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Spectral curves, variational problems, and the hermitian matrix model with external sourceJul 18 2019We consider a hermitian matrix model with external source and general polynomial potential, when the source has two distinct eigenvalues but is otherwise arbitrary. All such models studied so far have a common feature: an associated cubic equation (spectral ... More

An equilibrium problem on the sphere with two equal chargesJul 10 2019We study the equilibrium measure on the two dimensional sphere in the presence of an external field generated by two equal point charges. The support of the equilibrium measure is known as the droplet. Brauchart et al. showed that the complement of the ... More

Broken Detailed Balance and Non-Equilibrium Dynamics in Noisy Social Learning ModelsJun 27 2019We propose new Degroot-type social learning models with feedback in a continuous time, to investigate the effect of a noisy information source on consensus formation in a social network. Unlike the standard Degroot framework, noisy information models ... More

Some $n$-space $q$-binomial theorem extensions and similar identitiesJun 18 2019We give an $n$-space generalized $q$-binomial theorem, and some new $q$ series identities that resemble the traditional $q$ series partition generating functions. These identities enumerate stepping stone weighted vector partitions.

Well-posedness of compressible magneto-micropolar fluid equationsJun 17 2019Jun 18 2019We are concerned with compressible magneto-micropolar fluid equations (1.1)-(1.2). The global existence and large time behaviour of solutions near a constant state to the magneto-micropolar-Navier-Stokes-Poisson (MMNSP) system is investigated in $\mathbb{R}^3$. ... More

Well-posedness of compressible magneto-micropolar fluid equationsJun 17 2019We are concerned with compressible magneto-micropolar fluid equations (1.1)-(1.2). The global existence and large time behaviour of solutions near a constant state to the magneto-micropolar-Navier-Stokes-Poisson (MMNSP) system is investigated in $\mathbb{R}^3$. ... More

The $h^*$-polynomials of locally anti-blocking lattice polytopes and their $γ$-positivityJun 11 2019A lattice polytope $\mathcal{P} \subset \mathbb{R}^d$ is called a locally anti-blocking polytope if for any closed orthant $\mathbb{R}^d_{\varepsilon}$ in $\mathbb{R}^d$, $\mathcal{P} \cap \mathbb{R}^d_{\varepsilon}$ is unimodularly equivalent to an anti-blocking ... More

Note on an eigenvalue problem with applications to a Minkowski type regularity problem in $\mathbb{R}$^nJun 04 2019We consider existence and uniqueness of homogeneous solutions $ u > 0 $ to certain PDE of $p$-Laplace type, $ p $ fixed, $ n - 1 <p< \infty, n \geq 2, $ when $ u $ is a solution in $K(\alpha)\subset\mathbb{R}^n$ where \[ K (\alpha) := \{ x = (x_1, \dots, ... More

Harmonic Measure and the Analyst's Traveling Salesman TheoremMay 22 2019We study how generalized Jones $\beta$-numbers relate to harmonic measure. Firstly, we generalize a result of Garnett, Mourgoglou and Tolsa by showing that domains in $\mathbb{R}^{d+1}$ whose boundaries are lower $d$-content regular admit Corona decompositions ... More

Harmonic Measure and the Analyst's Traveling Salesman TheoremMay 22 2019Jul 08 2019We study how generalized Jones $\beta$-numbers relate to harmonic measure. Firstly, we generalize a result of Garnett, Mourgoglou and Tolsa by showing that domains in $\mathbb{R}^{d+1}$ whose boundaries are lower $d$-content regular admit Corona decompositions ... More

Sharp estimate on the inner distance in planar domainsMay 20 2019We show that the inner distance inside a bounded planar domain is at most the one-dimensional Hausdorff measure of the boundary of the domain. We prove this sharp result by establishing an improved Painlev\'e length estimate for connected sets and by ... More

Formation of Singularities and Existence of Global Continuous Solutions for the Compressible Euler EquationsMay 19 2019We are concerned with the formation of singularities and the existence of global continuous solutions of the Cauchy problem for the one-dimensional non-isentropic Euler equations for compressible fluids. For the isentropic Euler equations, we pinpoint ... More

Finite time stability for the Riemann problem with extremal shocks for a large class of hyperbolic systemsMay 10 2019In this paper on hyperbolic systems of conservation laws in one space dimension, we give a complete picture of stability for all solutions to the Riemann problem which contain only extremal shocks. We study stability of the Riemann problem amongst a large ... More

Criteria for the a-contraction and stability for the piecewise-smooth solutions to hyperbolic balance lawsApr 20 2019We show uniqueness and stability in $L^2$ and for all time for piecewise-smooth solutions to hyperbolic balance laws. We have in mind applications to gas dynamics, the isentropic Euler system and the full Euler system for a polytropic gas in particular. ... More

Plans on measures and AM-modulusApr 09 2019For measuring families of curves, or, more generally, of measures, $M_p$-modulus is traditionally used. More recent studies use so-called plans on measures. In their fundamental paper [4], Ambrosio, Di Marino and Savar\'e proved that these two approaches ... More

A Kaczmarz algorithm for sequences of projections, infinite products, and applications to frames in IFS $L^{2}$ spacesApr 09 2019We show that an idea, originating initially with a fundamental recursive iteration scheme (usually referred as "the" Kaczmarz algorithm), admits important applications in such infinite-dimensional, and non-commutative, settings as are central to spectral ... More

Clark measures on the complex sphereApr 08 2019Let $B_d$ denote the unit ball of $\mathbb{C}^d$, $d\ge 1$. Given a holomorphic function $\varphi: B_d \to B_1$, we study the corresponding family $\sigma_\alpha[\varphi]$, $\alpha\in\partial B_1$, of Clark measures on the unit sphere $\partial B_d$. ... More

Well-Rounded Lattices via PolynomialsApr 06 2019Well-rounded lattices have been a topic of recent studies with applications in wiretap channels and in cryptography. A lattice of full rank in Euclidean space is called well-rounded if its set of minimal vectors spans the whole space. In this paper, we ... More

The two-phase problem for harmonic measure in VMO via jump formulas for the Riesz transformApr 01 2019May 09 2019Let $\Omega^+\subset\mathbb R^{n+1}$ be an NTA domain and let $\Omega^-= \mathbb R^{n+1}\setminus \overline{\Omega^+}$ be an NTA domain as well. Denote by $\omega^+$ and $\omega^-$ their respective harmonic measures. Assume that $\Omega^+$ is a $\delta$-Reifenberg ... More

The two-phase problem for harmonic measure in VMO via jump formulas for the Riesz transformApr 01 2019Let $\Omega^+\subset\mathbb R^{n+1}$ be an NTA domain and let $\Omega^-= \mathbb R^{n+1}\setminus \overline{\Omega^+}$ be an NTA domain as well. Denote by $\omega^+$ and $\omega^-$ their respective harmonic measures. Assume that $\Omega^+$ is a $\delta$-Reifenberg ... More

The two-phase problem for harmonic measure in VMOApr 01 2019Jul 17 2019Let $\Omega^+\subset\mathbb R^{n+1}$ be an NTA domain and let $\Omega^-= \mathbb R^{n+1}\setminus \overline{\Omega^+}$ be an NTA domain as well. Denote by $\omega^+$ and $\omega^-$ their respective harmonic measures. Assume that $\Omega^+$ is a $\delta$-Reifenberg ... More

Recurrent Trajectories and Finite Critical Trajectories of Quadratic Differentials on the Riemann SphereMar 08 2019In this paper, the focus will be on both the existence and non-existence respectively of finite critical trajectories and recurrent trajectories of a quadratic differential on the Riemann sphere. We show the connection between these two items. More precisely, ... More

A logarithmic estimate for inverse source scattering problem with attenuation in a two-layered mediumMar 08 2019The paper aims a logarithmic stability estimate for the inverse source problem of the one-dimensional Helmholtz equation with attenuation factor in a two layer medium. We establish a stability by using multiple frequencies at the two end points of the ... More

A sequence of polynomials with optimal condition numberMar 04 2019We find an explicit sequence of univariate polynomials of arbitrary degree with optimal condition number. This solves a problem posed by Michael Shub and Stephen Smale in 1993.

Balayage of measures and subharmonic functions on a system of rays. III. Growth of entire functions of exponential typeMar 03 2019In the second part of this work was developed a technique of balayage of finite genus $q=0,1,2,\dots$ for measures (charges) and ($\delta$-) subharmonic functions of finite order to an arbitrary closed system of rays $S$ with vertex at origin on the complex ... More

Fatou-Type Theorems and Boundary Value Problems for Elliptic Systems in the Upper Half-SpaceFeb 21 2019We survey recent progress in a program aimed at proving general Fatou-type results and establishing the well-posedness of a variety of boundary value problems in the upper half-space ${\mathbb{R}}^n_{+}$ for second-order, homogeneous, constant complex ... More

Some Properties of Thinness and Fine Topology with Relative CapacityFeb 14 2019In this paper, we introduce a thinness in sense to a type of relative capacity for weighted variable exponent Sobolev space. Moreover, we reveal some properties of this thinness and consider the relationship with finely open and finely closed sets. We ... More

On some properties of relative capacity and thinness in weighted variable exponent Sobolev spacesFeb 13 2019In this paper, we define weighted relative $p(.)$-capacity and discuss properties of capacity in the space $W_{\vartheta }^{1,p(.)}(\mathbb{R}^{n}).$ Also, we investigate some properties of weighted variable Sobolev capacity. It is shown that there is ... More

Superposition, reduction of multivariable problems, and approximationFeb 07 2019We study reduction schemes for functions of "many" variables into system of functions in one variable. Our setting includes infinite-dimensions. Following Cybenko-Kolmogorov, the outline for our results is as follows: We present explicit reductions schemes ... More

Superposition, reduction of multivariable problems, and approximationFeb 07 2019Mar 06 2019We study reduction schemes for functions of "many" variables into system of functions in one variable. Our setting includes infinite-dimensions. Following Cybenko-Kolmogorov, the outline for our results is as follows: We present explicit reductions schemes ... More

Risk-Averse Models in Bilevel Stochastic Linear ProgrammingJan 31 2019We consider bilevel linear problems, where some parameters are stochastic, and the leader has to decide in a here-and-now fashion, while the follower has complete information. In this setting, the leader's outcome can be modeled by a random variable, ... More

Lemniscates as Trajectories of Quadratic Differentials (I)Jan 17 2019In this note, we study polynomial and rational lemniscates as trajectories of related quadratic differentials. Many classic results can be then proved easily...

The closed range property for the $\overline{\partial}$-operator on planar domainsJan 14 2019Let $\Omega\subset\mathbb{C}$ be an open set. We show that $\overline{\partial}$ has closed range in $L^{2}(\Omega)$ if and only if the Poincar\'e-Dirichlet inequality holds. Moreover, we give necessary and sufficient potential-theoretic conditions for ... More

A first approach to learning a best basis for gravitational field modellingJan 14 2019Jan 15 2019Gravitational field modelling is an important tool for inferring past and present dynamic processes of the Earth. Functions on the sphere such as the gravitational potential are usually expanded in terms of either spherical harmonics or radial basis functions ... More

Instability of unidirectional flows for the 2D $α$-Euler equationsJan 05 2019Jan 21 2019We study stability of unidirectional flows for the linearized 2D $\alpha$-Euler equations on the torus. The unidirectional flows are steady states whose vorticity is given by Fourier modes corresponding to a vector $\mathbf p \in \mathbb Z^{2}$. We linearize ... More

On the Distribution of Zero Sets of Holomorphic Functions. IV. A CriterionDec 31 2018Let $D$ be a proper domain in the extended complex plane ${\mathbb C}_{\infty}:={\mathbb C}\cup \{\infty\}$, $M=M_+-M_-\not\equiv \pm \infty$ be a difference of non-trivial subharmonic functions $M_{\pm}\not\equiv \mp \infty$ on $D$, $\text{Hol}(D,M)$ ... More

Decomposition of Gaussian processes, and factorization of positive definite kernelsDec 28 2018We establish a duality for two factorization questions, one for general positive definite (p.d) kernels $K$, and the other for Gaussian processes, say $V$. The latter notion, for Gaussian processes is stated via Ito-integration. Our approach to factorization ... More

Enriched chain polytopesDec 05 2018Jun 17 2019Stanley introduced a lattice polytope $\mathcal{C}_P$ arising from a finite poset $P$, which is called the chain polytope of $P$. The geometric structure of $\mathcal{C}_P$ has good relations with the combinatorial structure of $P$. In particular, the ... More

Enriched chain polytopesDec 05 2018Stanley introduced a lattice polytope $\mathcal{C}_P$ arising from a finite poset $P$, which is called the chain polytope of $P$. The geometric structure of $\mathcal{C}_P$ has good relations with the combinatorial structure of $P$. In particular, the ... More

Bernstein Polynomial Inequality on a Compact Subset of the Real LineNov 30 2018We prove an analogue of the classical Bernstein polynomial inequality on a compact subset $E$ of the real line. The Lipschitz continuity of the Green function for the complement of $E$ with respect to the extended complex plane and the differentiability ... More

On $C^1$-approximability of functions by solutions of second order elliptic equations on plane compact sets and $C$-analytic capacityNov 15 2018Criteria for approximability of functions by solutions of homogeneous second order elliptic equations (with constant complex coefficients) in the norms of the Whitney $C^1$-spaces on compact sets in $\mathbb R^2$ are obtained in terms of the respective ... More

Bi-parameter Potential theory and Carleson measures for the Dirichlet space on the bidiscNov 12 2018Dec 22 2018We characterize the Carleson measures for the Dirichlet space on the bidisc, hence also its multiplier space. Following Maz'ya and Stegenga, the characterization is given in terms of a capacitary condition. We develop the foundations of a bi-parameter ... More

Dimension drop for harmonic measure on Ahlfors regular boundariesNov 09 2018Jun 25 2019We show that given a domain $\Omega\subseteq \mathbb{R}^{d+1}$ with uniformly non-flat Ahlfors $s$-regular boundary and $s\geq d$, the dimension of its harmonic measure is strictly less than $s$.

Dimension drop for harmonic measure on Ahlfors regular boundariesNov 09 2018Dec 17 2018We show that given a domain $\Omega\subseteq \mathbb{R}^{d+1}$ with uniformly non-flat Ahlfors $s$-regular boundary and $s\geq d$, the dimension of its harmonic measure is strictly less than $s$.

Zeros of holomorphic functions in the unit disk and $ρ$-trigonometrically convex functionsNov 03 2018Let $M$\/ be a subharmonic function with Riesz measure $\mu_M$ on the unit disk $\mathbb D$ in the complex plane $\mathbb C$. Let $f$ be a nonzero holomorphic function on $\mathbb D$ such that $f$ vanishes on ${\sf Z}\subset \mathbb D$, and satisfies ... More

Balayage of measures and subharmonic functions on a system of rays. II. Balayage of finite genus and regularity of growth on one rayNov 03 2018We expand the classical balayage of measures and subharmonic functions on a system of rays $S$ with a common origin on the complex plane $\mathbb C$. This allows for an arbitrary subharmonic function $v$ of finite order on $\mathbb C$ build $\delta$-subharmonic ... More

Balayage of measures and subharmonic functions on a system of rays. I. Classic caseNov 03 2018We develop classical balayage (sweeping) measures and subharmonic functions on the ray system $S$ with a general origin on the complex plane $\mathbb C$. This allows for a subharmonic function $v$ on $\mathbb C$ to construct also a subharmonic function ... More

Some properties related to trace inequalities for the multi-parameter Hardy operators on poly-treesNov 02 2018In this note we investigate the multi-parameter Potential Theory on the weighted $d$-tree (Cartesian product of several copies of uniform dyadic tree), which is connected to the discrete models of weighted Dirichlet spaces on the polydisc. We establish ... More

Reflexive polytopes arising from bipartite graphs with $γ$-positivity associated to interior polynomialsOct 29 2018Apr 27 2019In this paper, we introduce polytopes ${\mathcal B}_G$ arising from root systems $B_n$ and finite graphs $G$, and study their combinatorial and algebraic properties. In particular, it is shown that ${\mathcal B}_G$ is a reflexive polytope with a regular ... More

Reflexive polytopes arising from bipartite graphs with $γ$-positivity associated to interior polynomialsOct 29 2018Nov 01 2018In this paper, we introduce polytopes ${\mathcal B}_G$ arising from root systems $B_n$ and finite graphs $G$, and study their combinatorial and algebraic properties. In particular, it is shown that ${\mathcal B}_G$ is a reflexive polytope with a regular ... More

Sensitivity of $\ell_{1}$ minimization to parameter choiceOct 29 2018Apr 02 2019The use of generalized LASSO is a common technique for recovery of structured high-dimensional signals. Each generalized LASSO program has a governing parameter whose optimal value depends on properties of the data. At this optimal value, compressed sensing ... More

Sensitivity of $\ell_{1}$ minimization to parameter choiceOct 29 2018Apr 01 2019The use of generalized LASSO is a common technique for recovery of structured high-dimensional signals. Each generalized LASSO program has a governing parameter whose optimal value depends on properties of the data. At this optimal value, compressed sensing ... More

On $n$-superharmonic functions and some geometric applicationsOct 24 2018In this paper we study asymptotic behavior of $n$-superharmonic functions at isolated singularity using the Wolff potential and $n$-capacity estimates in nonlinear potential theory. Our results are inspired by and extend those of Arsove-Huber and Taliaferro ... More

Rate of Growth of Distributionally Chaotic FunctionsOct 22 2018We investigate the permissible growth rates of functions that are distributionally chaotic with respect to differentiation operators. We improve on the known growth estimates for $D$-distributionally chaotic entire functions, where growth is in terms ... More

Self-improvement of pointwise Hardy inequalityOct 17 2018We prove the self-improvement of a pointwise $p$-Hardy inequality. The proof relies on maximal function techniques and a characterization of the inequality by curves.

$L^2$-boundedness of gradients of single layer potentials and uniform rectifiabilityOct 15 2018Nov 27 2018Let $A(\cdot)$ be an $(n+1)\times (n+1)$ uniformly elliptic matrix with H\"older continuous real coefficients and let $\mathcal E_A(x,y)$ be the fundamental solution of the PDE $\mathrm{div} A(\cdot) \nabla u =0$ in $\mathbb R^{n+1}$. Let $\mu$ be a compactly ... More

The Brunn-Minkowski inequality and a Minkowski problem for $\mathcal{A}$-harmonic Green's functionOct 08 2018In this article we study two classical problems in convex geometry associated to $\mathcal{A}$-harmonic PDEs, quasi-linear elliptic PDEs whose structure is modeled on the $p$-Laplace equation. Let $p$ be fixed with $2\leq n\leq p<\infty$. For a convex ... More

A positive lower bound for $\liminf_{N\to\infty} \prod_{r=1}^N \left| 2\sin πr \varphi \right|$Oct 04 2018Nearly 60 years ago, Erd\H{o}s and Szekeres raised the question of whether $$\liminf_{N\to \infty} \prod_{r=1}^N \left| 2\sin \pi r \alpha \right| =0$$ for all irrationals $\alpha$. Despite its simple formulation, the question has remained unanswered. ... More

Weak limits of the measures of maximal entropy for Orthogonal polynomialsOct 01 2018In this paper we study the sequence of orthonormal polynomials $\{P_n(\mu; z)\}$ defined by a probability measure $\mu$ with non-polar compact support $S(\mu)\subset\mathbb C$. We show that the support of any weak* limit of the sequence of measures of ... More

On Nontangential Limits and Shift Invariant SubspacesSep 26 2018In 1998, John B. Conway and Liming Yang wrote a paper in which they posed a number of open questions regarding the shift on $P^t(\mu)$ spaces. A few of these have been completely resolved, while at least one remains wide open. In this paper, we review ... More

The Diamond ensemble: a constructive set of points with small logarithmic energySep 25 2018We define a family of random sets of points, the Diamond ensemble, on the sphere $\mathbb{S}^{2}$ depending on several parameters. Its most important property is that, for some of these parameters, the asymptotic expected value of the logarithmic energy ... More

A new mixed potential representation for the equations of unsteady, incompressible flowSep 22 2018Sep 27 2018We present a new integral representation for the unsteady, incompressible Stokes or Navier-Stokes equations, based on a linear combination of heat and harmonic potentials. For velocity boundary conditions, this leads to a coupled system of integral equations: ... More

About the Jones-Wolff Theorem on the Hausdorff dimension of harmonic measureSep 21 2018Jul 11 2019These are the lecture notes of a seminar held at the Universitat Aut\`onoma de Barcelona where the Jones-Wolff theorem about the dimension of harmonic measure in the plane is explained in full detail, for non-expert readers.

About the Jones-Wolff Theorem on the Hausdorff dimension of harmonic measureSep 21 2018These are the lecture notes of a seminar held at the Universitat Aut\`onoma de Barcelona where the Jones-Wolff theorem about the dimension of harmonic measure in the plane is explained in full detail, for non-expert readers.

A generalization of the converse of Brolin's theoremAug 20 2018We prove a generalization of Lopes's theorem, that is, of the converse of Brolin's theorem.

Szegő's Condition on Compact subsets of $\mathbb{C}$Aug 17 2018Nov 19 2018Let $K$ be a non-polar compact subset of $\mathbb{C}$ and $\mu_K$ be its equilibrium measure. Let $\mu$ be a unit Borel measure supported on a compact set which contains the support of $\mu_K$. We prove that a Szeg\H{o} condition in terms of the Radon-Nikodym ... More

Optimal extension of Lipschitz embeddings in the planeAug 14 2018We prove that every bi-Lipschitz embedding of the unit circle into the plane can be extended to a bi-Lipschitz map of the unit disk with linear bounds on the constants involved. This answers a question raised by Daneri and Pratelli. Furthermore, every ... More

Physical-density integral equation methods for scattering from multi-dielectric cylindersAug 09 2018Feb 27 2019An integral equation-based numerical method for scattering from multi-dielectric cylinders is presented. Electromagnetic fields are represented via layer potentials in terms of surface densities with physical interpretations. The existence of null-field ... More

Physical-density integral equation methods for scattering from multi-dielectric cylindersAug 09 2018Oct 29 2018An integral equation-based numerical method for scattering from multi-dielectric cylinders is presented. Electromagnetic fields are represented via layer potentials in terms of surface densities with physical interpretations. The existence of null-field ... More

Singular Localised Boundary-Domain Integral Equations of Acoustic Scattering by Inhomogeneous Anisotropic ObstacleJul 27 2018Aug 22 2018We consider the time-harmonic acoustic wave scattering by a bounded {\it anisotropic inhomogeneity} embedded in an unbounded {\it anisotropic} homogeneous medium. The material parameters may have discontinuities across the interface between the inhomogeneous ... More

On Hilbert problem for Beltrami equations in quasihyperbolic domainsJul 24 2018Jan 09 2019It is studied the Hilbert boundary value problem for the nondegenerate Beltrami equations in domains $D$ of the complex plane $\mathbb C$ with the so--called quasihyperbolic boundary condition. It is proved the existence of solutions of this problem with ... More

Laplacian growth & sandpiles on the Sierpinski gasket: limit shape universality and exact solutionsJul 23 2018Jun 19 2019We establish quantitative spherical shape theorems for rotor-router aggregation and abelian sandpile growth on the graphical Sierpinski gasket ($SG$) when particles are launched from the corner vertex. In particular, the abelian sandpile growth problem ... More

Laplacian growth & sandpiles on the Sierpinski gasket: limit shape universality and exact solutionsJul 23 2018Sep 06 2018We establish quantitative spherical shape theorems for rotor-router aggregation and abelian sandpile growth on the graphical Sierpinski gasket ($SG$) when particles are launched from the corner vertex. In particular, the abelian sandpile growth problem ... More

Explicit inverse of nonsingular Jacobi matricesJul 17 2018We present here the necessary and sufficient conditions for the invertibility of tridiagonal matrices, commonly named Jacobi matrices, and explicitly compute their inverse. The techniques we use are related with the solution of Sturm-Liouville boundary ... More

Equality in Suita's conjectureJul 15 2018Nov 11 2018For any open Riemann surface $X$ admitting Green functions, Suita asked about the precise relations between the Bergman kernel and the logarithmic capacity. It was conjectured that the Gaussian curvature of the Suita metric is bounded from above by $-4$, ... More

On reproducing kernels, and analysis of measuresJul 11 2018Jul 17 2018Starting with the correspondence between positive definite kernels on the one hand and reproducing kernel Hilbert spaces (RKHSs) on the other, we turn to a detailed analysis of associated measures and Gaussian processes. Point of departure: Every positive ... More

On reproducing kernels, and analysis of measuresJul 11 2018Feb 23 2019Starting with the correspondence between positive definite kernels on the one hand and reproducing kernel Hilbert spaces (RKHSs) on the other, we turn to a detailed analysis of associated measures and Gaussian processes. Point of departure: Every positive ... More

The matching polynomials and spectral radii of uniform supertreesJul 03 2018We study matching polynomials of uniform hypergraph and spectral radii of uniform supertrees. By comparing the matching polynomials of supertrees, we extend Li and Feng's results on grafting operations on graphs to supertrees. Using the methods of grafting ... More

The classification of holomorphic $(m,n)$--subharmonic morphismsJun 20 2018Aug 27 2018We study the problem of classifying the holomorphic $(m,n)$-subharmonic morphisms in complex space. This determines which holomorphic mappings preserves $m$-subharmonicity in the sense that the composition of the holomorphic mapping with a $m$-subharmonic ... More

Approximating real-rooted and stable polynomials, with combinatorial applicationsJun 19 2018Let $p(x)=a_0 + a_1 x + \ldots + a_n x^n$ be a polynomial with all roots real and satisfying $x \leq -\delta$ for some $0<\delta <1$. We show that for any $0 < \epsilon <1$, the value of $p(1)$ is determined within relative error $\epsilon$ by the coefficients ... More

The lemniscate tree of a random polynomialJun 01 2018To each generic complex polynomial $p(z)$ there is associated a labeled binary tree (here referred to as a "lemniscate tree") that encodes the topological type of the graph of $|p(z)|$. The branching structure of the lemniscate tree is determined by the ... More

Caratheodory convergence and harmonic measureMay 22 2018We give several new characterizations of Caratheodory convergence of simply connected domains. We then investigate how different definitions of convergence generalize to the multiply-connected case.

The Hilbert transform and orthogonal martingales in Banach spacesMay 10 2018Let $X$ be a given Banach space and let $M$, $N$ be two orthogonal $X$-valued local martingales such that $N$ is weakly differentially subordinate to $M$. The paper contains the proof of the estimate $$ \mathbb E \Psi(N_t) \leq C_{\Phi,\Psi,X} \mathbb ... More

The Hilbert transform and orthogonal martingales in Banach spacesMay 10 2018Jul 02 2019Let $X$ be a given Banach space and let $M$, $N$ be two orthogonal $X$-valued local martingales such that $N$ is weakly differentially subordinate to $M$. The paper contains the proof of the estimate $$ \mathbb E \Psi(N_t) \leq C_{\Phi,\Psi,X} \mathbb ... More

Capacities, removable sets and $L^p$-uniqueness on Wiener spacesMay 10 2018We prove the equivalence of two different types of capacities in abstract Wiener spaces. This yields a criterion for the $L^p$-uniqueness of the Ornstein-Uhlenbeck operator and its integer powers defined on suitable algebras of functions vanishing in ... More

Universality and models for semigroups of operators on a Hilbert spaceMay 08 2018This paper considers universal Hilbert space operators in the sense of Rota, and gives criteria for universality of semigroups in the context of uniformly continuous semigroups and contraction semigroups. Specific examples are given. Universal semigroups ... More

Critical measures for vector energy: asymptotics of non-diagonal multiple orthogonal polynomials for a cubic weightMay 04 2018We consider the type I multiple orthogonal polynomials (MOPs) $(A_{n,m}, B_{n,m})$ and type II MOPs $P_{n,m}$, satisfying non-hermitian orthogonality with respect to the weight $e^{-z^3}$ on two unbounded contours on $\mathbb C$. Under the assumption ... More

On properties of the solutions to the $α$-harmonic equationApr 29 2018The aim of this paper is to establish properties of the solutions to the $\alpha$-harmonic equations: $\Delta_{\alpha}(f(z))=\partial{z}[(1-{|{z}|}^{2})^{-\alpha} \overline{\partial}{z}f](z)=g(z)$, where $g:\overline{\mathbb{ID}}\rightarrow\mathbb{C}$ ... More

The Dirichlet problem for semi-linear equationsApr 16 2018Mar 11 2019We study the Dirichlet problem for the semi--linear partial differential equations ${\rm div}\,(A\nabla u)=f(u)$ in simply connected domains $D$ of the complex plane $\mathbb C$ with continuous boundary data. We prove the existence of the weak solutions ... More

The Dirichlet problem for semi-linear equationsApr 16 2018Jan 03 2019We study the Dirichlet problem for the semi--linear partial differential equations in the simple connected domains $D$ in $\mathbb C$, the linear part of which is written in a divergence (anisotropic !) form. Thanking to a factorization theorem established ... More

The Dirichlet problem for semi-linear equationsApr 16 2018Apr 08 2019We study the Dirichlet problem for the semi--linear partial differential equations ${\rm div}\,(A\nabla u)=f(u)$ in simply connected domains $D$ of the complex plane $\mathbb C$ with continuous boundary data. We prove the existence of the weak solutions ... More

Exact asymptotic volume and volume ratio of Schatten unit ballsApr 10 2018The unit ball $B_p^n(\mathbb{R})$ of the finite-dimensional Schatten trace class $\mathcal S_p^n$ consists of all real $n\times n$ matrices $A$ whose singular values $s_1(A),\ldots,s_n(A)$ satisfy $s_1^p(A)+\ldots+s_n^p(A)\leq 1$, where $p>0$. Saint Raymond ... More

On capacity computation for symmetric polygonal condensersApr 04 2018Making use of two different analytical-numerical methods for capacity computation, we obtain matching to a very high precision numerical values for capacities of a wide family of planar condensers. These two methods are based respectively on the use of ... More

On capacity computation for symmetric polygonal condensersApr 04 2018Mar 04 2019Making use of two different analytical-numerical methods for capacity computation, we obtain matching to a very high precision numerical values for capacities of a wide family of planar condensers. These two methods are based respectively on the use of ... More

Binomial Inequalities of Chromatic, Flow, and Ehrhart PolynomialsMar 31 2018Nov 22 2018A famous and wide-open problem, going back to at least the early 1970's, concerns the classification of chromatic polynomials of graphs. Toward this classification problem, one may ask for necessary inequalities among the coefficients of a chromatic polynomial, ... More

Sharp pointwise gradient estimates for Riesz potentials with a bounded densityMar 30 2018We establish sharp inequalities for the Riesz potential and its gradient in $\mathbb{R}^{n}$ and indicate their usefulness for potential analysis, moment theory and other applications.

Harmonic measure and quantitative connectivity: geometric characterization of the $L^p$ solvability of the Dirichlet problem. Part IIMar 21 2018Jul 09 2018Let $\Omega\subset\mathbb R^{n+1}$ be an open set with $n$-AD-regular boundary. In this paper we prove that if the harmonic measure for $\Omega$ satisfies the so-called weak-$A_\infty$ condition, then $\Omega$ satisfies a suitable connectivity condition, ... More

Two-point distortion theorems and the Schwarzian derivatives of meromorphic functionsMar 13 2018Mar 27 2018For a meromorphic function $f$ in the unit disk $U=\{z:\;|z|<1\}$ and arbitrary points $z_1,z_2$ in $U$ distinct from the poles of $f$, a sharp upper bound on the product $|f'(z_1)f'(z_2)|$ is established. Further, we prove a sharp distortion theorem ... More

Non-conforming harmonic virtual element method: $h$- and $p$-versionsJan 02 2018Jul 27 2018We study the $h$- and $p$-versions of non-conforming harmonic virtual element methods (VEM) for the approximation of the Dirichlet-Laplace problem on a 2D polygonal domain, providing quasi-optimal error bounds. Harmonic VEM do not make use of internal ... More

On increasing stability in the two dimensional inverse source scattering problem with many frequenciesDec 23 2017In this paper, we will study increasing stability in the inverse source problem for the Helmholtz equation in the plane when the source term is assumed to be compactly supported in a bounded domain $\Omega$ with sufficiently smooth boundary. Using the ... More