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For Hill's equation on [0,infinity) we prove new characterizations of the spectral function rho(lambda) and the spectral density function f(lambda) based on analysis involving a companion system of first order differential equations in…

We show optimal Lipschitz regularity for very weak solutions of the (measure-valued) elliptic PDE $-\mathrm{div}(A(x) \nabla u) = Q \; \mathcal{H}^{n-1} \llcorner \Gamma$ in a smooth domain $\Omega \subset \mathbb{R}^n$. Here $\Gamma$ is a…

We present a Mathematica package for doing computations with gamma matrices, spinors, tensors and other objects, in any dimension and signature. The approach we use is based on defining the commutation relations of the relevant matrices,…

We consider the shape optimization problem $$\min\big\{{\mathcal E}(\Gamma)\ :\ \Gamma\in{\mathcal A},\ {\mathcal H}^1(\Gamma)=l\ \big\},$$ where ${\mathcal H}^1$ is the one-dimensional Hausdorff measure and ${\mathcal A}$ is an admissible…

In this paper we study the Steklov-Dirichlet eigenvalues $\lambda_k(\Omega,\Gamma_S)$, where $\Omega\subset \mathbb{R}^d$ is a domain and $\Gamma_S\subset \partial \Omega$ is the subset of the boundary in which we impose the Steklov…

We give a class of examples of $A$-hypergeometric systems that display integrality of mirror maps. Specifically, these systems have solutions $F(\lambda_1,\dots,\lambda_N) = 1$ and $\log\lambda^l + G(\lambda_1,\dots,\lambda_N)$ (for certain…

In this paper we show how to compute the $\Lambda_{\alpha}$ norm, $\alpha\ge 0$, using the dyadic grid. This result is a consequence of the description of the Hardy spaces $H^p(R^N)$ in terms of dyadic and special atoms.

We present a numerical scheme that can be combined with any fixed boundary finite element based Poisson or Grad-Shafranov solver to compute the first and second partial derivatives of the solution to these equations with the same order of…

We show that the isomorphism problem is solvable in the class of central extensions of word-hyperbolic groups, and that the isomorphism problem for biautomatic groups reduces to that for biautomatic groups with finite centre. We describe an…

Let $\Gamma$ be a locally finite graph, $L$ the normalized Laplacian of $\Gamma$. If $\Gamma$ is uniformy locally finite, i.e. if each vertex has no more than $d$ adjacent vertices, then the matrix of $L$ (with respect to the standard…

We study graphs coming from quadratic spaces over finite fields via orthogonality which generalize a recent result given by Bishnoi, Ihringer, and Pepe (2019). More precisely, we study the graph $\Gamma^{\square}(n,k,q)$ as follows: the…

We discuss certain facts involving a continuous local martingale $N$ and its supremum $\bar{N}$. A complete characterization of $(N,\bar{N})$-harmonic functions is proposed. This yields an important family of martingales, the usefulness of…

Stochastic Gradient Descent (SGD) stands as a cornerstone optimization algorithm with proven real-world empirical successes but relatively limited theoretical understanding. Recent research has illuminated a key factor contributing to its…

The dynamically defined measure (DDM) $\Phi$ arising from a finite measure $\phi_0$ on an initial $\sigma$-algebra on a set and an invertible map acting on the latter is considered. Several lower bounds for it are obtained and sufficient…

Let $\Gamma $ be a $C^\infty $ curve in $\Bbb{C}$ containing 0; it becomes $\Gamma_\theta $ after rotation by angle $\theta $ about 0. Suppose a $C^\infty $ function $f$ can be extended holomorphically to a neighborhood of each element of…

Let $\Gamma$ be a graph equipped with a Markov operator $P$. We introduce discrete fractional Littlewood-Paley square functionals and prove their $L^p$-boundedness under various geometric assumptions on $\Gamma$.

In this paper, we construct a Spectrum Generating Algebra (SGA) for a quantum system with purely continuous spectrum: the quantum free particle in a Lobachevski space with constant negative curvature. The SGA contains the geometrical…

A pair of graphs $(\Gamma,\Sigma)$ is called unstable if their direct product $\Gamma\times\Sigma$ admits automorphisms not from $\mathrm{Aut}(\Gamma)\times\mathrm{Aut}(\Sigma)$, and such automorphisms are said to be unexpected. The…

We characterize all fixed equilibrium point singularity distributions in the plane of logarithmic type, allowing for real, imaginary, or complex singularity strengths \Gamma . The dynamical system follows from the assumption that each of…

We study the problem of estimating the diagonal of an implicitly given matrix $A$. For such a matrix we have access to an oracle that allows us to evaluate the matrix vector product $Av$. For random variable $v$ drawn from an appropriate…