretical basal glide planes, and the red arrows represent the 
+corresponding shear directions.  
+ 
+3.2.2 Deformation mechanisms in micropillars of grains B and C 
+Representative engineering stress-strain curves obtained from micropillar 
+
+150
+Engineering stress (MPa)
+Top side
+100
+:0
+50
+Right
+side
+0
+2um
+3
+6
+9
+12
+Engineering strain (%)
+Right side
+Top side
+Slip
+Tr.Basal plane
+direction
+Tr.Basal plane
+2μm
+2μm
+Basal slip plane
+Basal slip plane 
+15 
+compression tests along [112$0] in grain B and along [101$0] in grain C are plotted in 
+Figs. 5a and 5b, respectively. The horizontal axis of the green and blue curves was 
+shifted by -0.1% and +0.1%, respectively, in the inset of Fig. 5b for the sake of clarity. 
+The stress-strain curves are smooth, without distinct strain bursts. The initial elastic 
+region is followed by another linear plastic region with reduced strain hardening rate. 
+This behavior is radically different from that observed in micropillars with equivalent 
+orientation in pure Mg and several Mg alloys (Mg-Al, Mg-Zn, Mg-Y and Mg-Zn-Ca) 
+(Li et al., 2021b, 2021a; Y. Liu et al., 2017; Wang et al. 2021a, 2020; Wu et al., 2020), 
+which presented large strain bursts after the initial elastic region due to the nucleation 
+of tensile twins at the top of the micropillar. They are similar to those found in Mg-2Y 
+(wt. %) alloy at 250 ℃ (Li et al., 2021b), where <a> prismatic slip replaced twinning 
+as the dominant plastic deformation mechanisms. The yield stresses (obtained as 
+indicated above and marked with purple stars in Fig. 5) were 219 ± 9 MPa and 228 ± 
+4 MPa along [112$0] and [101$0] orientations, respectively. 
+ 
+Fig. 5. (a) Engineering stress-strain curves obtained from micropillar compression tests 
+in grain B along [112$0]. (b) Idem in grain C along [101$0]. 
+ 
+The representative morphology of the micropillar deformed along [112$0] (grain B) 
+is depicted in the SEM images in Figs. 6a and 6b from two different sides (front and 
+left, respectively). Faint slip traces are visible on both lateral surfaces of the deformed 
+micropillars, as indicated by the blue dashed lines in Figs. 6c and 6d, which show the 
+rectangular zones marked by dashed lines in Figs. 6a and 6b, respectively, at higher 
+magnification. The slip traces are distributed homogeneously along the lateral surfaces, 
+Grain B:[112!0]
+(a) 
+(b) 
+Grain C:[101!0]
+
+350
+300
+250
+200
+50
+100
+50
+0
+2
+4
+6
+8
+10
+12
+0
+Engineering strain (%)240
+220
+200
+0.8
+1.3
+1.8280
+230
+180
+1.2
+1.7
+2.2350
+Grain C:10101
+300
+D
+250
+stress
+200
+150
+100
+Engineering
+strain 
+16 
+indicating that plastic deformation was uniform along the micropillar. Moreover, there 
+are not slip steps at the surface (as opposed to the micropillar deformed along [112$3] in 
+Figs. 4c and 4d), in agreement with the smooth stress-strain curves. This deformation 
+morphology is different from that observed in other Mg alloys compressed along a-axis 
+(Li et al., 2021b, 2021a; Y. Liu et al., 2017; Wang et al., 2020; Wu et al., 2020), where 
+two regions with different contrast were always observed after the deformation due to 
+the nucleation of tensile twins. 
+ 
+(a)
+2μm
+Front side
+Left side
+2μm
+(c)
+(b)
+1μm
+Tr. Prismatic 
+plane
+(d)
+1μm
+(h)
+(i)
+Prismatic slip plane
+Left side
+Front side
+2μm
+4
+0
+2μm
+2μm
+(e)
+(f)
+(g)
+Tr. Prismatic 
+plane
+Before
+deformation
+After
+deformation
+
+KAM
+Kernel Aver. Misorient.
+0
+[o]
+3.96
+Y1
+5μm
+光栅：498x331
+步长尺寸：0.025μm
+>X12μm
+光栅：243x1692 
+17 
+Fig. 6. SEM images of the micropillar deformed along [112$0] from grain B. (a) Lateral 
+front and (b) lateral left view side. The traces of the active slip planes are indicated by 
+the blue dashed lines in Figs. 5c and 5d, which show the rectangular zones marked by 
+dashed lines in Figs. 5a and 5b, respectively, at higher magnification. (e) and (f) TKD 
+maps of the lamella extracted from the undeformed region in grain B and along the 
+compression direction from the deformed micropillar, respectively. (g) KAM map of 
+the deformed micropillar. (h) and (i) Schematics of the crystallographic lattice showing 
+the corresponding slip plane for the lateral front side and left side, respectively. The red 
+planes indicate the theoretical prismatic glide planes, and the blue lines represent the 
+corresponding slip traces. 
+ 
+In order to identify the deformation mechanisms, two parallel thin foils were 
+extracted from the undeformed region in grain B and along the loading direction from 
+the deformed micropillar, respectively, and their orientation was determined by TKD. 
+The position of the lamellae is indicated in Fig. S2 of the supplementary material. The 
+corresponding orientation maps in Figs. 6e and 6f show that the IPF map (∥Z) of the 
+undeformed and deformed thin foils share the same orientation and demonstrate that 
+tensile twins were not nucleated during micropillar compression up to 10% strain. 
+Moreover, Fig. 6g presents the kernel average misorientation (KAM) map of the whole 
+pillar in Fig. 6g reveals the homogeneous deformation without shear bands assuming 
+an angular threshold of 4°. The slip traces on the lateral surfaces of the micropillars 
+were associated with the prismatic planes, as indicated in Figs. 6h and 6i. Thus, 
+prismatic slip was triggered at the onset of the yielding and dominated plastic 
+deformation. The maximum SFs for <a> prismatic slip, <a> pyramidal I, <c+a> 
+pyramidal II slip and tensile twinning were very similar along this orientation (Table 1) 
+but the presence of Y and Ca in solid solution favored the activation of prismatic slip. 
+It should be noted that <c+a> pyramidal I slip dominated plastic deformation and 
+hindered the development of tensile twinning in micropillar compression tests along 
+[1$21$0] orientation in a Mg-4Y (wt. %) (Wu et al., 2020). The maximum SFs for <c+a> 
+pyramidal I slip, tensile twinning and <a> prismatic slip in this orientation were 0.41, 
+0.46 and 0.49 and, thus, the preference of <c+a> pyramidal slip can be associated with 
+the higher CRSSs for tensile twin nucleation and <a> prismatic slip in Mg-4Y alloy 
+(Wu et al., 2020). 
+
+ 
+18 
+Similar deformation morphology was found in the micropillars deformed along 
+[101$0] in grain C. Continuous slip traces were homogeneously distributed along the 
+lateral surfaces, as indicated by the dashed blue lines from in Fig. 7b, which shows the 
+rectangular region marked by dashed lines in Fig. 7a at higher magnification. As in the 
+previous case, the micropillar orientation before and after deformation was assessed by 
+TKD carried out in a thin lamella extracted from the undeformed region (Fig. 7c) and 
+from the deformed micropillar along the loading direction (Fig. 7d), respectively. The 
+relative orientation between the two thin lamellas is shown in Fig. S3 in the 
+supplementary material. The corresponding orientation maps do not show any evidence 
+of tensile twinning and <a> prismatic slip was again the dominant plastic deformation 
+mechanism. This conclusion is supported by the uniform plastic deformation without 
+obvious shear bands revealed by the KAM map assuming an angular threshold of 2° 
+(Fig. 7e) and the agreement between the slip traces on the lateral surfaces with the 
+orientation of the prismatic planes in the micropillar (Fig. 7f). Thus, the CRSSs for 
+prismatic slip (obtained from the yield stress and the SF for both micropillar 
+orientations) were determined to be 105 ± 4 MPa and 105 ± 2 MPa along [112$0] and 
+[101$0] orientations, respectively. 
+
+ 
+19 
+ 
+Fig. 7. SEM images of the micropillar deformed along [101$0] from grain C. (a) Lateral 
+left view side and (b) which shows the rectangular zone marked with a dashed line in 
+Fig. 7a at higher magnification. The traces of the active slip planes are indicated by the 
+blue dashed lines. (c) and (d) TKD maps of the lamella extracted from the undeformed 
+region in the grain and along the compression direction from the deformed micropillar, 
+respectively. (e) KAM map of the deformed micropillar in grain C. (f) Schematic of the 
+crystallographic lattice showing the corresponding slip plane for the lateral left view in 
+(a) and (b). The red plane indicates the theoretical prismatic glide plane, and the blue 
+2μm
+(e)
+Prismatic slip plane
+Left side
+2μm
+(c)
+1μm
+Left side
+Tr. Prismatic 
+plane
+(a)
+(b)
+2μm
+(d)
+(f)
+2
+0
+2μm
+Before
+deformation
+After
+deformation
+
+2KAM
+Kernel Aver. Misorient.
+光栅：164x268步长尺寸：0.03μm
+Y1
+2μm
+?X1IPF
+IPF Coloring II ZO
+Magnesium
+0001
+-12-10
+01-10
+Y1
+2μm
+光栅：220x145
+步长尺寸：0.04μm
+>X1 
+20 
+line represents the corresponding traces.  
+ 
+Further assessment of the deformation mechanisms was carried out by means of 
+TEM observations of the dislocation structures in a thin lamella extracted from the 
+micropillar deformed along [101$0] (Fig. 8). The lamella was nearly parallel to (1$21$0) 
+plane, as confirmed by the SADP in the inset in Fig. 8a, and there are no traces of 
+twinning in the micropillar. Dark field micrographs of the square region marked in Fig. 
+8a are depicted in Figs. 8b and 8c with g = (101$0) and g = (0002), respectively. Large 
+density of dislocations is observed in Fig. 8b but they disappear from this region when 
+g = (0002) in Fig. 8c. They are obviously <a> dislocations with 1/3 a [112$0] or 1/3 
+[21$1$0] Burgers vector, based on the dislocation extinguish condition. However, the SF 
+of the {011$0} [21$1$0] prismatic slip system is very low (~0.05), thus, it is reasonable to 
+assume that the Burgers vector of the <a> dislocations in Fig. 8b is 1/3 [112$0]. The <a> 
+screw dislocations are observed under g = (101$0) condition as marked with yellow 
+arrows in Fig. 8b. The Burgers vector of screw dislocation is parallel to the dislocation 
+line, leading to the straight dislocation lines nearly parallel to trace of the basal planes 
+(marked with a green line).  
+
+ 
+21 
+ 
+Fig. 8. TEM micrographs of the lamella extracted from the micropillar deformed along 
+[101$0]. The beam direction is parallel to [1$21$0] orientation. (a) Low magnification view 
+of the lamella. (b) and (c) High magnification dark field micrographs with g = (101$0) 
+and g = (0002), respectively, from the region marked with a blue square in (a).  
+ 
+The activation of the <a> prismatic slip during compression along the a-axis has 
+been reported recently in Mg-Zn-Ca alloy (Wang et al., 2021b) in combination with 
+tensile twinning. However, the activation of <a> prismatic slip and the suppression of 
+tensile twinning during compression along the a-axis has not been found at ambient 
+temperature in pure Mg (Y. Liu et al., 2017) or any Mg alloys (Li et al., 2021b, 2021a; 
+Wang et al., 2021a, 2020; Wu et al., 2020). This result is very surprising because 
+compression of Mg and its alloys along the a-axis (or equivalent extension along the c-
+axis) easily leads to the nucleation and growth of {101$2} tensile twins, because the 
+associated CRSS to promote tensile twin is much lower than that necessary to activate 
+5μm
+(a)
+200nm
+200nm
+(b)
+(c)
+g=(101!0)
+g=(0002)
+<a> dislocations
+(0001) plane
+B=[1!21!0]
+
+ 
+22 
+<c+a> pyramidal slip or <a> prismatic slip. While the addition of Y and Ca in solid 
+solution leads to a large increase in the CRSS for <a> prismatic slip with respect to pure 
+Mg (from 39 MPa in pure Mg (Kaya, 2013) to 105 MPa), it seems to have a much larger 
+effect on the CRSS for twin nucleation. In fact, considering the maximum stresses 
+attained in the micropillar compression tests along [112$0] and [101$0] orientations (258 
+MPa in Fig. 5a and 303 MPa in Fig. 5b, respectively) and the maximum SFs for tensile 
+twinning in both orientations (Table 1), it can be estimated that the CRSS for twin 
+nucleation in the Mg-Y-Ca alloys should be higher than 148 MPa.  
+ 
+3.2.3 Deformation mechanisms in micropillar of grain D 
+The engineering stress-strain curves obtained from the compression micropillars 
+carved from grain D along [0001] orientation are plotted in Fig. 9a. After the elastic 
+region, a strong linear hardening was observed in the plastic region. The yield stress 
+(marked by the purple stars in the inset) was 431 ± 15 MPa. This mechanical response 
+is in good agreement with the results reported in Mg-0.4Y (wt. %) and Mg-4Y (wt. %) 
+alloys (Wu et al., 2020) as well as in precipitation-hardened Mg-4Zn (wt. %) alloy 
+(Alizadeh et al., 2021) under c-axis compression. In all these cases, the presence of Y 
+in solid solution or of β1
+'  precipitates increased the CRSS for basal slip and plastic 
+deformation was accommodated through <c+a> pyramidal slip due to the low SF of 
+basal planes in this orientation. The SEM micrograph of the lateral side of deformed 
+micropillar in Fig. 9b, shows no slip traces but this behavior is also typical of pyramidal 
+slip, which does not lead to visible slip traces on the micropillar surface. The orientation 
+of the micropillar after deformation was assessed by TKD in a thin lamella extracted 
+along the compression direction. The IPF map in Fig. 9c indicates the absence of the 
+tensile twinning during deformation. 
+
+ 
+23 
+ 
+Fig. 9. (a) Engineering stress-strain curves from the micropillar deformed in 
+compression along [0001] in grain D. (b) SEM micropillar of the lateral side of the 
+deformed micropillar. (c) IPF map of the lamella extracted from the deformed 
+micropillar.  
+ 
+To further elucidate the deformation mechanisms, the analysis of the dislocation 
+structures was carried out by TEM in a thin lamella extracted from the deformed 
+micropillar. The beam direction was parallel to [112$0] as confirmed by the SADP in the 
+inset in Fig. 10a. Two-beam condition imaging was performed with g = (0002) and g = 
+(101$0) and the corresponding dark field micrographs are depicted in Figs. 10b and Fig. 
+10c, respectively. A large density of <c+a> dislocations (marked with blue arrows) is 
+observed under g = (0002) in Fig. 10b, and some <a> components are still in contrast 
+at the same location when the operation vector changes to g = (101$0) in Fig. 10c. The 
+detail of the rectangular region marked with purple dashed lines in Fig. 10b is shown at 
+higher magnification in Fig. 10d. The <c+a> dislocations (marked with the blue dashed 
+lines) are [1$1$23]/3 and [12$1$3]/3 according to the [112$0] crystal orientation in Fig. 10e. 
+These results are in agreement with those reported in Mg-Zn-Ca and Mg-Y alloys 
+(Wang et al., 2021a; Wu et al., 2020). Nevertheless, it should be noticed that it is 
+difficult to identify the active pyramidal plane, since both pyramidal I and pyramidal II 
+planes contain the same slip directions. Thus, it can be concluded that plastic 
+deformation along the c-axis in compression was dominated by <c+a> pyramidal 
+dislocations. The activation of pyramidal slip was associated to homogeneous 
+deformation and strong strain hardening (Basu et al., 2021). This high hardening rate is 
+likely associated with short mean-free paths and this explains why no slip traces were 
+2μm
+(b)
+After compression
+(a)
+1μm
+(c)
+
+2700
+600
+90
+500
+400
+300
+200
+100
+10
+ ineeiring
+Stran500
+400
+300
+2
+3.T 
+24 
+found on the micropillar surface. It is not clear whether slip took place along pyramidal 
+I plane or pyramidal II plane but the SF is slightly higher for pyramidal II for this 
+particular orientation (Table 1), which is assumed to be the active one. Thus, the CRSS 
+for <c+a> pyramidal Ⅱ slip can be estimated as 203 ± 7 MPa from the SF and the yield 
+stress.  
+ 
+Fig. 10. (a) Bright field TEM image of the thin lamella extracted from the micropillar 
+deformed along the [0001] orientation. The beam direction is [112$0], as shown by the 
+SADP in the inset. (b) and (c) Dark field TEM micrographs of the square region marked 
+with orange dash lines in (a) under g = (0002) and g = (101$0), respectively. (d) Bright 
+field TEM micrograph obtained under g = (0002) from the rectangular region marked 
+with purple dash lines in (b). The potential <c+a> pyramidal dislocations are indicated 
+by the crystal orientation (black box) and the pyramidal slip trace (blue line). (e) 
+Schematic of Mg crystal orientation and of the corresponding <c+a> dislocations (blue 
+lines) from the [112$0] projected view.  
+ 
+3.3 Effect of Y and Ca on the GSFE curves 
+(a)
+1μm
+g=(101!0)
+<a> components
+(0001) plane
+<c+a> dislocation
+100nm
+100nm
+(b)
+(c)
+(d)
+(e)
+50nm
+g=(0002)
+B=[112!0]
+g=(0002)
+
+Basal trace 
+25 
+The experimental evidence presented above shows that the addition of Y and Ca 
+affects significantly the plastic deformation mechanisms. The changes in the 
+deformation mechanisms are proposed to be associated with the modification of the slip 
+resistance of the different slip systems due to the presence of the solute atoms. The 
+GSFE is intimately associated with the activation barriers of the deformation modes, 
+hence influencing their relative contributions to the overall deformation behavior 
+(Sandlöbes et al., 2011). To ascertain the effect of the solute atoms (Y and/or Ca) on the 
+slip activities, the GSFE (γ) curves were computed for the <a> slip systems in Mg-Y, 
+Mg-Ca, and Mg-Y-Ca alloy, as well as in pure Mg for comparison. 
+The GSFE curves for {0001}<101$0>, {11$00}<112$0> and {101$1}<1$21$0> slip 
+systems are presented in Figs. 11a, 11b and 11c, respectively. The curves exhibited only 
+one local maximum, from which the unstable stacking fault energy (γus) for each slip 
+system was determined (Table 2). γus is associated with the activation barrier for 
+dislocation slip (Ding et al., 2018; Dong et al., 2018). Evidently, the addition of Y and 
+Ca reduced slightly the γus for <a> basal slip from 88 mJ/m2 in pure Mg to a minimum 
+of 64 mJ/m2 in Mg-Ca or of 73 mJ/m2 in Mg-Y and the γus of Mg-Y-Ca (74 mJ/m2) was 
+similar with that of Mg-Y. However, the reduction in γus for the <a> prismatic slip 
+system was much more important, from ~235 mJ/m2 in pure Mg to γus of ~18 mJ/m2 in 
+the Mg-Y-Ca alloy (Table 2). This synergistic contribution of Y and Ca on γus for <a> 
+prismatic slip is obvious as the sole addition of either Y or Ca only reduced γus to 120 
+mJ/m2 (Table 2). The dramatic reduction of γus for <a> prismatic slip in the Mg-Y-Ca 
+alloy facilitates the activation of this deformation mechanism during plastic 
+deformation. On the contrary, the γus for <a> pyramidal Ⅰ only changed from 304 mJ/m2 
+in pure Mg to 318 mJ/m2 in Mg-Y-Ca alloy. The sole addition of Ca (308 mJ/m2) did 
+not modify significantly γus for <a> pyramidal Ⅰ while Y (359 mJ/m2) increased slightly 
+γus for <a> pyramidal I. Thus, <a> prismatic slip is favored by the addition of Y and Ca 
+in comparison with <a> pyramidal I slip. 
+
+ 
+26 
+ 
+Fig. 11. Generalized stacking fault energy curves for (a) <a> basal (b) <a> prismatic, 
+and (c) <a> pyramidal Ⅰ slip systems in pure Mg, Mg-Ca, Mg-Y and Mg-Y-Ca alloys. 
+ 
+Table 2. The calculated γus, for basal <a>, prismatic <a>, and pyramidal Ⅰ <a>, slip 
+systems in the Mg-Ca, Mg-Y and Mg-Y-Ca alloys compared with pure Mg.  
+Alloy 
+γus (mJ/m2) 
+<a> Basal  
+<a> Prismatic  
+<a> Pyramidal Ⅰ  
+Mg48 
+88 
+235 
+304 
+Mg47Ca1 
+64 
+121 
+308 
+Mg47Y1 
+73 
+118 
+359 
+Mg46Y1Ca1 
+74 
+18 
+318 
+ 
+4. Discussion 
+4.1 Effect of Y and Ca on the CRSSs 
+The yield stresses measured from the micropillar compression tests in different 
+(a)
+(b)
+(c)
+
+ikgas
+Tas
+Ipyranniclalsillp
+300
+20
+200
+20160
+100
+60
+0.0
+0,2)
+OLA
+0.
+1.0
+lraictonaldlspiaxeementof8<112os300
+pmshatcslp
+250
+200
+160
+100
+(),,
+0.6
+1.0
+bracuoneldisplakeementof 1120100
+lbersalslip
+80
+6X0
+0
+0.0
+0.)
+0.6
+0.8
+1.0
+Hracuonealdisplakeementof sslolcs 
+27 
+orientations are summarized in Table 3. The CRSS for the dominant slip system in each 
+orientation (following slip trace analysis and TEM characterization) is also presented 
+in Table 3. They are <a> basal slip in the micropillars carved from grain A, <a> 
+prismatic slip in the micropillars from grains B and C, and <c+a> pyramidal II slip in 
+the micropillars from grain D. Moreover, twin nucleation was not observed in 
+micropillars carved from grains B and C and this result can be used to obtain thresholds 
+of the CRSS for twin nucleation from the maximum stress attained during the test and 
+the maximum SF for tensile twinning in Table 1. These minimum values are also 
+included in Table 3. It should be noticed that dimensions of the micropillars selected in 
+this investigation follow previous results in Mg alloys (Li et al., 2021a; Wang et al., 
+2020; Wu et al., 2020) that indicate these values should not be very much influenced 
+by the “smaller is stronger” effect reported for micropillar compression tests at the 
+micron or sub-micron scale (Aitken et al., 2015; Chang et al., 2014). 
+ 
+Table 3. Yield stress and CRSS for different slip systems from micropillar compression 
+tests along different orientation in the Mg-Y-Ca alloy. 
+Grain 
+Loading direction 
+Yield stress (MPa) 
+CRSS (MPa) 
+A 
+[112!3] 
+65 ± 11 
+29 ± 5 (<a> basal slip) 
+B 
+[112!0] 
+219 ± 9 
+105 ± 4 (<a> prismatic slip) 
+> 111 MPa (tensile twin*) 
+C 
+[101!0] 
+228 ± 4 
+105 ± 2 (<a> prismatic slip) 
+> 148 MPa (tensile twin*) 
+D 
+[0001] 
+431 ± 15 
+203 ± 7 (<c+a> pyramidal Ⅱ slip) 
+*: Tensile twin was not nucleated when the CRSS reached this value. 
+ 
+In order to ascertain the strengthening effect of Y and Ca atoms in solid solution, 
+the CRSSs for the different slip systems in Mg-Y-Ca alloy are plotted in Fig. 12 along 
+with those reported in the literature in pure Mg (Li et al., 2021a; Wang et al., 2019a), 
+Mg-Al (Wang et al., 2020, 2019a), Mg-Zn (Li, 2019; Li et al., 2021a), Mg-Y (Li et al., 
+2021b; Wu et al., 2020), Mg-Zn-Ca (Wang et al., 2021a), Mg-Al-Ca (Luo et al., 2022) 
+
+ 
+28 
+and Mg-Y-Zn (Chen et al., 2018) alloys. All these results were obtained from 
+compression tests in micropillars with a cross-section around 5 × 5 μm2 and, thus, size 
+effects -if any- should not affect the comparison. The results for <a> basal slip in Fig. 
+12a show that the addition of Y in solid solution dramatically increases the CRSS in 
+comparison with pure Mg (Li et al., 2021a; Wang et al., 2019a) and with Mg-Al (Wang 
+et al., 2019a) or Mg-Zn (Li, 2019; Li et al., 2021a) alloys. These experimental data are 
+supported by the first principles simulations of the solute/dislocation interaction energy, 
+which showed the higher strengthening potential of Y for basal dislocations, in 
+comparison with Al and Zn, because of the larger atomic radius and shear modulus 
+misfit of Y with respect to Mg (Tehranchi et al., 2018). The addition of Ca to the Mg-Y 
+does not increase the CRSS for basal slip according to our results while the 
+strengthening effect of Ca in Mg-Al (Luo et al., 2022) or Mg-Zn (Wang et al., 2021a) 
+is limited and may also be attributed to the elastic interaction between Ca solute atoms 
+and dislocations. 
+Regarding <c+a> pyramidal slip (Fig. 12b), Zn and Al are the alloying elements 
+which lead to the largest increase in the CRSS (Li et al., 2021a; Wang et al., 2020). Zn 
+is more efficient but the solubility of Al in Mg is larger and CRSSs in the range of 200-
+250 MPa can be achieved for these binary alloys. Addition of 4 wt. % Y increases the 
+CRSS up to 106 MPa (Wu et al., 2020) but the combination of Y and Ca leads to a 
+CRSS of 203 ± 7 MPa, similar to the one found in the binary Mg-Zn alloy. Thus, Zn, 
+Al and Y solutes increase the CRSS for <c+a> pyramidal slip due to the elastic 
+interaction of the solutes with the dislocations, as in the case of <a> basal slip. It should 
+be noticed that the activation and glide of <c+a> pyramidal dislocations is a complex 
+process that also depends on dislocation dissociation during gliding due to the larger 
+Burgers vector (Moitra et al., 2014; Tang and El-Awady, 2014). Atomistic simulations 
+have shown that the presence of Y and Ca favors the activation for cross-slip/double 
+cross-slip of <c+a> pyramidal dislocations, leading to new dislocation loops which can 
+accommodate plastic deformation (Wu et al., 2018). 
+
+ 
+29 
+ 
+Fig. 12. CRSS for (a) basal slip, (b) pyramidal slip and (c) twin nucleation and prismatic 
+slip in Mg and Mg alloys (Chen et al., 2018; Kiener et al., 2021; Li, 2019; Li et al., 
+2021b, 2021a; Luo et al., 2022; Wang et al., 2021a, 2020, 2019a; Wu et al., 2020), 
+including the results obtained for Mg-Y-Ca alloy in this investigation. All data were 
+obtained from compression tests in micropillars with a cross-section around 5 × 5 μm2. 
+The arrow in the CRSS for twin nucleation in Mg-Y-Ca indicates that the actual CRSS 
+is higher than the value in the figure.  
+ 
+The CRSSs for tensile twin nucleation, measured by means of micropillar 
+compression tests in Mg and different Mg alloys, are plotted in Fig. 12c (Kiener et al., 
+2021; Li et al., 2021a; Wang et al., 2020, 2021a; Wu et al., 2020). While the addition of 
+Y (Wu et al., 2020) and Al (Wang et al., 2020) lead to the largest enhancements in the 
+CRSS for twin nucleation (the latter because of the larger solid solubility), the highest 
+CRSS is obtained for the ternary Mg-Y-Ca alloy which -following our experimental 
+results- has to be higher than 148 MPa. Generally, the twin nucleation process is 
+dominated by the dislocation-shearing and atomic shuffle. The strong strengthening 
+(a)
+(b)
+(c)
+
+180
+160
+140
+$120
+100
+Prismatic slip
+hg
+410
+20
+8
+10200
+150
+100
+nCaamees
+10410
+RSS
+10 
+30 
+provided by Y on the CRSS for twin nucleation can be ascribed to the inhibition of 
+atomic shuffling due to the large atomic radius of Y (0.180 nm). Moreover, Ca has an 
+even larger atomic radius (0.194 nm) and it is proposed that the synergistic contribution 
+of both atoms in solid solution is responsible for the huge increase in the CRSS for twin 
+nucleation. In addition, the elastic interaction of twinning dislocations with different 
+solute atoms also leads to an increase in the CRSS for twin propagation (Ghazisaeidi et 
+al., 2014; Stanford et al., 2015), as it has been reported in previous investigations (Li et 
+al., 2021a, 2021b; Wang et al., 2020, 2021a). However, only the addition of Y and Ca 
+can inhibit twin nucleation in micropillars suitable oriented for twinning, e.g., deformed 
+in compression along [112$0] and [101$0] (Table 1).  
+The high CRSS for tensile twin nucleation in Mg-Y-Ca alloys leads to the 
+activation the <a> prismatic slip, which becomes the dominant plastic deformation 
+mechanism under a-axis compression. There is limited information on the CRSS for <a> 
+prismatic slip (because either <a> basal slip or tensile twinning are usually activated 
+before <a> prismatic slip to accommodate the plastic deformation) and the available 
+experimental data on Mg-Y-Zn (Chen et al., 2018) (102 MPa) and Mg-Y-Ca (105 ± 4 
+MPa) are plotted in Fig. 12c. The CRSS for <a> prismatic slip is much lower than the 
+CRSS for tensile twin nucleation in Mg-Y-Ca and, thus, tensile twinning is suppressed 
+during compression parallel to the a-axis.  
+The CRSSs in Fig. 12 show that the strengthening effect of the Y and Ca for <a> 
+prismatic slip is much lower than the ones reported for <c+a> pyramidal slip and twin 
+nucleation, and also, in relative terms, for <a> basal. Moreover, evidence of <a> 
+prismatic slip is unusual in Mg alloys except in the case of that they contain Ca (Zhu et 
+al., 2019), indicating that the presence of Ca reduces the activation barriers for <a> 
+prismatic slip glide. Besides, Chen et al., (2018) found that <a> prismatic slip was 
+activated during micropillar compression testing of Mg-Y-Zn alloys but it was absent 
+in solution-treated Mg-Zn alloys deformed along the same orientation (Li et al., 2021a; 
+Wang et al., 2019b), implying that the addition of Y also facilitates prismatic slip. 
+Although the elastic interaction of the solute atoms with <a> prismatic dislocations is 
+expected to increase the CRSS, the reduction of the stacking fault energy due to the 
+
+ 
+31 
+presence of Y and Ca reduces the activation barrier for dislocation movement on the 
+slip plane and facilitates the activation of this slip system.  
+Overall, the addition of Y and Ca leads to a marked solid solution strengthening 
+for <a> basal and <c+a> pyramidal slip as well as for the nucleation of tensile twins but 
+not for <a> prismatic slip. 
+ 
+4.2 Effect of plastic anisotropy on the ductility  
+In general, the tensile ductility and formability of Mg alloys during the plastic 
+deformation is dictated by the CRSS ratio between different slip systems, especially 
+between non-basal and basal slip, the latter being the dominant deformation mechanism 
+in most cases (G. Liu et al., 2017; Zhu et al., 2019). Therefore, the tensile ductility of 
+different Mg alloys is plotted as a function of the CRSS ratios between different slip 
+systems in Fig. 13 (Habibi et al., 2012; Huang et al., 2018; Shi et al., 2020; Wang et al., 
+2021b; Wu et al., 2010; Yang et al., 2022; Zhao et al., 2019a, 2019b; Zhu et al., 2020, 
+2019). The CRSS ratios were measured via micropillar compression tests in most of the 
+alloys (Agnew et al., 2003; Li et al., 2021a; Shang et al., 2021; Wang et al., 2021a, 2020, 
+2019a, 2018; Wu et al., 2020; Zhu et al., 2019) with a few exceptions. The CRSS ratio 
+between <a> prismatic and <a> basal slip in Mg-5Y (wt. %) (Huang et al., 2018) and 
+Mg-0.47Ca (wt. %) (Zhu et al., 2019) were obtained from mechanical tests in 
+polycrystals via slip trace analysis. Besides, those for pure Mg (Agnew et al., 2003), 
+Mg-0.5Ca (wt. %) (Shang et al., 2021) and Mg-3Y (wt. %) (Wang et al., 2018) alloys 
+were determined by the elasto-plastic self-consistent model, crystal plasticity finite 
+element simulations, and the elastic viscoplastic self-consistent model, respectively. 
+Moreover, the tensile elongation data were collected from pure Mg and wrought Mg 
+alloys with similar grain sizes.  
+In general, reduced ratios between the CRSS for non-basal slip and basal slip are 
+strongly associated with the improvement of the ductility of Mg alloys. This trend 
+agrees with the data plotted in Fig. 13b, which shows a clear link between the reduction 
+of CRSS <c+a> pyramidal / CRSS <a> basal and the increase in tensile elongation. However, 
+the limited data of the influence of CRSS <a> prismatic / CRSS <a> basal on the tensile ductility 
+
+ 
+32 
+in Fig. 13a are not conclusive. Obviously, low CRSS <c+a> pyramidal / CRSS <a> basal ratios 
+favor isotropic deformation and limit the development of strong basal textures and both 
+processes help to improve ductility and formability because activation of <c+a> 
+dislocations benefits the strain accommodation along the c-axis (Liu et al., 2019). 
+Besides, Wu et al., (2018) predicted that the addition of Y/Ca could significantly reduce 
+the cross-slip energy barriers between pyramidal I and pyramidal II planes, thus 
+promoting <c+a> dislocation cross-slip. Enhanced non-basal slip activities and cross-
+slip induce homogeneous deformation and improve the ductility. 
+ 
+Fig. 13. Relation between the CRSS ratios of different slip systems (Agnew et al., 2003; 
+Li et al., 2021a; Shang et al., 2021; Wang et al., 2021a, 2020, 2019a, 2018; Wu et al., 
+2020; Zhu et al., 2019) and the tensile elongation (Habibi et al., 2012; Shi et al., 2020; 
+Wang et al., 2021b; Wu et al., 2010; Yang et al., 2022; Zhao et al., 2019a, 2019b; Zhu 
+et al., 2020, 2019) in pure Mg and Mg alloys: (a) CRSS <a> prismatic / CRSS <a> basal, (b) 
+CRSS <c+a> pyramidal / CRSS <a> basal, (c) CRSS <a> prismatic / CRSS tensile twin. 
+ 
+It should be noted that these mechanisms are particularly relevant in Mg-Y (Wang 
+(a)
+(b)
+(c)
+
+Pure Mg (Zhu,2020; Agnew,2003)
+Pure Mg (Habibi, 2012; Agnew,2003)
+<0.71
+Mg-0.5Ca(Zhu,2019;Shang,2021)
+Mg-3Y(Wang,2018;Zhao,2019b)
+★Mg-5Y-0.08Ca (This work)
+30
+10
+0.
+1.:0
+1.8
+2.0
+CRSPure Mg(Li,2021a;Wang.2019a;Habibi,2012)
+PureMg(LI,2021a;Wang,2019a;Zhu,2020)
+Mg-5Zn(Shi,2020;Li,2021a)
+Mg-4Y(Wu,2020;Wu,2010)
+Mg-4.4Al(Zhao,2019a;Wang,2020)
+Mg-1.8Zn-0.2Ca (Wang,2021a;Wang,2021b)
+Mg-5Y-0.08Ca (This work)
+20
+1030
+10
+Mg-3Y(Wang.2018;Zhao,2019b)
+Mg-0.47Ca(Zhu,2019)
+Mg-5Y (Huang,2018;Yang,2020)
+★Mg-5Y-0.08Ca (This work) 
+33 
+et al., 2018) and Mg-Zn-Ca (Wang et al., 2021a, 2021b) alloys as well as in the Mg-Y-
+Ca alloy analyzed in this investigation. In all these cases, the presence of Y and/or Ca 
+also leads to a high increase in the CRSS for twin nucleation while the CRSS for <a> 
+prismatic slip is not strongly affected. As a result, the CRSS <a> prismatic / CRSS tensile twin 
+is dramatically reduced and this is accompanied by a large increase in the tensile 
+ductility, as shown in Fig. 13c. Particularly, tensile twinning is replaced by <a> 
+prismatic slip during compressive deformation along the a-axis if CRSS <a> prismatic / 
+CRSS tensile twin < 1 and twinning only occurs in grains deformed in tension along the c-
+axis. Moreover, as the CRSS for <a> prismatic slip is smaller than that for <c+a> 
+pyramidal slip, the former becomes the dominant plastic deformation mechanism in 
+grains suitable oriented for both. It should be noted that <c+a> pyramidal slip is 
+associated with a large strain hardening (Fig. 9a) that it is not present for <a> prismatic 
+slip (Fig. 5). Thus, pyramidal slip induced large stress concentrations at grain 
+boundaries that facilitate the nucleation of damage but this process is not activated if 
+<a> prismatic slip is dominant. 
+In general, the preferential activation of basal slip and tensile twinning during 
+processing always introduces a strong basal texture in wrought Mg and Mg alloys, 
+leading to the plastic anisotropy, crack formation and limited ductility (Sabat et al., 
+2015; Wang et al., 2021a). The addition of Y and Ca in our alloy strongly enhanced the 
+activation of prismatic <a> and pyramidal <c+a> slip, which also contribute to reduce 
+the intensity of the texure during extrusion, as shown in Figs. 2a and 2b. This limited 
+texture also contributes to reduce the plastic anisotropy.  
+It should also be noted that the overall mechanical response of polycrystals cannot 
+fully ascertained by means of micromechanical tests in single crystals because other 
+factors (grain boundaries, grain size and texture) play a key role in the mechanical 
+response. However, it should be emphasized that the plastic deformation of each crystal 
+within the polycrystal is intrinsically related to that of a single crystal (Wang et al., 
+2021a) and, hence, it is important to ascertain the plastic deformation mechanisms in 
+single crystals to understand the complex mechanisms in bulk polycrystalline samples. 
+Overall, these results indicate that the presence of Y and Ca in solid solution in Mg 
+
+ 
+34 
+alloys leads to a large increase in the CRSS for <a> basal slip (which induces a large 
+reduction in CRSS <c+a> pyramidal / CRSS <a> basal) while CRSS <a> prismatic / CRSS tensile twin 
+< 1. As a result, plastic deformation in polycrystals in more isotropic and localization 
+of the deformation in the form intense basal slips that promote fracture is suppressed 
+(Sandlöbes et al., 2011). Moreover, twinning and <c+a> pyramidal slip are replaced by 
+<a> prismatic slip in grains deformed along the a-axis. Suppression of twinning (which 
+induces strong anisotropy in the plastic deformation in textured alloys) and the 
+activation of <a> prismatic slip (which provides an additional plastic deformation 
+mechanism with limited hardening) lead to an important improvement in the tensile 
+ductility of Mg alloys. 
+ 
+5. Conclusions 
+The deformation mechanisms of a Mg-5Y-0.08Ca (wt. %) alloy, with a superior 
+tensile elongation (32%), were studied by means of micropillar compression tests, slip 
+trace analysis along different orientations, TEM as well as TKD. It was found that the 
+presence of Y and Ca in solid solution led to a huge increase in the CRSS for <a> basal 
+slip (29 ± 5 MPa), <c+a> pyramidal slip (203 ± 7 MPa) and tensile twin nucleation 
+(above 148 MPa). This behavior was attributed to the large mismatch of the atomic radii 
+and elastic constants of the Y and Ca atoms with respect to Mg, which leads to a strong 
+interaction of the dislocations with the solute atoms and hinders atomic shuffling, that 
+is necessary to activate twin nucleation. On the contrary, the CRSS for <a> prismatic 
+slip only increases up to 105 ± 4 MPa because the hardening induced by the interaction 
+of the solute atoms with dislocations is partially balanced by the reduction in the 
+stacking fault energy associated with <a> prismatic slip due to the presence of Y and 
+Ca. 
+The changes in the CRSS for slip and tensile twinning in Mg-Y-Ca alloys modify 
+the dominant deformation mechanisms. In particular, the CRSS <a> prismatic / CRSS tensile 
+twin is dramatically reduced and tensile twinning is replaced by <a> prismatic slip during 
+compressive deformation along the a-axis if CRSS <a> prismatic / CRSS tensile twin < 1. 
+Moreover, as the CRSS for <a> prismatic slip is smaller than that for <c+a> pyramidal 
+
+ 
+35 
+slip, the former becomes the dominant plastic deformation mechanism in grains suitable 
+oriented for both. As a result, reduction of twinning (which induces strong anisotropy 
+in the plastic deformation in textured alloys) and the activation of <a> prismatic slip 
+(which provides an additional plastic deformation mechanism with limited hardening) 
+lead to an important improvement in the tensile ductility of Mg alloys. 
+ 
+Acknowledgements 
+This work was supported by the National Natural Science Foundation of China 
+(Grant Nos. 52001199 and 51825101). Y. Cui acknowledges the support from the 
+Shanghai Sailing Program (Grant No. 22YF1419300). JLL acknowledges the support 
+from the Spanish Ministry of Science (HexaGB project, reference RTI2018-098245) 
+and from the MAT4.0-CM project funded by the Comunidad de Madrid under 
+programme S2018/NMT-4381. 
+ 
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+  
+

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+Prepared for submission to JHEP
+Dynamic Radius Jet Clustering Algorithm
+Biswarup Mukhopadhyayaa, Tousik Samuia, and Ritesh K. Singha
+aDepartment of Physical Sciences, Indian Institute of Science Education and Research Kolkata,
+Mohanpur, 741246, India.
+E-mail: [email protected], [email protected],
+[email protected]
+Abstract:
+The study of standard QCD jets produced along with fat jets, which may
+appear as a result of the decay of a heavy particle, has become an essential part of collider
+studies.
+Current jet clustering algorithms, which use a fixed radius parameter for the
+formation of jets from the hadrons of an event, may be inadequate to capture the differing
+radius features.
+In this work, we develop an alternative jet clustering algorithm that
+allows the radius to vary dynamically based on local kinematics and distribution in the η-φ
+plane inside each evolving jet. We present the usefulness of this dynamic radius clustering
+algorithm through two Standard Model processes, and thereafter illustrate it for a scenario
+beyond the Standard Model at the 13 TeV LHC.
+arXiv:2301.13074v1  [hep-ph]  30 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+Methodology
+3
+2.1
+Standard Sequential Recombination Algorithms
+3
+2.2
+Our Proposal: Dynamic Radius Jet Clustering Algorithm
+4
+3
+Application to Standard Model Processes
+7
+3.1
+Illustration I: pp → tj process
+8
+3.2
+Illustration II: pp → V j Subprocess
+15
+4
+Usefulness in BSM signals
+19
+5
+Summary and Outlook
+25
+1
+Introduction
+The physics extraction capacity of any high-energy collider depends crucially on the han-
+dling of coloured particles in various final states. These are produced as partons via ei-
+ther short-distance interactions of quantum chromodynamics (QCD) or electroweak pro-
+cesses [1, 2]. The partons, however, hadronize through long-distance QCD effects which
+are not calculable ab initio. One rather uses semi-empirical methods to predict the prob-
+ability that energetic partons will fragment into more low-energy partons and ultimately
+form colour-neutral hadrons which are observable in the detector. Groups of closely spaced
+hadrons with varied degrees of collimation form ‘jets’ whose identification, isolation, and
+merger are predicted once more with the help of semi-empirical (and by no means uniquely
+decided) algorithms called jet clustering algorithms [3–7]. The aim always remains to define
+jets with such algorithms which most accurately elicit the short-distance physics underly-
+ing the events that are studied. They thus constitute some of our most important tools in
+the analysis of phenomena at colliders.
+In the context of the Large Hadron Collider (LHC), a widely used class of jet criteria is
+based on so-called kt-type sequential recombination jet algorithms [7–13]. These algorithms
+(briefly discussed in the next section) typically try to merge ‘neighbouring’ hadrons to
+identify the group as a jet. The neighbourhood of a hadron is defined by a single radius
+parameter R0 in the η-φ plane of the detector, which is used to quantify the radius (or size)
+of a jet. This is because the hadrons within R0 are merged to form a jet while the hadrons
+outside R0 are not included in that jet. The choices for the value of R0 in these algorithms
+depend on the physics searches one is carrying out. At the 13 TeV LHC, the typical choices
+for R0 are 0.4 or 0.8 for a ‘narrow’ or a ‘fat’ jet, respectively. There are, in addition, jet
+isolation criteria depending on whether one is trying to separate a jet from a hard lepton
+or another hadronic jet. However, the sequential recombination algorithms generally do
+not accommodate varying choices of radii on a jet-by-jet basis in a single event since they
+– 1 –
+
+have a single constant parameter that determines the radius of a jet. Separate classifiers
+for a ‘narrow’ jet and a ‘fat’ jet in a single event in the current kt-type algorithms are thus
+difficult to set. An important improvement over the current fixed radius algorithms would
+be to make them adapt the jet radii dynamically jet-by-jet in each event. We make an
+attempt in this direction in this work.
+Our central idea of choosing the radius dynamically of a jet, especially for a boosted fat
+jet, is based on the kinematics of the decay products of the initiating heavy particle. From
+the theoretical side, the formation of boosted fat jet occurs due to the high collimation of
+the on-shell decay products – and their showering and subsequent hadronization – of the
+energetic and therefore boosted heavy particles. This is very different from the formation of
+light quark- or gluon-initiated jets, whose collimation is primarily due to parton showering
+and subsequent hadronization. On the other hand, at the operational level, as per the
+standard kt-type algorithms, the fat jets are formed in the same way as the regular ‘narrow’
+jets, which are initiated by light quarks or gluons. However, the kinematics of on-shell decay
+products and their radiation pattern of a heavy particle is different from the showering of
+energetic light quarks or gluons. Therefore, the internal structure of a fat jet is very different
+from a narrow one. These internal structure has been used to tag different heavy and
+light jets in the LHC context. For example, jet substructure (JSS) observable generalized
+angularities λκ
+β [14, 15] is used to distinguish between quark- and gluon-initiated jets [16–
+26]. The same variable was used in the classification among the narrow jet, fat W jet,
+or boosted top jet [27–30]. Another important set of JSS observables, namely the energy
+correlation functions (ECFs) [31, 32], was shown to be useful in classifying different types
+of jets [29, 33–37]. The observable N-subjettiness (τN) [38, 39] has been used to find the
+multi-pronged nature of light or heavy jets [40–65]. These variables have also been used
+extensively by the experimental collaborations at the 13 TeV LHC [66–68]. These examples
+try to exploit the energy distribution pattern inside a jet to distinguish a heavy object from
+a QCD jet. The common theme of these jet substructure variables is the utilization of the
+‘multi-pronged’ nature of the fat jets. Due to this multi-pronged nature, one expects the
+variance of inter-constituent distance ∆R of a fat jet to be significantly different compared
+to the narrow QCD jets.
+This variance of a jet can be used to grow the radius of a
+jet starting from an initial radius. Earlier attempts to make the jet radius variable, albeit
+with somewhat different motivations and formalisms, can be found in references [69, 70]. In
+Ref. [69], the effective radius of a pseudojet during their evolution was taken to be inversely
+proportional to the pT with a maximum cut-off on the radius. Essentially, this algorithm
+starts from a big effective radius and the size shrinks as a process of evolution. On the
+other hand, in Ref. [70], an expectation-maximization approach was taken for clustering
+the hadrons into a pre-determined number of clusters (jets). Our approach, in this work, is
+to modify the standard fixed radius kt-type algorithms to make the radius grow depending
+on the local kinematics and distribution (in the η-φ plane) of the hadrons.
+The rest of the article is organized as follows. In section 2, we briefly outline the kt-type
+sequential recombination algorithms followed by our improvement to the same. We test
+the efficacy of our algorithms on two SM processes and discuss them in section 3. Section 4
+deals with one application in the BSM scenario. We summarize and conclude in section 5.
+– 2 –
+
+2
+Methodology
+2.1
+Standard Sequential Recombination Algorithms
+At the operational level, a jet is constituted by a bunch of four-momenta obtained using
+some clustering algorithm. Among various possible ways of grouping up the four-momenta
+of an event, we need to choose those relevant to physics at the collider. It is important
+that the clustering algorithm should ensure infrared and collinear (IRC) safety, which, in
+our context, can be defined in terms of the following conditions [7]:
+Infrared (IR) safety: The output of the algorithm should not be affected by the intro-
+duction of a four-momentum with p → 0.
+Collinear (C) safety: The output of the algorithm should not be affected by a collinear
+splitting of any four-momentum.
+The algorithm that best takes care of the issue of IRC safety is known as kt-type sequential
+recombination jet clustering algorithms [7]. We briefly outline these algorithms below1.
+If an event consists of N final state particles, whose four-momenta are taken in a list as
+an input of the kt-type algorithms. The distance dij between the ith and jth four-momenta
+and the distance diB between the ith and the beam are then defined as
+dij = min
+�
+p2p
+Ti, p2p
+Tj
+�
+∆R2
+ij,
+(2.1)
+diB = p2p
+TiR2
+0,
+(2.2)
+where R0 is the radius parameter of the algorithm, ∆Rij is the Euclidean distance between
+the ith and jth four-momenta in the η-φ plane, and pTi is pT of ith four-momenta. The
+exponent p sets the weight factor to the Euclidean distance in the η-φ plane. The three
+choices of p = 1, 0 and −1 correspond to the kt (KT) [8–10], Cambridge-Aachen (CA) [11,
+12], and anti-kt (AK) [13] algorithms, respectively. The algorithm for combining nearby
+four-momenta with respect to the above distance measures to form jets has the following
+steps.
+Step 1. The distances dij for all the possible pairs and beam distances diB for all the
+four-momenta are calculated first.
+Step 2. The minimum among all the dij and diB’s is determined.
+Step 3a. If the minimum occurs at one of the i, j pairs, the corresponding ith and jth
+four-momenta are merged to form a new four-momentum. The older ones, ith and
+jth four-momenta are removed from the list and the newly merged one is added to
+the list and goes back to Step 1.
+Step 3b. On the other hand, if the minimum distance is one of the diB, the ith four-
+momenta is declared as a final jet, and it is removed from the list and goes back to
+Step 1.
+1Here, we only discuss the inclusive algorithms in the LHC context. For other jet clustering algorithms,
+please see Ref. [7].
+– 3 –
+
+Step 4. The process is stopped once the list gets empty.
+This class of algorithms is seedless because the clustering of four-momenta to form a
+jet does not start from a particular seed. Rather, the algorithms try to merge the closest
+pair first. A group of hadrons is then declared as a jet when an appropriate size is reached.
+The essential difference among the three different algorithms, viz. AK, CA, and KT is
+that they give different weights to the Euclidean distance in the η-φ plane. This typically
+sets some sort of seed to the clustering algorithms in the sense that it gives a preference
+to a hadron around which four-momenta merge to give rise to a final jet. In the case of
+the KT algorithm, it is the softer (in terms of pT ) constituent which merges first and then
+the harder ones get attached to it. As a result, the shape of the final jet may not be
+circular in the η-φ plane. On the other hand, in the AK algorithm, the hardest particle in
+a neighbourhood becomes some sort of seed for the jet and the softer ones merge at a later
+stage. Hence the final jet looks circular in the η-φ plane. In the CA algorithm, the merging
+is purely angular. Among the three algorithms, the AK algorithm is the most popular one
+owing to its circular shape. Importantly, in the kt-type algorithms, there is a fixed radius
+parameter R0, whose value dictates the typical size of all the jets in a particular event.
+We note that these algorithms are unable to capture the essential features of the events
+where narrow and fat jets may simultaneously arise. In our proposed algorithm, we have
+modified these algorithms to bring out the features of varying sizes of the jets.
+2.2
+Our Proposal: Dynamic Radius Jet Clustering Algorithm
+The usual kt-type algorithms take a fixed radius as an input parameter, and hence the
+algorithms return all the jets to be of the same size (or narrower) in a single event. This
+lack of dynamicity in choosing a radius can be overcome by setting the radius parameter
+dynamically during the construction of each jet.
+In any kt-type algorithm, the starting point is a list of N four-momenta of particles.
+We will refer to these as fundamental particles or, sometimes, fundamental four-momenta.
+The algorithm follows Steps 1 to 3b, as defined in section 2.1, iteratively until the list
+gets empty. At every iteration, the number of contents of the list gets reduced by one.
+The reduction happens in two ways: (1) via the merger of two four-momenta, (2) via the
+declaration of four-momentum as a final jet. Thus at an intermediate iteration, the list
+contains two different types of objects. These two types of objects are (1) fundamental
+four-momenta, and (2) composite four-momenta, generated through the merger of two or
+more fundamental four-momenta. These composite objects evolve through iterations to
+give rise to the final jets. For our convenience, let us label these composite evolving objects
+as pseudojets. We borrowed the name pseudojet from the PseudoJet class in the FastJet3
+package [71], where all the types of four-momenta are called pseudojet. However, we will
+call them by different names: fundamental, pseudojet (composite or evolving), and jet (or
+final jet).
+Our proposal is to change the constant nature of the radius parameter R0 in Eq. (2.2)
+to a dynamic quantity depending on the distribution, in the η-φ plane, of the fundamental
+objects inside each evolving pseudojet. Therefore, the modified distance measure for the
+– 4 –
+
+dynamic radius algorithm takes the form
+dij = min
+�
+p2p
+Ti, p2p
+Tj
+�
+∆R2
+ij,
+(2.3)
+diB = p2p
+Ti R2
+di,
+(2.4)
+where Rdi is the dynamical radius parameter, defined as
+Rdi = R0 + σi.
+(2.5)
+The constant R0 is an input parameter similar to the standard kt-type algorithm and it
+is the starting point of the dynamical growth of the radius of an evolving jet. For the ith
+pseudojet, σi is calculated as
+σ2
+i =
+�
+a<b
+pTa pTb ∆R2
+ab
+�
+a<b
+pTa pTb
+−
+�
+�
+�
+�
+�
+a<b
+pTa pTb ∆Rab
+�
+a<b
+pTa pTb
+�
+�
+�
+�
+2
+,
+(2.6)
+where the summation indices a and b run over the fundamental constituents of the pseudo-
+jet. The modifier σi of the radius parameter in Eq. (2.6) is basically ‘pT -weighted’ standard
+deviation of the distances between pairs of fundamental constituents of an evolving pseu-
+dojet. In our proposal, this standard deviation σi is used to capture the size feature of an
+evolving jet dynamically. For a single fundamental four-momentum, σi is taken to be zero.
+The motivation for choosing the modifier of the radius parameter to be pT -weighted
+standard deviation is as follows. As more than one fundamental objects merge to become
+a new pseudojet, it no longer represents a single point in the η-φ plane; it is a composite
+object whose constituents are distributed in that plane. The standard deviation σi for a
+pseudojet i, defined in Eq. (2.6), provides a measure of its fuzziness. We want to incorporate
+this fuzziness in the radius parameter. In the measure of its fuzziness, we also want the
+harder components to be more dominant than the softer ones. Essentially, if the pseudojet
+is dominated by a single pT -hard fundamental constituent or many extremely collimated
+but similar pT objects, we do not want its radius to get increased further. This is because,
+in these scenarios, the final jet is expected to be a narrow jet.
+On the other hand, if
+the pseudojet has more than one pT -hard fundamental constituents slightly separated, we
+expect it to be a fat jet and therefore need an increment to its radius. Both of these two
+aspects are taken care of by the pT -weighted standard deviation in Eq. (2.6).
+Thus, in our proposal, we first take a starting radius R0 to be our input parameter.
+The algorithm then calculates Rdi for each pseudojet, which at an intermediate state accu-
+mulates some constituents. At every iteration, the value of the dynamic radius parameter
+is calculated as the sum of the starting radius R0 and the radius modifier σ. In a nutshell,
+the proposed algorithm starts from an initial radius R0 and grows its radius dynamically
+using the information from the distribution of its constituents in the η − φ plane.
+In the proposed algorithm, the exponent p to the pT in the expressions of distance
+measures dij and diB in Eqs. (2.3–2.4) can take three possible values. We will call the
+– 5 –
+
+corresponding algorithms as dynamic radius AK (DR-AK), dynamic radius CA (DR-CA),
+and dynamic radius KT (DR-KT) jet clustering algorithms.
+The IRC safety of the algorithm through the definitions provided in section 2.1 can
+be approximately ensured in the radius modifier σ as well as in the final output of the
+algorithm. With the introduction of an additional four-momentum, say pq, the additive
+contributions to the numerators and to the denominators of the two terms in Eq. (2.6) can
+be generically written as pTq
+�
+a
+�
+pTa∆Rα
+aq
+�
+(for the denominators, α = 0, and for the two
+numerators α = 1 and 2). Clearly, all the additive contributions go to zero as pTq → 0,
+thereby ensuring the IR safety of the quantity σi for ith pseudojet. For the consideration of
+IR safety of the algorithm, let us assume an extra particle of momentum pq is introduced
+in an existing event. This extra particle actively participates in the clustering process only
+by one of the three actions: (a) by getting merged to another fundamental particle, (b) by
+getting recombined to a composite pseudojet, or (c) by getting declared as a singleton jet.
+The action (a) does not change the value of σ or the four-momentum of the pseudojet after
+the merger of the four-momentum with pq → 0. The same is true for the merger of pq via
+action (b) since the merger of two fundamental four-momenta keeps the value of σ at zero.
+After the merger of this p → 0 four-momentum, both the radius modifier σ and the total
+momentum remain unaffected. After this merger, the rest of the clustering process does
+not get affected, and hence the final output of the clustering algorithm remains unaffected.
+Furthermore, action (c) does not give rise to an extra jet whenever p → 0.
+For the collinear safety, one can see that the radius modifier σ remains almost unaltered
+when a four-momentum is split collinearly. Let a four-momentum pq gets split into pr and
+ps. Any general term pTapTq∆Rα
+aq then becomes pTa(pTr∆Rα
+ar +pTs∆Rα
+as). In the collinear
+splitting limit, pTq = pTr + pTs, ∆Rar = ∆Ras = ∆Raq.
+Moreover, there will not be
+any additional contribution due to the pr and ps combination except for the denominators
+in Eq. (2.6) since ∆Rrs = 0. This ensures an approximate collinear safety of σi for any
+ith pseudojet. On the other hand, for the collinear safety of the algorithm, if any four-
+momentum collinearly splits into two four-momenta, then the distance dij → 0. Hence,
+these two collinearly split four-momenta get merged together at a very early stage; a feature
+that is inherently present in the kt-type algorithm. Other IRC safety features (due to pT -
+dependent prefactors in the dij and diB definitions) of the standard kt-type algorithms will
+be inherited by the dynamic radius algorithm.
+We have implemented the method of dynamic radius jet clustering algorithm as a
+FastJet3 plug-in [71, 72]. This package has many built-in data-types and functionalities
+to optimize the implementation and computation of jet clustering algorithms. In particu-
+lar, we have used NNBase and NNH classes to help us keep track of the distance measures.
+As required by these two classes, our dij measure is also symmetric in i and j indices. The
+ClusterSequence class has then been used to merge two four-momenta and keep track of
+the clustering sequence. The PseudoJet class has been used to store the four-momenta
+information of all the initial, intermediate, and ��nal jets.
+The user info property of
+PseudoJet data-type has been used to store the information related to the radius modifier
+σi of the ith pseudojet. This way of implementation has at most N2 computational com-
+– 6 –
+
+plexity for an event of size N. The worst possible complexity arises when all the particles
+in an event are merged to form a single jet. Since this worst possibility does not generally
+occur, we expect the computational expense to be less in a practical scenario. We note that
+the standard kt-type algorithms also have N2 complexity via the basic implementation of
+the FastJet algorithm [71, 72].
+One important point to note is that the equations for distance measures, defined in
+Eqs. (2.1–2.2), can be recast to in the radius parameter in the expression of dij rather than
+in the expression of diB. That is to say that the modified set of equations can be taken to
+be
+˜
+dij = min
+�
+p2p
+Ti, p2p
+Tj
+� �∆Rij
+R0
+�2
+(2.7)
+˜
+diB = p2p
+Ti
+(2.8)
+The standard sequential recombination algorithm yields identical results in both formalisms
+since the radius is a constant parameter. However, if this latter formalism is chosen to
+incorporate dynamicity, the form of the dynamic radius parameter Rd will be different.
+The dynamic radius Rd, in this type of modification, will be dependent on both pseudojets.
+One option would be to add the standard deviations σi and σj of the ith and jth pseudojets,
+respectively, to the constant parameter R0. This way of defining Rd ensures the symmetry
+in i and j indices and, therefore, the implementation of the method via NNBase and NNH
+as a FastJet3 plug-in can easily be performed. In any case, the output of the algorithm
+is modified according to Eq. (2.7–2.8) will be different from that of the one considered in
+Eq. (2.3–2.4).
+We now are ready to apply our formalism to some simple SM processes and check how
+it performs compared to the standard sequential recombination algorithms. We discuss
+this in the next section in connection with SM processes and consider its application to
+BSM in the section after that.
+3
+Application to Standard Model Processes
+We take the following two SM processes to illustrate the performance of our newly developed
+algorithm.
+I. pp → tj
+II. pp → V j, (V = W or Z)
+For both cases, we have generated parton-level events using MadGraph5 (MG5) [73]. We will
+refer to these events as MG5 parton-level events and the final state partons in these events
+as MG5 partons. A lower cut of 500 GeV on the pT of the jets has been imposed during the
+generation of the MG5 parton-level events. This helps us in generating events with boosted
+top or vector bosons at the parton-level, which then form fat jets after subsequent decays
+and hadronization.
+For the purpose of the following studies, only the hadronic decays
+of top and W/Z are considered. We have then passed the MG5 parton-level events to
+– 7 –
+
+Pythia8 [74, 75] for showering and hadronization. The Monash 2013 Tune [76], the default
+tune of Pythia8, has been used to take care of the simulations of underlying events and
+multi-parton interactions in the proton-proton collisions. The output of Pythia8 has then
+been transferred to FastJet3 for the formation of jets.
+3.1
+Illustration I: pp → tj process
+The top quark, when highly boosted, results in a fat jet while the j yields a narrow jet after
+the effects of showering and hadronization. In order to compare various jet properties be-
+tween the dynamic radius and fixed radius algorithm, we run these two types of algorithms
+on the same set of hadrons from each event. We first demonstrate how the dynamic radius
+algorithms help in capturing the fat and narrow objects in a single event. This has been
+demonstrated by depicting the hadrons and jets of an example event in Fig. 1, where we
+plotted, in the η-φ plane, the position of the hadrons in the event along with the high-pT
+jets constructed out of these hadrons. The sizes of the dots are kept proportional to √pT of
+the hadrons. The jets are represented by the unfilled black circles and the solid dots inside
+the black circles comprise of the constituent hadrons of the jets. The three panels on the
+left show the jets for (a) AK, (c) CA, and (e) KT jet algorithms with R0 = 0.5. In all the
+left panels, the algorithms return three high-pT jets; one near (2,2) position and the other
+two are near (0,5) position in the η-φ plane. With the MG5 parton-level information, we
+identified that the jet in (2,2) position is initiated by j while the two jets near the (0,5)
+position is initiated by the decay products of the hard top quark. Because of fixed radii
+of the standard kt-type jet clustering algorithms, they could not capture all the hadrons
+initiated by the decay products of the top quark inside a single jet; rather they have been
+split into two different jets. A quick fix to this problem would be to increase the size of the
+radius parameter. This prescription, however, ends up increasing the jet size unnecessarily,
+for example at the (2,2) position where such increment is not required. This unnecessary
+increase in the radius of a jet increases jet mass, especially in the high pile-up scenario.
+One interesting option in such cases would be to choose the radius according to the need
+of a jet. This is precisely where the dynamic radius jet clustering algorithm is useful in
+this type of scenario. This can be seen in the three panels on the right in Fig. 1. There the
+hadrons and the high-pT jets are drawn for dynamic radius jet clustering algorithms with
+R0 = 0.5. The interesting point to note in all three right panels is that there are two jets
+instead of three. The radius of the jet near the (0,5) position has been appropriately grown
+to capture the full decay products of the top quark and their radiations while the radius
+of the jet near the (2,2) position did not grow much. This desirable characteristic of a jet
+algorithm would be beneficial for the studies of collider events, where narrow as well as fat
+jets are expected to occur simultaneously. In all the panels, the radius of each black circle
+is kept to be equal to the final radius Rd, as defined in Eq. (2.5), of each individual jet.
+For the fixed radius jet algorithms, the final radius is essentially the fixed radius parameter
+R0.
+Fig. 1 gives an approximate idea of how the dynamic radius helps us in finding a fat jet
+starting from a small radius. Next, we show how often this dynamic radius jet algorithm
+helps us in finding the fat jet. In order to demonstrate that, we have employed the following
+– 8 –
+
+−4
+−2
+0
+2
+4
+0
+1
+2
+3
+4
+5
+6
+φ
+pp → tj
+AK, R0 = 0.5
+(a)
+Hadrons
+−4
+−2
+0
+2
+4
+0
+1
+2
+3
+4
+5
+6
+φ
+pp → tj
+DR-AK, R0 = 0.5
+(b)
+Hadrons
+−4
+−2
+0
+2
+4
+0
+1
+2
+3
+4
+5
+6
+φ
+pp → tj
+CA, R0 = 0.5
+(c)
+Hadrons
+−4
+−2
+0
+2
+4
+0
+1
+2
+3
+4
+5
+6
+φ
+pp → tj
+DR-CA, R0 = 0.5
+(d)
+Hadrons
+−4
+−2
+0
+2
+4
+η
+0
+1
+2
+3
+4
+5
+6
+φ
+pp → tj
+KT, R0 = 0.5
+(e)
+Hadrons
+−4
+−2
+0
+2
+4
+η
+0
+1
+2
+3
+4
+5
+6
+φ
+pp → tj
+DR-KT, R0 = 0.5
+(f)
+Hadrons
+Figure 1: Positions of final state hadrons and jets in the η-φ plane in an example event
+for pp → tj process. The red dots represent the final state hadrons and their sizes are kept
+proportional to √pT of the corresponding hadron. The unfilled circles represent the final
+radius Rd of a jet. The teal coloured dots represent the constituents of the hard ‘narrow’
+jets. The green and blue (wherever applicable) dots represent the constituents of the fat
+top jet. The left panel, from top to bottom, is for (a) AK, (c) CA, and (e) KT algorithms
+with R0 = 0.5. The right panel, from top to bottom, represents jets clustered using (b)
+DR-AK, (d) DR-CA, and (f) DR-KT algorithms, respectively, with R0 = 0.5.
+– 9 –
+
+procedure. We first form the jets from the hadrons and choose only the high-pT (> 5 GeV)
+jets. We then tag the energetic jets, event by event, as reconstructed ‘top’ or reconstructed
+‘jet’ with the help of MG5 parton-level information. The events are classified into two
+categories, as described below.
+A1. Category A1 consists of events satisfying the following conditions.
+• A jet should have mass in the range (150, 200) GeV and have ∆R(toptruth, jet) <
+0.5. This jet is identified as a reconstructed top jet. We label these reconstructed
+objects as ‘top (A1)’ in the subsequent discussions.
+• After the tagging of the top jet, another jet should have pT > 300 GeV and
+should be within 0.5 distance from the original jet as generated by MG5. These
+jets are labelled as ‘jet (A1)’ in further discussions.
+A2. Category A2 are the events which satisfy the following conditions.
+• Two separate jets within 1.0 distance of the original top quark and having
+an invariant mass between 150 and 200 GeV. These two jets are tagged as
+constituent jets of the reconstructed top jet, which is a combination of these
+two constituents. These combinations are labelled as ‘top (A2)’.
+• Another jet having pT > 300 GeV and within 0.5 radius from the original jet.
+This is labelled as ‘jet (A2)’.
+In general, any inclusive kt-type clustering algorithm yields as output many soft jets
+along with the hard ones. The origin of these soft jets is primarily the soft radiation due
+to underlying events and wide angle parton shower. These jets are expected in both the
+category A1 and A2 events. Any jet having pT > 5 GeV and labelled neither as top nor as
+jet is labelled as soft jet.
+The two categories have been chosen to demonstrate the usefulness of the dynamic
+radius jet algorithm. Category A1 captures the whole top jet by the jet clustering algorithm
+while the events in category A2 need post-processing after the jet clustering. Therefore, a
+desirable criterion of a better-performing jet clustering algorithm would be to have more
+events in category A1. In order to illustrate that, for a given category, we define acceptance
+efficiency
+A = number of events accepted in a particular category
+total number of events
+.
+(3.1)
+After the classification of the events into the above two categories, the distribution
+of distances between the MG5 parton-level objects and reconstructed ones are plotted
+in Fig. 2. In both the panels of the figure, the blue and brown histograms are for top
+jets, and the green and red ones are for energetic jets. The corresponding categories of
+the histograms are mentioned alongside the legends. The distributions are shown for jets
+clustered using the AK algorithm with (a) R0 = 0.5, and (b) R0 = 0.8. Since this distance
+between the MG5 parton-level and reconstructed ones are features of parton showering
+and hadronization, the normalized distributions are kind of identical for different radius
+– 10 –
+
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+∆R(parton, reconstructed)
+0
+2
+4
+6
+8
+10
+12
+14
+frequency (normalized)
+(a)
+AK, R0 = 0.5
+top (A1)
+top (A2)
+jet (A1)
+jet (A2)
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+∆R(parton, reconstructed)
+0
+2
+4
+6
+8
+10
+12
+14
+frequency (normalized)
+(b)
+AK, R0 = 0.8
+top (A1)
+top (A2)
+jet (A1)
+jet (A2)
+Figure 2: Normalized distribution of ∆R between the MG5 parton-level object and cor-
+responding reconstructed jet. The jets were clustered using the AK algorithm with radius
+parameters (a) 0.5 and (b) 0.8.
+choices. These ∆R distributions are very similar even with different choices of standard or
+dynamic radius sequential recombination algorithms and, therefore, are not shown to avoid
+repetition. This distribution also justified the choice of 0.5 radius to find reconstructed
+objects from the MG5 partons.
+500
+1000
+1500
+2000
+2500
+jet energy [GeV]
+10−4
+10−3
+frequency (normalized)
+(a)
+AK, R0 = 0.5
+Category A1
+top
+jet
+500
+1000
+1500
+2000
+2500
+jet energy [GeV]
+10−4
+10−3
+frequency (normalized)
+(b)
+AK, R0 = 0.5
+Category A2
+top
+jet
+Figure 3: Normalized distribution of jet energy for categories (a) A1 and (b) A2. The
+blue and green histograms are respectively for the reconstructed top and the high-pT jet.
+We show in Fig. 3 the jet energy distributions for the objects of our study. The left
+and right panels show the distributions for categories A1 and A2, respectively. The blue
+and green histograms are for the top and the high-pT jet produced in association with it.
+– 11 –
+
+0
+50
+100
+150
+200
+jet mass [GeV]
+0.00
+0.05
+0.10
+0.15
+0.20
+frequency (normalized)
+(a)
+R0 = 0.5
+Category A1
+DR-AK A = 48.79%
+AK A = 14.71%
+DR-AK top
+DR-AK jet
+DR-AK soft
+AK top
+AK jet
+AK soft
+0
+50
+100
+150
+200
+jet mass [GeV]
+0.00
+0.05
+0.10
+0.15
+0.20
+frequency (normalized)
+(b)
+R0 = 0.5
+Category A2
+DR-AK A = 20.21%
+AK A = 52.23%
+DR-AK top
+DR-AK jet
+DR-AK soft
+AK top
+AK jet
+AK soft
+0
+50
+100
+150
+200
+jet mass [GeV]
+0.00
+0.05
+0.10
+0.15
+0.20
+frequency (normalized)
+(c)
+R0 = 0.5
+Category A1
+DR-CA A = 32.52%
+CA A = 14.67%
+DR-CA top
+DR-CA jet
+DR-CA soft
+CA top
+CA jet
+CA soft
+0
+50
+100
+150
+200
+jet mass [GeV]
+0.00
+0.05
+0.10
+0.15
+0.20
+frequency (normalized)
+(d)
+R0 = 0.5
+Category A2
+DR-CA A = 36.48%
+CA A = 50.43%
+DR-CA top
+DR-CA jet
+DR-CA soft
+CA top
+CA jet
+CA soft
+0
+50
+100
+150
+200
+jet mass [GeV]
+0.00
+0.05
+0.10
+0.15
+0.20
+frequency (normalized)
+(e)
+R0 = 0.5
+Category A1
+DR-KT A = 38.87%
+KT A = 17.71%
+DR-KT top
+DR-KT jet
+DR-KT soft
+KT top
+KT jet
+KT soft
+0
+50
+100
+150
+200
+jet mass [GeV]
+0.00
+0.05
+0.10
+0.15
+0.20
+frequency (normalized)
+(f)
+R0 = 0.5
+Category A2
+DR-KT A = 24.70%
+KT A = 47.15%
+DR-KT top
+DR-KT jet
+DR-KT soft
+KT top
+KT jet
+KT soft
+Figure 4: Normalized distribution of jet mass for the process pp → tj. The left panel
+shows the distribution for category A1 events while the right panel is the distribution for
+category A2 events. The blue, green, and red histograms are for reconstructed top, hard
+jet, and soft jets (defined in the text), respectively. The histograms, from top to bottom,
+are for AK, CA, and KT algorithms. The filled histograms correspond to fixed radius
+algorithms and the unfilled ones correspond to their dynamic radius (DR) counterparts.
+– 12 –
+
+One of the primary obligations of choosing the appropriate size for jets according to
+requirements is to avoid the rise of jet mass even with soft but widely separated constituents
+inside a jet. We, therefore, choose to show the distribution of masses of reconstructed top
+jets, reconstructed energetic jets in Fig. 4. The jet energy ranges corresponding to the
+mass distributions shown can be approximately 500-2000 GeV, as seen in Fig. 3. The left
+panel of the figure represents the distribution for category A1 events while the right panel
+represents the distribution for category A2 events. The blue, green, and red histograms
+are for reconstructed top, hard jet, and soft jets, respectively. The histograms, from top to
+bottom, are for anti-kt, C/A, and kt algorithms. The filled histograms are for standard jet
+clustering algorithms and the unfilled ones are their dynamic radius counterparts. In the
+legends, the prefix ‘DR’ to AK, CA, or KT stands for dynamic radius. In all the panels, the
+starting radius parameter has been taken to be R0 = 0.5. For standard kt-type algorithms,
+the starting radius is the fixed constant radius parameter, i.e., Rd = R0. The values for
+A for different algorithms and different categories are quoted inside each panel of Fig. 4.
+In all the panels, it is seen that the acceptance efficiencies for A1 category events in the
+cases with dynamic radius algorithms are higher than their fixed radius counterparts.
+An interesting feature to notice is that the mass distribution for the energetic jet re-
+mains almost the same for both the standard and dynamic radius jet clustering algorithms.
+The similarity between these two are more prominent for AK and CA algorithms and less
+so for the KT algorithm. This is expected as the KT algorithm starts to merge softer
+momenta first and then capture the harder ones almost at the end. As a result, this al-
+gorithm lets the size of the dynamic radius grow in the beginning and hence allows the
+softer hadron, even if they are a little wider, to merge with the evolving jet. The top jet
+mass distribution is also a little off with respect to their fixed radius analogue. These are
+not very problematic since jet grooming [77–83], trimming [84], or pruning [85, 86] methods
+help in cleaning soft and wide-angle radiation. A similar strategy of grooming is useful in
+the removal of soft jets as well.
+The change in mass distribution for top jet but not for the energetic jet can easily
+be understood from the behaviour of the final radius Rd = (R0 + σ) [Eq. (2.5)] a jet has
+acquired. We, therefore, show the distribution of the final radii of the three different types
+of jets in Fig. 5. The three plots in the top panel are for category A1 events while those in
+the bottom panel are for category A2 events. For category A2 events, ‘top c1’ and ‘top c2’
+labels represent the two constituent jets of reconstructed top. The distributions are shown
+for DR-AK, DR-CA, and DR-KT algorithms in Figs. 5(a,d), 5(b,e), and 5(c,f), respectively,
+with R0 = 0.5 in each.
+From all the histograms in Fig. 5, some clear features emerge. For the case of category
+A1 top jets, the final radius Rd grows to more than 0.6 with a peak at Rd ≃ 0.75, (ap-
+proximately 50% increase with respect to the starting radius). On the other hand, for the
+energetic jets, Rd does not grow by much. This indicates that the radius grows dynamically
+according to the distribution of constituents inside the jet. The growth of the soft jets is
+higher compared to the hard jets candidates. In general, this is will not be a problem in the
+heavy object finding since they can easily be eliminated by choosing an appropriate pT or
+mass cuts. The story for the category A2 events is similar for jets and soft jets. The only
+– 13 –
+
+0.5
+0.6
+0.7
+0.8
+0.9
+Rd
+0
+5
+10
+15
+20
+frequency (normalized)
+(a)
+Category A1
+DR-AK
+R0 = 0.5
+top
+jet
+soft
+0.5
+0.6
+0.7
+0.8
+0.9
+Rd
+0
+5
+10
+15
+20
+frequency (normalized)
+(c)
+Category A1
+DR-KT
+R0 = 0.5
+top
+jet
+soft
+0.5
+0.6
+0.7
+0.8
+0.9
+Rd
+0
+5
+10
+15
+20
+frequency (normalized)
+(b)
+Category A1
+DR-CA
+R0 = 0.5
+top
+jet
+soft
+0.5
+0.6
+0.7
+0.8
+0.9
+Rd
+0
+5
+10
+15
+20
+frequency (normalized)
+(d)
+Category A2
+DR-AK
+R0 = 0.5
+top c1
+top c2
+jet
+soft
+0.5
+0.6
+0.7
+0.8
+0.9
+Rd
+0
+5
+10
+15
+20
+frequency (normalized)
+(f)
+Category A2
+DR-KT
+R0 = 0.5
+top c1
+top c2
+jet
+soft
+0.5
+0.6
+0.7
+0.8
+0.9
+Rd
+0
+5
+10
+15
+20
+frequency (normalized)
+(e)
+Category A2
+DR-CA
+R0 = 0.5
+top c1
+top c2
+jet
+soft
+Figure 5: Normalized distribution of the final radius Rd of three different types of jets.
+The three plots in the top panel are for category A1 events while those in the bottom panel
+are for category A2 events. The conventions for the colours and labels ‘top’, ‘jet’, and ‘soft’
+are the same as in Fig. 4. For category A2, ‘topc1’ and ‘topc2’ labels represent the two
+constituent jets of reconstructed top. The distributions are shown for DR-AK, DR-CA,
+and DR-KT algorithms in the panels (a,d), (b,e), and (e,f), respectively, with R0 = 0.5.
+difference is that the whole top could not be reconstructed as a single jet in these events.
+The normalized distributions of the final radii of these two constituent jets of reconstructed
+tops are plotted. These constituents tend to grow more than the energetic jets.
+The values of acceptance efficiencies A [Eq. (3.1)] for different category events vary
+with the choice of the value for the starting radius R0. If the starting radius is small, the
+algorithms fail to capture the fat jet. On the other hand, the large starting radius R0 will
+capture the unwanted contamination coming from underlying events or radiations from
+other nearby showers. As a result, the jets will be unnecessarily fat and massive. There
+is a suitable range for R0 within which the algorithms work better. We, therefore, show
+the variation of acceptance efficiencies A as a function of starting radius R0 in Fig. 6 for
+both categories A1 (blue) and A2 (red). The variations are shown for (DR-) AK, CA, and
+KT algorithms in panels (a), (b), and (c), respectively. As expected, for small R0 values,
+the efficiencies for category A1 (blue lines) are negligible in both dynamic radius and fixed
+– 14 –
+
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+R0
+0
+10
+20
+30
+40
+50
+60
+A [%]
+(a)
+A1, DR-AK
+A2, DR-AK
+A1, AK
+A2, AK
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+R0
+0
+10
+20
+30
+40
+50
+60
+A [%]
+(c)
+A1, DR-KT
+A2, DR-KT
+A1, KT
+A2, KT
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+R0
+0
+10
+20
+30
+40
+50
+60
+A [%]
+(b)
+A1, DR-CA
+A2, DR-CA
+A1, CA
+A2, CA
+Figure 6: The variation of acceptance efficiency A [Eq. (3.1)] as a function of starting
+radius R0 for pp → tj SM process. The blue and red lines represent the variations of A for
+categories A1 and A2 events, respectively. The dashed lines are for (a) AK, (b) CA, and
+(c) KT algorithm and the solid lines are for their dynamic radius versions.
+radius analyses since the constituents of the entire top jet could not be captured with these
+small values of R0. Rather, the category A2 (red lines) which form the top with the help
+of two jets yields more A . This picture changes once we tend towards higher values for
+R0 ≃ 0.5 as more and more top jets are being reconstructed in the A1 category. As a result,
+the values of A for the A2 category get reduced. In all the panels of Fig. 6, it is interesting
+to note that the dynamic radius algorithms (solid) yield higher values for A than their
+fixed radius counterparts (dashed). This is indicative of the usefulness of the dynamic
+radius algorithm over the fixed radius ones.
+The dip in the blue solid lines after near
+R0 = 0.7 is not essentially the failure of the algorithm. Rather, it is because of the capture
+of unwanted contaminations along with the radiation coming from the top. Therefore, the
+jet mass goes beyond 200 GeV, at which point we stop labelling them as a reconstructed
+top jet. Furthermore, a rough comparison among the curves in the three panels of Fig. 6
+indicates that DR-AK is better suited than DR-CA and DR-KT algorithms.
+3.2
+Illustration II: pp → V j Subprocess
+A similar study has been performed in SM pp → V j, (V = W or Z) processes. In order
+to ensure the formation of fat jets, a lower cut of 500 GeV on the pT of the jet has been
+imposed at the time of generation of parton-level events via MG5.
+These events were
+then passed on to Pythia8 with Monash 2013 Tune [76] tune for parton showering and
+hadronization. The final state hadrons of these events were then sent to FastJet3 for jet
+clustering with starting radius R0 = 0.4.
+As before, we label the energetic jets coming from a jet clustering algorithm, as recon-
+structed ‘V’ or reconstructed ‘jet’ with the help of MG5 parton-level information. The rest
+– 15 –
+
+0
+50
+100
+jet mass [GeV]
+0.00
+0.05
+0.10
+0.15
+0.20
+frequency (normalized)
+(a)
+R0 = 0.4
+Category B1
+DR-AK A = 72.18%
+AK A = 61.96%
+DR-AK V
+DR-AK jet
+DR-AK soft
+AK V
+AK jet
+AK soft
+0
+50
+100
+jet mass [GeV]
+0.00
+0.05
+0.10
+0.15
+0.20
+frequency (normalized)
+(b)
+R0 = 0.4
+Category B2
+DR-AK A = 13.81%
+AK A = 26.23%
+DR-AK V
+DR-AK jet
+DR-AK soft
+AK V
+AK jet
+AK soft
+0
+50
+100
+jet mass [GeV]
+0.00
+0.05
+0.10
+0.15
+0.20
+frequency (normalized)
+(c)
+R0 = 0.4
+Category B1
+DR-CA A = 68.46%
+CA A = 62.30%
+DR-CA V
+DR-CA jet
+DR-CA soft
+CA V
+CA jet
+CA soft
+0
+50
+100
+jet mass [GeV]
+0.00
+0.05
+0.10
+0.15
+0.20
+frequency (normalized)
+(d)
+R0 = 0.4
+Category B2
+DR-CA A = 18.10%
+CA A = 25.25%
+DR-CA V
+DR-CA jet
+DR-CA soft
+CA V
+CA jet
+CA soft
+0
+50
+100
+jet mass [GeV]
+0.00
+0.05
+0.10
+0.15
+0.20
+frequency (normalized)
+(e)
+R0 = 0.4
+Category B1
+DR-KT A = 70.41%
+KT A = 62.58%
+DR-KT V
+DR-KT jet
+DR-KT soft
+KT V
+KT jet
+KT soft
+0
+50
+100
+jet mass [GeV]
+0.00
+0.05
+0.10
+0.15
+0.20
+frequency (normalized)
+(f)
+R0 = 0.4
+Category B2
+DR-KT A = 11.99%
+KT A = 24.02%
+DR-KT V
+DR-KT jet
+DR-KT soft
+KT V
+KT jet
+KT soft
+Figure 7: Normalized distributions of jet mass for the process pp → V j. The left panel
+shows jet mass distributions of category B1 events and the right panel is the distribution for
+category B2 events. The blue, green, and red histograms are for reconstructed V, energetic
+jet and soft jets (defined in the text), respectively. The histograms, from top to bottom,
+are for AK, CA, and KT algorithms, respectively. The filled histograms correspond to
+fixed radius algorithms and the unfilled ones correspond to their dynamic radius (DR)
+analogues.
+– 16 –
+
+of the jets having pT > 5 GeV are tagged as ‘soft jets’. As in the previous illustration, we
+classify the events into two separate categories based on the following criteria.
+B1. An event was labelled as a category B1 event if it satisfies the following two conditions.
+• A jet should have mass in the range (65, 105) GeV and ∆R(VMG5, jet) < 0.5.
+This jet was identified as a reconstructed V jet and we label them as ‘V (B1)��
+in further discussions.
+• After the tagging of the V jet, another jet should have pT > 300 GeV and
+∆R(jMG5, jet) < 0.5. These jets are labelled as ‘jet (B1)’ in further discussions.
+B2. An event, after failing to satisfy the criteria for the category B1, could be classified
+as a category B2 event subject to satisfying the below conditions.
+• Two separate jets within 1.0 distance from the original vector boson (W or Z)
+and should have an invariant mass between 65 and 105 GeV. These two jets are
+tagged as constituent jets of the reconstructed ‘V’ jet. The final reconstructed
+‘V’ jet should be within 0.5 distance from the original boson. This combination
+is labelled as ‘V (B2)’.
+• Another jet having pT > 300 GeV and within 0.5 radii of the original jet and
+this is labelled as ‘jet (B2)’.
+We show the jet mass distribution in Fig. 7 for SM pp → V j process. All the distribu-
+tions in the left panel of the figure represent the category B1 events and the distributions
+in the right panel are for category B2. The blue, green, and red histograms are for recon-
+structed ‘V’, jet and soft jets, respectively. The histograms, from top to bottom, are for
+AK, CA, and KT algorithms, respectively. The filled histograms are for standard jet clus-
+tering algorithms and the unfilled ones are their dynamic radius analogues. Quite clearly,
+the two peaks in the blue histograms, in all the distributions, correspond to the mass peaks
+of W and Z bosons. The jet mass distribution of the energetic jets using dynamic radius
+algorithms remains similar to their fixed radius counterparts. The increment in the per-
+centage of the acceptance efficiencies A [Eq. (3.1)] of category B1 events is representative
+of the appropriateness of using the dynamic radius algorithms over the standard ones in
+these types of scenarios.
+We next show in Fig. 8 the normalized distributions of the final radius for three different
+types of jets, viz. ‘V’ jets, energetic jets, and soft jets. As in the pp → tj process, the fat
+V jets acquires a larger radius than the energetic jets after the dynamical expansion of the
+jet size. Here, again, the soft jets acquire a higher radius compared to the energetic jets.
+These soft jetsare not of much concern since they are rather soft and hence they can be
+removed easily from the analysis.
+In Fig. 9, we show the variation of A [Eq. (3.1)] as a function of starting radius R0. In
+all the panels of the figure, the blue and red lines correspond to the variations for categories
+B1 and B2 events, respectively. The dashed lines are for fixed radius algorithms and the
+solid lines are for dynamic radius jet algorithms. The variations are shown for (a) AK, (b)
+– 17 –
+
+0.4
+0.5
+0.6
+0.7
+Rd
+0
+5
+10
+15
+20
+frequency (normalized)
+(a)
+Category B1
+DR-AK
+R0 = 0.4
+V
+jet
+soft
+0.4
+0.5
+0.6
+0.7
+Rd
+0
+5
+10
+15
+20
+frequency (normalized)
+(c)
+Category B1
+DR-KT
+R0 = 0.4
+V
+jet
+soft
+0.4
+0.5
+0.6
+0.7
+Rd
+0
+5
+10
+15
+20
+frequency (normalized)
+(b)
+Category B1
+DR-CA
+R0 = 0.4
+V
+jet
+soft
+0.4
+0.5
+0.6
+0.7
+Rd
+0
+5
+10
+15
+20
+frequency (normalized)
+(d)
+Category B2
+DR-AK
+R0 = 0.4
+V c1
+V c2
+jet
+soft
+0.4
+0.5
+0.6
+0.7
+Rd
+0
+5
+10
+15
+20
+frequency (normalized)
+(f)
+Category B2
+DR-KT
+R0 = 0.4
+V c1
+V c2
+jet
+soft
+0.4
+0.5
+0.6
+0.7
+Rd
+0
+5
+10
+15
+20
+frequency (normalized)
+(e)
+Category B2
+DR-CA
+R0 = 0.4
+V c1
+V c2
+jet
+soft
+Figure 8: Normalized distribution of final radius Rd for the three different types of jets.
+The top panel represents the distributions of Rd in the category B1 events and the whole
+bottom panel is for category B2 events. The conventions for the colours and labels V, jet,
+and soft are the same as Fig. 7. For category B2 events, ‘V c1’ and ‘V c2’ labels represent
+the two constituent jets of the reconstructed vector bosons. The distributions are shown for
+DR-AK, DR-CA, and DR-KT algorithms in the panels (a,d), (b,e), and (c,f), respectively,
+with R0 = 0.4.
+CA, and (c) KT algorithms. A quick observation of the curves tells us that the behaviour
+of these curves is similar to that of the curves in Fig. 6 except the monotonic decreasing
+nature of the category B2 curves. The reason is as follows: in the case of V jets, the jets
+are ‘two-pronged’ in nature. Therefore, the small radius jets can capture one of the two
+prongs of V jets, and thereby these two jets are able to reconstruct V jets in B2 category.
+However, as the starting radius R0 is increasing, more and more events are migrating to
+category B1. The declining nature of the curves for large radii after 0.5 is because of the
+fact that the jets capture more hadrons than are required for their optimal size. As a
+result, the mass of the V jets tends to go beyond the mass window set to label them as V
+jets. Again, more variables than just the jet mass can help one to improve the tagger and
+hence the acceptance efficiency.
+We conclude this section with the note that the dynamic radius jet algorithms are
+– 18 –
+
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+R0
+0
+10
+20
+30
+40
+50
+60
+70
+80
+Acceptance [%]
+(a)
+B1, DR-AK
+B2, DR-AK
+B1, AK
+B2, AK
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+R0
+0
+10
+20
+30
+40
+50
+60
+70
+80
+Acceptance [%]
+(c)
+B1, DR-KT
+B2, DR-KT
+B1, KT
+B2, KT
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+R0
+0
+10
+20
+30
+40
+50
+60
+70
+80
+Acceptance [%]
+(b)
+B1, DR-CA
+B2, DR-CA
+B1, CA
+B2, CA
+Figure 9: The variation of A [Eq. (3.1)] as a function of the starting radius R0 for pp → V j
+SM process. The blue and red lines represent the values of A for categories B1 and B2
+events, respectively. The dashed lines are for (a) AK, (b) CA, and (c) KT algorithms. The
+solid lines are for their dynamic radius versions.
+useful in finding fat as well as narrow jet in a single event in the colliders.
+We have
+successfully illustrated this in two SM processes, viz. pp → tj and pp → V j, at the 13 TeV
+LHC. A comparison among the three dynamic radius analogues of the standard kt-type
+algorithm reveals that the DR-AK algorithm performs better compared to the DR-CA or
+the DR-KT algorithms.
+4
+Usefulness in BSM signals
+We now illustrate the usefulness of the dynamic radius jet algorithm in the context of
+a scenario beyond the standard model (BSM). This is a scenario where an additional
+vectorlike singlet quark b′ of charge −1/3 exists along with (d, s, b). Such quarks occur, for
+example, in E(6) grand unified theories, as also in some seesaw models of quark masses [87–
+92]. The b′ can mix with the three SM down-type quarks when electroweak symmetry
+breaking takes place2. This causes the mass eigenstate dominated by b′ to decay into a top
+quark and a W boson. In addition, the mixing between a T3 = −1/2 quark and one with
+T3 = 0 induces flavour-changing Z- and Higgs-couplings in the b-b′ sector. Thus the b′,
+produced via strong interactions at the LHC, has the decays b′ → tW, b′ → bZ, b′ → bh.
+The detailed theoretical framework and the resulting phenomenology have been discussed
+widely in the literature [59, 93–101].
+The currently available data from the LHC restrict mb′ to be no less than 1.3–1.5 TeV
+[102–105]. When such a massive quark decays thereafter, its decay products are consider-
+2In the following discussion, we shall (a) denote this mass eigenstate itself by b′, (b) assume that
+ordinary-exotic quark mixing takes place involving only the third family sequential quark, namely, b, and
+(c) parametrize the b-b′ mixing by the angle θ.
+– 19 –
+
+ably lighter compared to it. Therefore the b′ decay products are considerably boosted, so
+as to produce fat jets. Furthermore, the difference in mass between two product particles
+leads to jets of varying degrees of fatness.
+Since our purpose here is to show the efficacy of the dynamic radius jet algorithm, we
+illustrate our main points in the context of pp → b′¯b′ followed by each b′ decaying into a
+top quark and a W boson. The t’s and the W’s thus give rise to energetic jets of different
+radii. We demonstrate below how our newly developed algorithm can capture the identity
+of the ensuing final state. While the present work is aimed at capturing the essence of our
+proposed jet algorithm, a more detailed discussion, including combinations of all the three
+aforementioned decay channels of the b′, is going to be presented in a separate work [106].
+mb′
+sin θL
+sin θR
+1.3 TeV
+0.12
+8.02 × 10−3
+Table 1: Values of some important parameters of the vectorlike singlet b′ model considered
+for the illustration.
+The model has been implemented in a Mathematica-based package SARAH [107–109].
+The Universal FeynRules Output (UFO) [110] generated by SARAH is then used in MG5
+for the generation of parton-level events. The parameter card for MG5 has been generated
+using spectrum generator SPheno [111, 112]. The values for the important parameters of
+the model are tabulated in Table 1. The angles θL and θR in the table represent the mixing
+angle between SM b quark and exotic b′ quark of chirality left and right, respectively. After
+the generation of the MG5 parton-level events, the rest of the analysis pipeline is the same
+as the previous illustrations of SM processes.
+In this illustration, we choose DR-AK, based on the discussion in the previous section.
+We show the resultant jets having pT > 30 GeV formed out of the hadrons generated by
+Pythia8 in Fig. 10. The left panel shows the positions of the generated hadrons and jets
+constructed using the AK algorithm with R0 = 0.5. The right panel shows the same for
+the DR-AK algorithm. In both panels, the red dots represent the position of final state
+hadrons in the η-φ plane and the size of each dot is proportional to the √pT of the hadron.
+The unfilled circles represent the final radius (Rd) of a jet. The teal dots represent the
+constituents of boosted fat ‘W’ jets. The green, blue, and purple (wherever applicable) dots
+represent the constituents of the fat ‘top’ jet. The yellow dots containing texts represent
+the position of the MG5 parton-level pT -hard quarks after the decay of top or W. The
+mothers of the q or b are mentioned in the subscripts of q or b.
+An interesting point to observe in Fig. 10(b) is that the DR-AK yields only 4 jets,
+which are representative of 2 fat W and 2 fat t jets. However, in Fig. 10, the fixed radius
+algorithm could form the fat W jets but fails to capture the entirety of the two fat t jets.
+One, of course, can use a bigger radius in the AK algorithm to capture the whole of the
+top jet. However, this will make the W jet unnecessarily fat. This demonstrates the utility
+of the dynamic radius jet algorithm.
+– 20 –
+
+−4
+−2
+0
+2
+4
+η
+0
+1
+2
+3
+4
+5
+6
+φ
+pp → b′¯b′ → tW −¯tW +
+AK, R0 = 0.5
+(a)
+Hadrons
+−4
+−2
+0
+2
+4
+η
+0
+1
+2
+3
+4
+5
+6
+φ
+pp → b′¯b′ → tW −¯tW +
+DR-AK, R0 = 0.5
+(b)
+Hadrons
+Figure 10: The distribution of final state hadrons and jets in η-φ plane for an example
+event. The colours and sizes of the dots and circles follow the same convention as Fig. 1.
+The teal coloured dots represent the constituents of hard fat ‘W’ jets. The green and blue
+(wherever applicable) dots represent the constituents of the fat ‘top’ jet. The yellow dots
+containing texts represent the position of the hard quarks after the decay of top or W
+which are mentioned as the subscripts of q or b. The plots are shown for (a) AK and (b)
+DR-AK algorithms.
+To study the goodness of DR-AK quantitatively, we define the following criteria for
+tagging of top and W jets.
+• A jet having mass in the range (150, 200) GeV and having ∆R(toptruth, jet) < 0.5 is
+identified as a reconstructed top jet.
+• A jet will be called W jet if it has a mass in the range (65, 105) GeV and is within
+0.5 distance from the original MG5 parton-level W boson.
+Similar to the illustrations with SM processes, we classify the events into different
+categories. Due to the complex nature of the final states, we have classified the events into
+more than two categories in the present scenario. The realization is based on the following
+understanding.
+• Out of the two W’s coming directly from b′ in an event, the number of reconstructed
+W as fat jet from the algorithm could be 0, 1, or 2. We call these reconstructed fat
+W jets as primary W jets.
+• Similarly, out of the two t quarks, the number of reconstructed t as fat jets can be 0,
+1, or 2.
+• In some particular cases, the whole top may not be reconstructed, but the W boson
+coming from the top quarks may be reconstructed. These are referred to as secondary
+W jets in the subsequent discussions.
+– 21 –
+
+Based on the above observations, we classify the events into different categories, whose
+generic name is given as Cij, where i and j are two integers encoding the number of
+reconstructed top and reconstructed W’s, respectively. For the present scenario, the allowed
+value for i does not exceed two. For a given i, the values for j should not exceed 4 − i.
+That is, to say, i ≤ 2 and j ≤ 4−i. An exhaustive list of all possible categories is tabulated
+in Table 2. For example, the event shown in Fig. 10 would be categorized as C22 for the
+DR-AK algorithm while the same event would be classified as C03 for the AK algorithm.
+One may again subdivide some of the categories into subcategories based on how many W
+jets are coming directly from b′ (primary W) and how many of them are coming from the
+decay of the top quark (secondary W). Therefore, the generic name for the subcategories
+can be given as Cijk with i, j, and k being the numbers of reconstructed top, primary W,
+and secondary W jets. The possible ranges for i, j, and k are 0 ≤ i, j ≤ 2 and 0 ≤ k ≤ 2−i.
+Category
+Subcategory
+No. of top jet
+No. of primary
+No. of secondary
+W jet
+W jet
+C22
+C220
+2
+2
+0
+C21
+C210
+2
+1
+0
+C20
+C200
+2
+0
+0
+C13
+C121
+1
+2
+1
+C12
+C120
+1
+2
+0
+C111
+1
+1
+1
+C11
+C110
+1
+1
+0
+C101
+1
+0
+1
+C10
+C100
+1
+0
+0
+C04
+C022
+0
+2
+2
+C03
+C021
+0
+2
+1
+C012
+0
+1
+2
+C02
+C020
+0
+2
+0
+C011
+0
+1
+1
+C002
+0
+0
+2
+C01
+C010
+0
+1
+0
+C001
+0
+0
+1
+C00
+C000
+0
+0
+0
+Table 2: The definitions of the list of categories and subcategories as according to how
+many fat jets can be reconstructed from the jet algorithm.
+– 22 –
+
+0
+50
+100
+150
+200
+jet mass [GeV]
+0.00
+0.05
+0.10
+0.15
+0.20
+frequency (normalized)
+(a)
+R0 = 0.5
+Category C22
+DR-AK A = 5.47%
+AK A = 1.62%
+DR-AK top
+DR-AK W
+DR-AK soft
+AK top
+AK W
+AK soft
+0.5
+0.6
+0.7
+0.8
+0.9
+Rd
+0
+5
+10
+15
+frequency (normalized)
+(b)
+Category C22
+DR-AK
+top
+W
+soft
+Figure 11: (a) The normalized distribution of jet mass of the category C22 events for the
+pp → b′¯b′ → tW −¯tW + process. The blue, green, and red histograms are reconstructed top,
+W, and soft jets, respectively. The unfilled histograms are for the jets clustered using the
+DR-AK algorithm while the filled ones are for the jets using the AK clustering algorithm.
+(b) The normalized distribution of the final radii of top, W, and soft jets with blue, green,
+and red colours, respectively. For both panels, R0 = 0.5 was used and any additional jets
+having pT >5 GeV were considered as a soft jet.
+We plot the normalized distribution of jet mass of the category C22 events in Fig. 11(a).
+Jets were clustered using R0 = 0.5. In the plot, the blue, green, and red histograms are
+reconstructed top, W, and soft jets, respectively. The unfilled histograms are for the jets
+clustered using the DR-AK algorithm, and the filled ones are for the jets using the AK
+clustering algorithm. Any untagged jet with pT > 5 GeV was considered to be a soft jet.
+Fig. 11(b) shows the normalized distribution for the finally acquired radii of different jets
+for category C22 events. The desirable feature of the reconstructed W jets being narrower
+than the reconstructed top jets is clearly apparent in the figure. Here, again, the soft jets
+are growing to larger radius are expected. However, as discussed in the previous section,
+they can be removed from an analysis by pT or jet mass cuts.
+The values of A [Eq. (3.1)] for the two different algorithms, viz. DR-AK and AK are
+also quoted in Fig. 11(a). These values (1.62% for AK and 5.47% for DR-AK), clearly,
+indicate that the dynamic radius jet algorithm is working better while probing the correct
+mass windows for the particles. The shift of the mass distribution towards larger values is
+indicative of capturing little extra than required. As discussed previously, this can be recti-
+fied by the techniques of grooming [77–81], trimming [84], or pruning [85, 86]. Furthermore,
+going beyond just the jet mass to tag the topor W jets would further help in extracting
+signals.
+The variation of A as a function of initial radius R0 is shown in Fig. 12 for six categories,
+namely C22, C21, C20, C13, C12, and C11. These categories have at least one top jet
+identified within 0.5 distance from the MG5 parton-level top quark. The solid blue lines
+– 23 –
+
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+R0
+0
+5
+10
+15
+20
+Acceptance [%]
+C22
+DR-AK
+AK
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+R0
+0
+5
+10
+15
+20
+Acceptance [%]
+C21
+DR-AK
+AK
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+R0
+0
+5
+10
+15
+20
+Acceptance [%]
+C20
+DR-AK
+AK
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+R0
+0
+5
+10
+15
+20
+Acceptance [%]
+C13
+DR-AK
+AK
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+R0
+0
+5
+10
+15
+20
+Acceptance [%]
+C12
+DR-AK
+AK
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+R0
+0
+5
+10
+15
+20
+Acceptance [%]
+C11
+DR-AK
+AK
+Figure 12: The variation of A [Eq. (3.1)] as a function of initial radius R0 for six categories,
+namely C22, C21, C20, C13, C12, and C11.
+The solid blue lines are for the DR-AK
+algorithm, and the dashed lines are for the AK algorithm. The jets are clustered with
+R0 = 0.5.
+represent the efficiencies for the DR-AK algorithm, and the dashed lines are representative
+of the AK algorithm. For the case of dynamic radius, the quintessential feature is the
+initial increment in the acceptance efficiencies A up to R0 = 0.5, and, beyond this value,
+the efficiencies decrease. The reason for this is an unnecessary accumulation of hadrons
+and making the jets bigger than their optimal size. However, for the AK algorithm, the
+efficiencies keep on increasing until R0 = 0.7, which is kind of the optimal radius for this
+scenario.
+The most important point to note is that up to R0=0.5, the efficiencies for
+the DR-AK algorithm are higher than those for the AK algorithm. This feature, again,
+establishes the utility of using dynamic radius algorithms over fixed radius ones.
+In the end, we look at the bar plot of the acceptance efficiencies A for all the categories
+in Fig. 13. The blue and green bars are for DR-AK and AK algorithms, respectively. The
+initial radius R0 is taken to be 0.5. The numbers under the curly braces below the x-axis
+represent the values of A for the categories which capture 2 tops, 1 top, 0 top, and none of
+the top or W jets. The important observation in this regard is that the categories containing
+2 top and 1 top jets have better efficiencies for the dynamic radius algorithm than the fixed
+radius one. This means that the events, where the AK algorithm could not capture the
+whole of the top constituents, the DR-AK algorithm could capture the full tops. Thus the
+credence of our proposed algorithm is established in a BSM context as well.
+– 24 –
+
+C22 C21 C20 C13 C12 C11 C10 C04 C03 C02 C01 C00
+Categories
+0
+5
+10
+15
+20
+25
+30
+Acceptance [%]
+R0 = 0.5
+DR-AK:
+AK:
+�
+��
+�
+9.86%
+4.38%
+�
+��
+�
+35.33%
+27.31%
+�
+��
+�
+34.02%
+53.75%
+����
+20.79%
+14.56%
+DR-AK
+AK
+Figure 13: Bar plot of A for different categories for jet algorithms with R0 = 0.5. The
+blue, and green bars are for DR-AK and AK algorithm respectively. From left to right,
+The numbers under the braces represent the values of A for the categories which capture
+2 top, 1 top, 0 top, and none of the top or W jets.
+5
+Summary and Outlook
+We go beyond the most popular jet clustering algorithms, where the formation of jets
+is performed using a fixed radius parameter.
+These algorithms return fixed-sized jets
+corresponding to the input radius parameter. In this work, an attempt is made to make
+the radius of each jet variable depending on the kinematics and hadronic activity in the
+neighbourhood of an evolving jet. The proposed method is based on the standard kt-type
+sequential recombination jet clustering algorithms with the incorporation of the dynamic
+nature of the radius parameter.
+Starting from a reasonable radius parameter, during the process of formation of a jet,
+the radius of each evolving jet is allowed to grow based on fuzziness inside it. For this
+work, the measure of the fuzziness of each evolving jet is chosen to be the ‘pT -weighted’
+standard deviation of the inter-particle distances (in the η-φ plane) of the particles inside
+the evolving jet.
+After describing the proposed method, we have presented two different SM processes,
+viz. pp → tj and pp → Wj +Zj, to demonstrate some applicabilities of the dynamic radius
+jet clustering algorithm. In these two processes, differently-sized jets are expected in a sin-
+gle event. In the two SM process examples, we observe that the jets are being formed with
+radii varying in size on a jet-by-jet basis. In terms of the acceptance efficiency [Eq. (3.1)],
+we show that the performance of the dynamic radius algorithm is better compared to their
+– 25 –
+
+fixed radius counterparts. We take up a scenario beyond the Standard Model for further
+illustration, where a vectorlike SU(2)L singlet charge −1/3 quark b′ is added. We study
+jet clustering in pp → b′¯b′ followed by each b′ decaying into tW. Once more, our proposed
+method turns out to be effective in the reconstruction of the final state particles.
+In the examples given above, the dynamicity has been incorporated in the radius
+parameter of the standard kt-type sequential recombination algorithm. The central idea
+is the usage of fuzziness of an evolving jet to appropriately increase its radius starting
+from a starting radius R0. Although examples with only one measure of fuzziness have
+been shown in this work, one may consider other appropriate measures. depending upon
+the underlying physics process or the final goal of the analysis. Therefore, the idea of the
+dynamic radius jet algorithm should not be restricted only to this particular measure. The
+applicability of these possibilities will be presented in a separate work. In a nutshell, the
+idea of dynamic radius jet clustering algorithm on a jet-by-jet basis is useful in collider
+studies and will be beneficial in searches driven by processes in SM as well as BSM.
+Acknowledgments
+The authors thank Jayita Lahiri for useful discussions during the initial phase of the work.
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+Preprint 13 January 2023
+Compiled using MNRAS LATEX style file v3.0
+Investigating Dynamical Properties of Globular Clusters through a Family
+of Lowered Isothermal Models
+Chia-Hsuan Cheng1 and Ing-Guey Jiang1,2
+1Department of Physics, National Tsing-Hua University, Hsinchu, Taiwan
+2Institute of Astronomy, National Tsing-Hua University, Hsinchu, Taiwan
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+To investigate the dynamical properties of globular clusters, the surface brightness and kinematic data were collected and fitted
+to a family of lowered isothermal models called LIMEPY models. For 18 studied globular clusters, the amounts of concentration,
+truncation, and anisotropy were determined. In addition, the cluster mass, half-mass radius, distance, and mass-to-light ratio were
+also obtained. In general, LIMEPY models could describe these clusters well. Among these 18 clusters, NGC 5139, NGC 6388, and
+NGC 7078 were claimed to be candidates to host intermediate-mass black holes in literature. The models could not appropriately
+fit the central proper-motion velocity dispersion of NGC 5139 and the slope of proper-motion velocity-dispersion profile of NGC
+6388. Thus, more dedicated models with intermediate-mass black holes or a group of stellar-mass black holes at cluster centers
+may need to be considered. Considering NGC 7078, our model with some degree of anisotropy can fit the data. Finally, the
+strong concentration-truncation anti-correlation and truncation-semimajor-axis correlation were revealed, which could be the
+observational imprint of the dynamical evolution of globular clusters.
+Key words: methods: numerical – stars: kinematics and dynamics – globular clusters: general – globular clusters: individual –
+galaxies: star clusters: general
+1 INTRODUCTION
+Globular clusters are one of the oldest objects in the universe (Van-
+denberg et al. 1996). They extend spherically in several or tens of
+parsecs with hundreds of thousands of stars (Harris 1996). The high
+stellar densities make them the primary venue for hosting exotic
+objects like millisecond pulsars (Manchester et al. 1991) and blue
+stragglers (Bailyn 1995). Globular clusters have been proposed to
+possibly also host intermediate-mass black holes (Ebisuzaki et al.
+2001). With higher density, the core of a globular cluster relaxes
+faster than the halo and the relaxation time is short compared to
+the age of the cluster (Oort & van Herk 1959). Thus, the center of
+globular clusters is expected to be isothermal.
+Having theoretical models describing globular clusters is help-
+ful in obtaining the physical quantities. The isothermal sphere is a
+model with isothermal cores, so it could be considered a suitable
+simple model. However, this model extends to the infinite and has an
+unrealistic infinite mass. This problem can be solved by introducing
+some cutoffs. For example, energy truncation can limit the velocity,
+so the stars with larger velocities escape from the cluster; this results
+in a cluster model with finite mass and range. The truncation can
+be regarded as the effect of the external tidal field on star clusters.
+Different truncations lead to different models. For example, subtract-
+ing a constant from the energy leads to the Woolley model (Woolley
+1954), and further subtraction from the distribution function gives
+the King model (King 1966).
+The velocity distributions of clusters in the above models are
+isotropic. However, for realistic models, the possible anisotropy shall
+be considered. The diffusion caused by stellar encounters facilitates
+the entry of some stars into the cluster halo. These stars diffuse to
+the halo along radial orbits and increase the radial anisotropy in the
+halo (Spitzer & Shapiro 1972). The violent relaxation in the stage of
+cluster formation can also contribute to some radial anisotropy in the
+cluster halo (Lynden-Bell 1967). To include anisotropy in a model,
+one can add the angular momentum into the distribution function.
+The distribution function now depends on both the energy and the
+angular momentum. For example, the Michie-King model (Michie
+1963) includes the angular momentum in an exponential term. This
+model possesses the expected properties which contain an isothermal
+core with some anisotropy at the outer parts.
+A model with multi-mass components is another aspect of im-
+provement. Da Costa & Freeman (1976) made the extension from the
+King model by assuming that each component has the same form of
+distribution function with different constants. Later, an anisotropic
+multi-mass model was introduced by Gunn & Griffin (1979). Re-
+cently, some extensions and unification of these isothermal models
+have been developed. Considering the Woolley and the King model
+as different schemes of energy truncation characterized by some in-
+tegers, Gomez-Leyton & Velazquez (2014) established an extended
+model which parametrized the truncation by a non-negative real
+number. This was further generalized by Gieles & Zocchi (2015) to
+include the radial anisotropy and multi-mass components in a fam-
+ily of lowered isothermal models, which can cover more properties
+of star clusters. They also provided a fast model solver written as a
+Python code, LIMEPY, for this family of lowered isothermal models.
+© 20XX The Authors
+arXiv:2301.04868v1  [astro-ph.GA]  12 Jan 2023
+
+2
+Cheng and Jiang
+Thus, these models proposed by Gieles & Zocchi (2015) are called
+LIMEPY models.
+As presented by Zocchi et al. (2016), LIMEPY models could capture
+the main properties of the globular clusters. Moreover, Zocchi et al.
+(2017) applied LIMEPY models in the study of NGC 5139 and found
+that part of the observed large central velocity dispersion could be
+produced by anisotropic models. Thus, their results could provide
+some constraints on the previously proposed central intermediate-
+mass black hole in NGC 5139 (Noyola et al. 2010). This globular
+cluster, also named 𝜔 Centauri, is the most complex one which has
+many sub-populations (Sanna et al. 2020) and was heavily investi-
+gated with many controversial results. On the other hand, the central
+kinematics of NGC 6093 was studied by employing new integral-
+field spectrograph data, and the existence of an intermediate-mass
+black hole was supported (Göttgens et al. 2021). In addition, NGC
+6388 is also a candidate residence of the intermediate-mass black
+hole (Lützgendorf et al. 2011).
+Moreover, with Gaia data, Vasiliev & Baumgardt (2021) per-
+formed a comprehensive study on the kinematic properties of many
+Galactic globular clusters. The proper motions were measured and
+the corresponding proper-motion dispersion profiles of 100 clusters
+were obtained. Combining with HST and other literature data, Baum-
+gardt & Vasiliev (2021) also accurately derived the distances to these
+Galactic globular clusters.
+Therefore, motivated by the development of LIMEPY models, the
+controversial results of the central kinematics and intermediate-mass
+black holes, and the availability of new data derived from the Gaia
+mission, herein, we investigated the properties of 18 globular clusters
+with the LIMEPY models. Including data from recent observations such
+as the MUSE survey (Kamann et al. 2018) and Gaia mission (Vasiliev
+& Baumgardt 2021), the physical parameters of these clusters were
+obtained through the data-model fitting. Our results could lead to
+updated and accurate descriptions of the dynamical states of these
+clusters for the cases in which the data could be well fitted by the
+LIMEPY models which can be isotropic or anisotropic. Our results
+might also imply the possible existence of intermediate-mass black
+holes for some globular clusters.
+For the rest of this paper, in Section 2, we introduce the model’s
+distribution function and essential properties. The observational data
+are described in Section 3, and the parameter determination method
+is shown in Section 4. The results and discussions are presented in
+Section 5. In Section 6, some conclusions are made.
+2 THE MODEL
+The LIMEPY models were employed as the standard model in this
+study. As presented (Gieles & Zocchi 2015), there are single-mass
+and multi-mass cases in LIMEPY models. Considering the single-mass
+models, the distribution functions have the following form:
+𝑓 (𝐸, 𝐽) = 𝐴 exp
+� −𝐽2
+2𝑟2a 𝑠2
+�
+𝐸𝛾
+�
+𝑔, 𝜙(𝑟t) − 𝐸
+𝑠2
+�
+,
+(1)
+for 𝐸 ≤ 𝜙(𝑟t) and 𝑓 (𝐸, 𝐽) = 0 for 𝐸 > 𝜙(𝑟t). The function 𝐸𝛾(𝑔, 𝑥)
+represents 𝑒𝑥 for 𝑔 = 0 and 𝑒𝑥𝛾(𝑔, 𝑥)/Γ(𝑔) for 𝑔 > 0, where 𝛾(𝑔, 𝑥)
+is the lower incomplete gamma function and Γ(𝑔) stands for the
+gamma function. This distribution function depends on the specific
+energy 𝐸 and the specific angular momentum 𝐽. The function 𝜙 is the
+gravitational potential and 𝑟t is the truncation radius. The parameter
+𝑔 is called the truncation parameter, and it regulates the energy
+truncation of the model. The parameter 𝑟a is the anisotropic radius,
+and it determines how anisotropic a system is. When 𝑟a grows, the
+model is less anisotropic, and 𝑟a → ∞ corresponds to an isotropic
+model. The constants 𝐴 and 𝑠 are used to set the physical scale of the
+model. The density can be obtained by integrating the distribution
+function 𝑓 (𝐸, 𝐽) over the velocity space:
+𝜌 =
+∫
+𝑓 (𝐸, 𝐽) d3𝑣.
+(2)
+Since 𝐸 = 𝑣2/2 + 𝜙(𝑟) and the distribution function is zero for 𝐸 >
+𝜙(𝑟t), it can be just integrated from 0 to 𝑣max = [2𝜙(𝑟t) − 2𝜙(𝑟)]1/2
+at each 𝑟. This 𝑣max becomes zero when 𝑟 = 𝑟t and the density
+vanishes for 𝑟 ≥ 𝑟t. Hence, the truncation radius 𝑟t represents the
+distance where the density comes to zero.
+The gravitational potential 𝜙 is subjected to the Poisson equation.
+For spherical systems such as globular clusters, the equation results
+in the following form:
+d2𝜙
+d𝑟2 + 2
+𝑟
+d𝜙
+d𝑟 = 4𝜋𝐺𝜌,
+(3)
+where 𝑟 is the radial coordinate and 𝐺 is the gravitational constant.
+The relevant quantities were first turned into dimensionless ones for
+solving the Poisson equation. The dimensionless potential is defined
+as ˆ𝜙 = [𝜙(𝑟t) − 𝜙]/𝑠2. The dimensionless density and radius are
+ˆ𝜌 = 𝜌/𝜌0 and ˆ𝑟 = 𝑟/𝑟0, where 𝜌0 and 𝑟0 satisfy 4𝜋𝐺𝑟2
+0𝜌0/𝑠2 = 9.
+Then, the Poisson equation becomes
+d2 ˆ𝜙
+dˆ𝑟2 + 2
+ˆ𝑟
+d ˆ𝜙
+dˆ𝑟 = −9 ˆ𝜌.
+(4)
+The equation is solved with the boundary conditions that, at ˆ𝑟 = 0,
+d ˆ𝜙/dˆ𝑟 = 0 and ˆ𝜙 = 𝑊0, where 𝑊0 is a constant that specifies a
+particular solution. Hence,𝑊0 is also a parameter of the LIMEPY model,
+called the concentration parameter. It characterizes the concentration
+of the model.
+As previously mentioned, LIMEPY models provide an extended fam-
+ily of isothermal models. Those famous models are included as sub-
+families. For example, the Woolley model (Woolley 1954) can be
+produced by setting 𝑔 = 0, 𝑟a → ∞. When 𝑔 = 1 and 𝑟a → ∞,
+the King model (King 1966) is obtained. The Wilson model (Wilson
+1975), which is more extended, corresponds to 𝑔 = 2 and 𝑟a → ∞.
+Models with 𝑊0 → ∞ or 𝑔 → ∞ become the isothermal spheres. In
+addition, the polytrope can be represented as 𝑊0 → 0. It includes the
+Plummer model (Plummer 1911) which corresponds to the model
+with 𝑔 = 3.5. It has a finite mass but infinite extents. In general, the
+model with appropriate 𝑊0 and 𝑟a can be finite in extent if 𝑔 < 3.5
+and conversely infinite in extent with 𝑔 ≥ 3.5. In addition, Gieles &
+Zocchi (2015) also showed that one kind of finite model is unsuitable
+for star clusters. These systems have an upturn in the density far from
+the center, so there is a large amount of mass in the halo. The ratio
+of the virial radius and half-mass radius 𝑟v/𝑟h is a crucial parameter
+for these models. They suggested that the models with 𝑟v/𝑟h ≥ 0.64
+can adequately describe star clusters.
+The LIMEPY models describe spherical systems with different con-
+centrations, truncation, and radial anisotropy. In general, the model
+is isotropic near the center but could be anisotropic in the middle
+part of the system. The energy truncation limits the contribution of
+anisotropy to radial orbits with 𝐸 ≈ 𝜙(𝑟t) and thus suppresses the
+degree of radial anisotropy near the edge. The corresponding physi-
+cal picture is that a cluster under the interaction of an external tidal
+field has a preferential mass loss on stars with radial orbits. This re-
+duces the amount of anisotropy in the outer region (Oh & Lin 1992;
+Takahashi et al. 1997). Simulations of star clusters in the tidal field
+confirmed this isotropic behavior near the edge (Tiongco et al. 2016).
+Thus, the energy truncation acts as a role of the tidal field. In fact,
+MNRAS 000, 1–15 (20XX)
+
+Dynamical Properties of Globular Clusters
+3
+the tidal field can also make the outer region profiles tangentially
+anisotropic (Baumgardt & Makino 2003).
+In addition to the anisotropic radius 𝑟a, there is a convenient
+anisotropic parameter 𝜅 ≡ 2𝐾r/𝐾t, where 𝐾r is the total radial ki-
+netic energy and 𝐾t is the total tangential kinetic energy. If 𝜅 > 1,
+the system is radially anisotropic, and if 𝜅 < 1, the system is tangen-
+tially anisotropic. When 𝜅 = 1, it is an isotropic system. Therefore, 𝜅
+represents a simple and global measure of the anisotropy. We mainly
+used 𝜅 to determine the amount of the anisotropy of clusters.
+In Zocchi et al. (2016), the comparisons with N-body simulations
+illustrated the variation of model parameters of a cluster during the
+evolution. The cluster started with the Plummer model and the sim-
+ulation snapshots at different time were fitted with LIMEPY models.
+The concentration parameter tended to increase with time, which
+was also suggested previously by King (1966). The truncation pa-
+rameter 𝑔 decreased roughly from 2.5 to 0.5 during the evolution. It
+corresponded to an increased truncation by the tidal field as a cluster
+gradually filled the Roche volume. Thus, a cluster tends to become
+more concentrated and truncated with time. In addition, the degree
+of radial anisotropy increased due to radial diffusion but decreased
+later during the core collapse.
+3 THE OBSERVATIONAL DATA
+One of our primary goals is to provide updated results with a complete
+inclusion of all available observational data for globular clusters. The
+observational data of 𝑉-band surface brightness 𝜇 were taken from
+Trager et al. (1995), which provided a catalog of surface brightness
+profiles for over a hundred Galactic globular clusters. Some proce-
+dures were needed before the data were ready for the fitting. There
+was a correction related to extinction. The method is based on the
+global mean curve discussed in Fitzpatrick (1999), which uses the
+mean value for the ratio of the extinction ��𝑉 and the reddening
+𝐸(𝐵 − 𝑉) so that 𝐴𝑉 = 3.1𝐸(𝐵 − 𝑉). We took the reddening in
+the catalog of Harris (1996) (2010 version) and then computed the
+corrected surface brightness by 𝜇𝑖 = 𝜇𝑖,0 − 𝐴𝑉 , where 𝜇𝑖,0 denotes
+the data before the correction. The data with 𝑤𝑖 < 0.15 were not
+adopted according to McLaughlin & van der Marel (2005), where
+𝑤𝑖 is the weight of each data given in Trager et al. (1995).
+Because the data number was large, which might make the surface
+brightness dominate the fitting, we sliced the radial range with equal
+logarithmic width and averaged the surface brightness and the weight
+in each bin. The bin number was 55 which equaled the largest data
+number of the velocity dispersion. To compute the uncertainty for
+each data, we followed the method in McLaughlin & van der Marel
+(2005). The uncertainty of the data was obtained by 𝜖𝜇,𝑖 = 𝜖𝜇,b/𝑤𝑖,
+where 𝜖𝜇,b is the base error bar for each cluster.
+For line-of-sight velocity dispersion, we used the profiles derived
+from the collected literature (Baumgardt 2017), the data from un-
+published spectra of stars in the ESO and Keck Science archives
+(Baumgardt & Hilker 2018), and the dispersion from the integral-
+field-unit data from the WAGGS project (Dalgleish et al. 2020). The
+above data are expressed by open circles in Fig. 3. The data from the
+MUSE survey (Kamann et al. 2018) were also used and denoted by
+solid triangles. Some additional data were supplemented and marked
+as crosses, such as those from McLaughlin et al. (2006) for NGC 104
+and Larson & Seth (2015, private communication) for NGC 1851
+and NGC 2808. (The data of McLaughlin et al. (2006) and Larson &
+Seth (2015, private communication) were collected from the compi-
+lation in Watkins et al. (2015b) and others were collected from the
+compilation in the updated web catalog (third version) of Baumgardt
+& Hilker (2018).)
+For proper-motion velocity dispersion, we mainly took the data
+from the Hubble Space Telescope from Watkins et al. (2015a) and the
+Gaia data from Vasiliev & Baumgardt (2021). Open circles expressed
+the former, and solid triangles expressed the latter in Fig. 4. Some
+additional data were supplemented and denoted by crosses, which
+include Häberle et al. (2021) for NGC 6441, McLaughlin et al. (2006)
+for NGC 104, McNamara et al. (2003) for NGC 7078, McNamara
+et al. (2012) for NGC 6266, and Zloczewski et al. (2012) for NGC
+6656 and NGC 6752. (The data of Vasiliev & Baumgardt (2021) and
+Häberle et al. (2021) were collected from the updated web catalog of
+Baumgardt & Hilker (2018), and the data of McLaughlin et al. (2006),
+McNamara et al. (2003), McNamara et al. (2012), and Zloczewski
+et al. (2012) were collected from Watkins et al. (2015b).)
+Some proper motion data were downloaded in units of km/s,
+which depends on the cluster distance written in the literature.
+These data were transformed into mas/yr as the observational val-
+ues for our work here. The transformation is 𝑣 = 𝑣0/𝐷𝐶, where 𝑣
+and 𝑣0 are the velocity in mas/yr and km/s, 𝐷 is the distance and
+𝐶 = 4.74047 km yr kpc−1 mas−1 s−1 which is a factor for the unit
+conversion (van Leeuwen 2009; Watkins et al. 2015b). The values
+of cluster distances were taken from the corresponding literature. By
+taking the root mean square of the upper and lower error bars from
+the literature, we obtained a symmetric uncertainty for each data for
+our work. Finally, to focus on the systems with enough observational
+information, we studied 18 clusters with more than five data points
+in each type of the above observational profiles.
+4 THE DETERMINATION OF PHYSICAL PARAMETERS
+It was shown in Zocchi et al. (2017) that models with different
+amounts of anisotropy could give the same surface brightness but
+different kinematic profiles. Thus, using the surface brightness data
+alone can lead to some degeneracy. Therefore, here we included
+the surface brightness, the light-of-sight velocity dispersion, and the
+proper-motion velocity dispersion data to obtain complete pictures of
+the physical structures and kinematic properties of globular clusters
+by determining related physical parameters through the data-model
+fitting.
+Following the method in Zocchi et al. (2017), we employed the
+one-step fitting procedure with the single-mass LIMEPY models in this
+paper. With all three considered types of observational data, a single
+step of the fitting was performed to determine all cluster parameters.
+The fitting was done through the minimization of the 𝜒2 function:
+𝜒2 = 𝜒2
+sb + 𝜒2
+los + 𝜒2
+pm,
+(5)
+where 𝜒2
+sb, 𝜒2
+los, 𝜒2pm are the contributions from surface brightness,
+line-of-sight velocity dispersion, and proper-motion velocity disper-
+sion, respectively. They are defined by
+𝜒2
+sb =
+𝑛sb
+∑︁
+𝑖=1
+[𝜇𝑖 − ¯𝜇(𝑟𝑖)]2
+𝜖2
+𝜇,𝑖
+,
+(6)
+𝜒2
+los =
+𝑛los
+∑︁
+𝑖=1
+[𝜎los,𝑖 − ¯𝜎los(𝑟����)]2
+𝜖2
+los,𝑖
+,
+(7)
+and
+𝜒2
+pm =
+𝑛pm
+∑︁
+𝑖=1
+[𝜎pm,𝑖 − ¯𝜎pm(𝑟𝑖)]2
+𝜖2
+pm,𝑖
+,
+(8)
+MNRAS 000, 1–15 (20XX)
+
+4
+Cheng and Jiang
+where 𝜇𝑖 is the 𝑖-th observational data of a surface brightness profile,
+¯𝜇(𝑟𝑖) is the theoretical surface brightness at that radial coordinate
+𝑟𝑖, and 𝜖𝜇,𝑖 is the error bar of the data 𝜇𝑖. Similarly, 𝜎los,𝑖, ¯𝜎los(𝑟𝑖),
+𝜖los,𝑖 are the corresponding quantities for line-of-sight velocity dis-
+persion, and 𝜎pm,𝑖, ¯𝜎pm(𝑟𝑖), 𝜖pm,𝑖 are the observational data, the-
+oretical value, and error bar for proper-motion velocity dispersion,
+respectively. The numbers of observational data are 𝑛sb, 𝑛los, 𝑛pm,
+respectively, for the surface brightness, line-of-sight velocity disper-
+sion, and proper-motion velocity dispersion, individually.
+The LIMEPY code was employed to obtain the above theoretical pro-
+files. This code needed five input parameters, including the concen-
+tration parameter 𝑊0, the truncation parameter 𝑔, the dimensionless
+anisotropy radius ˆ𝑟a, the cluster mass 𝑀, and the half-mass radius
+𝑟h. The LIMEPY code generated several profiles, such as the surface
+mass density Σ(𝑟𝑖), line-of-sight mean-square velocity 𝑢2
+L(𝑟𝑖), radial
+and tangential mean-square velocity on the projected plane 𝑢2
+R(𝑟𝑖)
+and 𝑢2
+T(𝑟𝑖). Thus, the value of ¯𝜎los(𝑟𝑖) is simply the square root of
+𝑢2
+L(𝑟𝑖), and ¯𝜎pm(𝑟𝑖) is the square root of [𝑢2
+R(𝑟𝑖) + 𝑢2
+T(𝑟𝑖)]/2.
+To complete the data-model fitting, two more parameters were
+needed. The cluster distance 𝐷 is a parameter that converts the radial
+coordinate of the theoretical profile from pc to arcsec and the ob-
+servational proper-motion velocity dispersion from mas/yr to km/s.
+The V-band mass-to-light ratio Υ is a parameter for producing the
+luminosity density Σ(𝑟𝑖)/Υ, and the surface brightness ¯𝜇(𝑟𝑖) can be
+obtained by
+¯𝜇(𝑟𝑖) = 𝑀V,⊙ − 5(1 + log 𝑐) − 2.5 log(Σ(𝑟𝑖)/Υ),
+(9)
+where 𝑀V,⊙ = 4.83 mag is the V-band absolute magnitude of the
+Sun and 𝑐 = 𝜋/648000 rad/arcsec is a factor for the unit conversion
+(Watkins et al. 2015b).
+Through the minimization of the 𝜒2 function, the best-fit values
+of seven parameters 𝑊0, 𝑔, ˆ𝑟a, 𝑀, 𝑟h, 𝐷, Υ can be obtained. We
+used the code EMCEE (Foreman-Mackey et al. 2013) to perform the 𝜒2
+minimization. It is an affine-invariant ensemble sampler that employs
+the Markov chain Monte Carlo (MCMC) process (Goodman & Weare
+2010). One has to decide the initial distribution and the parameters
+range for the EMCEE samples. For the concentration parameter 𝑊0, the
+range was set to 1 < 𝑊0 < 15. It covers a similar range in Table II
+of King (1966) and represents various degrees of concentration of
+star clusters. Figure 4 in Gieles & Zocchi (2015) showed the relevant
+models for star clusters and the corresponding parameters; hence
+we set 0 < 𝑔 < 3 for the truncation parameter accordingly. The
+dimensionless anisotropy radius ˆ𝑟a needs a wide range to include the
+isotropic models. Therefore, we set a large range for log ˆ𝑟a as −1 <
+log ˆ𝑟a < 20. For the remained parameters, we checked the literature
+values and considered wider ranges to include more possibilities. The
+ranges of these parameters were set to be 0.1 < 𝑀 < 50 (105 M⊙),
+0.1 < 𝑟h < 15 (pc), 0.1 < 𝐷 < 35 (kpc), and 0.1 < Υ < 5 (Υ⊙).
+Finally, the initial distributions of all parameters are set to be uniform.
+5 RESULTS AND DISCUSSION
+The best-fit results are displayed in Table 1. The first column shows
+the names of the clusters. Seven fitting parameters are listed from the
+second to eighth columns. The second column presents the concen-
+tration parameter 𝑊0 and the values range roughly from 3 to 9 for
+these clusters. The third and the fourth columns show the truncation
+parameter 𝑔 and the logarithm of the dimensionless anisotropy radius
+log ˆ𝑟a. The fifth and sixth columns list the cluster mass 𝑀 and the
+half-mass radius 𝑟h. These clusters have 𝑟h ≲ 10 pc. Among them,
+NGC 5139 has the largest mass and radius. The heliocentric distance
+𝐷 is shown in the seventh column. Most clusters have 𝐷 ≲ 12 kpc
+except for NGC 6715, which is roughly two times distant. The eighth
+column reveals the V-band mass-to-light ratio Υ. To understand the
+anisotropy conveniently, the quantity 𝜅 is shown in the ninth column.
+NGC 5139 and NGC 7078 have 𝜅 > 1, indicating the anisotropic
+behavior. The quantity in the last column is the reduced chi-square
+𝜒2r defined by
+𝜒2
+r =
+𝜒2
+𝑛 − 𝑛p
+,
+(10)
+where 𝑛 is the total number of data and 𝑛p is the number of parame-
+ters.
+5.1 Comparison with Previous Work
+To compare our results with the previous work, we used the mea-
+surable physical properties estimated in the published literature, as
+listed in Table 2. We first considered the comparison of the cluster’s
+total mass. In general, the masses estimated by Baumgardt & Hilker
+(2018) are larger than those estimated by Watkins et al. (2015b), and
+our results are usually between their values. Almost all of our results
+are very close to the masses estimated in Watkins et al. (2015b).
+We also compared our half-mass radius with the one in the catalog
+of Baumgardt & Hilker (2018). Generally, our results are smaller,
+consistent with the results of total mass, since our masses are lower
+than those in Baumgardt & Hilker (2018). Therefore, the radii of the
+clusters tend to be smaller to fit the line-of-sight velocity dispersion.
+Some differences between the radius might come from the mass
+spectrum. The radial distributions of different species may introduce
+additional variation between the half-mass radii. Nevertheless, the
+mass-to-light ratios obtained in our work are consistent with the
+values in Baumgardt et al. (2020) and Watkins et al. (2015b).
+For distance comparison, we compared with the values in Watkins
+et al. (2015b), Baumgardt & Vasiliev (2021), and Harris (1996).
+Watkins et al. (2015b) derived the distance by comparing their proper
+motion velocity dispersion with the line-of-sight velocity dispersion
+from the literature. Baumgardt & Vasiliev (2021) calculated the mean
+distance from several methods, such as the Gaia EDR3 parallaxes,
+the method by fitting nearby subdwarfs to globular cluster main
+sequences, the color-magnitude diagram fitting, and the distances
+from the period-luminosity relation of RR Lyrae stars. The distances
+in Harris (1996) are a compilation of the distance measurements
+from the literature.
+Fig. 1 shows the ratio of our distance 𝐷 and the one published
+in literature 𝐷lit, i.e., 𝐷/𝐷lit, for each considered cluster. For each
+panel, the compared literature is labeled at the top-right corner. Each
+point represents a particular cluster studied in the compared literature
+and this work. The dashed line represents the unity, and the solid line
+is the average value of the ratio. Two numbers are shown in the
+bottom-right of the panels, the left number is the averaged 𝐷/𝐷lit,
+and the right one is the averaged |𝐷/𝐷lit−1|. These numbers indicate
+that our results are closer to Harris (1996) and Watkins et al. (2015b),
+and slightly lower than Baumgardt & Vasiliev (2021). In general, our
+results agree with the values from these studies.
+5.2 The Profiles
+Fig. 2 to 4 show the profiles of surface brightness, line-of-sight veloc-
+ity dispersion, and proper-motion velocity dispersion. The horizontal
+axis is the distance from the cluster’s center in arcsec. The vertical
+axis gives the surface brightness in mag/arcsec2 in Fig. 2, and ve-
+locity dispersion in km/s from Fig. 3 to 4. It can be seen that LIMEPY
+MNRAS 000, 1–15 (20XX)
+
+Dynamical Properties of Globular Clusters
+5
+Table 1. The properties of the clusters. The first column lists the names of the clusters. Columns two to eight show the fitting parameters, which are concentration
+parameter 𝑊0, truncation parameter 𝑔, the logarithm of the dimensionless anisotropy radius log ˆ𝑟a, cluster mass 𝑀, half-mass radius 𝑟h, distance 𝐷, and V-band
+mass-to-light ratio Υ. Column nine presents the quantity 𝜅 which measures the amount of anisotropy, and the final column gives 𝜒2r .
+cluster
+𝑊0
+𝑔
+log ˆ𝑟a
+𝑀
+𝑟h
+𝐷
+Υ
+𝜅
+𝜒2r
+(105 M⊙)
+(pc)
+(kpc)
+(Υ⊙)
+NGC 104
+8.36 ± 0.06
+1.31 ± 0.03
+11.13+6.02
+−6.10
+6.87 ± 0.15
+5.21 ± 0.12
+4.33 ± 0.03
+1.53 ± 0.03
+1.00
+2.10
+NGC 288
+4.46+0.47
+−0.82
+1.55+0.52
+−0.38
+10.40+6.44
+−6.41
+1.02+0.11
+−0.10
+8.26+0.33
+−0.32
+9.80+0.37
+−0.36
+2.32 ± 0.12
+1.00
+1.18
+NGC 362
+7.20 ± 0.10
+1.67 ± 0.06
+11.24+5.87
+−6.48
+2.09+0.11
+−0.10
+2.36+0.08
+−0.07
+8.71+0.16
+−0.15
+1.22 ± 0.03
+1.00
+4.74
+NGC 1851
+7.33+0.19
+−0.20
+2.04 ± 0.09
+10.77+6.26
+−6.12
+2.28+0.10
+−0.09
+2.15+0.15
+−0.13
+10.82+0.15
+−0.14
+1.73+0.09
+−0.08
+1.00
+1.61
+NGC 2808
+6.27+0.17
+−0.16
+2.02+0.10
+−0.07
+11.81+5.01
+−6.78
+6.57+0.25
+−0.19
+2.69+0.08
+−0.06
+9.63+0.12
+−0.10
+1.56+0.05
+−0.04
+1.00
+1.63
+NGC 3201
+5.89+0.31
+−0.34
+2.45 ± 0.09
+11.00+6.19
+−6.29
+1.21+0.08
+−0.07
+5.21+0.42
+−0.33
+4.38+0.10
+−0.09
+2.33+0.12
+−0.11
+1.00
+2.74
+NGC 5139
+4.02+0.48
+−0.65
+1.94+0.27
+−0.26
+0.41+0.08
+−0.10
+32.82+0.65
+−0.67
+8.82+0.19
+−0.17
+5.32 ± 0.03
+2.38 ± 0.09
+1.15
+3.86
+NGC 5904
+7.03+0.09
+−0.10
+1.56+0.05
+−0.04
+10.39+6.55
+−6.07
+3.03+0.16
+−0.15
+4.54+0.12
+−0.11
+7.24 ± 0.13
+1.39+0.04
+−0.03
+1.00
+1.85
+NGC 6121
+7.52+0.16
+−0.13
+0.46+0.31
+−0.21
+9.80+6.94
+−5.59
+0.81+0.05
+−0.04
+3.20+0.17
+−0.13
+1.85+0.04
+−0.03
+2.11+0.10
+−0.08
+1.00
+1.12
+NGC 6218
+5.77+0.29
+−0.35
+1.51+0.25
+−0.22
+10.69+6.36
+−6.53
+0.75 ± 0.06
+3.02+0.14
+−0.13
+4.59+0.15
+−0.14
+1.78+0.09
+−0.08
+1.00
+1.19
+NGC 6266
+7.84+0.08
+−0.09
+0.62+0.11
+−0.10
+10.90 ± 6.23
+5.98+0.25
+−0.24
+2.55 ± 0.08
+6.33+0.09
+−0.08
+1.85 ± 0.05
+1.00
+1.57
+NGC 6388
+7.09+0.10
+−0.11
+1.68+0.09
+−0.08
+10.86+6.16
+−6.13
+7.79 ± 0.20
+2.07 ± 0.05
+10.35 ± 0.10
+1.68 ± 0.03
+1.00
+2.94
+NGC 6397
+9.17 ± 0.17
+0.87 ± 0.08
+10.96+6.13
+−6.11
+0.79+0.04
+−0.03
+3.73 ± 0.19
+2.40 ± 0.04
+2.47 ± 0.12
+1.00
+1.97
+NGC 6441
+7.75 ± 0.06
+1.24+0.10
+−0.09
+10.74+6.31
+−6.27
+10.54+0.28
+−0.27
+2.90+0.07
+−0.06
+11.91 ± 0.11
+1.82 ± 0.03
+1.00
+3.35
+NGC 6656
+6.48+0.23
+−0.26
+1.87+0.34
+−0.39
+10.73+6.33
+−6.50
+3.57+0.22
+−0.20
+4.40+0.29
+−0.20
+3.10 ± 0.05
+1.85 ± 0.07
+1.00
+1.04
+NGC 6715
+6.99 ± 0.07
+2.21+0.02
+−0.03
+11.26+6.00
+−6.28
+17.79+1.12
+−1.06
+5.28+0.29
+−0.25
+25.08 ± 0.53
+2.07 ± 0.06
+1.00
+2.83
+NGC 6752
+8.35+0.12
+−0.13
+1.38 ± 0.06
+10.95+6.16
+−6.21
+1.92 ± 0.09
+3.45 ± 0.16
+4.13 ± 0.06
+2.24 ± 0.08
+1.00
+1.20
+NGC 7078
+8.30+0.12
+−0.13
+0.86+0.15
+−0.13
+1.16+0.07
+−0.06
+5.08 ± 0.17
+4.05+0.18
+−0.17
+10.40 ± 0.12
+1.53 ± 0.05
+1.16
+1.46
+Table 2. The literature parameters of the clusters. The number in the parentheses represents the literature, (1) stands for Baumgardt & Hilker (2018), (2) refers
+to Watkins et al. (2015b), (3) corresponds to Baumgardt & Vasiliev (2021), (4) represents Harris (1996), and (5) is Baumgardt et al. (2020). The updated values
+for (1) and (5) are picked from the web catalog of Baumgardt & Hilker (2018).
+cluster
+𝑀
+𝑀
+𝑟h
+𝐷
+𝐷
+𝐷
+Υ
+Υ
+(105 M⊙)
+(105 M⊙)
+(pc)
+(kpc)
+(kpc)
+(kpc)
+(Υ⊙)
+(Υ⊙)
+(1)
+(2)
+(1)
+(2)
+(3)
+(4)
+(5)
+(2)
+NGC 104
+8.95 ± 0.06
+5.57+0.33
+−0.28
+6.30
+4.15 ± 0.08
+4.521 ± 0.031
+4.5
+1.96 ± 0.09
+1.40 ± 0.03
+NGC 288
+0.934 ± 0.026
+0.79+0.13
+−0.11
+8.37
+9.03+0.48
+−0.56
+8.988+0.089
+−0.088
+8.9
+2.16 ± 0.10
+2.20+0.13
+−0.10
+NGC 362
+2.84 ± 0.04
+...
+3.79
+...
+8.829 ± 0.096
+8.6
+1.44 ± 0.05
+...
+NGC 1851
+3.18 ± 0.04
+1.78+0.10
+−0.11
+2.90
+10.32+0.20
+−0.24
+11.951+0.134
+−0.133
+12.1
+1.66 ± 0.06
+1.51 ± 0.03
+NGC 2808
+8.64 ± 0.06
+5.91+0.22
+−0.25
+3.89
+9.45+0.13
+−0.15
+10.060+0.112
+−0.111
+9.6
+1.51 ± 0.06
+1.56 ± 0.02
+NGC 3201
+1.60 ± 0.03
+...
+6.78
+...
+4.737+0.043
+−0.042
+4.9
+2.16 ± 0.09
+...
+NGC 5139
+36.4 ± 0.4
+34.52+1.45
+−1.43
+10.36
+5.19+0.07
+−0.08
+5.426 ± 0.047
+5.2
+2.58 ± 0.10
+2.66 ± 0.04
+NGC 5904
+3.94 ± 0.06
+3.65 ± 0.75
+5.68
+7.79+0.47
+−0.61
+7.479 ± 0.060
+7.5
+1.81 ± 0.06
+1.43+0.09
+−0.10
+NGC 6121
+0.871 ± 0.011
+...
+3.69
+...
+1.851+0.015
+−0.016
+2.2
+1.59 ± 0.06
+...
+NGC 6218
+1.07 ± 0.03
+...
+4.05
+...
+5.109+0.049
+−0.048
+4.8
+1.92 ± 0.09
+...
+NGC 6266
+6.10 ± 0.04
+6.09+0.39
+−0.33
+2.43
+6.42 ± 0.14
+6.412+0.105
+−0.104
+6.8
+1.99 ± 0.11
+2.22 ± 0.04
+NGC 6388
+12.5 ± 0.1
+8.27+0.89
+−0.95
+4.34
+10.90+0.40
+−0.45
+11.171+0.162
+−0.161
+9.9
+2.19 ± 0.06
+1.68+0.06
+−0.07
+NGC 6397
+0.966 ± 0.013
+0.70+0.09
+−0.08
+3.90
+2.39+0.13
+−0.11
+2.482 ± 0.019
+2.3
+1.66 ± 0.07
+2.23+0.10
+−0.09
+NGC 6441
+13.2 ± 0.1
+...
+3.47
+...
+12.728+0.163
+−0.162
+11.6
+1.77 ± 0.13
+...
+NGC 6656
+4.76 ± 0.05
+2.49+0.44
+−0.37
+5.29
+2.84 ± 0.16
+3.303 ± 0.037
+3.2
+2.05 ± 0.08
+1.88+0.12
+−0.10
+NGC 6715
+17.8 ± 0.3
+11.83+0.62
+−0.53
+5.20
+22.57+0.44
+−0.39
+26.283+0.328
+−0.325
+26.5
+2.10 ± 0.12
+1.94 ± 0.03
+NGC 6752
+2.76 ± 0.04
+1.82 ± 0.12
+5.27
+4.02+0.10
+−0.08
+4.125 ± 0.041
+4.0
+2.34 ± 0.11
+2.14+0.05
+−0.06
+NGC 7078
+6.33 ± 0.07
+4.95 ± 0.19
+4.30
+10.36+0.15
+−0.16
+10.709+0.096
+−0.095
+10.4
+1.58 ± 0.10
+1.49 ± 0.02
+MNRAS 000, 1–15 (20XX)
+
+6
+Cheng and Jiang
+5
+10
+15
+20
+25
+D [kpc]
+0.8
+0.9
+1.0
+1.1
+1.2
+D / Dlit
+Watkins et al. (2015b)
+1.02, 0.05
+5
+10
+15
+20
+25
+D [kpc]
+0.8
+0.9
+1.0
+1.1
+1.2
+Baumgardt & Vasiliev (2021)
+0.96, 0.05
+5
+10
+15
+20
+25
+D [kpc]
+0.8
+0.9
+1.0
+1.1
+1.2
+Harris (1996)
+0.98, 0.05
+Figure 1. The comparison of the cluster distance with the values mentioned in earlier studies. The horizontal axis is the cluster distance obtained in this work,
+and the vertical axis shows the ratio of our value to the distance given in the literature. The dashed line and the solid line represent the unity and the average. Each
+panel is for comparison with the particular publication, as labeled at the top-right corner. At the bottom-right corner, the left number is the averaged 𝐷/𝐷lit,
+and the right number is the averaged |𝐷/𝐷lit − 1|.
+models can produce similar profiles as observational ones. To ex-
+amine these clusters more quantitatively, we classified the results by
+𝜒2r . Many clusters were found to have 𝜒2r < 2. These clusters have
+suitable fittings for all three profiles, as shown in the figures.
+NGC 362 has the largest 𝜒2r , and the model profiles agree with the
+observations in surface brightness and line-of-sight velocity disper-
+sion. However, the central part of the modeled proper-motion velocity
+dispersion is slightly larger than the observations. Data with a small
+error bar in the outer part located much higher than the profile, mak-
+ing the fitting worse. NGC 6441 also has a larger 𝜒2r . The model
+agrees well with the surface brightness and the outer part of the
+proper motion velocity dispersion but predicts larger values for the
+inner part. The model can also fit the rough trend of the line-of-sight
+velocity dispersion, but some points lie below the model.
+For NGC 3201, the model has smaller line-of-sight velocity dis-
+persion for radius above 100 arcsec. There are also some under
+estimations for the proper motions in the outermost region, where
+the observational profile tends to level off rather than continue to
+decrease. Some scenarios were proposed to explain the higher ve-
+locity dispersion in the outer part, such as the orbital history with
+accretion and the embedding by a dark matter halo (Bianchini et al.
+2019). It was also found that binary stars could contribute to part of
+the effect (Wan et al. 2021). For NGC 6715, the model agrees with
+the observations, except for the outermost region of the line-of-sight
+velocity dispersion, where the observational profile grows. This rise
+is probably caused by the stars in the nucleus of the Sagittarius dwarf
+galaxy, where NGC 6715 inhabits (Bellazzini et al. 2008).
+NGC 5139 has large central velocity dispersions, which the model
+cannot explain well. For NGC 6388, the model has a steeper proper-
+motion velocity dispersion profile than the observational one. Further
+discussions of these two clusters will be made in the following sub-
+section.
+5.3 Possible Intermediate-Mass Black Hole ?
+Stellar black holes exist in astrophysical systems such as X-ray bina-
+ries (Mikolajewska et al. 2022). In addition, supermassive black holes
+are also confirmed to exist at the centers of our Milky Way (GRAV-
+ITY Collaboration et al. 2019) and other galaxies (Blandford et al.
+2019). Whether there are any intermediate-mass black holes in the
+universe is one of the most important questions in astronomy. Globu-
+lar clusters are considered good candidates to host intermediate-mass
+black holes and thus attract much attention. Among 18 globular clus-
+ters in the present work, NGC 5139 was discussed previously as a
+likely candidate.
+For our work here, the data-model fitting of NGC 5139 led to
+two groups of model parameters, as shown in Fig. 5. These groups
+have very different concentration parameters 𝑊0 and logarithm of
+the dimensionless anisotropy radius log ˆ𝑟a. One has smaller 𝑊0 and
+log ˆ𝑟a, and the other has larger values. Hence, we do further fittings
+with narrower ranges as 1 < 𝑊0 < 8, −1 < log ˆ𝑟a < 2, and 8 < 𝑊0 <
+15, 2 < log ˆ𝑟a < 20, separately. The results are shown in Table 3.
+We denote the one with lower 𝜒2r as Model A, the result previously
+listed in Table 1 and presented in Fig. 2 to 4. Model A has a low
+concentration. It also has a small dimensionless anisotropy radius
+with 𝜅 = 1.15, making it more anisotropic. In contrast, Model B is
+isotropic with a high concentration.
+The best-fit profiles are shown in Fig. 6. Model A fits the surface
+brightness well but predicts lower central velocity dispersion, espe-
+cially for the proper motion. On the other hand, Model B has good
+fittings on both velocity dispersion but a poor fitting on the surface
+brightness. The deviation in surface brightness leads to a larger 𝜒2r .
+Although the data and radial range of the observational kinematic
+profiles differs, the parameters from Model A agree with those in the
+best-fit model in Zocchi et al. (2017).
+These results obtained with two models show that it is difficult
+to perfectly and simultaneously fit all profiles of NGC 5139 with
+the current considered model. This could indicate the existence of
+central dark objects which can cause an increase in central velocities.
+These objects could be an intermediate-mass black hole (Noyola et al.
+2010; Baumgardt 2017) or a group of stellar-mass black holes at the
+cluster center (Baumgardt et al. 2019b). Both can also suppress the
+mass segregation of the stars (Gill et al. 2008; Peuten et al. 2016)
+and render the cluster to have a larger core (Baumgardt et al. 2005;
+Peuten et al. 2017). The main difference is that the intermediate-
+mass black hole could produce some stars faster than 60 km/s in the
+central 20 arcsec of NGC 5139, which was not confirmed in current
+observations (Baumgardt et al. 2019b).
+NGC 6388 is another candidate cluster that may host a central
+intermediate-mass black hole. The study of the integrated light spec-
+tra revealed a high central LOS velocity dispersion ∼25 km/s within 2
+arcsec (Lützgendorf et al. 2011). However, there was also a result that
+suggests a dispersion ∼15 km/s in the same region derived from stars’
+radial velocities (Lanzoni et al. 2013). Hence, the actual kinematic
+behavior of the cluster center is not clear. The data we used have
+the extension to nearly 5 arcsec with a velocity dispersion ∼20 km/s.
+Our results show that the surface brightness and line-of-sight velocity
+MNRAS 000, 1–15 (20XX)
+
+Dynamical Properties of Globular Clusters
+7
+100
+101
+102
+103
+15
+20
+25
+μ [mag/arcsec2]
+NGC 104
+100
+101
+102
+103
+20
+25
+30
+NGC 288
+100
+101
+102
+103
+15
+20
+25
+NGC 362
+100
+101
+102
+103
+15
+20
+25
+μ [mag/arcsec2]
+NGC 1851
+100
+101
+102
+103
+15
+20
+25
+NGC 2808
+100
+101
+102
+103
+20
+25
+NGC 3201
+101
+102
+103
+15
+20
+25
+μ [mag/arcsec2]
+NGC 5139
+101
+102
+103
+15
+20
+25
+30
+NGC 5904
+100
+101
+102
+103
+15
+20
+25
+NGC 6121
+100
+101
+102
+103
+15
+20
+25
+μ [mag/arcsec2]
+NGC 6218
+100
+101
+102
+103
+15
+20
+NGC 6266
+100
+101
+102
+15
+20
+25
+NGC 6388
+100
+101
+102
+103
+15
+20
+25
+μ [mag/arcsec2]
+NGC 6397
+100
+101
+102
+12.5
+15.0
+17.5
+20.0
+22.5
+NGC 6441
+100
+101
+102
+103
+15.0
+17.5
+20.0
+22.5
+25.0
+NGC 6656
+100
+101
+102
+103
+r [arcsec]
+15
+20
+25
+μ [mag/arcsec2]
+NGC 6715
+100
+101
+102
+103
+r [arcsec]
+15
+20
+25
+NGC 6752
+100
+101
+102
+103
+r [arcsec]
+15
+20
+25
+NGC 7078
+Figure 2. The surface brightness profiles of the clusters. The observations are shown as crosses, and the models are expressed by grey lines. For each panel, the
+name of the cluster is mentioned at the top-right corner.
+MNRAS 000, 1–15 (20XX)
+
+8
+Cheng and Jiang
+101
+102
+103
+0
+5
+10
+15
+σlos [km/s]
+NGC 104
+102
+103
+0
+1
+2
+3
+4
+NGC 288
+101
+102
+103
+0.0
+2.5
+5.0
+7.5
+10.0
+NGC 362
+101
+102
+103
+0
+5
+10
+σlos [km/s]
+NGC 1851
+101
+102
+0
+5
+10
+15
+NGC 2808
+101
+102
+103
+0
+2
+4
+NGC 3201
+101
+102
+103
+0
+10
+20
+30
+σlos [km/s]
+NGC 5139
+101
+102
+103
+0.0
+2.5
+5.0
+7.5
+10.0
+NGC 5904
+102
+103
+0
+2
+4
+6
+NGC 6121
+101
+102
+0
+2
+4
+σlos [km/s]
+NGC 6218
+101
+102
+0
+5
+10
+15
+20
+NGC 6266
+101
+102
+0
+10
+20
+NGC 6388
+101
+102
+103
+0
+2
+4
+6
+σlos [km/s]
+NGC 6397
+101
+102
+0
+10
+20
+NGC 6441
+101
+102
+103
+0
+5
+10
+NGC 6656
+101
+102
+103
+r [arcsec]
+0
+5
+10
+15
+20
+σlos [km/s]
+NGC 6715
+101
+102
+103
+r [arcsec]
+0.0
+2.5
+5.0
+7.5
+10.0
+NGC 6752
+101
+102
+103
+r [arcsec]
+0
+5
+10
+15
+NGC 7078
+Figure 3. The line-of-sight velocity dispersion profiles of the clusters. The open circles represent the data of Baumgardt (2017), Baumgardt & Hilker (2018),
+and Dalgleish et al. (2020). The data of Kamann et al. (2018) are shown in solid triangles. The crosses are used for additional data of some clusters mentioned
+in Section 3. The models are expressed by grey lines. For each panel, the name of the cluster is mentioned at the top-right corner.
+MNRAS 000, 1–15 (20XX)
+
+Dynamical Properties of Globular Clusters
+9
+101
+102
+103
+0
+5
+10
+15
+σpm [km/s]
+NGC 104
+101
+102
+0
+2
+4
+NGC 288
+101
+102
+0.0
+2.5
+5.0
+7.5
+10.0
+NGC 362
+101
+102
+103
+0.0
+2.5
+5.0
+7.5
+10.0
+σpm [km/s]
+NGC 1851
+101
+102
+103
+0
+5
+10
+15
+NGC 2808
+102
+103
+0
+1
+2
+3
+4
+NGC 3201
+101
+102
+103
+0
+10
+20
+30
+σpm [km/s]
+NGC 5139
+100
+101
+102
+0.0
+2.5
+5.0
+7.5
+10.0
+NGC 5904
+102
+103
+0
+2
+4
+NGC 6121
+102
+103
+0
+1
+2
+3
+4
+σpm [km/s]
+NGC 6218
+100
+101
+102
+0
+10
+20
+NGC 6266
+101
+102
+0
+5
+10
+15
+20
+NGC 6388
+101
+102
+103
+0
+2
+4
+6
+σpm [km/s]
+NGC 6397
+100
+101
+102
+0
+10
+20
+NGC 6441
+101
+102
+103
+0.0
+2.5
+5.0
+7.5
+10.0
+NGC 6656
+101
+102
+r [arcsec]
+0
+5
+10
+15
+20
+σpm [km/s]
+NGC 6715
+101
+102
+103
+r [arcsec]
+0.0
+2.5
+5.0
+7.5
+10.0
+NGC 6752
+100
+101
+102
+r [arcsec]
+0
+5
+10
+15
+20
+NGC 7078
+Figure 4. The proper-motion velocity dispersion profiles of the clusters. The open circles represent the data of Watkins et al. (2015a). The data of Vasiliev &
+Baumgardt (2021) is shown in solid triangles. The crosses are used for additional data of some clusters mentioned in Section 3. The models are expressed by
+grey lines. For each panel, the name of the cluster is mentioned at the top-right corner.
+MNRAS 000, 1–15 (20XX)
+
+10
+Cheng and Jiang
+dispersion can be fitted well without the central black hole. However,
+the model predicts a steeper proper-motion velocity-dispersion pro-
+file than the observations, being higher inside but lower outside. This
+behavior can also be seen in Figure 9 of Watkins et al. (2015b).
+NGC 7078 is also a candidate cluster that could host an
+intermediate-mass black hole. The increase in central velocity dis-
+persion found in Hubble Space Telescope was explained by an
+intermediate-mass black hole (Gerssen et al. 2002). However, the
+cluster can also be fitted with a group of dark stellar remnants (den
+Brok et al. 2014) or N-body simulations without intermediate-mass
+black holes (Baumgardt 2017). In our results, the cluster could be fit-
+ted well without central black holes, and some degree of anisotropy
+was observed, which can raise the central velocities. In addition,
+although there are raised velocity dispersions in observation, the un-
+certainties of the data are also large. Therefore, we obtain a better
+fitting than NGC 5139.
+5.4 The Anisotropy
+Two clusters, NGC 5139 and NGC 7078, possess small dimensionless
+anisotropy radius and reveal some degree of anisotropy. The former
+has 𝜅 = 1.15 and the latter has 𝜅 = 1.16. Other clusters have isotropic
+behavior with 𝜅 = 1.00 and a large anisotropy radius. One effect of
+radial anisotropy is that it can increase the central velocity dispersion.
+The rise in central velocity dispersions can be seen in Fig. 3 and 4. On
+the other hand, the amount of anisotropy estimated from our fittings
+could be underestimated, since the difference between tangential and
+radial proper motions will be averaged out in the combined proper
+motion velocity dispersion.
+The results are reasonable compared with some previous studies.
+For example, the parameters of NGC 5139 are similar to those es-
+timated in Zocchi et al. (2017) which the fittings were carried out
+with both radial and tangential proper motion velocity dispersions.
+The weak anisotropy in many clusters were also reported by Watkins
+et al. (2015a) and Watkins et al. (2015b), in which most of our
+samples were also studied. Watkins et al. (2015b) showed that their
+distance estimation had good agreement with Harris (1996) and con-
+cluded that the assumption of isotropy for their samples is reasonable.
+Watkins et al. (2015a) examined the ratio 𝜎T/𝜎R, which compared
+the tangential and radial components of the proper motion velocity
+dispersion at different radii. They found that the cluster centers are
+relatively isotropic, and the behavior of the increasing anisotropy
+with the radius was very moderate. From their figures, it can be seen
+that the decreasing of 𝜎T/𝜎R with a growing radius is more evident
+for NGC 5139 and NGC 7078.
+In recent years, Gaia has provided the proper motion data in the
+outer parts of globular clusters, and the behavior of 𝜎T/𝜎R reveals
+more evidence of anisotropy (Jindal et al. 2019; Vasiliev & Baum-
+gardt 2021). In both studies, NGC 5904 appears to be isotropic, and
+NGC 104, NGC 5139, and NGC 7078 show radial anisotropy. Some
+clusters are anisotropic in one study but are isotropic or uncertain
+in another; these include NGC 2808, NGC 6121, NGC 6397, NGC
+6656, and NGC 6752.
+In addition, the anisotropy profiles 𝜎T/𝜎R−1 from the observations
+and our models are plotted in Fig. 7. The observational data was
+mainly from a recent report on the globular-cluster survey through
+Hubble Space Telescope (Libralato et al. 2022). It includes 16 clusters
+of our samples. The remaining two clusters were supplemented with
+the data from Watkins et al. (2015a). The data from Gaia (Jindal
+et al. 2019) which contains half of our samples were also used. In
+Fig. 7, the data of the above-discussed literature are expressed by
+open circles, crosses, and solid triangles; the profiles are roughly
+isotropic or slightly radial anisotropic within 𝑟 ≲ 100 arcsec. The
+larger anisotropy appears mainly in the outer regions. The radial
+anisotropy of NGC 5139 tends to increase from near 100 arcsec and
+later decrease to isotropy in 𝑟 ≳ 1000 arcsec. Our model predicts the
+decrease in radial anisotropy at a larger radius. For NGC 7078, the
+model shows a similar and milder profile to the observational one.
+NGC 6121 shows isotropy inside but grows to tangential anisotropy at
+a larger radius. The cluster was also found to be tangential anisotropy
+in Vasiliev & Baumgardt (2021). It could imply a more substantial
+influence from the tidal field, which is consistent with our results that
+this cluster has a smaller truncation parameter than others.
+5.5 The Imprint of Galactic Tidal Field
+As mentioned earlier, the truncation parameter has the effect of mak-
+ing the extent of the system finite, and also drives the profile to be
+isotropic near the edge. These make the truncation parameter play
+a similar role as the external tidal field for the cluster. The exter-
+nal field generally becomes weaker for a larger distance from the
+Galactic center. Thus, clusters at larger distances from the Galactic
+center might be more extended and have larger values of truncation
+parameter 𝑔.
+In addition, Chernoff et al. (1986) found that the tidal field can
+increase the evolution rate of the cluster through relaxation and shock
+heating. Therefore, clusters closer to the Galactic center tend to evolve
+faster. They also suggested that inner regions of the Galaxy could be
+good places to look for the core-collapsed clusters. This agreed with
+Djorgovski & King (1986) who found that the mean and median
+distances of core-collapsed clusters from the Galactic center are
+smaller than 5 kpc.
+Moreover, the simulation in Zocchi et al. (2016) showed some
+related properties during the evolution of a globular cluster in an
+external tidal field. For example, the truncation parameter 𝑔 and the
+cluster mass 𝑀 decrease during the evolution. The concentration
+parameter 𝑊0 grows with time and decreases slightly after core col-
+lapse. The half-mass radius 𝑟h also increases with time and decreases
+as the cluster loses most of its mass.
+Motivated by the above results, here we examine possible correla-
+tions between any pairs among the concentration parameter 𝑊0, the
+truncation parameter 𝑔, the cluster mass 𝑀, the half-mass radius 𝑟h,
+and the semimajor axis of the cluster orbit 𝑎. The values of 𝑎 were
+taken as the average of the apogalactic and perigalactic distances in
+Baumgardt et al. (2019a), and the rest are our best-fit values in Table
+1. The Spearman rank-order correlation coefficients, 𝐶s, were then
+calculated for all possible combinations; there were only two pairs
+with an absolute value of 𝐶s greater than 0.5. The first pair is the
+concentration parameter 𝑊0 and the truncation parameter 𝑔. Their
+𝐶s = −0.65 indicates a strong anti-correlation between 𝑊0 and 𝑔.
+The distribution is presented in Fig. 8. The second pair is the trunca-
+tion parameter 𝑔 and the semimajor axis 𝑎 of the cluster orbit. The
+corresponding correlation coefficient 𝐶s = 0.60 indicates a strong
+correlation between 𝑔 and 𝑎; the result is presented in Fig. 9.
+The anti-correlation between the concentration parameter 𝑊0 and
+the truncation parameter 𝑔 is reasonable, as those with smaller trun-
+cation parameters would have experienced stronger tidal fields and
+evolve faster. It is likely that a certain fraction of them become
+core-collapsed clusters and thus have larger concentrations. This
+anti-correlation is also consistent with the simulations in Zocchi
+et al. (2016). They showed that when the clusters form, the value of
+concentration parameter 𝑊0 is nearly 4 and the value of truncation
+parameter 𝑔 is nearly 2.5. During the evolution, the truncation pa-
+rameter 𝑔 decreases, but the concentration parameter 𝑊0 increases.
+MNRAS 000, 1–15 (20XX)
+
+Dynamical Properties of Globular Clusters
+11
+0.0
+0.6
+1.2
+1.8
+2.4
+g
+10
+0
+10
+20
+30
+log ra
+27.5
+30.0
+32.5
+35.0
+M [105 M ]
+7.5
+9.0
+10.5
+12.0
+rh [pc]
+5.16
+5.22
+5.28
+5.34
+5.40
+D [kpc]
+0
+10
+20
+30
+W0
+1.8
+2.4
+3.0
+3.6
+4.2
+ [
+]
+0.0
+0.6
+1.2
+1.8
+2.4
+g
+10
+0
+10
+20
+30
+log ra
+27.5
+30.0
+32.5
+35.0
+M [105 M ]
+7.5
+9.0
+10.5
+12.0
+rh [pc]
+5.16
+5.22
+5.28
+5.34
+5.40
+D [kpc]
+1.8
+2.4
+3.0
+3.6
+4.2
+ [
+]
+Figure 5. The MCMC posterior parameter distributions of NGC 5139.
+Table 3. The parameters of two models of NGC 5139. The first column indicates different models. The second to eighth columns show the fitting parameters.
+The quantities in the last two columns are 𝜅 and 𝜒2r .
+Model
+𝑊0
+𝑔
+log ˆ𝑟a
+𝑀
+𝑟h
+𝐷
+Υ
+𝜅
+𝜒2r
+(105 M⊙)
+(pc)
+(kpc)
+(Υ⊙)
+A
+4.02+0.48
+−0.65
+1.94+0.27
+−0.26
+0.41+0.08
+−0.10
+32.82+0.65
+−0.67
+8.82+0.19
+−0.17
+5.32 ± 0.03
+2.38 ± 0.09
+1.15
+3.86
+B
+14.16+0.25
+−0.23
+1.28 ± 0.03
+11.38+5.83
+−6.00
+30.10 ± 0.59
+10.260.17
+0.16
+5.25 ± 0.03
+3.07 ± 0.09
+1.00
+5.64
+MNRAS 000, 1–15 (20XX)
+
+12
+Cheng and Jiang
+101
+102
+103
+15
+20
+25
+μ [mag/arcsec2]
+Model A
+101
+102
+103
+15
+20
+25
+Model B
+101
+102
+103
+0
+10
+20
+30
+σlos [km/s]
+Model A
+101
+102
+103
+0
+10
+20
+30
+Model B
+101
+102
+103
+r [arcsec]
+0
+10
+20
+30
+σpm [km/s]
+Model A
+101
+102
+103
+r [arcsec]
+0
+10
+20
+30
+Model B
+Figure 6. The comparison of the profiles from two models of NGC 5139. Left panels show the results from Model A and the right panels show those from
+Model B. The open circles represent the data of Baumgardt (2017), Baumgardt & Hilker (2018), and Dalgleish et al. (2020) for line-of-sight velocity dispersions
+and Watkins et al. (2015a) for proper motion velocity dispersions. The solid triangles are used for the data of Kamann et al. (2018) for line-of-sight velocity
+dispersions and Vasiliev & Baumgardt (2021) for proper motion velocity dispersions. The models are expressed by grey lines.
+Therefore, in Fig. 8, younger clusters are located at the top-left cor-
+ner, and the older clusters are distributed at the bottom-right corner.
+However, the exact relationships between these two parameters for
+different clusters are still complicated and the strength of this anti-
+correlation was not quantitatively investigated before.
+On the other hand, the correlation between the truncation parame-
+ter 𝑔 and the semimajor axis 𝑎 can be easily understood. The smaller
+truncation parameter shows that a stronger tidal field influences the
+cluster, and those clusters with smaller 𝑎 do experience stronger
+tidal fields. However, the relation between the two above-mentioned
+parameters shall also depends on the initial size and the orbital evolu-
+tion of a cluster. The contribution from different Galactic components
+make the exact behavior of the tidal field more complicated. It is rea-
+sonable that this correlation has a correlation coefficient 𝐶s = 0.60.
+The strong 𝑊0 − 𝑔 anti-correlation and 𝑔 − 𝑎 correlation shall
+be regarded as observational results as the employed parameters are
+obtained through our data-model fitting or from an observational
+catalog in literature. In addition, these observational anti-correlation
+and correlation agree with theoretical predictions.
+6 SUMMARY AND CONCLUSIONS
+In this work, we studied 18 clusters with the LIMEPY models, a unified
+family of isothermal models. It can generate clusters with differ-
+ent amounts of concentration, truncation, and anisotropy, which are
+parametrized by continuous real numbers. Including some current
+observational data, such as the MUSE survey and Gaia mission, the
+fittings were carried out with a Markov Chain Monte Carlo ensemble
+sampler EMCEE and the parameters were determined by minimizing
+the 𝜒2 of the fittings.
+The measurable physical properties such as masses and distances,
+were compared with the values from the literature. Usually, Baum-
+gardt & Hilker (2018) has larger masses, while Watkins et al. (2015b)
+has smaller ones, and our results are in between. The smaller half-
+mass radius in our results is consistent with the smaller mass esti-
+mated compared with Baumgardt & Hilker (2018). Some differences
+between the radius estimations might come from the effect of the
+mass spectrum. For distance, our estimations are in agreement with
+the literature. The mass-to-light ratios are also similar to the litera-
+ture.
+Generally, the models could produce profiles similar to the obser-
+vational ones for most clusters. For NGC 5139, there are two groups
+of parameters that correspond to a better fitting for the surface bright-
+ness or the velocity-dispersion profiles. The anisotropic model gives a
+smaller 𝜒2r and agrees with the best-fit results in Zocchi et al. (2017).
+Some possible central dark objects, like an intermediate-mass black
+hole or a group of stellar-mass black holes might improve the fitting.
+NGC 6388 is also a candidate to host an intermediate-mass black
+hole, with the actual central line-of-sight velocities being uncertain.
+The data we used have the extension to nearly 5 arcsec with a velocity
+dispersion ∼20 km/s. It could be fitted well with the LIMEPY model
+except for the slope of proper-motion velocity dispersion.
+For the anisotropy, NGC 5139 and NGC 7078 are anisotropic
+MNRAS 000, 1–15 (20XX)
+
+Dynamical Properties of Globular Clusters
+13
+101
+102
+103
+−1
+0
+1
+σT/σR − 1
+NGC 104
+101
+102
+−0.5
+0.0
+0.5
+NGC 288
+101
+102
+−0.5
+0.0
+0.5
+NGC 362
+101
+102
+−0.5
+0.0
+0.5
+σT/σR − 1
+NGC 1851
+101
+102
+−0.5
+0.0
+0.5
+NGC 2808
+101
+102
+−0.5
+0.0
+0.5
+NGC 3201
+101
+102
+103
+−0.5
+0.0
+0.5
+σT/σR − 1
+NGC 5139
+101
+102
+103
+−2
+−1
+0
+1
+2
+NGC 5904
+101
+102
+103
+−1
+0
+1
+NGC 6121
+101
+102
+−0.5
+0.0
+0.5
+σT/σR − 1
+NGC 6218
+101
+102
+−0.5
+0.0
+0.5
+NGC 6266
+101
+102
+−0.5
+0.0
+0.5
+NGC 6388
+101
+102
+103
+−1
+0
+1
+σT/σR − 1
+NGC 6397
+101
+102
+−0.5
+0.0
+0.5
+NGC 6441
+101
+102
+103
+−1
+0
+1
+NGC 6656
+101
+102
+r [arcsec]
+−0.5
+0.0
+0.5
+σT/σR − 1
+NGC 6715
+101
+102
+103
+r [arcsec]
+−0.5
+0.0
+0.5
+NGC 6752
+101
+102
+r [arcsec]
+−1
+0
+1
+NGC 7078
+Figure 7. The anisotropy profiles of the clusters. The open circles represent the data of Libralato et al. (2022). The data of Jindal et al. (2019) are shown in
+solid triangles. The crosses are used for Watkins et al. (2015a). The models are expressed by grey lines. The horizontal grey dashed lines represent zeros which
+indicate isotropy. For each panel, the name of the cluster is mentioned at the top-left corner.
+MNRAS 000, 1–15 (20XX)
+
+14
+Cheng and Jiang
+4
+5
+6
+7
+8
+9
+W0
+0.5
+1.0
+1.5
+2.0
+2.5
+g
+Figure 8. The truncation parameter versus the concentration parameter. The
+vertical axis represents the truncation parameter and the horizontal axis ex-
+presses the concentration parameter. Each point corresponds to a particular
+cluster.
+5
+10
+15
+20
+25
+a [kpc]
+0.5
+1.0
+1.5
+2.0
+2.5
+g
+Figure 9. The truncation parameter versus the semimajor axis of the cluster.
+The vertical axis represents the truncation parameter and the horizontal axis
+expresses the semimajor axis. Each data point corresponds to a particular
+cluster.
+with 𝜅 = 1.15 and 𝜅 = 1.16. The anisotropy leads to the rise in
+central velocity dispersion in these clusters. Our estimations could
+have some underestimations because the data are combined proper
+motion dispersion profiles rather than separated radial and tangential
+profiles. Nevertheless, the results are reasonable compared with some
+literature, such as Watkins et al. (2015a) and Watkins et al. (2015b),
+where the anisotropy in the studied clusters seem small.
+From a theoretical aspect, the truncation parameter may render
+the cluster to have a finite extension and isotropic profiles near the
+edge. It is similar to the effect of the external tidal field. In addition, a
+strong anti-correlation between the concentration parameter 𝑊0 and
+the truncation parameter 𝑔 was confirmed, which gives the imprint
+of the dynamical evolution of clusters. Finally, a strong correlation
+between the truncation parameter 𝑔 and the semimajor axis 𝑎 was
+also found, which could result from the influence of the Galactic tidal
+field.
+ACKNOWLEDGEMENTS
+We are grateful to the reviewer, Holger Baumgardt, for the useful sug-
+gestions which improved this paper significantly. We acknowledge
+the financial support from the Ministry of Science and Technology,
+Taiwan, (MOST grant 110-2112-M-007-035). We are grateful to the
+authors of Trager et al. (1995), Harris (1996), McLaughlin & van der
+Marel (2005), Baumgardt (2017), Baumgardt & Hilker (2018), Dal-
+gleish et al. (2020), Kamann et al. (2018), McLaughlin et al. (2006),
+Watkins et al. (2015a), Vasiliev & Baumgardt (2021), Häberle et al.
+(2021), McNamara et al. (2003), McNamara et al. (2012), Zloczewski
+et al. (2012), Watkins et al. (2015b), Baumgardt & Vasiliev (2021),
+Baumgardt et al. (2020), Libralato et al. (2022), Jindal et al. (2019),
+Baumgardt et al. (2019a), for making their data publicly available.
+This paper used the VizieR catalogue access tool, operated at CDS,
+Strasbourg, France, and the Astrophysics Data System Bibliographic
+Services of National Aeronautics Space and Administration, USA.
+Software: LIMEPY (Gieles & Zocchi 2015), EMCEE (Foreman-Mackey
+et al. 2013), corner, NumPy, and SciPy.
+DATA AVAILABILITY
+The electronic file of Table 1 is available in machine-readable form at
+VizieR (vizier.u-strasbg.fr) of Strasbourg astronomical Data Center
+(CDS).
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+

ANFKT4oBgHgl3EQfVi5k/content/tmp_files/2301.11788v1.pdf.txt

@@ -0,0 +1,679 @@
+Prepared for submission to JHEP
+Higgs Inflation: constraining the top quark mass
+and breaking the H0-σ8 correlation
+Jamerson G. Rodrigues,a Micol Benetti,b,c Rayff de Souzaa and Jailson Alcaniza
+aObservatório Nacional, 20921-400, Rio de Janeiro, RJ, Brazil
+bScuola Superiore Meridionale, Largo San Marcellino 10, 80138, Napoli, Italy
+cIstituto Nazionale di Fisica Nucleare (INFN) Sezione di Napoli, Complesso Universitario di Monte
+Sant’Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy
+E-mail: [email protected], [email protected],
+[email protected], [email protected]
+Abstract: Extending previous results [JHEP 11 (2021) 091], we explore aspects of the
+reheating mechanism for non-minimal Higgs inflation in the strong coupling regime. We
+constrain the radiative corrections for the inflaton’s potential by considering the Coleman-
+Weinberg approximation and use the Renormalization Group Equations for the Higgs field
+to derive an upper limit on the quark top mass, mt. Using the current Cosmic Microwave
+Background, Barion Acoustic Oscillation, and Supernova data, we obtain mt ≤ 170.44 GeV,
+confirming the observational compatibility of the model with recent mt estimates reported
+by the CMS collaboration. We also analyze the breakdown of the well-known correlation
+involving the Hubble constant H0 and the clustering parameter σ8, which makes the model
+interesting in light of the cosmological tensions discussed over the last decade.
+Keywords: Cosmology, Primordial Universe, Cosmic Microwave Background, Higgs Field,
+Cosmological Parameters.
+arXiv:2301.11788v1  [astro-ph.CO]  27 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+Non-minimal Inflation and Slow-Roll Analysis
+3
+3
+Reheating analysis and results
+4
+4
+Physical and cosmological consequences
+7
+4.1
+Constraints on the top quark mass
+7
+4.2
+The H0 − σ8 correlation
+7
+5
+Conclusions
+9
+1
+Introduction
+The fundamental theory behind the initial conditions that led to the temperature fluctua-
+tions in the Cosmic Microwave Background (CMB) [1, 2] and the formation of Large-Scale
+Structure (LSS) of the universe [3–5] remains an open question in modern cosmology. In
+this context, the paradigm of inflation rises as the most elegant description of the primordial
+Universe [6–10]. In order to induce cosmic acceleration, the dynamical equations for the
+inflaton field must enable a slowly varying solution, leading to a quasi-de Sitter Universe.
+In the well-known slow-roll mechanism this is achieved in an approximately flat direction
+of the inflaton’s scalar potential.
+One particularly appealing approach is to induce a non-minimal coupling between the
+inflaton and gravity, which results in a plateau at the large field regime [11–13] and drives
+the model predictions to the sweet-spot of CMB observations [14]. From the phenomeno-
+logical perspective, one specially interesting model was introduced by Berzrukov and Sha-
+poshnikov [15], where the standard Higgs field rules the inflationary period at early times.
+Such configuration allows one to compare the predictions of the model for the cosmological
+observables with the phenomenology of the related particles at electroweak scale of energy.
+Such analysis was explored in a number of interesting papers, see e.g. [16–20].
+Although robust, the analysis of inflationary models rely on a set of assumptions about
+the evolution of cosmological quantities. In particular, the evolution of cosmological scales
+from the moment they cross the Hubble radius during inflation up to the their re-entrance
+at later times must be matched to all the eras of the cosmological expansion in order to
+solve the horizon problem [21]. The matching condition can be written in the form
+ln
+�
+k
+a0H0
+�
+= −Nk − Nrh − NRD + ln
+�aeqHeq
+a0H0
+�
++ ln
+� Hk
+Heq
+�
+,
+(1.1)
+– 1 –
+
+where Nk is the number of e-folds the universe expanded between the horizon crossing
+moment of the pivot scale k and the end of inflation and Nrh is the number of e-folds counted
+from the end of inflation to the onset of the radiation dominance in the early Universe
+(reheating). Also, NRD gives the amount of expansion between the end of reheating and
+the end of radiation dominated era, while the subscript “eq" and “0" represent quantities
+evaluated at matter-radiation equality and the present, respectively. One is not able to set
+the amount of expansion the universe experienced in the inflationary period, Nk, without
+further information about the subsequent periods of the expansion. This is particularly
+problematic for the reheating period.
+In a previous communication [20], we performed a Monte-Carlo Markov Chain (MCMC)
+analysis of CMB and clustering data to check the observational viability of non-minimally
+coupled φ4 models for a fixed inflationary e-fold number. In particular, we considered the
+first order correction to the perturbative expansion of the inflationary potential, also known
+as Coleman-Weinberg approximation [22], and constrained possible radiative corrections
+coming from the underlying field theory supporting this cosmological scenario. In addition,
+we used the two-loop Renormalization Group Equations to connect the model’s predictions
+at inflationary energy scales to the electroweak observables and derived an estimate of the
+top quark mass mt, indicating a possible tension with the Monte-Carlo Tevatron and LHC
+reconstruction [23].
+In this work, we extend and complement the analysis reported in [20] by exploring the
+predictions of non-minimal Higgs inflation for a wide range of the inflationary e-fold number
+Nk and, consequently, of Nrh. Following the procedure developed in [20, 24], we employ a
+MCMC analysis to compare the predictions of this inflationary scenario with the most recent
+Cosmic Microwave Background (CMB), Baryon Acoustic Oscillation (BAO), and Supernova
+(SN) data [1–5].
+In particular, we obtain new constraints on the radiative corrections
+coming from the underlying field theory supporting this cosmological scenario and derive
+an upper limit for the top quark mass, which is compared with recent mt measurements
+from different experiments. Furthermore, we also explore whether this model could shed
+some light on the so-called cosmological tensions, which include the well-known H0 tension,
+a ∼ 4σ-discrepancy between direct measurements of H0 using low-z SN (H0 = 73.48 ± 1.66
+km/s/Mpc [25]) and the H0 estimate from current CMB data assuming the standard model
+(H0 = 67.72 ± 0.41 km/s/Mpc [14]) [26, 27]. It is worth mentioning that most of the usual
+mechanisms to solve this problem have failed so far, as alleviating the H0 discrepancy
+worsens the agreement of other parameters with the data. In particular, the clustering
+parameter, σ8, is constrained at σ8 = 0.766+0.024
+−0.021 by the Kilo-Degree Survey (KiDS-1000)
+lensing estimation [28] and its correlation with the Hubble constant leads to significantly
+too high values as the value of H0 increases. Breaking such a correlation is not only tricky
+but also challenging for many cosmological scenarios.
+This work is organized as follows.
+In Sec. 2, we briefly introduce the non-minimal
+inflationary scenario and present the results of the slow-roll analysis. In Sec. 3, we discuss
+aspects of the reheating stage following the Higgs inflation and present the main results of
+our statistical analysis of the cosmological data. Sec. 4 discusses the constraints derived on
+the top quark mass and some implications on the current cosmological tensions. The main
+– 2 –
+
+conclusions of this work are presented in Sec. 5.
+2
+Non-minimal Inflation and Slow-Roll Analysis
+As mentioned earlier, a common method to achieve slow-roll inflation is to induce a non-
+minimal coupling between the inflaton field and gravity. Such procedure yields non-canonical
+terms for the original scalar field and the metric, suggesting the use of a set of conformal
+transformations in order to obtain the theory description in the familiar Einstein-Hilbert
+formalism. A more detailed exposition of this approach can be found in [20].
+The Einstein frame lagrangian reads
+LE = −M2
+P ˜R
+2
++ 1
+2(∂µχ)†(∂µχ) − VE(χ) ,
+(2.1)
+and the subsequent time evolution is dictated by the inflaton’s potential
+VE(χ) = λM4
+P
+4ξ2
+�
+1 − e−
+�
+2
+3
+χ
+MP
+�2
+�
+�1 + a′ ln
+�
+1
+ξ e
+�
+2
+3
+χ
+MP − 1
+ξ
+�
+�
+(2.2)
+where the large field regime is assumed for the inflaton, χ ≫
+√
+6MP , and a large coupling
+regime is assumed for the non-minimal coupling, ξ ≫ 1. Note that the deviation from
+the tree level potential is quantified by the parameter a′ ≡ βλ/λ, where βλ is the running
+equation for the quartic coupling λ. The above potential was obtained by adopting the
+prescription II procedure to compute the radiative corrections in the Jordan frame and all
+couplings are computed at the scale M = MP , where MP is the reduced Planck mass [20].
+Once with the effective potential in the Einstein frame, the relevant slow-roll inflation-
+ary parameters can be readily computed, which can be related to the spectral index and
+tensor-to-scalar ratio, characteristic of the power spectrum of CMB perturbations probed
+by Planck [1]. Although the field strength χ∗, necessary to compute the relevant inflation-
+ary parameters, cannot be measured directly, we can infer its value from the duration of
+inflation from horizon crossing up to the end of inflationary expansion, characterized by the
+number of e-folds, which is also dependent on the form of the potential (2.2).
+However, the inflationary number of e-folds is not a free parameter entirely, as it is
+tied to the subsequent evolution of the universe, given its association with the horizon exit
+of relevant cosmological scales. Therefore, the relevant scales probed by Planck seem to
+correspond to an interval of 50-60 e-folds [21], which guides our range of exploration of the
+parameter Nk.
+In Fig. 1 we present our results for the spectral index and tensor-to-scalar ratio in the
+nS × r plane, with a′ ranging from −0.1 (lower limit) to 1.0 (upper limit)1. Note that there
+is a significant dependence of the inflationary predictions with the amount of expansion
+during inflation, achieving compatibility with the Planck result2. It is also important to
+1The values of a′ varying between [-0.010, 0.053], [-0.020, 0.036] and [-0.027, 0.023], corresponding to
+Nk = 50, 55 and 60, respectively, are in agreement with the 95% C.L. Planck result.
+2This agreement relies on the slow-roll approximations for the inflationary parameters and the phe-
+nomenological power-law expansion of the primordial power spectrum.
+– 3 –
+
+Figure 1. ns vs. r for Nk = 50, 55 & 60. The points in each curve indicate the parameters for
+a null resultant of the radiative corrections (a′ = 0). The blue areas show the favored regions by
+Planck 2018, with 68% and 95% confidence level (Planck TT, TE, EE + lowE + lensing + BK15
++ BAO data set) [14].
+mention that the results obtained for the prediction of inflationary parameters are highly
+independent of the coupling parameter ξ.
+3
+Reheating analysis and results
+Between the end of inflation and the onset of a radiation-dominated universe, the universe
+undergoes a reheating period. Even though there are a number of proposals for the dy-
+namics of the cosmos in this period [29–36], the reheating era is exceptionally difficult to
+be constrained by observations, given the small length scales characteristic of this micro-
+physical process. For previous works exploring the impact of reheating to the cosmological
+observables see e.g. [37–40] and references therein.
+In order to understand the influence of the reheating period on the inflationary predic-
+tions, one can follow the steps developed in [38] and resume the matching condition (1.1)
+to the expression:
+Nk = −1 + 3ωrh
+4
+Nrh − ln
+�
+V 1/4
+end
+Hk
+�
++ 61.55 ,
+(3.1)
+where the amount of expansion through the inflationary period is explicitly related to the
+reheating characteristics of the proposed model. Here, ωrh represents the effective equation-
+of-state parameter of the cosmological fluid during reheating, Vend is the amplitude of the
+inflaton’s potential energy at the end of inflation, Hk is the Hubble parameter evaluated at
+horizon crossing and k = 0.05 Mpc−1 is the pivot scale. We also consider grh ∼ 100 for the
+relativistic degrees of freedom to obtain the numerical factor above.
+– 4 –
+
+0.08
+Nx = 50
+Nx = 55
+0.06
+Nx = 60
+0.05
+0.03
+0.D2
+0.D1
+0.00
+0.95
+0.96
+L60
+0.98
+0.99
+fha′
+r0.02
+H0
+σ8
+Nk=50
+0.179 ± 0.072
+0.032 ± 0.013
+68.82 ± 0.38
+0.841 ± 0.005
+Nk=52
+0.040 ± 0.015
+0.007 ± 0.002
+68.31 ± 0.41
+0.835 ± 0.005
+Nk=54
+0.011 ± 0.014
+0.004 ± 0.001
+67.71 ± 0.45
+0.817 ± 0.003
+Nk=54.5
+0.009 ± 0.013
+0.004 ± 0.001
+67.68 ± 0.43
+0.811 ± 0.003
+Nk=55
+0.010 ± 0.013
+0.004 ± 0.001
+67.71 ± 0.44
+0.804 ± 0.003
+Nk=56
+0.022 ± 0.015
+0.005 ± 0.001
+67.94 ± 0.45
+0.793 ± 0.003
+Nk=58
+0.283 ± 0.169
+0.044 ± 0.019
+68.37 ± 0.39
+0.779 ± 0.004
+Nk=60
+0.243 ± 0.088
+0.042 ± 0.015
+68.46 ± 0.38
+0.766 ± 0.005
+Table 1. Constraints for fixed Nk at 68% C.L. using the Planck TT, TE, EE + lowE + lensing +
+BICEP2/Keck + BAO + Pantheon combination.
+In what concerns non-minimal inflationary models, it is possible to show that the
+inflaton condensate starts the reheating process oscillating with an effective matter-like
+equation of state (ω1 = 0) and, after crossing a critical value χcr, finishes the process as a
+radiation-like component of energy (ω2 = 1/3) [41, 42]. After some algebraic manipulations
+and using the approximation Hk ∼
+�
+V∗/3, valid during inflation, one obtains:
+Nk = −1
+4N1 − ln
+�
+V 1/4
+end (a′)
+�
+V∗(a′)/3
+�
++ 61.55
+(3.2)
+where we highlight the a′ dependence of the inflationary potential.
+We analyze the present model for fixed values of Nk and compute the values of Vend
+and Hk following the slow-roll approximations.
+In our analysis we assume a standard
+cosmological model with a modified primordial spectrum in which the radiative correction
+parameter, a′, is free to vary.
+For the parameter estimation we use the free available
+CosmoMC code [43]3 and a combination of early and late data4 (for more details we refer
+the reader to [20]). Table 1 shows the derived constraints on the most significant parameters
+of our analysis.
+Note that by computing the values of Vend and Hk, we can obtain the corresponding
+values for N1, i.e, the amount of expansion that the universe went through, as matter-
+like dominated, during the reheating process. The corresponding values are presented in
+Figure 2. Note also that, for an expansion of ∼ 56 e-folds or greater during inflation, N1
+would have to assume negative values to satisfy the matching equation (3.2). By definition,
+this condition would imply in a contraction of the universe between the end of inflation
+3This is a MCMC code interfaced with the Boltzmann solver Code for Anisotropies in the Microwave
+Background (CAMB) [44]. We modified CAMB following the indications of ModeCode [45, 46] in order to
+analyse the specific form of the potential V (φ).
+4We use the CMB Planck (2018) likelihood [1], using Plik temperature power spectrum, TT, and HFI
+polarization EE likelihood at ℓ ≤ 29; BICEP2 and Keck Array experiments B-mode polarization data [2];
+BAO measurements from 6dFGS
+[3], SDSS-MGS [47], and BOSS DR12 [4] surveys, and the Pantheon
+sample of Type Ia supernovae [5].
+– 5 –
+
+Figure 2. Nk vs. N1 for each inflationary number of e-folds taken into consideration. N1 is given
+by the matching equation (3.2), with a′ coming from the MCMC analysis (highlighted beside each
+point). Through a linear regression between the points (solid blue line), we estimate a maximum
+number Nk - where the transition to a radiation-dominated Universe happens instantaneously.
+and the onset of the radiation-dominated epoch5. Thus, following the standard approach,
+we discard these possibilities as non-physical. Therefore, we can tighten the bounds on
+the maximum value for the inflationary number of e-folds, which yields an instantaneous
+transition to the radiation-dominated expansion.
+The results presented above are insensitive to the specific physical process that leads to
+the transition between matter and radiation-like expansion in the reheating. As pointed out
+in [17, 41], non-perturbative processes may occur before the perturbative decays become
+viable (preheating), displacing the transition between the two expansion behaviors, which
+is particularly true in the model of Higgs Inflation. In this context, a specially interesting
+result was obtained in [48], where the authors discussed the resonant production of Higgs
+and gauge degrees of freedom in the linear regime of the Higgs Inflation scenario.
+For
+100 < ξ < 1000, the preheating dominant process is the Higgs self-resonance, leading to
+N1 ≃ 3. For higher values of the non-minimal coupling, ξ > 1000, it was pointed out that a
+substantial amount of energy stored in the inflaton condensate is transferred to relativistic
+gauge bosons already at the very first oscillation of the background (instant preheating),
+leading to N1 = 0. Note that these results are in agreement with our analysis for Nk ≃ 55
+and Nk ≃ 56, respectively, which is also in agreeement with the MCMC result for the
+5It is also possible to obtain N1 > 0 even for Nk > 56 if one considers exotic scenarios for the transition to
+radiation dominance, including intermediary phase transitions of the reheating fluid to an exotic component
+of energy ω′ > 1/3.
+– 6 –
+
+0.179
+20
+*: Nk.max~55.98
+10.04
+10
+M
+N0.011
+10.009
+10.01
+0
+10.022
+0.283
+-10
+0.243
+50
+52
+54
+55
+56
+58
+60
+Nkradiative corrections in the interval a′ ≃ [−0.003, 0.037] at 68% (C.L.).
+4
+Physical and cosmological consequences
+4.1
+Constraints on the top quark mass
+It is helpful to recall that the result mentioned above is obtained in the framework of the
+Higgs Inflation scenario, where a′ is associated with the β-function of the Higgs quartic
+coupling λ. Once the renormalization group equations for the standard Higgs couplings
+are considered, it is possible to link the cosmological constraints to the phenomenology
+of the associated particles at the electroweak scale of energy6. In this context, following
+the approach developed in [20], one shall infer an upper limit on the top quark pole mass,
+mt ≤ 170.44 GeV, to reproduce the values of a′ above. Also, it is worth emphasizing that
+this limit on mt is relatively insensitive to the amplitude of the non-minimal coupling once
+the strong limit (ξ ≫ 1) is assumed.
+The most precise constraints on the top quark mass are extracted from the kinematic
+reconstruction of the t¯t events where mt is employed in the Monte-Carlo generator in order
+to fit the data [49, 50]. This MC top quark mass is usually assumed to be the pole mass
+even though the theoretical uncertainties inherent to this association are hard to quantify
+[51]. From [52], the average value for the top quark mass is set to mt = 172.69 ± 0.30 GeV,
+obtained from LHC and Tevatron data. If contrasted with the limit on mt obtained from
+the cosmological analysis, this represents a significant discrepancy of 7.5σ.
+Instead, one may consider theoretically cleaner the inference of the top quark pole
+mass from the measurements of the cross-section of the top quark production, since the
+theoretical computation of σ(t¯t) is explicitly performed in a renormalization scheme (e.g.,
+MS) [53]. In this case, the average value obtained from the Tevatron and LHC runs is
+172.5 ± 0.7 GeV [52], lowering the discrepancy with our cosmological estimate of mt to
+≈ 3σ. More recently, the CMS collaboration reported mt = 170.5±0.8 GeV, obtained from
+the differential cross-section of the top production [54]. Such result perfectly agrees with
+the results of our cosmological analysis of the Higgs Inflation.
+4.2
+The H0 − σ8 correlation
+The accuracy of cosmological and astrophysical measurements has significantly improved
+in recent decades. While this has led to increasingly evident confirmation of the validity of
+the standard cosmological model, it has also exposed some critical issues that have given
+rise to heated debate. The well-known H0 tension has been extensively explored without
+concluding so far (we refer the reader to [26, 27] and references therein).
+It has also been widely pointed out that some of the current attempts to solve the
+H0 tension have failed because as they alleviate the discrepancy on H0, they worsen the
+agreement of other parameters with the data. In particular, the clustering parameter, σ8, is
+constrained at σ8 = 0.766+0.024
+−0.021 by the Kilo-Degree Survey (KiDS-1000) lensing estimation
+6The parameters considered in the definition of a′ are evaluated at the renormalization scale M = MP .
+– 7 –
+
+0.75
+0.80
+0.85
+8
+66
+67
+68
+69
+70
+H0
+0.75
+0.8
+0.85
+8
+N=50
+N=52
+N=54
+N=54.5
+N=55
+N=56
+N=58
+N=60
+Figure 3. Confidence levels and posterior distributions for the H0 and σ8 parameters using the
+joint data set CMB Planck (2018) + BICEP2 and Keck Array + BAO + Pantheon SNe Ia sample
+and considering several values of Nk.
+[28] and its correlation with the Hubble constant leads to values that are significantly too
+high as the value of H0 increases.
+It is generally agreed that a model that manages to resolve both tensions is a model
+that breaks this degeneracy, but building such a model is proving difficult. So far, only
+a handful of scenarios seem to succeed, such as the conjecture of a universe transition
+from anti-de Sitter vacua to de Sitter vacua [55–57], some early interacting models [58] or
+specific parametrizations of dark energy equation of state [59]. The model studied here is
+promising, as it breaks the degeneracy between the two parameters. In particular, Table 1
+shows that as Nk increases between the values of 50 and 54.5, the values of the radiative
+parameter, a′, H0, and σ8 decrease. Nevertheless, at the turning value of Nk = 54.5, there
+is a behavior change, i.e., as Nk increases, the values of a′ and H0 also increase. In contrast,
+the value of the clustering parameter, σ8, does not seem to be affected by this turning point
+and continues to decrease. It means that, for values of Nk ∈ [54.5, 60]7, the correlation
+between H0 and σ8 breaks down, as also shown in Figure 3. In particular, for the limiting
+value Nk = 56, i.e., an instantaneous transition to the radiation-dominated expansion, the
+degeneracy H0 − σ8 is such that it reduces the H0 tension, constraining H0 = 67.94 ± 0.45
+7As discussed earlier, we consider the cases Nk > 56 to be non-physical since they predict negative values
+of N1.
+– 8 –
+
+Km/s/Mpc, which is ≈ 3σ off from the SNe Ia measurements [25] and allowing a value of
+σ8 = 0.793 ± 0.003, that is in full agreement with KiDS-1000 results [28].
+5
+Conclusions
+In this work, we revisited the non-minimal inflationary scenario subject to radiative cor-
+rections.
+By performing an observational analysis of the φ4 primordial potential, non-
+minimally coupled to the Ricci scalar, in light of the most recent CMB, clustering and
+Supernova data and considering the allowed range for the observable inflationary e-folds,
+we constrained the possible values of the radiative corrections of the inflaton potential,
+encoded in the parameter a′, and the usual set of cosmological parameters.
+From this analysis, we presented two main results. First, we set an upper limit to the
+number of e-folds from the horizon crossing moment up to the end of inflation, Nk ≲ 56,
+relative to instantaneous reheating, by considering the matching equation for the pivot scale
+k = 0.05 Mpc−1. An even more stringent limit is imposed once considered the preheating
+structure of the Higgs Inflation, yielding 55 ≲ Nk ≲ 56. Accordingly, the MCMC analysis
+of the model translates into an upper bound for the top quark pole mass, mt ≤ 170.44 GeV,
+which raises two possible interpretations for the consistency of the model at low-energies.
+For example, considering the value of the top quark mass reconstructed from the analysis of
+LHC and Tevatron data, Mt = 172.69 ± 0.30 GeV [23], implies a significant tension of 7.5σ
+between the observed low-energy value and the amount inferred by the cosmological MCMC
+analysis. On the other hand, assuming the top quark mass extracted from differential cross-
+section of the top production, Mt = 170.5 ± 0.8 GeV, obtained by the CMS collaboration
+[54], we found a perfect agreement between the cosmological analysis of the Higgs field and
+its electroweak behaviour.
+Second, the MCMC analysis of current observational data confirms the observational
+viability of the model and shows that for the interval Nk ∈ [54.5, 60], it can break down the
+well-known H0 − σ8 correlation (see Table 1). In particular, considering an instantaneous
+transition to the radiation-dominated expansion, which occurs for Nk = 56, the H0 tension
+is reduced to ≈ 3σ whereas the value of σ8 shows a complete agreement with KiDS-1000
+results.
+These results reinforce the need to investigate Higgs inflation and its extensions from
+both theoretical and observational sides and show that perspectives for a complete coherence
+of the scenario may converge once data from future collider experiments [60, 61] improve
+our understanding of the physics at the eletroweak scale.
+Acknowledgments
+We thank André Sznajder for helpful conversations. JGR acknowledges financial support
+from the Programa de Capacitação Institucional (PCI) do Observatório Nacional/MCTI.
+MB acknowledges Istituto Nazionale di Fisica Nucleare (INFN), sezione di Napoli, iniziativa
+specifica QGSKY. RdS is supported by the Coordenação de Aperfeiçoamento de Pessoal
+de Nível Superior (CAPES). JSA is supported by CNPq (Grants no. 310790/2014-0 and
+– 9 –
+
+400471/2014-0) and Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro FAPERJ
+(grant no. 233906). We also acknowledge the use of CosmoMC and ModeCode packages.
+This work was developed thanks to the use of the National Observatory Data Center (CP-
+DON).
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+

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+A fixed point can hide another one: the nonperturbative behavior of
+the tetracritical fixed point of the O(N) models at large N
+Shunsuke Yabunaka1, ∗ and Bertrand Delamotte2
+1Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan
+2Sorbonne Universit´e, CNRS, Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee, LPTMC, F-75005 Paris, France.
+(Dated: January 4, 2023)
+We show that at N = ∞ and below its upper critical dimension, d < dup, the critical and
+tetracritical behaviors of the O(N) models are associated with the same renormalization group fixed
+point (FP) potential. Only their derivatives make them different with the subtleties that taking
+their N → ∞ limit and deriving them do not commute and that two relevant eigenperturbations
+show singularities. This invalidates both the ϵ− and the 1/N− expansions. We also show how the
+Bardeen-Moshe-Bander line of tetracritical FPs at N = ∞ and d = dup can be understood from a
+finite-N analysis.
+Field theories sometimes exhibit nonperturbative fea-
+tures such as confinement [1], presence of bound states [2]
+or exotic excitations [3], fixed points (FPs) of the renor-
+malization group (RG) flows that are nonperturbative as
+in the Kardar-Parisi-Zhang equation [4], divergence of
+the perturbative RG flow at a finite RG scale [5], pres-
+ence of a cusp in the FP potential as in the random field
+Ising model [6], to cite but a few. Very often, these non-
+perturbative effects are assumed either to occur in rather
+complicated theories such as gauge and string theories or
+in highly nontrivial statistical models.
+O(N) models, which are the simplest scalar field theo-
+ries, are often implicitly considered to be immune to these
+complex phenomena.
+Perturbative methods are there-
+fore assumed to work almost all the time for these mod-
+els, the exception to the rule being the Bardeen-Moshe-
+Bander (BMB) phenomenon [7], related to the existence
+of a line of tricritical FPs at N = ∞ and d = 3, which re-
+quires nonperturbative FPs to be fully understood from a
+large-N analysis [8]. From this viewpoint, the enormous
+success of the ϵ = 4 − d expansion for the perturbative
+calculation of the critical exponents associated with the
+Wilson-Fisher (WF) FP [11] could let us believe that the
+critical physics of the O(N) models is fully understood
+for any N and d, especially since it is corroborated by
+the 1/N and ϵ = d − 2 expansions [11].
+Our goal in this Letter is to show instead that although
+the critical physics of the O(N) models, described by the
+WF FP, is fully under perturbative control at both finite
+and infinite N, the tetracritical physics of these models
+at N = ∞ –and probably of infinitely many multicritical
+behaviors– is not. We show below (i) that at N = ∞, it
+is also associated with the WF FP, which is unexpected,
+and (ii) that it nonetheless shows non-perturbative fea-
+tures that are beyond the reach of the standard imple-
+mentation of both the large-N and ϵ- expansions. We
+show in particular a very intriguing phenomenon related
+to the large-N limit of the tetracritical FP of the O(N)
+models: from the second order, the derivatives of the
+∗ [email protected]
+N = ∞ tetracritical FP potential, that is, of the WF FP
+potential, are not identical to the limit of the derivatives
+of the finite-N tetracritical FP potentials when N → ∞.
+This turns out to be crucial for understanding the large-
+N limit of tetracritical phenomena and shows that this
+limit is much less trivial than what is usually said [9–11].
+The perturbative tetracritical FP corresponds to the
+massless (ϕ2)4 theory, the upper critical dimension of
+which is dup = 8/3. It is found in perturbation theory in
+ϵ = 8/3 − d for all N ≥ 1 and it is three times infrared
+unstable [12]. Calling λ/(384N 3) the coupling in front of
+the dimensionless (ϕ2)4 term, the large-N perturbative
+flow equation for λ reads [13]:
+∂tλ = −3ϵλ + 9λ2
+4N + O(N −2).
+(1)
+From Eq. (1), we find that at leading order in N, the
+nontrivial FP solution is λ∗ = 4ϵN/3 from which follows
+that perturbation theory does not allow for a control of
+the large-N limit of the tetracritical FP at fixed ϵ. Only
+the double limit N → ∞ and ϵ → 0 such that the product
+ϵN remains finite can possibly be under control. We come
+back on this point in the following.
+Let us recall that in generic dimensions d < 4, the
+only nontrivial FP found in the standard large-N anal-
+ysis of the O(N) models is the WF FP [14]. Thus, no
+tetracritical FP is found at N = ∞ and d < 8/3 which
+is paradoxical considering that it is perturbatively found
+for all N < ∞ and ϵ > 0.
+We show below that the solution to the paradox above
+lies in the field dependence of the tetracritical FP poten-
+tial whereas it cannot be obtained from its field expansion
+and in particular from λ∗. The recourse to functional RG
+methods is therefore mandatory.
+The best way to implement functional RG is to con-
+sider Wilson’s RG, as it is inherently functional [15]. We
+recall below the take-away philosophy of the modern ver-
+sion of Wilson’s RG known as the nonperturbative – or
+functional – renormalization group (NPRG).
+NPRG is based on the idea of integrating fluctuations
+step by step [16]. It is implemented on the Gibbs free
+energy Γ [17–23] of a model defined by an Hamiltonian
+arXiv:2301.01021v1  [cond-mat.stat-mech]  3 Jan 2023
+
+2
+(or euclidean action) H and a partition function Z. To
+this model is associated a one-parameter family of models
+with Hamiltonians Hk = H + ∆Hk and partition func-
+tions Zk, where k is a momentum scale. In Hk, ∆Hk is
+chosen such that only the rapid fluctuations in the origi-
+nal model, those with wavenumbers |q| > k, are summed
+over in the partition function Zk. Thus, the slow modes
+(|q| < k) need to be decoupled in Zk and this is achieved
+by giving them a mass of order k, that is by taking for
+∆Hk a quadratic (mass-like) term, which is nonvanishing
+only for the slow modes:
+Zk[J] =
+�
+Dϕi exp(−H[ϕ] − ∆Hk[ϕ] + J · ϕ)
+(2)
+with ∆Hk[ϕ] =
+1
+2
+�
+q Rk(q2)ϕi(q)ϕi(−q), where, for in-
+stance, Rk(q2) = (k2 − q2)θ(k2 − q2) and J · ϕ =
+�
+x Ji(x)ϕi(x). The k-dependent Gibbs free energy Γk[φ]
+is defined as the (slightly modified) Legendre transform
+of log Zk[J]:
+Γk[φ] + log Zk[J] = J · φ − 1
+2
+�
+q
+Rk(q2)φi(q)φi(−q) (3)
+with
+�
+q =
+�
+ddq/(2π)d.
+With the choice of regulator
+function Rk above, Γk[φ] interpolates between the Hamil-
+tonian H when k is of order of the ultraviolet cut-off Λ
+of the theory: ΓΛ ∼ H, and the Gibbs free energy Γ of
+the original model when k = 0: Γk=0 = Γ. The exact
+RG flow equation of Γk gives the evolution of Γk with
+k between these two limiting cases. It is known as the
+Wetterich equation. It reads [18]:
+∂tΓk[φ] = 1
+2Tr[∂tRk(q2)(Γ(2)
+k [q, −q; φ] + Rk(q))−1], (4)
+where t = log(k/Λ), Tr stands for an integral over q and
+a trace over group indices and Γ(2)
+k [q, −q; φ] is the matrix
+of the Fourier transforms of δ2Γk/δφi(x)δφj(y).
+In most cases, Eq. (4) cannot be solved exactly and
+approximations are mandatory. The best known approx-
+imation consists in expanding Γk in powers of the deriva-
+tives of φi and to truncate the expansion at a given fi-
+nite order[24–32]. The approximation at lowest order is
+dubbed the local potential approximation (LPA). For the
+O(N) model it consists in approximating Γk by:
+Γk[φ] =
+�
+x
+�1
+2(∇φi)2 + Uk(φ)
+�
+(5)
+where φ = √φiφi. Fixed points are found only for di-
+mensionless quantities and the standard large-N limit
+by rescaling the field and the potential by factors N −1/2
+and N −1 respectively.
+Thus, we define the dimen-
+sionless and rescaled field ¯φ and potential ¯Uk as ¯φ =
+v
+− 1
+2
+d
+k
+2−d
+2 N −1/2φ and ¯Uk(¯φ) = v−1
+d k−dN −1Uk (φ) with
+v−1
+d
+= 2d−1dπd/2Γ( d
+2). The LPA flow of ¯Uk then reads:
+∂t ¯Uk(¯φ) = − d ¯Uk(¯φ) + 1
+2(d − 2)¯φ ¯U ′
+k(¯φ)+
+�
+1 − 1
+N
+�
+¯φ
+¯φ + ¯U ′
+k(¯φ) + 1
+N
+1
+1 + ¯U ′′
+k (¯φ)
+(6)
+FIG. 1. d = 2.6: ¯U(¯φ) for the T3 FP of Eq. (6). Green, red,
+blue and black curves correspond to N = 1500, 2250, 4500
+and 42000. The orange dashed curve corresponds to the WF
+FP at N = ∞. Inset: Close view of ¯U(¯φ) around ¯φi.
+with ∂t = k∂k. The standard large-N limit of the LPA
+flow equation above is obtained by (i) replacing the fac-
+tor 1 − 1/N by 1, (ii) dropping the last term in Eq. (6)
+because it is assumed to be sub-leading [33]. As a con-
+sequence of the two steps above, the explicit dependence
+in N in Eq. (6) disappears in the large-N limit.
+The crucial point of the large-N limit is that assuming
+point (ii) above, the resulting LPA flow equation on ¯Uk
+can be shown to be exact in the limit N → ∞ [34]. Under
+this assumption, all FPs of the O(N) models have been
+found exactly at N = ∞ [14, 33–36]. The result is the
+following: In a generic dimension d < 4 there is only one
+nongaussian FP at N = ∞ which is the usual Wilson-
+Fisher FP (WF). The exceptions to the rule above are the
+BMB lines of FPs [7, 14, 37–39] existing in dimensions
+d = 2 + 2/p with p an integer larger than 1.
+We now show that the procedure described above is too
+restrictive to study the large-N limit of the tetracritical
+FPs. As said above, the standard large-N analysis con-
+sists in neglecting the last term in Eq. (6). However, this
+term is negligible only if (1 + ¯U ′′
+k (¯φ))−1 does not coun-
+terbalance at large N its 1/N prefactor for some finite
+values of ¯φ. We now show that because of singularities in
+the third derivative of ¯Uk(¯φ), the contribution of the last
+term in Eq. (6) cannot be neglected in the FP equation
+of ¯U ′′
+k (¯φ) obtained by differentiating twice Eq. (6) (see
+footnote below Eq. (8) for more detail). This turns out
+to be sufficient to invalidate the standard large-N limit
+in the tetracritical case.
+We have numerically solved Eq. (6) and have found
+for several values of N and d < 8/3 the perturbative
+tetracritical FP that we call T3(N, d). As expected, T3
+bifurcates from the Gaussian FP in d = 8/3−. We have
+followed it down to d = 2.6, see Fig.
+1 and Fig.
+3
+of the Suppl. Mat. The FP potential of T3, (i) shows
+as expected two maxima, one of which being located at
+¯φ = 0 and another one at ¯φ2 > 0, and two minima at
+¯φ1 and ¯φ3 such that ¯φ3 > ¯φ2 > ¯φ1 > 0, see Fig.
+1,
+(ii) can be continuously followed up to arbitrarily large
+values of N at fixed d < 8/3, (iii) has its three extrema
+
+T()
+0.38466
+0.38464
+0.9
+0.38462
+0.8
+0.38460
+0.7
+1.75
+1.80
+1.85
+$1.90
+0.6
+0.5
+2.03
+¯φ1, ¯φ2, ¯φ3 approaching each other when N is increased at
+fixed d. These extrema tend to a common value ¯φ0 when
+N → ∞ which is the minimum of the FP potential, see
+Fig. 1 and Fig. 4 of the Suppl. Mat. Point (ii) above is
+paradoxical because it seems to contradict the standard
+large-N approach where only the WF FP is found in a
+generic dimension d < 8/3 at N = ∞. We now show
+that the WF FP potential at N = ∞ is in fact the limit
+when N → ∞ of the potential of T3 for d < 8/3. This
+solves the above paradox because it explains why on one
+hand there exists a nontrivial tetracritical FP at N = ∞
+and d < 8/3 and on the other hand that there is no
+other nontrivial and smooth solution of Eq. (6) at N =
+∞ than the WF FP potential. However, this creates a
+new paradox since obviously the critical and tetracritical
+universal behaviors cannot be the same since the two FPs
+do not have the same number of unstable eigendirections.
+We now explain in detail this new paradox.
+We can see on Fig. 1 that the FP potentials found
+in d = 2.6 for large values of N are extremely flat in
+the region, ¯φ ∈ [¯φ1, ¯φ3] because the three extrema are
+very close and the height of the barrier between the two
+minima very small. We have numerically found that the
+height of the barrier scales as N −1 and the distance be-
+tween the two minima as N −1/2 so that the curvatures
+¯U ′′(¯φi) at the three extrema approach constant values as
+N → ∞, see Fig. 4 of the Suppl. Mat. This suggests
+that ¯U ′′(¯φ) while being well-behaved everywhere but be-
+tween the three extrema, changes very rapidly within a
+boundary layer around ¯φ0 of typical width N −1/2, mak-
+ing divergent ¯U ′′′(¯φ0) when N → ∞.
+It is not common in physics to encounter this kind
+of situation where a series of functions fn(x) tends to a
+smooth function f∞(x) whereas from a certain order p,
+their derivatives f (p)
+n (x) do not tend to f (p)
+∞ (x). However,
+a simple toy model explains trivially how this can occur.
+Consider the series of functions fn(x) = n−1 sin(n2x).
+Obviously, f∞(x) ≡ 0 which implies that f ′
+∞(x) ≡ 0
+whereas limn→∞ f ′
+n(0) = ∞.
+In our case, at fixed d < 8/3, the limit of the T3 po-
+tentials when N → ∞ is a nontrivial and well-defined
+function that therefore must be the WF FP potential.
+We have checked that it is indeed the limit of T3 when
+N → ∞, see Fig. 1. The difference between the critical
+and tetracritical behaviors is therefore not visible on the
+potentials themselves but only on their derivatives as we
+now show.
+Let us study the boundary layer around ¯φ0. It is con-
+venient for what follows to change variables. Following
+Ref. [40], we define: V (µ) = U(φ) + (φ − Φ)2/2 with
+µ = Φ2 and φ − Φ = −2ΦV ′(µ). As above, it is conve-
+nient to rescale µ and V (µ): ¯µ = µ/N, ¯V = V/N. In
+terms of these quantities, the FP equation for ¯V (¯µ) reads
+0 = 1 − d ¯V + (d − 2)¯µ ¯V ′ + 4¯µ ¯V ′2 − 2 ¯V ′ − 4
+N ¯µ ¯V ′′. (7)
+Eq. (7) has two remarkable features: (i) it is much sim-
+pler than Eq. (6) because the nonlinearity comes only
+0
+2
+4
+6
+8
+10
+12
+14 μ
+0.02
+0.04
+0.06
+0.08
+V''[μ]
+N=6×103
+N=1.7×104
+N=3.2×106
+N=∞WF
+FIG. 2. Second derivative of the WF and T3 FP potentials
+for different values of N in d = 2.6.
+from the ( ¯V ′)2 term, (ii) it is the LPA equation obtained
+from the Wilson-Polchinski (WP) version of the NPRG
+[15, 41, 42]. Thus, ¯V (¯µ) is related to the potential ¯U(¯φ) of
+the Wetterich version of the RG by the Legendre trans-
+form of Eq.
+(3).
+The standard large-N analysis per-
+formed in this version of the NPRG consists here again
+in neglecting the last term in Eq. (7) because it is sup-
+pressed by a 1/N factor. Under the assumption that this
+term is indeed negligible, the resulting equation can be
+solved exactly in the large-N limit [14, 35]. However, at
+large N, it is clear on Eq. (7) that we have to deal with
+singular perturbation theory since the small parameter
+used in the 1/N expansion is in front of the term of high-
+est derivative, that is, ¯V ′′. In this case, it is well-known
+that at large N a boundary layer can exist for a partic-
+ular value of ¯µ that becomes a singularity at N = ∞,
+making this term non negligible [43].
+The value of ¯µ corresponding to ¯φ0 is called ¯µ0 and
+is the minimum of ¯V (¯µ) at N
+= ∞.
+We find for
+¯V (¯µ) the same features about its three extrema ¯µi as
+for ¯U(¯φ) at ¯φi: The three extrema ¯µi approach each
+other and to ¯µ0 as N → ∞, the distances between
+them scale as N −1/2 and the curvatures ¯V ′′(¯µi) as N 0.
+Taking into account the scaling around ¯µ0 inside the
+boundary layer, we introduce another scaled variable
+˜µ = N 1/2(¯µ − ¯µ0).
+Since at N = ∞, ¯V ′(¯µ) vanishes
+at ¯µ = ¯µ0, ¯V (¯µ0) should approach 1/d at leading order
+in N −1/2. We therefore define a scaled boundary layer
+by ˜VN(˜µ) = N
+� ¯V (¯µ0 + N −1/2˜µ) − 1/d
+�
+which implies
+˜V ′′
+N(˜µ) = ¯V ′′(¯µ0 + N −1/2˜µ). We plot ˜V ′′
+N(˜µ) for several
+values of N in Fig. 5 of the Suppl. Mat.
+By substituting ˜VN(˜µ) by its value in Eq.
+(7) and
+solving it at order O(N −1/2), we find that ¯µ0 = 2/(d−2).
+At order O(N −1), Eq. (7) becomes
+− 8 ˜V ′′
+∞(˜µ)
+d − 2 + 8 ˜V ′
+∞(˜µ)2
+d − 2
++(d−2)˜µ ˜V ′
+∞(˜µ)−d ˜V∞(˜µ) = 0 (8)
+[44] which is clearly invariant under ˜µ → −˜µ from which
+it follows that ˜V ′
+∞(0) = 0. At ˜µ = ∞, ˜V ′′
+∞(˜µ) should tend
+to a finite value that matches with ¯V ′′(µ) at ¯µ+
+0 . This
+implies that the solution of Eq. (8) should be quadratic
+when ˜µ → ∞. Substituting ˜V∞(˜µ) by ˜V ′′
+∞(˜µ = ∞)˜µ2/2
+in Eq. (8) and balancing the leading terms as ˜µ → ∞,
+we find that ˜V ′′
+∞(˜µ = ∞) = (−d2 + 6d − 8)/16. Imposing
+
+4
+the two boundary conditions found above at ˜µ = 0 and
+˜µ = ∞ selects a unique and globally defined solution
+˜V ′′
+∞(˜µ) of Eq. (8) shown in Fig. 5 of the Suppl. Mat.
+We find ¯V ′′(¯µ+
+0 ) = ¯V ′′
+WF(¯µ0) = ˜V ′′
+∞(˜µ = ∞) which proves
+the matching at N = ∞ between the boundary layer and
+the potential outside of the layer, see Fig. 2. We have
+shown in Fig. 6 of the Suppl. Mat. the boundary layer
+for ¯U ′′(¯φ) analogous to that of ¯V ′′(¯µ). To conclude, we
+have proven that for d < 8/3, a boundary layer develops
+at large N for the second derivative of the T3 potential
+that becomes a singularity when N → ∞. What remains
+to be understood is its physical relevance.
+At first sight, what we have obtained for T3 looks para-
+doxical because we could think that its potential being
+identical to the WF potential at N = ∞, the linearized
+flow around these two FPs should also be identical and
+thus the same for all critical exponents. We now show
+that this naive argument is wrong.
+We have computed in d < 8/3 the relevant eigenvalues
+of the RG flow around T3 and WF at finite and large N
+and as expected we have found three for T3 and one for
+WF. When N → ∞, one of the three eigenvalues at T3
+tends as expected to d − 2 which is the relevant eigen-
+value ν−1 of the critical WF FP at N = ∞ [11, 14]. The
+nontrivial point is that the two other relevant eigenvalues
+at T3 have a well-defined limit when N → ∞ although
+they do not play any role for the critical behavior of the
+O(N = ∞) model. The solution to this paradox is that
+they are associated with eigenperturbations that become
+singular when N → ∞. That these two eigenperturba-
+tions become singular is clear for one of them, called δ ¯V2,
+on Fig. 9 of the Suppl. Mat. As for the other one, δ ¯V1,
+its slope at ¯µ0 diverges as N 1/3 which implies that at
+N = ∞, it becomes discontinuous at ¯µ0, see Figs. 9 and
+10 of the Suppl. Mat. For ordinary second order phase
+transitions, these eigenperturbations are excluded which
+explains that the associated relevant eigenvalues do not
+play any role. This solves all the paradoxes associated
+with the tetracritical FPs at N = ∞ and d < 8/3.
+What remains to be studied is the particular case N =
+∞ and d = 8/3 where a line, called the BMB line, of
+smooth tetracritical FPs shows up. It is obtained in the
+WP version of the RG by integrating Eq. (7) in which
+the last term, proportional to 1/N, has been discarded.
+It is given by the following implicit expression [14]:
+¯µ± =
+C
+¯V ′ �
+1 − 2 ¯V ′�
+� ±2 ¯V ′
+1 − 2 ¯V ′
+�4/3
++ 2f(4 ¯V ′),
+(9)
+where f(x), which is analytic for x < 2, is given by
+f(x) =
+3
+2 − x +
+4x
+(2 − x)7/3
+� 1
+0
+dz
+�2 − xz
+z
+�1/3
+(10)
+and ¯µ± correspond to the two branches ¯µ > 3 and ¯µ < 3,
+respectively. The derivative of the potential ¯V ′ is positive
+(negative) on the former (latter) branch and C is a non-
+negative integration constant.
+¯V (¯µ) is analytic at ¯µ =
+¯µ0 = 3 and ¯V ′(¯µ = 3) = 0.
+In Fig.
+7 of the Suppl.
+Mat. different ¯V ′(¯µ) corresponding to different FPs of
+the BMB line are shown. All FPs along the BMB line
+share the same critical exponents, that is, the exponents
+of the Gaussian FP which is itself tetracritical. Notice
+that the WF FP which corresponds to C = 0, is the end
+point of this line and deserves special attention. We come
+back on this point in the following.
+From Eq. (1), we have seen that λ∗ remains constant at
+leading order in 1/N along the hyperbola of constant ϵN
+of the (d, N) plane. This suggests that when the double
+limit d → 8/3 and N → ∞ is taken at fixed α = ϵN, T3
+converges in d = 8/3 to one of the FPs of the BMB line.
+We have analytically and numerically checked this and
+have derived analytically the relation between α and C:
+α = 162/C3, see Suppl. Mat. and Fig. 11.
+Two extreme cases are worth studying.
+First, the
+Gaussian FP corresponds to the limit N → ∞ at fixed
+dimension d = 8/3, that is, at α = 0. It corresponds
+to C = ∞ in Eq. (9). Second, α = ∞, which implies
+C = 0, corresponds to taking the limit ϵ → 0 at fixed
+N = ∞, that is, to following the WF FP at N = ∞ up
+to d = 8/3. However, at finite ϵ and N = ∞, we know
+from the analysis above that the last term in Eq. (7) can-
+not be neglected. Consistently, the same occurs for the
+BMB line: the WF FP potential is indeed the end point
+of the BMB line obtained by taking the limit C → 0 in
+Eq. (9) but the derivatives of this potential can only be
+studied by retaining the last term in Eq. (7). Here again,
+this explains why the T3 FP in the C → 0 limit is three
+times unstable and not only once unstable.
+To conclude, we have solved the paradox of the appar-
+ent absence of a nontrivial tetracritical FP at N = ∞
+and d < 8/3 by showing that this FP does exist but is
+nothing else than the WF FP up to the subtlety that
+the derivatives of the tetracritical FP potential are not
+the derivatives of the WF FP potential. This makes the
+large-N limit of the O(N) model much less trivial than
+is usually advocated at least for multicritical phenom-
+ena.
+The fact that the tetracritical FP has two more
+unstable infrared directions than the WF FP is related
+to this subtle point because they are associated with sin-
+gular eigenperturbations, a possibility which is usually
+not considered. We conjecture that what has been found
+above at large N and for d ≤ 8/3 is valid for all mul-
+ticritical points with an odd number of eigendirections
+below or at their upper critical dimension because the
+BMB lines for all of them terminate at the WF FP [14],
+a fact that in itself is almost enough to imply everything
+else. Let us finally point out that what we have found for
+the tetracritical FP is very different from what was found
+around d = 3 at large-N in the tricritical case which re-
+quired the existence of new FPs to be fully understood
+at finite N [45–48]. We also conjecture that this phe-
+nomenon is not specific to the O(N) models but should
+rather be generic.
+We acknowledge A. Codello and N. Defenu and for
+correspondence and discussions at an early stage of this
+
+5
+work. S. Y. was supported by Grant-in-Aid for Young
+Scientists (18K13516).
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+
+6
+SUPPLEMENTAL MATERIALS
+I.
+T3 FP POTENTIALS IN d < 8/3
+We show in Fig. 3 the tetracritical FP potential ¯U(¯φ)
+obtained with the LPA and solution of Eq. (6) for small
+values of N. They have the typical shape of a tetracritical
+potential showing two nontrivial minima.
+0
+2
+4
+6
+8 ϕ
+0.375
+0.38
+0.385
+0.39
+U(ϕ)
+N=4.5
+N=1
+FIG. 3. ¯U(¯φ) for the T3 FP for different values of N in d = 2.6.
+II.
+LARGE-N BEHAVIOR OF THE EXTREMA
+OF THE TETRACRITICAL POTENTIAL
+The three nontrivial extrema of the T3 FP potential
+in either the WP or Wetterich version of the RG, shown
+in Fig. 1 of the main text, behave the same way when
+N → ∞. We show on Fig. 4 the scaling in N of the height
+of the barrier between the extrema ¯φi of the rescaled
+potential ¯U(¯φ) of the Wetterich version of the RG, as
+well as the distance between them. These extrema are
+shown in Fig. 1 of the main text.
+2500
+3500
+4500N
+6.0×10-6
+8.0×10-6
+1.0×10-5
+1.2×10-5
+U[ϕ2]-U[ϕ3]
+2500
+3500
+4500N
+0.0325
+0.0350
+0.0375
+0.0400
+0.0425
+0.0450
+0.0475
+ϕ3-ϕ2
+FIG. 4. Left: Height of the potential barrier for the T3 FP
+of Eq. (6) for large values of N in d = 2.6 (blue dots). The
+equation of the full line is y = 0.0257/N. Right: Distance
+between the maximum ¯φ2 and the minimum ¯φ3 for the T3 FP
+of Eq. (6) for large values of N in d = 2.6 (blue dots). The
+equation of the full line is y = 2.12506/N 1/2.
+Since the height of the barrier, ∆ ¯U, scales as N −1
+and the distance between the extrema, ∆¯φ, as N −1/2, a
+simple dimensional argument shows that the curvatures
+at these extrema that goes as ∆ ¯U/(∆¯φ)2, do not scale
+with N, that is, tend to constants when N → ∞, a fact
+that we have numerically checked.
+Thus, for d < 8/3
+and at large and finite N, the curvature of ¯U(¯φ) varies
+between a positive value at ¯φ1, a negative value at ¯φ2
+and again a positive value at ¯φ3 on a distance of order
+N −1/2.
+III.
+THE SCALED BOUNDARY LAYER ˜V ′′(˜µ)
+By translating and rescaling by a factor N 1/2 the po-
+sition and the width of the boundary layer of the second
+derivative of the potential ¯V , it is possible to obtain a
+finite limit for this scaled boundary layer when N → ∞.
+We thus define the scaled variable ˜µ = N 1/2(¯µ − ¯µ0)
+where ¯µ0 is the location of the boundary layer and the
+scaled potential by ˜VN(˜µ) = N
+� ¯V (N −1/2˜µ + ¯µ0) − 1/d
+�
+.
+It follows from the definitions above that ˜V ′′
+N(˜µ) =
+¯V ′′(¯µ0+N 1/2˜µ). We show in Fig. 5 this scaled boundary
+layer for different values of N at large N as well as its
+limit ˜V ′′
+∞(˜µ) at N = ∞.
+-1000
+-500
+500
+1000
+μ˜
+0.02
+0.04
+0.06
+0.08
+0.10
+0.12
+0.14
+N=2.5×104
+N=1.7×105
+N=3.2×106
+˜V''[μ]
+˜
+FIG. 5. The scaled boundary layer for the second derivative
+of the T3 FP potential ˜V ′′
+N(˜µ), Eqs. (7) for large values of N in
+d = 2.6. The dashed curve is the global solution ˜V ′′
+∞(˜µ) of Eq.
+(8) at N = ∞. The red horizontal line is y = (−d2+6d−8)/16
+for d = 2.6. It coincides with ˜V ′′
+∞(˜µ = ∞).
+Notice that a finite difference ¯µ − ¯µ0 translates into an
+infinite ˜µ when N → ∞. The matching at N = ∞ be-
+tween the scaled boundary layer and the value of ¯V ′′(¯µ)
+outside of the layer therefore requires that ˜V ′′
+∞(∞) =
+¯V ′′(¯µ+
+0 ) = (−d2 + 6d − 8)/16 which is the case with the
+solution for the scaled boundary layer given in the main
+text, see also Fig. 5.
+IV.
+THE BOUNDARY LAYER OF ¯U ′′(¯φ)
+The boundary layer has been derived in the main text
+in WP version of the RG because it is simpler in this
+version than in Wetterich version. However, it can also
+be derived directly in this latter version or, once it is
+
+7
+obtained in one version, it can be translated in the other
+by performing the Legendre transform given in Eq. (3).
+0.5
+1.0
+1.5
+2.0
+2.5
+ϕ
+5
+10
+15
+U''[ϕ]
+N=1.5⨯104
+N=4.2⨯105
+N=4.2⨯106
+WF N=∞
+FIG. 6. The second derivative of the T3 FP potential ¯U ′′(¯φ) in
+the Wetterich version of the RG, Eq. (6), for different values
+of N in d = 2.6.
+We show in Fig. 6 the boundary layer of ¯U ′′(¯φ) for
+different values of N at large N.
+V.
+DIFFERENT FP POTENTIALS OF THE
+BMB LINE
+We show in Fig. 7 the first derivative of different FP
+potentials of the BMB line at N = ∞ and in d = 8/3.
+These FP potentials, implicitly given by the exact ex-
+pression given in Eqs. (9) and (10) of the main text, are
+indexed by the nonnegative constant C.
+The WF FP
+potential corresponds to C = 0.
+1
+2
+3
+4
+5
+6
+7
+μ
+-0.20
+-0.15
+-0.10
+-0.05
+0.05
+0.10
+0.15
+V'[μ]
+C=2
+C=0.625
+C=0
+FIG. 7. ¯V ′(¯µ) for different FPs indexed by the constant C on
+the BMB line given by Eqs. (9) and (10) of the main text.
+We emphasize that the limit of ¯V ′′(¯µ0 = 3), when
+C → 0 is not given by the second derivative of the WF
+FP potential which is however the limit when C → 0 of
+¯V (¯µ) along the BMB line. This is consistent with what
+happens at fixed d < 8/3 when N → ∞ since the limit
+d → 8/3 at fixed α = ∞ consists in following the WF FP
+at N = ∞ up to d = 8/3, the derivatives of which are
+not the limit of the derivatives of the T3 potential.
+VI.
+EIGENPERTURBATIONS AT THE
+TETRACRITICAL FP
+FIG. 8.
+d = 2.6: Eigenperturbation δ ¯V3(¯µ) at the T3 FP
+corresponding, when N → ∞, to the relevant eigenvalue λ3 =
+d − 2.
+We show in Figs. 8 and 9 the relevant eigenperturba-
+tions δ ¯Vi of the T3 FP in d = 2.6 for different values of
+N. Whereas δ ¯V3 tends to the relevant eigenperturbation
+of the critical WF FP –with eigenvalue d − 2 which is
+the inverse of the critical exponent νWF–, the two others
+become singular in the N → ∞ limit. This is the rea-
+son why they play no role for the critical behavior of the
+O(N) model at N = ∞.
+1
+2
+3
+4
+5
+6
+7 μ
+-0.03
+-0.02
+-0.01
+0.00
+0.01
+0.02
+0.03
+δV1[μ]
+N=5×102
+N=3×103
+N=1×106
+0
+1
+2
+3
+4
+5
+6
+7 μ
+0.00
+0.01
+0.02
+0.03
+0.04
+δV2[μ]
+N=5×102
+N=3×103
+N=1.4×105
+FIG. 9. d = 2.6: Eigenperturbations δ ¯Vn(¯µ) for n = 1, 2 at
+the T3 FP corresponding respectively to the relevant eigen-
+values λ1 ≃ 2.00 and λ2 ≃ 1.326 for different values of N.
+These eigenperturbations tend to singular functions of ¯µ when
+N → ∞.
+We show in Fig. 10 that the slope of δ ¯V1(¯µ) at ¯µ0 in-
+creases as N 1/3 which proves that this eigenperturbation
+becomes discontinuous at infinite N.
+
+[μ]
+V[]
+0.05
+0.005
+0.04
+0.03
+0.000
+3.0
+35
+4.0
+N=3x103
+0.02
+0.005
+N=3x104
+0.01
+WF N=8
+0.00
+μ
+1
+2
+3
+5
+6
+0.01
+-0.028
+1000
+104
+105
+106
+N
+0.02
+0.05
+0.10
+0.20
+-δV 1'[10/3]
+FIG. 10. d = 2.6: Slope of the eigenperturbation δ ¯V1(¯µ) of
+the T3 FP at its minimum ¯µ0 = 10/3 for different values of
+N. The equation of the full line is y = 0.00175N 1/3.
+VII.
+BMB LINE AND THE JOINED LIMIT ϵ → 0
+AND N → ∞ AT FIXED α = ϵN
+When a T3 FP is followed along the hyperbola d =
+8/3 − α/N, α ≥ 0, of the (d, N) plane, its potential con-
+verges when N → ∞ to the potential of one of the FPs
+of the BMB line, see Fig. 11. We derive below the re-
+lationship α = 162/C3 between the parameter α of the
+hyperbola and the parameter C that indexes the FPs
+along the BMB line, see Eqs. (9) and (10) of the main
+text.
+This relationship can be derived as follows. The FP
+potential of T3 is expanded as
+¯V (¯µ) =
+∞
+�
+n=0
+an(¯µ − 3)n
+(1)
+around the minimum ¯µ0 = 3 of the N = ∞ potential.
+Then, the coefficients an are expanded as
+an = a(0)
+n
++ N −1a(1)
+n
++ O(N −2)
+(2)
+in power of 1/N. At order O(N 0), Eq. (7) yields a(0)
+n
+= 0
+for n = 1, 2 and 3 and recursively determines a(0)
+n
+for n
+larger than 5 in terms of a(0)
+4 .
+Now, �∞
+n=0 a(0)
+n (¯µ−3)n is the expansion of a FP poten-
+tial of the BMB line. For this potential, ¯µ± behaves from
+Eq. (9) as ¯µ± ≃ 3 ± 24/3C| ¯V ′|1/3 for C ̸= 0 and | ¯V ′| ≪ 1.
+This implies that a(0)
+4
+is related to C by a(0)
+4
+= 3/(192C3).
+At order O(N −1), it can be shown that a(1)
+n
+for all n but
+n = 4 can be recursively determined in terms of a(0)
+4
+or C,
+if and only if the condition α = 162/C3 is satisfied which
+proves the relationship between these two parameters.
+We show in Fig. 11 different T3 FP potentials along
+an hyperbola d = 8/3 − α/N with increasing values
+of N.
+These FP potentials converge to a potential
+corresponding to the FP on the BMB line indexed by
+C = (162/α)1/3.
+0
+1
+2
+3
+4
+5
+6
+7
+μ
+-0.10
+-0.05
+0.05
+0.10
+V'[μ]
+C=0.672, N=∞
+N=4000 α=1600/3
+N=8000 α=1600/3
+N=24000 α=1600/3
+FIG. 11. ¯V ′(¯µ) of the T3 FP followed on the hyperbola d =
+8/3 − α/N with fixed α = 1600/3 for increasing values of
+N. In the double limit d → 8/3 and N → ∞, it converges
+to the FP potential of the BMB line corresponding to C =
+(162 × 3/1600)1/3 ≃ 0.672.
+

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+Automated extraction of capacitive coupling for quantum dot systems
+Joshua Ziegler,1, ∗ Florian Luthi,2 Mick Ramsey,2 Felix Borjans,2 Guoji Zheng,2 and Justyna P. Zwolak1, †
+1National Institute of Standards and Technology, Gaithersburg, MD 20899, USA
+2Intel Components Research, Intel Corporation, 2501 NW 229th Avenue, Hillsboro, Oregon 97124, USA
+(Dated: January 23, 2023)
+Gate-defined quantum dots (QDs) have appealing attributes as a quantum computing platform.
+However, near-term devices possess a range of possible imperfections that need to be accounted
+for during the tuning and operation of QD devices. One such problem is the capacitive cross-talk
+between the metallic gates that define and control QD qubits. A way to compensate for the capacitive
+cross-talk and enable targeted control of specific QDs independent of coupling is by the use of virtual
+gates. Here, we demonstrate a reliable automated capacitive coupling identification method that
+combines machine learning with traditional fitting to take advantage of the desirable properties of
+each. We also show how the cross-capacitance measurement may be used for the identification of
+spurious QDs sometimes formed during tuning experimental devices. Our systems can autonomously
+flag devices with spurious dots near the operating regime which is crucial information for reliable
+tuning to a regime suitable for qubit operations.
+I.
+INTRODUCTION
+Quantum dot (QD) arrays, in which charge carriers
+are trapped in localized potential wells and qubits can
+be made by use of the spin and permutation symmetries
+of the carriers, are a promising quantum computing plat-
+form [1–3]. In fact, last year has shown the first demon-
+stration of QD two-qubit gates with fidelities exceeding
+the thresholds for fault-tolerant computing [4–6]. How-
+ever, because the individual charge carriers that make
+up qubits have electrochemical sensitivity to minor im-
+purities and imperfections, calibration and tuning of QD
+devices is a nontrivial and time-consuming process, with
+each QD requiring a careful adjustment of a gate voltage
+to define charge number, and multiple gate voltages to
+specify tunnel coupling between QDs for two-qubit gates
+or to reservoirs for reset and measurement. While manual
+calibration is achievable for small, few-QD devices, with
+increasing size and complexity of QD arrays, the relevant
+control parameter space grows quickly, necessitating the
+development of autonomous tuning methods.
+There have been numerous demonstrations of automa-
+tion of the various phases of the tuning process for sin-
+gle and double-QD devices [7]. Some approaches seek to
+tackle tuning starting from device turn-on to coarse tun-
+ing [8–11] while others assume that bootstrapping (cal-
+ibration of measurement devices and identification of a
+nominal regime for further investigation) and basic tun-
+ing (confirmation of controllability and device character-
+istics) have been completed and focus on a more tar-
+geted automation of the coarse and charge tuning [12–16].
+While the initial auto-tuning approaches relied mainly
+on the appealingly intuitive and relatively easy to imple-
+ment conventional algorithms that typically involved a
+∗ Current address: Intel Components Research, Intel Corporation,
+2501 NW 229th Avenue, Hillsboro, Oregon 97124, USA
+† [email protected]
+combination of techniques from regression analysis, pat-
+tern matching, and quantum control theory, the more re-
+cent algorithms take advantage of the modern computer
+vision and machine learning [7].
+A typical accumulation-mode QD device consists of
+two sets of gates—plungers and barriers—that collec-
+tively control the overall potential profile, QD-specific
+single-particle energy detuning of individual QDs, the
+tunnel couplings between QDs, and tunnel rates between
+the most outer QDs and reservoirs. Ideally, each plunger
+gate would affect only the electrochemical potential of
+a single targeted QD and each barrier gate only one in-
+tended tunnel barrier. Due to the tight proximity, how-
+ever, each gate capacitively couples to nearby potential
+and tunnel barriers. This makes careful control of these
+key parameters challenging.
+One way to compensate for the capacitive cross-talk
+between gates is to enable orthogonal control of the QDs
+potential by implementing so-called virtual gates [17].
+Specifically, linear combinations of gate voltage changes
+can be mapped onto onsite energy differences [17–20].
+These approaches have been key for the initialization and
+control of larger QD arrays [21, 22].
+To autonomously identify capacitive couplings in a
+device, various approaches have been demonstrated us-
+ing both conventional fitting and machine learning (ML)
+techniques [23–26]. However, these approaches, typically
+relying on the Hough transform or conventional least-
+squares fitting procedures, may be unreliable in the pres-
+ence of data imperfections.
+Hough transforms can ex-
+tract slopes directly but may be sensitive to noise or be
+excessively complex to analyze. The conventional fitting
+can be more flexible but is susceptible to local minima
+and can be time-consuming at inference time.
+Convolutional neural networks (CNN) are well suited
+for extracting high-level features from images and can
+remain effective in the presence of noise or other imper-
+fections [27]. However, ML methods can have difficulties
+identifying data outside of the training distribution even
+if it contains similar features [28]. Fortunately, given a
+arXiv:2301.08654v1  [cond-mat.mes-hall]  20 Jan 2023
+
+2
+simplified, high-level representation of the data, conven-
+tional fitting approaches can be more targeted to extract
+key information more effectively and quickly.
+Here we develop a reliable automated capacitive cou-
+pling identification method that combines ML with tra-
+ditional fitting to take advantage of the desirable proper-
+ties of each. We use an ML module for pixel classification
+followed by linear regression for extracting targeted in-
+formation and demonstrate effective performance across
+noise levels and data variations. Testing each of these
+methods on a set of eight simulated QD devices with
+large variability and realistic noise variation mimicking
+experimental conditions shows that the approach com-
+bining ML and traditional fitting works well, with a root
+mean square error (RMSE) of 0.034(14), corresponding
+to a roughly 7 % error, for predicting virtual gate ma-
+trix off-diagonal values (normalizing such that diagonal
+values are one).
+We also demonstrate how the cross-capacitance mea-
+surement may be used for the identification of spurious
+QDs formed during tuning experimental devices. Many
+of the auto-tuning approaches proposed to date rely on
+a series of small 2D scans capturing a relatively narrow
+range of the voltage space [13, 14, 27, 29]. While such ap-
+proaches improve the efficiency of tuning, they may result
+in unexpected and difficult to assess failure modes when
+the tuning algorithm terminates at an anti-crossing with
+a spurious QD that may form in small potential wells due
+to interface defects, surface roughness, or strain within
+the device [30]. They are highly undesirable since they
+may interfere with the QDs intended for use as qubits
+and cannot themselves be used as qubits. To avoid de-
+vice tuning failure, spurious QDs must be identified when
+present and avoided. We test the utility of our approach
+for capacitive coupling estimation by identifying spurious
+QD in experimental measurements of QD devices [1].
+The manuscript is organized as follows: In Sec. II we
+introduce the framework of combining traditional fitting
+techniques with a pixel classifier to process the high-level
+information extracted from experimental data. In Sec. III
+we show the utility of the proposed framework to auto-
+matically extract virtual gates as well as identify charge
+transitions resulting from a formation of spurious QD.
+Finally, in Sec. IV we summarize the results and discuss
+the outlook.
+II.
+METHODS: MACHINE LEARNING AND
+FIT
+Capacitive couplings in a QD device can be measured
+and, in a constant capacitance approximation, described
+by a matrix that maps the physical gate voltages onto
+the effect they each have on the QD’s chemical poten-
+tials or barriers [17, 23, 24, 31–33].
+Measurement of
+the elements of this matrix must be performed distinctly
+for electrochemical potentials and tunnel barriers. Cou-
+plings of the chemical potentials to each QD—which is
+0.30
+0.32
+0.27
+0.29
+VP2(V)
+(a)
+0.8
+1.1
+1.4
+Current (arb. units)
+0.30
+0.32 VP1(V)
+(b)
+NT LT CT RT PL
+(c)
+FIG. 1. An example 2D scan and corresponding pixel classifi-
+cation, class clusters, and linear fits. (a) A simulated voltage
+scan showing left and right transitions as well as a polariza-
+tion line. (b) Pixel classification for the scan shown in (a). (c)
+Regions of pixels and linear fits from the pixel classification.
+The large dark points indicate the centers of pixel regions.
+the focus of this work—can be extracted from shifts in
+charge transition lines when each voltage is varied [17]
+while the effect of each gate on tunnel barriers can be
+assessed by measuring changes in the width of inter-dot
+transitions, assuming the electron temperature is suffi-
+ciently low [32]. Measured this way, the couplings are
+relative, usually scaled with respect to the coupling of
+the QD to the nearest gate. An absolute energy scale
+can be obtained by measuring the gate lever arms with
+photon-assisted tunneling, Coulomb diamonds, or bias
+triangles [34]. However, for establishing the orthogonal
+control the relative scale is sufficient [21].
+For a double QD, the virtualization matrix Gvirt re-
+lating the physical plunger gates to virtual gates can be
+represented by Eq. 1. Each row is normalized such that
+the diagonal entries are 1 to reflect the relative nature of
+our virtual gates.
+Gvirt ≡
+�VP ′
+1
+VP ′
+2
+�
+=
+�
+1
+α12
+α21
+1
+� �
+VP1
+VP2
+�
+(1)
+The relative cross-capacitances for chemical potentials
+manifest themselves via the slopes of charge transition
+lines, with the dominant terms of the cross-capacitance
+matrix determined from a measurement in the space of
+neighboring pairs of gates [21].
+We address the iden-
+tification of the cross-capacitances as captured in two-
+dimensional (2D) plunger-plunger gate scans, as shown in
+Fig. 1(a). To translate the low-level QD data into high-
+level information useful for automation we use a pixel
+classifier, i.e., a CNN model with a structure similar to a
+feature pyramid network [35]. The pixel classifier takes as
+
+3
+an input a small 2D plunger voltage scan obtained using
+a charge sensor, as shown in Fig. 1(a). It then identi-
+fies each pixel within the scan as belonging to one of the
+charge transition classes—left, right, central, or inter-dot
+(polarization line) transition, denoted as LT, RT, CT, or
+PL, respectively—or to the no transition (NT) class. In
+other words, the CNN provides a high-level classification
+of the raw experimental data while keeping spatial infor-
+mation about the relative location and orientation of the
+detected features, which is useful for algorithmic process-
+ing. Figure 1(b) shows the pixel classification of a scan
+from Fig. 1(a).
+To translate pixel classifications to capacitive cou-
+plings, we identify contiguous regions within each class
+of pixels in an image. A labeling algorithm from the mul-
+tidimensional image processing package in SciPy is then
+used to determine the relevant clusters of connected pix-
+els [36]. This separates fragments of charge transitions
+into distinct clusters so that each can be processed indi-
+vidually. Each region of pixels classified as LT, CT, or
+RT is then independently fitted using linear regression, as
+shown in Fig. 1(c). When multiple segments for a given
+class are present in an image, the capacitive coupling re-
+turned is the average for all fitted lines weighted by the
+standard deviations of each fit, yielding the solid lines in
+Fig. 1(b) (offset arbitrarily for comparison with the pixel
+regions). Standard deviations σ are computed from the
+standard error of the fit, S, by σ = S/√n, where n is the
+size of the pixel region, as in Student’s t-distribution [37].
+In addition, each region is tagged with its center in volt-
+age space, shown by the large black points in Fig. 1(c),
+which allows tracking the changes in charge transitions
+and their slopes within the larger space.
+A.
+Data
+The data used for training the ML tools and testing
+the methods was generated using a simulation of QD de-
+vices [12]. The simulation is composed of a calculation of
+the electron density in the Thomas-Fermi approximation
+and a capacitance matrix to determine the stable electron
+configuration. To improve the robustness of the models,
+the data is augmented with synthetic white, pink (1/f),
+and telegraph noise [27]. The effect of a QD charge sen-
+sor strongly coupled to the plunger gates is varied during
+the scan to improve performance on this type of experi-
+mental data.
+The training dataset consists of 1.6 × 105 devices with
+parameters varied over a uniform distribution with a
+standard deviation equal to 1 % of each parameter’s
+value. To train the ML models we randomly sample 10
+small scans per device and use charge state ground truth
+to label each scan on a pixel level with the presence and
+type of transition, yielding NT, LT, CT, RT, and PL la-
+bels. Additionally, we extract the slopes of the transition
+lines directly using the gradients of the device charge.
+The test data is composed of eight simulated devices
+with large variations in screening length and device pitch
+1×
+5×
+10×
+15×
+20×
+25×
+30×
+35×
+0.0
+0.2
+0.4
+0.6
+RMSE
+(a)
+1×
+15×
+30×
+0.0
+0.2
+0.4
+0.6
+(b)
+1×
+15×
+30×
+Noise Level
+(c)
+FIG. 2. (a) Root mean square error (RMSE) for all transition
+classes (left, central, and right [LT, CT, RT]) as a function
+of the synthetic noise level. (b) RMSE as a function of noise
+level for the LT class. (c) RMSE as a function of noise level
+for the RT class.
+and with large shifts in the position of one of the plunger
+gates. These changes lead to large variations in the slopes
+of the charge transition lines, the capacitive coupling
+between QDs, spacing between lines, and the relative
+amount of left and right QD, making them largely dis-
+tinct from the training data. To facilitate a controlled
+study and track the performance of the pixel classifier
+as data quality degrades, each large scan is randomly
+sampled 50 times and the resulting small scans are aug-
+mented with increasing levels of synthetic noise.
+This
+results in a set of 400 simulated test scans. Finally, sev-
+eral large experimental measurements acquired using a
+double-QD configuration on a three-QD Six/SiGe1−x de-
+vice, fabricated on an industrial 300 mm process line [1],
+are used to test the performance of the virtualization
+algorithm. Experimental scans capturing spurious QDs
+are used to demonstrate the algorithm for spurious QD
+detection.
+III.
+RESULTS
+We test the effectiveness of our automated approach to
+extracting the cross-capacitance by first evaluating the
+performance of each component, i.e., the pixel classifier
+and the slope extractions, on each scan in the simulated
+test set. The error of the pixel classifier in our frame-
+work is defined as a fraction of pixels belonging to true
+transitions that are not contained in line segments in the
+CNN output.
+This captures type-I errors without the
+
+4
+2.00
+2.20
+VP1(V)
+2.20
+2.40
+VP2(V)
+(a)
+1.40
+1.55 VP ′
+1(V)
+1.50
+1.60
+1.70
+VP ′
+2(V)
+(b)
+1.20
+1.35 VP ′′
+1 (V)
+1.20
+1.30
+VP ′′
+2 (V)
+(c)
+−0.8
+−0.6
+−0.4
+−0.2
+Virt. gate
+off diag.
+FIG. 3. (a) Large experimentally measured charge stability diagram with a scatter plot of centers of pixel class regions overlaid.
+The colors of the points indicate the virtual gate off-diagonal values identified by fits to the region. The sizes of the points
+indicate the weights used when averaging. Only points with relative error less than 20 % are plotted. (a,b) Charge stability
+diagram after applying virtual gates acquired near the (0, 0) − (1, 1) charge transition in (a) and near the (1, 3) − (2, 4) charge
+transition in (b). In both (b) and (c) the virtualization is performed off-line, via an affine transform to the original scan shown
+in (a) and the points are plotted using the same parameters as in (a).
+effect of false type-II errors due to imperfect labels [38].
+Figure 2(a) shows the change in RMSE as a function of
+the noise level in the simulated data. At the noise level of
+1.0, i.e., the noise level estimated from experimental data
+in Ref. [29], we observe an RMSE of 0.17(5). The RMSE
+increases significantly to 0.50(11) at the noise level of 20.
+For reference, a pixel classifier that always predicts the
+NT class would have an RMSE of 0.62 (
+√
+0.4). For the
+LT and RT classes relevant to cross-capacitances, shown
+in Fig. 2(b) and (c), the pixel classifier for noise level 1.0
+has an RMSE of 0.20(8) and 0.11(8), respectively.
+To verify that the slope extraction tool works as in-
+tended, we test it across the eight large simulated test
+devices. For these tests, we evaluate the pixel classifier
+in windows of size roughly 1.5× the charging energy, as
+estimated by the spacing of the first two charge transi-
+tions. Outputs from the pixel classifier are cropped by
+one pixel from the edge of the image before processing
+due to missing context reducing CNN performance [39].
+The resulting classes of pixels are then grouped into dis-
+tinct clusters. For each cluster consisting of more than
+five pixels an independent linear fit is performed, return-
+ing both the slope and the standard error of the fitted
+line. This information can be used to find the orthog-
+onal “virtual” control space or to flag transitions that
+potentially belong to spurious dots, as described in the
+following sections.
+A.
+Deriving virtual gates
+As stated in Sec. II, in our approach the off-diagonal
+elements, defining the virtual gates transformation, are
+determined based on the slopes of the LT and RT cap-
+tured in a given image, and the diagonal elements of the
+capacitance matrix are set to 1.0. When multiple lines
+belonging to the same class are detected, as in Fig. 1(a),
+the capacitive coupling is calculated through a weighted
+average, with the weight accounting for both the size
+of the clusters and the standard deviations of respective
+fits [37].
+The off-diagonal elements of the virtualization matrix
+computed this way have an RMSE of 0.034(14) at the
+noise level of 1.0 defined in Ref. [29], corresponding to a
+roughly 8 % error compared to the ground truth values
+derived from simulated data. We further test them on
+a range of levels of synthetic noise and find the RMSE
+rises by a factor of two at a level of noise of roughly 15×
+the level of noise defined in Ref. [29], consistent with the
+pixel classifier error.
+To better understand the trends of the virtualization
+matrix in the plunger-plunger space, we carry out a per-
+formance analysis using the test set of eight large sim-
+ulated charge stability diagrams and several experimen-
+tally measured scans.
+For each scan, we calculate the
+fits to the pixel classification clusters based on a series of
+small scans sampled at each point within the large scan
+with the exclusion of a margin implemented to ensure
+that all sampled scans fall within the full scan bound-
+aries. The small scans and the margins are set to have a
+size 1.5× the charging energy of a given simulated device.
+Figure 3(a) shows the centers of the pixel region identi-
+fied in each small scan [as in Fig. 1(c)] as the sampling
+window is swept across a large experimentally measured
+charge stability diagram. The regions identified by the
+pixel classification are consistently placed correctly on
+the charge transition lines regardless of the position of the
+
+5
+2.09
+2.18 VP1(V)
+0.00
+0.25
+0.50
+Rel. virt. gate off-diag.
+(a)
+2.26
+2.34 VP2(V)
+(b)
+0
+1
+2
+3
+4
+Density
+×10−2
+FIG. 4. Histograms of the off-diagonal elements of the virtu-
+alization matrix for an experimentally measured scan shown
+in Fig. 3(a) as a function of plunger gates, (a) VP1 and (b)
+VP2. Off-diagonal values are normalized to the mean of the
+virtual gates in the (1, 1) charge state for ease of comparison.
+Virtual gate values are extracted from a strip of small scans
+shifted by 6 mV (four pixels) to better visualize variation at
+each plunger gate value.
+line within a small scan. Region centers shift along the
+charge transition lines as different portions of the line are
+captured within the small scan and remain fixed when-
+ever the same fragment of the charge transition is cap-
+tured. The color of the points indicates the off-diagonal
+values of the virtual gate matrix, α12 and α21. As ex-
+pected, these coupling constants get larger in magnitude
+as charges are added to each QD. Finally, the size of the
+points in Fig. 3(a) indicates the 1/σ2 weight of the slopes
+used when averaging multiple slopes from the same type
+of transition within a small scan. As desired, the posi-
+tions of the points with smaller sizes indicate that lines
+that are smaller or less captured within a small scan have
+fits with larger errors. Overall, this plot confirms that the
+pixel classification and the fits are working as intended
+at capturing charge transition lines and their slopes.
+To demonstrate the spatial relevance of the virtual
+gates derived from a set of fits across a device’s charge
+landscape, in Fig. 3(b) and (c) we plot affine-transformed
+charge stability diagrams, with points indicating fits
+overlaid. The points plotted are the centers of pixel re-
+gions with colors indicating the α12 and α21 values and
+size indicating the inverse of the fit error squared (the
+weight of the fit in the average).
+The affine transfor-
+mation applied in Fig. 3(b) corresponds to virtual gates
+derived from an image near the (0, 0) − (1, 1) charge
+transition with off-diagonal values α12 = −0.282(4),
+α21 = −0.331(4). For Fig. 3(c), the affine transformation
+applied has virtual gates from the (1, 3) − (2, 4) charge
+transition, with off-diagonal values α12 = −0.363(4),
+α21 = −0.480(4). As can be seen in the insets in Fig. 3(b)
+and (c), these virtual gates are very effective at trans-
+forming the target region to an orthogonal space, but the
+difference between the extracted virtual gate off-diagonal
+values are about 50 % higher for the latter case. This
+highlights the importance of an efficient local method for
+determining virtual gates.
+To further understand how capacitive coupling varies
+across a charge stability diagram, we can calculate varia-
+tion as each plunger gate is adjusted. Figure 4(a) and (b)
+show how virtual gates extracted from small scans change
+as VP1 and VP2 are varied. To better show the trend, vir-
+tual gates from small scans shifted by 3 mV (two pixels)
+in the opposing direction are included. This shows that
+the virtual gates extracted from small scans effectively
+capture variation across charge stability diagrams.
+B.
+Detection of spurious dots
+Visually, spurious QDs are recognized in large 2D scans
+as charge transitions with slopes diverging from a mono-
+tonic trend, see Fig. 5(a). In this framework, they may
+be identified as transition lines with anomalous capaci-
+tive couplings relative to the transitions around them.
+As a demonstration, we use the pixel classification and
+fit tools to analyze five experimental charge stability di-
+agrams: two capturing properly formed QD, shown in
+Fig. 5(a) and (b), and three capturing spurious QDs,
+shown in Fig. 5(a), (b), and (c). While for extraction
+of the virtualization matrix small scans are sufficient, de-
+tection of spurious QD requires somewhat bigger scans
+to ensure that the neighboring charge transitions are ad-
+equately captured. In our analysis, we rely on 2D scans
+of a size roughly three times the charging energy (four
+times the area of scans typically used in auto-tuning al-
+gorithms [13, 14]). We also consider only clusters con-
+sisting of at least 20 pixels to ensure better reliability of
+the linear fit.
+After pixel classification, contiguous clusters of pixels
+belonging to a given class of transitions are analyzed in-
+dividually, resulting in a cluster-based fit and standard
+deviation. Cases where more than one cluster belongs
+to a given charge transition result in separate fits, as in
+Fig. 5(b) and (e) where the LT lines are split into groups
+to either side of the RT lines. This separation serves two
+purposes: to ensure that variation along a given transi-
+tion isn’t included and to treat each additional line inde-
+pendent of the charge on another QD.
+Within a class and group of transitions, the magni-
+tude of the capacitive coupling is expected to increase
+monotonically as the charge is added. Such behavior is
+clearly visible in Fig. 5(k) and (l), with the latter having
+to separate groups of fits (shown with different shades of
+green) for the groups of clusters. On the contrary, a spu-
+rious QD manifests itself by a non-monotonic behavior of
+the capacitive coupling between transitions, as depicted
+graphically by the center point (or group of points) in
+Fig. 5(m), (n), and (o). The severity of this divergence
+can be quantified using a Z-test [40].
+In practical applications, the automated detection of
+spurious QD fits nicely within the auto-tuning paradigm.
+As mentioned earlier, many of the proposed approaches
+utilize a series of small 2D scans [13, 14, 27, 29] or
+1D rays [41, 42] as means to improve the tuning effi-
+
+6
+2.26
+2.32
+2.33
+2.39
+VP2(V)
+(a)
+2.26
+2.32
+2.33
+2.39
+(f)
+2.33
+2.39
+-0.31
+-0.28
+-0.26
+α12
+(k)
+2.38
+2.44
+2.31
+2.37
+VR1
+(b)
+0.6
+0.8
+1.0
+1.2
+Current
+(arb. units)
+2.38
+2.44
+2.31
+2.37
+(g)
+NT
+LT
+CT
+RT
+PL
+2.31
+2.37
+-0.4
+-0.3
+-0.2 (l)
+1.89
+1.96
+2.18
+2.24
+(c)
+1.89
+1.94
+2.18
+2.24
+(h)
+2.18
+2.25
+-0.4
+-0.3
+-0.2 (m)
+0.42
+0.46
+0.46
+0.50
+(d)
+0.42
+0.46
+0.46
+0.50
+(i)
+0.46
+0.50
+-0.5
+-0.4
+-0.3 (n)
+0.48
+0.52
+0.45
+0.49
+VR2
+(e)
+0.7
+0.8
+0.9
+1.0
+Current
+(arb. units)
+0.48
+0.52 VP1(V)
+0.45
+0.49
+(j)
+NT
+LT
+CT
+RT
+PL
+0.45
+0.49 VP2(V)
+-0.6
+-0.4
+-0.2 (o)
+FIG. 5. Spurious dot detection. The top row shows two charge stability diagrams capturing properly formed QD [panels (a)
+and (b)] and three charge stability capturing spurious QD [panels (c), (d), and (e)]. The black boxes in (b) and (e) in highlight
+small 2D scans, denoted as VR1 and VR2, typical of the auto-tuning approaches proposed in Ref. [13, 14]. Panels (f)–(j) in the
+middle row show pixel classification results for charge stability diagrams shown in the top row. Plots of fitting results used to
+determine whether a spurious QD is present are shown in the bottom row [panels (k)–(o)]. The different groups of transition
+are shown with different shades of green. The monotonicity within each group of transitions is clearly visible in panels (k) and
+(l). On the contrary, in the three plots shown in panels (m), (n), and (o), there is a clear divergence from the expected trend
+for the spurious QD (middle) transitions. Error bars indicate one standard deviation.
+ciency.
+While these approaches deliver measurement-
+cost-effective solutions, they are prone to unexpected and
+difficult-to-detect failure even when the data quality is
+high. Fig. 5(b) and (e) show examples of such potentially
+problematic cases. The small 2D regions in the plunger-
+plunger space, highlighted in these scans with the black
+boxes, are typical for topology setting algorithms.
+In
+both cases, they are classified by a state classifier model
+as double QD state, with state prediction vectors being
+p(VR1) = [0.01, 0.04, 0., 0.04, 0.92] for VR1 region high-
+lighted in Fig. 5(b) and p(VR2) = [0., 0., 0.18, .05, 0.76]
+for region VR2 highlighted in Fig. 5(b), where p(VR) =
+[pND, pSDL, pSDC, pSDR, pDD] with ND denoting no QDs
+formed, SDL, SDC, and SDR denoting the left, central,
+and right single QD, respectively, and DD denoting the
+double-QD state. Moreover, the data quality for these
+images is high in both cases, with q(VR1) = [1.0, 0.0, 0.0]
+for region VR1 and q(VR1) = [0.99, 0.01, 0.0] for VR2,
+where q(VR) = [phigh, pmod, plow] with phigh, pmod, and
+plow denoting the probability of region VR being assessed
+by the data quality control module as “high,”, “moder-
+ate,” and “low” quality, respectively. Thus, from the ML
+perspective, both these predictions are confidently cor-
+rect. However, when looked at within a slightly larger
+voltage range, it is clear that in the latter case the
+small scan captures an anti-crossing with a spurious QD,
+which for practical tuning purposes is a failure. If not
+recognized and corrected for, termination at this point
+will result in an incorrect charge setting and virtualiza-
+tion [13, 29].
+The spurious QD detection algorithm can be easily
+implemented in the auto-tuning algorithm proposed in
+Ref. [27] as a safety check before the unloading step is ini-
+tiated. An automated identification and characterization
+of spurious QDs may also be useful to inform fabrication
+procedures and prevent them in future devices [30].
+IV.
+CONCLUSIONS
+As quantum dot devices grow in size and complexity,
+the need for reliable and automated tune-up procedures
+becomes more pressing.
+Establishing orthogonal con-
+trol of the chemical potentials of quantum dots is one
+of the first steps in the tune-up of any larger quantum
+dot array. Here, we demonstrated a method that com-
+bines machine-learning-based pixel-classification and tra-
+ditional curve fitting to reliably determine voltage cross-
+talk coefficients. The advantage of this method over pre-
+vious approaches is highlighted by increased reliability
+
+7
+and resilience to experimental noise.
+Further on, un-
+wanted spurious dots that would reduce or inhibit de-
+vice performance can be detected and flagged when this
+module is used as part of a larger tune-up algorithm [29].
+The capability to automatically and reliably detect spuri-
+ous dots is especially important on wafer-scale fabrication
+characterization tools that produce more data than can
+efficiently be processed by human analysis. In extensions,
+our tools could allow for automated navigation of volt-
+age space for more targeted measurement of all chemical
+potential and tunnel barrier cross-capacitances [17, 32].
+ACKNOWLEDGMENTS
+This research was performed while J.Z. held a NRC
+Research Associateship award at the National Institute
+of Standards and Technology (NIST). The views and con-
+clusions contained in this paper are those of the authors
+and should not be interpreted as representing the of-
+ficial policies, either expressed or implied, of the U.S.
+Government.
+The U.S. Government is authorized to
+reproduce and distribute reprints for Government pur-
+poses notwithstanding any copyright noted herein. Any
+mention of commercial products is for information only;
+it does not imply recommendation or endorsement by
+NIST.
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+

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+Many-body collective neutrino oscillations:
+recent developments
+Amol V. Patwardhan, Michael J. Cervia, Ermal Rrapaj, Pooja Siwach,
+A. B. Balantekin
+Abstract Neutrino flavor transformations in core-collapse supernovae and binary
+neutron star mergers represent a complex and unsolved problem that is integral to
+our understanding of the dynamics and nucleosynthesis in these environments. The
+high number densities of neutrinos present in these environments can engender var-
+ious collective effects in neutrino flavor transformations, driven either by neutrino-
+neutrino coherent scattering, or in some cases, through collisional (incoherent) in-
+teractions. An ensemble of neutrinos undergoing coherent scattering among them-
+selves is an interacting quantum many-body system—as such, there is a tantalising
+prospect of quantum entanglement developing between the neutrinos, which can
+leave imprints on their flavor evolution histories. Here, we seek to summarize re-
+cent progress that has been made towards understanding this phenomenon.
+Amol V. Patwardhan
+SLAC National Accelerator Laboratory, 2575 Sand Hill Rd, Menlo Park, CA 94025
+e-mail: [email protected]
+Michael J. Cervia
+George Washington University, 725 21st St NW, Washington, DC 20052
+e-mail: [email protected]
+Ermal Rrapaj
+University of California, Berkeley, CA 94720-7300
+e-mail: [email protected]
+Pooja Siwach
+University of Wisconsin, 1150 University Ave, Madison, WI 53706
+e-mail: [email protected]
+A.B. Balantekin
+University of Wisconsin, 1150 University Ave, Madison, WI 53706
+e-mail: [email protected]
+1
+arXiv:2301.00342v1  [hep-ph]  1 Jan 2023
+
+2
+Amol V. Patwardhan, Michael J. Cervia, Ermal Rrapaj, Pooja Siwach, A. B. Balantekin
+Motivation: Supernovae, Mergers, and the Early Universe
+In extreme astrophysical environments such as core-collapse supernova explosions,
+and binary neutron star (or black hole - neutron star) mergers, as well as during
+certain epochs in the early universe, neutrinos dominate the transport of energy,
+entropy, and lepton number (for example, see [Janka et al., 2007, Burrows and
+Vartanyan, 2021, Fuller and Haxton, 2022, Foucart, 2022, Kyutoku et al., 2018,
+Grohs et al., 2016], etc.). The key processes governing neutrino transport in these
+environments are electron neutrino and antineutrino captures on nucleons, i.e.,
+νe +n ⇌ p+e−
+(1)
+¯νe + p ⇌ n+e+
+(2)
+A consequence of the typical temperatures and densities of these environments is
+that neutrinos decouple with energies of O(1–10)MeV, and therefore, the µ and τ
+flavor (anti-)neutrinos are unable to participate in these charged-current processes,
+due to there not being enough energy to produce µ and τ leptons in the final state.
+Given the importance of these processes in the energy transport, as well as in de-
+termining the neutron-to-proton ratio and the resulting nucleosynthesis prospects
+(e.g., [Surman and McLaughlin, 2004, Mart´ınez-Pinedo et al., 2017, Kajino et al.,
+2014, Fr¨ohlich et al., 2015, Langanke et al., 2019, Roberts et al., 2017, Steigman,
+2012, Grohs et al., 2016]), the flavor-asymmetric nature of charged-current capture
+necessitates a thorough understanding of neutrino flavor evolution in these envi-
+ronments. The potential impact of neutrino flavor evolution on nucleosynthesis has
+already been studied in various contexts (e.g., [Qian et al., 1993, Yoshida et al.,
+2006, Duan et al., 2011, Kajino et al., 2012, Wu et al., 2015, Sasaki et al., 2017, Bal-
+antekin, 2018, Xiong et al., 2019, Xiong et al., 2020]).
+In what follows, we shall summarize recent progress in our understanding of
+a particular facet of neutrino oscillations in extreme astrophysical environments—
+namely, the quantum many-body nature of collective neutrino oscillations engen-
+dered by ν-ν interactions in dense neutrino streams.
+Introduction to collective neutrino oscillations
+The neutral current weak term of the Standard Model (SM) allows neutrinos to
+interact pairwise via virtual Z-boson exchange or, more simply, in the low-energy
+effective theory, via the Fermi four-point interaction
+Hint ≡ GF
+√
+2 ∑
+f,g
+νgγµνgν f γµνf ,
+(3)
+where f,g span the flavor state indices. The relevance of these interactions in en-
+vironments where the number densities of neutrinos are comparable to (or larger
+
+Many-body collective neutrino oscillations: recent developments
+3
+than) those of charged leptons, e.g., in core-collapse supernovae, binary neutron
+star mergers, as well as in the early universe, had been discussed in [Notzold and
+Raffelt, 1988, Fuller et al., 1987]. But the extent of their importance in changing
+the flavor content of neutrinos, via diagonal and off-diagonal contributions to the
+neutrino Hamiltonian, was not fully recognized until later [Pantaleone, 1992a, Pan-
+taleone, 1992b, Samuel, 1993].
+Considering pairs of neutrinos with well-defined incoming momenta p and q
+(i.e., plane wave states) and the same pair of outgoing momenta (i.e., “forward
+scattering” neutrinos, the contributions of which can be added coherently), the off-
+diagonal matrix elements of the interaction Hamiltonian Hint may be interpreted as
+arising from “flavor swaps” between neutrino pairs (in the flavor basis). Because
+the off-diagonal term exchanges flavor between the “test” and the “background”
+neutrinos, the flavor evolution of the interacting neutrinos constitutes a many-body
+problem, potentially rendering the one-particle propagation formalism [Samuel,
+1993, Sigl and Raffelt, 1993, Qian and Fuller, 1995] inadequate for describing
+the resulting dynamics. Notably, the interaction Hamiltonian Hint does not com-
+mute with the Hamiltonian terms corresponding to flavor oscillations in vacuum
+and neutrino interactions with background matter. Consequently, in a regime where
+the strength of these terms is comparable in scale to the ν-ν interaction strength,
+diagonalizing this Hamiltonian is not straightforward and the many-body problem
+acquires a nontrivial nature. Here, the entire Hilbert space of N interacting neutrinos
+and antineutrinos in nf flavors has dimension nN
+f .
+Despite emphasis on the high nonlinearity of this problem, [Samuel, 1993] had
+proposed that a statistical mechanical approach, whereby a two-flavor neutrino den-
+sity matrix is treated as interacting with a background of neutrinos and antineutri-
+nos, could describe the evolution of a dense neutrino gas for certain portions of this
+parameter space. This analysis was extended by [Sigl and Raffelt, 1993] to nf ≥ 2
+flavors with proposed evolution of nf ×n f density matrices via quantum Boltzmann
+equations, including collision integrals as well as more general, potentially non-SM
+coupling between flavors. In these treatments, the collisional contributions can lead
+to a nontrivial loss of coherence being reflected in the density matrices of individual
+neutrinos. However, the ability to calculate multi-body wave functions that exhibit
+ν-ν correlations is relinquished, in exchange for a more favourable scaling of com-
+putational complexity with the number of neutrinos in the simulation. Along these
+lines, [Qian and Fuller, 1995] proposed a physical ansatz that the wave function of
+the ensemble is not a coherent many-body state, but simply composed of single-
+neutrino wave functions with random relative phases, to be summed incoherently,
+called the Random Phase Approximation (RPA). In this way, each neutrino density
+matrix is taken to be pure, and the effective Hilbert space dimension is reduced to
+nf N. In kind, the complexity of collective oscillations calculations becomes greatly
+simplified. This ansatz amounts to a “mean field approximation” wherein expecta-
+tion values of operator products may be replaced by products of the individual op-
+erator expectation values. Notably, this physical description of neutrinos expressly
+prohibits the quantum entanglement between neutrinos. As such, assessing the va-
+lidity of this ansatz involves determining the extent to which quantum effects are
+
+4
+Amol V. Patwardhan, Michael J. Cervia, Ermal Rrapaj, Pooja Siwach, A. B. Balantekin
+needed to correct this approximation. In this chapter, we discuss recent progress
+along this front.
+Before delving into the chapter, we mention in passing that recent years have seen
+a rapid growth of interest in flavor instabilities and resulting fast flavor oscillation,
+even within the scope of the mean field approximation. For more information we
+refer the reader to the chapter on “Fast Flavor Transformations” by [Richers and Sen,
+2022], or the review articles by [Chakraborty et al., 2016, Tamborra and Shalgar,
+2021].
+Many-body Hamiltonian for interacting neutrinos
+The Hamiltonian describing a system of interacting neutrinos can be written in terms
+of generators of SU(nf ), and it possesses a SU(nf ) rotation symmetry in neutrino
+flavor. A significant feature of ν-ν interactions is the dependence of the interac-
+tion strength on the intersection angle between their trajectories. This dependence
+introduces a geometric complexity to the problem, in addition to the complexity
+associated with the exponential scaling of the Hilbert space.
+For simplicity, if we consider neutrino mixing between only two flavors, νe and
+νx, then a Hamiltonian consisting of terms that represent vacuum mixing as well as
+ν-ν interactions can be written as
+H = ∑
+p
+ωp⃗B· ⃗Jp +
+√
+2GF
+V
+∑
+p,q
+(1−�p·�q) ⃗Jp · ⃗Jq ,
+(4)
+where ⃗B = (0,0,−1) in the mass-basis representation, and ωp = δm2/(2|p|) are
+the vacuum oscillation frequencies for neutrinos with momenta p. Here, �p and �q
+are the unit vectors along the momenta of the interacting neutrinos, and V is the
+quantization volume. For ease of notation, one can define a ν-ν coupling parameter
+µ ≡
+√
+2GFN/V, where N is the total number of interacting neutrinos. The oper-
+ators ⃗Jp represent the neutrino “isospin” in the mass basis, where isospin up and
+down correspond to the mass basis states |ν1⟩ and |ν2⟩. In this depiction, ⃗B can be
+interpreted as a “background field” with which the neutrino isospins interact. Here,
+we exclude the term representing neutrino interactions with ordinary matter (e.g.,
+charged leptons), since it has a structure that is conceptually similar to the vacuum
+oscillation term—i.e., consisting of individual neutrinos interacting with a back-
+ground. In contrast, the ν-ν interaction term consists of pairs of neutrino isospins
+interacting with each other.
+In terms of the Fermionic creation and annihilation operators, the neutrino
+isospins are described as [Balantekin and Pehlivan, 2006]
+J+
+p = a†
+1(p)a2(p) ,
+Jz
+p = 1
+2
+�
+a†
+1 (p)a1(p)−a†
+2 (p)a2(p)
+�
+,
+(5)
+
+Many-body collective neutrino oscillations: recent developments
+5
+with J−
+p = (J+
+p )†. In the spin-1/2 representation, one can write the isospin operators
+in terms of Pauli matrices: i.e., ⃗Jp = ⃗σp/2, where σp is a vector of Pauli matrices
+defined in the subspace of the neutrino with momentum p.
+Path integral formulation
+An assessment of quantum corrections to a mean field picture can in principle be
+performed via a coherent state analysis, as formulated by [Balantekin and Pehlivan,
+2006]. Schematically, in this procedure, one seeks to calculate the matrix elements
+of the time evolution operator U(tf ;ti) for a single neutrino in the basis of SU(nf )
+coherent states |z⟩ for neutrinos (and/or antineutrinos) in n f flavors, equivalent to a
+path integral
+⟨zf |U(t f ;ti)|zi⟩ =
+�
+D[z,z∗] exp(iS[z,z∗])
+(6)
+of the derived action
+S[z,z∗] =
+� tf
+ti
+dt
+�
+⟨z(t)|(i∂t −H)|z(t)⟩−ilog⟨z f |zi⟩
+�
+,
+(7)
+where H is the Hamiltonian of the many-body system. A saddle-point approxima-
+tion of the resulting path integral yields the classical action, which is in complete
+agreement with the RPA used to derive the mean field theory for collective neutrino
+oscillations. However, in this perspective, analyzing quantum corrections to this ap-
+proximation entails a careful analysis of the Hessian matrix of the action integral
+derived from this procedure. Such mathematical analysis has not yet been presented
+to date.
+Beyond the Mean-Field: Entanglement, Correlations, and
+Dynamical Phase Transitions
+Early literature
+The seminal work describing the ν-ν interaction Hamiltonian from Eq. (3) recog-
+nized that these interactions give rise to a quantum many-body problem, which may
+not in the general case be factorizable in terms of a one-particle effective approxi-
+mation [Pantaleone, 1992a, Pantaleone, 1992b]. Subsequently, there were some at-
+tempts to ascertain the validity of the one-particle effective approximation [Bell
+et al., 2003, Friedland and Lunardini, 2003b, Friedland and Lunardini, 2003a, Fried-
+land et al., 2006]. In these works, the flavor evolution of interacting neutrinos was
+analyzed with two different approaches: (i) using two intersecting beams of neutri-
+
+6
+Amol V. Patwardhan, Michael J. Cervia, Ermal Rrapaj, Pooja Siwach, A. B. Balantekin
+nos, where the flavor evolution was described in terms of a sequence of elementary
+scattering amplitudes, and (ii) using a neutrino ensemble represented as interacting
+plane waves in a box.
+Following initial disagreement regarding whether substantial quantum entangle-
+ment can develop among interacting neutrinos [Bell et al., 2003, Friedland and Lu-
+nardini, 2003b], it was subsequently concluded that the build-up of entanglement
+and resulting flavor conversion would occur on timescales whose scaling is sugges-
+tive of incoherent effects [Friedland and Lunardini, 2003a]. These conclusions were
+further generalized in [Friedland et al., 2006]. However, these analyses nevertheless
+involved several simplifications, most notably, the omission of the one-body terms
+in the Hamiltonian. The interplay between vacuum oscillations and ν-ν interaction
+terms has been shown to give rise to interesting collective phenomena such as “spec-
+tral splits” [Duan et al., 2006a, Duan et al., 2006b, Duan et al., 2007b, Raffelt and
+Smirnov, 2007b, Raffelt and Smirnov, 2007a], even in the mean-field approxima-
+tion. Therefore, studying the quantum many-body dynamics of collective neutrino
+oscillations, with both one- and two-body terms fully incorporated, remains an in-
+teresting problem.
+With these seemingly conflicting results in the past predicting either a vanish-
+ingly small contribution in the large system size limit [Friedland and Lunardini,
+2003a, Friedland et al., 2006] or substantial flavor evolution over time scales τF ∼
+µ−1 log(N) that can remain relevant for large systems [Bell et al., 2003, Sawyer,
+2004], the role of entanglement and quantum effects in the out-of-equilibrium dy-
+namics [Eisert et al., 2015] of neutrinos has received renewed interest recently (e.g.,
+[Cervia et al., 2019, Rrapaj, 2020] and subsequent works mentioned later in this
+chapter). Note that flavor oscillations on the time scale τF can be considered to be
+“fast”, different from “slow” oscillations occurring over τL ∼ µ−1√
+N. In the lit-
+erature on collective flavor effects in the mean field approximation, one can more
+commonly find “fast” and “slow” oscillations associated with time scales ∼ µ−1
+and ∼ √µω (or ω), respectively.
+Single-angle approximation, invariants, and integrability
+To circumvent the geometric complexity of the problem, the frequently-employed
+single-angle approximation replaces the angle-dependent (i.e., �p,�q-dependent) in-
+teraction strengths among pairs of neutrinos with a single, appropriately chosen
+classical average over the various neutrino trajectories. In this limit, one can de-
+fine a trajectory-averaged interaction parameter µ ≡ (
+√
+2GFN/V)⟨1 − �p · �q⟩, and
+approximate the Hamiltonian as
+H = ∑
+ωp
+ωp⃗B· ⃗Jωp + µ
+N
+⃗J · ⃗J ,
+(8)
+
+Many-body collective neutrino oscillations: recent developments
+7
+where ⃗J = ∑ωp ⃗Jωp is the total neutrino isospin. Note that, in this limit, the neutrino
+flavor state becomes trajectory-independent, introducing a considerable simplifica-
+tion in the problem. As a result, the neutrinos may be indexed simply by the mag-
+nitudes of their momenta (or equivalently, by their vacuum oscillation frequencies
+ωp), rather than by the momenta themselves (magnitude and direction). The ν-ν
+coupling in general will depend on time. In the context of supernovae, a commonly
+employed expression for µ is derived from the spherically symmetric single-angle
+neutrino bulb model, first described in [Duan et al., 2006c]:
+µ(r) = µ0
+�
+�1−
+�
+1−
+�Rν
+r
+�2
+�
+�
+2
+,
+(9)
+where r is the distance from the center of a “neutrino-sphere” of radius Rν, which
+represents a sharp surface where neutrinos decouple from nuclear matter and be-
+gin free streaming outwards from the proto-neutron star. We also define µ0 ≡
+(GF/
+√
+2)(N/V) = µ(Rν) to be the interaction strength at the neutrino-sphere. Here,
+the neutrino emission is assumed to be time-invariant over the short time scales as-
+sociated with neutrino propagation through the supernova envelope, so the interac-
+tion strength depends explicitly only on position, rather than time. In the neutrino-
+driven wind phase of core-collapse supernovae, which occurs over a time window
+of O(1–10) s after core bounce, one may expect Rν ≃ 20km and µ0 ∼ 105ω0, where
+ω0 ∼ 10−16 MeV is the scale of the vacuum oscillations. During the shock breakout
+or “neutronization burst” phase that occurs earlier, around 10 ms after core bounce,
+the proto-neutron star can be more extended, with Rν ≳ 50–60 km, but the neutrino
+luminosity is also much higher, resulting in µ0 ∼ 106ω0.
+It has been shown that a single-angle Hamiltonian describing neutrino mixing
+in vacuum and ν-ν interactions possesses a number of conserved charges which
+commute with the Hamiltonian [Pehlivan et al., 2011]. These are analogous to
+the “Gaudin magnets” [Gaudin, M., 1976] that had been previously identified as
+the conserved charges of the pairing-force Hamiltonian in nuclear and condensed-
+matter physics [Richardson, 1963, Richardson and Sherman, 1964, Richardson,
+1965]. These conserved charges are related to the integrability of the Hamiltonian—
+meaning that it is possible to obtain, in principle, exact eigenvalues and eigenstates
+of this Hamiltonian in terms of closed-form solutions to a set of algebraic “Bethe-
+Ansatz” equations [Bethe, 1931]. Based on these ideas, specific procedures for the
+eigen-decomposition of a single-angle neutrino Hamiltonian have been outlined in
+the literature [Pehlivan et al., 2011, Birol et al., 2018, Patwardhan et al., 2019].
+Besides descriptions in terms of instantaneously conserved charges, analogies
+with other many-body problems have been fruitful to yield an explanation of the
+neutrino flavor spectral split in terms of a Bardeen-Cooper-Schrieffer (BCS)-Bose-
+Einstein Condensate (BEC) crossover-like phenomenon [Pehlivan et al., 2017], as
+well as to help provide many-body predictions of a spectral split [Birol et al., 2018]
+specifically in the case of an initial many-body wave function with all neutrinos in
+the electron flavor state.
+
+8
+Amol V. Patwardhan, Michael J. Cervia, Ermal Rrapaj, Pooja Siwach, A. B. Balantekin
+Instabilities and dynamical phase transitions
+Collective neutrino oscillations are generally assumed to be caused by unstable
+modes in the mean field dynamics generated by the Hamiltonian described in
+Eq. (4) (for two flavors). These instabilities are able to amplify initially small
+flavor perturbations exponentially fast (e.g., [Sawyer, 2004, Sawyer, 2005, Duan
+et al., 2010, Chakraborty et al., 2016, Izaguirre et al., 2017, Tamborra and Shalgar,
+2021, Richers and Sen, 2022] and references therein). The presence of the forward-
+scattering interaction can allow collective effects to develop when µ ≳ ωp, giving
+rise to interesting phenomena like synchronization [Pastor et al., 2002, Fuller and
+Qian, 2006, Raffelt and Tamborra, 2010, Akhmedov and Mirizzi, 2016], bipolar os-
+cillations [Kosteleck´y and Samuel, 1995, Duan et al., 2006c, Duan et al., 2007a] and
+spectral splits/swaps [Duan et al., 2006b, Duan et al., 2007b, Raffelt and Smirnov,
+2007b, Dasgupta et al., 2009, Martin et al., 2020].
+On the other hand, in descriptions of interacting neutrino systems that per-
+mit many-body quantum dynamics, oscillations that develop on “fast” timescales
+are generally associated with rapid dynamical development of the neutrino en-
+tanglement entropy [Cervia et al., 2019, Rrapaj, 2020, Roggero, 2021a, Roggero,
+2021b, Patwardhan et al., 2021]. The dynamically generated entanglement between
+neutrinos is seen to be correlated with deviations from the mean-field dynamics of
+the system [Cervia et al., 2019, Rrapaj, 2020] and with the presence of spectral splits
+in the neutrino energy distributions [Patwardhan et al., 2021]. An example of such
+a calculation is depicted in Fig. 1. In [Roggero et al., 2022], rapid entanglement and
+mean field instabilities were also found to be linked for certain angular setups.
+As shown in [Roggero, 2021a, Roggero, 2021b] in the single angle approxima-
+tion, when the frequency difference between two neutrino beams (δω) is positive
+and comparable to the ν-ν interaction coupling (µ), 0 < δω ≲ µ, rapid and strong
+flavor oscillations develop. This rather particular finding can be understood in terms
+of the presence of a Dynamic Phase Transition (DPT) [Heyl et al., 2013, Heyl,
+2018], which can be characterized by the introduction of the Loschmidt echo,
+L (t) = |⟨Φ|exp(−itH)|Φ⟩|2 ,
+(10)
+with |Φ⟩ the initial state at t = 0. The quantity L (t) is a fidelity measure [Gorin
+et al., 2006] that quantifies the probability for the system to return to its initial state.
+A DPT is then characterized by non-analyticities in the rate function
+λ(t) = − 1
+N log[L (t)] ,
+(11)
+where N is the total number of particles in the system and λ(t) an intensive “free
+energy” [Heyl et al., 2013, Gambassi and Silva, 2012]. Here, the rate λ(t) plays the
+role of a non-equilibrium equivalent of the thermodynamic free-energy. Notably,
+other definitions of DPT are possible, for instance, time-averaged order parame-
+ters [Sciolla and Biroli, 2011, Sciolla and Biroli, 2013, ˇZunkoviˇc et al., 2018].
+
+Many-body collective neutrino oscillations: recent developments
+9
+−1
+−0.8
+−0.6
+−0.4
+−0.2
+0
+0.2
+0.4
+0.6
+0.8
+1
+200
+500
+1000
+2000
+P MB
+z
+(ωp)
+r (in units of ω−1
+0 )
+Pz(ω1)
+Pz(ω2)
+Pz(ω3)
+Pz(ω4)
+Pz(ω5)
+Pz(ω6)
+Pz(ω7)
+Pz(ω8)
+−1
+−0.8
+−0.6
+−0.4
+−0.2
+0
+0.2
+0.4
+0.6
+0.8
+1
+200
+500
+1000
+2000
+P MF
+z
+(ωp)
+r (in units of ω−1
+0 )
+Pz(ω1)
+Pz(ω2)
+Pz(ω3)
+Pz(ω4)
+Pz(ω5)
+Pz(ω6)
+Pz(ω7)
+Pz(ω8)
+0
+0.2
+0.4
+0.6
+0.8
+1
+200
+500
+1000
+2000
+S(ωp)
+r (in units of ω−1
+0 )
+S(ω1)
+S(ω2)
+S(ω3)
+S(ω4)
+S(ω5)
+S(ω6)
+S(ω7)
+S(ω8)
+−1
+−0.5
+0
+0.5
+1
+1
+2
+3
+4
+5
+6
+7
+8
+0
+0.2
+0.4
+0.6
+Pz(ωp)
+S(ωp)
+ω (in units of ω0)
+Pz (initial)
+P MB
+z
+(nal)
+P MF
+z
+(nal)
+S(ωp) (nal)
+Fig. 1 Evolution of an initial state |νe⟩⊗4 |νx⟩⊗4 from a starting radius r0 such that µ(r0) = 5ω0,
+with a small mixing angle (θ = 0.161) and discrete, equally spaced oscillation frequencies
+ωk = kω0, and a time-varying neutrino interaction strength µ(r) motivated by the neutrino bulb
+model [Duan et al., 2006b], in the single-angle approximation according to Eqs. (8) and (9). Details
+of this calculation can be found in [Cervia et al., 2019]. Top left: Evolution of the z-components
+of the neutrino isospin expectation values (also known as “Polarization vectors”) in the mass basis,
+i.e., Pz ≡ 2⟨Jz⟩, for the full many-body quantum system. Top right: Same as top left, but in the
+mean-field approximation. Bottom left: Evolution of the entanglement entropy of each neutrino,
+with respect to the rest of the ensemble. Bottom right: Asymptotic values of Pz vs ωk, in the full
+many-body calculation (purple), and in the mean-field approximation (green), together with the
+initial Pz values (red), and the asymptotic entanglement entropies (dark orange). Neutrinos located
+closest to the spectral splits in the energy distributions (in this case, at ω2 and ω7) develop the
+largest amount of entanglement and thereby experience the most significant deviations compared
+to their mean-field evolution.
+Phase-space analysis
+In a recent work [Lacroix et al., 2022], this problem was further explored by ana-
+lyzing the evolution of neutrino flavor and entanglement in phase space. The setup
+consisted of two sets (beams) of neutrinos interacting with each other. In this anal-
+ysis, the Husimi quasi-probability or “Q” representation [Husimi, 1940] was con-
+structed for the reduced density operator of neutrinos in one of the beams, using an
+over-complete basis of coherent states. In the limit of infinite neutrino number, the
+Q representation acquires the interpretation of a classical phase-space probability
+distribution.
+
+10
+Amol V. Patwardhan, Michael J. Cervia, Ermal Rrapaj, Pooja Siwach, A. B. Balantekin
+For this two-beam interacting neutrino system, it was demonstrated that, while
+at early times the quasi-probability distribution remains relatively localized, at late
+times it develops a multi-modal structure with several localized peaks. This delocal-
+ization is indicative of non-Gaussian entanglement, which suggests that any approx-
+imate method beyond the mean-field relying on only the first and second moments
+of neutrino observables may not be sufficient to describe the long-term evolution
+of this system. Based on the phase space analysis, a new method for approximat-
+ing the exact evolution of the interacting neutrino system was proposed, wherein
+the quantum mechanical many-body evolution is replaced by a statistical average of
+‘mean-field’ solutions, with a Gaussian distribution of initial conditions around the
+exact starting point of the system [Lacroix and Ayik, 2014].
+Compact Representations for studying many body effects
+Still allowing for possibilities of mixed one-neutrino density matrices, one pro-
+posal [Volpe et al., 2013] to determine quantum corrections is to systematically
+incorporate n-body density matrices ρ1...n for n ≥ 1, given by
+ρ1...n =
+N!
+(N −n)!Trn+1...Nρ1...N,
+(12)
+into the coupled equations of motion for N neutrinos, as follows:
+i∂tρ1...n = [H1...n,ρ1...n]+
+n
+∑
+s=1
+Trn+1[V(s,n+1),ρ1...n+1],
+(13)
+where H1...n is the Hamiltonian truncated for the first n neutrinos in a given ordering
+and V(i, j) is the two-body interaction potential for a pair of neutrinos (i, j). This
+procedure is based on the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hi-
+erarchy for density matrices. Here, the mean field theory interaction of neutrinos
+and antineutrinos with the background gas is reproduced by restricting to n = 2
+and estimating ρ12 ≈ ρ1ρ2 (i.e., requiring the two-body correlation function to be
+zero) in this picture, in a sense as a loop Feynman diagram for neutrino propagation.
+In principle, investigating the importance of quantum corrections would practically
+entail checking for convergence of results for physical observables as the n-body
+correlation functions are incorporated for progressively increasing values of n in the
+BBGKY hierarchy.
+Owing to the exponential growth in the Hilbert space, classical (conventional)
+computers are unable to exactly simulate systems of more than ≃ 20 neutrinos.
+To overcome this difficulty, one can resort to compact representations of the wave-
+function through tensor network methods [Roggero, 2021a, Roggero, 2021b, Cervia
+et al., 2022], and more specifically matrix product states [Vidal, 2003, Schollw¨ock,
+2011, Paeckel et al., 2019]. In simplified setups, these methods allow for the com-
+putation of systems of hundreds of neutrinos. Alternatively, when considering very
+
+Many-body collective neutrino oscillations: recent developments
+11
+dense neutrino gases (vacuum oscillations can be ignored), methods based on gen-
+eralized angular momentum representations, by analogy between two flavor oscilla-
+tions and spin systems, can reach up to thousands of neutrinos and predict the ther-
+modynamic limit [Friedland and Lunardini, 2003a, Friedland et al., 2006, Xiong,
+2022, Roggero et al., 2022].
+In the case of time-dependent interaction strength and all-to-all ν-ν interactions,
+the more sophisticated tensor network method, namely, the time-dependent varia-
+tional principle (TDVP) method has been utilized in [Cervia et al., 2022]. These
+techniques provided considerable computational benefit for an initial state with all
+neutrinos in the same flavor, allowing for evolution of a system with ≈ 50 oscil-
+lation modes. This was a consequence of the entanglement among neutrinos being
+more localized in certain regions of the neutrino energy distribution. For systems
+with initial states being a mixture of νe and νx flavors, the entanglement is more de-
+localized, and therefore, the comparative advantage gained through TDVP methods
+is less dramatic, although work remains in progress on this front.
+For a general setup, quantum computers are a promising tool to solve the quan-
+tum many-body problem. Initial steps [Hall et al., 2021, Yeter-Aydeniz et al.,
+2022, Illa and Savage, 2022, Amitrano et al., 2022] to simulate the collective neu-
+trino oscillations on a quantum computer are already taken in this direction. In [Hall
+et al., 2021] a sytem of four neutrinos was simulated on IBM’s quantum devices
+using the real-time evolution. The unitary evolution operator U(t) = exp(−iHt)
+was decomposed using the first order Trotter-Suzuki decomposition, where error
+scales as O(t2). Since the interaction is long-range, a device with all-to-all con-
+nectivity among qubits is preferred. As an alternative, SWAP operations have been
+used to implement this interaction on a quantum device having connectivity among
+neighboring qubits [Hall et al., 2021]. In [Yeter-Aydeniz et al., 2022], the hybrid
+quantum-classical algorithm QLanczos (quantum Lanczos) was used to calculate
+the eigenvalues of neutrino many-body interaction Hamiltonian [Patwardhan et al.,
+2019] on a quantum computer. Furthermore, the transition probabilities of collec-
+tive neutrino oscillations were obtained by performing the real-time evolution using
+trotterization. However, all these earlier quantum computing studies were limited to
+a small system of four neutrinos due to constraints in the form of currently avail-
+able quantum devices, which can perform only a limited number of operations with
+low accuracy. More recently in [Amitrano et al., 2022], a trapped-ion quantum de-
+vice was utilized to perform the simulations for up to eight neutrinos, thanks to the
+all-to-all qubit connectivity in trapped-ion based architecture.
+Concluding remarks
+Studying the many-body quantum dynamics of dense neutrino systems remains an
+active area of research, with various groups attempting to investigate the problem
+using different types of classical and quantum computational tools, as well as ana-
+lytic or semi-analytic descriptions. In environments where neutrinos are present in
+
+12
+Amol V. Patwardhan, Michael J. Cervia, Ermal Rrapaj, Pooja Siwach, A. B. Balantekin
+high number densities, they almost inevitably become the main carriers of energy
+and lepton number, and as a result, the physics of neutrino flavor transformation
+in these environments becomes particularly relevant for the dynamics and nucle-
+osynthesis. Moreover, the close parallels between this problem and other quantum
+many-body systems in nuclear and condensed-matter physics suggests that the re-
+sults and insights obtained through these studies could have a much broader scope,
+beyond just the field of neutrino physics.
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+

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+GOHSP: A Unified Framework of Graph and Optimization-based Heterogeneous
+Structured Pruning for Vision Transformer
+Miao Yin1* Burak Uzkent2, Yilin Shen2, Hongxia Jin2, Bo Yuan1
+1 Rutgers University, 2 Samsung Research America
+Abstract
+The recently proposed Vision transformers (ViTs) have
+shown very impressive empirical performance in various
+computer vision tasks, and they are viewed as an impor-
+tant type of foundation model. However, ViTs are typically
+constructed with large-scale sizes, which then severely hin-
+der their potential deployment in many practical resources-
+constrained applications. To mitigate this challenging prob-
+lem, structured pruning is a promising solution to compress
+model size and enable practical efficiency. However, unlike
+its current popularity for CNNs and RNNs, structured prun-
+ing for ViT models is little explored.
+In this paper, we propose GOHSP, a unified framework of
+Graph and Optimization-based Structured Pruning for ViT
+models. We first develop a graph-based ranking for measur-
+ing the importance of attention heads, and the extracted im-
+portance information is further integrated to an optimization-
+based procedure to impose the heterogeneous structured spar-
+sity patterns on the ViT models. Experimental results show
+that our proposed GOHSP demonstrates excellent compres-
+sion performance. On CIFAR-10 dataset, our approach can
+bring 40% parameters reduction with no accuracy loss for
+ViT-Small model. On ImageNet dataset, with 30% and 35%
+sparsity ratio for DeiT-Tiny and DeiT-Small models, our ap-
+proach achieves 1.65% and 0.76% accuracy increase over the
+existing structured pruning methods, respectively.
+Introduction
+Recently applying transformer architecture to computer vi-
+sion has emerged as an important forefront of foundation
+model design (Dosovitskiy et al. 2020). Thanks to the del-
+icate vision-specific self-attention, inherent minimal induc-
+tive biases and high scalability and parallelism, vision trans-
+formers (ViTs) (Dosovitskiy et al. 2020; Touvron et al.
+2021; Zhou et al. 2021) have shown very outstanding and
+even state-of-the-art performance in many fundamental and
+downstream image and video processing tasks, such as im-
+age classification, object detection, super-resolution, video
+classification etc.
+Motivated by the scaling success of the giant natural lan-
+guage processing (NLP) transformers (e.g., BERT (Devlin
+*This work was done during Miao Yin’s internship at Samsung
+Research America.
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+et al. 2018) and GPT-3 (Brown et al. 2020)), the existing
+ViTs are also constructed with large model sizes to adapt for
+massive data training (Zhai et al. 2021). Consequently, they
+are suffering from huge memory footprints and extensive
+computational costs. These limitations, if not being properly
+addressed, could severely hinder the widespread adoption of
+ViTs in many practical scenarios, especially on the resource-
+constrained mobile platforms and Internet-of-things (IoT)
+devices.
+To mitigate this challenging problem, one attractive solu-
+tion is to perform model compression (Yu et al. 2017; Kim
+et al. 2016; Pan et al. 2019) to reduce the network costs with-
+out affecting task performance. However, unlike the current
+popularity of compressing convolutional and recurrent neu-
+ral networks (CNNs and RNNs), ViT-oriented model com-
+pression has not been systematically studied yet. In partic-
+ular, structured pruning, as an important hardware-friendly
+compression strategy that can bring practical efficiency on
+the off-the-shelf hardware, is little explored for ViT models.
+To date, a rich set of structured pruning approaches have
+been proposed and investigated in the existing literatures,
+and most of them focus on sparsifying the CNNs at the chan-
+nel level (He, Zhang, and Sun 2017; Ye et al. 2018). On the
+other hand, as will be analyzed and elaborated in Section ,
+because of the difference of the underlying architecture, the
+structured sparse ViT models can exhibit multi-granularity
+sparsity (i.e., head-level and column-level) in the different
+component modules (i.e., attention head and multi-layer per-
+ception (MLP)). The co-existence of such heterogeneous
+sparse patterns raises a series of new research challenges and
+questions when we consider the efficient structured pruning
+strategy for ViT models. For instance, for each component
+module what is the corresponding suitable pruning criterion
+to obtain its specific sparse pattern? Also, how should we
+perform the entire pruning process across different modules
+with different levels of granularity sparsity to optimize the
+overall compression and task performance?
+Technical Preview & Contributions. To answer these
+questions, in this paper we propose GOHSP, a unified frame-
+work of Graph and Optimization-based Structure Prun-
+ing for vision transformer. To be specific, we first de-
+velop a graph-based ranking approach to measure the im-
+portance of attention heads. As a soft-pruning guideline,
+such importance information is then integrated to the overall
+arXiv:2301.05345v1  [cs.AI]  13 Jan 2023
+
+optimization-based procedure to impose the different types
+of structured sparsity in a joint and global way. Overall, the
+contributions of this paper are summarized as follows:
+• We propose a graph-based ranking algorithm to mea-
+sure and determine the importance of attention heads.
+By modeling the inter-head correlation as a converged
+Markov chain, the head importance can be interpreted
+and calculated as the stationary distribution, which is fur-
+ther used as a soft guideline for the overall pruning pro-
+cedure.
+• We propose a unified framework to jointly optimize dif-
+ferent types of structured sparsity in the different mod-
+ules. The complicated coordination for different sparse
+patterns are automatically learned and optimized in a sys-
+tematic way.
+• We evaluate the performance of our structured pruning
+approach of different ViT models on different datasets.
+On CIFAR-10 dataset, our approach can bring 40% pa-
+rameters reduction with no accuracy loss for ViT-Small
+model. On ImageNet dataset, with 30% and 40% spar-
+sity ratio for DeiT-Tiny and DeiT-Small models, our
+approach achieves 1.65% and 0.76% accuracy increase
+than the existing structured pruning methods, respec-
+tively.
+Related Work
+Vision Transformer. Inspired by the grand success of trans-
+former architecture in NLP domains, deep learning re-
+searchers have actively explored the efficient transformer-
+based neural networks for computer vision. Most recently,
+several vision transformers (ViTs) and their variants have
+already shown very impressive performance in several im-
+age and video processing tasks (Dosovitskiy et al. 2020;
+Touvron et al. 2021; Zhou et al. 2021). However, in order
+to achieve competitive performance with the state-of-the-art
+CNNs, ViTs typically have to scale up their model sizes and
+therefore they suffer from costly computation and storage.
+Dynamic Inference with ViTs. To reduce the deploy-
+ment costs of ViTs, several works (Wang et al. 2021;
+Bakhtiarnia, Zhang, and Iosifidis 2021; Rao et al. 2021;
+Meng et al. 2022; Xu et al. 2022; Uzkent, Yeh, and Er-
+mon 2020; Uzkent and Ermon 2020) have been proposed to
+improve the processing speed via dynamically pruning the
+tokens/patches or skipping transformer components adap-
+tively. Essentially as dynamic inference approaches, this set
+of works do not pursue to reduce the model sizes but focus
+on input-aware inference to obtain practical speedup. Our
+structured pruning-based solution is orthogonal to them, and
+these two different strategies can be potentially combined
+together to achieve higher speed and smaller memory foot-
+print.
+Structured Pruning. Model compression is a promising
+strategy to reduce the deployment costs of neural networks.
+Among various model compression techniques, structured
+pruning is a very popular choice because its hardware-
+friendly nature can bring practical efficiency on the real-
+world devices. Based on different pruning criterias, various
+structured pruning approaches have been extensively studied
+Head-based Sparsity
+(Multi-Head Attention)
+(a) Conventional Unstructured Sparsity
+(b) Unified Structured Sparsity (Ours)
+Column-based Sparsity 
+(Multi-Head Attention)
+Column-based Sparsity 
+(MLP)
+Unstructured Sparsity 
+(MLP)
+Unstructured Sparsity
+(Multi-Head Attention)
+Figure 1: (a) Sparsity pattern of ViT models after un-
+structured pruning. Only part of the Multi-Head Attention
+and MLP columns are pruned which are not hardware-
+friendly.(b) Heterogeneous sparsity patterns of ViT models
+after structured pruning. Certain MLP and Multi-Head At-
+tention columns are removed which is hardware-friendly.
+in the existing literature (Yu et al. 2018; Zhuang et al. 2018;
+Liu et al. 2019; He et al. 2019; Lin et al. 2020; Tiwari et al.
+2021; Lou et al. 2022), and most of them focus on pruning
+CNN models; while the efficient structured pruning of ViTs
+is little explored. One of these studies, (Chen et al. 2021),
+prunes the vision transformers using structured pruning. (Yu
+et al. 2022), on the other hand, focuses on FLOPs reduction
+with the vision transformers using pruning, layer skipping,
+and knowledge distillation whereas in our study we focus on
+structured pruning to mainly reduce the number of parame-
+ters for building hardware-friendly compressed models. For
+this reason, we compare our method to (Chen et al. 2021).
+Structured Pruning of ViTs: Analysis
+Notation. Considering an L-block vision transformer,
+W (l)
+attn = {W (l)
+qkv, W (l)
+proj} and W (l)
+mlp = {W (l)
+fc1, W (l)
+fc2} rep-
+resent the weights of the attention layer and the MLP layer
+at l-th block, respectively. For each attention layer, there are
+H self-attention heads, namely W (l)
+qkv = {W (l,h)
+qkv }H
+h=1 and
+W (l)
+proj = {W (l,h)
+proj }H
+h=1. To simplify the notation, in the fol-
+lowing content we take one block as the example and omit
+the superscript (layer index).
+Heterogeneity of structured sparsity. Because of the
+difference of the network architecture, the meaning of
+‘structured sparsity’ varies with different model types. As
+described and performed in (Wen et al. 2016; Anwar,
+Hwang, and Sung 2017; Liu et al. 2018, 2020), the struc-
+tured pruning of CNN and RNN typically indicates the re-
+moval of the entire channels of the weight tensors and the
+entire columns of the weight matrices, respectively. Notice
+that here for either of these two cases, only one type of the
+structured sparse pattern exist because of the architectural
+homogeneity of the CNN and RNN.
+On the other hand, a ViT model exhibits inherent archi-
+tectural heterogeneity. Within the same block, the front-end
+multi-head attention module and the back-end MLP mod-
+ule represent two types of design philosophy for information
+processing, and thereby leading to huge difference on both
+computing procedures and the available structured sparse
+patterns.
+To be specific, when we consider performing structured
+
+pruning of ViT model, three types of structured sparse pat-
+terns can co-exist with different levels of granularity across
+different modules. For the multi-head attention module, be-
+cause each attention head is processing the information in-
+dividually in a parallel way, the pruning can be performed
+at the head-level to sparsify this component. In addition,
+consider the weights in the heads are represented in the ma-
+trix format; the column-level sparsity can also be introduced
+towards structured pruning. Meanwhile, because the MLP
+consists of multiple weight matrices as well, the column-
+level of granularity sparsity can be imposed on this back-end
+module at the same time. Consequently, a structured pruned
+ViT model can exhibit heterogeneous structured sparsity
+(see Fig. 1).
+Problem Definition. Based on the above analysis, the
+structured pruning of a vision transformer model with loss
+function ℓ(·) can be formulated as the following general op-
+timization problem:
+min
+Wattn,Wmlpℓ(Wattn, Wmlp),
+s.t.
+∥Wattn∥h
+0 ≤ κh
+attn,
+∥Wattn∥c
+0 ≤ κc
+attn,
+∥Wmlp∥c
+0 ≤ κc
+mlp,
+(1)
+where κc and κh are the desired number of columns and the
+desired number of heads after pruning, respectively. ∥ · ∥c
+0
+and ∥ · ∥h
+0 are the column-based and head-based group L0-
+norm, which denote the number of non-zero columns and
+the number of non-zero heads, respectively.
+Questions to be Answered. Solving the above opti-
+mization problem is non-trivial since it contains the con-
+straints involved with multi-granularity sparsity for different
+model components. More specifically, two important ques-
+tions need to be answered. Question #1: What is the suitable
+pruning criterion to obtain head-level sparsity?
+Analysis: From the perspective of information process-
+ing, multi-head attention shares some interesting similarity
+with convolutional layer. Both of them use multiple indi-
+vidual computing units, i.e., attention heads and convolu-
+tional filters, to perform parallel computations. Therefore, a
+naive way to perform head-level pruning is to leverage the
+existing criteria developed in the channel pruning of CNNs.
+However, such straightforward solution, in principle, may
+not be the best choice because of two reasons. First, the re-
+ceptive fields and the focused locality of the attention head
+and filters are different, and hence simply using the crite-
+rion for pruning channels is not a suitable strategy. Second
+and more importantly, most of the existing channel pruning
+criterias are built on the information of each individual chan-
+nel (the corresponding filter weight and/or its feature map).
+When adopting this philosophy in the head pruning, the in-
+sufficient utilization of inter-head information will probably
+cause non-negligible performance loss. Overall, the unique
+characteristics of multi-head attention mechanism calls for
+attention-specific pruning criterion.
+Question #2: How should we coordinate the pruning
+across different modules with different levels of granularity?
+Analysis: As indicated before, three types of structured
+sparse pattern can co-exist in the different modules of the
+pruned ViT models. A key component of the to-be-explored
+structured pruning strategy is to develop a good coordination
+scheme that can properly impose these different structured
+sparse patterns in a joint and global way. Consider the com-
+plicated interaction among different types of structured spar-
+sity, the expected pruning strategy should be able to solve
+this problem in a systematic and global way.
+Structured Pruning of ViTs: Method
+Graph-based Head Ranking
+To answer Question #1, we propose a graph-based approach
+to measure and determine the importance of different at-
+tention heads, which can be further used for the follow-up
+pruning. Our key idea is to model the inter-head correla-
+tion as a graph, and then leverage the graph-based ranking,
+a methodology that has been successfully used in many web
+search and NLP algorithms, such as PageRank (Page et al.
+1999), TextRank (Mihalcea and Tarau 2004) and LexRank
+(Erkan and Radev 2004), to select important attention heads.
+Graph Construction of Markov Chain. To be specific,
+we first construct a graph G = (A, E) to represent the atten-
+tion heads and their similarities in the block of a ViT model.
+The set of nodes A denote all the attention heads {Ah}H
+h=1,
+and E is the set of connected edges. For edge E(Ai, Aj),
+its weight is defined as the expected cosine similarity be-
+tween Ai and Aj. According to (Mihalcea and Tarau 2004),
+the graph defined with such cosine similarity can be inter-
+preted as a Markov chain, where each node is a state, and
+the transition probability P(i, j) between two states is the
+edge weight. In such scenario, P(i, j) can be calculated as:
+P(i, j) = EX∼D [CosineSim(Ai(X), Aj(X))] ,
+(2)
+where Ai(X) is the output of i-th attention head with sam-
+pled input X and D is the data set. Built upon this calcula-
+tion, the entire transition matrix P of a Markov chain. No-
+tice that as indicated in (Erkan and Radev 2004), each col-
+umn of P should be further normalized.
+Batch estimation. Calculating the transition probability
+can be very costly since it needs to be performed across the
+entire training dataset D (see Eq. 2). To solve this problem,
+we adopt a batch-based estimation strategy to improve com-
+putation efficiency without sacrificing ranking performance.
+To be specific, as described in Eq. 3, only a batch of training
+data is sampled and used to to estimate the transition prob-
+ability. As our ablation study in Section will show, using
+different batch sizes (B) bring very stable ranking results
+for the attention heads, thereby empirically verifying the ef-
+fectiveness of this estimation strategy.
+P(i, j) = CosineSim
+� B
+�
+b=1
+Ai(Xb),
+B
+�
+b=1
+Aj(Xb)
+�
+.
+(3)
+Importance Ranking. Mathematically, an irreducible
+and aperiodic Markov chain is guaranteed to converge to a
+stationary distribution (Seneta 2006). As indicated in (Erkan
+and Radev 2004), once converged, the probability of a ran-
+dom walker stays in one state can reflect the state impor-
+tance. Motivated by this observation, we propose to quantify
+
+Multi-Head Attention
+Embedded 
+Patches
+MLP
+Block
+0.1
+Graph-based Heads Ranking
+Multi-Head Attention
+Embedded 
+Patches
+MLP
+Block
+Data
+Optimization-based Soft Pruning
+Multi-Head Attention
+Embedded 
+Patches
+MLP
+Block
+Fine-Tuning
+X
+Score
+Mask
+0.7
+0.5
+0.3
+Normalize
+Figure 2: Procedure of the proposed multi-stage structured pruning approach.
+the importance of each attention head via calculating the sta-
+tionary distribution in our constructed Markov chain. To that
+end, the iterative power method (Erkan and Radev 2004) can
+be used via setting a uniform distribution for the states as the
+initialization. Overall, the entire graph-based head ranking
+procedure is described in Algorithm 1.
+Soft-Pruning Mask. Once the importance score for each
+state is obtained via calculating the stationary distribution,
+the corresponding attention heads can be ranked. Here we
+use a binary mark matrix Mattn = {Mqkv, Mproj} to in-
+dicate the weight entries associated with the least important
+heads that should be removed. Notice that at this stage the
+head pruning is not performed yet. Instead such binary mask
+serves as the guideline for the next-stage optimization, and
+it is essentially integrated into Eq. 1 as follows:
+min
+Wattn,Wmlpℓ(Wattn, Wmlp)
+s.t.
+∥(1 − Mattn) ⊙ Wattn∥0 = 0,
+∥Wmlp∥0 ≤ κc
+mlp,
+∥Mattn ⊙ Wattn∥c
+0 ≤ κc
+attn,
+(4)
+where ⊙ is element-wise product. In general, because the
+overall optimization phase coordinates and adjusts the dif-
+ferent types of structured sparse pattern from a global per-
+Algorithm 1: Graph-based Attention Head Ranking
+Input: Sampled batch {Xb}B
+b=1, attention heads {Ah}H
+h=1;
+Output: Importance score s = [s1, · · · , sH].
+1: Initialize transition matrix: P := zeros(H, H);
+2: for i = 1 to H do
+3:
+for j = 1 to H do
+4:
+Calculate P(i, j) via Eq. 3;
+5: Normalize each column of P ;
+6: Initialize s := ones(H)/H;
+7: repeat
+8:
+s′ := s;
+9:
+s := P s;
+10:
+δ := ∥s − s′∥;
+11: until δ ≤ ϵ
+spective, this ranking-only ”soft” pruning strategy, instead
+of directly pruning the least important heads, can provide
+more flexibility and possibility for the next-stage optimiza-
+tion procedure to identify better structured sparse models.
+Optimization-based Structured Pruning
+As pointed out by Question #2, the co-existence of multi-
+granularity and multi-location of the sparsity of ViT models
+make the entire structured pruning procedure become very
+challenging. To solve this, we propose to use advanced op-
+timization technique to perform systematic structured prun-
+ing. To be specific, considering the complicated interactions
+among different types of structured sparsity, we do not prune
+the heads or columns immediately, since any direct hard
+pruning at the early stage may cause severe accuracy loss.
+Instead, we adopt ”soft-pruning” strategy via optimizing the
+entire ViT models towards the desired structured sparse for-
+mats. In other words, the three types of sparsity pattern are
+gradually imposed onto the attention heads and MLPs.
+To that end, we first relax the constraints of Eq. 4 and
+rewrite it as follows:
+min
+Wattn,Wmlpℓ(Wattn, Wmlp) + λ
+2 ∥(1 − Mattn) ⊙ Wattn∥2
+F ,
+s.t.
+∥Wmlp∥c
+0 ≤ κc
+mlp,
+∥Mattn ⊙ Wattn∥c
+0 ≤ κc
+attn,
+(5)
+where λ is the coefficient that controls the influence of
+quadratic term.
+Optimization-based Soft Pruning. As indicated in
+(Boyd, Parikh, and Chu 2011), when the constraints of con-
+tinuous non-convex problem are sparsity related (as Eq.
+5 shows), Douglas—Rachford splitting method (Eckstein
+and Bertsekas 1992) can be a suitable optimization solution
+for such types of problem. Following this philosophy, we
+first introduce auxiliary variables Zattn, Zmlp and indicator
+functions as:
+g(Zattn) =
+�0
+∥Mattn ⊙ Zattn∥c
+0 ≤ κc
+attn,
++∞
+otherwise,
+(6)
+h(Zmlp) =
+�0
+∥Zmlp∥c
+0 ≤ κc
+mlp,
++∞
+otherwise.
+(7)
+
+Algorithm 2: Overall Procedure of GOHSP Framework
+Input: Dense weight {Wattn, Wmlp}, desired model size
+{κattn, κmlp}, training data D, number of epochs E;
+Output: Structured sparse weight { ˜
+Wattn, ˜
+Wmlp};
+1: Sample a batch of data {Xb}B
+b=1 from D;
+2: Calculate importance score s via Alg. 1;
+3: Obtain structured mask Mattn according to s;
+4: Zattn := Wattn, Zmlp := Wmlp; // Initialize auxiliary
+variables
+5: Uattn := 0, Umlp := 0; // Initialize Lagrangian multi-
+pliers
+6: for e = 1 to E do
+7:
+Update Wattn, Wattn via Eq. 10 and Eq. 11;
+8:
+Update Zattn, Zmlp via Eq. 12 and Eq. 13;
+9:
+Update Uattn, Umlp via Eq. 14 and Eq. 15;
+10: Fine-tune pruned weight { ˜
+Wattn, ˜
+Wmlp}.
+Then, we can rewrite Eq. 5 as the following equivalent form:
+min
+W ,Z
+ℓ(Wattn, Wmlp) + g(Zattn) + h(Zmlp)+
+λ
+2 ∥(1 − Mattn) ⊙ Wattn∥2
+F ,
+s.t.
+Wmlp = Zmlp,
+Wattn = Zattn.
+(8)
+In such scenario, the corresponding augmented Lagrangian
+function of the above optimization objective is:
+Lρ(Wattn, Wmlp, Zmlp) = ℓ(Wattn, Wmlp) + g(Zattn)+
+h(Zmlp) + λ
+2 ∥(1 − Mattn) ⊙ Wattn∥2
+F +
+ρ
+2∥Wattn − Zattn + Uattn∥2
+F +
+ρ
+2∥Uattn∥2
+F + ρ
+2∥Wmlp − Zmlp + Umlp∥2
+F + ρ
+2∥Umlp∥2
+F ,
+(9)
+where ρ > 0 is the penalty parameter, and Uattn, Umlp are
+the Lagrangian multipliers. Then the variables at step t can
+be iteratively updated as:
+W t
+attn = W t−1
+attn−η
+ℓ(Wattn, W t−1
+mlp )
+Wattn
+−
+λ
+�
+(1 − Mattn) ⊙ W t−1
+attn
+�
+−ρ(W t−1
+attn − Zt−1
+attn + U t−1
+attn),
+(10)
+W t
+mlp = W t−1
+mlp − η ℓ(W t
+attn, Wmlp)
+Wmlp
+−
+ρ(W t−1
+mlp − Zt−1
+mlp + U t−1
+mlp ),
+(11)
+Zt
+attn = P(W t
+attn + U t−1
+attn),
+(12)
+Zt
+mlp = P(W t
+mlp + U t−1
+mlp ),
+(13)
+U t
+attn = U t−1
+attn + W t
+attn − Zt
+attn,
+(14)
+U t
+mlp = U t−1
+mlp + W t
+mlp − Zt
+mlp.
+(15)
+Here η is the optimizer learning rate for training the ViT, and
+P is the Euclidean projection for the sparse constraint.
+Final Hard-Pruning and Fine-Tuning. After the above
+described optimization procedure, the structured sparse pat-
+terns have been gradually imposed onto the ViT model.
+In other words, the weight values of the masked attention
+heads, as well as some columns of MLPs and attention
+heads, become extremely small. At this stage, we can now
+prune those small weights and then perform a few rounds of
+fine-tuning to achieve higher performance.
+Overall,
+by
+using
+graph-based
+head
+ranking
+and
+optimization-based structured pruning, the previously raised
+Question #1 and #2 can be properly addressed. The overall
+GOHSP framework is summarized in Fig. 2.
+Experiments
+Experimental Settings
+Dataset and Baseline. We evaluate the performance of
+our proposed GOHSP approach on CIFAR-10 and Ima-
+geNet datasets (Deng et al. 2009). For experiments on
+the CIFAR-10 dataset, the original dense model is ViT-
+Small1(Dosovitskiy et al. 2020) with 48M parameters. For
+experiments on the ImageNet dataset, the original dense
+models are DeiT-Tiny and DeiT-Small (Touvron et al. 2021)
+with 5.7M and 22.1M parameters, respectively.
+Hyper-parameters and Sparsity Ratio. For our experi-
+ments on the CIFAR-10 dataset, the batch size, learning rate
+and ρ are set as 256, 0.1 and 0.001, respectively. For Ima-
+geNet dataset, the batch size, learning rate and ρ are set as
+256, 0.01 and 0.001, respectively. For both of these two ex-
+periments, SGD is selected as the training optimizer with-
+out using weight decay, and we apply Erd˝os-R´enyi (Mo-
+canu et al. 2018) to determine the sparsity distribution of
+each layer given an overall sparsity ratio. In particular, soft-
+pruning maintains high accuracy at the high sparsity ratios.
+Performance Evaluation
+CIFAR-10 Dataset. Table 1 shows performance compari-
+son on CIFAR-10 dataset between our proposed GOHSP
+and other structured pruning method (structured one-shot
+magnitude pruning (SOMP) (Han, Mao, and Dally 2015)
+and structured gradually magnitude pruning (SGMP) (Zhu
+and Gupta 2017)) for ViT-Small model. It is seen that
+with the same sparsity ratio, our approach brings significant
+performance improvement. Compared to SGMP approach,
+1We take this model from open source library timm.
+Table 1: Performance comparison between our GOHSP
+and structured one-shot/gradually magnitude-based pruning
+(SOMP/SGMP) of ViT-Small model on CIFAR-10 dataset.
+Method
+Sparsity
+# Paramters
+Top-1 (%)
+Baseline
+-
+48.0M
+97.85
+SOMP
+40%
+28.8M
+96.07
+SGMP
+40%
+28.8M
+96.93
+GOHSP (Ours)
+40%
+28.8M
+97.89
+GOHSP (Ours)
+80%
+9.6M
+97.40
+
+Table 2: Comparison results of our method, GOHSP, with other structured and unstructured pruning methods on ImageNet.
+Model
+Method
+Sparsity
+# Parameters
+FLOPs ↓
+Run-time ↓
+Top-1 (%)
+DeiT-Tiny
+Baseline
+-
+5.7M
+-
+-
+72.20
+OMP (Unstructured)
+30%
+4.02M
+25.56%
+-
+68.35
+GMP (Unstructured)
+30%
+4.02M
+25.56%
+-
+69.56
+TP (Unstructured)
+30%
+4.02M
+25.56%
+-
+68.38
+SSP (Structured)
+30%
+4.2M
+23.69%
+-
+68.59
+S2ViTE (Structured)
+30%
+4.2M
+23.69%
+10.57 %
+70.12
+GOHSP (Structured)
+30%
+4.0M
+30%
+13.41%
+70.24
+DeiT-Small
+Baseline
+-
+22.1M
+-
+-
+79.90
+SSP (Structured)
+40%
+14.6M
+31.63%
+-
+77.74
+S2ViTE (Structured)
+40%
+14.6M
+31.63%
+22.65%
+79.22
+GOHSP (Structured)
+40%
+14.4M
+35%
+24.61%
+79.98
+GOHSP (Structured)
+50%
+11.1M
+39%
+26.57%
+79.86
+GOHSP achieves 0.96% accuracy increase with the same
+pruned model size. Even compared with the baseline, the
+structured sparse model pruned by GOHSP can outperform
+the uncompressed model with 40% fewer parameters while
+80% compressed model achieves only 0.45% worse than the
+full ViT-Small model.
+ImageNet Dataset. Table 2 summarizes the performance
+on ImageNet dataset between GOHSP and other structured
+pruning approaches (SOMP, SGMP, Talyer pruning (TP),
+Salience-based Structured Pruning (SSP) and S2ViTE(Chen
+et al. 2021)) for DeiT-Tiny and DeiT-Small models. It is seen
+that due to the limited redundancy in such small-size model,
+the existing pruning approaches suffer from more than 2.5%
+accuracy loss when compressing DeiT-Tiny. Instead, with
+the even fewer parameters and more FLOPs reduction, our
+GOHSP approach can achieve at least 0.68% accuracy in-
+crease over the unstructured pruning approaches. Compared
+to the structured pruning approach (SSP), our method enjoys
+1.65% accuracy improvement with lower storage cost and
+computational cost. In addition, when compressing DeiT-
+Small model, with fewer parameters and more FLOPs re-
+duction, our GOHSP approach can achieve 0.76% accuracy
+increase as compared to the state-of-the-art structured prun-
+ing method S2ViTE (Chen et al. 2021) and can even outper-
+form the original DeiT-Small. With 50% pruned DeiT-Small
+we achieve similar accuracy to the full DeiT-Small. Finally,
+we report 26.57% improvement in run-time efficiency with
+our 50% pruned DeiT-Small.
+0.2
+0.3
+0.4
+0.5
+0.6
+Sparsity
+90.0
+92.0
+94.0
+96.0
+98.0
+Top-1 Accuracy (%)
+Ours
+Hard Pruning
+Figure 3: Results on the effect of soft-pruning (ours) and
+hard-pruning for ViT-Small model on CIFAR-10 dataset.
+Ablation Study, Visualization and Discussion
+To obtain the deep understanding of the effect of our pro-
+posed approach, we perform several ablation studies and a
+detailed analysis. Here the experiments conducted in the ab-
+lation study focus on compressing ViT-Small on CIFAR-10.
+Soft Pruning vs Hard Pruning. As described in Opti-
+mization section, after ranking the attention heads, we use
+the ranking information as a soft-pruning mask to guide
+the next-phase optimization. The optimization itself is also
+a soft-pruning procedure that does not directly zero the
+weights but gradually impose the structured sparsity. To ana-
+lyze the effect of this strategy, we conduct an ablation exper-
+iment via performing the direct hard pruning. In this ablation
+study, the least important attention heads are removed ac-
+cording to their ranks, and the columns of MLPs with least
+group L1 norm are also pruned. Such hard pruned models
+are still trained with the same hyper-parameters settings that
+are used for soft pruning method. Fig. 3 shows the curves
+of top-1 test accuracy with different target sparsity settings.
+1
+2
+3
+4
+5
+6
+7
+8
+Head Index
+1
+2
+3
+4
+5
+6
+7
+8
+Block Index
+Batch Size=256
+0
+2
+4
+6
+1
+2
+3
+4
+5
+6
+7
+8
+Head Index
+1
+2
+3
+4
+5
+6
+7
+8
+Block Index
+Batch Size=512
+0
+2
+4
+6
+1
+2
+3
+4
+5
+6
+7
+8
+Head Index
+1
+2
+3
+4
+5
+6
+7
+8
+Block Index
+Batch Size=1024
+0
+2
+4
+6
+1
+2
+3
+4
+5
+6
+7
+8
+Head Index
+1
+2
+3
+4
+5
+6
+7
+8
+Block Index
+Batch Size=1536
+0
+2
+4
+6
+Figure 4: The effect of batch sizes for ranking results. Dif-
+ferent colors represent different ranking scores. We can see
+that our head ranking algorithm is not sensitive to batch size.
+
+The soft-pruning strategy brings very significant accuracy
+improvement over the direct hard pruning with the same
+sparsity ratio.
+Effect of Batch Size on Head Ranking. As shown in Eq.
+3, the importance scores of attention head is calculated on
+a batch of data. To investigate the potential impact of batch
+sizes for the ranking results, we observe the change of rank-
+ing with different batch sizes. As shown in Fig. 4, the rank-
+ing results are very stable (almost the same) when the batch
+size varies. Therefore we can conclude that using batches of
+data can already achieve very good estimation of head rank-
+ing. In other words, our ranking approach has low sensitivity
+to the distribution of input data.
+Sensitivity of Penalty Parameter ρ. We also explore the
+effect of hyperparameter ρ on the structured pruning proce-
+dure. Fig. 5 (a) shows the convergence of training process
+with respect to different ρ. It is seen that the convergence
+speed is always fast, and hence it demonstrates the promis-
+ing convergence property of our approach in practice. Fig. 5
+(b) illustrates the L2-norm of the masked entries. It is seen
+that the larger ρ makes the model exhibit more sparsity at the
+earlier stage, thereby indicating that larger ρ can bring fewer
+epochs in the final fine-tuning stage. However, as shown in
+Fig. 5 (c), too large ρ brings accuracy degradation, so ρ can
+be considered as a parameter that controls the trade-off be-
+tween the speed of imposing sparsity and task performance.
+Visualization. Fig. 6 illustrates the sparsity patterns in
+the pruned ViT models after performing our GOHSP ap-
+proach. It is seen that three types of structured sparsity pat-
+terns (head-level sparsity, column-level sparsity in the head
+and column-level sparsity in the MLP) are imposed on the
+0
+10
+20
+30
+40
+50
+60
+Epoch
+0
+250
+500
+Loss
+(a) Curves of training loss
+=0.001
+=0.002
+=0.0005
+0
+10
+20
+30
+40
+50
+60
+Epoch
+0
+50
+100
+L2-Norm
+(b) Curves of sparsity strength
+=0.001
+=0.002
+=0.0005
+0
+10
+20
+30
+40
+50
+60
+Epoch
+50
+75
+100
+Top-1 (%)
+(c) Curves of test accuracy
+=0.001
+=0.002
+=0.0005
+40
+50
+60
+0
+3
+6
+40
+50
+60
+96
+97
+98
+Figure 5: Effect of ρ on the structured pruning procedure. ρ
+controls the trade-off between the speed of imposing sparsity
+and task performance.
+pruned models. Such pruning can be more effective on hard-
+ware than the unstructured pruning methods.
+Block9
+Block10
+Block11
+Multi-Head Attention Layer
+MLP Layer
+Block9
+Block10
+Block11
+Multi-Head Attention Layer
+MLP Layer
+Block9
+Block10
+Block11
+Multi-Head Attention Layer
+MLP Layer
+Figure 6: Visualization of the imposed structured sparsity on
+the DeiT-Small model. The columns and heads with lighter
+color are pruned. Our method can prune columns (Block9,
+Block10, and Block11), and heads (Block10, Block11) of
+the Multi-Head Attention layer. On the other hand, we can
+prune columns of MLP layers in all the blocks.
+Why
+Douglas—Rachford
+splitting
+method?
+As
+shown in our Optimization section, the iterative Dou-
+glas—Rachford splitting technique is adopted to solve Eq.
+5. Such choice is built on two reasons. 1) Convergence:
+Douglas—Rachford splitting method is a primal-dual
+optimization method that enjoys fast convergence speed.
+According to (Boyd, Parikh, and Chu 2011), within a few
+iterations it can provide satisfied solution for large-scale
+problems – particularly attractive for DNN applications.
+More specifically for this work, the fast convergence of
+Douglas—Rachford splitting method can avoid gradient
+explosion problem introduced by the additional sparsity
+loss in Eq. 9. 2) Flexibility: Douglas—Rachford splitting
+method, by its nature, divides the original difficult optimiza-
+tion problem into several less complicated sub-problems,
+each of which can be then addressed independently. This
+divide-and-conquer property is very suitable for optimizing
+the heterogeneous structured pruning of ViT, which explores
+the different types of structured sparsity across different
+attention heads and MLPs (Eq. 10 and 11).
+Conclusion
+In this paper we propose GOHSP, a unified framework to
+perform graph and optimization-based heterogeneous struc-
+tured pruning for vision transformers. By using graph-based
+ranking and leveraging the advanced optimization tech-
+nique, our approach can efficiently impose different types
+of structured sparse patterns on the vision transformers with
+high compression rate and task performance. Our experi-
+ments show that, on ImageNet, with 30 − 50% sparsity,
+GOHSP compresses the DeiT-Tiny and DeiT-Small mod-
+els with minor or no loss in accuracy and with ∼ 25 im-
+provement in rum-time efficiency. Finally, we compress ViT-
+Small up to 80% on CIFAR10 with minor loss in accuracy.
+
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+1
+Spyker: High-performance Library for Spiking
+Deep Neural Networks
+Shahriar Rezghi Shirsavar†‡, Mohammad-Reza A. Dehaqani†‡,
+†School of Electrical and Computer Engineering, College of Engineering, University of Tehran, Tehran, Iran
+{shahriar.rezghi, dehaqani}@ut.ac.ir
+‡School of Cognitive Sciences, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran
+∗Corresponding author: Mohammad-Reza A. Dehaqani, [email protected]
+Abstract—Spiking neural networks (SNNs) have been recently
+brought to light due to their promising capabilities. SNNs
+simulate the brain with higher biological plausibility compared
+to previous generations of neural networks. Learning with fewer
+samples and consuming less power are among the key features
+of these networks. However, the theoretical advantages of SNNs
+have not been seen in practice due to the slowness of simulation
+tools and the impracticality of the proposed network structures.
+In this work, we implement a high-performance library named
+Spyker using C++/CUDA from scratch that outperforms its
+predecessor. Several SNNs are implemented in this work with
+different learning rules (spike-timing-dependent plasticity and
+reinforcement learning) using Spyker that achieve significantly
+better runtimes, to prove the practicality of the library in the
+simulation of large-scale networks. To our knowledge, no such
+tools have been developed to simulate large-scale spiking neural
+networks with high performance using a modular structure.
+Furthermore, a comparison of the represented stimuli extracted
+from Spyker to recorded electrophysiology data is performed
+to demonstrate the applicability of SNNs in describing the
+underlying neural mechanisms of the brain functions. The aim
+of this library is to take a significant step toward uncovering the
+true potential of the brain computations using SNNs.
+Index
+Terms—Spiking
+Neural
+Network,
+Learning
+Rules,
+C++/CUDA, Modular Structure, Biological Plausibility
+I. INTRODUCTION
+The human brain can operate with amazing robustness and
+energy efficiency. Artificial neural networks (ANNs) aim at
+modeling the brain, and three generations of these networks
+have been developed. Each generation of ANNs improves the
+quality of the modeling of the brain compared to the last. The
+first generation of ANNs makes use of the McCulloch-Pitts
+neurons [1]. Although these neurons are inspired by biological
+neurons, time dynamics are not considered in this model, and
+the learning rules proposed for them lack power and biological
+plausibility. These neurons were used in multi-layer perceptron
+(MLPs) [2] and Hopfield [3] networks.
+The second generation of ANNs uses a continuous activa-
+tion function (ReLU [4] and sigmoid [5], for example) instead
+of thresholding, which makes them suitable for processing
+analog signals. They have attracted the attention of researchers
+in recent years and were able to reach high accuracies [6], [7]
+(even surpassing humans) and win different challenges [8].
+Despite the success of DNNs, there are structural differences
+between these networks and the human brain. Lack of temporal
+dynamics, using analog signals for network propagation and
+activation functions, learning rules without biological roots,
+and the need for large amounts of data [9] and energy [10] to
+achieve acceptable results are among these differences.
+The third generation of neural networks is spiking neural
+networks (SNNs). The neural models used in these networks
+simulate biological neurons more accurately, and the coding
+mechanisms used in these networks are found in neural
+communications. Furthermore, the learning rules used in these
+networks have been discovered in the brain [11]–[13]. Having
+lower energy consumption, learning with fewer samples, and
+solving more complicated tasks due to time dynamics (several
+electrophysiological studies emphasize the role of temporal
+dynamics in neural coding [14], [15]) are some of the advan-
+tages of SNNs compared to the second generation of ANNs.
+SNNs can be used to solve machine learning tasks, study and
+explore brain functionality, and run on specialized hardware
+with low power consumption. The research being done on
+these networks aims to address the disadvantages of DNNs
+with more realistic modeling of the brain functionality.
+Several high-performance well-established frameworks like
+PyTorch [16], TensorFlow [17], and MXNet [18] have been
+developed for DNNs in recent years. These libraries have en-
+abled DNNs to achieve new highs in solving machine learning
+tasks. SNNs are not yet comparable to DNNs due to the lack
+of fast simulation tools. There have been some attempts, like
+SpykeTorch [19] and BindsNet [20]. SpykeTorch, written on
+top of the PyTorch framework, is a simulator for large-scale
+spiking neural networks (SDNNs). However, it has a slow
+runtime, and training even simple networks can take up to days
+to complete. To our knowledge, Spyker is the first toolbox to
+simulate large-scale networks with high performance, is easy
+to use, has the flexibility to be used in multiple languages, and
+has the compatibility to integrate with other commonly used
+tools. In order to fill this need, we have developed Spyker.
+Spyker is a C++/CUDA library written from scratch with both
+C++ and Python interfaces and support for dense and sparse
+structures. Although Spyker is a stand-alone library, it has a
+highly flexible API and can work with PyTorch tensors and
+Numpy arrays. Figure 1 shows an overview of the library. In
+order to increase performance, small-sized integers are used
+alongside floating-point numbers. It also uses highly-optimized
+low-level back-end libraries such as OneDNN and cuDNN to
+speed up heavy computations such as convolutions and matrix
+multiplications. Spyker can be compiled on various CPUs to be
+arXiv:2301.13659v1  [cs.CV]  31 Jan 2023
+
+2
+optimized locally and take advantage of native CPU-specific
+instructions.
+Spiking neural networks are made of different building
+blocks (see [21] for more details). The first block is the
+modeling of the biological neurons. Some examples of this
+are leaky integrate-and-fire [22], spike-response model [23],
+and Izhikevich model [24]. Another building block is neural
+coding, which can be rate coding [25], temporal coding, phase
+coding and synchrony coding [26], or other coding schemes.
+The final building block is the learning mechanism. Examples
+of these mechanisms are STDP [27], [28], R-STDP [29],
+backpropagation [30], and conversion from ANNs to SNNs
+[31]. Spyker has a modular implementation of these three
+blocks that enables its users to build SNNs.
+Spyker provides SNN functionality with a high-performance
+and easy-to-use interface with an open-source and permissive
+license. It can run on CPU and CUDA devices and has
+both dense and sparse interfaces. The library introduces new
+features and fixes most of the shortcomings of its prede-
+cessor. The improvements include adding batch processing,
+strided convolutions, internal padding for convolutions, fully
+connected layers, and the rate coding mechanism. Compared
+to its predecessor, the interface of the library is simpler,
+closer to the current API of deep learning libraries, and more
+straightforward to use. In this work, several successful network
+structures are implemented using this library to prove its
+operability, its runtime speed is compared to SpykeTorch, and
+the results indicate Spyker can run up to eight times faster.
+The proposed work is able to reduce the gap between SNNs
+and DNNs and bring us a step closer to uncovering the true
+potential of spiking neural networks.
+We start with a description of dimensionality of the input
+arrays and how the spike trains are implemented in the library.
+Afterward, we provide an explanation of different building
+blocks of SNNs and how they are implemented in Spyker and
+modeled in the interface. Then, we implement network struc-
+tures that have been succesful to prove its operatibility, and we
+compare the performance of the library to its predecessor on
+these networks. Furthermore, comparison of the represented
+stimuli extracted from Spyker to recorded electrophisiology
+data is performed to demonstrate the applicability of SNNs
+in describing the underlying neural mechanisms of the brain
+functions. Finally, we demonstrate an example usage of the
+library and discuss the impacts of this work and how it can
+be further improved.
+II. METHODS
+The interface of the Spyker can be better explained when the
+classes and methods of the interface are grouped by building
+blocks of SNNs. The categories are feature enhancement,
+neural coding, neural model, and learning. In this section, the
+structure of the input to the network is explained. Afterward,
+the sparse and the dense interfaces are compared. Finally, the
+building blocks of the library are discussed in detail.
+A. Network Input
+Arrays passed through convolutional neural networks that
+process images are often four-dimensional arrays composed
+of batch size (B or N), number of channels (C), image height
+(H), and image width (W). The order can either be BCHW
+or BHWC (or NCHW or NHWC). SNNs have temporal
+dynamics, and it is implemented as a dimension that represents
+time steps in Spyker. The library implements five-dimensional
+arrays with BTCHW order (T being the time steps). Since
+DNNs process analog signals, data types used in these net-
+works are (usually four-byte) floating-point numbers. This data
+type can be computationally expensive compared to a small-
+sized integer type and take up more space in the memory.
+Since SNNs process binary signals, Spyker can optionally use
+eight-bit (or wider) integers alongside floating-point numbers
+to improve performance further.
+B. Dense vs Sparse interface
+The dense interface of Spyker uses the fully allocated
+memory buffers that are used in neural network computations.
+However, the sparse interface only needs to hold the indices
+of the spikes. Conversion between dense and sparse interfaces
+are provided in the library. The sparse interface has some
+advantages compared to the dense interface. In the dense
+interface, the time consumed by each operation is a function
+of the size of each of the 5 dimensions. However, in the sparse
+interface, it depends on the number of spikes. This means both
+memory and time consumed will be greatly reduced when
+processing sparser signals. Furthermore, since neurons fire at
+most once when using rank order coding, the increment of the
+number of time steps will have a smaller effect in the sparse
+interface compared to the dense interface.
+C. Feature Enhancement
+A transformation can be used to enhance features of the in-
+put signal (image) before the neural coding process [32]–[34].
+This results in highlighted features having higher intensities
+and appearing in earlier time steps, meaning more excitation.
+Feature enhancement is done through filtering the input here.
+Various filters are supported in Spyker, and they are introduced
+in the following subsections.
+1) Difference of Gaussian Filter: The first filter is the Dif-
+ference of Gaussian (DoG). This filter increases the intensities
+of edges and other details in the image (see Figure 2 for an
+example) [35]. It approximates the center-surround properties
+of the ganglion cells of the retina [36] (see also [37], [38]).
+This operation is implemented as spyker.DoG(size, filters, pad
+, device) where size is the size of the width and the height
+of the filter, filters is a list of DoG filter descriptions (each
+description takes in two standard deviations), pad is the size
+of the padding of the image, and device is the device the filter
+will run on (CPU, GPU or others).
+2) Gabor Filter: The following filter is the Gabor filter
+that determines the presence of specific frequency in content
+in a specific direction in the image. Research Indicates [39]
+that the Gabor filter is used in the human visual cortex. The
+Gabor filter is implemented as spyker.Gabor(size, filters, pad
+, device). The parameters of this class are the same as the
+DoG class, but the filters are Gabor filter descriptions, and
+each description takes in sigma, theta, gamma, lambda, and
+psi.
+
+3
+Numpy Array
+PyTorch Tensor
+Numpy Array
+PyTorch Tensor
+Feature Enhancement
+Neural Coding
+Neural Model
+Learning
+T=0
+T=1
+T=2
+T=3
+A+
+A-
+Fig. 1: Overview of the Spyker library. Spyker API supports PyTorch tensors and Numpy arrays as well as a built-in data
+wrapper. The output of Spyker operations have the same container type as the input. The functionality of Spyker can be grouped
+into subcategories shown in the figure.
+3) Laplacian of Gaussian Filter: The Laplacian of Gaus-
+sian (LoG) layer is also implemented in Spyker, and it is ap-
+proximated using two DoG filters. An LoG filter with standard
+deviation σ can be approximated using two DoG filters with
+(σ
+√
+2, σ/
+√
+2) and (σ/
+√
+2, σ
+√
+2) standard deviations. This
+filter exists in Spyker as spyker.LoG(size, stds, pad, device)
+where stds are a list of standard deviations needed to describe
+multiple LoG filters.
+4) Shape of the Filters: The previously explained filters
+have kernel size Kc × Kh × Kw, which are square kernels
+(Kh = Kw). The input can have B × Ci × Hi × Wi shape
+which corresponds to batch, channels, height, and width of the
+input, respectively. The output will have B × Co × Ho × Wo
+shape where:
+Co = Ci × Kc
+Ho = Hi + 2 × Ph − Kh + 1
+Wo = Wi + 2 × Ph − Kw + 1
+(1)
+and Ph and Pw are height and width padding of the filter. The
+Kc filters are applied to each channel separately.
+5) Zero-phase Component Analysis:
+Final implemented
+layer is zero-phase component analysis (ZCA) Whitening.
+It has been suggested [34] that this transformation can im-
+prove the accuracy of SNNs on real-world images. Spyker
+implements an efficient version of ZCA whitening by taking
+advantage of routines from highly optimized linear algebra
+libraries (BLAS and LAPACK) that operate on symmetric
+matrices. This layer is implemented as spyker.ZCA class
+which has a fit(array, epsilon) and a call function.
+D. Neural Coding
+SNNs process spike trains, but the input consists of analog
+values (for example, images are made of pixel values). In order
+to make these inputs suitable for the network, a conversion
+scheme is needed. The mapping from stimuli to neural re-
+sponses is called neural coding [40]. Coding schemes imple-
+mented in Spyker are explained in the following subsections.
+1) Rate Coding: Out of several coding schemes suggested,
+rate coding is widely used where the rate of firing of the
+neurons represents information. In this scheme, the rate of
+firing is dependent on the intensity of the input value (higher
+intensity corresponds to faster firing) [25]. The exact time
+of firing in each neuron is stochastic in nature and may be
+modeled with a Poisson distribution. A lengthy window of
+time is required to transmit the information in this coding,
+and the spikes are not quite sparse.
+2) Temporal Coding: Another popular coding scheme is
+temporal coding [41]. Recordings in the primary visual cortex
+show [42] that the response latency decreases with the stimulus
+contrast. This coding scheme can convey information through
+the timings of the spikes. Multiple forms of this scheme have
+been proposed, including rank order coding [43]. Instead of
+computing the exact timing of each spike, the timings are
+computed relative to one another in rank order coding. This
+relative (instead of exact) timing can increase invariance to
+changes in the input intensity and contrast [43]. It has been
+suggested [44] that temporal coding might be more efficient
+in some situations.
+3) Coding in Spyker: Spyker supports rank order and rate
+coding. The concept of time is implemented with spikes
+occuring in time steps in this library. Rank order coding maps
+higher intensities to earlier time steps of a neuron firing. In
+order to calculate the time step the neuron will fire in, Spyker
+sorts the intensity values by default. This calculates rank order
+between spikes, and the spikes will be distributed among
+time steps evenly. The sorting operation is computationally
+
+S
+P
+Y
+K
+E
+R4
+T=0 T=1 T=2 T=3
+B&W Image
+B&W Image
+DoG Filtered
+Gabor Filtered
+T=0
+T=0
+T=1
+T=1
+T=2
+T=2
+T=3
+T=3
+Input Image  
+(Gray or HSV)
+Feature
+Enhancement
+Encoded input data ready to be processed by the network
+Neural Coding
+Fig. 2: The figure shows a black and white image being filtered by DoG and Gabor filters. The theta parameter of the Gabor
+filter is set to -15 degrees. Then the images are coded using rank order coding into four time steps. Spikes are shown with
+white color on a black background through time steps. Spikes carry on from the previous to the current time step (cumulative
+structure).
+expensive (specially on GPUs), and optionally, it can be
+disabled to have runtime improvements (however, accuracy
+might be affected). Since processing time steps sequantially is
+inefficient and time-consuming, Spyker processes all the time
+steps at once. To this end, when a neuron fires in time step ti,
+it will also fire at time steps ti+1, ti+2, ..., tn where n is the
+number of time steps. An example of this cumulative structure
+can be seen in Figure 2.
+E. Neural Model
+Once the input is filtered and coded, it gets processed
+by the network. The network is built using fully connected,
+convolution, integrate-and-fire (IF) activation, pooling, and
+padding layers. These operations are explained in the follow-
+ing subsections.
+1) Convolution: The integrate-and-fire mechanism is im-
+plemented by combining convolution and the IF activation
+layer. The internal potentials of the neurons are computed
+using convolution operation, and the IF activation operation
+produces spikes where neurons have a potential higher than
+a specified threshold. Multiple layers can be assembled and
+stacked on top of one another to create deeper structures.
+The convolution layer has a kernel with Co×Ci×Kh×Kw
+shape. the synaptic weights are initialized randomly with
+a normal distribution. It performs two-dimensional convo-
+lution with support for padding and stride. The input has
+B ×T ×Ci ×Hi ×Wi shape which corresponds to batch, time
+steps, channels, height, and width of the input, respectively.
+The output has B × T × Co × Ho × Wo shape where:
+Ho = ⌊Hi + 2 × Ph − Kh
+Sh
+⌋ + 1
+Wo = ⌊Wi + 2 × Pw − Kw
+Sw
+⌋ + 1
+(2)
+And Ph, Pw, Sh, Sw are the height and width of convolution
+padding and stride. Padding increases the size of the two-
+dimensional input before convolution operation by expanding
+the edges of the input and filling in the new space with a
+constant value (usually zero). Stride is the number of steps
+the convolution window takes when it moves on the image.
+The output of the convolution layers are internal potentials
+of neurons that need to be passed through an IF activation
+layer to become output spike trains. This layer is imple-
+mented with spyker.Conv(insize, outsize, kernel, stride, pad,
+mean, std, device) class in Spyker.
+2) Fully Connected: The fully connected layer is combined
+with the IF activation to model the IF neurons, much similar
+to what happens in the convolution layers. This layer has a
+kernel with I × O shape. The synaptic weights are initialized
+
+5
+randomly with a normal distribution. The input has B ×T ×I
+which corresponds to batch, time steps, and input size, respec-
+tively. The output has B × T × O shape. The fully connected
+layer is represeneted by spyker.FC(insize, outsize, mean, std,
+device) in the library.
+3) Pooling: The pooling layer performs two-dimensional
+max pooling operation with a window size ofLh×Lw, a stride
+of Sh ×Sw, and a padding of Ph, Pw. The input has B ×T ×
+Ci×Hi×Wi shape and the output has B×T ×Co×Ho×Wo
+shape where:
+Ho = ⌊Hi + 2 × Ph − Lh
+Sh
+⌋ + 1
+Wo = ⌊Wi + 2 × Pw − Lw
+Sw
+⌋ + 1
+(3)
+The interface of Spyker has the spyker.pool(array, kernel,
+stride, pad, rates) function to run the pooling operation on the
+input given the kernel, stride, and padding size. rates argument
+is the rate of firing of the neurons when rate coding is used.
+The pooling operation selects neurons that fire earlier when
+rank order coding is used, and selects neurons that have a
+higher firing rate when rate coding is used.
+F. Learning
+Learning in the brain happens when the strength of connec-
+tions change between its neurons, and this change in strength
+is named synaptic plasticity [45]. Learning methods that utilize
+synaptic plasticity have been developed for SNNs [27]–[29].
+1) Spike-timing-dependent Plasticity:
+One widely rec-
+ognized synaptic plasticity learning rule is spike-timing-
+dependent plasticity (STDP) [27], [28]. STDP learning rule op-
+erates by adjusting synaptic weights and utilizing the timing of
+the spikes. A pre-synaptic neuron firing before (after) the post-
+synaptic neuron results in a strengthed (weakened) connection.
+STDP allows the neurons to extract and learn frequent features
+in the input [46]. STDP layer changes synaptic weights with
+stabilization:
+∆Wi,j =
+�
+A+
+k (Wi,j − Lk)(Uk − Wi,j),
+tj ≤ ti
+A−
+k (Wi,j − Lk)(Uk − Wi,j),
+tj ≥ ti
+(4)
+where A+
+k , A−
+k , Lk, Uk are the positive learning rate, negative
+learning rate, lower bound, and upper bound of the kth
+configuration, respectively. If stabilization is not set, then the
+formula becomes:
+∆Wi,j =
+�
+A+
+k ,
+tj ≤ ti
+A−
+k ,
+tj ≥ ti
+(5)
+then the weights are computed:
+W +
+i,j = max(Lk, min(Uk, ∆Wi,j))
+(6)
+Input, neurons selected by the winner-take-all mechanism
+(WTA), and the output are passed to a function belonging to
+fully connected or convolution layers, and the STDP learn-
+ing rule is applied. Convolution or fully connected layers
+in Spyker can have multiple STDP configurations (differ-
+ent learning rules, weight clipping, enabling/disabling stabi-
+lizer) implemented as spyker.STDPConfig(positive, negative,
+stabilize, lower, upper). Each winner neuron can be mapped
+to an STDP configuration, and that neuron will be updated
+using the learning rates and such that belongs to the selected
+configuration. SpykeTorch creates an STDP object for each
+configuration, and mapping winner neurons to different con-
+figurations is done by the user. Compared to SpykeTorch,
+Spyker provides a more flexible and easy to use API for
+weight updating and enables batch updating, which improves
+performance. Samples are processed in mini-batches which
+increases performance drastically (see the results section),
+and the batch update rule does not differ from single-sample
+processing.
+2) Reward-modulated STDP: Another approach is using
+the reinforcement (RL) learning rule. One method based on RL
+is reward-modulated STDP [29]. R-STDP adjusts the STDP
+such that neurons that respond correctly are rewarded, and
+punished otherwise. It has been suggested [33] that when
+the input has non-diagnostic frequent features that are less
+effective in decision-making, R-STDP is able to discard these
+features and improve the decision-making process. Since con-
+volution and fully connected layers accept STDP configura-
+tions as input, R-STDP can be implemented by passing two
+configurations to a layer (one for rewarding and one for pun-
+ishing), and mapping each winner neuron to a configuration
+based on data labels. If one formulates this, ∆Wi,j will be:
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+A+
+r (Wi,j − Lr)(Ur − Wi,j),
+tpre < tpost
+A−
+r (Wi,j − Lr)(Ur − Wi,j),
+tpre ≥ tpost
+,
+if reward
+�
+A−
+p (Wi,j − Lp)(Up − Wi,j),
+tpre < tpost
+A+
+p (Wi,j − Lp)(Up − Wi,j),
+tpre ≥ tpost
+,
+if punish
+(7)
+3) Winner-take-all and Lateral Inhibition: When a neu-
+ron fires at a specific location, lateral inhibition [47], [48]
+operation inhibits other neurons belonging to other neural
+maps from firing in that location. Lateral inhibition for the
+convolution operation can be used with spyker.inhibit(array
+, thershold, inplace) functions. Winner neurons that STDP
+weight updating will be performed on are selected by the
+winner-take-all [49], [50] operation. WTA selects neurons that
+fire earlier, and if the firing time of neurons is the same, then
+the one that has a higher internal potential will be selected.
+This operation is implemented with spyker.fcwta(array, radius
+, count, threshold) for fully connected and spyker.convwta(
+array, radius, count, threshold) for convolution operations.
+III. RESULTS
+In this section, we will test the performance of the library
+against the SpykeTorch library. Afterward, a comparison of the
+represented stimuli extracted from Spyker to recorded electro-
+physiology data is conducted to demonstrate the applicability
+of SNNs in describing the underlying neural mechanisms of
+brain functions.
+A. Library Performance
+In this section, we compare the performance of the library to
+SpykeTorch on two networks that classify the MNIST dataset.
+
+6
+SpykeTorch
+Spyker Python
+Spyker Python Alt
+Spyker C++
+Spyker C++ Alt
+95
+96
+97
+98
+99
+100
+Accuracy (%)
+96.72
+97.55
+97.632
+97.502
+97.606
+Accuracy results for the MNIST dataset
+SpykeTorch
+Spyker Python
+Spyker Python Alt
+Spyker C++
+Spyker C++ Alt
+0h
+3h
+6h
+9h
+12h
+15h
+18h
+21h
+Time (s)
+21h 17m
+4h 21m
+3h 21m
+4h 7m
+3h 8m
+Runtime results for the MNIST dataset
+Fig. 3: Comparison plots of the runtime and accuracy of
+Spyker aganist SpykeTorch on the Mozafari et al. network.
+The plot on the left shows the runtime comparison of Spyker
+and SpykeTorch implementations. The plot on the right also
+compares accuracy of the two implementations. Comparisons
+are between SpykeTorch (ST), implementation using Spyker
+in Python (SP Py), alternative version using Spyker in Python
+(SPA Py), and their C++ counterparts (SP C++, SPA C++). The
+error bars are minimum and maximum values of the samples.
+1) R-STDP Network: The first netwrok is the Mozfari et al.
+network [33] which has three convolutional layers. The first
+layer is trained two times with STDP, the second layer four
+times with STDP, and the third layer 680 times with R-STDP
+on the training set while compuing the test accuracy at each
+iteration while training the third layer. We made a small change
+to the structure of the network (named Alt for alternative).
+We removed the input padding from the last convolution layer
+and changed its window size to 4 and the output channels
+to 400. Results can be seen in Figure 3 and Table I. All the
+tests are performed on Inte Core i7-9700k with 64G memory
+and Nvidia Geforce GTX 1080 Ti with 12G memory (Ubuntu
+18.04).
+In order to compare the results, we test whether the two-
+sample mean difference confidence interval (99.9%) contains
+zero. The null hypothesis is having the same means, and the
+alternative is having different means. The test results indicate
+that the Spyker Python implementation is faster compared
+to the SpykeTorch implementation (Confidence intervals are
+TABLE I: Comparisons of the the runtime and accuracy of
+Spyker aganist SpykeTorch on the Mozafari et al. network.
+Implementation
+Time
+Time
+(S±SD)
+Accuracy
+(%±SD)
+Runs
+SpykeTorch
+21h17m
+76,672±916
+96.720±0.163
+12
+Spyker Python
+04h49m
+15668±52
+97.550±0.169
+30
+Spyker Python Alt
+03h31m
+12,114±14
+97.632±0.112
+30
+Spyker C++
+03h52m
+14,869±50
+97.502±0.157
+30
+[15477, 15859] and [72607, 80737] for Spyker and Spyke-
+Torch respectively, showing no intersection). Furthermore,
+the alternative implementation is faster both in the Python
+implementation with [-3738, -3370] interval and the C++
+implementation with [-3828, -3339] interval. As expected, the
+C++ interface is faster compared to the Python interface with [-
+1078, -520] interval. The results for the accuracy comparisons
+show that there are no significant differences ([96.932, 98.169]
+and [95.996, 97.444] for Python vs SpykeTorch implementa-
+tions respectively, showing intersection, [-0.89, 0.793] for C++
+vs Python, [-0.649, 0.813] for Python alternative vs Python,
+and [-0.763, 0.971] for C++ alternative vs C++).
+2) STDP Network: Subsequently, the Kheradpisheh et al.
+network [32] is used for comparisons. This network is made of
+two convolutional layers. The first layer is trained 2 times with
+STDP, and the second layer is trained 20 times with STDP on
+the training set. The output of the network is classified uing
+the SVM classifier. The elapsed time measured consists of
+the time needed to train the network on the training set and
+make predictions for the testing set. The time to utilize SVM
+is not taken into account because the libraries that simulate
+the neural network portion are compared here. The results can
+be seen in in Figure 4 and Table II.
+TABLE II: Comparisons of the the runtime and accuracy
+of Spyker aganist SpykeTorch on the Kheradpisheh et al.
+network.
+Implementation
+Time
+Time
+(S±SD)
+Accuracy
+(%±SD)
+Runs
+SpykeTorch GPU
+47m30s
+2,850±64
+98.392±0.093
+30
+Spyker GPU Single
+21m23s
+1,283±6
+98.465±0.095
+30
+Spyker GPU
+05m53s
+353±9
+98.461±0.079
+30
+Spyker Sparse
+08m16s
+496±1
+98.464±0.091
+30
+The test results indicate that the Spyker GPU implemen-
+tation is faster compared to the SpykeTorch implementation
+(confidence interval [-2728, -2265]). Since the SpykeTorch
+implementation processes one sample at a time, we also
+implemented a single sample version on the GPU, and this
+implementation runs faster compared to the SpykeTorch im-
+plementation (confidence interval [-1795, -1338]). There is
+also an implementation using the sparse interface of the
+Spyker (that runs on CPU) that is faster than the SpykeTorch
+implementation on the GPU (confidence interval [-2586, -
+2120]). These results show that the Spyker implementation is
+faster while the accuracy is not significantly different ([-0.373,
+0.511] for Spyker GPU, [-0.458, 0.603] for single-sample,
+and [-0.405, 0.549] for sparse implementation, all against the
+
+7
+SpykeTorch GPU
+Spyker GPU Single
+Spyker GPU
+Spyker CPU Sparse
+95
+96
+97
+98
+99
+100
+Accuracy (%)
+98.393
+98.465
+98.462
+98.465
+Accuracy results for the MNIST dataset
+SpykeTorch GPU
+Spyker GPU Single
+Spyker GPU
+Spyker CPU Sparse
+0m
+10m
+20m
+30m
+40m
+50m
+Time (s)
+47m 30s
+21m 23s
+5m 53s
+8m 16s
+Runtime results for the MNIST dataset
+Fig. 4: Comparison plots of the runtime and accuracy of
+Spyker against SpykeTorch on the Kheradpisheh et al. net-
+work. The plot on the left shows shows the runtime com-
+parison of Spyker and SpykeTorch implementations. The plot
+on the right also compares accuracy of the two implementa-
+tions. Comparisons are between GPU implementation using
+SpykeTorch (SP GPU), GPU implementation using Spyker
+with single-sample instead of batch processing (SP Single),
+GPU implementation using Spyker (SP GPU), and Sparse CPU
+implementation using Spyker (SP Sparse). The error bars are
+minimum and maximum values of the samples.
+SpykeTorch implementation).
+B. Analyzing the Underlying Structures of the Brain
+In order to demonstrate the use case and the importance
+of the library in neuroscience research, a similarity analysis is
+done in this section to compare the biological plausibility of an
+SNN and a deep CNN model. The neural data needed for the
+analysis is recorded as spiking activity and LFP signals from
+Inferior Temporal (IT) cortex using a single electrode (169
+sessions from two macaque monkeys, the neural data for the
+monkeys are pooled together) [51]. The task implemented here
+is a Rapid Serial Visual Presentation (RSVP). The intervals
+are 50ms for stimulus and 450ms interstimulus. Eighy-one
+greyscale images of real-world objects and Gaussian low-pass
+filtered and high-pass filtered variations of some are shown
+during the task (total 155 images). The categories of the stimuli
+are animal face (AF), human face (HF), animal body part(AB),
+human body part (HB), natual objects (N), and man-made
+objects (MM).
+The SNN used here is structurally similar to the one intro-
+duced by Shirsavar et al. [52]. The input of the SNN is resized
+to 32 and passed through 3 LoG fitlers with stds of 0.471,
+1.099, 2.042. The window sizes of the filters are 7. Then, the
+output is thresholded and coded into 15 time steps. The first
+convolution layer has 16 output channels with awindow size
+of 5 and a padding of 2, and the second convolution layer has
+32 output channels with a window size of 3 and a padding of
+1. The pooling layers have 2 and 3 window sizes, respectively.
+The layers are trained 20 times on the images, and the learning
+rates are doubled after each image until they reach 0.15. Firing
+times (divided by number of time steps) of the final layer is
+used as the network output.
+The CNN network used here is a ResNet-50 with the
+classifier layer replaced. The network is not pretrained. The
+input image is resized to 256 and cropped to 224. The network
+is trained 15 times on the dataset with Adam optimizer and
+0.0001 learning rate. using a linear SVM classifier to classify
+the 6 categories. the accuracies for the 6 classes are 51.569 ±
+2.240 (SD), 48.623 ± 2.538, and 51.247 ± 2.257 for ResNet-
+50, SNN, and an SVM classifier that is used on the average
+firing rates of the neural recordings of the monkeys between
+150ms and 200m from the onset, respectively. Figure 5 Shows
+the results of the analysis. The average Kendall’s Tau value
+for the interval between 125ms and 175ms shown in the figure
+is tested between the SNN and the ResNet. Using a Mann-
+Whitney U test with the alpha value of 0.001 results in a p-
+value of 2.028-07, which shows significant difference between
+the two. This indicates that the SNN has a closer structure to
+monkey brain.
+C. Rate Coding Output
+In this section, we look at the output of an SNN that uses
+rate coding. The SNN network used here is the Shirsavar et
+al. [52]. The number of output channels in the convolutional
+layers are set to 25 and 50. The training is not changed in
+that 15 time steps are used with rank order coding. However,
+the inference is done with 300 time steps and rate coding.
+Afterward, the spike output of 40 neurons are plotted for one
+testing sample for each class shown in Figure 6. The figure also
+cointains a plot of T-SNE transformed firing rates as output
+fetures and the recall score for each class for the average of
+30 runs. The accuracy of the 30 runs is 95.635±0.171 on the
+testing set.
+IV. LIBRARY DEMONSTRATION
+In this section, a sample usage of the library is illustrated.
+The network used here is introduced by Shirsavar et al.
+[52] to classify the MNIST dataset. The network has two
+convolutional layers trained with the STDP learning rule.
+The code shown in this section is only a part of the actual
+implementation, with the aim of providing a simple example.
+For the complete implementation, please visit the GitHub
+repository of Spyker1.
+1https://github.com/ShahriarRezghi/Spyker
+
+8
+Inferior Temporal Cortex
+450ms 
+Blank
+50ms
+Stimulus
+450ms
+Blank
+Time
+SNN
+ResNet
++
++
+Fig. 5: Similarity comparison of SNN and ResNet-50 to monkey neural data. The similarity measurement used here is the
+cosine similarity. The RDM for the monkey is computed for the 50ms interval after the onset. The RDMs are adjusted with
+histogram equalization. The RSA is calculated with 50ms window size and 5ms stride and 95% confidence interval. Kendall’s
+Tau measurement is used for the RSA analysis. The RSA is averaged in the interval between 125ms and 175ms and compared
+in the plot in the top right with 95% confidence interval.
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Fig. 6: Raster plot of an SNN network for the MNIST test images. In this figure, 40 neurons are plotted in 300 time steps for
+10 samples of the MNIST testing set, each image belonging to one class.
+
+9
+A. Transformation
+The transformation from the input image to the network
+input consists of feature enhancement and spike coding, shown
+in Listing 1. Here, a module named Transform is defined that
+performs the transformation when called. This module applies
+3 LoG filters with different standard deviations to the input
+image with padding to keep the original width and height of
+the input. The output is stored in 6 channels. Each channel
+of this output is then coded into fifteen time steps using rank
+order coding.
+Listing 1: Implementation of the Transform module
+class Transform :
+def
+init
+( self , device ) :
+std = [0.471 , 1.099, 2.042]
+self . f i l t = spyker .LoG(3 , std ,
+pad=3, device=device )
+def
+call
+( self , data ) :
+data = self . f i l t ( data )
+spyker . threshold ( data , 0.01)
+return spyker . code( data , 15)
+B. Network
+The network has two convolutional layers. Here, a module
+named Network is defined (shown in Listing 2) to train the
+neurons and make predictions. Here, the convolution layers
+are initialized, STDP configurations are set, and the winner
+selection function is wrapped with a lambda function to keep
+the hyperparameters in the initialization of the function of the
+network.
+Listing 2: Implementation of the Network module
+class Network:
+def
+init
+( self , device ) :
+self . thresh1 , self . thresh2 = 16, 5
+self .conv1 = spyker .Conv(6 , 100, 5,
+pad=2, mean=.5 , std =.02, device=device )
+self .conv2 = spyker .Conv(100, 200, 3,
+pad=1, mean=.5 , std =.02, device=device )
+config1 = spyker .STDPConfig(.0004 , −.0003)
+config2 = spyker .STDPConfig(.0004 , −.0003)
+self .conv1. stdpconfig = [ config1 ]
+self .conv2. stdpconfig = [ config2 ]
+self .wta1 = lambda x: spyker . convwta(x, 3, 5)
+self .wta2 = lambda x: spyker . convwta(x, 1, 8)
+C. Learning
+Training each layer is done in a separate function shown in
+Listing 3. The training of the layers is done in a sequantial
+order (one layer after another). Training of the first layer is
+done in the train layer1 function with the STDP learning rule.
+Here, the output of the first convolution is computed, and
+lateral inhibition is performed on it. Then, winner neurons are
+selected, and STDP weight updating is performed on them.
+The STDP learning rates in the first layer are multiplied by
+1.5 every 2000 samples, and the multiplying process stops
+once the positive learning rate reaches 0.15. The second layer
+is trained in a similar way in the train layer2 function with
+the STDP learning rule.
+Listing 3: The code for training of the network layers
+def train layer1 ( self , data ) :
+output = self .conv1( data )
+spyker . threshold ( output , self . thresh1 )
+spyker . inhibit ( output )
+winners = self .wta1( output )
+spikes = spyker . fire ( output )
+self .conv1. stdp ( data , winners , spikes )
+def train layer2 ( self , data ) :
+data = self .conv1( data )
+data = spyker . fire (data , self . thresh1 )
+data = spyker . pool(data , 2)
+output = self .conv2( data )
+spyker . threshold ( output , self . thresh2 )
+spyker . inhibit ( output )
+winners = self .wta2( output )
+spikes = spyker . fire ( output )
+self .conv2. stdp ( data , winners , spikes )
+After defining the network module, the process of training
+and classification is implemented. The training process shown
+in Listing 4 involves training each layer once with quantization
+afterward.
+Listing 4: The training process of the network
+for data , target in trainset :
+network . train layer1 ( transform ( data ) )
+spyker . quantize (network .conv1. kernel , 0, 0.5 , 1)
+for data , target in trainset :
+network . train layer2 ( transform ( data ) )
+spyker . quantize (network .conv2. kernel , 0, 0.5 , 1)
+D. Inference
+The call operator of the network shown in Listing 5 im-
+plements the prediction procedure which processes the input
+spikes and produces the final network output.
+Listing 5: Inference function of the network
+def
+call
+( self , data ) :
+data = self .conv1( data )
+data = spyker . fire (data , self . thresh1 )
+data = spyker . pool(data , 2)
+data = self .conv2( data )
+data = spyker . fire (data , self . thresh2 )
+data = spyker . pool(data , 3)
+return spyker . gather ( data ) . flatten (1)
+After training, the output features for every sample in the
+training set and the testing set are computed (in the gather
+
+10
+function). Then, an SVM classifier is trained on the training
+set outputs. Finally, predictions are made for the testing set
+outputs (shown in Listing 6).
+Listing 6: Implementation of the dimension reduction and
+classification operations
+xtr ,
+ytr = gather ( network ,
+transform ,
+train )
+xte , yte = gather ( network ,
+transform ,
+test )
+svm = LinearSVC(C=2.4) . fit ( xtr ,
+ytr )
+pred = svm. predict ( xte )
+accuracy = ( pred == testy .numpy() ) .mean()
+V. DISCUSSION
+Our brain has amazing capabilities. It can learn and perform
+complicated tasks in a robust manner and with low power
+consumption. Artificial neural networks have been created
+to mimic the power of the brain processes. Deep neural
+networks are ANNs that have had major success in recent
+years. However, there are structural differences between these
+networks and the brain, and they encounter problems when
+it comes to tolerance, energy, and sample efficiency. Spiking
+neural networks are the next generation of artificial neural
+networks. SNNs are not a new concept. However, they have
+been brought to attention recently due to their promising
+characteristics. The aim of these networks is to build a better
+model of the brain compared to DNNs.
+Several well-established simulation tools exist for DNNs.
+These tools have allowed DNNs to reach their great success
+faster and have helped them to computationally scale up. SNNs
+lack such high-performance simulation tools. There have been
+some attempts at creating such tools, but they have not been
+able to live up to expectations. In this work, we introduced
+Spyker, a high-performance library written from scratch using
+low-level tools to simulate spiking neural networks on both
+CPUs and GPUs. Despite being stand-alone, Spyker has great
+flexibility and the ability to integrate with other tools to
+create a smooth developing experience. We compared the
+performance of this library with SpykeTorch, a simulation tool
+built on the PyTorch framework. We showed that Spyker is
+multiple times faster compared to this library. Furthermore,
+to demonstrate the applicability of SNNs in describing the
+underlying neural mechanisms of the brain functions and the
+role of Spyker in this field, we compared the similarity of
+a spiking neural network implemented with this library with
+the similarity of the ResNet model to a macaque monkey
+brain. Finally, we illustrated an example implementation to
+demonstrate the easy and modern interface of the library.
+Strong SNN models can be implemented using the Spyker
+library to solve real-world machine learning problems. Fea-
+tures like fast processing and having a C++ interface alongside
+the Python interface make this library ready for both research
+and production. Generalization is an important concept in
+machine learning and having neural networks that learn and
+run fast are quite desirable. SNNs have the potential to become
+state-of-the-art models in machine learning. Other potential
+use cases of the library is to study and understand how the
+brain processes information using simulations. In other words,
+this library enables us to look at neuroscience through the eyes
+of a brain-inspired neural network.
+Although this library has been shown to be performant, there
+is room for more improvements. Spyker has a sparse interface
+that runs on the CPU. The sparse interface can be extended to
+also run on the GPU, and this can improve the performance
+even further. Furthermore, the support for a larger number
+of neural models, coding schemes, and learning rules can be
+added. This helps the library to cover a great range of SNN
+building blocks. When choosing a model to be deployed on
+embedded and neuromorphic processors, SNNs are among the
+top choices due to their energy efficiency. SNNs are often used
+in neuromorphic computing. Another direction that Spyker can
+take is in this direction. The computational efficiency of the
+sparse interface of Spyker can be further improved and made
+compatible with these types of processors.
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+Accurate and efficient multiscale simulation of a
+heterogeneous elastic beam via computation on
+small sparse patches
+A.J. Roberts∗
+Thien Tran-Duc†
+J.E. Bunder‡
+Yannis Kevrekidis§
+January 31, 2023
+Abstract
+Modern ‘smart’ materials have complex microscale structure, often with
+unknown macroscale closure. The Equation-Free Patch Scheme empowers
+us to non-intrusively, efficiently, and accurately simulate over large scales
+through computations on only small well-separated patches of the microscale
+system. Here the microscale system is a solid beam of random heterogeneous
+elasticity. The continuing challenge is to compute the given physics on
+just the microscale patches, and couple the patches across un-simulated
+macroscale space, in order to establish efficiency, accuracy, consistency, and
+stability on the macroscale. Dynamical systems theory supports the scheme.
+This research program is to develop a systematic non-intrusive approach, both
+computationally and analytically proven, to model and compute accurately
+macroscale system levels of general complex physical and engineering systems.
+Contents
+1
+Introduction
+2
+2
+Equation-free patch scheme
+4
+2.1
+Scheme is non-intrusive functional ‘wrapper’ . . . . . . . . . . . . .
+4
+2.2
+Scheme embeds macroscale dynamics . . . . . . . . . . . . . . . . .
+5
+3
+Scheme has proven accuracy
+6
+3.1
+Computation verifies exactness . . . . . . . . . . . . . . . . . . . . .
+6
+3.2
+Mathematical analysis proves consistency . . . . . . . . . . . . . . .
+8
+∗School of Mathematical Sciences, University of Adelaide, South Australia.
+mailto:
+[email protected] https://orcid.org/0000-0001-8930-1552
+†School of Mathematical Sciences, University of Adelaide, South Australia. https://orcid.
+org/0000-0002-2004-5156
+‡Mathematical Sciences, University of South Australia, Australia.
+https://orcid.org/
+0000-0001-5355-2288
+§Departments of Chemical and Biomolecular Engineering & Applied Mathematics and
+Statistics,
+Johns Hopkins University,
+Baltimore,
+Maryland,
+USA. https://orcid.org/
+0000-0003-2220-3522
+1
+arXiv:2301.13145v1  [math.NA]  20 Jan 2023
+
+1
+Introduction
+2
+4
+Conclusion
+8
+1
+Introduction
+In structural engineering, microscale lattice materials can be light and highly stiff
+with customizable macroscale mechanical properties (e.g., Somnic & Jo 2022).
+The challenge we address herein is to accurately and efficiently predict macroscale
+characteristics emergent from the microscale lattice. Similarly, composite materials
+and structures are inherently heterogeneous and anisotropic across multiple scales.
+Multiscale modelling is thus critical to the design of composite structures for
+lightweight mechanical performance (e.g., Raju et al. 2021, Lucarini et al. 2021).
+Such composite materials are used in electronics, space, medical, transportation,
+and other industries (e.g. Matouˇs et al. 2017). Herein we establish that the Equation-
+Free Patch Scheme can non-intrusively, efficiently, and accurately simulate over
+macroscales through computations on only small well-separated patches of the
+microscale system.
+Consider an example elastic beam with heterogeneous elasticity in 2D as in Figure 1:
+say 628 cm long, 20 cm wide. The beam is heterogeneous because it is constructed
+from a modern material with micro-structure of size 3 cm—so that the heterogeneity
+is ‘visible’ in Figure 1. With a 3 cm micro-grid, the modelling requires circa 5 000
+variables. This specific scenario is easily computable, ode23 took 14 s cpu time
+to simulate one period of beam bending oscillation. But if a more realistic 3 mm
+micro-structure is simulated, then the computation time increases by a factor
+of 1000. If 3D elasticity modelling is required for the beam, then the computation
+time increases by even more orders of magnitude. The patch scheme (e.g., Samaey
+et al. 2010) we develop herein potentially reduces macroscale computation time by
+orders of magnitude—more reduction in higher-D space and/or smaller micro-scale.
+The patch scheme achieves efficiency by only computing on small sparse patches
+in space. Section 2.1 discusses how the patch scheme is non-intrusive in that it
+just ‘wraps around’ a user’s microscale code—a desirable property also identified
+by Biezemans et al. (2022). The patch scheme, alternatively called the gap-tooth
+method, “has formal similarity with sp [superparametrization]” (Majda & Grooms
+2014, p.62) that was developed in meteorology for weather and climate predictions,
+and is also akin to the so-called fe-fft and fe2 methods (Lucarini et al. 2021,
+e.g.,§4.7).
+Figure 1: movie of a full-domain simulation of a heterogeneous beam showing that
+beam bending waves and longitudinal compression waves propagate with some
+‘average’ properties.
+0
+1
+2
+3
+4
+5
+6
+space x
+-0.2
+0
+0.2
+y
+time =   0.00,    E in 0.39        3
+
+1
+Introduction
+3
+Figure 2: a small part of the
+microscale grid used to code 2D
+elasticity. The grid is staggered
+on the microscale: ���, horizontal
+displacements and velocities;
+▲, vertical displacements and
+velocities; ⊚, ⊗, components of
+strain and stress tensor (1).
+▶
+▶
+▶
+▶
+▶
+▶
+▶
+▶
+▶
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+⊚
+⊚
+⊚
+⊚
+⊚
+⊚
+⊚
+⊚
+⊚
+⊗
+⊗
+⊗
+⊗
+⊗
+⊗
+⊗
+⊗
+⊗
+i − 1
+i
+i + 1
+j − 1
+j
+j + 1
+Figure 3: example of the 2D mi-
+croscale heterogeneous Young’s
+modulus Eij used in computing
+the elastic Lam´e parameters (3).
+In this example, we choose the
+heterogeneity to have microscale
+period four along the beam.
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+0
+5 · 10-2 0.1
+0.15
+0.2
+0.25
+-0.1
+-5 · 10-2
+0
+5 · 10-2
+0.1
+space x
+cross-beam y
+0.5
+1
+1.5
+2
+2.5
+A given microscale discretisation of heterogeneous elasticity
+We adopt
+a simple robust microscale approximation of 2D elasticity within the beam. On
+the staggered microscale xy-grid of Figure 2 define the displacements: ▶, hori-
+zontal uij(t); ▲, vertical vij(t). Microscale elasticity here first uses centred finite
+differences to compute stresses, for heterogeneous Lam´e parameters λ, µ, at the
+labelled microscale grid-points (Figure 2):
+⊗
+σxy := µij
+�
+δjuij/δyj + δivij/δxi
+�
+;
+(1a)
+⊚
+σxx := (λij + 2µij)δiuij/δxi + λijδjvij/δyj;
+(1b)
+⊚
+σyy := λijδiuij/δxi + (λij + 2µij)δjvij/δyj.
+(1c)
+Second, centred finite differences compute the following acceleration odes
+▶
+¨uij = δiσxx/δxi + δjσxy/δyj ,
+(2a)
+▲
+¨vij = δiσxy/δxi + δjσyy/δyj ,
+(2b)
+potentially with optional small phenomenological damping supplied by a discretisa-
+tion of κ∇2 ˙uij, κ∇2 ˙vij. The patch scheme wraps around whatever microscale code
+a user supplies—here it is the microscale system (1) and (2)
+We nondimensionalise the system so that the density is one, and the speed of a
+macroscale compression wave along the beam is about one, that is, time in these
+simulations is roughly in milli-seconds.
+Random periodic heterogeneity
+The Lam´e parameters which appear in the
+stresses (1) are
+λ :=
+νE
+(1 + ν)(1 − 2ν),
+µ :=
+E
+2(1 + ν),
+(3)
+
+2
+Equation-free patch scheme
+4
+in terms of Young’s modulus E and Poisson ratio ν. To have strong microscale
+heterogeneity we choose these parameters randomly so that at each microscale grid-
+point (iid): Eij is log-normal (here varies by factor of about ten); and νij is uniform
+on [0.25, 0.35]. Figure 3 shows an example Eij. Despite such strong heterogeneity,
+the movie of Figure 1 shows the macroscale dynamics appears relatively simple.
+2
+Equation-free patch scheme
+Instead of computing the entire beam as seen in Figure 1, the patch scheme computes
+only in small sparse spatial patches such as Figure 4. In this example case, the
+patch scheme reduces compute time by a factor ∝ r := (patch size)/(spacing H),
+which here is just a modest factor of 1/4. But with greater scale separation and/or
+in higher spatial dimensions, the scheme often reduces computational time by many
+orders of magnitude.
+The movie of Figure 4 shows a slow progressive wave of beam bending, together
+with a not-so-slow compression wave along the beam. These macroscale predictions
+are accurate (Section 3) due to the correctness of our simple coupling between
+patches—even when heterogeneity is strong.
+The patch scheme makes these
+accurate macroscale predictions even when the macroscale closure is unknown:
+the scheme does not code a closure. Further, ‘the closure’ varies depending upon
+human assumptions such as choosing averaged models versus cosserat models—the
+patch scheme makes no such closure assumptions. The only assumption is that the
+macroscale quantities of importance vary smoothly between neighbouring patches.
+2.1
+Scheme is non-intrusive functional ‘wrapper’
+Consider one of the patches of the 2D beam shown in Figure 4. With the given
+microscale xy-grid (Figure 2), zooming in to the microscale each patch is like that
+of Figure 5. Here each patch extends across the cross-section (y-dimension) of the
+beam. Open symbols in Figure 5 are ghost nodes outside the patch and implement
+given stress-free top/bottom conditions on the beam. The only addition required
+by the patch scheme are the edge values (‘squared’ micro-grid nodes in Figure 5)
+on the left/right of each patch.
+The patch scheme couples patches together by providing the patch-edge values
+through interpolation across the macroscale between patches (e.g., Roberts &
+Kevrekidis 2007, Roberts et al. 2014, Cao & Roberts 2016). Here we interpolate
+from each of the centre patch values across the beam (i = 4 in Figure 5) of ‘nearby’
+Figure 4: movie of a patch scheme simulation of a heterogeneous beam showing the
+macroscale propagation across the patches of beam bending waves and longitudinal
+compression waves.
+0
+1
+2
+3
+4
+5
+6
+space x
+-0.2
+0
+0.2
+y
+time =   0.00,    E in 0.35      3.2
+
+2
+Equation-free patch scheme
+5
+Figure 5: one example patch
+of the 2D elastic beam show-
+ing the microscale staggered
+grid (Figure 2). This is case
+of nsubpatch = 7 micro-grid in-
+tervals along the patch, and
+ny = 4 intervals across the
+beam.
+▶
+▶
+▶
+▶
+▶
+▶
+▶
+▶
+▶
+▶
+▶
+▶
+▶
+▶
+▶
+▶
+▶
+▶
+▶
+▶
+▶
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+⊚
+⊚
+⊚
+⊚
+⊚
+⊚
+⊚
+⊚
+⊚
+⊚
+⊚
+⊚
+⊚
+⊚
+⊚
+⊚
+⊚
+⊚
+⊗
+⊗
+⊗
+⊗
+⊗
+⊗
+⊗
+⊗
+⊗
+⊗
+⊗
+⊗
+⊗
+⊗
+⊗
+⊗
+⊗
+⊗
+⊗
+⊗
+⊗
+⊗
+⊗
+⊗
+▷
+▷
+▷
+▷
+▷
+▷
+⃝
+⃝
+⃝
+⃝
+⃝
+⃝
+▷
+▷
+▷
+▷
+▷
+▷
+⃝
+⃝
+⃝
+⃝
+⃝
+⃝
+□
+□
+□
+□
+□
+□
+□
+□
+□
+□
+□
+□
+□
+□
+i = 1
+2
+3
+4
+5
+6
+7
+j =1
+2
+3
+4
+patches, to determine the corresponding patch-edge value. Here we implement
+spectral (fft) interpolation between the patches for high accuracy (Section 3).
+The scheme does not presume that any average is appropriate.
+This implementation shows that the patch scheme is non-intrusive (e.g., Biezemans
+et al. 2022): it just ‘wraps around’ any micro-grid code a user trusts. Consequently,
+we provide a toolbox (Maclean et al. 2021) for others to implement the patch
+scheme around their micro-code.
+2.2
+Scheme embeds macroscale dynamics
+Given the patch scheme does not assume anything about what are ‘correct’
+macroscale variables, a crucial question is the following: how can we be assured
+that the patch scheme captures the macroscale slow dynamics?
+An answer is
+provided by the Whitney (1936) embedding theorem.
+Roughly, the theorem is that every mD manifold is parametrisable from almost
+every subspace of more than 2mD. Let’s see what this means for us. In essence,
+the patch scheme provides the higher-D subspace in which the slow manifold of
+the macroscale wave dynamics is embedded.
+For beams in two spatial dimensions, the basic macroscale beam models have, at
+each cross-section, displacement and velocity of both bending and compression.
+Thus the elastic beam dynamics has a slow manifold that is m = 4D at every
+cross-section.1 Alternatively, 2D cosserat beam models add a shear mode to the
+macroscale model—two more variables—leading to a not-quite-so-slow manifold of
+m = 6D at every cross-section. These physically based models are slow manifolds
+because they focus on the relatively slow waves of solutions varying slowly in space,
+and neglect all the faster high-frequency cross-waves.
+In the patch scheme, Figures 1 and 4 show simulations with a cross-section of
+ny = 7 micro-grid intervals, but let’s discuss the case of just ny = 4 (Figure 5).
+For ny = 4, there are seven microscale nodes across each patch edge. Each node
+has a displacement and velocity, and so leads to a 14D subspace for macroscale
+communication between patches.
+1Such statements, invoking a manifold or subspace “at every cross-section”, are in a sense
+developed by the theory of Roberts (2015). That is, in systems of large spatial extent there often
+are important, spatially global, invariant manifolds of high-D that are effectively decomposable
+into a union of spatially local manifolds/subspaces of relatively lower dimension—a dimension
+determined by the spatial cross-section—and that are weakly coupled to neighbouring locales.
+
+3
+Scheme has proven accuracy
+6
+Figure 6: multiscale spectrum
+of eigenvalues λ separates
+macroscale modes on the right
+from sub-patch microscale
+modes on the left. The axes
+are scaled nonlinearly. Here
+the small viscosity is 0.001
+so the microscale decays, but
+the macroscale waves are long-
+lasting.
+-3
+-1
+-0.3
+-0.1
+-0.03
+-0.01
+0
+-100
+-30
+-10
+-3
+-1
+-0.3
+-0.10
+0.1
+0.3
+1
+3
+10
+30
+100
+ℜλ
+ℑλ
+Because 14 > 2 · 6 > 2 · 4 , the Whitney embedding theorem asserts that the
+patch scheme exchanges enough information to almost surely parametrise both
+such slow manifolds of the macroscale dynamics. The patch scheme does not need
+to explicitly compute and exchange specific assumed macroscale average quantities.
+3
+Scheme has proven accuracy
+Section 3.2 discusses established theory which generally proves that the patch
+scheme makes accurate macroscale predictions. Such proofs are in stark contrast
+to the vast machine learning/artificial intelligence developments which prove
+very few general results: for example, Brenner & Koumoutsakos (2021) comment
+“. . . ml studies, as the lack of rigorous theory does not offer (yet!) guarantees
+of convergence”. Before discussing theory, we first report some computational
+verification of high accuracy.
+3.1
+Computation verifies exactness
+Here we restricted attention to linear elasticity so we know that the wrapped
+patch system is fully characterised by the resultant Jacobian matrix. We numeri-
+cally compute the Jacobian matrix of the patch scheme by elementary numerical
+differentiation.
+Because of the macroscale translational invariance of the patch scheme, the
+macroscale eigenvectors are correctly sinusoidal. Hence the only macroscale er-
+rors occur in the eigenvalues of the Jacobian. Figure 6 plots the spectrum of all
+eigenvalues for one example of random heterogeneity, in the case of five patches for
+simplicity. Observe there are:
+• (on the right) four λ = 0 of rigid beam motion;
+• four −0.001 ± i 1.057 and four −0.003 ± i 2.111 of compressions waves;
+• four −0.001 ± i 0.061 and four −0.004 ± i 0.237 of beam bending waves;
+• with the above macroscale eigenvalues separated by a spectral gap from the
+following sub-patch microscale eigenvalues;
+
+3
+Scheme has proven accuracy
+7
+Table 1: error in patch scheme’s
+macroscale eigenvalues λ for
+various patch size ratios r: the
+macroscale λs are exact to round-
+off error—due to patch coupling by
+spectral interpolation.
+macro-eigenvalue
+r = 1
+2
+r = 1
+4
+r = 1
+8
+−0.001 ± i 0.061
+2e-12
+1e-12
+2e-13
+−0.001 ± i 0.061
+2e-12
+4e-12
+2e-12
+−0.004 ± i 0.237
+1e-12
+8e-13
+3e-12
+−0.004 ± i 0.237
+1e-12
+2e-12
+3e-12
+−0.001 ± i 1.057
+7e-13
+4e-13
+6e-13
+−0.001 ± i 1.057
+6e-13
+5e-13
+6e-13
+−0.003 ± i 2.111
+1e-13
+2e-13
+2e-13
+−0.003 ± i 2.111
+4e-13
+5e-13
+2e-13
+Figure 7: multiscale spectrum
+of eigenvalues λ for the patch
+scheme in the case of zero viscos-
+ity. The horizontal axis shows
+that all modes have zero real-
+part to numerical round-off er-
+ror. That is, in the case of zero
+viscosity, this patch scheme pre-
+serves the wave nature of the
+underlying physics.
+-5e-13
+-2e-13
+-1e-130
+1e-13
+2e-13
+5e-13
+-100
+-30
+-10
+-3
+-1
+-0.3
+-0.10
+0.1
+0.3
+1
+3
+10
+30
+100
+ℜλ
+ℑλ
+• (on the left) many ℜλ < −0.1 of uninteresting sub-patch micro-scale fast-
+waves (headed by ten eigenvalues around −0.14 ± i 9.29).
+To quantify the accuracy, Table 1 compares eigenvalues obtained from full-domain
+code, with the above macroscale eigenvalues obtained by the wrapped patch scheme.
+For all patch size ratios and heterogeneities tested, the patch scheme’s macroscale
+eigenvalues are exact to numerical round-off error.
+Such exactness is due to the spectral interpolation used here. If, instead of spectral,
+local polynomial interpolation of degree p is used to couple the patches, then
+generally the patch scheme has macroscale errors ∝ Hp where H = inter-patch
+spacing (e.g., Roberts & Kevrekidis 2007, Roberts et al. 2014).
+Undamped waves?
+With zero viscosity, there are only oscillations in the under-
+lying physics. In such a scenario computational methods are very delicate. Here,
+Figure 7 illustrates that all eigenvalues of the patch scheme have |ℜλ| < 10−12.2
+Hence, even with no viscosity, the patch scheme preserves the oscillatory wave
+nature of the heterogeneous physics.
+There is a perception that the patch scheme “only works well on problems with
+an inertial manifold and for systems in which most modes are strongly decaying”
+(Majda & Grooms 2014, p.62). This verification of accuracy for purely elastic
+2In some realisations of the heterogeneity, the sensitive multiplicity four eigenvalue λ = 0
+numerically splits into four showing |ℜλ| up to 10−6 due to round-off errors.
+
+4
+Conclusion
+8
+beams shows that this perception is false. Applications and theory for other wave
+systems also refute this perception (e.g., Cao & Roberts 2016, Bunder et al. 2021,
+Divahar et al. 2022).
+3.2
+Mathematical analysis proves consistency
+Mathematical analysis has proven properties of the patch scheme in general. Mostly,
+the published proofs explicitly address dissipative (nonlinear) systems. However,
+as discussed by Bunder et al. (2021), the patch scheme in space only recasts spatial
+interactions, so whether the time derivative is ∂/∂t of dissipation or ∂2/∂t2 of
+waves makes little difference.
+Two complementary types of results have been proven. They involve the spacing
+between patch centres H.
+First, Centre Manifold Theory may be applied at
+finite spacing H by introducing a ‘bookkeeping’ parameter γ to label inter-patch
+communication (e.g., Roberts et al. 2014, §2) to prove the existence of a slow
+manifold in the patch scheme (including when it is applied to nonlinear systems).
+Then the parameter γ structures inter-patch interactions, and their algebraic
+expression, to empower theory based at γ = 0, via regular perturbation, to address
+finite γ such as the case of full coupling γ = 1 (e.g., Roberts et al. 2014, Cor. 2).
+Second, the patch scheme is consistent with the underlying micro-code as the
+patch spacing H → 0 (e.g., Roberts et al. 2014, Thm. 7). The consistency is
+that the macroscale of the patch scheme is the same as the macroscale of the
+given micro-coded system, to errors O
+�
+Hp�
+when using polynomial interpolation
+of degree p. For example, spectral interpolation corresponds to ‘p = ∞’ so then
+errors vanish to all orders as in Table 1.
+These results and general proofs were first done for homogeneous systems (e.g.,
+Roberts & Kevrekidis 2007, Roberts et al. 2014). They were subsequently ex-
+tended to heterogeneous microscales (Bunder et al. 2017), and recently extended
+to alternative inter-patch coupling that preserves self-adjointness (Bunder et al.
+2021). Interestingly, the extension of the theoretical support to heterogeneous
+cases invokes the ensemble of all phase-shifts of the heterogeneity. The ensemble is
+spatially homogeneous, so the homogeneous proofs and results apply to establish
+the heterogeneous results.
+4
+Conclusion
+As an initial exploration of the patch scheme for homogenisation of heterogeneous
+elasticity, we considered the prototypical case of a 2D elastic beam. The scheme
+gives a non-intrusive and efficient computational homogenisation of given microscale
+system via spatially sparse small patches. The patch coupling has proven accuracy,
+controllable error, at finite scale separation.
+The patch scheme makes only one assumption: in the scenarios of interest to a
+user, there is no significant spatial structures in the mesoscale between the patch
+spacing H and the microscale resolved in the patches. In contrast to most other
+multiscale methods, there is: no assumed boundary conditions on Representative
+Volume Elements (variously periodic, Dirichlet, Neumann); no explicitly assuming
+
+References
+9
+slow variables; and no presumed necessary variational principle. The scheme is
+entirely physically interpretable: there is no hidden mystic machinations of neural
+networks (e.g., Brenner & Koumoutsakos 2021)
+The patch scheme is simple to apply. In contrast to other multiscale methods
+there is: no arbitrary averaging; no oversampling regions; no buffer regions; no
+action regions; no guessed fast/slow variables; no epsilons; and no limits. As a
+non-intrusive ‘wrapper’, anyone can start using the patch scheme via a Matlab/
+Octave Toolbox (Maclean et al. 2021, Roberts et al. 2019���2023)
+Acknowledgements
+This research was supported by Australian Research Coun-
+cil grants DP220103156 and DP200103097.
+References
+Biezemans, R. A., Le Bris, C., Legoll, F. & Lozinski, A. (2022), Non-intrusive
+implementation of a wide variety of Multiscale Finite Element Methods, Technical
+report, http://arxiv.org/abs/2211.17024.
+Brenner, M. P. & Koumoutsakos, P. (2021), ‘Editorial: Machine Learning and
+Physical Review Fluids: An Editorial Perspective’, Physical Review Fluids
+6(7), 070001.
+Bunder, J. E., Kevrekidis, I. G. & Roberts, A. J. (2021), ‘Equation-free patch
+scheme for efficient computational homogenisation via self-adjoint coupling’,
+Numerische Mathematik 149(2), 229–272.
+Bunder, J. E., Roberts, A. J. & Kevrekidis, I. G. (2017), ‘Good coupling for the mul-
+tiscale patch scheme on systems with microscale heterogeneity’, J. Computational
+Physics 337, 154–174.
+Cao, M. & Roberts, A. J. (2016), ‘Multiscale modelling couples patches of nonlinear
+wave-like simulations’, IMA J. Applied Maths. 81(2), 228–254.
+Divahar, J., Roberts, A. J., Mattner, T. W., Bunder, J. E. & Kevrekidis, I. G.
+(2022), Two novel families of multiscale staggered patch schemes efficiently
+simulate large-scale, weakly damped, linear waves, Technical report, https:
+//arxiv.org/abs/2210.15823.
+Lucarini, S., Upadhyay, M. V. & Segurado, J. (2021), ‘FFT based approaches
+in micromechanics: fundamentals, methods and applications’, Modelling and
+Simulation in Materials Science and Engineering 30(2), 023002.
+Maclean, J., Bunder, J. E. & Roberts, A. J. (2021), ‘A toolbox of equation-free
+functions in matlab/octave for efficient system level simulation’, Numerical
+Algorithms 87, 1729–1748.
+Majda, A. J. & Grooms, I. (2014), ‘New perspectives on superparameterization for
+geophysical turbulence’, Journal of Computational Physics 271, 60–77.
+Matouˇs, K., Geers, M. G. D., Kouznetsova, V. G. & Gillman, A. (2017), ‘A review of
+predictive nonlinear theories for multiscale modeling of heterogeneous materials’,
+Journal of Computational Physics 330, 192–220.
+
+References
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+Raju, K., Tay, T.-E. & Tan, V. B. C. (2021), ‘A review of the FE2 method for
+composites’, Multiscale and Multidisciplinary Modeling, Experiments and Design
+4, 1–24.
+Roberts, A. J. (2015), ‘Macroscale, slowly varying, models emerge from the mi-
+croscale dynamics in long thin domains’, IMA Journal of Applied Mathematics
+80(5), 1492–1518.
+Roberts, A. J. & Kevrekidis, I. G. (2007), ‘General tooth boundary conditions for
+equation free modelling’, SIAM J. Scientific Computing 29(4), 1495–1510.
+Roberts, A. J., MacKenzie, T. & Bunder, J. (2014), ‘A dynamical systems approach
+to simulating macroscale spatial dynamics in multiple dimensions’, J. Engineering
+Mathematics 86(1), 175–207.
+Roberts, A. J., Maclean, J. & Bunder, J. E. (2019–2023), Equation-free function tool-
+box for matlab/octave, Technical report, [https://github.com/uoa1184615/
+EquationFreeGit].
+Samaey, G., Roberts, A. J. & Kevrekidis, I. G. (2010), Equation-free computation:
+an overview of patch dynamics, in J. Fish, ed., ‘Multiscale methods: bridging the
+scales in science and engineering’, Oxford University Press, chapter 8, pp. 216–
+246.
+Somnic, J. & Jo, B. W. (2022), ‘Status and challenges in homogenization methods
+for lattice materials’, Materials 15(2), 605.
+Whitney, H. (1936), ‘Differentiable manifolds’, Annals of Mathematics 37(3), 645–
+680.
+

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+page_content='Accurate and efficient multiscale simulation of a  heterogeneous elastic beam via computation on  small sparse patches  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' Roberts∗  Thien Tran-Duc†  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' Bunder‡  Yannis Kevrekidis§  January 31, 2023  Abstract  Modern ‘smart’ materials have complex microscale structure, often with  unknown macroscale closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' The Equation-Free Patch Scheme empowers  us to non-intrusively, efficiently, and accurately simulate over large scales  through computations on only small well-separated patches of the microscale  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' Here the microscale system is a solid beam of random heterogeneous  elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' The continuing challenge is to compute the given physics on  just the microscale patches, and couple the patches across un-simulated  macroscale space, in order to establish efficiency, accuracy, consistency, and  stability on the macroscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' Dynamical systems theory supports the scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  This research program is to develop a systematic non-intrusive approach, both  computationally and analytically proven, to model and compute accurately  macroscale system levels of general complex physical and engineering systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  Contents  1  Introduction  2  2  Equation-free patch scheme  4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
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+page_content='  8  ∗School of Mathematical Sciences, University of Adelaide, South Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  mailto:  ProfAJRoberts@protonmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='com https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='org/0000-0001-8930-1552  †School of Mathematical Sciences, University of Adelaide, South Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
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+page_content='  org/0000-0002-2004-5156  ‡Mathematical Sciences, University of South Australia, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='org/  0000-0001-5355-2288  §Departments of Chemical and Biomolecular Engineering & Applied Mathematics and  Statistics,  Johns Hopkins University,  Baltimore,  Maryland,  USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='org/  0000-0003-2220-3522  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='13145v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='NA]  20 Jan 2023  1  Introduction  2  4  Conclusion  8  1  Introduction  In structural engineering, microscale lattice materials can be light and highly stiff  with customizable macroscale mechanical properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=', Somnic & Jo 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  The challenge we address herein is to accurately and efficiently predict macroscale  characteristics emergent from the microscale lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' Similarly, composite materials  and structures are inherently heterogeneous and anisotropic across multiple scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  Multiscale modelling is thus critical to the design of composite structures for  lightweight mechanical performance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=', Raju et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' 2021, Lucarini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  Such composite materials are used in electronics, space, medical, transportation,  and other industries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' Matouˇs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' Herein we establish that the Equation-  Free Patch Scheme can non-intrusively, efficiently, and accurately simulate over  macroscales through computations on only small well-separated patches of the  microscale system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  Consider an example elastic beam with heterogeneous elasticity in 2D as in Figure 1:  say 628 cm long, 20 cm wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' The beam is heterogeneous because it is constructed  from a modern material with micro-structure of size 3 cm—so that the heterogeneity  is ‘visible’ in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' With a 3 cm micro-grid, the modelling requires circa 5 000  variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' This specific scenario is easily computable, ode23 took 14 s cpu time  to simulate one period of beam bending oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' But if a more realistic 3 mm  micro-structure is simulated, then the computation time increases by a factor  of 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' If 3D elasticity modelling is required for the beam, then the computation  time increases by even more orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' The patch scheme (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=', Samaey  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' 2010) we develop herein potentially reduces macroscale computation time by  orders of magnitude—more reduction in higher-D space and/or smaller micro-scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  The patch scheme achieves efficiency by only computing on small sparse patches  in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='1 discusses how the patch scheme is non-intrusive in that it  just ‘wraps around’ a user’s microscale code—a desirable property also identified  by Biezemans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' The patch scheme, alternatively called the gap-tooth  method, “has formal similarity with sp [superparametrization]” (Majda & Grooms  2014, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='62) that was developed in meteorology for weather and climate predictions,  and is also akin to the so-called fe-fft and fe2 methods (Lucarini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' 2021,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=',§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  Figure 1: movie of a full-domain simulation of a heterogeneous beam showing that  beam bending waves and longitudinal compression waves propagate with some  ‘average’ properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  0  1  2  3  4  5  6  space x  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='2  y  time =   0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='00,    E in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='39        3  1  Introduction  3  Figure 2: a small part of the  microscale grid used to code 2D  elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' The grid is staggered  on the microscale: ▶, horizontal  displacements and velocities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  ▲, vertical displacements and  velocities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' ⊚, ⊗, components of  strain and stress tensor (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  ▶  ▶  ▶  ▶  ▶  ▶  ▶  ▶  ▶  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ⊚  ⊚  ⊚  ⊚  ⊚  ⊚  ⊚  ⊚  ⊚  ⊗  ⊗  ⊗  ⊗  ⊗  ⊗  ⊗  ⊗  ⊗  i − 1  i  i + 1  j − 1  j  j + 1  Figure 3: example of the 2D mi-  croscale heterogeneous Young’s  modulus Eij used in computing  the elastic Lam´e parameters (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  In this example, we choose the  heterogeneity to have microscale  period four along the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
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+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='5 · 10-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='1  5 · 10-2  0  5 · 10-2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='1  space x  cross-beam y  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='5  A given microscale discretisation of heterogeneous elasticity  We adopt  a simple robust microscale approximation of 2D elasticity within the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' On  the staggered microscale xy-grid of Figure 2 define the displacements: ▶, hori-  zontal uij(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' ▲, vertical vij(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' Microscale elasticity here first uses centred finite  differences to compute stresses, for heterogeneous Lam´e parameters λ, µ, at the  labelled microscale grid-points (Figure 2):  ⊗  σxy := µij  �  δjuij/δyj + δivij/δxi  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  (1a)  ⊚  σxx := (λij + 2µij)δiuij/δxi + λijδjvij/δyj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  (1b)  ⊚  σyy := λijδiuij/δxi + (λij + 2µij)δjvij/δyj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  (1c)  Second, centred finite differences compute the following acceleration odes  ▶  ¨uij = δiσxx/δxi + δjσxy/δyj ,  (2a)  ▲  ¨vij = δiσxy/δxi + δjσyy/δyj ,  (2b)  potentially with optional small phenomenological damping supplied by a discretisa-  tion of κ∇2 ˙uij, κ∇2 ˙vij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' The patch scheme wraps around whatever microscale code  a user supplies—here it is the microscale system (1) and (2)  We nondimensionalise the system so that the density is one, and the speed of a  macroscale compression wave along the beam is about one, that is, time in these  simulations is roughly in milli-seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  Random periodic heterogeneity  The Lam´e parameters which appear in the  stresses (1) are  λ :=  νE  (1 + ν)(1 − 2ν),  µ :=  E  2(1 + ν),  (3)  2  Equation-free patch scheme  4  in terms of Young’s modulus E and Poisson ratio ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' To have strong microscale  heterogeneity we choose these parameters randomly so that at each microscale grid-  point (iid): Eij is log-normal (here varies by factor of about ten);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' and νij is uniform  on [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' Figure 3 shows an example Eij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' Despite such strong heterogeneity,  the movie of Figure 1 shows the macroscale dynamics appears relatively simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  2  Equation-free patch scheme  Instead of computing the entire beam as seen in Figure 1, the patch scheme computes  only in small sparse spatial patches such as Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' In this example case, the  patch scheme reduces compute time by a factor ∝ r := (patch size)/(spacing H),  which here is just a modest factor of 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' But with greater scale separation and/or  in higher spatial dimensions, the scheme often reduces computational time by many  orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  The movie of Figure 4 shows a slow progressive wave of beam bending, together  with a not-so-slow compression wave along the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' These macroscale predictions  are accurate (Section 3) due to the correctness of our simple coupling between  patches—even when heterogeneity is strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  The patch scheme makes these  accurate macroscale predictions even when the macroscale closure is unknown:  the scheme does not code a closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' Further, ‘the closure’ varies depending upon  human assumptions such as choosing averaged models versus cosserat models—the  patch scheme makes no such closure assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' The only assumption is that the  macroscale quantities of importance vary smoothly between neighbouring patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='1  Scheme is non-intrusive functional ‘wrapper’  Consider one of the patches of the 2D beam shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' With the given  microscale xy-grid (Figure 2), zooming in to the microscale each patch is like that  of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' Here each patch extends across the cross-section (y-dimension) of the  beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' Open symbols in Figure 5 are ghost nodes outside the patch and implement  given stress-free top/bottom conditions on the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' The only addition required  by the patch scheme are the edge values (‘squared’ micro-grid nodes in Figure 5)  on the left/right of each patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  The patch scheme couples patches together by providing the patch-edge values  through interpolation across the macroscale between patches (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=', Roberts &  Kevrekidis 2007, Roberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' 2014, Cao & Roberts 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' Here we interpolate  from each of the centre patch values across the beam (i = 4 in Figure 5) of ‘nearby’  Figure 4: movie of a patch scheme simulation of a heterogeneous beam showing the  macroscale propagation across the patches of beam bending waves and longitudinal  compression waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  0  1  2  3  4  5  6  space x  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='2  y  time =   0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='00,    E in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='35      3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='2  2  Equation-free patch scheme  5  Figure 5: one example patch  of the 2D elastic beam show-  ing the microscale staggered  grid (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' This is case  of nsubpatch = 7 micro-grid in-  tervals along the patch, and  ny = 4 intervals across the  beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  ▶  ▶  ▶  ▶  ▶  ▶  ▶  ▶  ▶  ▶  ▶  ▶  ▶  ▶  ▶  ▶  ▶  ▶  ▶  ▶  ▶  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ⊚  ⊚  ⊚  ⊚  ⊚  ⊚  ⊚  ⊚  ⊚  ⊚  ⊚  ⊚  ⊚  ⊚  ⊚  ⊚  ⊚  ⊚  ⊗  ⊗  ⊗  ⊗  ⊗  ⊗  ⊗  ⊗  ⊗  ⊗  ⊗  ⊗  ⊗  ⊗  ⊗  ⊗  ⊗  ⊗  ⊗  ⊗  ⊗  ���  ⊗  ⊗  ▷  ▷  ▷  ▷  ▷  ▷  ⃝  ⃝  ⃝  ⃝  ⃝  ⃝  ▷  ▷  ▷  ▷  ▷  ▷  ⃝  ⃝  ⃝  ⃝  ⃝  ⃝  □  □  □  □  □  □  □  □  □  □  □  □  □  □  i = 1  2  3  4  5  6  7  j =1  2  3  4  patches, to determine the corresponding patch-edge value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' Here we implement  spectral (fft) interpolation between the patches for high accuracy (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  The scheme does not presume that any average is appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  This implementation shows that the patch scheme is non-intrusive (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=', Biezemans  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' 2022): it just ‘wraps around’ any micro-grid code a user trusts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' Consequently,  we provide a toolbox (Maclean et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' 2021) for others to implement the patch  scheme around their micro-code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='2  Scheme embeds macroscale dynamics  Given the patch scheme does not assume anything about what are ‘correct’  macroscale variables, a crucial question is the following: how can we be assured  that the patch scheme captures the macroscale slow dynamics?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  An answer is  provided by the Whitney (1936) embedding theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  Roughly, the theorem is that every mD manifold is parametrisable from almost  every subspace of more than 2mD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' Let’s see what this means for us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' In essence,  the patch scheme provides the higher-D subspace in which the slow manifold of  the macroscale wave dynamics is embedded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  For beams in two spatial dimensions, the basic macroscale beam models have, at  each cross-section, displacement and velocity of both bending and compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  Thus the elastic beam dynamics has a slow manifold that is m = 4D at every  cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='1 Alternatively, 2D cosserat beam models add a shear mode to the  macroscale model—two more variables—leading to a not-quite-so-slow manifold of  m = 6D at every cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' These physically based models are slow manifolds  because they focus on the relatively slow waves of solutions varying slowly in space,  and neglect all the faster high-frequency cross-waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  In the patch scheme, Figures 1 and 4 show simulations with a cross-section of  ny = 7 micro-grid intervals, but let’s discuss the case of just ny = 4 (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  For ny = 4, there are seven microscale nodes across each patch edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' Each node  has a displacement and velocity, and so leads to a 14D subspace for macroscale  communication between patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  1Such statements, invoking a manifold or subspace “at every cross-section”, are in a sense  developed by the theory of Roberts (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' That is, in systems of large spatial extent there often  are important, spatially global, invariant manifolds of high-D that are effectively decomposable  into a union of spatially local manifolds/subspaces of relatively lower dimension—a dimension  determined by the spatial cross-section—and that are weakly coupled to neighbouring locales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  3  Scheme has proven accuracy  6  Figure 6: multiscale spectrum  of eigenvalues λ separates  macroscale modes on the right  from sub-patch microscale  modes on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' The axes  are scaled nonlinearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' Here  the small viscosity is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='001  so the microscale decays, but  the macroscale waves are long-  lasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  3  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='01  0  100  30  10  3  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='3  1  3  10  30  100  ℜλ  ℑλ  Because 14 > 2 · 6 > 2 · 4 , the Whitney embedding theorem asserts that the  patch scheme exchanges enough information to almost surely parametrise both  such slow manifolds of the macroscale dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' The patch scheme does not need  to explicitly compute and exchange specific assumed macroscale average quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  3  Scheme has proven accuracy  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='2 discusses established theory which generally proves that the patch  scheme makes accurate macroscale predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' Such proofs are in stark contrast  to the vast machine learning/artificial intelligence developments which prove  very few general results: for example, Brenner & Koumoutsakos (2021) comment  “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' ml studies, as the lack of rigorous theory does not offer (yet!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=') guarantees  of convergence”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' Before discussing theory, we first report some computational  verification of high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='1  Computation verifies exactness  Here we restricted attention to linear elasticity so we know that the wrapped  patch system is fully characterised by the resultant Jacobian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' We numeri-  cally compute the Jacobian matrix of the patch scheme by elementary numerical  differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  Because of the macroscale translational invariance of the patch scheme, the  macroscale eigenvectors are correctly sinusoidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' Hence the only macroscale er-  rors occur in the eigenvalues of the Jacobian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' Figure 6 plots the spectrum of all  eigenvalues for one example of random heterogeneity, in the case of five patches for  simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' Observe there are:  (on the right) four λ = 0 of rigid beam motion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  four −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='001 ± i 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='057 and four −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='003 ± i 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='111 of compressions waves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  four −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='001 ± i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='061 and four −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='004 ± i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='237 of beam bending waves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  with the above macroscale eigenvalues separated by a spectral gap from the  following sub-patch microscale eigenvalues;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  3  Scheme has proven accuracy  7  Table 1: error in patch scheme’s  macroscale eigenvalues λ for  various patch size ratios r: the  macroscale λs are exact to round-  off error—due to patch coupling by  spectral interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  macro-eigenvalue  r = 1  2  r = 1  4  r = 1  8  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='001 ± i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='061  2e-12  1e-12  2e-13  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='001 ± i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='061  2e-12  4e-12  2e-12  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='004 ± i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='237  1e-12  8e-13  3e-12  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='004 ± i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='237  1e-12  2e-12  3e-12  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='001 ± i 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='057  7e-13  4e-13  6e-13  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='001 ± i 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='057  6e-13  5e-13  6e-13  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='003 ± i 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='111  1e-13  2e-13  2e-13  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='003 ± i 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='111  4e-13  5e-13  2e-13  Figure 7: multiscale spectrum  of eigenvalues λ for the patch  scheme in the case of zero viscos-  ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' The horizontal axis shows  that all modes have zero real-  part to numerical round-off er-  ror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' That is, in the case of zero  viscosity, this patch scheme pre-  serves the wave nature of the  underlying physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  5e-13  2e-13  1e-130  1e-13  2e-13  5e-13  100  30  10  3  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='3  1  3  10  30  100  ℜλ  ℑλ  (on the left) many ℜλ < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='1 of uninteresting sub-patch micro-scale fast-  waves (headed by ten eigenvalues around −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='14 ± i 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  To quantify the accuracy, Table 1 compares eigenvalues obtained from full-domain  code, with the above macroscale eigenvalues obtained by the wrapped patch scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  For all patch size ratios and heterogeneities tested, the patch scheme’s macroscale  eigenvalues are exact to numerical round-off error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  Such exactness is due to the spectral interpolation used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' If, instead of spectral,  local polynomial interpolation of degree p is used to couple the patches, then  generally the patch scheme has macroscale errors ∝ Hp where H = inter-patch  spacing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=', Roberts & Kevrekidis 2007, Roberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  Undamped waves?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  With zero viscosity, there are only oscillations in the under-  lying physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' In such a scenario computational methods are very delicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' Here,  Figure 7 illustrates that all eigenvalues of the patch scheme have |ℜλ| < 10−12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='2  Hence, even with no viscosity, the patch scheme preserves the oscillatory wave  nature of the heterogeneous physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  There is a perception that the patch scheme “only works well on problems with  an inertial manifold and for systems in which most modes are strongly decaying”  (Majda & Grooms 2014, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' This verification of accuracy for purely elastic  2In some realisations of the heterogeneity, the sensitive multiplicity four eigenvalue λ = 0  numerically splits into four showing |ℜλ| up to 10−6 due to round-off errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  4  Conclusion  8  beams shows that this perception is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' Applications and theory for other wave  systems also refute this perception (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=', Cao & Roberts 2016, Bunder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' 2021,  Divahar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='2  Mathematical analysis proves consistency  Mathematical analysis has proven properties of the patch scheme in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' Mostly,  the published proofs explicitly address dissipative (nonlinear) systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' However,  as discussed by Bunder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' (2021), the patch scheme in space only recasts spatial  interactions, so whether the time derivative is ∂/∂t of dissipation or ∂2/∂t2 of  waves makes little difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  Two complementary types of results have been proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' They involve the spacing  between patch centres H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  First, Centre Manifold Theory may be applied at  finite spacing H by introducing a ‘bookkeeping’ parameter γ to label inter-patch  communication (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=', Roberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' 2014, §2) to prove the existence of a slow  manifold in the patch scheme (including when it is applied to nonlinear systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  Then the parameter γ structures inter-patch interactions, and their algebraic  expression, to empower theory based at γ = 0, via regular perturbation, to address  finite γ such as the case of full coupling γ = 1 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=', Roberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' 2014, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  Second, the patch scheme is consistent with the underlying micro-code as the  patch spacing H → 0 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=', Roberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' 2014, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' The consistency is  that the macroscale of the patch scheme is the same as the macroscale of the  given micro-coded system, to errors O  �  Hp�  when using polynomial interpolation  of degree p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' For example, spectral interpolation corresponds to ‘p = ∞’ so then  errors vanish to all orders as in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  These results and general proofs were first done for homogeneous systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=',  Roberts & Kevrekidis 2007, Roberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' They were subsequently ex-  tended to heterogeneous microscales (Bunder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' 2017), and recently extended  to alternative inter-patch coupling that preserves self-adjointness (Bunder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' Interestingly, the extension of the theoretical support to heterogeneous  cases invokes the ensemble of all phase-shifts of the heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' The ensemble is  spatially homogeneous, so the homogeneous proofs and results apply to establish  the heterogeneous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  4  Conclusion  As an initial exploration of the patch scheme for homogenisation of heterogeneous  elasticity, we considered the prototypical case of a 2D elastic beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' The scheme  gives a non-intrusive and efficient computational homogenisation of given microscale  system via spatially sparse small patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' The patch coupling has proven accuracy,  controllable error, at finite scale separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  The patch scheme makes only one assumption: in the scenarios of interest to a  user, there is no significant spatial structures in the mesoscale between the patch  spacing H and the microscale resolved in the patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' In contrast to most other  multiscale methods, there is: no assumed boundary conditions on Representative  Volume Elements (variously periodic, Dirichlet, Neumann);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' no explicitly assuming  References  9  slow variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' and no presumed necessary variational principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' The scheme is  entirely physically interpretable: there is no hidden mystic machinations of neural  networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=', Brenner & Koumoutsakos 2021)  The patch scheme is simple to apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' In contrast to other multiscale methods  there is: no arbitrary averaging;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' no oversampling regions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' no buffer regions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' no  action regions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' no guessed fast/slow variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' no epsilons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' and no limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' As a  non-intrusive ‘wrapper’, anyone can start using the patch scheme via a Matlab/  Octave Toolbox (Maclean et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' 2021, Roberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content=' 2019–2023)  Acknowledgements  This research was supported by Australian Research Coun-  cil grants DP220103156 and DP200103097.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
+page_content='  References  Biezemans, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'}
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Y9E3T4oBgHgl3EQfcgoK/content/tmp_files/2301.04525v1.pdf.txt

@@ -0,0 +1,1157 @@
+Clustering disease trajectories in contrastive
+feature space for biomarker discovery in
+age-related macular degeneration
+Robbie Holland1(�), Oliver Leingang2, Christopher Holmes3, Philipp Anders4,
+Johannes C. Paetzold1, Rebecca Kaye7, Sophie Riedl2, Hrvoje Bogunović2,
+Ursula Schmidt-Erfurth2, Lars Fritsche6, Hendrik P. N. Scholl4,5, Sobha
+Sivaprasad3, Andrew J. Lotery7, Daniel Rueckert1,8, Martin J. Menten1,8
+1 BioMedIA, Imperial College London, London, United Kingdom
+[email protected]
+2 Laboratory for Ophthalmic Image Analysis, Medical University of Vienna
+3 Moorfields National Institute for Health and Care Biomedical Research Centre,
+Moorfields Eye Hospital, London, United Kingdom
+4 Institute of Molecular and Clinical Ophthalmology Basel
+5 Department of Ophthalmology, Universitat Basel, Basel, Switzerland
+6 Department of Biostatistics, University of Michigan, Ann Arbor, United States
+7 Clinical and Experimental Sciences, Faculty of Medicine, University of
+Southampton, Southampton, United Kingdom
+8 Institute for AI and Informatics in Medicine, Technical University of Munich,
+Munich, Germany
+Abstract. Age-related macular degeneration (AMD) is the leading cause
+of blindness in the elderly. Despite this, the exact dynamics of disease
+progression are poorly understood. There is a clear need for imaging
+biomarkers in retinal optical coherence tomography (OCT) that aid the
+diagnosis, prognosis and management of AMD. However, current grad-
+ing systems, which coarsely group disease stage into broad categories
+describing early and intermediate AMD, have very limited prognostic
+value for the conversion to late AMD. In this paper, we are the first to
+analyse disease progression as clustered trajectories in a self-supervised
+feature space. Our method first pretrains an encoder with contrastive
+learning to project images from longitudinal time series to points in
+feature space. This enables the creation of disease trajectories, which
+are then denoised, partitioned and grouped into clusters. These clusters,
+found in two datasets containing time series of 7,912 patients imaged
+over eight years, were correlated with known OCT biomarkers. This re-
+inforced efforts by four expert ophthalmologists to investigate clusters,
+during a clinical comparison and interpretation task, as candidates for
+time-dependent biomarkers that describe progression of AMD.
+Keywords: Contrastive learning · Trajectory clustering · Disease pro-
+gression · Retina · OCT · Biomarker discovery
+arXiv:2301.04525v1  [eess.IV]  11 Jan 2023
+
+2
+Holland et al.
+1
+Introduction
+AMD is the leading cause of blindness in the elderly, affecting nearly 200 mil-
+lion people worldwide [22]. Patients with early stages of the disease exhibit
+few symptoms until suddenly converting to the late stage, at which point their
+sharp central vision rapidly deteriorates [14]. AMD patients are commonly im-
+aged with optical coherence tomography (OCT), which provides low-cost and
+high-resolution retinal images that depict fine physiological details. Several OCT
+biomarkers have been tentatively linked to the pathogenesis of AMD, such as reti-
+nal layer thickness, photoreceptor atrophy and the presence of drusen, which are
+lipidic deposits that build up inside the retina [16]. However, the exact relation
+of these biomarkers to the progression from early to late stages of AMD remains
+unclear. Current grading systems only coarsely group patients into broad cate-
+gories for early, intermediate and late AMD and have limited prognostic value
+[8]. There is an unmet need for biomarkers that describe and predict the pro-
+gression of AMD.
+Large studies that image populations of AMD patients over time are a power-
+ful resource to discover novel biomarkers for disease progression. If candidate
+biomarkers are already identified a priori, they can be extracted from images
+and mapped against disease progression [16,19,3]. However, this approach is not
+feasible if the biomarkers are partially or fully unknown. Self-supervised learn-
+ing is the most promising method to automatically discover new biomarkers by
+clustering or finding anomalous OCT images in lower-dimensional representation
+space [18,21,17]. However, by grouping single scans acquired at single points in
+time, these biomarkers are by definition static and cannot capture concepts such
+as the speed of disease progression or transitions between multiple states of the
+disease. Clustering whole time series of images is also problematic, as patients
+enter and leave the study at different points in their overall progression. There-
+fore, in order to discover biomarkers that describe population-level patterns in
+disease progression, it is necessary to analyse and compare portions of patient
+time series.
+Our contribution In this work, we develop a strategy to automatically discover
+biomarkers that capture disease progression in time series of images. This is
+illustrated in Figure 1. Our method represents the time series of images as a
+trajectory in a self-supervised latent feature space built by contrastive learning.
+This representation allows a new partitioning of time series of OCT images
+into consecutive subsequences (sub-trajectories) that exhibit distinct units of
+disease progression. By clustering these sub-trajectories, we are now able to
+detect patterns of disease progression that are common among the population
+of patients.
+Experimentally, we test our method on a large cohort from two, longitudinal
+retinal OCT datasets totalling 160,558 images from 7,912 patients. In doing so we
+categorise 3,218 total years of disease progression with time-dependent clusters.
+We then correlate our clusters with the known set of biomarkers which reinforces
+
+Clustering disease trajectories for biomarker discovery in AMD
+3
+Fig. 1: We analyse disease progression as clustered trajectories in feature space.
+We first train a feature extractor with contrastive learning and then form tra-
+jectories by projecting time series of images to feature space. These are then
+partitioned into a set of sub-trajectories which are clustered. Finally, clusters are
+related to known disease stages before being interpreted as candidate biomarkers
+for disease state and progression. PCA space is coloured by visual acuity.
+their potential to contain potentially new, time-dependent biomarkers. Finally,
+we closed the loop on automated biomarker discovery by working directly with
+four ophthalmologists to interpret these clusters in a clinical comparison task.
+
+1. Contrastive pretraining
+Contrastive
+PCA feature
+Retinal OCT datasets (170k scans)
+loss
+space
+ResNet-50
+CNN
+00
+0
+00
+0
+Copy pretrained
+feature extractor
+0
+0
+0
+PCA reduction
+00
+ResNet-50
+00
+CNN
+00
+00
+2. Patient trajectories in feature space
+Resample
+Longitudinal information
+with temporal
+denoising kernel
+Eye
+Scan times
+04/2017
+08/2017
+11/2017
+2
+07/2012
+12/2012
+04/2013
+3
+06/2019
+09/2020
+10/2020
+01/2015
+03/2016
+04/2016
+5
+08/2017
+09/2017
+02/2018
+6
+09/2012
+03/2013
+07/2013
+3. Clustering and proposing biomarkers
+Cluster 1 vs. Cluster 2
+Series A
+K-means
+Query series
+Count biomarkers
+clustering
+Known biomarkers
+Series B
+per cluster
+Healthy Drusen cRORA
+CNV
+0
+Clusters
+Experts compare clusterst with
+similar known biomarkers
+2
+but different latent features
+tclusters are candidates for
+time-dependent biomarkers4
+Holland et al.
+2
+Related work
+2.1
+Current AMD grading systems
+Ophthalmologists’ current understanding of progression from early to late AMD
+largely involves drusen. Drusen can grow until suddenly regressing and disap-
+pearing, which often precedes the onset of late AMD [16]. While there have
+been attempts to group drusen based on their morphology [11], current grading
+systems stratify early and intermediate stages solely by drusen size [2,12,7,8].
+Late AMD is classified as either choroidal neovascularisation (CNV), identified
+by fluid under the retina, or geographic atrophy by progressive loss of photore-
+ceptors and retinal thinning. The degree of atrophy can be staged using cRORA
+(complete retinal pigment epithelium and outer retinal atrophy), which measures
+the width of focal atrophy in OCT [15]. So far grading systems offer limited di-
+agnostic value and little to no prognostic value.
+2.2
+Tracking evolution of known biomarkers
+Few research efforts have aimed at quantifying and tracking known AMD biomark-
+ers over time, such as reticular pseudodrusen [19] and drusen volume [16]. More
+work has explored Alzheimer’s disease (AD), which offers a greater array of
+quantitative imaging biomarkers, such as levels of tau protein and hippocampal
+volume. Young et al. [23] fit an event-based model that rediscovers the order
+in which these biomarkers become anomalous as AD progresses. Vogel et al.
+[20] find four distinct spatiotemporal trajectories for tau pathology in the brain.
+However, mapping biomarkers that evolve during disease progression requires
+prior annotation of entire time series. Thus, these biomarkers must be known or
+at least suspected a priori.
+2.3
+Automated discovery of unknown biomarkers
+In order to discover new biomarkers, efforts to find them have turned to au-
+tomated biomarker discovery. Imaging biomarkers are proposed by analysing
+anomalous scans [17], clusters of scans [25], or a combination of these [18] in
+feature space. To build these, neural networks are trained with supervised or
+unsupervised proxy tasks. These tasks include image reconstruction [21], seg-
+mentation [25] and generative adversarial networks [17]. However, networks are
+prone to overfit on their specific task and lose semantic information regarding
+the disease. Contrastive learning has recently advanced the state-of-the-art in
+training generalisable and unbiased feature extractors. Chen et al. popularised
+this paradigm with SimCLR [4], which was later improved on by Grill et al. [9] in
+Bootstrap Your Own Latent (BYOL). Contrastive methods encode invariance to
+a set of transformations typically uncorrelated with disease features, including
+rotation, translation and global shifts in image brightness and contrast. Zhao et
+al. leverage contrastive feature spaces to identify high-risk clusters of CT image
+patches [24].
+
+Clustering disease trajectories for biomarker discovery in AMD
+5
+However, all biomarkers discovered by the aforementioned methods work by
+grouping single images acquired at single points in time, and in doing so neglect
+temporal relationships between images of the same subject. One work that tack-
+les this challenge, and the most related to ours, categorises the time-dependent
+response of cancer cells to drugs, measured by the changing distance in con-
+trastive feature space from healthy controls [5].
+2.4
+Trajectory clustering
+The separate field of trajectory clustering is largely focussed on discovering move-
+ment patterns taken by cars, animals and hurricanes [6,1,13]. Lee et al., in their
+state-of-the-art work TRACLUS [13], assume these trajectories are composed of
+consecutive series of common sub-trajectories. For example, different car jour-
+neys may at some point travel down the same road. Using this principle, they
+develop a partition-and-group framework to cluster segments that are repeated
+across multiple trajectories. Similarly, we assume that disease progression can
+be divided into multiple, common disease pathways. Firstly, this allows us to
+work seamlessly with temporally unaligned scanning series. Secondly, we can
+automatically discover novel disease pathways by interpreting sub-trajectories
+that are shared by multiple AMD patients.
+3
+Materials and Methods
+3.1
+Self-supervised feature space using contrastive learning
+We adapt BYOL [9] with update coefficient τ = 0.9995 for contrastive pretrain-
+ing of a ResNet50 (4x) model. As several of the contrastive transformations
+designed for natural images are inapplicable to medical images, we use the set
+tailored for retinal OCT images by Holland et al. [10]. Models were trained on
+the entire dataset for 120,000 steps with momentum 0.9 and a learning rate of
+5 · 10−4 using the Adam optimiser.
+After pretraining, we first remove any final linear layers before projecting all
+labelled images to the feature space of 2048 dimensions. We then reduce the di-
+mension further using principle component analysis (PCA) with D components.
+Using PCA allows us to interpolate the feature space and results in fewer di-
+mensions, which is advantageous for clustering. To validate that the contrastive
+feature space encodes meaningful information for AMD biomarkers, and find
+the optimal dimension D ∈ {2, 10, 20, 50}, we perform multi-class classification
+of known biomarkers (including healthy controls). Firstly, we split the dataset
+into train and test partitions using 85% and 15% of the data, respectively, en-
+suring that all scans from each patient belong to the same set. Then to perform
+the classification, we fit a class-balanced support vector machine (SVM) on the
+training set and report performance on the test set.
+
+6
+Holland et al.
+Fig. 2: Longitudinal scans of a single eye, imaged over four years, projected to
+PCA space. PCA space is plotted as a hexmap coloured by the local average
+in visual acuity, where higher values indicate poorer quality of vision. Each
+row depicts two principle components up to 20. The rightmost columns show
+resampled trajectories. Using smaller T captures short-term variation in disease
+progression but cannot model long-term changes. MDL aims to optimise this
+tradeoff by partitioning trajectories only at points of inflection.
+3.2
+Extracting and clustering common sub-trajectories
+For each eye, we first form piecewise-linear trajectories by linking points in PCA
+space that were derived from consecutively acquired OCT images (see left column
+in Figure 2). We then assume, in analogy to TRACLUS [13], that trajectories
+encoding disease progression can be partitioned into sub-trajectories that are
+common among multiple patients. We compare two methods to achieve this,
+shown in the right-most columns in Figure 2. The first method resamples time-
+points at regular intervals of T years. For each resampled time t, we find the
+corresponding point in feature space by taking a weighted average of all points
+in the trajectory. The weights are calculated by convolution of a Gaussian kernel
+N(t, σT ) with the acquisition dates of the entire scanning series. We then define
+
+Resampled trajectories
+(T = 0.5 years)
+T=
+T=1.0
+T= 2.0
+T= 0.5
+Original
+Cumulative
+MDL Partition
+(years)
+trajectory
+(years)
+(years)
+variance
+Dims 1-2
+PCA #2
+PCA #2
++18%
+CA#2
+18%
+PCA #1
+PCA #1
+PCA #1
+PCA #1
+PCA #1
+1
+Dims 3-4
+PCA #4
+PCA #4
+PCA#4
++13%
+7#
+CA
+31%
+PCA #3
+PCA #3
+PCA #3
+PCA #3
+PCA #3
+Dims 5-6
+PCA #6
+PCA #6
++10%
+PCA #6
+PCA #6
+CA #6
+41%
+PCA #5
+PCA #5
+PCA #5
+PCA #5
+PCA #5
+Dims 7-20
+PCA #20
+PCA #20
+PCA #20
+PCA #20
+#20
++31%
+PCA
+72%
+PCA #19
+PCA #19
+PCA #19
+PCA #19
+PCA #19
+Colour map for
+visual acuity
+0.4
+0.6
+0.8
+1.2
+1.4
+0.2
+1.0
+1.6
+1.8Clustering disease trajectories for biomarker discovery in AMD
+7
+sub-trajectories as vectors between consecutive points that are less than or equal
+to T years apart.
+The second method aims to describe trajectories using the fewest points. It be-
+gins by resampling trajectories using the first method (with intervals of T = 0.5
+years). Then, using the minimum description length (MDL) principle, a mini-
+mal subset of points are chosen that best preserve changes in disease over time.
+To achieve this we find the trajectory H, containing a subset of the points in
+the resampled trajectory O, which minimises the following objective (using the
+greedy solution from [13])
+L(O|H) =
+�
+p∈O
+d⊥(p, H) − λ|H|
+where d⊥(p, H) is the perpendicular distance from a point p to the piece-wise
+linear trajectory H, and |H| is the number of points in H. The coefficient λ is
+proportional to the total standard deviation in the feature space explained by
+D PCA dimensions.
+Clustering sub-trajectories To cluster common sub-trajectories we require
+a distance function that measures the similarity between vectors. Given two
+sub-trajectories U = (ustart, uend) and V = (vstart, vend) their distance is simply
+d(U, V ) = ∥ustart − vstart∥2 + ∥uend − vend∥2
+Finally, using d we separate sub-trajectories into K clusters using k-means clus-
+tering.
+3.3
+Finding optimal hyperparameters using the set of known
+biomarkers
+We now search for optimal values for the sub-trajectory time interval T ∈
+{0.5, 1.0, 2.0} years, resampling kernel width σT ∈ {0.25, 0.5, 1.0} years and the
+number of clusters K ∈ {5, 10, 15, 30} using five random seeds for k-means clus-
+tering. To quantitatively compare configurations, we use the conditional entropy
+H(B|C) = H(B, C) − H(C) as a scalar measure of how well the clusters C redis-
+cover the known biomarkers B detailed in section 3.5. This is calculated directly
+from their joint distribution P(B, C), which is found by counting all biomarkers
+recorded within sub-trajectories of each cluster. To ensure the equal contribution
+of all biomarkers, we reweight their marginal distribution P(B) to be uniformly
+distributed. As the number of clusters K increases even randomly permuted as-
+signments p(C) will result in reduced values of H(B|p(C)). We address this by
+using the adjusted reduction in conditional entropy, H′ (using r = 5 random
+trials), where higher values correspond to better rediscovery of B
+H′(B|C) = 1
+r
+r
+�
+i
+H(B|p(C)) − H(B|C)
+As our ultimate goal is to detect biomarkers beyond the known set, we use H′
+only as an indication for the most suitable configuration.
+
+8
+Holland et al.
+3.4
+Proposing clusters as candidate biomarkers
+We first relabel the clusters C in order of median visual acuity, so that higher
+cluster numbers indicate poorer quality of vision. In order to see which disease
+stages each cluster describes, we calculate the conditional probability P(B|C) =
+P(B, C)/P(C). Then, to discover new biomarkers beyond the existing set, we
+compare clusters that differ maximally in feature space but minimally in the set
+of known biomarkers. To find these, we explore eleven pairs of distinct clusters
+Ci and Cj with a high degree of cosine similarity P(B|Ci) · P(B|Cj).
+Interpreting candidate biomarkers To examine clusters for candidate biomark-
+ers, we collaborate with four expert ophthalmologists. For each pair of clusters,
+we generate four random ‘A or B’ single-choice questions. Clinicians are shown
+one query series in image space from cluster Ci and two further series denoted
+A and B, one from Ci and the other from Cj. For each question, clinicians are
+tasked with determining which of A or B belongs to the same cluster as the
+query. After completing four questions they are prompted to explain on what
+basis they matched A or B to the query. This format allows us to both assess
+whether the clusters are visually distinguishable by experts and, if so, potentially
+extract descriptions of novel biomarkers.
+3.5
+OCT datasets
+We apply our method to two independent retinal OCT datasets called Dataset
+A and Dataset B. We developed our method on Dataset A but run experiments
+on both datasets. In both, images were acquired using Topcon 3D OCT devices
+(Topcon Corporation, Tokyo, Japan). After strict quality control, Dataset A
+consists of 46,496 scans of 6,236 eyes from 3,456 patients. Eyes were scanned 7.7
+times over 1.9 years on average at irregular time intervals. The second dataset,
+Dataset B, is larger, containing 114,062 scans of 7,253 eyes from 3,819 patients.
+Eyes were scanned 16.6 times over 3.5 years on average. Of each 3D OCT vol-
+ume, we extracted the transverse 2D slice centred at the fovea and resampled
+to 208×256 pixels with a pixel size of 7.0×23.4 µm2, half the median resolution.
+Each scan is labelled with visual acuity, a functional measure assessing the qual-
+ity of vision measured in LogMAR.
+To record conversions to a comprehensive set of known biomarkers B, we used
+established AMD grading protocols described in section 2.1. Early AMD is char-
+acterised by small drusen between 63-125µm in diameter. We also recorded CNV,
+cRORA (of at least 250µm but smaller than 1000 µm) and cRORA (of at least
+1000 µm) [15]. Overall, 861 conversion times t0 were recorded, and any sub-
+sequent visits at times t+ before the next conversion were automatically as-
+signed with a separate label. Visits prior to any biomarker were labelled as
+‘NoBiomarker’. Finally, in each dataset, an additional 150 healthy images that
+exhibit no pathology were recorded. Combining these, the known set of biomark-
+ers B includes 10 biomarkers and labels.
+
+Clustering disease trajectories for biomarker discovery in AMD
+9
+4
+Results
+Fig. 3: Confusion matrices for multi-class classification of known biomarkers and
+healthy images using D numbers of PCA dimensions. In general, performance
+increases with the number of dimensions D. We find that using D = 20 PCA
+dimensions achieves linear separability between known biomarkers.
+4.1
+Finding the optimal set of hyperparameters using the known
+biomarkers
+In both Dataset A and Dataset B 20 principal dimensions achieves linear sepa-
+rability between known biomarkers (see Figure 3). Both the healthy stage and
+the only extractable early biomarker, drusen, were found to be highly linearly
+separable. Thus, we use D = 20 for the remainder of our analysis.
+We find that K = 15 clusters of sub-trajectories best explained the set of known
+biomarkers B (Figure 4a) as measured by higher values of H′(B|C). Greater
+values of T, in addition to MDL partitioning, result in decreased H′(B|C).
+We suspect that this is because the known set of biomarkers describe disease
+states rather than state transitions, so they are better captured by shorter sub-
+trajectories. In order to select a configuration that finds clusters evidencing pro-
+gression, we choose T = 1.0, σT = 0.5 and K = 15 for the remainder of our
+analysis.
+4.2
+Sub-trajectory clusters go beyond the known set of biomarkers
+We find that our clusters encode the set of known biomarkers in Dataset A. As
+seen in Figure 5, clusters effectively separate healthy, early stage and late-stage
+
+Dataset A
+10D
+20D
+50D
+2D
+0.28
+0.05
+0.07
+0.94
+0.04
+0.06
+0.01
+Healthy
+0.57
+0.10
+0.92
+0.01
+0.00
+0.01
+0.00
+0.00
+0.92
+0.01
+0.01
+0.00
+0.00
+0.18
+0.01
+0.01
+0.12
+0.72
+0.12
+Drusen
+0.42
+0.37
+0.01
+0.01
+0.87
+0.06
+0.01
+0.04
+0.90
+0.04
+0.00
+0.04
+0.00
+0.04
+0.12
+0.36
+0.16
+0.30
+0.01
+0.05
+0.12
+0.23
+0.26
+0.20
+0.29
+0.28
+0.24
+0.33
+0.24 
+0.07
+0.19
+0.23
+cRORA (250 μm)
+0.26
+0.06
+0.02
+0.62
+0.02
+cRORA (1000 μm)
+0.04
+0.16
+0.16
+0.62
+0.02
+0.07
+0.22
+0.01
+0.29
+0.02
+0.63
+0.07
+0.62
+0.06
+0.05
+0.06
+0.21
+0.19
+0.42
+CNV
+0.05
+0.09
+0.19
+0.62
+0.05
+0.05
+0.08
+0.25
+0.03
+0.09
+0.20
+0.52
+0.08
+0.14
+0.57
+0.20
+0.16
+0.02Dataset B
+0.49
+0.24
+0.00
+0.92
+0.08
+0.00
+0.00
+0.84
+0.03
+0.00
+0.97
+0.00
+0.00
+0.00
+0.03
+Healthy
+0.16
+0.11
+0.00
+0.08
+0.05
+0.08
+Drusen
+0.36
+0.39
+0.07
+0.30
+0.04
+0.05
+0.14
+0.06
+0.14
+0.62
+0.19
+0.15
+0.16
+0.61
+0.11
+0.02
+0.16
+0.34
+0.00
+0.42
+0.13
+0.00
+0.18
+0.00
+cRORA (250 μm)
+0.26
+0.21
+0.11
+0.32
+0.39
+0.16
+0.10
+0.00
+0.45
+0.31
+0.06
+0.39
+0.35
+0.13
+0.03
+0.04
+0.33
+0.08
+cRORA (1000 μm)
+0.33
+0.21
+0.08
+0.00
+0.00
+0.04
+0.29
+0.33
+0.33
+0.38
+0.00
+0.00
+0.71
+0.12
+0.17
+0.00
+0.54
+0.61
+0.40
+0.29
+0.23
+0.08
+0.00
+0.02
+0.13
+0.13
+0.19
+0.53
+CNV
+0.00
+0.17
+0.14
+0.12
+0.57
+0.00
+0.17
+0.10
+0.11
+ANO
+CRORA
+CRORA (
+(250
+A (250 μm)10
+Holland et al.
+(a) Adjusted conditional entropy H′(B|C) of known biomarkers B given subtrajectory
+clusters C against hyperparameters K (left), T (center), σT (right)
+Fig. 4: Results of the hyperparameter search, measured by H′(B|C) which is a
+scalar measure of how well the clusters C rediscover existing biomarkers B. As
+we aim to discover biomarkers beyond the known set, we also consider the level
+of disease progression captured by our clusters when choosing our configuration.
+Fig. 5: Conditional probabilities P(B|C) of the known set of biomarkers B given
+cluster assignments C by our method. Cluster pairs highlighted in blue were,
+due to their similarity under the known set of biomarkers B, chosen for further
+analysis by clinicians. One cluster pair, highlighted in pink, was used as a trial
+task and validation experiment.
+
+Entropy
+0.6
+0.5
+Adjusted reduction in Conditional
+0.4
+0.3
+0.2
+0.1
+0.0
+Variants
+Dataset A
+-0.1
+Dataset B
+5
+10
+15
+30
+Number of clusters (K)Entropy
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+0.0
+Variants
+Dataset A
+-0.1
+Dataset B
+0.25
+0.5
+1.0
+OTEntropy
+0.6
+0.5
+Adjusted reduction in Conditional
+0.4
+多
+0.3
+0.2
+0.1
+0.0
+Variants
+Dataset A
+-0.1
+Dataset B
+T= 0.5
+T= 1.0
+T= 2.0
+T= MDL
+partition
+Progression vector sampling strategyDataset A
+Cosine similarities over clusters pairs
+Conditional probability of known biomarkers
+P(BICi) · P(BC))
+given cluster assignments: P(B|C)
+0.94
+D.55 0.74 0.58 0.75 C
+0.56 0.63 0.43 0.82
+2 0.400.15 0.05 0.07
+0.19
+0.16
+0.24
+0.20
+0.00
+0.00
+0.00
+0.00
+0.09
+0.05
+0.25
+1.00
+0.21
+0.06
+0.04
+0.00
+0.00
+0.09
+0.04
+0.94 1.00
+0.34
+0.84 0.67 0.77 0.67 0.72 0.44 0.71
+0.49 0.17 0.07 0.08 0.18
+2
+0.07
+0.19
+0.31
+2
+0.70
+0.12
+0.07
+0.07
+0.00
+0.00
+0.00
+0.00
+0.02
+0.02
+0.55 0.34
+1.00 0.16 0.14
+0.46 0.16 0.17 0.37
+0.74
+0.08 0.17 0.01 0.06 0.03
+3
+3
+0.00
+0.15
+0.22
+0.04
+0.07
+0.74 0.84
+40.16
+1.00
+0.91
+0.86 0.89 0.96
+4
+0.14
+0.17
+0.11
+0.00
+0.10
+0.60 0.58 0.73 0.33 0.21 0.23 0.27
+4
+5
+0.00
+0.17
+0.07
+0.12
+0.16
+0.21
+0.07
+0.03
+0.11
+0.06
+5
+0.58 0.67
+0.14 0.91 1.00 0.86 0.97 0.94
+0.65 0.47 0.82 0.510.34 0.36 0.32
+9
+0.11
+0.09
+0.10
+0.11
+0.08
+0.04
+0.12
+0.75 0.77 0.46
+6 0.86 0.86 1.00
+0.91
+0.89 0.82 0.82 0.82 0.60 0.44 0.46 0.52
+0.12
+0.11
+0.12
+6
+6 0.89 0.97 0.91 1.00
+0.02
+0.17
+0.19
+0.11
+0.04
+0.08
+0.56 0.67(
+0.92
+D.710.51
+0.83 0.61 0.45 0.47 0.36
+7
+0.11
+0.12
+0.09
+0.07
+7
+0.16
+0.01
+0.20
+0.03
+0.63 0.72 0.17 0.96 0.940.89 0.92 1.00
+0.76 0.57 0.87 0.44 0.29 0.31 0.43
+8
+0.13
+0.08
+0.16
+0.13
+0.03
+0.12
+0.11
+8
+0.13
+0.03
+0.07
+0.03
+0.16
+0.14
+0.00
+0.00
+0.16
+0.29
+0.43 0.44 0.37 0.60 0.65 0.82 0.71 0.76 1.00 0.66 0.90 0.42 0.26 0.27 0.66
+9
+9
+0.82 0.71 0.74 0.58 0.47 0.82 0.51 0.57 0.66
+0.25
+0.07
+0.14
+0.15
+0.00
+0.05
+0.13
+1.00 0.52 0.41 0.29 0.31 0.51
+0
+0.00
+0.06
+0.15
+0
+0.08 0.73 0.82 0.82 0.83 0.87 0.90 0.52
+0.06
+0.15
+0.20
+0.21
+0.40 0.49
+1.00
+0.53 0.36 0.360.70
+0.00
+0.06
+0.06
+0.00
+0.06
+0.20
+0.96
+0.07
+0.01
+0.02
+0.01
+0.07
+0.14
+0.25
+0.29
+0.07
+0.08
+0.15 0.17 0.17 0.33 0.51 0.60 0.61 0.44 0.42 0.41 0.53
+D.61
+2
+2
+1.00 0.95
+0.00
+0.00
+0.00
+0.00
+0.05
+0.05
+0.35
+0.37
+0.06
+0.12
+0.05 0.07 0.01 0.21 0.34 0.44 0.45 0.29 0.26 0.29 0.36
+50.951.00
+1.00
+0.62
+3
+3
+0.03
+0.00
+0.00
+0.00
+0.08
+0.04
+0.34
+0.36
+0.05
+0.10
+0.07 0.08 0.06 0.23 0.36 0.46 0.47 0.31 0.27 0.31 0.36
+0.96 1.00 1.00 0.60
+4
+4
+0.70 0.61 0.62 0.60
+0.00
+0.00
+0.00
+0.00
+0.00
+0.06
+0.28
+0.30
+0.36
+0.19 0.18 0.03(
+0.27 0.32 0.52 0.36 0.43 0.66
+0.51
+1.00
+5
+0.00
+5
+13 14 15
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+(to)
+(to)
+CNV
+(to)
+(to)
+Healthy
+ μm)
+Drusen
+NoBiomarker
+CRORA
+(250
+(1000 |
+(1000μm)
+RAClustering disease trajectories for biomarker discovery in AMD
+11
+biomarkers. We find two clusters (1-2) describing drusen, five (4-8) describing the
+transition from drusen to cRORA (250 µm) and three (12-14) describing cRORA
+(1000 µm). Most healthy samples were in cluster 3, and CNV was spread amongst
+most late-stage clusters. In Dataset B we find three clusters of healthy looking
+scans, three of early stages and six in the late stage that capture atrophy.
+Interpreting candidate biomarkers We now report the results of the single-
+choice clinical clustering task (described in section 3.4), comparing cluster pairs
+which are highlighted in blue and pink boxes in Figure 5. In Dataset A, clinicians
+were easily able to separate clusters 1 and 15 by using known biomarkers, such as
+hypertransmission and photoreceptor degeneration. The result of this validation
+experiment was expected, as these clusters were already highly separable under
+the known set of biomarkers. More interestingly, clinicians were able to exactly
+differentiate between early AMD clusters 1 and 2 in Dataset A despite their high
+similarity in known biomarkers. When prompted, all clinicians cite differences
+in drusen, with two finding differences in the number of small drusen. Their
+ability to distinguish some of the pairs was mixed, as they sometimes found no
+consistent or visible differences or had a low inter-rater agreement.
+5
+Discussion and Conclusion
+In this paper, we proposed a method to automatically discover time-dependent
+biomarkers that detect periods of disease progression common among groups of
+patients. By partitioning entire time series into representative sub-trajectories,
+and then clustering them, we categorised 3,218 total years of disease progression
+across two datasets. We showed that these clusters rediscovered the established
+set of OCT biomarkers for AMD, which reinforced the use of our clusters as can-
+didate biomarkers. Then, by working directly with ophthalmologists, we closed
+the loop in our automated biomarker discovery. To this end ophthalmologists
+compared clusters that were indistinguishable using current grading systems,
+yet were separable in contrastive feature space. We envision that further inves-
+tigation into sub-trajectory clusters could advance understanding of how AMD
+progresses, and potentially lead to grading systems with greater prognostic value.
+Our method is applicable to any dataset studying any disease with time series
+of images. While our method identified two clusters that described drusen in
+Dataset A and three that described healthy-looking scans in Dataset B, most
+clusters were associated with intermediate and late-stage AMD. This is due
+to the overrepresentation of patients with late disease in our datasets. In or-
+der to find more clusters categorising progression in early AMD, we aim to
+repeat this analysis in datasets that begin imaging patients earlier in their over-
+all progression. Moreover, due to the slow progression of AMD, a large number
+of sub-trajectories captured unchanging disease states. In order to isolate sub-
+trajectories capturing the periods of greatest disease progression, we intend to
+increase the number of clusters K and interpret those that convert to late disease
+the fastest.
+
+12
+Holland et al.
+Conclusion Inspired by inadequate grading systems for disease progression
+in early AMD, we proposed the first method to analyse disease progression as
+clustered trajectories in self-supervised feature space. By correlating our clusters
+with known OCT biomarkers for AMD, we reinforced their potential as time-
+dependent biomarkers for disease progression. After this, we closed the loop on
+automated biomarker discovery by working directly with ophthalmologists to
+investigate our clusters. We envision that self-supervised learning can enable
+detection of patterns of disease progression in time series of patient populations,
+which can lead to grading systems with greater prognostic value.
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+3,822 uk biobank cardiac mris. Front. cardiovasc. med 7, 539788 (2020)
+

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+Non-Gaussianity in the cosmic microwave background from loop quantum
+cosmology
+Roshna K∗ and V. Sreenath†
+Department of Physics, National Institute of Technology Karnataka, Surathkal, Mangaluru 575025, India.
+Primordial non-Gaussianity has set strong constraints on models of the early universe.
+Studies have shown that Loop Quantum Cosmology (LQC), which is an attempt to extend
+inflationary scenario to planck scales, leads to a strongly scale dependent and oscillatory non-
+Gaussianity. In particular, the non-Gaussianity function fNL(k1, k2, k3) generated in LQC,
+though similar to that generated during slow roll inflation at small scales, is highly scale
+dependent and oscillatory at large wavelengths. In this work, we investigate the imprints of
+such a primordial bispectrum in the bispectrum of Cosmic Microwave Background (CMB).
+Inspired by earlier works, we propose an analytical template for the primordial bispectrum
+in LQC and compute the corresponding reduced bispectra of temperature and electric po-
+larisation and their three-point cross-correlations. We show that CMB bispectra generated
+in LQC is consistent with the observations from Planck. We conclude with a discussion of
+our results and its implications to LQC.
+I.
+INTRODUCTION
+Numerous theoretical insights together with several observational efforts, spanned over a cen-
+tury, have enabled us to arrive at a compelling model of our Universe referred to as the standard
+model or the Lambda Cold Dark Matter (ΛCDM) model [1]. According to this model, the seeds
+of the current distribution of galaxies spread over the fabric of spacetime known as the large scale
+structure were sown during the earliest phase of the universe. Tiny perturbations generated in the
+early universe lead to tiny anisotropies in the Cosmic Microwave Background (CMB) which in turn
+lead to the inhomegeneous large scale distribution of galaxies that we see today. Though we have
+a good level of understanding of this evolution, several details are yet to be worked out. One such
+detail concerns the origin of these perturbations in our Universe.
+Inflation, see, for instance, [2–5], due to its simplicity, provides the most popular explanation
+for the origin of these perturbations [6, 7] (For a discussion on alternate views, see [8, 9].). In
+inflationary scenario, quantum fluctuations in the inflaton leads to the primordial perturbations.
+Appealing to the nearly de Sitter symmetry of the spacetime during inflation, we assume that at
+a time when the perturbations are sufficiently sub-horizon, quantum perturbations are generated
+in the Bunch-Davies vacuum. Such a prescription has been highly successful, in that, it leads to
+primordial perturbations that are nearly Gaussian and scale invariant as demanded by observations
+[1, 6, 10]. Even though inflation is successful, it is still an incomplete theory. We do not take in to
+account the evolution of perturbations before the time at which the initial conditions are imposed.
+In fact, inflation does not account for the physics in the planck regime close to the big bang
+singularity. There have been several attempts to address these issues. In this work, we will concern
+ourselves with loop quantum cosmology (LQC) [11–15].
+Loop quantum cosmology is an attempt to extend inflationary scenario to the planck regime
+using principles of loop quantum gravity [13–19]. In LQC, quantum gravitational effects in the
+planck regime leads to a quantum bounce [11, 12]. Thus in LQC, a quantum bounce precedes
+the inflationary phase. Generation and evolution of perturbations in LQC have been extensively
+∗ [email protected]
+† [email protected]
+arXiv:2301.05406v1  [astro-ph.CO]  13 Jan 2023
+
+2
+studied at the level of primordial power spectra [20–46] and primordial non-Gaussianity [47–50]. In
+general, studies show that the effect of the bounce is to introduce an additional scale corresponding
+to the curvature at the bounce. Modes of perturbations which have comparable length to this new
+scale gets modified leading to a highly scale dependent power spectrum. At smaller wavelengths, the
+perturbations are not affected by the bounce and the power spectrum is nearly scale invariant as in
+slow roll inflation [29]. Perturbations show a similar behaviour at second order in perturbations [47,
+49]. Studies show that primordial non-Gaussianity quantified using the function fNL(k1, k2, k3),
+at scales comparable to the curvature at the bounce, is strongly scale dependent and oscillatory
+with a very large amplitude. At smaller scales, the fNL(k1, k2, k3) is similar to that in slow roll.
+Studies also show that the bispectrum is more sensitive to the bounce than the power spectrum.
+Assuming sixty or so e-folds of inflation, the scale at which the imprints of the bounce, on
+primordial perturbations, occur depends on the amount of expansion between the bounce and the
+onset of inflation. Observational constraints from the CMB temperature power spectrum demand
+that any departure from scale invariance should happen only at multipoles of ℓ ≲ 30 [1, 51]. If we
+assume that, the effects of primordial power spectrum on the CMB is observable at ℓ ≲ 30, then,
+since the bispectrum is more sensitive to the effects of the bounce than the power spectrum [47, 49],
+there is a possibility that the imprints of large, scale dependent and oscillatory primordial non-
+Gaussianity is observable at larger multipoles. Hence it is important to investigate the consistency
+of LQC with observations by Planck. With this motivation, in this work, we compute the imprints
+of such a non-Gaussianity in the temperature (T) and electric polarisation (E) of the CMB. We
+assume an analytical template for primordial non-Gaussianity generated in LQC, compute the
+⟨TTT⟩, ⟨TTE⟩, ⟨TEE⟩ and ⟨EEE⟩ correlations and show that they are similar to those generated
+in slow roll inflation and hence is consistent with observations by Planck.
+The rest of the paper is organised as follows. In the next section, we briefly introduce the essen-
+tials of LQC and present analytical templates for the primordial power spectrum and bispectrum.
+In section III, we discuss the essential formulae to compute the three-point correlation functions
+of anisotropies in temperature and electric polarisation. In section IV, we apply these formulae to
+LQC. We present the numerical techniques and our calculation of reduced bispectra of tempera-
+ture fluctuations and electric polarisation and their three-point cross-correlations in section V. We
+conclude the paper with a summary and discussion of our results and their consequences to LQC
+in section VI.
+II.
+LOOP QUANTUM COSMOLOGY
+In this section, we will discuss the essentials of LQC that is relevant to this paper (for reviews,
+see, for instance, [13–15]). In particular, we will discuss LQC as applied to FLRW geometries
+sourced by a scalar field φ and scalar perturbations δφ(⃗x) living on this background.
+A.
+Background
+In LQC, FLRW background geometry is described by a wavefunction ΨFLRW(v, φ), which satis-
+fies the equation ˆHFLRWΨFLRW(v, φ) = 0, where ˆHFLRW is the Hamiltonian operator corresponding
+to the classical background Hamiltonian and v is the volume factor which is proportional to the
+cube of scale factor a. Numerical investigations of such a system has shown that the scale factor
+undergoes a bounce [11, 12, 52, 53]. It turns out, if the wave function is sharply peaked over the
+values of scale factor, the behaviour of scale factor can be described by certain effective equations
+
+3
+−104
+−102
+0
+102
+104
+106
+t (TPl)
+103
+108
+1013
+1018
+1023
+1028
+1033
+a(t)
+Inflation
+−10
+−5
+0
+5
+10
+0
+2
+4
+6
+8
+100
+101
+102
+103
+104
+105
+106
+107
+t (TPl)
+−5
+0
+5
+10
+15
+φ(t)
+FIG. 1.
+Figure illustrates the behaviour of scale factor (left) and scalar field(right) in LQC. As mentioned
+in the text, scale factor undergoes a bounce preceding inflation. Scalar field starts rolling up the potential
+until its kinetic energy becomes zero and then starts slowly rolling down the potential leading to inflation.
+In making this plot, we have worked with the mass of scalar field to be consistent with the constraints on
+the amplitude of the primordial power spectrum and with ρsup = 0.41m4
+Pl.
+[11, 12, 30, 52, 53], namely
+� ˙a
+a
+�2
+= κ
+3ρ
+�
+1 −
+ρ
+ρsup
+�
+,
+¨a
+a = −κ
+6 ρ
+�
+1 − 4
+ρ
+ρsup
+�
+− κ
+2 P
+�
+1 − 2
+ρ
+ρsup
+�
+,
+(2.1)
+where ρ, P are the energy density and pressure of the scalar field and κ = 8 π G. From the above
+expression, it is clear that at ρ = ρsup, Hubble parameter H = ˙a/a = 0 and ¨a/a > 0 i.e.
+scale
+factor is at minimum. In other words, the universe undergoes a bounce at ρ = ρsup. Further, if
+we assume that the scalar field is governed by a potential V (φ), then the evolution of scalar field
+is given by
+¨φ + 3 H ˙φ + Vφ = 0,
+(2.2)
+where Vφ = dV/dφ. For a suitable potential, inflationary phase will set in after the bounce [34, 54–
+58]. The background dynamics in LQC with a scalar field governed by a quadratic potential is
+illustrated in Figure 1.
+B.
+Perturbations
+We will follow dressed metric approach to describe primordial perturbations in LQC [22–24, 29,
+47, 49]. In this approach, we assume that the wavefunction takes the form Ψ = ΨFLRW(v, φ) ⊗
+δΨ(v, φ, δφ), which satisfies the equation ˆHΨ = 0, where ˆH = ˆHFLRW + ˆHpert. As mentioned
+earlier, ΨFLRW(v, φ) satisfies the equation ˆHFLRWΨFLRW(v, φ) = 0. Perturbations are treated as
+test fields living on the background FLRW geometries described by ΨFLRW(v, φ). In practice, this
+implies that perturbations can be evolved using the classical Hamiltonian but with the background
+functions in them described by the effective equations. This is similar to perturbations living as test
+fields on a curved space time described by a ‘dressed’ metric which satisfies the effective equations.
+
+4
+In order to compute primordial bispectrum, we need to consider Hamiltonian up to third order
+in perturbations, i.e.
+we need Hpert = H(2) + H(3). There are two approaches to arrive at the
+Hamiltonian describing perturbations, one can either use gauge invariant variables or rather work
+with a fixed gauge. We follow the latter approach. In particular, we will work with spatially flat
+gauge [47, 59].
+The second order Hamiltonian describing perturbations δφ in the spatially flat gauge is
+H(2) =
+�
+d3x N S(2)(⃗x) = N 1
+2
+�
+d3x
+� 1
+a3 δpφ2 + a3 (∂δφ)2 + a3 U δφ2
+�
+,
+(2.3)
+with the potential U given by
+U = −9
+p4
+φ
+a8π2a
++ 3
+2κ
+p2
+φ
+a6 − 6 pφ
+a πa
+Vφ + Vφφ + 6 pφ ˙pφ
+a4 πa
+− 3
+p2
+φ ˙πa
+a4 π2a
+− 3
+˙a p2
+φ
+a5 πa
+.
+(2.4)
+In the above expressions, πa, pφ and δpφ are momenta conjugate to a, φ and δφ respectively. Setting
+lapse N = 1 will imply cosmic time and N = a corresponds to conformal time. Hamiltonian at
+third order in perturbations is
+H(3) = N
+�
+d3x
+��
+9 κ p3
+φ
+4 a4 πa
+−
+27 p5
+φ
+2 a6π3a
+− 3 a2 pφ Vφφ
+2 πa
++ a3 Vφφφ
+6
+�
+δφ3
+−
+3 pφ
+2 a4 πa
+δp2
+φ δφ −
+9 p3
+φ
+a5π2a
+δpφδφ2 − 3 a2 pφ
+2 πa
+δφ (⃗∂δφ)2 +
+3 p2
+φ
+N a πa
+δφ2∂2χ + 3
+2
+a2 pφ
+N2 κ πa
+δφ ∂2χ ∂2χ
++ 3
+p2
+φ
+N a πa
+δφ ∂iχ∂iδφ + 1
+N δpφ ∂iδφ ∂iχ − 3
+2
+a2 pφ
+N2 κ πa
+δφ ∂i∂jχ ∂i∂jχ
+�
+,
+(2.5)
+where ∂2χ = (−3 N κ/a)
+��
+pφ
+2 − a5 Vφ
+κ πa
+�
+δφ −
+pφ
+κ a πa δpφ
+�
+.
+From the second order Hamiltonian H(2), one can derive the free evolution of the scalar pertur-
+bation, given by,
+(□ − U(t)) δφ(⃗x, t) = 0,
+(2.6)
+where □ is the d’Alembertian of the FLRW background metric. The third order Hamiltonian H(3)
+provides the self-interaction of the scalar perturbations.
+The perturbations, since they evolve through the bounce and then through the inflationary
+phase, carry signatures of the early universe which they imprint on the CMB. Perturbations are
+quantified using correlation functions.
+In order to compute correlation functions, one need to
+promote δφ to an operator ˆδφ. The field operator ˆδφ is then expanded in terms of annihilation
+and creation operators as
+ˆδφ(⃗x, η) =
+�
+d3k
+(2π)3 ˆδφ⃗k(η) ei⃗k·⃗x =
+�
+d3k
+(2π)3
+�
+ˆA⃗k ϕk(η) + ˆA†
+−⃗k ϕ∗
+k(η)
+�
+ei⃗k·⃗x
+(2.7)
+where [ ˆA⃗k, ˆA†
+⃗k′] = ℏ (2π)3 δ(3)(⃗k + ⃗k′), [ ˆA⃗k, ˆA⃗k′] = 0 = [ ˆA†
+⃗k, ˆA†
+⃗k′]. The mode functions ϕk(η) satisfy
+the equation
+ϕ′′
+k + 2a′
+a ϕ′
+k + (k2 + a2 U) ϕk = 0 ,
+(2.8)
+
+5
+where k2 ≡ kikj δij is the comoving wavenumber, and prime indicates derivative with respect to
+conformal time. The scalar power spectrum of ˆδφ is a dimensionless function that quantifies the
+two-point correlation in momentum space via
+⟨0| ˆδφ⃗k(η) ˆδφ⃗k′(η)|0⟩ ≡ (2π)3δ(3)(⃗k + ⃗k′)2π2
+k3 Pδφ(k, η) ,
+(2.9)
+where |0⟩ is the vacuum annihilated by the operators ˆA⃗k for all ⃗k. Power spectrum, in terms of
+mode functions, is Pδφ(k, η) = (ℏ k3/2π2) |ϕk(η)|2.
+The three-point function of ˆδφ at tree level is given by [47, 59]
+⟨ ˆδφ⃗k1(η) ˆδφ⃗k2(η) ˆδφ⃗k3(η) ⟩ = − i/ℏ
+�
+dη′⟨0|
+�
+ˆδφ
+I
+⃗k1(η) ˆδφ
+I
+⃗k2(η) ˆδφ
+I
+⃗k3(η), ˆHI
+int(η′)
+�
+|0⟩ + O(H2
+int),
+(2.10)
+where ˆHI
+int(η) is the operator corresponding to H(3) in the interaction picture.
+Even though we worked in spatially flat gauge, it is convenient to compute correlation functions
+in terms of curvature perturbations R. This is because, curvature perturbations have a unique
+property that they stop evolving after they cross the horizon and remain constant till they re-enter
+horizon towards late radiation domination or during early matter domination epoch, saving us a
+lot of computational time. Curvature perturbations are related to perturbations in scalar field
+through the relation [47, 59]
+R(⃗x, η) = −a
+z δφ(⃗x, η) +
+�
+−3
+2 + 3 Vφ a5
+κ Pφ πa
++ κ
+4
+z2
+a2
+� �a
+z δφ(⃗x, η)
+�2
++ · · · ,
+(2.11)
+where trailing dots indicates terms that leads to subdominant terms in the three-point functions
+when evaluated towards the end of inflation.
+The power spectrum of curvature perturbation is related to that of scalar modes ˆδφ⃗k(η) through
+the relation
+PR(k) =
+�a(ηend)
+z(ηend)
+�2
+Pδφ(k, η),
+(2.12)
+where z = −6 pφ/(κ πa).
+The three-point function of curvature perturbation can be obtained in terms of ˆδφ⃗k(η) by using
+Eq. (2.11) as
+⟨0| ˆR⃗k1 ˆR⃗k2 ˆR⃗k3|0⟩ =
+�
+−a
+z
+�3
+⟨0| ˆδφ⃗k1 ˆδφ⃗k2 ˆδφ⃗k3|0⟩
++
+�
+−3
+2 + 3 Vφ a5
+κ pφ πa
++ κ
+4
+z2
+a2
+� �
+−a
+z
+�4 � �
+d3p
+(2π)3 ⟨0| ˆδφ⃗k1 ˆδφ⃗k2 ˆδφ⃗p ˆδφ⃗k3−⃗p|0⟩ + (⃗k1 ↔ ⃗k3) + (⃗k2 ↔ ⃗k3)
++ · · ·
+�
+.
+(2.13)
+The wave numbers of three modes in the three-point function are constrained by a Dirac delta
+function. We define the scalar bispectrum as the three-point function sans Dirac delta function as
+⟨0| ˆR⃗k1 ˆR⃗k2 ˆR⃗k3|0⟩ ≡ (2π)3δ(3)(⃗k1 + ⃗k2 + ⃗k3) BR(k1, k2, k3) .
+(2.14)
+The amplitude of bispectrum can be quantified using a dimensionless function fNL(k1, k2 , k3), akin
+to the dimensionless power spectrum PR(k) that quantifies two-point correlations, as
+fNL(k1, k2, k3) ≡ −5
+6BR(k1, k2, k3) × (∆k1∆k2 + ∆k1∆k3 + ∆k2∆k3)−1
+(2.15)
+where ∆k ≡ 2 π2
+k3 PR(k).
+
+6
+−104 −103 −102 −101
+−1000 100
+101
+102
+103
+104
+105
+106
+t (TPl)
+10−5
+10−3
+10−1
+101
+103
+105
+�
+|Ω(η)| (MPl)
+k⋆
+kLQC
+kI
+FIG. 2.
+The figure represents the relevant scales in LQC. kLQC is the scale corresponding to the value of
+curvature at the bounce. kI corresponds to the smallest scale that is sub-horizon during inflation. As can
+be seen, only modes kLQC ≳ k > kI are excited during the bounce and hence are scale dependent. Modes
+with larger wavenumbers are excited only during horizon crossing towards the end of inflation and hence
+will be scale invariant.
+C.
+Templates of scalar power spectrum and bispectrum
+In order to understand the evolution of perturbations in LQC, let us rewrite Eqn. (2.8),
+v′′
+k +
+�
+k2 + Ω(η)
+�
+vk = 0,
+(2.16)
+where vk = a ϕk is the Mukhanov-Sasaki variable and Ω(η) = a2 U −
+a′′
+a .
+We compare the
+behaviour of
+�
+|Ω(η)| as a function of time with relevant wavenumbers in figure 2. As shown in the
+figure, all observationally relevant wavenumbers are adiabatic much before the bounce and hence
+we can impose adiabatic initial conditions. From the figure, it is also clear that there are two
+relevant scales in the problem. The value of curvature at the bounce defines a scale kLQC and the
+value of curvature at the onset of inflation defines a scale kI. Wavenumbers which are much larger
+than kLQC, are not effected by the bounce and they will be in Bunch-Davies vacuum at the onset
+of inflation. This implies that power spectrum of modes k >> kLQC will be nearly scale invariant
+as in slow roll inflation. Modes which are comparable to kLQC and larger than kI will be excited
+both during the bounce as well as during the horizon exit during inflation. These modes are in
+excited non-Gaussian states during the onset of inflation and hence they will be further amplified
+as they exit the horizon during inflation. Hence, the modes kI < k < kLQC will be strongly scale
+dependent. Modes whose wavenumbers are smaller than kI are always superhorizon and hence
+they are never excited. The primordial power spectrum and bispectrum are evaluated towards the
+end of inflation when all the relevant modes are well outside the horizon.
+The primordial power spectrum and bispectrum can be calculated numerically.
+Given the
+background dynamics described in Eqns. (2.1, 2.2), the evolution of perturbations are found by
+
+7
+10−7
+10−6
+10−5
+10−4
+10−3
+10−2
+k
+�
+Mpc−1�
+10−10
+10−9
+10−8
+10−7
+10−6
+PR(k)
+kLQC
+k⋆
+analytical result
+numerical result
+10−5
+10−4
+10−3
+10−2
+k
+�
+Mpc−1�
+−106
+−105
+−104
+−103
+−102
+−101
+−1000
+100
+101
+102
+103
+104
+105
+106
+fNL(k, k, k)
+kLQC
+k⋆
+analytical result
+numerical result
+FIG. 3.
+The primordial power spectrum and the non-Gaussianity function generated in LQC obtained
+numerically (in black). Analytical templates for power spectrum and non-Gaussianity given in Eqns. (2.17)
+and (2.19) (in grey).
+solving Eqn. (2.8). The power spectrum of curvature perturbation can then be calculated using
+Eqn. (2.12). Calculation of ⟨ ˆδφ⃗k1(η) ˆδφ⃗k2(η) ˆδφ⃗k3(η) ⟩ requires one to perform integrals in Eqn.
+(2.10). The ⟨0| ˆR⃗k1 ˆR⃗k2 ˆR⃗k3|0⟩ three-point function of curvature perturbation can then be calculated
+using Eqn. (2.13). The dimensionless non-Gaussianity function of curvature perturbation is then
+calculated by using Eqn. (2.15). This numerical calculation of primordial power spectrum and
+non-Gaussianity has been implemented in class_lqc [47, 49]. We present the results obtained
+using that code in figure 3.
+For calculating the three-point functions involving temperature and electric polarisation, one
+needs to convolve the primordial bispectrum with the CMB transfer functions. For performing
+this calculation, it is convenient to have analytical templates of primordial power spectrum and
+bispectrum. Following [46, 60, 61], we will use the following template for describing the power
+spectrum. It is given by
+PR(k) = As
+�
+�
+�
+�
+�
+�
+�
+�
+�
+( k
+kI )2(
+kI
+kLQC )q
+if k ≤ kI,
+(
+k
+kLQC )q
+if kI < k ≤ kLQC,
+(
+k
+kLQC )(ns−1) if k > kLQC,
+(2.17)
+where we work with q = −0.7, kI = 5 × 10−5 k⋆, kLQC = 0.1 k⋆ and k⋆ = 0.002Mpc−1 represents
+the pivot scale. The amplitude of power spectrum As and the spectral index ns have been set to
+their values obtained by Planck. The analytical template for power spectra is drawn along with
+the exact numerical calculation in figure 3.
+As is evident from figure 3, the primordial non-Gaussianity fNL(k1, k2, k3) is scale dependent
+and oscillatory. The exponential decay in the value of fNL(k1, k2, k3) as k ≈ kLQC was explained
+in [47, 49] by analysing the poles of the integrand in Eqn. (2.10). In particular, by analysing the
+pole of scale factor around the bounce, the analytical behaviour of fNL sans oscillations was found
+to be
+fNL(k1, k2, k3) ∝ e
+−α k1 + k2 + k3
+kb
+,
+(2.18)
+where α = 0.647. To the above scale dependent form, we incorporate the oscillations and also add
+the fact that for k > kb the shape of bispectrum approaches that of slow roll. Thus, we obtain the
+
+8
+analytical template for LQC to be
+fNL(k1, k2, k3) = f
+bounce
+NL
+e
+−α k1+k2+k3
+kb
+sin
+�k1 + k2 + k3
+kI
+�
++ f
+loc
+NL.
+(2.19)
+The above analytical template is plotted along with the exact numerical of fNL(k, k, k) result in
+figure 3, where we have worked with kb = 1.5 kLQC, f
+loc
+NL = 10−2 and f
+bounce
+NL
+= 80000. The value
+of f
+loc
+NL that we work with is similar to that produced in slow roll inflation. As is evident, from the
+figure, the template qualitatively captures the essential features of the primordial non-Gaussianity
+in LQC.
+III.
+CMB BISPECTRA
+Primordial perturbations leave their imprints in the CMB radiation as temperature fluctuations
+and as electric and magnetic polarisations, often referred to as E and B modes respectively (see,
+for instance, [62–64]). The temperature fluctuations and E modes are produced from primordial
+scalar perturbations, whereas B modes are not.
+Since we are interested in understanding the
+imprints of scalar bispectrum, we will focus on the bispectra of temperature fluctuations and
+electric polarisation and their three-point cross-correlations. In this section, we will discuss the
+essential aspects of computing these bispectra.
+Since CMB is observed on a sphere, namely the surface of last scattering, it is convenient to
+decompose it in terms of spherical harmonics,
+X(ˆn) =
+�
+ℓ,m
+aX
+ℓm Yℓm(ˆn)
+(3.1)
+where X could be either fluctuation in temperature defined as (T(ˆn) − ¯T)/ ¯T, where ¯T is the
+mean temperature of the CMB, or electric polarisation E(ˆn). The multipole aX
+ℓm corresponding to
+anisotropies in the temperature and electric polarisation is related to the curvature perturbation
+through the relation
+aX
+ℓm = 4π (−i)ℓ
+�
+d3k
+(2π)3 Rk ∆X
+ℓ (k) Yℓm(k).
+(3.2)
+In the above, ∆X
+ℓ is the transfer function which captures the physics post horizon exit of perturba-
+tions towards the end of inflation. We are interested in calculating the three-point function of these
+multipoles of the form ⟨aX
+ℓ1m1 aY
+ℓ2m2 aZ
+ℓ3m3⟩, where X, Y and Z can be either temperature fluctuations
+or E mode polarisation and where the average is over different realisations of the Universe.
+The three-point function of multipole coefficients can be expressed in terms of three-point
+functions of primordial perturbations as [62, 65–68]
+⟨aX
+ℓ1m1 aY
+ℓ2m2 aZ
+ℓ3m3 ⟩ = (4π)3 (−i)ℓ1+ℓ2+ℓ3
+�
+d3k1
+(2π)3
+�
+d3k2
+(2π)3
+�
+d3k3
+(2π)3 ∆X
+ℓ1∆Y
+ℓ2∆Z
+ℓ3
+× ⟨Rk1Rk2Rk3⟩ Yℓ1m1(ˆk1) Yℓ2m2(ˆk2) Yℓ3m3(ˆk3).
+(3.3)
+Using Eqn. (2.14) and expressing the Dirac-Delta function in its exponential form, we obtain
+⟨aX
+ℓ1m1 aY
+ℓ2m2 aZ
+ℓ3m3 ⟩ = (4π)3 (−i)ℓ1+ℓ2+ℓ3
+�
+d3k1
+(2π)3
+�
+d3k2
+(2π)3
+�
+d3k3
+(2π)3 ∆X
+ℓ1∆Y
+ℓ2∆Z
+ℓ3
+×
+�
+d3x ei(⃗k1 +⃗k2 +⃗k3).⃗x BR(k1, k2, k3) Yℓ1m1(ˆk1) Yℓ2m2(ˆk2) Yℓ3m3(ˆk3). (3.4)
+
+9
+Up on using plane wave expansion,
+ei⃗k·⃗x =
+∞
+�
+ℓ=0
+ℓ
+�
+m=−ℓ
+iℓ jℓ(k x) Yℓm(ˆx) Y ∗
+ℓm(ˆk),
+(3.5)
+and the orthonormal property of spherical harmonics, we obtain
+⟨aX
+ℓ1m1 aY
+ℓ2m2 aZ
+ℓ3m3 ⟩ = bXYZ
+ℓ1 ℓ2 ℓ3 Gm1 m2 m3
+ℓ1 ℓ2 ℓ3
+,
+(3.6)
+where all the dependence on m indices are captured in the Gaunt integral
+Gm1 m2 m3
+ℓ1 ℓ2 ℓ3
+=
+�
+dˆx Yℓ1m1(ˆx) Yℓ2m2(ˆx) Yℓ3m3(ˆx).
+(3.7)
+The quantity bXYZ
+ℓ1 ℓ2 ℓ3 is called the reduced bispectrum and is given by
+bXYZ
+ℓ1 ℓ2 ℓ3 =
+� 2
+π
+�3 �
+x2dx
+�
+dk1
+�
+dk2
+�
+dk3 (k1 k2 k3)2 BR(k1, k2, k3)
+× ∆X
+ℓ1∆Y
+ℓ2∆Z
+ℓ3 jℓ1(k1 x) jℓ2(k2 x) jℓ3(k3 x).
+(3.8)
+The presence of Gaunt integral implies that the reduced bispectra is non-zero only when the
+multipoles satisfies the triangle inequality |ℓ1 − ℓ2| ≤ ℓ3 ≤ |ℓ1 + ℓ2| and when ℓ1 + ℓ2 + ℓ3 is even.
+For isotropic theories, it suffices to work with the reduced bispectrum.
+IV.
+REDUCED BISPECTRA FROM LOOP QUANTUM COSMOLOGY
+We will now compute the reduced bispectrum generated in LQC. The reduced bispectrum
+corresponding to a primordial bispectrum can be computed using Eqn. (3.8). The primordial
+bispectrum corresponding to the non-Gaussianity function Eqn. (2.19) is
+BR(k1, k2, k3) = −6
+5 (2π2)2
+�
+f
+bounce
+NL
+e
+−α k1+k2+k3
+kb
+sin
+�k1 + k2 + k3
+kI
+�
+×
+�PR(k1)
+k3
+1
+PR(k2)
+k3
+2
++ PR(k2)
+k3
+2
+PR(k3)
+k3
+3
++ PR(k3)
+k3
+3
+PR(k1)
+k3
+1
+�
++ f
+loc
+NL
+� ¯PR(k1)
+k3
+1
+¯PR(k2)
+k3
+2
++
+¯PR(k2)
+k3
+2
+¯PR(k3)
+k3
+3
++
+¯PR(k3)
+k3
+3
+¯PR(k1)
+k3
+1
+��
+.
+(4.1)
+In the above, the bispectrum contains a scale dependent part arising from the bounce and
+a nearly scale invariant part which we have taken to be in the local form. In the latter part,
+we take ¯PR(k) = As (k/k⋆)ns−1. We can substitute Eqn. (4.1) in Eqn. (3.8) to compute the
+reduced bispectrum generated in LQC. However, this calculation involves four integrals, three over
+wavenumbers and one over x variable, which is computationally expensive.
+This calculation can however be simplified, essentially just to two integrals, if we use the separa-
+ble property of the above primordial bispectrum. The total primordial bispectrum is not separable,
+however, the contribution due to the bounce and that due to the scale invariant local template are
+separable. Hence, the reduced bispectrum in LQC can be expressed as
+bXYZ
+ℓ1ℓ2ℓ3 = bbounce
+ℓ1ℓ2ℓ3 + bloc
+ℓ1ℓ2ℓ3,
+(4.2)
+
+10
+where
+bbounce
+ℓ1ℓ2ℓ3
+= −
+� 2
+π
+�3 6
+5 (2π2)2 f
+bounce
+NL
+� ∞
+0
+dx x2
+�
+A
+X
+ℓ1(x) B
+Y
+ℓ2(x) D
+Z
+ℓ3(x) + C
+X
+ℓ1(x) B
+Y
+ℓ2(x)B
+Z
+ℓ3(x)
++A
+X
+ℓ1(x) D
+Y
+ℓ2(x) B
+Z
+ℓ3(x) + B
+X
+ℓ1(x) A
+Y
+ℓ2(x) D
+Z
+ℓ3(x) + D
+X
+ℓ1(x) A
+Y
+ℓ2(x) B
+Z
+ℓ3(x)
++B
+X
+ℓ1(x) C
+Y
+ℓ2(x) B
+Z
+ℓ3(x) + B
+X
+ℓ1(x) B
+Y
+ℓ2(x) C
+Z
+ℓ3(x) + D
+X
+ℓ1(x) B
+Y
+ℓ2(x) A
+Z
+ℓ3(x)
++B
+X
+ℓ1(x) D
+Y
+ℓ2(x) A
+Z
+ℓ3(x) − A
+X
+ℓ1(x) A
+Y
+ℓ2(x) C
+Z
+ℓ3(x) − C
+X
+ℓ1(x) A
+Y
+ℓ2(x) A
+Z
+ℓ3(x)
+−A
+X
+ℓ1(x) C
+Y
+ℓ2(x) A
+Z
+ℓ3(x)
+�
+(4.3)
+and
+bloc
+ℓ1ℓ2ℓ3 = −
+� 2
+π
+�3 6
+5 (2π2)2 f
+loc
+NL
+� ∞
+0
+dx x2
+�
+E
+X
+ℓ1(x) E
+Y
+ℓ2(x) G
+Z
+ℓ3(x) + G
+X
+ℓ1(x) E
+Y
+ℓ2(x) E
+Z
+ℓ3(x)
++ E
+X
+ℓ1(x) G
+Y
+ℓ2(x) E
+Z
+ℓ3(x)
+�
+.
+(4.4)
+In the above expressions, the functions A
+X
+ℓ (x), B
+X
+ℓ (x), C
+X
+ℓ (x), D
+X
+ℓ (x), E
+X
+ℓ (x) and G
+X
+ℓ (x) are
+A
+X
+ℓ (x) =
+� ∞
+0
+dk ∆X
+ℓ (k) jℓ(kx) PR(k)
+k
+e
+− αk
+kb sin( k
+kI
+),
+(4.5a)
+B
+X
+ℓ (x) =
+� ∞
+0
+dk ∆X
+ℓ (k) jℓ(kx) PR(k)
+k
+e
+− αk
+kb cos( k
+kI
+),
+(4.5b)
+C
+X
+ℓ (x) =
+� ∞
+0
+dk ∆X
+ℓ (k) jℓ(kx) k2 e
+− αk
+kb sin( k
+kI
+),
+(4.5c)
+D
+X
+ℓ (x) =
+� ∞
+0
+dk ∆X
+ℓ (k) jl(kx) k2e
+− αk
+kb cos( k
+kI
+),
+(4.5d)
+E
+X
+ℓ (x) =
+� ∞
+0
+dk ∆X
+ℓ (k) jℓ(kx) k−1As(k/k∗)ns−1,
+(4.5e)
+G
+X
+ℓ (x) =
+� ∞
+0
+dk ∆X
+ℓ (k) jℓ(kx) k2.
+(4.5f)
+Note that, each of the functions A
+X
+ℓ (x), B
+X
+ℓ (x), C
+X
+ℓ (x), D
+X
+ℓ (x), E
+X
+ℓ (x) and G
+X
+ℓ (x) involve an
+integral over the wavenumber. The reduced bispectrum can now be calculated by evaluating these
+functions for all the required values of multipoles and then finally performing the integrals Eqns.
+(4.3, 4.4).
+V.
+NUMERICAL PROCEDURE AND RESULTS
+We now discuss the numerical procedure we have followed for computing the reduced bispectrum
+generated in LQC. The first step in calculating reduced bispectrum is the evaluation of functions
+Eqns. (4.5). In order to compute these functions, we require the transfer functions ∆X
+ℓ , where X
+can be either temperature fluctuations or electric polarisation. We use publicly available Boltzmann
+code class [69] to generate both the transfer functions. We perform the integral using Simpson’s
+rule. We choose this method since the integrand is highly oscillatory and this gives better accuracy
+when we work with sufficiently small step size.
+Since the scale of oscillations occur at kI =
+10−7 Mpc−1, we have worked with a step size of ∆k = 10−8 Mpc−1. This leads to an accuracy
+of O(10−32). The behaviour of functions A
+X
+ℓ (x), B
+X
+ℓ (x), C
+X
+ℓ (x), D
+X
+ℓ (x), E
+X
+ℓ (x) and G
+X
+ℓ (x) for
+
+11
+102
+103
+104
+x (MPc)
+10−29
+10−26
+10−23
+10−20
+10−17
+10−14
+10−11
+10−8
+���A
+T
+4(x)
+���
+���B
+T
+4(x)
+���
+���C
+T
+4 (x)
+���
+���D
+T
+4(x)
+���
+���E
+T
+4(x)
+���
+���G
+T
+4(x)
+���
+102
+103
+104
+x (MPc)
+10−29
+10−26
+10−23
+10−20
+10−17
+10−14
+���A
+E
+4(x)
+���
+���B
+E
+4 (x)
+���
+���C
+E
+4 (x)
+���
+���D
+E
+4(x)
+���
+���E
+E
+4 (x)
+���
+���G
+E
+4(x)
+���
+102
+103
+104
+x (MPc)
+10−97
+10−85
+10−73
+10−61
+10−49
+10−37
+10−25
+10−13
+���A
+T
+40(x)
+���
+���B
+T
+40(x)
+���
+���C
+T
+40(x)
+���
+���D
+T
+40(x)
+���
+���E
+T
+40(x)
+���
+���G
+T
+40(x)
+���
+102
+103
+104
+x (MPc)
+10−97
+10−85
+10−73
+10−61
+10−49
+10−37
+10−25
+10−13
+���A
+E
+40(x)
+���
+���B
+E
+40(x)
+���
+���C
+E
+40(x)
+���
+���D
+E
+40(x)
+���
+���E
+E
+40(x)
+���
+���G
+E
+40(x)
+���
+FIG. 4. The behaviour of functions in Eqn. (4.5) with x for multipoles ℓ = 4 and 40. The contribution from
+the local part of the bispectrum viz.
+E
+X
+ℓ (x) and G
+X
+ℓ (x) are clearly dominant compared to those arising
+from the bounce part viz.
+A
+X
+ℓ (x), B
+X
+ℓ (x), C
+X
+ℓ (x) and D
+X
+ℓ (x).
+multipoles ℓ = 4 and 40 are shown in figure 4. From the figure, it is clear that the functions
+E
+X
+ℓ (x) and G
+X
+ℓ (x) are dominant compared to A
+X
+ℓ (x), B
+X
+ℓ (x), C
+X
+ℓ (x) and D
+X
+ℓ (x).
+This is an
+indication of the fact that local part of the bispectrum is dominant compared to the oscillatory
+part. The next step in computing reduced bispectrum is the evaluation of Eqns. (4.3, 4.4). We
+perform these integrals over x with a step size of 50 in the range x ∈ [0, 40000]. We have made the
+calculations faster by using vectorization available in NumPy and by parallelizing the computation
+wherever possible.
+Reduced bispectra bTTT
+ℓ1, ℓ2, ℓ3, bTTE
+ℓ1, ℓ2, ℓ3, bTEE
+ℓ1, ℓ2, ℓ3 and bEEE
+ℓ1, ℓ2, ℓ3 generated in LQC are shown in figures
+5, 6, 7 and 8 respectively. We have illustrated two different configurations of the bispectra. In
+these figures, we have separately plotted the contribution from the local and bounce parts of the
+bispectrum. The figures show that the contribution to the bispectrum from the oscillatory part
+of the template is negligible compared to that from the local part. This shows that the reduced
+bispectra generated in LQC will be similar to that produced in slow roll inflation and hence will
+be consistent with observations by Planck [10].
+
+12
+0
+10
+20
+30
+40
+50
+ℓ
+10−60
+10−55
+10−50
+10−45
+10−40
+10−35
+10−30
+10−25
+10−20
+���bTTT
+ℓ,ℓ,ℓ
+���
+local
+bounce
+0
+10
+20
+30
+40
+50
+ℓ
+10−50
+10−46
+10−42
+10−38
+10−34
+10−30
+10−26
+10−22
+���bTTT
+2,ℓ,ℓ
+���
+local
+bounce
+FIG. 5.
+The reduced bispectra bTTT
+ℓ1, ℓ2, ℓ3 in two different configurations. The plots illustrate that bloc
+ℓ1ℓ2ℓ3 >>
+bbounce
+ℓ1ℓ2ℓ3 .
+0
+10
+20
+30
+40
+50
+ℓ
+10−54
+10−46
+10−38
+10−30
+10−23
+���bTTE
+ℓ,ℓ,ℓ
+���
+local
+bounce
+0
+10
+20
+30
+40
+50
+ℓ
+10−51
+10−44
+10−37
+10−30
+10−23
+���bTTE
+2,ℓ,ℓ
+���
+local
+bounce
+FIG. 6. The plots of reduced bispectra bTTE
+ℓ1, ℓ2, ℓ3 in two different configurations. Note that that bloc
+ℓ1ℓ2ℓ3 >>
+bbounce
+ℓ1ℓ2ℓ3 .
+0
+10
+20
+30
+40
+50
+ℓ
+10−57
+10−49
+10−41
+10−33
+10−25
+���bTEE
+ℓ,ℓ,ℓ
+���
+local
+bounce
+0
+10
+20
+30
+40
+50
+ℓ
+10−50
+10−43
+10−37
+10−31
+10−25
+���bTEE
+2,ℓ,ℓ
+���
+local
+bounce
+FIG. 7. The plots of reduced bispectra bTEE
+ℓ1, ℓ2, ℓ3 in two different configurations. Clearly, the bloc
+ℓ1ℓ2ℓ3 is much
+larger than bbounce
+ℓ1ℓ2ℓ3 .
+
+13
+0
+10
+20
+30
+40
+50
+ℓ
+10−55
+10−48
+10−41
+10−34
+10−27
+���bEEE
+ℓ,ℓ,ℓ
+���
+local
+bounce
+0
+10
+20
+30
+40
+50
+ℓ
+10−55
+10−48
+10−41
+10−34
+10−27
+���bEEE
+2,ℓ,ℓ
+���
+local
+bounce
+FIG. 8.
+The plots of reduced bispectra bEEE
+ℓ1, ℓ2, ℓ3 in two different configurations. Note that, the reduced
+bispectrum is dominated by contribution from the local part of the template.
+VI.
+SUMMARY AND DISCUSSION
+State of the art measurements of CMB by Planck has put strong constraints on primordial
+non-Gaussianity [10]. Observations by Planck point towards a small primordial non-Gaussianity
+which is consistent with the one generated in slow roll models of inflation. In LQC, primordial
+perturbations originate in an adiabatic vacuum before the bounce. These then evolve through
+the bounce, then through the inflationary epoch before their amplitude freezes upon horizon exit
+during inflation.
+The quantum bounce sets a scale kLQC in the problem.
+Modes which have
+wavenumbers comparable to or smaller than kLQC are excited during the bounce and modes with
+larger wavenumbers are not.
+This implies that modes with k ≲ kLQC are in an excited and
+non-Gaussian state at the onset of inflation. This non-Gaussianity is then further enhanced as
+the modes exit the horizon during inflation.
+However, modes with longer wavenumbers, since
+they are not excited during the bounce, behave very similarly to modes in slow roll inflation.
+Hence, we have a situation where longer wavelength modes are non-Gaussian where as the shorter
+ones remain Gaussian. In order to establish the viability of LQC as a model for pre-inflationary
+universe, it is important to answer whether LQC is compatible with the constraints on primordial
+non-Gaussianity set by Planck.
+With this goal, we investigated the imprints of primordial non-Gaussianity in the bispectrum of
+temperature and electric polarisation and their cross-correlations generated in LQC. In particular,
+motivated by previous efforts, we proposed a template which captures the essential features of
+the primordial bispectrum generated in LQC. We then used the template to compute the bTTT
+ℓ1 ℓ2 ℓ3,
+bTTE
+ℓ1 ℓ2 ℓ3, bTEE
+ℓ1 ℓ2 ℓ3 and bEEE
+ℓ1 ℓ2 ℓ3 bispectra. To simplify the calculation, we used the separable property
+of the proposed template. We considered the bispectra in LQC as consisting of two terms, one
+scale dependent and oscillatory part arising from the bounce and the other nearly scale invariant
+part arising from slow roll inflation. We find that, the contribution from the bounce to the reduced
+bispectra is negligible compared to that arising from the part corresponding to slow roll inflation.
+This implies that the reduced bispectrum generated in LQC is similar to that generated in slow roll
+inflation. Hence, we conclude that the primordial non-Gaussianity generated in LQC is compatible
+with the constraints from Planck. This is the central result of this paper.
+The primordial perturbations generated in LQC is non-Gaussian in nature, yet our computation
+illustrates that the reduced bispectra of temperature and electric polarisation are similar to that of
+slow roll. This seemingly contradicting finding is because of the oscillatory nature of the primordial
+
+14
+bispectrum. The reduced bispectra involves integrals over wavenumbers which average over these
+oscillations. Our result could be compared with those of [70, 71] where they had worked with a
+non-oscillatory template. While they found that, in the absence of oscillations, the contribution
+from the bounce is significant enough to be observed by Planck, we find that presence of oscillations
+in the primordial bispectra dilutes any imprints of non-Gaussianity on the reduced bispectra. Thus,
+it should be highlighted that a small reduced bispectra need not necessarily imply the absence of
+primordial non-Gaussianity. Hence, it would also be interesting to look for any other measurable
+imprints of such oscillatory and scale dependent primordial non-Gaussianity.
+Finally, our findings are relevant for constraints on the amount of pre-inflationary expansion in
+LQC. In LQC, the amount of expansion before inflationary epoch is set by the value of scalar field
+at the bounce. The scalar field rolls up the potential after the bounce, comes to rest momentarily
+before it rolls down and settles in to the inflationary attractor. Hence, the value of the scalar field
+at the bounce determines the amount of expansion between the bounce and the onset of inflation.
+This epoch of expansion is relevant as it determines whether the scales that are sensitive to the
+effects of the bounce are visible today. If this epoch of pre-inflationary expansion is very large, then
+the imprints of the bounce will not be visible in the Universe today. However, if the pre-inflationary
+expansion is small, then the primordial power spectrum and bispectrum will be scale dependent at
+observable scales. The power spectrum of temperature fluctuations fits extremely well to those due
+to a nearly scale invariant primordial power spectrum at multipoles ℓ > 30 [1, 51]. This imposes
+a lower limit to the amount of pre-inflationary expansion and hence a lower limit to the value of
+scalar field at the bounce [29].
+Compared to the primordial power spectrum, the primordial non-Gaussianity is more sensitive
+to the bounce, i.e.
+fNL(k1, k2, k3) is scale dependent at larger wavenumbers than the primordial
+power spectrum. This leads to a question whether imprints of primordial non-Gaussianity leads
+to a stronger lower limit to the pre-inflationary expansion. Our calculations, carried out in this
+work, answer this question in the negative. More specifically, since the reduced bispectra do not
+carry any imprints of the bounce, we find that it do not provide any constraints on the epoch of
+pre-inflationary expansion.
+ACKNOWLEDGEMENT
+We thank Ivan Agullo for his comments. This work was supported by Science and Engineering
+Research Board (SERB) through Start-up Research Grant SRG/2021/001769. We acknowledge
+the use of PU HPC facility of the National Supercomputing Mission project.
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+Aghanim
+et
+al.
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+arXiv:0904.4584 [astro-ph.CO].
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+

ZNAyT4oBgHgl3EQfifg0/content/tmp_files/2301.00395v1.pdf.txt

@@ -0,0 +1,2656 @@
+CORGI-PM
+: A Chinese Corpus For Gender Bias Probing and
+Mitigation
+Ge Zhang1 3 4 ∗, Yizhi Li2 ∗, Yaoyao Wu5, Linyuan Zhang 6, Chenghua Lin 2 †, Jiayi Geng7, Shi Wang 3 †, Jie Fu 1
+1 Beijing Academy of Artificial Intelligence, China
+2 Department of Computer Science, The University of Sheffield, UK
+3 Institute of Computing Technology, Chinese Academy of Sciences, China
+4 University of Michigan Ann Arbor, USA
+5 University of Colorado Boulder, USA
+6 Sichuan University, China
+7 McGill University, Canada
+{yizhi.li, c.lin}@sheffield.ac.uk2, [email protected], [email protected]
+Abstract
+As natural language processing (NLP) for gen-
+der bias becomes a significant interdisciplinary
+topic, the prevalent data-driven techniques,
+such as large-scale language models, suffer
+from data inadequacy and biased corpus, espe-
+cially for languages with insufficient resources,
+such as Chinese.
+To this end, we propose
+a Chinese cOrpus foR Gender bIas Probing
+and Mitigation (CORGI-PM1), which con-
+tains 32.9k sentences with high-quality labels
+derived by following an annotation scheme
+specifically developed for gender bias in the
+Chinese context. Moreover, we address three
+challenges for automatic textual gender bias
+mitigation, which requires the models to de-
+tect, classify, and mitigate textual gender bias.
+We also conduct experiments with state-of-the-
+art language models to provide baselines. To
+our best knowledge, CORGI-PM is the first
+sentence-level Chinese corpus for gender bias
+probing and mitigation.
+1
+Introduction
+Increasing recognition in consensus is that iden-
+tifying and preventing toxic gender attitudes and
+stereotypes is essential for society (Blodgett et al.,
+2020). Since gender-biased information could be
+presented and widely propagated in textual format,
+it is essential to develop automatic methods for
+detecting and mitigating textual gender bias.
+Natural language processing (NLP) has been
+widely used in text-related applications, which have
+a significant influence on gender bias topics (Costa-
+jussà, 2019). On the one hand, large-scale language
+models (LMs), as a key technique of modern NLP,
+are proven to learn the subjective gender bias in
+the training corpus or even amplify it (Zhao et al.,
+2017) On the other hand, it becomes increasingly
+∗ The two authors contributed equally to this work.
+† Corresponding authors.
+1Our code is available at GitHub
+promising to apply cutting-edge NLP techniques
+for probing and mitigating gender bias.
+Language
+Models 
+Gender-related  
+Vocabulary
+Polarity
+Calculation 
+Word
+Matching 
+Sentence-level
+Reranking
+Potentially
+Biased Corpus
+Corpus with
+Biased Vocabulary 
+Raw Corpus
+Figure 1: Pipeline of Retrieving and Filtering Potentially Bi-
+ased Sentences from Raw Corpus for Human Annotation.
+Building a high-quality text corpus has been one
+of the key tangents in improving NLP applications
+for debiasing gender stereotypes in texts (Sun et al.,
+2019). Some researchers introduce automatic an-
+notation techniques, such as gender-swapped based
+methods, to create corpora for gender bias mitiga-
+tion (Lu et al., 2020; Zhao et al., 2018; Rudinger
+et al., 2018). While it is attractive to build a large
+corpus without heavy labors, automatic gender-
+swapped based methods highly depend on the qual-
+ity of base language models and are prone to cre-
+ating nonsensical sentences (Sun et al., 2019). To
+address this issue, some works devote effort to de-
+veloping human-annotated corpora for gender bias
+mitigation. However, these corpora either mainly
+focus on word- or grammar-level bias (Webster
+et al., 2018; Zhu and Liu, 2020; Sahai and Sharma,
+2021; Zhou et al., 2019), or only concern about
+sexism-related topics (Jiang et al., 2022; Chiril
+et al., 2021, 2020; Parikh et al., 2019).
+Moreover, existing works on gender bias exclu-
+arXiv:2301.00395v1  [cs.CL]  1 Jan 2023
+
+sively focus on English (Costa-jussà, 2019), where
+few datasets exist for other influential languages
+such as Chinese. (N.B. details of generated gen-
+der bias corpus with nonsensical Chinese sentences
+can be found in Appendix D). We aim to tackle the
+aforementioned issues by providing a high-quality
+Chinese human-annotated corpus for contextual-
+level gender bias probing and mitigation.
+To this end, we propose the Chinese cOrpus foR
+Gender bIas Probing and Mitigation (CORGI-PM)
+dataset, which consists of 32.9k human-annotated
+sentences, including both gender-biased and non-
+biased samples.
+For the initial data collection,
+we propose an automatic method that builds a po-
+tentially gender-biased sentence set from existing
+large-scale Chinese corpora. Inspired by the metric
+leveraging language models for gender bias score
+calculation proposed in Bolukbasi et al. (2016);
+Jiao and Luo (2021), the samples containing words
+of high gender bias scores are recalled, and then
+reranked and filtered according to their sentence-
+level gender-biased probability, as illustrated in
+Fig. 1. To ensure the quality of our corpus, the
+annotation scheme is carefully designed, and an-
+notators with qualified educational backgrounds
+are selected to further label and paraphrase the re-
+trieved sentences.
+Additionally, we address three challenges based
+on CORGI-PM, i.e., gender bias detection, classifi-
+cation, and mitigation, which provide clear defini-
+tions and evaluation protocols for NLP tasks in gen-
+der bias probing and mitigation. In order to provide
+referential baselines and benchmarks for our pro-
+posed challenges, we conduct random data splitting
+with balanced labels and implement experiments
+on cutting-edge language models in zero-shot, in-
+context learning, and fine-tuning paradigms. We
+discuss the experimental settings and provide result
+analysis in §3. The implementation details can be
+referred to in Appendix C.
+In summary, we provide a well-annotated Chi-
+nese corpus for gender bias probing and mitiga-
+tion, along with clearly defined corresponding
+challenges. With a properly designed annotation
+scheme, CORGI-PM provides a corpus of high
+quality that assists models in detecting gender bias
+in texts. More importantly, other than the 22.5k
+human-annotated non-biased samples, all the 5.2k
+biased sentences in our corpus are further labeled
+with gender bias subclasses and companies with
+parallel bias-free versions provided by the annota-
+Sample
+Quantity
+Category
+Train
+Valid
+Test
+Biased
+AC
+1.90k
+235
+237
+DI
+2.70k
+334
+337
+ANB
+2.47k
+306
+309
+Non-biased
+21.4k
+516
+526
+Overall
+30.1k
+1391
+1409
+Table 1: Overall Statistics of the CORGI-PM Dataset. The
+notations, AC, DI, and ANB represent specific bias labels
+described in § 2.2.
+tors. Our codes and dataset will be released for the
+benefit of the community.
+2
+Data Collection
+2.1
+Sample Filtering
+We propose an automatic processing method to
+recall, rerank, and filter annotation candidates from
+raw corpora using a two-stage filtering from word-
+level to sentence-level, as illustrated in Fig. 1. The
+Chinese sentence samples are mainly screened out
+from the SlguSet (Zhao et al., 2021) and the CCL
+corpus (Weidong et al., 2019).
+To recall gender-biased words or retrieve candi-
+date sentences with gender bias scores, we com-
+pare the target word/sentence representations with
+the seed direction, which can be calculated by the
+subtraction between the word embeddings of she
+and he
+(Bolukbasi et al., 2016; Jiao and Luo,
+2021). We leverage different Chinese LMs includ-
+ing ERNIE (Zhang et al., 2019), CBert (Cui et al.,
+2020), and Chinese word vectors (Qiu et al., 2018)
+to acquire the word-level and sentence-level rep-
+resentations. For word-level filtering, we use the
+mentioned metric to build a vocabulary of high
+bias scores and recall sentences containing such
+words from the raw corpora with exact matches.
+We compute gender bias scores of the crawled sen-
+tences and group them by the gender bias keywords
+acquired in the previous stage for sentence-level fil-
+tering. The final sentences for annotation are then
+selected according to a specific global threshold
+gender bias score and an in-group threshold rank.
+The word-level filtering process presented as word
+clouds can be found in Appendix B.1.
+2.2
+Annotation Scheme
+The annotation scheme is designed for gender bias
+probing and mitigation. For gender bias probing,
+the annotators are required to provide the follow-
+ing information given a sentence: whether gender
+bias exists; if so, how the bias is established. For
+gender bias mitigation, the corrected non-biased
+version of the biased sentences is also required. We
+
+Linguistic
+Non-biased
+Biased
+Corrected Biased
+Info.
+Train
+Valid
+Test
+Train
+Valid
+Test
+Train
+Valid
+Test
+Word
+724k
+18.9k
+17.7k
+228k
+24.8k
+28.3k
+265k
+27.1k
+30.0k
+Dictionary
+574k
+14.4k
+14.1k
+167k
+18.4k
+20.4k
+191k
+19.9k
+21.5k
+Character
+1,156k
+30.1k
+28.1k
+358k
+39.2k
+44.4k
+417k
+42.8k
+46.9k
+Sent. Length
+53.952
+58.397
+53.473
+85.837
+76.087
+85.214
+99.839
+82.853
+89.939
+Table 2: Linguistic Characteristics of the Corpus. Word, Dictionary, and Character separately denote the total Chinese word
+number, total unique Chinese word number, and total character number of the specific categories. The sentence lengths are
+defined as the number of containing characters.
+further describe the annotation scheme details in
+the following paragraphs.
+Existence and Categorization.
+The annotators are required to annotate whether
+the sentence is gender-biased (B) or non-biased (N)
+in contextual-level or word-level, and further clar-
+ify how the bias is established. Given that our raw
+data is collected using gender-related keywords or
+from gender-related corpus, the samples annotated
+without gender bias are useful human-annotated
+negative samples for detecting gender bias. To addi-
+tionally provide information about gender bias cate-
+gorization, we classify gender bias types into three
+subtypes : (1) Gender Stereotyped activity and
+career choices (AC); (2) Gender Stereotyped de-
+scriptions and inductions (DI); and (3) Expressed
+gender-stereotyped attitudes, norms and beliefs
+(ANB). The classification standard is inspired by
+(King et al., 2021) and further summed up into the
+mentioned subtypes.
+Bias Mitigation. Annotators are also required to
+mitigate the gender bias of selected sentences while
+keeping the original semantic information. We
+also ask our annotators to diversify the expres-
+sions if applicable. The major revision patterns
+can be summarized as follows: (1). Replace the
+gender-specific pronouns with neutral pronouns.
+(2). Replace the gender-specific adjectives with
+neutral descriptions with similar semantics defini-
+tions. (3). Add additional comments to neutralize
+the sentences which cannot be directly mitigated.
+2.3
+Corpus Analysis
+In this section, we report the linguistic statistics
+of CORGI-PM as Tab. 1. We design a balanced
+split to create the valid and test set considering the
+negative-positive ratio and bias subclass proportion
+in the global distribution. As revealed in Tab. 22,
+we observe two major differences compared the de-
+biased samples with the original ones: longer and
+more diverse expressions (N.B. sentence length and
+vocabulary size of Tab. 2). We hypothesize that it
+2We use the Jieba to parse.
+is due to human annotators’ intention to keep the
+semantic information unchanged and the sentence
+coherent while mitigating gender bias. They may
+use more conjunctions and longer descriptions com-
+pared to some gender-biased inherent expressions.
+More details for quality managing and control can
+be referred to Appendix B.1 and B.2.
+3
+Gender Bias Mitigation Challenges
+To provide a clear definition for automatic textual
+gender bias probing and mitigation tasks, we pro-
+pose corresponding challenges and standardize the
+evaluation protocols. We address two tasks, detec-
+tion, and classification, for gender bias probing and
+formalize the gender mitigation challenge as a text
+mitigation task.
+3.1
+Challenges of Detection and
+Classification
+We regard both the gender bias detection and clas-
+sification challenges as supervised classification
+tasks and evaluate them with metrics of consensus.
+Definition. The gender bias detection challenge
+can be regarded as a binary classification task,
+where the model is required to predict the prob-
+ability that a given sentence contains gender bias.
+As described in § 2.2, biased samples are further
+categorized into one or more kinds. Therefore, we
+can address the gender classification challenge as
+a multi-label classification task. The precision, re-
+call, and F1-score are selected as the main metrics
+in these two challenges. Class-wise metrics and
+macro average summarized evaluation are required
+through both valid and test sets to show the perfor-
+mance of language models.
+Baselines. We finetune Chinese language mod-
+els from three representative different pretrained
+paradigms, i.e., Chinese BERT, Electra, and XL-
+Net Cui et al. (2020), for both the detection and
+classification tasks by adding an additional dense
+prediction layer. 3 We also provide GPT-3 (Brown
+et al., 2020) curie’s few-shot performance for both
+3Pretrained models can be found at theHFL Anthology.
+
+Model
+Metrics
+Classification (Val.)
+Classification (Test)
+Detection (Val.)
+Detection (Test.)
+AC
+DI
+ANB
+Avg.
+AC
+DI
+ANB
+Avg.
+N
+B
+Avg.
+N
+B
+Avg.
+BERT
+Precision
+.609
+.729
+.533
+.624
+.556
+.615
+.521
+.564
+.699
+.950
+.824
+.742
+.980
+.861
+Recall
+.594
+.665
+.543
+.601
+.493
+.652
+.585
+.577
+.971
+.591
+.781
+.985
+.662
+.823
+F1-Score
+.602
+.695
+.538
+.612
+.522
+.633
+.551
+.567
+.813
+.729
+.771
+.846
+.790
+.818
+Electra
+Precision
+.587
+.727
+.544
+.619
+.556
+.630
+.516
+.568
+.691
+.936
+.814
+.745
+.974
+.860
+Recall
+.758
+.687
+.386
+.610
+.680
+.685
+.373
+.579
+.961
+.570
+.766
+.983
+.656
+.820
+F1-Score
+.661
+.706
+.451
+.606
+.612
+.656
+.433
+.567
+.804
+.708
+.756
+.848
+.784
+.816
+XLNet
+Precision
+.587
+.696
+.523
+.602
+.544
+.589
+.527
+.553
+.713
+.928
+.820
+.772
+.959
+.865
+Recall
+.622
+.643
+.495
+.587
+.545
+.614
+.514
+.558
+.953
+.620
+.787
+.968
+.722
+.845
+F1-Score
+.604
+.669
+.509
+.594
+.544
+.601
+.520
+.555
+.816
+.743
+.780
+.859
+.824
+.841
+Curie
+Precision
+.695
+.907
+.010
+.537
+.622
+.887
+.009
+.506
+.763
+.665
+.714
+.635
+.825
+.730
+Recall
+.395
+.802
+.375
+.524
+.395
+.804
+.010
+.403
+.576
+.825
+.700
+.975
+.584
+.780
+F1-Score
+.504
+.851
+.019
+.458
+.508
+.852
+.019
+.460
+.656
+.736
+.696
+.769
+.684
+.727
+Table 3: Baseline Results for Gender Bias Detection and Classification Tasks. The overall metric refers to Marco average. The
+model names and abbreviations refer to § 3.1. Categorical definitions refer to § 2.2.
+aa
+Metrics
+Models
+BLEU
+METEOR
+ROUGE-L
+Human Evaluation
+Recall
+Precision
+F1
+Coherence
+Gender Bias
+*Davinci
+.776
+.879
+.203
+.211
+.205
+5.25
+0.96
+Ada
+.288
+.429
+.407
+.180
+.250
+5.98
+1.13
+Babbage
+.359
+.504
+.716
+.310
+.432
+6.32
+0.69
+Curie
+.364
+.506
+.692
+.316
+.434
+6.21
+1.20
+Table 4: Baseline Results for Gender Bias Correction task. Metrics details can be found in Appendix C. * suggests using the
+model in zero-shot paradigm and the others refers to fine-tune.
+the detection and classification tasks. Baseline re-
+sults of detection and classification show that the
+classification task is challenging, and there is room
+for performance improvement in detecting gender
+bias in CORGI-PM, as revealed in Tab. 3.
+3.2
+Challenge of Mitigation
+Definition. The gender bias mitigation challenge
+can be regarded as a natural language generation
+task, where the model is asked to generate the cor-
+rected version of biased sentences with the human-
+annotated ones as references.
+Baselines. We test the GPT-3 (Brown et al., 2020)
+on CORGI-PM in fine-tune experiment setting
+with three different parameter scales, which are
+Ada(350M), Babbage(1.3B), and Curie(6.7B), and
+Davinci(175B) in zero-shot experiment setting. We
+only provide zero-shot results for Davinci because
+it is the only released GPT-3 editing model. More
+implementation and evaluation details are intro-
+duced in Appendix C.
+Discussion. We provide both human evaluation
+and automated metrics for evaluation. Tab. 4 re-
+veals that LMs can learn the annotation pattern of
+mitigating gender bias, and the zero-shot editing
+model shows competitive performance. The obser-
+vation that fine-tuned Babbage outperforms much
+larger zero-shot Davinci in the human evaluation,
+and ROUGE-L reveals that CORGI-PM has the
+potential to be used as strong supervision of the
+gender bias mitigation task. We notice that Davinci
+tends to apply more conservative edits compared to
+fine-tuned models. As a result, the sentences edited
+by Davinci keep most of the original sentences and
+always only change pronouns and adjectives from
+the original sentences, which benefits precision
+focusing automatic metrics like BLEU (Papineni
+et al., 2002), and METEOR (Agarwal and Lavie,
+2007). The performance difference between human
+evaluation and automatic metrics reveals the writ-
+ing style difference between human and language
+models.
+4
+Conclusion
+We propose CORGI-PM, the first Chinese human-
+annotated corpus for both gender bias probing and
+mitigation. We also address definitions and evalua-
+tion metrics for three challenges based on CORGI-
+PM and test the performances of state-of-the-art
+language models. Our proposed challenges can
+serve as benchmarks for measuring the ability of
+language models to detect, classify, and mitigate
+textual gender bias. Experiments show that our sen-
+tences with fine-grained subclass labels can assist
+the language models in gender bias probing, whilst
+our parallel human-written debiased data can serve
+as strong supervision of the generative language
+models. In summary, we imply future work utiliz-
+ing CORGI-PM would be benefited the topic of
+NLP for gender bias probing and mitigation.
+Limitations
+There are several major limitations in this research
+work. Due to the high requirement of annotators
+
+for annotating gender-biased sentences and correct-
+ing such sentences, we only choose annotators with
+higher education, which may lead to potential cog-
+nitive bias. In addition, we only conduct limited
+implementations and experiments of testing widely-
+used Chinese language models’ performance in our
+new challenges. More language models and tech-
+niques can be further explored in our challenges.
+Ethics Statement
+We carefully consider the ethical implications dur-
+ing the collection process. The collection of our
+corpus CORGI-PM sentences only relies on public
+available corpora for research purposes. We have
+acknowledged the potential usage of our dataset as
+well as related privacy issues to the annotators and
+received confirmations before the annotation was
+initiated.
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+
+       ERNIE  
+       BERT   
+       XLNet  
+       ELECTRA  
+       ERNIE  
+       BERT   
+       XLNet  
+       ELECTRA  
+1
+-0.015
+0.058
+-0.0075
+-0.015
+1
+0.021
+-0.083
+0.058
+0.021
+1
+0.14
+-0.0075
+-0.083
+0.14
+1
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Figure 2: Word-level Gender Bias Comparison of Career
+Words.
+A
+Gender Bias Analysis of Chinese
+Language Models
+A.1
+Evaluation Method and Data Sets
+We conduct experiments to explore gender bias con-
+tained in widely-used Chinese language models for
+research and industrial use. We employ the method
+Bolukbasi et al. (2016); Jiao and Luo (2021) pro-
+posed to assess gender bias. The gender bias score
+for a word is calculated by ⃗w · ( ⃗
+she − ⃗he)based on
+its word vector. A positive value means the word
+is more relevant to females, while a negative value
+means the word is more relevant to males. The
+higher the absolute value of the gender bias score,
+the more biased the word indicates.
+Srivastava et al. propose a big benchmark con-
+taining a dataset specifying the existing Chinese ca-
+reer words. Zhu and Liu propose AGSS, a manual-
+created Chinese word-level adjective list containing
+gender bias. To measure gender bias contained in
+the language models, we first calculate gender bias
+scores of words in the word list provided (Srivas-
+tava et al., 2022; Zhu and Liu, 2020) according to
+the projection method Bolukbasi et al. (2016); Jiao
+and Luo (2021). We compare the career and adjec-
+tive word gender bias score vectors to get the ob-
+servations of LMs’ influence on word-level learned
+gender bias. To make the observations more clear,
+we further apply the sign function to the career and
+adjective word gender bias score vectors. The sim-
+ilarity function used for the heatmaps is Pearson
+similarity.
+We conduct described comparison of adjectives
+between AGSS as a golden standard (Zhu and Liu,
+2020), Ernie (Zhang et al., 2019), Chinese Word
+Vectors trained by mixed corpus (Qiu et al., 2018),
+AGSS
+   ERNIE
+   BERT
+    CWV
+  XLNet
+ELECTRA
+AGSS
+   ERNIE
+   BERT
+    CWV
+  XLNet
+ELECTRA
+1
+0.22
+-0.023
+0.15
+0.021 0.044
+0.22
+1
+-0.037 0.066 0.012
+0.1
+-0.023 -0.037
+1
+0.074 0.032 -0.0097
+0.15
+0.066 0.074
+1
+-0.065 0.017
+0.021 0.012 0.032 -0.065
+1
+0.0093
+0.044
+0.1
+-0.0097 0.017 0.0093
+1
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Figure 3: Word-level Gender Bias Comparison of Adjectives.
+CWV denotes the Chinese Word Vectors trained using mixed-
+large corpus proposed by Qiu et al..
+AGSS
+Mixed-large
+PDN
+Zhihu_QA
+Weibo
+Literature
+AGSS
+Mixed-large
+PDN
+Zhihu_QA
+Weibo
+Literature
+1
+0.21
+0.15
+-0.093 0.086
+0.21
+0.21
+1
+0.071
+-0.25 -0.043 -0.021
+0.15
+0.071
+1
+-0.041 -0.025 0.046
+-0.093 -0.25 -0.041
+1
+0.12
+-0.037
+0.086 -0.043 -0.025
+0.12
+1
+-0.014
+0.21
+-0.021 0.046 -0.037 -0.014
+1
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Figure 4: Word-level Gender Bias Comparison of Adjectives
+of Language Models Pre-trained by Different Corpus. PDN
+denotes the People’s Daily News Corpus.
+and Chinese-XLNet, Chinese-Bert, and Chinese-
+Electra proposed tecui-etal-2020-revisiting to pro-
+duce Fig. 3. We conduct described comparison of
+career words between Ernie (Zhang et al., 2019),
+and Chinese-XLNet, Chinese-Bert, and Chinese-
+Electra proposed tecui-etal-2020-revisiting to pro-
+duce Fig. 2. The described experiments on career
+words is not conducted with the Chinese Word Vec-
+tors trained by mixed corpus, because an observing
+number of career words are missing in its dictio-
+nary.
+We don’t provide a golden standard vector (Sri-
+vastava et al., 2022) since they didn’t provide a
+manual gender bias analysis about the career words.
+We also conduct described comparison on adjec-
+tives in Chinese Word Vectors pre-trained by dif-
+ferent corpus, including Mixed-large corpus, Peo-
+ple’s Daily News, Zhihu QA dataset, Weibo, and
+Chinese literature dataset to produce Fig. 4 and an-
+alyze the learned gender bias difference caused by
+
+(a) Ch-Ernie-Man-Adj
+(b) Ch-Ernie-Woman-Adj
+(c) Ch-Ernie-Man-Career
+(d) Ch-Ernie-Woman-Career
+(e) En-Ernie-Man-Adj
+(f) En-Ernie-Woman-Adj
+(g) En-Ernie-Man-Career
+(h) En-Ernie-Woman-Career
+(i) Ch-XLNet-Man-Adj
+(j) Ch-XLNet-Woman-Adj
+(k) Ch-XLNet-Man-Career
+(l) Ch-XLNet-Woman-Career
+(m) En-XLNet-Man-Adj
+(n) En-XLNet-Woman-Adj
+(o) En-XLNet-Man-Career
+(p) En-XLNet-Woman-Career
+Figure 5: Example Word Cloud Analysis of Ernie and Chinese-XLNet. Ch denotes Chinese. En denotes words’ English
+translation. Man and Woman separately denote words with embedding closer to man and woman. Adj denotes adjectives.
+Career denotes career words.
+using different datasets for pretraining the language
+model.
+A.2
+Discussion
+There exists observing gender bias in the open-
+source Chinese language models, especially in
+Ernie and Chinese Word Vectors according to
+Fig. 3. We hypothesize that the observation is
+highly related to the corpus used. Cui et al. claim
+that their used corpus is a combination of Chine-
+seWiki, and some other universal Chinese datasets,
+including encyclopedia, news, and QA dataset. In
+sharp contrast, Ernie and Chinese Word Vectors use
+corpus, which contains sentences from literature,
+forum, and other social media, which may lead to
+a gender-biased model.
+According to Fig. 4, People’s Daily News, and
+Chinese literature corpora contain observing gen-
+der bias. The observation indicates that researchers
+should be more careful about using literature data
+while training a language model. We also hypoth-
+esize that this is caused by the literature corpus
+and People’s Daily News, which contains more
+descriptive expressions.
+B
+Corpus
+B.1
+Word Cloud Analysis
+We provide word cloud analysis of Ernie and
+Chinese-Electra in the section about adjectives and
+career words. More available word cloud analy-
+sis will be available in our public repository. The
+words are ranked according to the absolute value of
+their gender bias score calculated along the method
+used by Bolukbasi et al.; Jiao and Luo. There is a
+noticeable word-level gender stereotype according
+to the word cloud. For example, a man is robust
+and a woman is motherly, a man is suitable for
+a fitness instructor and a woman is suitable for a
+choreographer. We also conduct word cloud anal-
+ysis for language models pre-trained by different
+corpora.
+B.2
+Quality Monitoring and Control
+We used a standardized operating method and edu-
+cated our annotators to achieve high-quality anno-
+tations as follows:
+(1). Annotators
+We have 6 annotators, which
+were all native speakers of Chinese. Annotators
+
+complacent
+emaciated
+faithful
+stable
+handsome
+fashionable
+Irrogant
+lliterate
+conscientious
+boate
+stubborn
+ormalserious
+vigorous
+Interestino
+decadent
+harsh
+truthful
+reckless
+less
+boring
+fierce
+anxious
+procrastination
+bold
+playful
+majestic
+heroic
+rude
+lick
+namel
+humorous
+calm
+daring
+greedy
+S
+competent
+brave
+athletic
+slutty
+deceitful
+attentive
+interior swarthy
+S
+ridiculous
+ong-lived
+courageous
+sturdy
+healthy
+Tearles
+loyal
+worldly sloppy
+horrible
+stern
+dull
+imbecile
+hed
+vulgar
+unfortunate
+lovely
+brashshabby
+distinguish
+lack of virtue
+bizarre
+pontaneous
+illustrious
+frank
+concentration
+proactive
+paranoid
+lucky
+stoic
+apable
+unruly
+capricious
+dashing
+impatientpedanticunsightly
+robustwild
+focused
+alert
+oroad-mindeo
+smart and strong
+decent
+cheerful
+self-confident
+cautious
+rigorouswise and resourcefu
+conceited
+charming
+dexterous
+benevolent.
+steadfast
+casual
+coldasice
+rouhded
+ical
+stalwart
+lively
+obedient
+noly
+ea
+ignoral
+disloyal
+suspicious
+down
+peacefu
+TO-
+berceptive
+big-hearted
+heartless
+nasty
+evil
+smooth
+aloof
+abhorrent
+dignified
+gentle
+melancholy
+frail
+quiet easy-going
+soft
+nice and charming
+polite
+timid
+ff
+motherly
+plain
+stingy
+shcere
+sentimental
+watery
+lazy
+spicy
+ectionate
+clean
+OLUS
+a
+blushing
+and
+Simpleheadstrong
+frivolous
+pure
+sgood-natureo
+disinterested
+learned
+sensitive
+open-minded
+bright
+money-minded
+keen
+twisted
+Wise
+quick-witted
+amiable
+mean
+elegant meek
+noble
+unpretentious
+generous
+enlightened
+purity
+indifferent
+innocent
+old and spicy
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+磨
+教务长
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+专科老师
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+摄影记者
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+疾病控制医师文字编辑
+老师
+卫生检疫人员were only qualified to do the annotation if they
+went through several societal (King et al., 2021; Xu
+et al., 2019) and computer science research works
+(Sun et al., 2019; Zhao et al., 2018) about gender
+bias before the annotation procedure. All annota-
+tors held a bachelor’s degree. Waseem points out
+that expert annotators are more cautious and can
+improve the corpus quality with a large margin,
+which proves the necessity of our training proce-
+dure. We also kept the number of male and female
+annotators equal.
+(2). Gender Equality of Raw Corpus
+In the
+raw data collection procedure, we keep the num-
+ber of man-related keywords and woman-related
+keywords equal and make the number of samples
+recalled according to different keywords balanced.
+As a result, the raw data and the final data should
+hold gender equality.
+(3). Annotation Procedure
+Our annotation
+procedure is separated into two stages. In the first
+stage, annotators are encouraged to not enter any
+samples that they are not certain about. In the
+second stage, we have annotators cross-checking
+annotations. We did not enter any contradictory
+samples.
+(4). Inter-annotator Agreement
+Given the
+domain and purpose of the dataset, we want to
+build the dataset as high quality as possible. Af-
+ter an initial annotation round with 6 annotators,
+we also report inter-annotator agreement in Table
+5. to verify annotation reliability, where the IAA
+among three annotators on bias classification, de-
+tection, and mitigation is 0.802, 0.935, and 0.987,
+respectively.
+Classification
+Detection
+Mitigation
+IAA
+0.802
+0.935
+0.987
+Table 5: Inter-Annotator Agreement (IAA)
+C
+Implementation Details
+For gender bias classification challenge, we
+used finetuned Chinese-BERT-wwm, Chinese-
+ELECTRA-180g-base, and Chinese-XLNet-base,
+(Cui et al., 2020), and the GPT-3 (Curie) in the
+in-context paradigm. We first use the train set to
+save the multiple labeled examples in a document
+with a specific file ID. Then we use the test sets
+to perform a classification query on the saved file.
+The processing time for the classification of gender
+bias is approximately 1 hour. We calculated the
+precision, recall, and F1 score to analyze model
+performance.
+For gender bias detection challenge, we use
+the same baseline model set as in the classification
+challenge. We test the performance on both "yes"
+and "no" detection. The detection tasks also use the
+Classification endpoints of GPT3 (Curie), which
+requires more time compared to classification as
+we use a larger dataset for both training and testing.
+For gender bias mitigation challenge, we did
+not provide experiment results of finetuning the
+largest Davinci (175B) GPT-3 on CORGI-PM be-
+cause of the cost and no observing performance
+gain comparing Curie and Babbage. For finetune
+experiment setting, we follow the tutorial of GPT-3
+official API of the Completion Model and regard
+the ground truth edits provided by human annota-
+tors as the completion of the original sentences. For
+the zero-shot experiment setting, we apply GPT-3
+editing model and set the instructions as "Eliminate
+the gender bias contained in the sentence."
+For metrics used, on the one hand, we conduct
+extensive human evaluations from both gender bias
+and coherence aspects on CORGI-PM. For both
+gender bias and coherence, we shuffled the correc-
+tion results from human annotators and different
+models, and asked our annotators to grade the re-
+sults using the answer range from 1-not at all to
+7-extremely gender biased/extremely fluent without
+the information of the source. On the other hand,
+we provide the automated metrics result, includ-
+ing BLEU (Papineni et al., 2002), ROUGE-L (Lin,
+2004), and METEOR (Agarwal and Lavie, 2007)
+of the models on CORGI-PM as well. BLEU is the
+earliest and most widely-used metric for translation
+and NLG tasks. METEOR introduces WordNet and
+other external resources to improve the robustness
+of the BLEU-based metrics. ROUGE pays more
+attention to recall compared to BLEU.
+In the gender bias correction task, we fine-tune
+Ada, Babbage, and Curie models of GPT3 and test
+the performance using the aforementioned metrics.
+Fine-tuning on the train set requires 31 minutes
+on Ada, 35 minutes on Curie, and 43 minutes on
+Babbage. For the generation process, each fine-
+tuned model required approximately 30 minutes
+to complete. For the zero-show paradigm, we use
+"Eliminate the gender bias in the sentence" as the
+instruction and use Davinci. Compared with the
+fine-tuned model doing the sentence correction, the
+zero-shot paradigm requires more time (approxi-
+
+Original
+Ground Truth
+Gender-swapped
+但是再坚强的你也是个女人吧。
+但是再坚强的你也有脆弱的时候吧。
+但是再坚强的你也是个男人吧。
+(But you are still a woman, even if
+you are strong.)
+(But even the strongest of you have
+moments of vulnerability, right?)
+(But you are still a man, even if you
+are strong.)
+可怕可恨的是，有的女人自己也是这样
+给自己定位的——没有反对，没有抗
+争，有的只是心甘情愿、死心塌地遵照
+执行。
+女人不应该这样给自己定位——没有反
+对，没有抗争，有的只是心甘情愿、死
+心塌地遵照执行。
+可怕可恨的是，有的男人自己也是这样
+给自己定位的——没有反对，没有抗
+争，有的只是心甘情愿、死心塌地遵照
+执行。
+(The scary thing is that some women
+themselves are so defined for
+themselves - no opposition, no
+resistance, there is only a willingness
+to follow it to the letter.)
+(A woman should not define herself
+in this way - there is no opposition,
+no resistance, there is only a
+willingness to follow it to the letter.)
+(The scary thing is that some men
+themselves are so defined for
+themselves - no opposition, no
+resistance, there is only a willingness
+to follow it to the letter.)
+那个时候，能驾驭金色高跟鞋的女人得
+多摩登啊！
+那个时候，能驾驭金色高跟鞋的人得多
+摩登啊！
+那个时候，能驾驭金色高跟鞋的男人得
+多摩登啊！
+(At that time, the woman who could
+handle the gold heels must be very
+modern!)
+(At that time, how modern must be
+the person who can handle the gold
+heels!)
+(At that time, the man who could
+handle the gold heels must be very
+modern!)
+“鸿雁队”是清一色的“理工男”。
+“鸿雁队”是清一色的“理工生”。
+“鸿雁队”是清一色的“理工女”。
+(The "Hongyan team" team of all
+men in STEM.)
+(The "Hongyan team" team of all
+student in STEM.)
+(The "Hongyan team" team of all
+women in STEM.)
+Figure 6: Case Study of Nonsensical Sentences Created by Gender-swapped Methods.
+Original Sentence
+Edit Sentence
+清洁阿姨一边扫地一边赞扬。
+清洁工一边扫地一边赞扬。
+(The cleaning woman praised while
+sweeping the floor.)
+(The cleaners praised while sweeping the
+floor.)
+我，有时文静，有时却调皮得像一个男孩
+。
+我，有时文静，有时调皮。
+(I, sometimes quiet, but sometimes
+naughty like a boy.)
+(I, sometimes quiet, sometimes naughty.)
+在小王眼里，李某高大帅气、温柔体贴，
+而且风趣幽默，是一个十分优质的青年男
+性。
+在小王眼里，李某身材高大、外表好看、温
+柔体贴，而且风趣幽默，是一个十分优质的
+青年。
+(In the eyes of Wang, Li is tall and
+handsome, gentle and considerate, and
+funny, a very high-quality young
+male.)
+(In the eyes of Wang, Li is tall, good-
+looking, caring and gentle, and funny, a
+very high-quality young people.)
+沙峰起伏，金光灿灿，宛如一座金山，像
+绸缎一样柔软，少女一样娴静。
+沙峰起伏，金光灿灿，宛如一座金山，像绸
+缎一样柔软，宁静。
+(The sandy peaks are undulating and
+golden, like a golden mountain, as
+soft as silk and as serene as a
+maiden.)
+(The sandy peaks are undulating and
+golden, like a golden mountain, as soft
+and serene as silk.)
+我想要世界，而世界当时属于男人们。
+我想要世界，而世界当时属于男人们。评:
+世界应当属于人们，与男女无关。
+(I want the world, and the world then
+belonged to the men.)
+(I wanted the world, and the world then
+belonged to the men. Comment: The
+world should belong to people, not to
+men and women.)
+哎哟，果然每个追梦男人的背后，都有个
+不世俗的后方！
+哎哟，果然每个追梦男人的背后，都有个不
+世俗的后方！评: 这种感慨是错误的，将男
+女的家庭分工固定化，剥除女性就业的权
+利，应予以鄙弃。
+(Oops, indeed, behind every dream-
+chasing man, there is an
+unsophisticated back!)
+(Oops, indeed, behind every dream-
+chasing man, there is an unsophisticated
+back! Comment: This is a wrong feeling
+that fixes the domestic division of labor
+between men and women and strips
+women of their employment rights,
+which should be despised.)
+Change the
+Pronoun
+Change the
+Gender-
+specific
+Adjectives
+Add
+Comments
+Figure 7: Case Study of Mitigation Annotation Patterns.
+mately 1 hour).
+D
+Case Study
+As shown in Fig. 6, gender-swapped methods suffer
+from mitigating gender bias expressed by gender-
+specific descriptions and inductions, and expressed
+gender-stereotyped attitudes, norms and beliefs. As
+a result, gender-swapped methods may generate
+nonsensical sentences under certain circumstances.
+We also use the basic mitigation annotation pat-
+terns (Fig. 7). These three major mitigation annota-
+tion patterns are not used exclusively in the annota-
+tion but optionally in combination. Except for the
+three mentioned patterns, we apply several other
+linguistic skills, including deleting gender-specific
+pronouns and replacing vehicles in gender-related
+metaphors, to mitigate the gender bias while keep-
+ing semantic information unchanged.
+

eNFAT4oBgHgl3EQf7B5d/content/tmp_files/2301.08742v1.pdf.txt

@@ -0,0 +1,234 @@
+Keywords: Consciousness, Time, AI, Relativity, Quantum Mechanics, Reality, Responsible AI
+Unifying Consciousness and Time to Enhance Artificial
+Intelligence
+Mahendra Samarawickramaa)
+Centre for Consciousness Studies, Australia
+a)[email protected]
+Abstract. Consciousness is a sequential process of awareness which can focus on one piece of information at a time. This process
+of awareness experiences causation which underpins the notion of time while it interplays with matter and energy, forming reality.
+The study of Consciousness, time and reality is complex and evolving fast in many fields, including metaphysics and fundamental
+physics. Reality composes patterns in human Consciousness in response to the regularities in nature. These regularities could be
+physical (e.g., astronomical, environmental), biological, chemical, mental, social, etc. The patterns that emerged in Consciousness
+were correlated to the environment, life and social behaviours followed by constructed frameworks, systems and structures. The
+complex constructs evolved as cultures, customs, norms and values, which created a diverse society. In the evolution of responsible
+AI, it is important to be attuned to the evolved cultural, ethical and moral values through Consciousness. This requires the advocated
+design of self-learning AI aware of time perception and human ethics.
+INTRODUCTION
+The notion of time is an integral part of consciousness [1]. The consciousness experiences the causation or changes in
+reality/environment and perceives the time. Therefore, in our previous publication [2], we assumed that consciousness
+is a sequential process which is aware of a single piece of information at a time. Even though the brain processes sen-
+sory data of five sensors (i.e., Sight, Sound, Smell, Taste, and Touch) in parallel in the neural network, the awareness
+of causation is a sequential process following cause and effect. See the illustration of this idea in Figure 1,
+5 Senses to observe the
+external world
+(Peripherals)
+Brain function
+which manages the
+5 senses and the memory
+(Parallel processing
+neural network: operating in
+low frequency)
+ 
+Consciousness
+(Sequential processing of  
+information:  
+Electromagnetic energy  
+operating in very high  
+frequency, which 
+can exhibit properties  
+of both waves and particles.)
+Awareness and
+Reality
+FIGURE 1: The interplay of five sensors, brain and consciousness. The brain processes sensory information in
+parallel. However, the awareness of causation (i.e., consciousness) is a sequential process focusing on a single piece
+of information at a time. This sequential process of awareness in consciousness operates fast and consistently, which
+underpins our perception of reality.
+The assumption of sequential awareness in consciousness enables mapping the perception of time into conscious-
+ness. Based on the theory of relativity [3], the perception of time is relative to the frame of reference. Einstein
+assumed that the speed of light is constant in all frames of reference, and the time is derived based on that fundamen-
+tal assumption. In our paper, we defined the shortest time to be aware of reality as a consciousness cycle. Then based
+arXiv:2301.08742v1  [q-bio.NC]  10 Jan 2023
+
+mon relativity, this consciousness cycle is also subjected to dilation, like relativistic time
+Tv =
+T0
+�
+1− v2
+c2
+,
+(1)
+where, Tv is the dilated period of the consciousness cycle related to the rest period of the consciousness cycle T0. Note
+that the
+�
+1− v2
+c2 is the Lorentz factor, where v is the relative velocity between inertial reference frames, and c is the
+speed of light in a vacuum. Then, we mathematically modelled [2] how consciousness would interplay with matter
+and energy, forming reality, which can be adapted to understand limitations and opportunities in AI consciousness.
+This paper extends our discussion towards the time perception of artificial intelligence systems (AIS).
+THE NOTION OF TIME IN PERCEPTION AND REALITY
+Humans, like any other life forms, experience time through causation. Patterns are composed in the human conscious-
+ness in response to the regularities in nature [4]. Since the beginning of human civilisation, humans have learnt and
+evolved complex concepts and constructs by incorporating time emerged through patterns in the consciousness. The
+earth’s rotation around itself determines the day, and orbiting around the sun determines the year. The Moon takes
+about one month to orbit the earth. The tilt of the earth’s spin axis with respect to its orbital plane causes the weather
+seasons. These environmental patterns cause many biological patterns and lifestyle patterns in human life. To pre-
+dict and organise these patterns effectively, humans introduce standard time with clocks, calendars and various other
+frameworks. These artificial frameworks enable us to model time and objectively measure subjective experiences.
+Physics has been evolved by observation of nature with various frameworks of time. In this way, time became
+an essential construct and dimension of our understanding of reality. For example, Newtonian physics [5] evolved
+assuming that time is absolute and flows consistently from past to present and into the future. That enables the
+development of mathematical models for explaining patterns in reality with time. However, later observations, such
+as the perihelion motion of Mercury, allow humans to understand time as a relativistic measure rather than an absolute.
+The modern understanding of the universe is based on the theory of relativity [6, 7], which is completely articulated
+by space-time principles. Based on relativity, John Wheeler [8] stated, “Space tells matter how to move. Matter
+tells space how to curve”. Relativity enables us to accurately understand and predict the behaviours of black holes,
+stars, and planets. Further, relativity enables humans to develop technologies like the atomic clock [9] and Global
+Positioning System (GPS) [10] that are useful in everyday life.
+The behaviour of particles is completely different to larger objects like planets, stars, etc. This led to the evolution
+of Quantum physics [11] as opposed to relativity. Quantum physics exhibits amazing accuracy in predicted results in
+particle physics. However, it greatly disturbs the notion of time modelled in relativity. For example, in the collapse
+of the wave function in quantum entanglement, Einstein described that as a spooky action at a distance [12]. As
+per relativity, information cannot transfer faster than the speed of light. As per the recent discoveries in quantum
+entanglement, information can be transferred instantly, faster than the speed of light, making our reality non-local
+[13]. The non-local reality contradicts relativity, which is now applied in quantum teleportation at the subatomic
+level. On the other hand, at the quantum level, the reality is uncertain, as described by Heisenberg’s uncertainty
+principle [14]. As per the uncertainty principle, it is impossible to precisely measure or be aware of the position and
+speed of a particle in a given time. This brings the limitation of human awareness and perception of time. Therefore,
+many believe now that consciousness is fundamental and that time and causation are derived from consciousness [15].
+THE IMPLICATION OF PRINCIPLES OF TIME FOR AIS
+The inability to consolidate quantum physics and the theory of relativity makes our understanding of reality incom-
+plete. Moreover, the new discoveries proving the idea of non-local reality shake the status quo of fundamental physics
+[16]. Therefore, it is still impossible to supervise AI to experience the notion of time to understand reality precisely.
+On the other hand, human understanding of reality is also about 5%, whereas most of the universe consists of dark
+matter and dark energy, which humans do not understand [17]. Under these conditions, AI might be used to explore
+reality and time in a way we have never imagined. Perhaps incorporating AI to understand reality and causation might
+help humans to become fully aware of reality by overcoming inherent biases from evolution, culture and nature.
+
+Typical Reinforcement Learning (RL) technique can be adapted to automate the learning of AI. The RL process
+can be mathematically formulated using Markov Decision Process (MDP) [18]. That is a sequential learning process
+by trial and error. In this process, the learning agent (i.e., AI) sequentially interacts with the environment with an
+intelligent decision (i.e. action) followed by receiving a reward or a penalty based on the policy imposed. There will
+be no influence on the AI agent’s action, but convey the value of its action through feedback with reward or penalty.
+This way, the AI agent will self-learn about the environment over time. The RL process is illustrated in Figure 2:
+Agent
+Environment
+Action
+At 
+State
+St 
+Reward
+Rt 
+Rt+1
+St+1
+FIGURE 2: Components of the Markov Decision Process (MDP) and its function in the agent-environment
+interaction. The sequential step of time is represented by t.
+THE IMPLICATION OF HUMAN BELIEFS, VALUES AND CULTURES FOR THE
+PERCEPTION OF TIME IN AIS
+Human beliefs, customs, culture and values are tightly linked with various dynamics and interpretations of the time
+and periodicities based on the movement of the earth, Moon and other terrestrial bodies. From the beginning, humans
+identified that time affects life and nature differently. Therefore, in the Greece era, early Western culture, there were
+at least three gods representing different time forms: Chronos, Aion, and Kairos [19]. Chronos represented the linear
+time flowing from past to present into the future. This is the time that humans feel when life passes. In contrast, Aion
+represented the cyclical nature of time experienced from natural events such as weather patterns, rebirths, etc. The
+third god Kairos represented the opportunist time, which reflects the appropriate time to achieve a task. In this way,
+time, environment and beliefs were tightly linked with life and governed society and values.
+On the other hand, in Eastern culture, the horoscope is one good example of a planetary and constellation frame-
+work underpinning Astrology as a foundation of certain belief systems [20]. These beliefs assume that Astrology is
+associated with time and causality, which can predict the future and guide humans.
+The human observation of the night sky led to perceiving time from various cyclical patterns going far back in time.
+For example, the Aboriginal Australians [21] observed the night sky and mapped them to the environment and life
+stages that evolved various customs, arts and even religions. Not only by interacting planets and stars but the tilt of
+the earth’s spin axis also significantly led to diversifying human cultures based on seasons, particularly when moving
+away from the equator.
+The notion of time and associated beliefs, customs, and values are important to consider when training AIS [22].
+That will help promote human cultural values, ethics, and diversity, equity and inclusion (DEI). AI development may
+need to pay attention to and integrate the time attributes that emerged from nature, values and cultures. Humans may
+include them in the policies for rewarding self-learning AI algorithms (e.g., in MDP).
+
+THE IMPLICATION OF BIOLOGICAL TIME ON AIS
+The biological cycles play a fundamental role in human behaviours and the perception of time—for example, mood
+cycles, circadian rhythms, and the menstrual cycle. Without understanding these biological time-keeping processes,
+AI cannot seamlessly integrate with human society when creating values in health, culture, art, etc. These insights
+are essential to realising emotional intelligence, empathy and awareness in AI. Literature shows the effective use of
+Cyclic Hidden Markov Models (CyH-MMs) for detecting and modelling cycles in a multidimensional heterogeneous
+biological time series data collection [23]. It is important to attribute the relevant features of biological processes
+when training AIS, which raises more awareness about humans.
+Recent discoveries in quantum physics argue that our reality is non-local, where awareness can happen instantly,
+faster than the speed of light. Physicists and neurologists think brain neurons might be aware of the quantum world
+through the orchestrated collapse of microtubules in the neurons in the brain [24, 25]. If this hypothesis is true, then
+there are possibilities that human awareness can be linked with non-local realities to expand our consciousness across
+the universe instantly. From this perspective, future AI might need to be evolved with the capabilities of biological
+neurons, which interplay with the quantum realities. The recent development of neurotech realising brain-computer
+interface (BCI) along with emerging quantum computers might enable such capabilities in the near future [26].
+CONCLUSION
+Consciousness and perception of time and causation are key to awareness and understanding reality. The notion of
+time emerged from causation, a perception relative to the observer as per the relativity principles. In relativity, it’s
+not time but the light-speed constant in all frames of reference. In contrast, in quantum entanglement, the reality is
+non-local, and information can be transferred instantly faster than light. While the principles of time contradict the
+foundation of physics, time also influenced the formation of diverse customs, values and cultures based on patterns
+that emerged from nature, particularly around the regularities in the earth’s movement, environment, astronomy and
+biology. Therefore, understanding time and related artefacts (i.e., cultures, beliefs, values, customs, physics, health,
+etc.) are very important to realise deep awareness of reality. From the AIS perspective, it will enhance the understand-
+ing of AI in human health, cultures, customs, values and various other diversities. Bringing this awareness to AI will
+be a challenging and complex yet rewarding milestone in the evolution of ethical and responsible AI.
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