  We consider a periodic magnetic Schr\"odinger operator $H^h$, depending on
the semiclassical parameter $h>0$, on a noncompact Riemannian manifold $M$ such
that $H^1(M, {\mathbb R})=0$ endowed with a properly discontinuous cocompact
isometric action of a discrete group. We assume that there is no electric field
and that the magnetic field has a periodic set of compact magnetic wells. We
suppose that the magnetic field vanishes regularly on a hypersurface $S$.
First, we prove upper and lower estimates for the bottom $\lambda_0(H^h)$ of
the spectrum of the operator $H^h$in $L^2(M)$. Then, assuming the existence of
non-degenerate miniwells for the reduced spectral problem on $S$, we prove the
existence of an arbitrary large number of spectral gaps for the operator $H^h$
in the region close to $\lambda_0(H^h)$, as $h\to 0$. In this case, we also
obtain upper estimates for the eigenvalues of the one-well problem.


  The phase transitions between the XY, the dimer and the Haldane phases of the
spin-1 bond-alternating XXZ chain are studied by the numerical diagonalization.
We determine the phase diagram at $T=0$ and also identify the universality
class with the level spectroscopy. We find that exactly on $\Delta=0$ there is
a Berezinskii-Kosterlitz-Thouless transition line which separates the XY and
the Haldane phase and there exists a multicritical point of the XY, the dimer
and the Haldane phases. We discuss that the critical properties of this model
is of the 2-D Ashkin-Teller type reflecting the hidden $Z_2\times Z_2$
symmetry.


  Cardiac development is complex, multiscale process encompassing cell fate
adoption, differentiation and morphogenesis. To elucidate pathways underlying
this process, a recently developed algorithm to reverse engineer gene
regulatory networks was applied to time-course microarray data obtained from
the developing mouse heart. The algorithm generates many different putative
network topologies that are capable of explaining the experimental data via
model simulation. To cull specious network interactions, thousands of
topologies are merged and filtered to generate a scale-free, hierarchical
network. The network is validated with known gene interactions and used to
identify regulatory pathways critical to the developing mammalian heart. The
predicted gene interactions are prioritized using semantic similarity and gene
profile uniqueness metrics. Using these metrics, the network is expanded to
include all known mouse genes to form the most likely cardiogenic gene
regulatory network. The method outlined herein provides an informative approach
to network inference and leads to clear testable hypotheses related to gene
regulation.


  Due to the importance of understanding the formation of the nonlinear
structures in the universe and stellar systems, the nonlinear Jeans instability
which plays the key role in these formation problems is a long-standing open
problem presented by astrophysics more than one hundred years. In this article,
we focus on a reduced model of the nonlinear Jeans instability for the
expanding Newtonian universe which is characterized by a class of second order
nonlinear hyperbolic equations
  \begin{equation*}
  \Box \varrho(x^\mu) +\frac{\mathcal{a} }{t} \partial_{t}\varrho(x^\mu) -
  \frac{\mathcal{b}}{t^2} \varrho(x^\mu) (1+ \varrho(x^\mu) )
-\frac{\mathcal{c}-\mathcal{k}}{1+\varrho(x^\mu)}
(\partial_{t}\varrho(x^\mu))^2= \mathcal{k} F(t).
  \end{equation*} We establish a family of nonlinear self-increase blowup
solution (the solution itself becomes infinite, stable ODE-type blowup) to this
equation and give estimates on the growth rate of $\varrho$ which may explain
the much faster growth rate of the nonlinear structures of universe in
astrophysical observations than the rate given by the classical Jeans
instability.


  Let K be an algebraically closed field of characteristic 0. Following
Medvedev-Scanlon, a polynomial of degree d > 1 is said to be disintegrated if
neither f nor -f is linearly conjugate to x^d or T_d(x) where T_d is the
Chebyshev polynomial of degree d. Let m and n be integers greater than 1, we
prove that there exists an effectively computable constant c(m,n) depending
only on m and n such that the following holds. Let f_1,...,f_n be polynomials
with coefficients in K, which are disintegrated polynomials of degree at most m
and let F be the induced coordinate-wise self-map of the n-th dimensional
affine space, i.e. F(x_1,..,x_n)=(f_1(x_1),...,f_n(x_n)). Then the period of
every irreducible F-periodic subvariety of the n-th dimensional affine space
with non-constant projection to each coordinate axis is at most c(m,n). As an
immediate application, we prove an instance of the dynamical Mordell-Lang
problem following recent work of Xie. The main technical ingredients are
Medvedev-Scanlon classification of invariant subvarieties together with
classical and more recent results in Ritt's theory of polynomial decomposition.


  The acquisition of objects outside the Line-of-Sight of cameras is a very
intriguing but also extremely challenging research topic. Recent works showed
the feasibility of this idea exploiting transient imaging data produced by
custom direct Time of Flight sensors. In this paper, for the first time, we
tackle this problem using only data from an off-the-shelf indirect Time of
Flight sensor without any further hardware requirement. We introduced a Deep
Learning model able to re-frame the surfaces where light bounces happen as a
virtual mirror. This modeling makes the task easier to handle and also
facilitates the construction of annotated training data. From the obtained data
it is possible to retrieve the depth information of the hidden scene. We also
provide a first-in-its-kind synthetic dataset for the task and demonstrate the
feasibility of the proposed idea over it.


  We have investigated the electronic properties of elliptical quantum dots in
a perpendicular external magnetic field, and in the presence of the Rashba
spin-orbit interaction. Our work indicates that the Fock-Darwin spectra display
strong signature of Rashba spin-orbit coupling even for a low magnetic field,
as the anisotropy of the quantum dot is increased. An explanation of this
pronounced effect with respect to the anisotropy is presented. The strong
spin-orbit coupling effect manifests itself prominently in the corresponding
dipole-allowed optical transitions, and hence is susceptible to direct
experimental observation.


  In this note we show that a mutation theory of species with potential can be
defined so that a certain class of skew-symmetrizable integer matrices have a
species realization admitting a non-degenerate potential. This gives a partial
affirmative answer to a question raised by Jan Geuenich and Daniel
Labardini-Fragoso. We also provide an example of a class of skew-symmetrizable
$4 \times 4$ integer matrices, which are not globally unfoldable nor strongly
primitive, and that have a species realization admitting a non-degenerate
potential.


  Common wisdom asserts that bound excitons cannot form in high-dimensional
(d>1) metallic structures because of their overwhelming screening and
unavoidable resonance with nearby continuous bands. Strikingly, here we
illustrate that this prevalent assumption is not quite true. A key ingredient
that has been overlooked is that of viable decoherence that thwarts the
formation of resonances. As an example of this general mechanism, we focus on
an experimentally relevant material and predict bound excitons in twisted
bilayer graphene, which is a two-dimensional gapless structure exhibiting
metallic screening. The binding energies calculated by first-principles
simulations are surprisingly large. The low-energy effective model reveals that
these bound states are produced by a unique destructive coherence between two
alike subband resonant excitons. In particular, this destructive coherent
effect is not sensitive to the screening and dimensionality, and hence may
persist as a general mechanism for creating bound excitons in various metallic
structures, opening the door for excitonic applications based on metallic
structures.


  Understanding the gas content of high redshift halos is crucial for studying
the formation of the first generation of galaxies and reionization. Recently,
Tseliakhovich & Hirata showed that the relative "stream" velocity between the
dark matter and baryons at the time of recombination - formally a second order
effect, but an unusually large one - can influence the later structure
formation history of the Universe. We quantify the effect of the stream
velocity on the so-called "characteristic mass" - the minimum mass of a dark
matter halo capable of retaining most of its baryons throughout its formation
epoch - using three different high-resolution sets of cosmological simulations
(with separate transfer functions for baryons and dark matter) that vary in box
size, particle number, and the value of the relative velocity between the dark
matter and baryons. In order to understand this effect theoretically, we
generalize the linear theory filtering mass to properly account for the
difference between the dark matter and baryonic density fluctuation evolution
induced by the stream velocity. We show that the new filtering mass provides an
accurate estimate for the characteristic mass, while other theoretical ansatzes
for the characteristic mass are substantially less precise.


  The mixed-size placement benchmarks described in this book chapter directly
address several of the shortcomings in previously published suboptimality
benchmarks. Two new sets of placement examples are constructed, one targeting
the role of nonlocal nets in suboptimality, and another targeting the role of
white space and large variations in module sizes. The first set, PEKO-MC, is a
set of standard-cell circuits with nonlocal nets in known optimal placements. A
given netlist is modified so as to render a given placement for it optimal for
the new netlist. Cell dimensions and locations are not changed, net-degree
statistics are matched exactly, and over 60% of the original netlist is left
unchanged. The second set, PEKO-MS, incorporates a parametrized percentage of
white space into a mixed-size placement which precisely matches given macro
dimensions and locations as well as the net-degree distributions of the ISPD
2005 benchmark suite [21]. HPWL for the placements generated for the PEKO-MS
circuits are proven to be less than 3% above optimal for most cases and within
8% of optimal on all cases.


  We present new spectroscopic and polarimetric observations of the first
discovered gravitational lens Q0957+561 obtained with the 6m telescope of the
Special Astrophysical Observatory (SAO, Russia). We explore spectropolarimetric
parameters of Q0957+561 A,B components to investigate the innermost structure
of the quasar, and explore the nature of polarization in lensed quasars.
Additionally, we compare their present-day spectral characteristics with
previous observations in order to study long-term spectral changes. We analyze
spectral characteristics of lensed quasar comparing spectra of A and B images,
as well as comparing previously observed image spectra with present-day ones.
The polarization parameters of A-B images are compared. We also model the
macro-lens influence on the polarization of the images representing the
gravitational lens with a singular isothermal elliptical potential.
  We find that the brightness and SED ratio of components A and B changed
during a long period. Polarization in broad lines of components A and B showed
that the equatorial scattering cannot be detected in this quasar. We find
wavelength-dependent polarization that may be explained as a combination of the
polarization from the disc and outflowing material. There is a significant
difference between polarization parameters of the A and B images: the B
component shows a higher polarization degree and polarization angle. However,
both polarization vectors are nearly perpendicular to the observed radio jet
projection. It indicates that the polarization in the continuum is coming from
the accretion disc. Our simple lensing model of a polarized source showed that
macro-lens can cause the observed differences in polarization parameters of
Q0957+561 A,B images. Using Mg II broad line and luminosity of component A we
estimated that the Q0957+561 black hole mass is M~(4.8-6.1) $10^8$ M$\odot$


  Ab initio molecular dynamics simulation is used to study the structure and
electronic properties of the liquid Ga-Se system at the three compositions
Ga$_2$Se, GaSe and Ga$_2$Se$_3$, and of the GaSe and Ga$_2$Se$_3$ crystals. The
calculated equilibrium structure of GaSe crystal agrees well with available
experimental data. The neutron-weighted liquid structure factors calculated
from the simulations are in reasonable agreement with recent neutron
diffraction measurements. Simulation results for the partial radial
distribution functions show that the liquid structure is closely related to
that of the crystals. A close similarity between solid and liquid is also found
for the electronic density of states and charge density. The calculated
electronic conductivity decreases strongly with increasing Se content, in
accord with experimental measurements.


  In the present work, we study dark solitons in dynamical lattices with the
saturable nonlinearity and compare them with those in lattices with the cubic
nonlinearity. This comparison has become especially relevant in light of recent
experimental developments in the former context. The stability properties of
the fundamental waves, for both on-site and inter-site modes, are examined
analytically and corroborated by numerical results. Furthermore, their
dynamical evolution when they are found to be unstable is obtained through
appropriately crafted numerical experiments.


  We solve the Schr\"odinger equation from first principles to investigate the
many-body effects in the expansion dynamics of one-dimensional repulsively
interacting bosons released from a harmonic trap. We utilize the
multiconfigurational time-dependent Hartree method for bosons (MCTDHB) to solve
the many-body Schr\"odinger equation at high level of accuracy. The MCTDHB
basis sets are explicitly time-dependent and optimised by variational
principle. We probe the expansion dynamics by three key measures; time
evolution of one-, two- and three-body densities. We observe when the
mean-field theory results to unimodal expansion, the many-body calculation
exhibits trimodal expansion dynamics. The many-body features how the initially
fragmented bosons independently spreads out with time whereas the mean-field
pictures the expansion of the whole cloud. We also present the three different
time scale of dynamics of the inner core, outer core and the cloud as a whole.
We analyze the key role played by the dynamical fragmentation during expansion.
A Strong evidence of the many-body effects is presented in the dynamics of two-
and three-body densities which exhibit correlation hole and pronounced
delocalization effect.


  The order, shape and critical point for the phase transition between the
hadronic matter and quark-gluon plasma are considered in a thermodynamical
consistent approach. The hadronic phase is taken as Van der Waals gas of all
the known hadronic mass spectrum particles $m_H\le 2.0$ GeV as well as Hagedorn
bubbles which correspond hadronic states with mass spectrum $m_H> 2.0$ GeV. The
density of states for Hagedorn bubbles is derived by calculating the
microcanonical ensemble for a bag of quarks and gluons with specific internal
color-flavor symmetry. The mixed-grand and microcanonical ensembles are derived
for massless and massive flavors. We find Hagedorn bubbles are strongly
suppressed in the dilute hadronic matter and they appear just below the line of
the phase transition. The order of the phase transition depends on Hagedorn
bubble's internal color-flavor structure and the volume fluctuation as well. On
the other hand, the highly compressed hadronic matter undergoes a smooth phase
transition from the gas of known mass spectrum hadrons to another one dominated
by Hagedorn bubbles with specific internal color-flavor structure before the
phase transition to quark-gluon plasma takes place at last. The phase
transition is found a first order for the intermediate and large chemical
potentials. The existence of the tri-critical point depends on the modification
of the bubble's internal structure specified by a phenomenological parameter
$\gamma\propto\mu_B$ in the medium.


  Far and mid infrared optical pulses have been shown to induce non-equilibrium
unconventional orders in complex materials, including photo-induced
ferroelectricity in quantum paraelectrics, magnetic polarization in
antiferromagnets and transient superconducting correlations in the normal state
of cuprates and organic conductors. In the case of non-equilibrium
superconductivity, femtosecond drives have generally resulted in electronic
properties that disappear immediately after excitation, evidencing a state that
lacks intrinsic rigidity. Here, we make use of a new optical device to drive
metallic K$_3$C$_{60}$ with mid-infrared pulses of tunable duration, ranging
between one picosecond and one nanosecond. The same superconducting-like
optical properties observed over short time windows for femtosecond excitation
are shown here to become metastable under sustained optical driving, with
lifetimes in excess of ten nanoseconds. Direct electrical probing becomes
possible at these timescales, yielding a vanishingly small resistance. Such a
colossal positive photo-conductivity is highly unusual for a metal and, when
taken together with the transient optical conductivities, it is rather
suggestive of metastable light-induced superconductivity.


  We propose a protocol for secret sharing, called dual quantum information
splitting (DQIS), that reverses the roles of state and channel in standard
quantum information splitting. In this method, a secret is shared via
teleportation of a fiducial input state over an entangled state that encodes
the secret in a graph state basis. By performing a test of violation of a Bell
inequality on the encoded state, the legitimate parties determine if the
violation is sufficiently high to permit distilling secret bits. Thus, the code
space must be maximally and exclusively nonlocal. To this end, we propose two
ways to obtain code words that are degenerate with respect to a Bell operator.
The security of DQIS comes from monogamy of nonlocal correlations, which we
illustrate by means of a simple single-qubit attack model. The nonlocal basis
of security of our protocol makes it suitable for security in general
monogamous theories and in the more stringent, device-independent cryptographic
scenario.


  The binding energies of n < 100 carbon clusters are calculated using the
ab-initio density functional theory code Quantum Espresso. Carbon cluster
geometries are determined using several levels of classical techniques and
further refined using density functional theory. The resulting energies are
used to compute the work of cluster formation and the nucleation rate in a
saturated, hydrogen-poor carbon gas. Compared to classical calculations that
adopt the capillary approximation, we find that nucleation of carbon clusters
is enhanced at low temperatures and depressed at high temperatures. This
difference is ascribed to the different behavior of the critical cluster size.
We find that the critical cluster size is at n = 27 or n = 8 for a broad range
of temperatures and saturations, instead of being a smooth function of such
parameters. The results of our calculations can be used to follow carbonaceous
cluster/grain formation, stability, and growth in hydrogen poor environments,
such as the inner layers of core-collapse supernova and supernova remnants.


  The closed symmetrized tridisc $\Gamma_3$ and its distinguished boundary
$b\Gamma_3$ are the sets $\Gamma_3=\{
(z_1+z_2+z_3,z_1z_2+z_2z_3+z_3z_1,z_1z_2z_3): \,|z_i|\leq 1, i=1,2,3
\}\subseteq \mathbb C^3$ $b\Gamma_3=\{
(z_1+z_2+z_3,z_1z_2+z_2z_3+z_3z_1,z_1z_2z_3): \,|z_i|= 1, i=1,2,3 \}\subseteq
\Gamma_3.$ A triple of commuting operators $(S_1,S_2,P)$ defined on a Hilbert
space $\mathcal H$ for which $\Gamma_3$ is a spectral set is called a
$\Gamma_3$-contraction. In this article we show by a counter example that there
are $\Gamma_3$-contractions which do not dilate. It is also shown that under
certain conditions a $\Gamma_3$-contraction can have normal $b\Gamma_3$
dilation. We determine several classes of $\Gamma_3$-contractions which dilate
and show explicit construction of their dilations. A concrete functional model
is provided for the $\Gamma_3$-contractions which dilate. Various
characterizations for $\Gamma_3$-unitaries and $\Gamma_3$-isometries are
obtained; the classes of $\Gamma_3$-unitaries and $\Gamma_3$-isometries are
analogous to the unitaries and isometries in one variable operator theory. Also
we find out a model for the class of pure $\Gamma_3$-isometries. En route we
study the geometry of the sets $\Gamma_3$ and $b\Gamma_3$ and provide variety
of characterizations for them.


  This paper investigates the convergence time of log-linear learning to an
$\epsilon$-efficient Nash equilibrium (NE) in potential games. In such games,
an efficient NE is defined as the maximizer of the potential function. Existing
results are limited to potential games with stringent structural assumptions
and entail exponential convergence times in $1/\epsilon$. Unaddressed so far,
we tackle general potential games and prove the first finite-time convergence
to an $\epsilon$-efficient NE. In particular, by using a problem-dependent
analysis, our bound depends polynomially on $1/\epsilon$. Furthermore, we
provide two extensions of our convergence result: first, we show that a variant
of log-linear learning that requires a factor $A$ less feedback on the utility
per round enjoys a similar convergence time; second, we demonstrate the
robustness of our convergence guarantee if log-linear learning is subject to
small perturbations such as alterations in the learning rule or noise-corrupted
utilities.


  In the pursuit of advanced energy storage solutions, the crystal structure of
ionic conductors plays a pivotal role in facilitating ion transport. The
conventional structural design principle that compounds with the body-centered
cubic (BCC) anionic frameworks have high ionic conductivity is well known. We
have extended the conventional design principle by uncovering that many of the
anionic frameworks of Ag-ion conductors are characterized by tetrahedrally
packed (TP) structures. Leveraging our findings, we have virtually screened TP
framework compounds, uncovering their intrinsic potential for superior ionic
conductivity through first-principles molecular dynamics simulations. Our
design principle is applicable to Ag$^+$ and other mobile ions, including
Li$^+$ and F$^-$. We proposed the Met2Ion method to generate ionic crystal
structures using metal crystal structures as templates and demonstrated that
new ionic conductors with TP frameworks can be discovered. This work paves the
way for the discovery and development of next-generation energy storage
materials with enhanced performance.


  We investigate how a symmetric penetrable object immersed in an active fluid
becomes motile due to a negative drag acting in the direction of its velocity.
While similar phenomena have been reported only for active fluids that posses
polar or nematic order, we demonstrate that such motility can occur even in
active fluids without any preexisting order. The emergence of object motility
is characterized by both continuous and discontinuous transitions associated
with the symmetry-breaking bifurcation of the object's steady-state velocity.
Furthermore, we also discuss the relevance of the transitions to the
nonmonotonic particle-size dependence of the object's diffusion coefficient.


  Existing fine-grained visual categorization methods often suffer from three
challenges: lack of training data, large number of fine-grained categories, and
high intraclass vs. low inter-class variance. In this work we propose a generic
iterative framework for fine-grained categorization and dataset bootstrapping
that handles these three challenges. Using deep metric learning with humans in
the loop, we learn a low dimensional feature embedding with anchor points on
manifolds for each category. These anchor points capture intra-class variances
and remain discriminative between classes. In each round, images with high
confidence scores from our model are sent to humans for labeling. By comparing
with exemplar images, labelers mark each candidate image as either a "true
positive" or a "false positive". True positives are added into our current
dataset and false positives are regarded as "hard negatives" for our metric
learning model. Then the model is retrained with an expanded dataset and hard
negatives for the next round. To demonstrate the effectiveness of the proposed
framework, we bootstrap a fine-grained flower dataset with 620 categories from
Instagram images. The proposed deep metric learning scheme is evaluated on both
our dataset and the CUB-200-2001 Birds dataset. Experimental evaluations show
significant performance gain using dataset bootstrapping and demonstrate
state-of-the-art results achieved by the proposed deep metric learning methods.


  Transformation optics gives rise to numerous unusual optical devices, such as
novel metamaterial lenses and invisibility cloaks. Very recently Mattheakis et
al. have suggested theoretical design of an optical waveguide based on a
network of Luneburg lenses, which may be useful in sensing and nonlinear optics
applications. Here we report the first experimental realization of such
Luneburg waveguides. We have studied wavelength and polarization dependent
performance of the waveguides.


  [abridged] The role of magnetic fields in the process of star formation is a
matter of continuous debate. Clear observational proof of the general influence
of magnetic fields on the early phase of cloud collapse is still pending. First
results on Bok globules with simple structures indicate dominant magnetic
fields across large spatial scales (Bertrang+2014).
  The aim of this study is to test the magnetic field influence across Bok
globules with more complex density structures. We apply near-infrared
polarimetry to trace the magnetic field structure on scales of 10^4-10^5au in
selected Bok globules. The combination of these measurements with archival data
in the optical and sub-mm wavelength range allows us to characterize the
magnetic field on scales of 10^3-10^6au.
  We present polarimetric data in the near-infrared wavelength range for the
three Bok globules CB34, CB56, and [OMK2002]18, combined with archival
polarimetric data in the optical wavelength range for CB34 and CB56, and in the
sub-millimeter wavelength range for CB34 and [OMK2002]18. We find a strong
polarization signal (P>2%) in the near-infrared and strongly aligned
polarization segments on large scales (10^4-10^6au) for all three globules.
  This indicates dominant magnetic fields across Bok globules with complex
density structures.
  To reconcile our findings in globules, the lowest mass clouds known, and the
results on intermediate (e.g., Taurus) and more massive (e.g., Orion) clouds,
we postulate a mass dependent role of magnetic fields, whereby magnetic fields
appear to be dominant on low and high mass but rather sub-dominant on
intermediate mass clouds.