St-Donat, Bas-St-Laurent, Québec

St-Donat, Bas-St-Laurent, Québec

Charts usually show values as visual properties, like the length in a bar chart, the location in a scatterplot, the area in a bubble chart, etc. Unit charts show values as multiples instead. One famous example of these charts is called ISOTYPE, and you may have seen them in information graphics as well. They’re an […]

How do we read pie charts? This seems like a straightforward question to answer, but it turns out that most of what you’ve probably heard is wrong. We don’t actually know whether we use angle, area, or arc length. In a short paper at the VIS conference this week I’m presenting a study I ran […]

We all use data all the time, but what exactly is data? How do different programs know what to do with our data? How is visualizing data different from other uses of data? And isn’t everything inside a computer data in the end? The latest episode of eagereyesTV looks at what data is and what […]

This book from 1945 contains a very interesting mix of different charts made by the ISOTYPE Institute, some classic and some quite unusual. As a book about labor and unemployment, it also makes extensive use of Gerd Arntz’s famous unemployed man icon. Michael Young and Theodor Prager’s There’s Work for All is part of a […]

Visualization turns data into images, but are images themselves data? There are often claims that they are, but then you mostly see the images themselves without much additional data. In this video, I look at image browsers, a project classifying selfies along a number of criteria, and the additional information stored in HEIC that makes […]

We provide an explicit construction for a class of commutative, non-associative algebras for each of the simple Chevalley groups of simply laced type. Moreover, we equip these algebras with an associating bilinear form, which turns them into Frobenius algebras. This class includes a 3876-dimensional algebra on which the Chevalley group of type E8 acts by automorphisms. We also prove that these algebras admit the structure of (axial) decomposition algebras.

A topological dynamical system $(X,f)$ induces two natural systems, one is on the probability measure spaces and other one is on the hyperspace.

We introduce a concept for these two spaces, which is called entropy order, and prove that it coincides with topological entropy of $(X,f)$. We also consider the entropy order of an invariant measure and a variational principle is established.

In this paper, for a family of second-order parabolic equation with rapidly oscillating and time-dependent periodic coefficients, we are interested in an approximate two-sphere one-cylinder inequality for these solutions in parabolic periodic homogenization, which implies an approximate quantitative propagation of smallness. The proof relies on the asymptotic behavior of fundamental solutions and the Lagrange interpolation technique.

We introduce the conormal fan of a matroid M, which is a Lagrangian analog of the Bergman fan of M. We use the conormal fan to give a Lagrangian interpretation of the Chern-Schwartz-MacPherson cycle of M. This allows us to express the h-vector of the broken circuit complex of M in terms of the intersection theory of the conormal fan of M. We also develop general tools for tropical Hodge theory to prove that the conormal fan satisfies Poincar'e duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. The Lagrangian interpretation of the Chern-Schwartz-MacPherson cycle of M, when combined with the Hodge-Riemann relations for the conormal fan of M, implies Brylawski's and Dawson's conjectures that the h-vectors of the broken circuit complex and the independence complex of M are log-concave sequences.

Applying the equivariant moving frames method, we construct convergent normal forms for real-analytic 5-dimensional totally nondegenerate CR submanifolds of C^4. These CR manifolds are divided into several biholomorphically inequivalent subclasses, each of which has its own complete normal form. Moreover it is shown that, biholomorphically, Beloshapka's cubic model is the unique member of this class with the maximum possible dimension seven of the corresponding algebra of infinitesimal CR automorphisms. Our results are also useful in the study of biholomorphic equivalence problem between CR manifolds, in question.

The $(q, mathbf{Q})$-current algebra associated with the general linear Lie algebra was introduced by the second author in the study of representation theory of cyclotomic $q$-Schur algebras. In this paper, we study the $(q, mathbf{Q})$-current algebra $U_q(mathfrak{sl}_n^{langle mathbf{Q} angle}[x])$ associated with the special linear Lie algebra $mathfrak{sl}_n$. In particular, we classify finite dimensional simple $U_q(mathfrak{sl}_n^{langle mathbf{Q} angle}[x])$-modules.

Consider that $u_0$ nodes are aware of some piece of data $d_0$. This note derives the expected time required for the data $d_0$ to be disseminated through-out a network of $n$ nodes, when communication between nodes evolves according to a graphical Markov model $overline{ mathcal{G}}_{n,hat{p}}$ with probability parameter $hat{p}$. In this model, an edge between two nodes exists at discrete time $k in mathbb{N}^+$ with probability $hat{p}$ if this edge existed at $k-1$, and with probability $(1-hat{p})$ if this edge did not exist at $k-1$. Each edge is interpreted as a bidirectional communication link over which data between neighbors is shared. The initial communication graph is assumed to be an Erdos-Renyi random graph with parameters $(n,hat{p})$, hence we consider a emph{stationary} Markov model $overline{mathcal{G}}_{n,hat{p}}$. The asymptotic "$u_0$-expected flooding time" of $overline{mathcal{G}}_{n,hat{p}}$ is defined as the expected number of iterations required to transmit the data $d_0$ from $u_0$ nodes to $n$ nodes, in the limit as $n$ approaches infinity. Although most previous results on the asymptotic flooding time in graphical Markov models are either emph{almost sure} or emph{with high probability}, the bounds obtained here are emph{in expectation}. However, our bounds are tighter and can be more complete than previous results.

We derive the non-relativistic $c oinfty$ and ultra-relativistic $c o 0$ limits of the $kappa$-deformed symmetries and corresponding spacetime in (3+1) dimensions, with and without a cosmological constant. We apply the theory of Lie bialgebra contractions to the Poisson version of the $kappa$-(A)dS quantum algebra, and quantize the resulting contracted Poisson-Hopf algebras, thus giving rise to the $kappa$-deformation of the Newtonian (Newton-Hooke and Galilei) and Carrollian (Para-Poincar'e, Para-Euclidean and Carroll) quantum symmetries, including their deformed quadratic Casimir operators. The corresponding $kappa$-Newtonian and $kappa$-Carrollian noncommutative spacetimes are also obtained as the non-relativistic and ultra-relativistic limits of the $kappa$-(A)dS noncommutative spacetime. These constructions allow us to analyze the non-trivial interplay between the quantum deformation parameter $kappa$, the curvature parameter $eta$ and the speed of light parameter $c$.

Inserting renewable energy in the electric grid in a decentralized manneris a key challenge of the energy transition. However, at local scale, both production and demand display erratic behavior, which makes it delicate to match them. It is the goal of Energy Management Systems (EMS) to achieve such balance at least cost. We present EMSx, a numerical benchmark for testing control algorithms for the management of electric microgrids equipped with a photovoltaic unit and an energy storage system. EMSx is made of three key components: the EMSx dataset, provided by Schneider Electric, contains a diverse pool of realistic microgrids with a rich collection of historical observations and forecasts; the EMSx mathematical framework is an explicit description of the assessment of electric microgrid control techniques and algorithms; the EMSx software EMSx.jl is a package, implemented in the Julia language, which enables to easily implement a microgrid controller and to test it. All components of the benchmark are publicly available, so that other researchers willing to test controllers on EMSx may reproduce experiments easily. Eventually, we showcase the results of standard microgrid control methods, including Model Predictive Control, Open Loop Feedback Control and Stochastic Dynamic Programming.

This paper considers the distributed nonconvex optimization problem of minimizing a global cost function formed by a sum of local cost functions by using local information exchange. We first propose a distributed first-order primal-dual algorithm. We show that it converges sublinearly to the stationary point if each local cost function is smooth and linearly to the global optimum under an additional condition that the global cost function satisfies the Polyak-{L}ojasiewicz condition. This condition is weaker than strong convexity, which is a standard condition for proving the linear convergence of distributed optimization algorithms, and the global minimizer is not necessarily unique or finite. Motivated by the situations where the gradients are unavailable, we then propose a distributed zeroth-order algorithm, derived from the proposed distributed first-order algorithm by using a deterministic gradient estimator, and show that it has the same convergence properties as the proposed first-order algorithm under the same conditions. The theoretical results are illustrated by numerical simulations.

Starting from the Hamiltonian formulation of ${cal N}{=},2$ supersymmetric Calogero models associated with the classical $A_n, B_n, C_n$ and $D_n$ series and their hyperbolic/trigonometric cousins, we provide their superspace description. The key ingredients include $n$ bosonic and $2n(n{-}1)$ fermionic ${cal N}{=},2$ superfields, the latter being subject to a nonlinear chirality constraint. This constraint has a universal form valid for all Calogero models. With its help we find more general supercharges (and a superspace Lagrangian), which provide the ${cal N}{=},2$ supersymmetrization for bosonic potentials with arbitrary repulsive two-body interactions.

Yang-Mills gauge theory on Minkowski space supports a Batalin-Vilkovisky-infinity algebra structure, all whose operations are local. To make this work, the axioms for a BV-infinity algebra are deformed by a quadratic element, here the Minkowski wave operator. This homotopy structure implies BCJ/color-kinematics duality; a cobar construction yields a strict algebraic structure whose Feynman expansion for Yang-Mills tree amplitudes complies with the duality. It comes with a `syntactic kinematic algebra'.

We study the dependence of the first eigenvalue of the Finsler $p$-Laplacian and the corresponding eigenfunctions upon perturbation of the domain and we generalize a few results known for the standard $p$-Laplacian. In particular, we prove a Frech'{e}t differentiability result for the eigenvalues, we compute the corresponding Hadamard formulas and we prove a continuity result for the eigenfunctions. Finally, we briefly discuss a well-known overdetermined problem and we show how to deduce the Rellich-Pohozaev identity for the Finsler $p$-Laplacian from the Hadamard formula.

This work has been motivated by recent papers that quantify the density of values of generic quadratic forms and other polynomials at integer points, in particular ones that use Rogers' second moment estimates. In this paper we establish such results in a very general framework. Namely, given any subhomogeneous function (a notion to be defined) $f: mathbb{R}^n o mathbb{R}$, we derive a necessary and sufficient condition on the approximating function $psi$ for guaranteeing that a generic element $fcirc g$ in the $G$-orbit of $f$ is $psi$-approximable; that is, $|fcirc g(mathbf{v})| le psi(|mathbf{v}|)$ for infinitely many $mathbf{v} in mathbb{Z}^n$. We also deduce a sufficient condition in the case of uniform approximation. Here, $G$ can be any closed subgroup of $operatorname{ASL}_n(mathbb{R})$ satisfying certain axioms that allow for the use of Rogers-type estimates.

We prove the following result: Let $(M,g_0)$ be a compact manifold of dimension $ngeq 12$ with positive isotropic curvature. Then $M$ is diffeomorphic to a spherical space form, or a compact quotient manifold of $mathbb{S}^{n-1} imes mathbb{R}$ by diffeomorphisms, or a connected sum of a finite number of such manifolds. This extends a recent work of Brendle, and implies a conjecture of Schoen in dimensions $ngeq 12$. The proof uses Ricci flow with surgery on compact orbifolds with isolated singularities.

This paper addresses the problem of identifying the graph structure of a dynamical network using measured input/output data. This problem is known as topology identification and has received considerable attention in recent literature. Most existing literature focuses on topology identification for networks with node dynamics modeled by single integrators or single-input single-output (SISO) systems. The goal of the current paper is to identify the topology of a more general class of heterogeneous networks, in which the dynamics of the nodes are modeled by general (possibly distinct) linear systems. Our two main contributions are the following. First, we establish conditions for topological identifiability, i.e., conditions under which the network topology can be uniquely reconstructed from measured data. We also specialize our results to homogeneous networks of SISO systems and we will see that such networks have quite particular identifiability properties. Secondly, we develop a topology identification method that reconstructs the network topology from input/output data. The solution of a generalized Sylvester equation will play an important role in our identification scheme.

In this work, we study dynamic properties of classical solutions to a homogenous Neumann initial-boundary value problem (IBVP) for a two-species and two-stimuli chemotaxis model with/without chemical signalling loop in a 2D bounded and smooth domain. We successfully detect the product of two species masses as a feature to determine boundedness, gradient estimates, blow-up and $W^{j,infty}(1leq jleq 3)$-exponential convergence of classical solutions for the corresponding IBVP. More specifically, we first show generally a smallness on the product of both species masses, thus allowing one species mass to be suitably large, is sufficient to guarantee global boundedness, higher order gradient estimates and $W^{j,infty}$-convergence with rates of convergence to constant equilibria; and then, in a special case, we detect a straight line of masses on which blow-up occurs for large product of masses. Our findings provide new understandings about the underlying model, and thus, improve and extend greatly the existing knowledge relevant to this model.

In a multitype branching process, it is assumed that immigrants arrive according to a nonhomogeneous Poisson or a generalized Polya process (both processes are formulated as a nonhomogeneous birth process with an appropriate choice of transition intensities). We show that the renormalized numbers of objects of the various types alive at time $t$ for supercritical, critical, and subcritical cases jointly converge in distribution under those two different arrival processes. Furthermore, some transient moment analysis when there are only two types of particles is provided. AMS 2000 subject classifications: Primary 60J80, 60J85; secondary 60K10, 60K25, 90B15.

This work presents a novel data-driven framework for constructing eigenfunctions of the Koopman operator geared toward prediction and control. The method leverages the richness of the spectrum of the Koopman operator away from attractors to construct a rich set of eigenfunctions such that the state (or any other observable quantity of interest) is in the span of these eigenfunctions and hence predictable in a linear fashion. The eigenfunction construction is optimization-based with no dictionary selection required. Once a predictor for the uncontrolled part of the system is obtained in this way, the incorporation of control is done through a multi-step prediction error minimization, carried out by a simple linear least-squares regression. The predictor so obtained is in the form of a linear controlled dynamical system and can be readily applied within the Koopman model predictive control framework of [12] to control nonlinear dynamical systems using linear model predictive control tools. The method is entirely data-driven and based purely on convex optimization, with no reliance on neural networks or other non-convex machine learning tools. The novel eigenfunction construction method is also analyzed theoretically, proving rigorously that the family of eigenfunctions obtained is rich enough to span the space of all continuous functions. In addition, the method is extended to construct generalized eigenfunctions that also give rise Koopman invariant subspaces and hence can be used for linear prediction. Detailed numerical examples with code available online demonstrate the approach, both for prediction and feedback control.

We study the geometry and topology of exotic Springer fibers for orbits corresponding to one-row bipartitions from an explicit, combinatorial point of view. This includes a detailed analysis of the structure of the irreducible components and their intersections as well as the construction of an explicit affine paving. Moreover, we compute the ring structure of cohomology by constructing a CW-complex homotopy equivalent to the exotic Springer fiber. This homotopy equivalent space admits an action of the type C Weyl group inducing Kato's original exotic Springer representation on cohomology. Our results are described in terms of the diagrammatics of the one-boundary Temperley-Lieb algebra (also known as the blob algebra). This provides a first step in generalizing the geometric versions of Khovanov's arc algebra to the exotic setting.

The two-dimensional directed spanning forest (DSF) introduced by Baccelli and Bordenave is a planar directed forest whose vertex set is given by a homogeneous Poisson point process $mathcal{N}$ on $mathbb{R}^2$. If the DSF has direction $-e_y$, the ancestor $h(u)$ of a vertex $u in mathcal{N}$ is the nearest Poisson point (in the $L_2$ distance) having strictly larger $y$-coordinate. This construction induces complex geometrical dependencies. In this paper we show that the collection of DSF paths, properly scaled, converges in distribution to the Brownian web (BW). This verifies a conjecture made by Baccelli and Bordenave in 2007.

The article is devoted to the expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity $k$ $(kinmathbb{N})$ based on the generalized iterated Fourier series. The case of Fourier-Legendre series as well as the case of trigonotemric Fourier series are considered in details. The obtained expansion provides a possibility to represent the iterated Stratonovich stochastic integral in the form of iterated series of products of standard Gaussian random variables. Convergence in the mean of degree $2n$ $(nin mathbb{N})$ of the expansion is proved. Some modifications of the mentioned expansion were derived for the case $k=2$. One of them is based of multiple trigonomentric Fourier series converging almost everywhere in the square $[t, T]^2$. The results of the article can be applied to the numerical solution of Ito stochastic differential equations.

High dimensional expanders is a vibrant emerging field of study. Nevertheless, the only known construction of bounded degree high dimensional expanders is based on Ramanujan complexes, whereas one dimensional bounded degree expanders are abundant.

In this work, we construct new families of bounded degree high dimensional expanders obeying the local spectral expansion property. This property has a number of important consequences, including geometric overlapping, fast mixing of high dimensional random walks, agreement testing and agreement expansion. Our construction also yields new families of expander graphs which are close to the Ramanujan bound, i.e., their spectral gap is close to optimal.

The construction is quite elementary and it is presented in a self contained manner; This is in contrary to the highly involved previously known construction of the Ramanujan complexes. The construction is also very symmetric (such symmetry properties are not known for Ramanujan complexes) ; The symmetry of the construction could be used, for example, in order to obtain good symmetric LDPC codes that were previously based on Ramanujan graphs.

The main tool that we use for is the theory of coset geometries. Coset geometries arose as a tool for studying finite simple groups. Here, we show that coset geometries arise in a very natural manner for groups of elementary matrices over any finitely generated algebra over a commutative unital ring. In other words, we show that such groups act simply transitively on the top dimensional face of a pure, partite, clique complex.

We study flows on C*-algebras with the Rokhlin property. We show that every Kirchberg algebra carries a unique Rokhlin flow up to cocycle conjugacy, which confirms a long-standing conjecture of Kishimoto. We moreover present a classification theory for Rokhlin flows on C*-algebras satisfying certain technical properties, which hold for many C*-algebras covered by the Elliott program. As a consequence, we obtain the following further classification theorems for Rokhlin flows. Firstly, we extend the statement of Kishimoto's conjecture to the non-simple case: Up to cocycle conjugacy, a Rokhlin flow on a separable, nuclear, strongly purely infinite C*-algebra is uniquely determined by its induced action on the prime ideal space. Secondly, we give a complete classification of Rokhlin flows on simple classifiable $KK$-contractible C*-algebras: Two Rokhlin flows on such a C*-algebra are cocycle conjugate if and only if their induced actions on the cone of lower-semicontinuous traces are affinely conjugate.

In this review paper, we survey Hardy type inequalities from the point of view of Folland and Stein's homogeneous groups. Particular attention is paid to Hardy type inequalities on stratified groups which give a special class of homogeneous groups. In this environment, the theory of Hardy type inequalities becomes intricately intertwined with the properties of sub-Laplacians and more general subelliptic partial differential equations. Particularly, we discuss the Badiale-Tarantello conjecture and a conjecture on the geometric Hardy inequality in a half-space of the Heisenberg group with a sharp constant.

Positive geometries encode the physics of scattering amplitudes in flat space-time and the wavefunction of the universe in cosmology for a large class of models. Their unique canonical forms, providing such quantum mechanical observables, are characterised by having only logarithmic singularities along all the boundaries of the positive geometry. However, physical observables have logarithmic singularities just for a subset of theories. Thus, it becomes crucial to understand whether a similar paradigm can underlie their structure in more general cases. In this paper we start a systematic investigation of a geometric-combinatorial characterisation of differential forms with non-logarithmic singularities, focusing on projective polytopes and related meromorphic forms with multiple poles. We introduce the notions of covariant forms and covariant pairings. Covariant forms have poles only along the boundaries of the given polytope; moreover, their leading Laurent coefficients along any of the boundaries are still covariant forms on the specific boundary. Whereas meromorphic forms in covariant pairing with a polytope are associated to a specific (signed) triangulation, in which poles on spurious boundaries do not cancel completely, but their order is lowered. These meromorphic forms can be fully characterised if the polytope they are associated to is viewed as the restriction of a higher dimensional one onto a hyperplane. The canonical form of the latter can be mapped into a covariant form or a form in covariant pairing via a covariant restriction. We show how the geometry of the higher dimensional polytope determines the structure of these differential forms. Finally, we discuss how these notions are related to Jeffrey-Kirwan residues and cosmological polytopes.

Let $X$ (resp. $Y$) be a curve of genus 1 (resp. 2) over a base field $k$ whose characteristic does not equal 2. We give criteria for the existence of a curve $Z$ over $k$ whose Jacobian is up to twist (2,2,2)-isogenous to the products of the Jacobians of $X$ and $Y$. Moreover, we give algorithms to construct the curve $Z$ once equations for $X$ and $Y$ are given. The first of these involves the use of hyperplane sections of the Kummer variety of $Y$ whose desingularization is isomorphic to $X$, whereas the second is based on interpolation methods involving numerical results over $mathbb{C}$ that are proved to be correct over general fields a posteriori. As an application, we find a twist of a Jacobian over $mathbb{Q}$ that admits a rational 70-torsion point.

In cite{CC} the authors, investigating a model of population dynamics, find the following result. Let $Omegasubset mathbb{R}^N$, $Ngeq 1$, be a bounded smooth domain. The weighted eigenvalue problem $-Delta u =lambda m u $ in $Omega$ under homogeneous Dirichlet boundary conditions, where $lambda in mathbb{R}$ and $min L^infty(Omega)$, is considered. The authors prove the existence of minimizers $check m$ of the principal positive eigenvalue $lambda_1(m)$ when $m$ varies in a class $mathcal{M}$ of functions where average, maximum, and minimum values are given. A similar result is obtained in cite{CCP} when $m$ is in the class $mathcal{G}(m_0)$ of rearrangements of a fixed $m_0in L^infty(Omega)$. In our work we establish that, if $Omega$ is Steiner symmetric, then every minimizer in cite{CC,CCP} inherits the same kind of symmetry.

We present a systematic derivation of the abelianity conditions for the $q$-deformed $W$-algebras constructed from the elliptic quantum algebra $mathcal{A}_{q,p}(widehat{gl}(N)_{c})$. We identify two sets of conditions on a given critical surface yielding abelianity lines in the moduli space ($p, q, c$). Each line is identified as an intersection of a countable number of critical surfaces obeying diophantine consistency conditions. The corresponding Poisson brackets structures are then computed for which some universal features are described.

Axial algebras are non-associative algebras generated by semisimple idempotents whose adjoint actions obey a fusion law. Axial algebras that are generated by two such idempotents play a crucial role in the theory. We classify all primitive 2-generated axial algebras whose fusion laws have two eigenvalues and all graded primitive 2-generated axial algebras whose fusion laws have three eigenvalues. This represents a significant broadening in our understanding of axial algebras.

Let $Gamma subset operatorname{PU}(1,n)$ be a lattice, and $S_Gamma$ the associated ball quotient. We prove that, if $S_Gamma$ contains infinitely many maximal totally geodesic subvarieties, then $Gamma$ is arithmetic. We also prove an Ax-Schanuel Conjecture for $S_Gamma$, similar to the one recently proven by Mok, Pila and Tsimerman. One of the main ingredients in the proofs is to realise $S_Gamma$ inside a period domain for polarised integral variations of Hodge structures and interpret totally geodesic subvarieties as unlikely intersections.

We propose a general procedure to construct noncommutative deformations of an algebraic submanifold $M$ of $mathbb{R}^n$, specializing the procedure [G. Fiore, T. Weber, Twisted submanifolds of $mathbb{R}^n$, arXiv:2003.03854] valid for smooth submanifolds. We use the framework of twisted differential geometry of [Aschieri et al.,Class. Quantum Gravity 23 (2006), 1883], whereby the commutative pointwise product is replaced by the $star$-product determined by a Drinfel'd twist. We actually simultaneously construct noncommutative deformations of all the algebraic submanifolds $M_c$ that are level sets of the $f^a(x)$, where $f^a(x)=0$ are the polynomial equations solved by the points of $M$, employing twists based on the Lie algebra $Xi_t$ of vector fields that are tangent to all the $M_c$. The twisted Cartan calculus is automatically equivariant under twisted $Xi_t$. If we endow $mathbb{R}^n$ with a metric, then twisting and projecting to normal or tangent components commute, projecting the Levi-Civita connection to the twisted $M$ is consistent, and in particular a twisted Gauss theorem holds, provided the twist is based on Killing vector fields. Twisted algebraic quadrics can be characterized in terms of generators and $star$-polynomial relations. We explicitly work out deformations based on abelian or Jordanian twists of all quadrics in $mathbb{R}^3$ except ellipsoids, in particular twisted cylinders embedded in twisted Euclidean $mathbb{R}^3$ and twisted hyperboloids embedded in twisted Minkowski $mathbb{R}^3$ [the latter are twisted (anti-)de Sitter spaces $dS_2,AdS_2$].

In this article, we prove that if $R$ is a finitely generated ring over $mathbb{Z}$ of dimension $d, dgeq2, frac{1}{d!}in R$, then any unimodular row over $R[X]$ of length $d+1$ can be mapped to a factorial row by elementary transformations.

We consider two special cases of the connection problem for the second Painlev'e equation (PII) using the method of uniform asymptotics proposed by Bassom et al.. We give a classification of the real solutions of PII on the negative (positive) real axis with respect to their initial data. By product, a rigorous proof of a property associate with the nonlinear eigenvalue problem of PII on the real axis, recently revealed by Bender and Komijani, is given by deriving the asymptotic behavior of the Stokes multipliers.

In this paper, we derive a formula for the sums of powers of the first $n$ positive integers that involves the hyperharmonic numbers and the Stirling numbers of the second kind. Then, using an explicit representation for the hyperharmonic numbers, we generalize this formula to the sums of powers of an arbitrary arithmetic progression. Moreover, as a by-product, we express the Bernoulli polynomials in terms of the hyperharmonic polynomials and the Stirling numbers of the second kind.

This is the second of two papers in which we construct formal power series solutions in external parameters to the vacuum Einstein equations, implementing one bounce for the Belinskii-Khalatnikov-Lifshitz (BKL) proposal for spatially inhomogeneous spacetimes. Here we show that spatially inhomogeneous perturbations of spatially homogeneous elements are unobstructed. A spectral sequence for a filtered complex, and a homological contraction based on gauge-fixing, are used to do this.

We construct a smooth moduli stack of tuples consisting of genus two nodal curves, line bundles, and twisted fields. It leads to a desingularization of the moduli of genus two stable maps to projective spaces. The construction of this new moduli is based on systematical application of the theory of stacks with twisted fields (STF), which has its prototype appeared in arXiv:1906.10527 and arXiv:1201.2427 and is fully developed in this article. The results of this article are the second step of a series of works toward the resolutions of the moduli of stable maps of higher genera.

We give a simple proof of the strong maximum principle for viscosity subsolutions of fully nonlinear elliptic PDEs on the form $$ F(x,u,Du,D^2u) = 0 $$ under suitable structure conditions on the equation allowing for non-Lipschitz growth in the gradient terms. In case of smooth boundaries, we also prove the Hopf lemma, the boundary Harnack inequality and that positive viscosity solutions vanishing on a portion of the boundary are comparable with the distance function near the boundary. Our results apply to weak solutions of an eigenvalue problem for the variable exponent $p$-Laplacian.

We observe that the DeTurck Laplacian flow of G2-structures introduced by Bryant and Xu as a gauge fixing of the Laplacian flow can be regarded as a flow of G2-structures (not necessarily closed) which fits in the general framework introduced by Hamilton in [4].

Let $Gamma$ be a discrete group acting properly discontinuously and isometrically on the three-dimensional anti-de Sitter space $mathrm{AdS}^{3}$, and $square$ the Laplacian which is a second-order hyperbolic differential operator. We study linear independence of a family of generalized Poincar'{e} series introduced by Kassel-Kobayashi [Adv. Math. 2016], which are defined by the $Gamma$-average of certain eigenfunctions on $mathrm{AdS}^{3}$. We prove that the multiplicities of $L^{2}$-eigenvalues of the hyperbolic Laplacian $square$ on $Gammaackslashmathrm{AdS}^{3}$ are unbounded when $Gamma$ is finitely generated. Moreover, we prove that the multiplicities of extit{stable $L^{2}$-eigenvalues} for compact anti-de Sitter $3$-manifolds are unbounded.

Our goal in this article is to study the global Lorentz estimates for gradient of weak solutions to $p$-Laplace double obstacle problems involving the Schr"odinger term: $-Delta_p u + mathbb{V}|u|^{p-2}u$ with bound constraints $psi_1 le u le psi_2$ in non-smooth domains. This problem has its own interest in mathematics, engineering, physics and other branches of science. Our approach makes a novel connection between the study of Calder'on-Zygmund theory for nonlinear Schr"odinger type equations and variational inequalities for double obstacle problems.

In information theory, the so-called log-sum inequality is fundamental and a kind of generalization of the non-nagativity for the relative entropy. In this paper, we show the generalized log-sum inequality for two functions defined for scalars. We also give a new result for commutative matrices. In addition, we demonstrate further results for general non-commutative positive semi-definite matrices.

Let $Nge 2$ and $ hoin(0,1/N^2]$. The homogenous Cantor set $E$ is the self-similar set generated by the iterated function system

[

left{f_i(x)= ho x+frac{i(1- ho)}{N-1}: i=0,1,ldots, N-1 ight}.

]

Let $s=dim_H E$ be the Hausdorff dimension of $E$, and let $mu=mathcal H^s|_E$ be the $s$-dimensional Hausdorff measure restricted to $E$. In this paper we describe, for each $xin E$, the pointwise lower $s$-density $Theta_*^s(mu,x)$ and upper $s$-density $Theta^{*s}(mu, x)$ of $mu$ at $x$. This extends some early results of Feng et al. (2000). Furthermore, we determine two critical values $a_c$ and $b_c$ for the sets

[

E_*(a)=left{xin E: Theta_*^s(mu, x)ge a ight}quad extrm{and}quad E^*(b)=left{xin E: Theta^{*s}(mu, x)le b ight}

] respectively, such that $dim_H E_*(a)>0$ if and only if $a<a_c$, and that $dim_H E^*(b)>0$ if and only if $b>b_c$. We emphasize that both values $a_c$ and $b_c$ are related to the Thue-Morse type sequences, and our strategy to find them relies on ideas from open dynamics and techniques from combinatorics on words.