pinecone/core-2020-05-10-deduplication huggingface dataset:
On projective modules over finite quantum groups

Let $\mathcal{D}$ be the Drinfeld double of the bosonization ${\mathfrak
B}(V)\#\Bbbk G$ of a finite-dimensional Nichols algebra ${\mathfrak B}(V)$ over
a finite group $G$. It is known that the simple $\mathcal{D}$-modules are
parametrized by the simple modules over $\mathcal{D}(G)$, the Drinfeld double
of $G$. This parametrization can be obtained by considering the head
$\mathsf{L}(\lambda)$ of the Verma module $\mathsf{M}(\lambda)$ for every
simple $\mathcal{D}(G)$-module $\lambda$. In the present work, we show that the
projective $\mathcal{D}$-modules are filtered by Verma modules and the BGG
Reciprocity
$[\mathsf{P}(\mu):\mathsf{M}(\lambda)]=[\mathsf{M}(\lambda):\mathsf{L}(\mu)]$
holds for the projective cover $\mathsf{P}(\mu)$ of $\mathsf{L}(\mu)$. We use
graded characters to proof the BGG Reciprocity and obtain a graded version of
it. Also, we show that a Verma module is simple if and only if it is
projective. We also describe the tensor product between projective modules.Comment: 17 pages. v2: We add information about the tensor product between
  projective modules [Theorem 4.10]. We show that $\mathcal{D}$ is graded
  symmetric [Lemma 3.1] and hence a shift of grading is not needed in the
  graded version of the BGG Reciprocity [Corollary 3.6]. Minor changes in
  Section
LOWER BOUNDS ON THE NOETHER NUMBER

The best known method to give a lower bound for the Noether number of a given finite group is to use the fact that it is greater than or equal to the Noether number of any of the subgroups or factor groups. The results of the present paper show in particular that these inequalities are strict for proper subgroups or factor groups. This is established by studying the algebra of coinvariants of a representation induced from a representation of a subgroup. © 2018 Springer Science+Business Media, LLC, part of Springer Natur
Spectral uniqueness of bi-invariant metrics on symplectic groups

In this short note, we prove that a bi-invariant Riemannian metric on
$\mathrm{Sp}(n)$ is uniquely determined by the spectrum of its Laplace-Beltrami
operator within the class of left-invariant metrics on $\mathrm{Sp}(n)$. In
other words, on any of these compact simple Lie groups, every left-invariant
metric which is not right-invariant cannot be isospectral to a bi-invariant
metric. The proof is elementary and uses a very strong spectral obstruction
proved by Gordon, Schueth, and Sutton.Comment: The proof of Lemma 3.1 was modifie
Line Congruences of Low Order

A line congruence is an irreducible subvariety of dimension n−1 in the Grassmannian of lines in Pn. There are two numerical invariants associated to a line congruence: the order, which is the number of lines passing through a general point of Pn, and the class, which is the number of lines of the congruence contained in a general hyperplaneH and meeting a general line inH. The paper reviews the classification of line congruences of order 0 and 1, and then gives some new results online congruences of order 2 in P3, which is a work in progress. The last section states some open questions
Continuum thermodynamics and phase-ﬁeld models

Phase transitions between two phases are modelled as space regions where a phase-field changes smoothly. The two phases are separated by a thin transition layer, the so-called diffuse interface. All thermodynamic quantities are allowed to vary inside this layer, including the pressure and the mass density. A thermodynamic approach is developed by allowing for the nonlocal character of the continuum. It is based on an extra entropy flux which is proved to be non vanishing inside the transition layer, only. The phase-field is regarded as an internal variable and the kinetic or evolution equation is viewed as a constitutive equation of rate type.  Necessary and sufficient restrictions placed by thermodynamics are derived for the constitutive equations and, furthermore, a general form of the evolution equation for the phase-field is obtained within the schemes of both a non-conserved and a conserved phase-field. Based on the thermodynamic restrictions, a  phase-field model for the ice-water  transition is established which allows for superheating and undercooling. A model ruling the liquid-vapor phase transition is also provided which accounts for both temperature and pressure variations during the evaporation process. The explicit expression of the Gibbs free enthalpy, the Clausius-Clapeyron formula and the customary form of the vapor pressure curve are recovered
Why and how much the Brouwer's Fixed Point Theorem fails in noncompact setting?

The celebrated Brouwer’s Fixed Point Theorem is dated in 1912. Its extension to compact set setting in Banach spaces due to Schauder appeared in 1930. Immediately it raised the question whether the Theorem can be extended to noncompact setting. The works of Kakutani, Klee, Benyamini and Sterfeld, Sternfeld and Lim solved the qualitative part of the problem. Lack of compactness makes the statement of the theorem false. However, there are some quantitative aspects of the question. The two basic are called minimal displacement problem, and optimal retraction problem. The aim of this article is to present the historical back ground and possibly, up to date state of investigations in this field. A list of open problems with comments will be discussed
Evolution models for mass transportation problems

We present a survey on several mass transportation problems, in which a given
mass dynamically moves from an initial configuration to a final one. The
approach we consider is the one introduced by Benamou and Brenier in [5], where
a suitable cost functional $F(\rho,v)$, depending on the density $\rho$ and on
the velocity $v$ (which fulfill the continuity equation), has to be minimized.
Acting on the functional $F$ various forms of mass transportation problems can
be modeled, as for instance those presenting congestion effects, occurring in
traffic simulations and in crowd motions, or concentration effects, which give
rise to branched structures.Comment: 16 pages, 14 figures; Milan J. Math., (2012
Mean curvature, threshold dynamics, and phase field theory on finite
  graphs

In the continuum, close connections exist between mean curvature flow, the
Allen-Cahn (AC) partial differential equation, and the Merriman-Bence-Osher
(MBO) threshold dynamics scheme. Graph analogues of these processes have
recently seen a rise in popularity as relaxations of NP-complete combinatorial
problems, which demands deeper theoretical underpinnings of the graph
processes. The aim of this paper is to introduce these graph processes in the
light of their continuum counterparts, provide some background, prove the first
results connecting them, illustrate these processes with examples and identify
open questions for future study. We derive a graph curvature from the graph cut
function, the natural graph counterpart of total variation (perimeter). This
derivation and the resulting curvature definition differ from those in earlier
literature, where the continuum mean curvature is simply discretized, and bears
many similarities to the continuum nonlocal curvature or nonlocal means
formulation. This new graph curvature is not only relevant for graph MBO
dynamics, but also appears in the variational formulation of a discrete time
graph mean curvature flow. We prove estimates showing that the dynamics are
trivial for both MBO and AC evolutions if the parameters (the time-step and
diffuse interface scale, respectively) are sufficiently small (a phenomenon
known as "freezing" or "pinning") and also that the dynamics for MBO are
nontrivial if the time step is large enough. These bounds are in terms of graph
quantities such as the spectrum of the graph Laplacian and the graph curvature.
Adapting a Lyapunov functional for the continuum MBO scheme to graphs, we prove
that the graph MBO scheme converges to a stationary state in a finite number of
iterations. Variations on this scheme have recently become popular in the
literature as ways to minimize (continuum) nonlocal total variation.Comment: 76 pages, 7 figures (consisting of 25 subfigures in total) v2 has a
  typo in Lemma 5.2, isolated, but important for potential future
  use/reference. Corrected in v3. Some ambiguities in Section 3.3 are clarified
  in v
Non-simultaneous quenching in a system of heat equations coupled at the boundary

We study the solutions of a parabolic system of heat equations coupled at the boundary through a nonlinear flux. We characterize in terms of the parameters involved when nonsimultaneous quenching may appear. Moreover, if quenching is non-simultaneous we find the quenching rate, which surprisingly depends on the flux associated to the other component.Fil: Ferreira, Raúl. Universidad Carlos III de Madrid; EspañaFil: de Pablo, Arturo. Universidad Carlos III de Madrid; EspañaFil: Quirós, Fernando. Universidad Autónoma de Madrid; EspañaFil: Rossi, Julio Daniel. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin
Instability of solitary waves on Euler's elastica

Stability of solitary waves in a thin inextensible and unshearable rod of
infinite length is studied. Solitary-wave profile ofthe elastica of such a rod
without torsion has the form of a planar loop and its speed depends on a
tension in the rod. The linear instability of a solitary-wave profile subject
to perturbations escaping from the plane of the loop is established for a
certain range of solitary-wave speeds. It is done using the properties of the
Evans function, an analytic function on the right complex half-plane, that has
zeroes if and only if there exist the unstable modes of the linearization
around a solitary-wave solution. The result follows from comparison of the
behaviour of the Evans function in some neighbourhood of the origin with its
asymptotic at infinity. The explicit computation of the leading coefficient of
the Taylor series of the Evans function near the origin is performed by means
of the symbolic computer language.Comment: 19 pages, 2 figure
Symmetry analysis for uniaxial compression of a hypoplastic\ud
granular material

A variety of modelling approaches currently exist to describe and predict the diverse\ud
behaviours of granular materials. One of the more sophisticated theories is hypoplasticity, which is a stress-rate theory of rational continuum mechanics with a constitutive law expressed in a single tensorial equation. In this paper, a particular version of hypoplasticity, due to Wu [2], is employed to describe a class of one-dimensional granular deformations. By combining the constitutive law with the conservation laws of continuum mechanics, a system of four nonlinear partial differential equations is derived for the axial and lateral stress, the velocity and the void ratio. Under certain restrictions, three of the governing equations may be combined to yield\ud
ordinary differential equations, whose solutions can be calculated exactly. Several new analytical results are obtained which are applicable to oedometer testing. In general this approach is not possible, and analytic progress is sought via Lie symmetry analysis. A complete set or "optimal system" of group-invariant solutions is identified using the Olver method, which involves the adjoint representation of the symmetry group on its Lie algebra. Each element in the optimal system is governed by a system of nonlinear ordinary differential equations which in general must be solved numerically. Solutions previously considered in the literature are noted, and their relation to our optimal system identified. Two illustrative examples are examined and the variation of various functions occuring in the physical variables is shown graphically
Birkhoffian formulation of the dynamics of LC circuits

We present a formulation of general nonlinear LC circuits within the
framework of Birkhoffian dynamical systems on manifolds. We develop a
systematic procedure which allows, under rather mild non-degeneracy conditions,
to write the governing equations for the mathematical description of the
dynamics of an LC circuit as a Birkhoffian differential system. In order to
illustrate the advantages of this approach compared to known Lagrangian or
Hamiltonian approaches we discuss a number of specific examples. In particular,
the Birkhoffian approach includes networks which contain closed loops formed by
capacitors, as well as inductor cutsets. We also extend our approach to the
case of networks which contain independent voltage sources as well as
independent current sources. Also, we derive a general balance law for an
associated "energy function".Comment: 26 pages, 2 figures. Z. Angew. Math. Phys. (ZAMP), accepted for
  publicatio
Dark solitons in external potentials

We consider the persistence and stability of dark solitons in the
Gross-Pitaevskii (GP) equation with a small decaying potential. We show that
families of black solitons with zero speed originate from extremal points of an
appropriately defined effective potential and persist for sufficiently small
strength of the potential. We prove that families at the maximum points are
generally unstable with exactly one real positive eigenvalue, while families at
the minimum points are generally unstable with exactly two complex-conjugated
eigenvalues with positive real part. This mechanism of destabilization of the
black soliton is confirmed in numerical approximations of eigenvalues of the
linearized GP equation and full numerical simulations of the nonlinear GP
equation with cubic nonlinearity. We illustrate the monotonic instability
associated with the real eigenvalues and the oscillatory instability associated
with the complex eigenvalues and compare the numerical results of evolution of
a dark soliton with the predictions of Newton's particle law for its position.Comment: 39 pages, 10 figure
Exact Fourier expansion in cylindrical coordinates for the
  three-dimensional Helmholtz Green function

A new method is presented for Fourier decomposition of the Helmholtz Green
Function in cylindrical coordinates, which is equivalent to obtaining the
solution of the Helmholtz equation for a general ring source. The Fourier
coefficients of the Helmholtz Green function are split into their half
advanced+half retarded and half advanced-half retarded components. Closed form
solutions are given for these components in terms of a Horn function and a
Kampe de Feriet function, respectively. The systems of partial differential
equations associated with these two-dimensional hypergeometric functions are
used to construct a fourth-order ordinary differential equation which both
components satisfy. A second fourth-order ordinary differential equation for
the general Fourier coefficent is derived from an integral representation of
the coefficient, and both differential equations are shown to be equivalent.
Series solutions for the various Fourier coefficients are also given, mostly in
terms of Legendre functions and Bessel/Hankel functions. These are derived from
the closed form hypergeometric solutions or an integral representation, or
both. Numerical calculations comparing different methods of calculating the
Fourier coefficients are presented
Some remarks on the isoperimetric problem for the higher eigenvalues of
  the Robin and Wentzell Laplacians

We consider the problem of minimising the $k$th eigenvalue, $k \geq 2$, of
the ($p$-)Laplacian with Robin boundary conditions with respect to all domains
in $\mathbb{R}^N$ of given volume $M$. When $k=2$, we prove that the second
eigenvalue of the $p$-Laplacian is minimised by the domain consisting of the
disjoint union of two balls of equal volume, and that this is the unique domain
with this property. For $p=2$ and $k \geq 3$, we prove that in many cases a
minimiser cannot be independent of the value of the constant $\alpha$ in the
boundary condition, or equivalently of the volume $M$. We obtain similar
results for the Laplacian with generalised Wentzell boundary conditions $\Delta
u + \beta \frac{\partial u}{\partial \nu} + \gamma u = 0$.Comment: 16 page
Stable standing waves for a class of nonlinear Schroedinger-Poisson
  equations

We prove the existence of orbitally stable standing waves with prescribed
$L^2$-norm for the following Schr\"odinger-Poisson type equation \label{intro}
%{%{ll} i\psi_{t}+ \Delta \psi - (|x|^{-1}*|\psi|^{2}) \psi+|\psi|^{p-2}\psi=0
\text{in} \R^{3}, %-\Delta\phi= |\psi|^{2}& \text{in} \R^{3},%. when $p\in
\{8/3\}\cup (3,10/3)$. In the case $3<p<10/3$ we prove the existence and
stability only for sufficiently large $L^2$-norm. In case $p=8/3$ our approach
recovers the result of Sanchez and Soler \cite{SS} %concerning the existence
and stability for sufficiently small charges. The main point is the analysis of
the compactness of minimizing sequences for the related constrained
minimization problem. In a final section a further application to the
Schr\"odinger equation involving the biharmonic operator is given
Explicit asymptotic modelling of transient Love waves propagated along a thin coating

The official published version can be obtained from the link below.An explicit asymptotic model for transient Love waves is derived from the exact equations of anti-plane elasticity. The perturbation procedure relies upon the slow decay of low-frequency Love waves to approximate the displacement field in the substrate by a power series in the depth coordinate. When appropriate decay conditions are imposed on the series, one obtains a model equation governing the displacement at the interface between the coating and the substrate. Unusually, the model equation contains a term with a pseudo-differential operator. This result is confirmed and interpreted by analysing the exact solution obtained by integral transforms. The performance of the derived model is illustrated by numerical examples.This work is sponsored by the grant from Higher Education of Pakistan and by the Brunel University’s “BRIEF” research award
Heat flow method to Lichnerowicz type equation on closed manifolds

In this paper, we establish existence results for positive solutions to the
Lichnerowicz equation of the following type in closed manifolds -\Delta
u=A(x)u^{-p}-B(x)u^{q},\quad in\quad M, where $p>1, q>0$, and $A(x)>0$,
$B(x)\geq0$ are given smooth functions. Our analysis is based on the global
existence of positive solutions to the following heat equation {ll} u_t-\Delta
u=A(x)u^{-p}-B(x)u^{q},\quad in\quad M\times\mathbb{R}^{+}, u(x,0)=u_0,\quad
in\quad M with the positive smooth initial data $u_0$.Comment: 10 page
Finite--dimensional global attractor for a system modeling the 2D
  nematic liquid crystal flow

We consider a 2D system that models the nematic liquid crystal flow through
the Navier--Stokes equations suitably coupled with a
transport-reaction-diffusion equation for the averaged molecular orientations.
This system has been proposed as a reasonable approximation of the well-known
Ericksen--Leslie system. Taking advantage of previous well-posedness results
and proving suitable dissipative estimates, here we show that the system
endowed with periodic boundary conditions is a dissipative dynamical system
with a smooth global attractor of finite fractal dimension
Discrete spectrum of a model Schrödinger operator on the half-plane with Neumann conditions

International audienceWe study the eigenpairs of a model Schrödinger operator with a quadratic potential and Neumann boundary conditions on a half-plane. The potential is degenerate in the sense that it reaches its minimum all along a line which makes the angle \theta with the boundary of the half-plane. We show that the first eigenfunctions satisfy localization properties related to the distance to the minimum line of the potential. We investigate the densification of the eigenvalues below the essential spectrum in the limit \theta \to 0 and we prove full asymptotic expansion for these eigenvalues and their associated eigenvectors. We conclude the paper by numerical experiments obtained by a finite element method. The numerical results confirm and enlighten the theoretical approach
Further mathematical results concerning Burgers fluids and their generalizations

In this paper, we extend the earlier work by Quintanilla and Rajagopal (Math Methods Appl Sci 29: 2133–2147,\ud
2006) and establish qualitative new results for a proper generalization of Burgers’ original work that stems form a general\ud
thermodynamic framework. Such fluids have been used to describe the behavior of several geological materials such as\ud
asphalt and the earth’s mantle as well as polymeric fluids. We study questions concerning stability, uniqueness and continuous\ud
dependence on initial data for the solutions of the flows of these fluids. We show that if certain conditions are not\ud
satisfied by the material moduli, the solutions could be unstable. The spatial behavior of the solutions is also analyzed.Peer ReviewedPostprint (published version
The critical state in thin superconductors as a mixed boundary value
  problem: analysis and solution by means of the Erd\'elyi-Kober operators

With this paper we provide an effective method to solve a large class of
problems related to the electromagnetic behavior of thin superconductors. Here
all the problems are reduced to finding the weight functions for the Green
integrals that represent the magnetic field components; these latter must
satisfy the mixed boundary value conditions that naturally arise from the
critical state assumptions. The use of the Erd\'elyi-Kober operators and of the
Hankel transforms (and mostly the employment of their composition properties)
is the keystone to unify the method toward the solution. In fact, the procedure
consists always of the same steps and does not require any peculiar invention.
For this reason the method, here presented in detail for the simplest cases
that can be handled in analytical way (two parts boundary), can be directly
extended to many other more complex geometries (three or more parts), which
usually will require a numerical treatment. In this paper we use the operator
technique to derive the current density and field distributions in perfectly
conducting and superconducting thin discs and tapes subjected to a uniform
magnetic field or carrying a transport current. Although analytical expressions
for the field and current distributions have already been found by other
authors in the past by using several other methods, their derivation is often
cumbersome or missing key details, which makes it difficult for the reader to
fully understand the derivation of the analytical formulas and, more
importantly, to extend the same methods to solve similar new problems. On the
contrary, the characterization of these cases as mixed boundary conditions has
the advantage of referring to an immediate and na\"ive translation of physics
into a consistent mathematical formulation whose possible extension to other
cases is self-evident.Comment: The final publication is available at
  http://link.springer.com/article/10.1007/s00033-011-0185-
On entropy weak solutions of Hughes' model for pedestrian motion

International audienceWe consider a generalized version of Hughes' macroscopic model for crowd motion in the one-dimensional case. It consists in a scalar conservation law accounting for the conservation of the number of pedestrians, coupled with an eikonal equation giving the direction of the flux depending on pedestrian density. As a result of this non-trivial coupling, we have to deal with a conservation law with space-time discontinuous flux, whose discontinuity depends non-locally on the density itself. We propose a definition of entropy weak solution, which allows us to recover a maximum principle. Moreover, we study the structure of the solutions to Riemann-type problems and we construct them explicitly for small times, depending on the choice of the running cost in the eikonal equation. In particular, aiming at the optimization of the evacuation time, we propose a strategy that is optimal in the case of high densities. All results are illustrated by numerical simulations