  We review the state of the art on the delocalized non-ergodic regime of the
Anderson model on Bethe lattices. We also present new results using Belief
Propagation, which consists in solving the self-consistent recursion relations
for the Green's functions directly on a given sample. This allows us to
numerically study very large system sizes and to directly access observables
related to the eigenfunctions and energy level statistics. In agreement with
recent works, we establish the existence of a delocalized non-ergodic phase on
Cayley trees. On random regular graphs instead our results indicate that
ergodicity is recovered when the system size is larger than a cross-over scale
$N_c (W)$, which diverges exponentially fast approaching the localization
transition. This scale corresponds to the size at which the mean-level spacing
becomes smaller than the Thouless energy $E_{Th} (W)$. Such energy scale, which
vanishes exponentially fast approaching the localization transition, is the one
below which ergodicity in the level statistics is restored in the thermodynamic
limit. Remarkably, the behavior of random regular graphs below $N_c (W)$
coincides with the one found close to the root of loop-less infinite Cayley
trees, {\it i.e.} only above $N_c (W)$ the effects of loops emerge and random
regular graphs behave differently from Cayley trees. Our results indicate that
ergodicity is recovered in the thermodynamic limit on random regular graph.
However, all observables probing volumes smaller than $N_c(W)$ and times
smaller than $\hbar/E_{Th} (W)$ are expected to behave as if there were an
intermediate phase. Given the very fast divergence of $N_c(W)$ and
$\hbar/E_{Th} (W)$ these non-ergodic effects are very pronounced in a large
region preceding the localization transition, and they can be related to the
intermediate phase present on Cayley trees.


  We study the notion of formal duality introduced by Cohn, Kumar, and
Sch\"urmann in their computational study of energy-minimizing particle
configurations in Euclidean space. In particular, using the Poisson summation
formula we reformulate formal duality as a combinatorial phenomenon in finite
abelian groups. We give new examples related to Gauss sums and make some
progress towards classifying formally dual configurations.


  While the last five years have seen considerable progress in understanding
the internal representations of deep learning models, many questions remain.
This is especially true when trying to understand the impact of model design
choices, such as model architecture or training algorithm, on hidden
representation geometry and dynamics. In this work we present a new approach to
studying such representations inspired by the idea of a frame on the tangent
bundle of a manifold. Our construction, which we call a neural frame, is formed
by assembling a set of vectors representing specific types of perturbations of
a data point, for example infinitesimal augmentations, noise perturbations, or
perturbations produced by a generative model, and studying how these change as
they pass through a network. Using neural frames, we make observations about
the way that models process, layer-by-layer, specific modes of variation within
a small neighborhood of a datapoint. Our results provide new perspectives on a
number of phenomena, such as the manner in which training with augmentation
produces model invariance or the proposed trade-off between adversarial
training and model generalization.


  This paper considers the topological degree of $G$-shifts of finite type for
the case where $G$ is a nonabelian monoid. Whenever the Cayley graph of $G$ has
a finite representation and the relationships among the generators of $G$ are
determined by a matrix $A$, the coefficients of the characteristic polynomial
of $A$ are revealed as the number of children of the graph. After introducing
an algorithm for the computation of the degree, the degree spectrum, which is
finite, relates to a collection of matrices in which the sum of each row of
every matrix is bounded by the number of children of the graph. Furthermore,
the algorithm extends to $G$ of finite free-followers.


  We study a large class of 1D reaction diffusion models with quenched disorder
using a real space renormalization group method (RSRG) which yields exact
results at large time. Particles (e.g. of several species) undergo diffusion
with random local bias (Sinai model) and react upon meeting. We obtain the
large time decay of the density of each specie, their associated universal
amplitudes, and the spatial distribution of particles. We also derive the
spectrum of exponents which characterize the convergence towards the asymptotic
states. For reactions with several asymptotic states, we analyze the dynamical
phase diagram and obtain the critical exponents at the transitions. We also
study persistence properties for single particles and for patterns. We compute
the decay exponents for the probability of no crossing of a given point by,
respectively, the single particle trajectories ($\theta$) or the thermally
averaged packets ($\bar{\theta}$). The generalized persistence exponents
associated to n crossings are also obtained. Specifying to the process $A+A \to
\emptyset$ or A with probabilities $(r,1-r)$, we compute exactly the exponents
$\delta(r)$ and $\psi(r)$ characterizing the survival up to time t of a domain
without any merging or with mergings respectively, and $\delta_A(r)$ and
$\psi_A(r)$ characterizing the survival up to time t of a particle A without
any coalescence or with coalescences respectively. $\bar{\theta}, \psi, \delta$
obey hypergeometric equations and are numerically surprisingly close to pure
system exponents (though associated to a completely different diffusion
length). Additional disorder in the reaction rates, as well as some open
questions, are also discussed.


  In this paper a nonlocal generalization of field quantization is suggested.
This quantization principle presupposes the assumption that the commutator
between a field operator an the operator of the canonical conjugated variable
referring to other space-time points does not vanish as it is postulated in the
usual setting of quantum field theory. Based on this presupposition the
corresponding expressions for the field operators, the eigenstates and the path
integral formula are determined. The nonlocal quantization principle also leads
to a generalized propagator. If the dependence of the commutator between
operators on different space-time points on the distance of these points is
assumed to be described by a Gaussian function, one obtains that the propagator
is damped by an exponential. This leads to a disappearance of UV divergences.
The transfer of the nonlocal quantization principle to canonical quantum
gravity is considered as well. In this case the commutator has to be assumed to
depend also on the gravitational field, since the distance between two points
depends on the metric field.


  In this paper we enumerate the necessary and sufficient conditions for the
weak modular product of two simple graphs to be perfect. The weak modular
product differs from the direct product by also encoding non-adjacencies of the
factor graphs in its edges. This work is motivated by the following: a 1978
theorem of Kozen states that two graphs on $n$ vertices are isomorphic if and
only if there is a clique of size $n$ in the weak modular product between the
two graphs. Furthermore, a straightforward corollary of Kozen's theorem and
Lov\'{a}sz's sandwich theorem is if the weak modular product between two graphs
is perfect, then checking if the graphs are isomorphic is polynomial in $n$.
Interesting cases include complete multipartite graphs and disjoint unions of
cliques. All perfect weak modular products have factors that fall into classes
of graphs for which testing isomorphism is already known to be polynomial in
the number of vertices.


  In formal concept analysis, the collection of protoconcepts of any context
forms a double Boolean algebra (dBa) which is fully contextual. Semiconcepts of
a context form a pure dBa. The present article is a study on topological
representation results for dBas, and in particular, those for fully contextual
and pure dBas. In this work, a context on topological spaces (CTS) is
considered, and the focus is on a special kind of CTS in which the relation
defining the context as well as the converse of the relation are continuous
with respect to the topologies. Such CTS are denoted as CTSCR. It is observed
that clopen object oriented protoconcepts of a CTSCR form a fully contextual
dBa, while clopen object oriented semiconcepts form a pure dBa. Every dBa is
shown to be quasi-embeddable into the dBa of clopen object oriented
protoconcepts of a particular CTSCR. The quasi-embedding turns into an
embedding in case of a contextual dBa, and into an isomorphism, when the dBa is
fully contextual. For pure dBas, one obtains an isomorphism with the algebra of
clopen object oriented semiconcepts of the CTSCR. Representation of finite dBas
and Boolean algebras are also addressed in the process. Abstraction of
properties of this CTSCR leads to the definition of Stone contexts. Stone
contexts and CTSCR- homeomorphisms are seen to form a category, denoted Scxt.
Furthermore, correspondences are observed between dBa isomorphisms and
CTSCR-homeomorphisms. This motivates a study of categorical duality of dBas,
constituting the second part of the article. Pure dBas and fully contextual
dBas along with dBa isomorphisms form categories, denoted PDBA and FCDBA
respectively. It is established that PDBA is equivalent to FCDBA, while FCDBA
and PDBA are dually equivalent to Scxt.


  We study the dipolar spin-ice model at fixed density of single excitations,
$\rho$, using a Monte Carlo algorithm where processes of creation and
annihilation of such excitations are banned. In the limit of $\rho$ going to
zero, this model coincides with the usual dipolar spin-ice model at low
temperatures, with the additional advantage that a negligible number of
monopoles allows for equilibration even at the lowest temperatures. Thus, the
transition to the ordered fundamental state found by Melko et al. in 2001 is
reached using simple local spin flip dynamics. As the density is increased, the
monopolar nature of the excitations becomes apparent: the system shows a rich
$\rho$ vs. $T$ phase diagram with "charge" ordering transitions analogous to
that observed for Coulomb charges in lattices. A further layer of complexity is
revealed by the existence of order both within the charges and their associated
vacuum, which can only be described in terms of spins --the true microscopic
degrees of freedom of the system.


  We adapted a recently proposed framework to characterize the optical response
of interacting electrons in solids in order to expedite its computation without
compromise in accuracy at the microscopic level. Our formulation is based on
reliable parameterizations of Hamiltonians and Coulomb interactions, which
allows economy and flexibility in obtaining response functions. It is suited to
computing the optical response to fields of arbitrary temporal shape and
strength, to arbitrary order in the field, and natively accounts for excitonic
effects. We demonstrate the approach by computing the frequency-dependent
susceptibilities of MoS2 and hexagonal BN monolayers up to the third-harmonic.
Grounded on a generic non-equilibrium many-body perturbation theory, this
framework allows extensions to handle generic interaction models or to describe
electronic processes taking place at ultrafast time scales.


  We show how to construct the nonstandard hull of certain infinite-dimensional
Lie algebras in order to generalize a theorem of Pestov on the enlargeability
of Banach-Lie algebras. In the process, we consider a nonstandard smoothness
condition on functions between locally convex spaces to ensure that the induced
function between the nonstandard hulls is smooth. We also discuss some
conditions on a function between locally convex spaces which guarantee that its
nonstandard extension maps finite points to finite points.


  We present a realistic, simple and natural model of strongly-interacting dark
matter based on the neutrino-portal paradigm. The strong interactions at small
velocities are generated by the exchange of dark photons, and produce the
observed core-like DM distribution in galactic centers; this effect could be
spoiled by the formation of DM bound states (also due to dark-photon effects),
which we avoid by requiring the DM candidates to be light, with masses below
O(10 GeV). The mixing of the dark photon with the Z and ordinary photon is
strongly suppressed by introducing a softly-broken discrete symmetry similar to
charge conjugation, which also ensures that the dark photon life-time is short
enough to avoid restrictions derived form big-bang nucleosynthesis and
large-scale structure formation. Other constraints are accommodated without the
need of fine tuning, in particular nucleon scattering occurs only at one loop,
so direct detection cross sections are naturally suppressed. Neutrino masses
are generated through the inverse see saw.


  A significant number of oil paintings produced by Georgia O'Keeffe
(1887-1986) show surface protrusions of varying width, up to several hundreds
of microns. These protrusions are similar to those described in the art
conservation literature as metallic soaps. Since the presence of these
protrusions raises questions about the state of conservation and long-term
prospects for deterioration of these artworks, a 3D-imaging technique,
photometric stereo using ultraviolet illumination, was developed for the
long-term monitoring of the surface-shape of the protrusions and the
surrounding paint. Because the UV fluorescence response of painting materials
is isotropic, errors typically caused by non-Lambertian (anisotropic)
specularities when using visible reflected light can be avoided providing a
more accurate estimation of shape. As an added benefit, fluorescence provides
additional contrast information contributing to materials characterization. The
developed methodology aims to detect, characterize, and quantify the
distribution of micro-protrusions and their development over the surface of
entire artworks. Combined with a set of analytical in-situ techniques, and
computational tools, this approach constitutes a novel methodology to
investigate the selective distribution of protrusions in correlation with the
composition of painting materials at the macro-scale. While focused on
O'Keeffe's paintings as a case study, we expect the proposed approach to have
broader significance by providing a non-invasive protocol to the conservation
community to probe topological changes for any relatively flat painted surface
of an artwork, and more specifically to monitor the dynamic formation of
protrusions, in relation to paint composition and modifications of
environmental conditions, loans, exhibitions and storage over the long-term.


  In 2010, Tom Halverson and Georgia Benkart introduced the Motzkin algebra, a
generalization of the Temperley-Lieb algebra, whose elements are diagrams that
can be multiplied by stacking one on top of the other. Halverson and Benkart
gave a diagrammatic algorithm for decomposing any Motzkin diagram into diagrams
of three subalgebras: the Right Planar Rook algebra, the Temperley-Lieb
algebra, and the Left Planar Rook algebra. We first explore the Right and Left
Planar Rook monoids, by finding presentations for these monoids by generators
and relations, using a counting argument to prove that our relations suffice.
We then turn to the newly-developed Motzkin monoid, where we describe
Halverson's decomposition algorithm algebraically, find a presentation by
generators and relations, and use a counting argument but with a much more
sophisticated algorithm.


  Estimating quantiles of an outcome conditional on covariates is of
fundamental interest in statistics with broad application in probabilistic
prediction and forecasting. We propose an ensemble method for conditional
quantile estimation, Quantile Super Learning, that combines predictions from
multiple candidate algorithms based on their empirical performance measured
with respect to a cross-validated empirical risk of the quantile loss function.
We present theoretical guarantees for both iid and online data scenarios. The
performance of our approach for quantile estimation and in forming prediction
intervals is tested in simulation studies. Two case studies related to solar
energy are used to illustrate Quantile Super Learning: in an iid setting, we
predict the physical properties of perovskite materials for photovoltaic cells,
and in an online setting we forecast ground solar irradiance based on output
from dynamic weather ensemble models.


  Artificial Intelligence (AI) applications based on Deep Neural Networks (DNN)
or Deep Learning (DL) have become popular due to their success in solving
problems likeimage analysis and speech recognition. Training a DNN is
computationally intensive and High Performance Computing(HPC) has been a key
driver in AI growth. Virtualisation and container technology have led to the
convergence of cloud and HPC infrastructure. These infrastructures with diverse
hardware increase the complexity of deploying and optimising AI training
workloads. AI training deployments in HPC or cloud can be optimised with
target-specific libraries, graph compilers, andby improving data movement or
IO. Graph compilers aim to optimise the execution of a DNN graph by generating
an optimised code for a target hardware/backend. As part of SODALITE (a Horizon
2020 project), MODAK tool is developed to optimise application deployment in
software defined infrastructures. Using input from the data scientist and
performance modelling, MODAK maps optimal application parameters to a target
infrastructure and builds an optimised container. In this paper, we introduce
MODAK and review container technologies and graph compilers for AI. We
illustrate optimisation of AI training deployments using graph compilers and
Singularity containers. Evaluation using MNIST-CNN and ResNet50 training
workloads shows that custom built optimised containers outperform the official
images from DockerHub. We also found that the performance of graph compilers
depends on the target hardware and the complexity of the neural network.


  By extending the dipole Pomeron (DP) model, successful in describing elastic
nucleon-nucleon scattering, to proton single diffractive dissociation (SD), we
predict a dip-bump structure in the squared four-momentum transfer ($t$)
distribution of proton's SD. Structures in the $t$ distribution of single
diffractive dissociation are predicted around $t=-4$ GeV$^2$ at LHC energies in
the range of 3 GeV$^2$ $\lesssim|t|\lesssim$ 7 GeV$^2$. Apart from the
dependence on $s$ (total energy squared) and $t$ (squared momentum transfer),
we predict also a dependence on missing masses. We include the minimum set of
Regge trajectories, namely the Pomeron and the Odderon, indispensable at the
LHC. Further generalization, e.g., by the inclusion of non-leading Regge
trajectories, is straightforward. The present model contains two types of Regge
trajectories: those connected with $t$-channel exchanges (the Pomeron, the
Odderon, and non-leading (secondary) reggeons) appearing at small and moderate
$-t$, where they are real and nearly linear, as well as direct-channel
trajectories $\alpha(M^2)$ related to missing masses. In this paper, we
concentrate on structures in $t$ neglecting (for the time being) resonances in
$M^2$.


  A quantum scar - an enhancement of a quantum probability density in the
vicinity of a classical periodic orbit - is a fundamental phenomenon connecting
quantum and classical mechanics. Here we demonstrate that some of the
eigenstates of the perturbed two-dimensional anisotropic (elliptic) harmonic
oscillator are strongly scarred by the Lissajous orbits of the unperturbed
classical counterpart. In particular, we show that the occurrence and geometry
of these quantum Lissajous scars are connected to the anisotropy of the
harmonic confinement, but unlike the classical Lissajous orbits the scars
survive under a small perturbation of the potential. This Lissajous scarring is
caused by the combined effect of the quantum (near) degeneracies in the
unperturbed system and the localized character of the perturbation.
Furthermore, we discuss experimental schemes to observe this
perturbation-induced scarring.


  The spin Hall angle in Pt is evaluated in Pt/NiFe bilayers by spin torque
ferromagnetic resonance (ST-FMR) measurements, and is found to increase with
increasing the NiFe thickness. To extract the intrinsic spin Hall angle in Pt
by estimating the total spin current injected into NiFe from Pt, the NiFe
thickness dependent measurements are performed and the spin diffusion in the
NiFe layer is taken into account. The intrinsic spin Hall angle of Pt is
determined to be 0.068 at room temperature, and is found to be almost constant
in the temperature range 13 - 300 K.


  We propose an extension to the Pauli stabiliser formalism that includes
fractional $2\pi/N$ rotations around the $Z$ axis for some integer $N$. The
resulting generalised stabiliser formalism - denoted the XP stabiliser
formalism - allows for a wider range of states and codespaces to be
represented. We describe the states which arise in the formalism, and
demonstrate an equivalence between XP stabiliser states and 'weighted
hypergraph states' - a generalisation of both hypergraph and weighted graph
states. Given an arbitrary set of XP operators, we present algorithms for
determining the codespace and logical operators for an XP code. Finally, we
consider whether measurements of XP operators on XP codes can be classically
simulated.


  Brilliance of novel gamma-ray Crystal-based Light Sources (CLS) that can be
constructed through exposure of oriented crystals to beams of
ultra-relativistic charged particles is calculated basing on the atomistic
scale numerical modeling of the channeling process. In an exemplary case study,
the brilliance of radiation emitted in a diamond-based Crystalline Undulator LS
by a 10 GeV positron beam available at present at the SLAC facility is
computed. Intesity of CU radiation in the photon energy range 10^0 - 10^1 MeV,
which is inaccessible to conventional synchrotrons, undulators and XFELs,
greatly exceeds that of laser-Compton scattering LSs and can be higher than
predicted in the Gamma Factory proposal to CERN. Construction of novel CLSs is
a challenging task which constitutes a highly interdisciplinary field
entangling a broad range of correlated activities. CLSs provide a low-cost
altenative to conventional LSs and have enomorous number of applications.


  We present an a posteriori error analysis for one-dimensional random
hyperbolic systems of conservation laws. For the discretization of the random
space we consider the Non-Intrusive Spectral Projection method, the
spatio-temporal discretization uses the Runge--Kutta Discontinuous Galerkin
method. We derive an a posteriori error estimator using smooth reconstructions
of the numerical solution, which combined with the relative entropy stability
framework yields computable error bounds for the space-stochastic
discretization error. Moreover, we show that the estimator admits a splitting
into a stochastic and deterministic part.


  The search for extraterrestrial intelligence at radio frequencies has largely
been focused on continuous-wave narrowband signals. We demonstrate that
broadband pulsed beacons are energetically efficient compared to narrowband
beacons over longer operational timescales. Here, we report the first extensive
survey searching for such broadband pulsed beacons towards 1883 stars as a part
of the Breakthrough Listen's search for advanced intelligent life. We conducted
233 hours of deep observations across 4 to 8 GHz using the Robert C. Byrd Green
Bank Telescope and searched for three different classes of signals with
artificial (or negative) dispersion. We report a detailed search -- leveraging
a convolutional neural network classifier on high-performance GPUs -- deployed
for the very first time in a large-scale search for signals from
extraterrestrial intelligence. Due to the absence of any signal-of-interest
from our survey, we place a constraint on the existence of broadband pulsed
beacons in our solar neighborhood: $\lesssim$1 in 1000 stars have transmitter
power-densities $\gtrsim$10$^5$ W/Hz repeating $\leq$500 seconds at these
frequencies.


  Ultrasound computed tomography (USCT), as an emerging technology, can provide
multiple quantitative parametric images of human tissue, such as sound speed
and attenuation images, distinguishing it from conventional B-mode (reflection)
ultrasound imaging. Full waveform inversion (FWI) is acknowledged as a
technique with the greatest potential for reconstructing high-resolution sound
speed images in USCT. However, traditional FWI for sound speed image
reconstruction suffers from high sensitivity to the initial model caused by its
strong non-convex nonlinearity, resulting in poor performance when ultrasound
signals are at high frequencies. This limitation significantly restricts the
application of FWI in the USCT imaging field. In this paper, we propose an
untrained neural network (UNN) that can be integrated into the traditional
iteration-based FWI framework as an implicit regularization prior. This
integration allows for seamless deployment as a plug-and-play module within
existing FWI algorithms or their variants. Notably, the proposed UNN method can
be trained in an unsupervised fashion, a vital aspect in medical imaging where
ground truth data is often unavailable. Evaluations of the numerical simulation
and phantom experiment of the breast demonstrate that the proposed UNN improves
the robustness of image reconstruction, reduces image artifacts, and achieves
great image contrast. To the best of our knowledge, this study represents the
first attempt to propose an implicit UNN for FWI in reconstructing sound speed
images for USCT.


  We discuss a classically conformal radiative neutrino model with gauged B$-$L
symmetry, in which the B$-$L symmetry breaking can occur through the
Coleman-Weinberg mechanism. As a result, Majorana mass term is generated and EW
symmetry breaking also occurs. We show some allowed parameters to satisfy
several theoretical and experimental constraints. Theoretical constraints are
inert conditions and Coleman-Weinberg condition. Experimental bounds are lepton
flavor violation(especially \mu -> e \gamma), the current bound on the $Z'$
mass at LHC, in additions to the neutrino oscillations.


  A tutorial introduction is given to general Hopf algebras and to general
compact quantum groups. In the definition and further treatment of compact
quantum groups C*-algebras are avoided. Contact with Woronowicz's compact
matrix quantum groups is made at a later stage.


  We report a detailed magnetic and structural analysis of highly reduced Co
doped rutile TiO2 films displaying an anomalous Hall effect (AHE). The
temperature and field dependence of magnetization, and transmission electron
microscopy clearly establish the presence of nano-sized superparamagnetic
cobalt clusters of 8-10 nm size in the films at the interface. The
co-occurrence of superparamagnetism and AHE raises questions regarding the use
of the AHE as a test of the intrinsic nature of ferromagnetism in diluted
magnetic semiconductors.


  A review is given of phase properties in molecular wave functions, composed
of a number of (and, at least, two) electronic states that become degenerate at
some nearby values of the nuclear configuration. Apart from discussing phases
and interference in classical (non-quantal) systems, including light-waves, the
review looks at the constructability of complex wave functions from observable
quantities ("the phase problem"), at the controversy regarding quantum
mechanical phase-operators, at the modes of observability of phase and at the
role of phases in some non-demolition measurements. Advances in experimental
and (especially) theoretical aspects of Aharonov-Bohm and topological (Berry)
phases are described, including those involving two-electron and relativistic
systems. Several works in the phase control and revivals of molecular
wave-packets are cited as developments and applications of complex-function
theory. Further topics that this review touches on are: coherent states,
semiclassical approximations and the Maslov index. The interrelation between
time and the complex state is noted in the contexts of time delays in
scattering, of time-reversal invariance and of the existence of a molecular
time-arrow.


  The complexity of a block of a symmetric algebra can be measured by the
notion of defect, a numerical datum associated with each of the simple modules
contained in the block. Geck showed that the defect is a block invariant for
Iwahori-Hecke algebras of finite Coxeter groups in the equal parameter case,
and speculated that a similar result should hold in the unequal parameter case.
We prove that the defect is a block invariant for all cyclotomic Hecke algebras
associated with the complex reflection groups of the infinite series
$G(l,p,n)$, which include the Weyl groups of type $B_n$ in the unequal
parameter case. In particular, for the groups $G(l,1,n)$, we show that the
defect corresponds to the notion of weight in the sense of Fayers. We thus also
obtain a new way of computing the weight, which uses a generalisation of the
notion of hook lengths. We further show computationally that the defect is a
block invariant for all cyclotomic Hecke algebras of exceptional type for which
the blocks are known, and we conjecture that the result should hold for all
complex reflection groups. Finally, we obtain that the defect is also a block
invariant for cyclotomic Yokonuma-Hecke algebras.


  We present a survey of ergodic theorems for actions of algebraic and
arithmetic groups recently established by the authors, as well as some of their
applications. Our approach is based on spectral methods employing the unitary
representation theory of the groups involved. This allows the derivation of
ergodic theorems with a rate of convergence, an important phenomenon which does
not arise in classical ergodic theory. Quantitative ergodic theorems give rise
to new and previously inaccessible applications, and we demonstrate the
remarkable diversity of such applications by presenting several
number-theoretic results. These include, in particular, general uniform error
estimates in lattice points counting problems, explicit estimates in sifting
problems for almost-prime points on symmetric varieties, bounds for exponents
of intrinsic Diophantine approximation, and results on fast distribution of
dense orbits on homogeneous spaces.


  An energy-efficient voltage controlled domain wall device for implementing an
artificial neuron and synapse is analyzed using micromagnetic modeling in the
presence of room temperature thermal noise. By controlling the domain wall
motion utilizing spin transfer or spin orbit torques in association with
voltage generated strain control of perpendicular magnetic anisotropy in the
presence of Dzyaloshinskii-Moriya interaction, different positions of the
domain wall are realized in the free layer of a magnetic tunnel junction to
program different synaptic weights. The feasibility of scaling of such devices
is assessed in the presence of thermal perturbations that compromise
controllability. Additionally, an artificial neuron can be realized by
combining this DW device with a CMOS buffer. This provides a possible pathway
to realize energy efficient voltage controlled nanomagnetic deep neural
networks that can learn in real time.


  The games research community has developed substantial knowledge on designing
engaging experiences that draw players in. Surprisingly, less is known about
player \textit{dis}engagement, with existing work predominantly addressing
disengagement from the perspective of problematic play, and research exploring
player disengagement from a constructive designer perspective is lacking. In
this paper, we address this gap and argue that disengagement from games should
be constructively designed, allowing players to exit play sessions in a
self-determined way. Following a two-phase research approach that combines an
interview study (n=16) with a follow-up online survey (n=111), we
systematically analyze player perspectives on exiting play sessions. Our work
expands the existing notion of disengagement through a characterization of exit
experiences, a lens on disengagement as a process, and points for reflection
for the design of games that seek to address player disengagement in a
constructive way.


  We introduce a notion of pattern occurrence that generalizes both classical
permutation patterns as well as poset containment. Many questions about pattern
statistics and avoidance generalize naturally to this setting, and we focus on
functional complexity problems -- particularly those that arise by constraining
the order dimensions of the pattern and text posets. We show that counting the
number of induced, injective occurrences among dimension 2 posets is #P-hard;
enumerating the linear extensions that occur in realizers of dimension 2 posets
can be done in polynomial time, while for unconstrained dimension it is
GI-complete; counting not necessarily induced, injective occurrences among
dimension 2 posets is #P-hard; counting injective or not necessarily injective
occurrences of an arbitrary pattern in a dimension 1 text is #P-hard, although
it is in FP if the pattern poset is constrained to have bounded intrinsic
width; and counting injective occurrences of a dimension 1 pattern in an
arbitrary text is #P-hard, while it is in FP for bounded dimension texts. This
framework easily leads to a number of open questions, chief among which are (1)
is it #P-hard to count the number of occurrences of a dimension 2 pattern in a
dimension 1 text, and (2) is it #P-hard to count the number of texts which
avoid a given pattern?


  This paper presents a new approach to generate samples from conditional
belief functions for a restricted but non trivial subset of conditional belief
functions. It assumes the factorization (decomposition) of a belief function
along a bayesian network structure. It applies general conditional belief
functions.


  The Z method is a popular atomistic simulation method for determining the
melting temperature where a sequence of molecular dynamics runs are carried out
to target the lowest system energy where the solid always melts. Homogeneous
melting at the limit of critical superheating, Th, is accompanied by a drop in
temperature as kinetic energy is converted to potential energy and the
equilibrium melting temperature, Tm, can be calculated directly from the liquid
state. Implementation of the Z method interfaced with modern ab initio
electronic structure packages use Hellmann-Feynman forces to propagate the ions
in the microcanonical(NVE) ensemble where the Mermin free energy plus the ionic
kinetic energy is conserved. The electronic temperature, Tel, is kept fixed
along the trajectory which may introduce some spurious ion-electron
interactions in MD runs with large temperature changes such as often seen in
homogeneous melting and freezing processes in the NVE ensemble. We estimate
systematic errors in the calculated melting temperature to choice of Tel for
two main mantle components, SiO2 and CaSiO3 at high pressure. Comparison of the
calculated melting temperature from runs where the Tel=Th and Tel=Tm
representing reasonable upper and lower boundaries respectively to choice of
Tel shows that the difference in melting temperature is 200-300 K for our two
test systems. The melting temperature decreases with increasing Tel due to the
increasing entropic stabilisation of the liquid and the systems melts typically
about 3 times faster in MD runs with Tel = Th compared to runs where Tel = Tm.
A careful choice of electron temperature in BOMD simulations where the ions are
propagated using Hellmann-Feynamn forces with the Mermin free energy + the
ionic kinetic energy being conserved is therefore essential for the critical
evaluation of the Z method and in particular at very high temperatures.


  Performing homodyne detection at one port of squeezed-state light
interferometer and then binarzing measurement data are important to achieve
super-resolving and super-sensitive phase measurements. Here we propose a new
data-processing technique by dividing the measurement quadrature into three
bins (equivalent to a multi-outcome measurement), which leads to a higher
improvement in the phase resolution and the phase sensitivity under realistic
experimental condition. Furthermore, we develop a new phase-estimation protocol
based on a combination of the inversion estimators of each outcome and show
that the estimator can saturate the Cramer-Rao lower bound, similar to
asymptotically unbiased maximum likelihood estimator.


  I make some critical comments on the article "An experimental test of
non-local realism", by S. Groblacher, T. Paterek, R. Kaltenbaek, C. Brukner, M.
Zukowski, M. Aspelmeyer, & A. Zeilinger, Nature 446, p. 871-875 (2007). This
article makes incorrect claims concerning Bell's theorem. Moreover, I point to
the fact that a hypothesis referred to as "realism" in the article (Groblacher
et al.) is not used in the deduction of Leggett's inequality.


  Educational Institutions have an essential role in promoting the teaching and
learning process, within universities, colleges, and communities. Due to the
recent coronavirus COVID 19 pandemic, many educational institutions adopted
hybrid learning (HL), which is a combination of classic and online learning. It
integrates the advantages of both, and it is a fundamental factor to ensure
continued learning. Technological innovations such as HL are changing the
teaching process, and how students, lecturers, and administrators interact.
Based on this, the Community College of Qatar (CCQ) focused on researching the
structures and elements related to the adoption of HL. Thus, the goal of this
research paper is to reveal the impact of HL on the learning process in CCQ,
and the effective instructional technology (INST) tools required for a
successful HL program. Our research questions for the survey were designed to
measure the opinions of the students, instructors, and administrators about the
HL program. It is observed from the results that the majority of students,
instructors, and administrators showed a positive attitude toward HL, but some
had negative views and experienced challenges. The results were analyzed and
discussed to better utilize HL to meet the growing demands of the community.


  For internal gauge forces, the result of locally gauging, i.e., of performing
the substitution $\partial \rightarrow D$, is physically the same whether
performed on the action or on the corresponding Euler-Lagrange equations of
motion. Rather unsettling, though, such commutativity fails for the standard
way of coupling a Dirac fermion to the gravitational field in the setting of a
local Lorentz gauge theory of general relativity in the vierbein formalism, the
equivalence principle thus seemingly being here violated. This paper will
present a formalism in which commutativity holds for the gravitational force as
well, the action for the gravitational field itself being still the
Einstein-Hilbert one. Notably, in this formalism, the spinor field will carry a
world/coordinate index, rather than a Lorentz spinor index as it does
standardly. More generally, no Lorentz indices will figure, neither vector
indices nor spinor indices, which from a parsimonious point of view seems quite
satisfactory.


  Applying gyrokinetic simulations for theoretical turbulence and transport
studies to the plasma edge and scrape-off layer (SOL) presents significant
challenges. To in particular account for steep density and temperature
gradients in the SOL, the "full-f" code PICLS was developed. PICLS is a
gyrokinetic particle-in-cell (PIC) code and is based on an electrostatic model
with a linearized field equation and uses kinetic electrons. In previously
published results we were applying PICLS to the well-studied 1D parallel
transport problem during an edge-localized mode (ELM) in the SOL without
collisions. As an extension to this collision-less case and in preparation for
3D simulations, in this work a collisional model will be introduced. The
implemented Lenard-Bernstein collision operator and its Langevin discretization
will be shown. Conservation properties of the collision operator as well as a
comparison of the collisional and non-collisional case will be discussed.


  It is pointed out that a collider experiment involves a local contribution to
the energy-momentum tensor, a circumstance which not a common feature of the
current state of the Universe at large characterized by the cosmological
constant $\Lambda_0$. This contribution may be viewed as a change in the
structure of space-time from its large scale form governed by $\Lambda_0$ to
one governed by a $\Lambda$ peculiar to the scale of the experiment. Possible
consequences of this effect are explored by exploiting the asymptotic symmetry
of space-time for non-vanishing $\Lambda$ and its relation to vacuum energy.


  For three constrained Brownian motions, the excursion, the meander, and the
reflected bridge, the densities of the maximum and of the time to reach it were
expressed as double series by Majumdar, Randon-Furling, Kearney, and Yor
(2008). Some of these series were regularized by Abel summation. Similar
results for Bessel processes were obtained by Schehr and Le Doussal (2010)
using the real space renormalization group method. Here this work is reviewed,
and extended from the point of view of one-dimensional diffusion theory to some
other diffusion processes including skew Brownian bridges and generalized
Bessel meanders. We discuss the limits of the application of this method for
other diffusion processes.


  We present an estimate of the partial width of X(3872) into p p-bar under the
assumption that it is a weakly-bound hadronic molecule whose constituents are a
superposition of the charm mesons D^{*0} D-bar^0 and D^{0} D-bar^{*0}. The p
p-bar partial width of X is therefore related to the cross section for p p-bar
to D^{*0} D-bar^0 near the threshold. That cross section at an energy well
above the threshold is estimated by scaling the measured cross section for p
p-bar to K^{*-} K^+. It is extrapolated to the D^{*0} D-bar^0 threshold by
taking into account the threshold resonance in the 1^{++} channel. The
resulting prediction for the p p-bar partial width of X(3872) is proportional
to the square root of its binding energy. For the current central value of the
binding energy, the estimated partial width into p p-bar is comparable to that
of the P-wave charmonium state chi_{c1}.


  It is well known that the sixth Painlev\'e equation $\PVI$ admits a group of
B\"acklund transformations which is isomorphic to the affine Weyl group of type
$\mathrm{D}_4^{(1)}$. Although various aspects of this unexpectedly large
symmetry have been discussed by many authors, there still remains a basic
problem yet to be considered, that is, the problem of characterizing the
B\"acklund transformations in terms of Riemann-Hilbert correspondence. In this
direction, we show that the B\"acklund transformations are just the pull-back
of very simple transformations on the moduli of monodromy representations by
the Riemann-Hilbert correspondence. This result gives a natural and clear
picture of the B\"acklund transformations. Key words: B\"acklund
transformation, the sixth Painlev\'e equation, Riemann-Hilbert correspondence,
isomonodromic deformation, affine Weyl group of type $\mathrm{D}_4^{(1)}$.


  Multi-label zero-shot learning strives to classify images into multiple
unseen categories for which no data is available during training. The test
samples can additionally contain seen categories in the generalized variant.
Existing approaches rely on learning either shared or label-specific attention
from the seen classes. Nevertheless, computing reliable attention maps for
unseen classes during inference in a multi-label setting is still a challenge.
In contrast, state-of-the-art single-label generative adversarial network (GAN)
based approaches learn to directly synthesize the class-specific visual
features from the corresponding class attribute embeddings. However,
synthesizing multi-label features from GANs is still unexplored in the context
of zero-shot setting. In this work, we introduce different fusion approaches at
the attribute-level, feature-level and cross-level (across attribute and
feature-levels) for synthesizing multi-label features from their corresponding
multi-label class embedding. To the best of our knowledge, our work is the
first to tackle the problem of multi-label feature synthesis in the
(generalized) zero-shot setting. Comprehensive experiments are performed on
three zero-shot image classification benchmarks: NUS-WIDE, Open Images and MS
COCO. Our cross-level fusion-based generative approach outperforms the
state-of-the-art on all three datasets. Furthermore, we show the generalization
capabilities of our fusion approach in the zero-shot detection task on MS COCO,
achieving favorable performance against existing methods. The source code is
available at https://github.com/akshitac8/Generative_MLZSL.


  Based on the framework of the quantum-inspired evolutionary algorithm, a
cuckoo quantum evolutionary algorithm (CQEA) is proposed for solving the graph
coloring problem (GCP). To reduce iterations for the search of the chromatic
number, the initial quantum population is generated by random initialization
assisted by inheritance. Moreover, improvement of global exploration is
achieved by incorporating the cuckoo search strategy, and a local search
operation, as well as a perturbance strategy, is developed to enhance its
performance on GCPs. Numerical results demonstrate that CQEA operates with
strong exploration and exploitation abilities, and is competitive to the
compared state-of-the-art heuristic algorithms.


  We propose a novel method to test the consistency of the multipole moments of
compact binary systems with the predictions of General Relativity (GR). The
multipole moments of a compact binary system, known in terms of symmetric and
trace-free tensors, are used to calculate the gravitational waveforms from
compact binaries within the post-Newtonian (PN) formalism. For nonspinning
compact binaries, we derive the gravitational wave phasing formula, in the
frequency domain, parametrizing each PN order term in terms of the multipole
moments which contribute to that order. Using GW observations, this
{\it{parametrized multipolar phasing}} would allow us to derive the bounds on
possible departures from the multipole structure of GR and hence constrain the
parameter space of alternative theories of gravity. We compute the projected
accuracies with which the second generation ground-based detectors, such as
Advanced Laser Interferometer Gravitational-wave Observatory (LIGO), the third
generation detectors such as Einstein Telescope and Cosmic Explorer, as well as
space-based detector Laser Interferometer Space Antenna (LISA) will be able to
measure these multipole parameters. We find that while Advanced LIGO can
measure the first two or three multipole coefficients with good accuracy,
Cosmic Explorer and Einstein Telescope may be able to measure the first four
multipole coefficients which enter the phasing formula. Intermediate mass ratio
inspirals, with mass ratio of several tens, in the frequency band of planned
space-based LISA mission should be able to measure all the seven multipole
coefficients which appear in the 3.5PN phasing formula. Our finding highlights
the importance of this class of sources for probing the strong-field gravity
regime. The proposed test will facilitate the first probe of the multipolar
structure of Einstein's general relativity.


  The peaks of 30 optical afterglows and 14 X-ray light-curves display a good
anticorrelation of the peak flux with the peak epoch: F_p ~ t_p^{-2.0} in the
optical, F_p ~ t_p^{-1.6} in the X-ray, the distributions of the peak epochs
being consistent with each other. We investigate the ability of two
forward-shock models for afterglow light-curve peaks -- an observer location
outside the initial jet aperture and the onset of the forward-shock
deceleration -- to account for those peak correlations. For both models, the
slope of the F_p - t_p relation depends only on the slope of the afterglow
spectrum. We find that only a conical jet seen off-aperture and interacting
with a wind-like medium can account for both the X-ray peak relation, given the
average X-ray spectral slope beta_x = 1.0, and for the larger slope of the
optical peak relation. However, any conclusion about the origin of the peak
flux - peak epoch correlation is, at best, tentative, because the current
sample of X-ray peaks is too small to allow a reliable measurement of the F_p -
t_p relation slope and because more than one mechanism and/or one afterglow
parameter may be driving that correlation.


  This paper examines the traffic flows on a two-dimensional stochastic lattice
model that comprises a junction of two traveling routes: the domestic route and
the international route each of which has parking sites. In our model, the
system distributes the arrived particles to either of the two routes and
selects one of the parking sites in the route for each particle, which stops at
the parking site once during its travel. Because each particle has antennas in
the back and front directions to detect other approaching particles, the effect
of the volume exclusion of each particle extends in the moving direction. The
system displays interesting behavior; remarkably, the dependence of the
throughput on the distribution ratio of particles to the domestic route reduces
after reaching the maximum parking capacity of the domestic route. Our
simulations and analysis with the queueing model describe this phenomenon and
suggest the following fact: As the distribution ratio of particles to the
international route decreases, the throughput of the international route
reduces, and simultaneously, that of the domestic route saturates. The
simultaneous effect of the decrease and saturation causes a reduction in the
throughput of the entire system.


  We analyze the possibility of superluminal neutrino propagation delta v = (v
- c)/c > 0 as indicated by OPERA data, in view of previous phenomenological
constraints from supernova SN1987a and gravitational Cerenkov radiation. We
argue that the SN1987a data rule out delta v ~ (E_\nu/M_N)^N for N \le 2 and
exclude, in particular, a Lorentz-invariant interpretation in terms of a
'conventional' tachyonic neutrino. We present two toy Lorentz-violating
theoretical models, one a Lifshitz-type fermion model with superluminality
depending quadratically on energy, and the other a Lorentz-violating
modification of a massless Abelian gauge theory with axial-vector couplings to
fermions. In the presence of an appropriate background field, fermions may
propagate superluminally or subluminally, depending inversely on energy, and on
direction. Reconciling OPERA with SN1987a would require this background field
to depend on location.


  In this article, we characterize the Beurling and Model subspaces of the
Hardy-Hilbert space $H^2(\mathbb{D})$ invariant under the composition operator
$C_{\phi_a}f=f\circ\phi_a$, where $\phi_a(z) = az + 1 - a$ for $a \in (0,1)$ is
an affine self-map of the open unit disk $\mathbb{D}$. These operators have
universal translates (in the sense of Rota) and have attracted attention
recently due to their connection with the Invariant Subspace Problem (ISP) and
the classical Ces\`aro operator.


  With his Clifford algebra of differential forms, Kaehler's algebra addresses
the overlooked manifestation of symmetry in the solutions of exterior systems.
In this algebra, solutions with a given symmetry are members of left ideals
generated by corresponding idempotents. These combine with phase factors to
take care of all the dependence of the solution on $x^{i}$ and $% dx^{i}$ for
each $\partial /\partial x^{i}=0$ symmetry. The maximum number of idempotents
(each for a one-parameter group) that can, therefore, go into a solution is the
dimensionality $n$ of the space. This number is further limited by
non-commutativity of idempotents.
  We consider the tensor product of Kaehler algebra with tangent Clifford
algebra, i.e. of valuedness. It has a commutative subalgebra of "mirror
elements", i.e. where the valuedness is dual to the differential form (like in
$dx\mathbf{i}$, but not in $dx\mathbf{j}$). It removes the aforementioned
limitation in the number of idempotent factors that can simultaneously go into
a solution.
  In the Kaehler calculus, which is based on the Kaehler algebra, total angular
momentum is linear in its components, even it is not valued in tangent algebra.
In particular, it is not a vector operator.
  We carry Kaehler's treatment over to the aforementioned commutative algebra.
Non-commutativity thus ceases to be a limiting factor in the formation of
products of idempotents representing each a one-parameter symmetry. The
interplay of factors that emerges in the expanded set of such products is
inimical to operator theory. For practical reasons, we stop at products
involving three one-parameter idempotents ---henceforth call ternary--- even
though the dimensionality of the all important spacetime manifold allows for
four of them.


  Early-type dwarfs are the most common galaxy in the local universe, yet their
origin and evolution remain a mystery. Various cosmological scenarios predict
that dwarf-like galaxies in dense areas are the first to form and hence should
be the oldest stellar systems in clusters. By using radial velocities of
early-type dwarfs in the Virgo cluster we demonstrate that these galaxies are
not an old cluster population but have signatures of production from the infall
of field galaxies. Evidence of this includes the combined large dispersions and
sub-structure in spatial and kinematic distributions for Virgo early-type
dwarfs, and a velocity dispersion ratio with giant ellipticals expected for
virialized and accreted populations. We also argue that these galaxies cannot
originate from accreted field dwarfs, but must have physically evolved from a
precursor population, of different morphology, that fell into Virgo some time
in the past.


  The renormalization group flows of the coupling constants for the gauged U(N)
vector model, with N_f massless fermions in the defining representation, are
studied in the large N limit, to all orders in the scalar coupling lambda,
leading order in 1/N, and lowest two orders in the gauge coupling g^2. It is
shown that the restrictions of asymptotic freedom, and the reality of the
coupling constants throughout the flows, places important restrictions on
N_f/N. For the case with massless mesons, these conditions are sufficiently
restrictive to imply the existence of an infrared fixed-point (g_*,lambda_*) in
both couplings. Thus, the consistent massless theory is scale invariant, and in
a non-abelian Coulomb phase. The case of massive mesons, and of spontaneously
broken symmetry is also discussed, with similar, but not identical,
conclusions. Speculations related to the possibility that there is a
non-perturbative (in g^2) breakdown of chiral symmetry are presented.


  We study classical, real Klein-Gordon theory on Lorentzian Anti de Sitter
(AdS_{1,d}) spacetimes with spatial dimension d. We give a complete list of
well defined and bounded Klein-Gordon solutions for three types of regions on
AdS: slice (time interval times all of space), rod hypercylinder (all of time
times solid ball in space), and tube hypercylinder (all of time times solid
shell in space). Hypercylinder regions are of natural interest for AdS since
the neighborhood of the AdS-boundary is a tube. For the solution spaces of our
regions we find the actions induced by the AdS isometry group SO(2,d). For all
three regions we find one-to-one correspondences between initial data and
solutions on the regions. For rod and tube regions this initial data can also
be given on the AdS boundary. We calculate symplectic structures associated to
the solution spaces, and show their invariance under the isometry actions. We
compare our results to the corresponding expressions for (3+1)-dimensional
Minkowski spacetime, arising from AdS_{1,3} in the limit of large curvature
radius.


  We investigate the entanglement properties of resonating-valence-bond states
on two and higher dimensional lattices, which play a significant role in our
understanding of various many-body systems. We show that these states are
genuinely multipartite entangled, while there is only a negligible amount of
two-site entanglement. We comment on possible physical implications of our
findings.


  We employ the matching method to analytically investigate the holographic
superconductors with Lifshitz scaling in an external magnetic field. We discuss
systematically the restricted conditions for the matching method and find that
this analytic method is not always powerful to explore the effect of external
magnetic field on the holographic superconductors unless the matching point is
chosen in an appropriate range and the dynamical exponent $z$ satisfies the
relation $z=d-1$ or $z=d-2$. From the analytic treatment, we observe that
Lifshitz scaling can hinder the condensation to be formed, which can be used to
back up the numerical results. Moreover, we study the effect of Lifshitz
scaling on the upper critical magnetic field and reproduce the well-known
relation obtained from Ginzburg-Landau theory.


  In recent years, many publications showed that convolutional neural network
based features can have a superior performance to engineered features. However,
not much effort was taken so far to extract local features efficiently for a
whole image. In this paper, we present an approach to compute patch-based local
feature descriptors efficiently in presence of pooling and striding layers for
whole images at once. Our approach is generic and can be applied to nearly all
existing network architectures. This includes networks for all local feature
extraction tasks like camera calibration, Patchmatching, optical flow
estimation and stereo matching. In addition, our approach can be applied to
other patch-based approaches like sliding window object detection and
recognition. We complete our paper with a speed benchmark of popular CNN based
feature extraction approaches applied on a whole image, with and without our
speedup, and example code (for Torch) that shows how an arbitrary CNN
architecture can be easily converted by our approach.


  A virtual string can be defined as an equivalence class of planar diagrams
under certain kinds of diagrammatic moves. Virtual strings are related to
virtual knots in that a simple operation on a virtual knot diagram produces a
diagram for a virtual string.
  In this paper we consider three operations on a virtual string or virtual
strings which produce another virtual string, namely covering, composition and
cabling. In particular we study virtual strings unchanged by the covering
operation. We also show how the based matrix of a composite virtual string is
related to the based matrices of its components, correcting a result by Turaev.
Finally we investigate what happens under cabling to some invariants defined by
Turaev.


  Compulsory Dirac multipoles in the bilayer perovskite Ca3Ru2O7 are absent in
published analyses of experimental data. In a first step at correcting
knowledge of the magnetic structure, we have analysed existing Bragg
diffraction patterns gathered on samples held well below the N\'eel temperature
at which A-type antiferromagnetic order of axial dipoles spontaneously
develops. Patterns were gathered with neutrons, and linearly polarized x-rays
tuned in energy to a ruthenium atomic resonance. Neutron diffraction data
contains solid evidence of Dirac dipoles (anapoles or toroidal moments). No
such conclusion is reached with existing x-ray diffraction data, which instead
is ambiguous on the question. To address this shortcoming by future
experiments, we calculated additional diffraction patterns. Chiral order of
Dirac multipoles is allowed by magnetic space-group PCna21, and it can be
exposed in Bragg diffraction using circularly polarized x-rays. Likewise, a
similar experiment can expose a chiral order of axial dipoles. A magnetic field
applied parallel to the b-axis creates a ferrimagnetic structure in which bulk
magnetization arises from field-induced nonequivalent Ru sites (magnetic
space-group Pm'c'21).


  Let $k$ be a field of characteristic $p>0$. Denote by $W_r(k)$ the ring of
truntacted Witt vectors of length $r \geq 2$, built out of $k$. In this text,
we consider the following question, depending on a given profinite group $G$.
  $Q(G)$: Does every (continuous) representation $G\longrightarrow GL_d(k)$
lift to a representation $G\longrightarrow GL_d(W_r(k))$?
  We work in the class of cyclotomic pairs (Definition 4.3), first introduced
in [DCF] under the name "smooth profinite groups". Using Grothendieck-Hilbert'
theorem 90, we show that the algebraic fundamental groups of the following
schemes are cyclotomic: spectra of semilocal rings over
$\mathbb{Z}[\frac{1}{p}]$, smooth curves over algebraically closed fields, and
affine schemes over $\mathbb{F}_p$. In particular, absolute Galois groups of
fields fit into this class. We then give a positive partial answer to $Q(G)$,
for a cyclotomic profinite group $G$: the answer is positive, when $d=2$ and
$r=2$. When $d=2$ and $r=\infty$, we show that any $2$-dimensional
representation of $G$ stably lifts to a representation over $W(k)$: see Theorem
6.1. \\When $p=2$ and $k=\mathbb{F}_2$, we prove the same results, up to
dimension $d=4$.
  We then give a concrete application to algebraic geometry: we prove that
local systems of low dimension lift Zariski-locally (Corollary 6.3).


  Selectivity is an important phenomenon in chemical reaction dynamics. This
can be quantified by the branching ratio of the trajectories that visit one or
the other wells to the total number of trajectories in a system with a
potential with two sequential index-1 saddles and two wells (top well and
bottom well). In our case, the branching ratio is 1:1 because of the symmetry
of our potential energy surface. The mechanisms of transport and the behavior
of the trajectories in this kind of systems have been studied recently. In this
paper we study the time evolution after the selectivity as energy varies using
periodic orbit dividing surfaces. We investigate what happens after the first
visit of a trajectory to the region of the top or the bottom well for different
values of energy. We answer the natural question, what is the destiny of these
trajectories?


  In the era of large all-sky surveys, there will be a need for rapid,
automatic classifications of newly discovered transient objects. Our focus here
is the classification of supernovae (SNe). We consider random forest machine
learning algorithm applied to the classification of Type Ia and core-collapse
supernovae (CCSNe) by full light curves. We also quantitatively show the
potential of early-epoch classification using the same machine learning
techniques. The algorithm uses the shape and magnitude of the light curve peak
to determine the classification. This an initial study to essentially determine
SN type with as few data points as possible. New all-sky surveys will discover
new transients at a rapid rate and decisions will have to be made on very
little data which transients to follow-up. Here we present an initial study
where as few as five data points near peak, we an identify whether it is a Type
Ia or CCSNe with better than 80\% accuracy, but with several biases in the data
that will require addressing in future work.


  This paper deals with the finite-time blow-up phenomena of classical
solutions for Vlasov/Navier-Stokes equations under suitable assumptions on the
initial configurations. We show that a solution to the coupled kinetic-fluid
system may be initially smooth, however, it can become singular in a finite
period of time. We provide a simple idea of showing the finite time blow up of
classical solutions to the coupled system which has not been studied so far. We
also obtain analogous results for related systems, such as isentropic
compressible Navier-Stokes equations, two-phase fluid equations consisting of
pressureless Euler equations and Navier-Stokes equations, and thick sprays
model.


  In our companion work \cite{Stojnicl1RegPosasymldp} we revisited random
under-determined linear systems with sparse solutions. The main emphasis was on
the performance analysis of the $\ell_1$ heuristic in the so-called asymptotic
regime, i.e. in the regime where the systems' dimensions are large. Through an
earlier sequence of work
\cite{DonohoPol,DonohoUnsigned,StojnicCSetam09,StojnicUpper10}, it is now well
known that in such a regime the $\ell_1$ exhibits the so-called \emph{phase
transition} (PT) phenomenon. \cite{Stojnicl1RegPosasymldp} then went much
further and established the so-called \emph{large deviations principle} (LDP)
type of behavior that characterizes not only the breaking points of the
$\ell_1$'s success but also the behavior in the entire so-called
\emph{transition zone} around these points. Both of these concepts, the PTs and
the LDPs, are in fact defined so that one can use them to characterize the
asymptotic behavior. In this paper we complement the results of
\cite{Stojnicl1RegPosasymldp} by providing an exact detailed analysis in the
non-asymptotic regime. Of course, not only are the non-asymptotic results
complementing those from \cite{Stojnicl1RegPosasymldp}, they actually are the
ones that ultimately fully characterize the $\ell_1$'s behavior in the most
general sense. We introduce several novel high-dimensional geometry type of
strategies that enable us to eventually determine the $\ell_1$'s behavior.


  Planets intermediate in size between the Earth and Neptune, and orbiting
closer to their host stars than Mercury does the Sun, are the most common type
of planet revealed by exoplanet surveys over the last quarter century. Results
from NASA's Kepler mission have revealed a bimodality in the radius
distribution of these objects, with a relative underabundance of planets
between 1.5 and 2.0 $R_{\oplus}$. This bimodality suggests that sub-Neptunes
are mostly rocky planets that were born with primary atmospheres a few percent
by mass accreted from the protoplanetary nebula. Planets above the radius gap
were able to retain their atmospheres ("gas-rich super-Earths"), while planets
below the radius gap lost their atmospheres and are stripped cores ("true
super-Earths"). The mechanism that drives atmospheric loss for these planets
remains an outstanding question, with photoevaporation and core-powered mass
loss being the prime candidates. As with the mass-loss mechanism, there are two
contenders for the origins of the solids in sub-Neptune planets: the migration
model involves the growth and migration of embryos from beyond the ice line,
while the drift model involves inward-drifting pebbles that coagulate to form
planets close-in. Atmospheric studies have the potential to break degeneracies
in interior structure models and place additional constraints on the origins of
these planets. However, most atmospheric characterization efforts have been
confounded by aerosols. Observations with upcoming facilities are expected to
finally reveal the atmospheric compositions of these worlds, which are arguably
the first fundamentally new type of planetary object identified from the study
of exoplanets.


  We have conducted a deep ($15 \la r \la 23$), 20 night survey for transiting
planets in the intermediate age open cluster M37 (NGC 2099) using the Megacam
wide-field mosaic CCD camera on the 6.5m Multiple Mirror Telescope (MMT). In
this paper we describe the observations and data reduction procedures for the
survey and analyze the stellar content and dynamical state of the cluster. By
combining high resolution spectroscopy with existing $BVI_{C}K_{S}$ and new
$gri$ color magnitude diagrams we determine the fundamental cluster parameters:
$t = 485 \pm 28$ Myr without overshooting ($t = 550 \pm 30 {\rm Myr}$ with
overshooting), $E(B-V) = 0.227 \pm 0.038$, $(m-M)_{V} = 11.57 \pm 0.13$ and
$[M/H] = +0.045 \pm 0.044$ which are in good agreement with, though more
precise than, previous measurements. We determine the mass function down to
$0.3 M_{\odot}$ and use this to estimate the total cluster mass of $3640 \pm
170 M_{\odot}$.


  Let $F$ be a non-Archimedean local field whose residue characteristic is odd.
In this paper we develop a theory of newforms for $U(1,1)(F)$, building on
previous work on $SL_2(F)$. This theory is analogous to the results of
Casselman for $GL_2(F)$ and Jacquet, Piatetski-Shapiro, and Shalika for
$GL_n(F)$. To a representation $\pi$ of $U(1,1)(F)$, we attach an integer
$c(\pi)$ called the conductor of $\pi$, which depends only on the $L$-packet
$\Pi$ containing $\pi$. A newform is a vector in $\pi$ which is essentially
fixed by a congruence subgroup of level $c(\pi)$. We show that our newforms are
always test vectors for some standard Whittaker functionals, and, in doing so,
we give various explicit formulae for newforms.


  In this work, we propose an optimization framework for estimating a sparse
robust one-dimensional subspace. Our objective is to minimize both the
representation error and the penalty, in terms of the l1-norm criterion. Given
that the problem is NP-hard, we introduce a linear relaxation-based approach.
Additionally, we present a novel fitting procedure, utilizing simple ratios and
sorting techniques. The proposed algorithm demonstrates a worst-case time
complexity of $O(n^2 m \log n)$ and, in certain instances, achieves global
optimality for the sparse robust subspace, thereby exhibiting polynomial time
efficiency. Compared to extant methodologies, the proposed algorithm finds the
subspace with the lowest discordance, offering a smoother trade-off between
sparsity and fit. Its architecture affords scalability, evidenced by a 16-fold
improvement in computational speeds for matrices of 2000x2000 over CPU version.
Furthermore, this method is distinguished by several advantages, including its
independence from initialization and deterministic and replicable procedures.
Furthermore, this method is distinguished by several advantages, including its
independence from initialization and deterministic and replicable procedures.
The real-world example demonstrates the effectiveness of algorithm in achieving
meaningful sparsity, underscoring its precise and useful application across
various domains.


  The production at central rapidity of K0s, Lambda, Xi and Omega particles in
Pb-Pb collisions at 158 A GeV/c has been measured by the NA57 experiment over a
centrality range corresponding to the most central 53% of the inelastic Pb-Pb
cross section. In this paper we present the rapidity distribution of each
particle in the central rapidity unit as a function of the event centrality.
The distributions are analyzed based on hydrodynamical models of the
collisions.


  We study the convergence of the log-linear non-Bayesian social learning
update rule, for a group of agents that collectively seek to identify a
parameter that best describes a joint sequence of observations. Contrary to
recent literature, we focus on the case where agents assign decaying weights to
its neighbors, and the network is not connected at every time instant but over
some finite time intervals. We provide a necessary and sufficient condition for
the rate at which agents decrease the weights and still guarantees social
learning.


  We consider conditional McKean-Vlasov stochastic differential equations
(SDEs), such as the ones arising in the large-system limit of mean field games
and particle systems with mean field interactions when common noise is present.
The conditional time-marginals of the solutions to these SDEs satisfy
non-linear stochastic partial differential equations (SPDEs) of the second
order, whereas the laws of the conditional time-marginals follow Fokker-Planck
equations on the space of probability measures. We prove two superposition
principles: The first establishes that any solution of the SPDE can be lifted
to a solution of the conditional McKean-Vlasov SDE, and the second guarantees
that any solution of the Fokker-Planck equation on the space of probability
measures can be lifted to a solution of the SPDE. We use these results to
obtain a mimicking theorem which shows that the conditional time-marginals of
an Ito process can be emulated by those of a solution to a conditional
McKean-Vlasov SDE with Markovian coefficients. This yields, in particular, a
tool for converting open-loop controls into Markovian ones in the context of
controlled McKean-Vlasov dynamics.


  We present a gluing construction which adds, via a localized deformation,
exactly Delaunay ends to generic metrics with constant positive scalar
curvature. This provides time-symmetric initial data sets for the vacuum
Einstein equations with positive cosmological constant with exactly
Kottler-Schwarzschild-de Sitter ends, extending the results in arXiv:0710.3365
[gr-qc].


  In massive multiple-input multiple-output (MIMO) systems, hybrid
analog-digital (AD) beamforming can be used to attain a high directional gain
without requiring a dedicated radio frequency (RF) chain for each antenna
element, which substantially reduces both the hardware costs and power
consumption. While massive MIMO transceiver design typically relies on the
conventional mean-square error (MSE) criterion, directly minimizing the symbol
error rate (SER) can lead to a superior performance. In this paper, we first
mathematically formulate the problem of hybrid transceiver design under the
minimum SER (MSER) optimization criterion and then develop a MSER-based
gradient descent (GD) iterative algorithm to find the related stationary
points. We then propose a deep-unfolding neural network (NN), in which the
iterative GD algorithm is unfolded into a multi-layer structure wherein a set
of trainable parameters are introduced for accelerating the convergence and
enhancing the overall system performance. To implement the training stage, the
relationship between the gradients of adjacent layers is derived based on the
generalized chain rule (GCR). The deep-unfolding NN is developed for both
quadrature phase shift keying (QPSK) and for $M$-ary quadrature amplitude
modulated (QAM) signals and its convergence is investigated theoretically.
Furthermore, we analyze the transfer capability, computational complexity, and
generalization capability of the proposed deep-unfolding NN. Our simulation
results show that the latter significantly outperforms its conventional
counterpart at a reduced complexity.


  A general market model with memory is considered in terms of stochastic
functional differential equations. We aim at representation formulae for the
sensitivity analysis of the dependence of option prices on the memory. This
implies a generalization of the concept of delta.


  In the present paper, we construct and investigate a variant of modified
$(p,q)$-Baskakov operators, which reproduce the test function $x^2$. The order
of approximation of the operators via K-functional and second order, usual
modulus of continuity, weighted approximation properties are disscussed. In the
end some graphical results, comparison with $(p,q)$-Baskakov operators is
explained.


  We use a function field version of the circle method to prove that a positive
proportion of elements in $\mathbb{F}_q[t]$ are representable as a sum of three
cubes of minimal degree from $\mathbb{F}_q[t]$, assuming a suitable form of the
Ratios Conjecture and that the characteristic is greater than 3. The analogue
of this conjecture for quadratic Dirichlet $L$-functions is known for large
fixed $q$, via recent developments in homological stability.


  Astrobiology is the field of science devoted to searching for life elsewhere
in the Universe. It is inherently interdisciplinary, integrating results from
multiple fields of science, and in this respect has strong synergies with 'big
history'. I argue that big history and astrobiology are both acting to widen
human perspectives in intellectually and socially beneficial directions,
especially by enhancing public awareness of cosmic and evolutionary worldviews.
I will further argue that these perspectives have important implications for
the social and political organisation of humanity, including the eventual
political unification of our planet. Astrobiology and big history are also
concerned with the future of humanity, and I will argue that this future will
be culturally and intellectually enriched if it includes the exploration of the
universe around us.


  We present a statistical analysis of the associated, high ionization narrow
absorption line (NAL) systems in a sample of 59 QSOs defined from the HST QSO
Absorption Line Key Project. We have compiled the QSO luminosities at 2500 A, 5
GHz, and 2 keV, spectral indices at 2500 A and 5 GHz, the H-beta emission line
FWHM, the CIV emission line FWHM, and the radio core fraction. We find 17
associated NALs toward 15 QSOs, of which ~10 are statistically expected to be
intrinsic. From a multivariate clustering analysis, the QSOs seem to group
together based primarily on radio luminosity, followed by radio spectral index,
CIV emission line FWHM, and soft X-ray luminosity. Radio-loud QSOs which have
compact radio morphologies, flat radio spectra, and mediocre CIV FWHM (<6000
km/s) do not have detectable associated NALs, down to REW(CIV)=0.35 A. We find
that BALQSOs have an enhanced probability of hosting detectable NAL gas and
that the velocities of associated NALs are peaked around the emission
redshifts, rather than the systemic redshifts of the QSOs. There is only one
strong NAL [REW(CIV)>1.5 A] in our low redshift sample which implies evolution
in the number of strong NALs with redshift. We interpret these results in the
context of an accretion-disk model. In this framework, we suggest that the
observational clues result from differences in orientation and wind properties,
the latter presumably associated with the QSO radio properties. (abridged)


  We report the detection of sixteen binary systems from the Anglo-Australian
Planet Search. Solutions to the radial velocity data indicate that the stars
have companions orbiting with a wide range of masses, eccentricities and
periods. Three of the systems potentially contain brown-dwarf companions while
another two have eccentricities that place them in the extreme upper tail of
the eccentricity distribution for binaries with periods less than 1000 d. For
periods up to 12 years, the distribution of our stellar companion masses is
fairly flat, mirroring that seen in other radial velocity surveys, and
contrasts sharply with the current distribution of candidate planetary masses,
which rises strongly below 10MJ. When looking at a larger sample of binaries
that have FGK star primaries as a function of the primary star metallicity, we
find that the distribution maintains a binary fraction of ~43$\pm$4% between
-1.0 to +0.6 dex in metallicity. This is in stark contrast to the giant
exoplanet distribution. This result is in good agreement with binary formation
models that invoke fragmentation of a collapsing giant molecular cloud,
suggesting this is the dominant formation mechanism for close binaries and not
fragmentation of the primary star's remnant proto-planetary disk.


  We investigate electron transport inside a ring system composed of a quantum
dot (QD) coupled to two Majorana bound states confined at the ends of a
one-dimensional topological superconductor nanowire. By tuning the magnetic
flux threading through the ring, the model system we consider can be switched
into states with or without zero-energy modes when the nanowire is in its
topological phase. We find that the Fano profile in the conductance spectrum
due to the interference between bound and continuum states exhibits markedly
different features for these two different situations, which consequently can
be used to detect the Majorana zero-energy mode. Most interestingly, as a
periodic function of magnetic flux, the conductance shows $2\pi$ periodicity
when the two Majorana bound states are nonoverlapping (as in an infinitely long
nanowire) but displays $4\pi$ periodicity when the overlapping becomes nonzero
(as in a finite length nanowire). We map the model system into a QD--Kitaev
ring in the Majorana fermion representation and affirm these different
characteristics by checking the energy spectrum.


  In this paper we provide a pinching condition for the characterization of the
totally geodesic disk and the rotational annulus among minimal surfaces with
free boundary in geodesic balls of three-dimensional hyperbolic space and
hemisphere. The pinching condition involves the length of the second
fundamental form, the support function of the surface, and a natural potential
function in hyperbolic space and hemisphere.


  We consider the resonant tunneling through a multi-level system. It is
demonstrated that the resonant current displays quantum interference effects
due to a possibility of tunneling through different levels. We show that the
interference effects are strongly modulated by a relative phase of states
carrying the current. This makes it possible to use these effects for measuring
the phase difference between resonant states in quantum dots. We extend our
model for a description of magnetotransport through the Zeeman doublets. It is
shown that, due to spin-flip transitions, the quantum interference effects
generate a distinct peak in the shot-noise power spectrum at the frequency of
Zeeman splitting. This mechanism explains modulation in the tunneling current
at the Larmor frequency observed in scanning tunneling microscope experiments
and can be utilized for a single spin measurement.


  Level set methods are widely used for image segmentation because of their
capability to handle topological changes. In this paper, we propose a novel
parametric level set method called Disjunctive Normal Level Set (DNLS), and
apply it to both two phase (single object) and multiphase (multi-object) image
segmentations. The DNLS is formed by union of polytopes which themselves are
formed by intersections of half-spaces. The proposed level set framework has
the following major advantages compared to other level set methods available in
the literature. First, segmentation using DNLS converges much faster. Second,
the DNLS level set function remains regular throughout its evolution. Third,
the proposed multiphase version of the DNLS is less sensitive to
initialization, and its computational cost and memory requirement remains
almost constant as the number of objects to be simultaneously segmented grows.
The experimental results show the potential of the proposed method.


  Much uncertainty surrounds the origin of super-luminous supernovae (SNe).
Motivated by the discovery of the Type Ic SN2007bi, we study its proposed
association with a pair-instability SN (PISN). We compute stellar-evolution
models for primordial ~200Msun stars, simulating the implosion/explosion due to
the pair-production instability, and use them as inputs for detailed non-LTE
time-dependent radiative-transfer simulations that include non-local energy
deposition and non-thermal processes. We retrieve the basic morphology of PISN
light curves from red-supergiant, blue-supergiant, and Wolf-Rayet (WR) star
progenitors. Although we confirm that a progenitor 100Msun helium core (PISN
model He100) fits well the SN2007bi light curve, the low ratios of its kinetic
energy and 56Ni mass to the ejecta mass, similar to standard core-collapse SNe,
conspire to produce cool photospheres, red spectra subject to strong line
blanketing, and narrow line profiles, all conflicting with SN2007bi
observations. He-core models of increasing 56Ni-to-ejecta mass ratio have bluer
spectra, but still too red to match SN2007bi, even for model He125 -- the
effect of 56Ni heating is offset by the associated increase in blanketing. In
contrast, the delayed injection of energy by a magnetar represents a more
attractive alternative to reproduce the blue, weakly-blanketed, and broad-lined
spectra of super-luminous SNe. The extra heat source is free of blanketing and
is not explicitly tied to the ejecta. Experimenting with a ~9Msun WR-star
progenitor, initially exploded to yield a ~1.6B SN Ib/c ejecta but later
influenced by tunable magnetar-like radiation, we produce a diversity of blue
spectral morphologies reminiscent of SN2007bi, the peculiar Type Ib SN2005bf,
and super-luminous SN2005ap-like events.


  We study the distribution of occupation times for a one-dimensional random
walk restricted to a finite interval by reflecting boundary conditions. At
short times the classical bimodal distribution due to L\'evy is reproduced with
walkers staying mostly either left or right to the initial point. With
increasing time, however, the boundaries suppress large excursions from the
starting point, and the distribution becomes unimodal converging to a
$\delta$-distribution in the long time limit. An approximate spectral analysis
of the underlying Fokker-Planck equation yields results in excellent agreement
with numerical simulations.


  High-density cores (HDCs) of galaxy superclusters that embed rich clusters
and groups of galaxies are the earliest large objects to form in the cosmic
web, and the largest objects that may collapse in the present or future. We
study the dynamical state and possible evolution of the HDCs in the BOSS Great
Wall (BGW) superclusters at redshift $z \approx 0.5$ in order to understand the
growth and evolution of structures in the Universe. We derived the density
contrast values for the spherical collapse model in a wide range of redshifts
and used these values to study the dynamical state and possible evolution of
the HDCs of the BGW superclusters. The masses of the HDCs were calculated using
stellar masses of galaxies in them. We found the masses and radii of the
turnaround and future collapse regions in the HDCs and compared them with those
of local superclusters. We determined eight HDCs in the BGW superclusters. The
masses of their turnaround regions are in the range of $M_{\mathrm{T}} \approx
0.4 - 3.3\times~10^{15}h^{-1}M_\odot,$ and radii are in the range of
$R_{\mathrm{T}} \approx 3.5 - 7 h^{-1}$Mpc. The radii of their future collapse
regions are in the range of $R_{\mathrm{FC}} \approx 4 - 8h^{-1}$Mpc. Distances
between individual cores in superclusters are much larger: of the order of $25
- 35h^{-1}$Mpc. The richness and sizes of the HDCs are comparable with those of
the HDCs of the richest superclusters in the local Universe. The BGW
superclusters will probably evolve to several poorer superclusters with masses
similar to those of the local superclusters. This may weaken the tension with
the $\Lambda$CDM model, which does not predict a large number of very rich and
large superclusters in our local cosmic neighbourhood, and explains why there
are no superclusters as elongated as those in the BGW in the local Universe.


  The Palomar Transient Factory (PTF) is systematically charting the optical
transient and variable sky. A primary science driver of PTF is building a
complete inventory of transients in the local Universe (distance less than 200
Mpc). Here, we report the discovery of PTF10fqs, a transient in the luminosity
"gap" between novae and supernovae. Located on a spiral arm of Messier 99, PTF
10fqs has a peak luminosity of Mr = -12.3, red color (g-r = 1.0) and is slowly
evolving (decayed by 1 mag in 68 days). It has a spectrum dominated by
intermediate-width H (930 km/s) and narrow calcium emission lines. The
explosion signature (the light curve and spectra) is overall similar to thatof
M85OT2006-1, SN2008S, and NGC300OT. The origin of these events is shrouded in
mystery and controversy (and in some cases, in dust). PTF10fqs shows some
evidence of a broad feature (around 8600A) that may suggest very large
velocities (10,000 km/s) in this explosion. Ongoing surveys can be expected to
find a few such events per year. Sensitive spectroscopy, infrared monitoring
and statistics (e.g. disk versus bulge) will eventually make it possible for
astronomers to unravel the nature of these mysterious explosions.


  If an open quantum system is initially uncorrelated from its environment,
then its dynamics can be written in terms of a Lindblad-form master equation.
The master equation is divided into a unitary piece, represented by an
effective Hamiltonian, and a dissipative piece, represented by a
hermiticity-preserving superoperator; however, the division of open system
dynamics into unitary and dissipative pieces is non-unique. For
finite-dimensional quantum systems, we resolve this non-uniqueness by
specifying a norm on the space of dissipative superoperators and defining the
canonical Hamiltonian to be the one whose dissipator is minimal. We show that
the canonical Hamiltonian thus defined is equivalent to the Hamiltonian
initially defined by Lindblad, and that it is uniquely specified by requiring
the dissipator's jump operators to be traceless, extending a uniqueness result
known previously in the special case of Markovian master equations. For a
system weakly coupled to its environment, we give a recursive formula for
computing the canonical effective Hamiltonian to arbitrary orders in
perturbation theory, which we can think of as a perturbative scheme for
renormalizing the system's bare Hamiltonian.


  We study the following fundamental hypothesis testing problem, which we term
Gaussian mean testing. Given i.i.d. samples from a distribution $p$ on
$\mathbb{R}^d$, the task is to distinguish, with high probability, between the
following cases: (i) $p$ is the standard Gaussian distribution,
$\mathcal{N}(0,I_d)$, and (ii) $p$ is a Gaussian $\mathcal{N}(\mu,\Sigma)$ for
some unknown covariance $\Sigma$ and mean $\mu \in \mathbb{R}^d$ satisfying
$\|\mu\|_2 \geq \epsilon$. Recent work gave an algorithm for this testing
problem with the optimal sample complexity of $\Theta(\sqrt{d}/\epsilon^2)$.
Both the previous algorithm and its analysis are quite complicated. Here we
give an extremely simple algorithm for Gaussian mean testing with a one-page
analysis. Our algorithm is sample optimal and runs in sample linear time.


  In this article, we construct a $2$-category of Lagrangians in a fixed
shifted symplectic derived stack S. The objects and morphisms are all given by
Lagrangians living on various fiber products. A special case of this gives a
$2$-category of $n$-shifted symplectic derived stacks $Symp^n$. This is a
$2$-category version of Weinstein's symplectic category in the setting of
derived symplectic geometry. We introduce another $2$-category $Symp^{or}$ of
$0$-shifted symplectic derived stacks where the objects and morphisms in
$Symp^0$ are enhanced with orientation data. Using this, we define a partially
linearized $2$-category $LSymp$. Joyce and his collaborators defined a certain
perverse sheaf on any oriented $(-1)$-shifted symplectic derived stack. In
$LSymp$, the $2$-morphisms in $Symp^{or}$ are replaced by the hypercohomology
of the perverse sheaf assigned to the $(-1)$-shifted symplectic derived
Lagrangian intersections. To define the compositions in $LSymp$ we use a
conjecture by Joyce, that Lagrangians in $(-1)$-shifted symplectic stacks
define canonical elements in the hypercohomology of the perverse sheaf over the
Lagrangian. We refine and expand his conjecture and use it to construct $LSymp$
and a $2$-functor from $Symp^{or}$ to $LSymp$. We prove Joyce's conjecture in
the most general local model. Finally, we define a $2$-category of $d$-oriented
derived stacks and fillings. Taking mapping stacks into a $n$-shifted
symplectic stack defines a $2$-functor from this category to $Symp^{n-d}$.


  Cross sections for \nuc{6}{Li}($\gamma$,$\gamma$)\nuc{6}{Li} have been
measured at the High Intensity Gamma-Ray Source (\HIGS) and the sensitivity of
these cross sections to the nucleon isoscalar polarizabilities was studied.
Data were collected using a quasi-monoenergetic 86 MeV photon beam at photon
scattering angles of 40$^{\circ}$--160$^{\circ}$. These results are an
extension of a previous measurement at a lower energy. The earlier work
indicated that the \nuc{6}{Li}($\gamma$,$\gamma$)\nuc{6}{Li} reaction at 60 MeV
provides a means of extracting the nucleon polarizabilities; this work
demonstrates that the sensitivity of the cross section to the polarizabilities
is increased at 86 MeV. A full theoretical treatment is needed to verify this
conclusion and produce values of the polarizabilities.


  It has been suggested that the dark halos of galaxies are constituted by
cloudlets of cold (< 10 K) H_2 and dense (> 10^7$ cm^{-3}) molecular gas. Such
gas is extremely difficult to detect, because the classical tracers of
molecular gas, CO and/or dust grains, have very low abundances and their
emission is exceedingly weak. For this reason, the cloudlet hypothesis remains
so far substantially unproven. In this Letter we propose a new method to probe
the presence of cold H_2 clouds in galactic halos: the ground transition of
ortho-H_2D^+ at 372 GHz. We discuss why the H_2D^+ is abundant under the
physical conditions appropriate for the cloudlets, and present a chemical model
that predicts the H_2D^+ abundance as function of four key parameters: gas
density and metallicity, cosmic ray ionization rate and dust grain size. We
conclude that current ground-based instruments might detect the ortho-H_2D^+
line emitted by the cloudlets halo, and prove, therefore, the existence of
large quantities of dark baryonic matter around galaxies.


  We study the number of limit cycles bifurcating from a piecewise quadratic
system. All the differential systems considered are piecewise in two zones
separated by a straight line. We prove the existence of 16 crossing limit
cycles in this class of systems. If we denote by $H_p(n)$ the extension of the
Hilbert number to degree $n$ piecewise polynomial differential systems, then
$H_p(2)\geq 16.$ As fas as we are concerned, this is the best lower bound for
the quadratic class. Moreover, all the limit cycles appear in one nest
bifurcating from the period annulus of some isochronous quadratic centers.


  We study the process of quantum telecloning of $d$-dimensional pure quantum
states using partially entangled pure states as quantum channel. This process
efficiently mixes optimal universal symmetric cloning with quantum
teleportation. It is shown that it is possible to implement universal symmetric
telecloning in a probabilistic way using unambiguous state discrimination and
quantum state separation schemes. It is also shown that other strategies, such
as minimum error discrimination, lead to a decrease in the fidelity of the
copies and that certain partially entangled pure states with maximal Schmidt
rank lead to an average telecloning fidelity which is always above the optimal
fidelity of measuring and preparation of quantum states. We also discuss the
case of partially entangled pure states with non-maximal Schmidt rank. The
results presented here are valid for arbitrary numbers of copies of a single
input qudit state of any dimension.


  We consider a variant of the path cover problem, namely, the
$k$-fixed-endpoint path cover problem, or kPC for short, on interval graphs.
Given a graph $G$ and a subset $\mathcal{T}$ of $k$ vertices of $V(G)$, a
$k$-fixed-endpoint path cover of $G$ with respect to $\mathcal{T}$ is a set of
vertex-disjoint paths $\mathcal{P}$ that covers the vertices of $G$ such that
the $k$ vertices of $\mathcal{T}$ are all endpoints of the paths in
$\mathcal{P}$. The kPC problem is to find a $k$-fixed-endpoint path cover of
$G$ of minimum cardinality; note that, if $\mathcal{T}$ is empty the stated
problem coincides with the classical path cover problem. In this paper, we
study the 1-fixed-endpoint path cover problem on interval graphs, or 1PC for
short, generalizing the 1HP problem which has been proved to be NP-complete
even for small classes of graphs. Motivated by a work of Damaschke, where he
left both 1HP and 2HP problems open for the class of interval graphs, we show
that the 1PC problem can be solved in polynomial time on the class of interval
graphs. The proposed algorithm is simple, runs in $O(n^2)$ time, requires
linear space, and also enables us to solve the 1HP problem on interval graphs
within the same time and space complexity.


  We consider the rationally extended exactly solvable Eckart potentials which
exhibit extended shape invariance property. These potentials are isospectral to
the conventional Eckart potential. The scattering amplitude for these
rationally ex- tended potentials is calculated analytically for the generalized
mth (m = 1, 2, 3, ...) case by considering the asymptotic behavior of the
scattering state wave functions which are written in terms of some new
polynomials related to the Jacobi polyno- mials. As expected, in the m = 0
limit, this scattering amplitude goes over to the scattering amplitude for the
conventional Eckart potential.


  Employing heuristic susceptibility equations in conjunction with the
well-known critical exponents, the magnetization and partition function for
two-dimensional nearest neighbour Ising models are formulated in terms of the
Gauss hypergeometric functions. The isomorphism existing between the
Bragg-Williams approximation and the exact solution of Onsager is pointed out.
The precise manner in which the critical exponents influence the partition
functions is pointed out.