Revisiting Landauʼs density theorems for Paley–Wiener spaces

We present a surprisingly simple approach to Landau?s density theorems for sampling and interpolation, which provides stronger versions of these results. In particular, we extend the interpolation theorem to unbounded spectra.     

Approximation formulas for Landauʼs constants

In this paper, we establish approximation formulas for evaluating Landau?s constants.     

Multivariate Hörmander-Type Multiplier Theorem for the Hankel transform

Let $\mathcal{H}(f)(x)=\int_{(0,\infty)^{d}} f(\lambda) E_{x}(\lambda) d\nu(\lambda )$ , be the multivariate Hankel transform, where $E_{x}(\lambda)=\prod_{k=1}^{d} (x_{k} \lambda_{k})^{-\alpha _{k}+1/2}J_{\alpha_{k}-1/2}(x_{k} \lambda_{k})$ , with dν(λ)=λ ^2α dλ, α=(α ₁,…,α _d ). We give sufficient conditions on a bounded function m(λ) which guarantee that the operator $\mathcal{H}(m\mathcal{H} f)$ is bounded on L ^p (dν) and of weak-type (1,1), or bounded on the Hardy space H ¹((0,∞) ^d ,dν) in the sense of Coifman-Weiss.     

Hardy’s Inequality for the n-Dimensional Hankel Transform

Analysis Mathematica - The purpose of this paper is to prove a Hardy type inequality associated with the n-dimensional Hankel transform (n ≥ 1) for the exponent σ0 = n(2 ? p) +...     

Interior estimates for the n-dimensional Abreuʼs equation

We study the Abreu?s equation in n-dimensional polytopes and derive interior estimates of solutions under the assumption of the uniform K-stability.     

Eigenfunctions of the Airyʼs integral transform: Properties,numerical computations and asymptotic behaviors

This paper is devoted to the study of the spectrum of the integral operator with Airy?s kernel. We provide the reader with some efficient methods of computing the eigenfunctions and the eigenvalues of this operator. The first method is based on the use of a differential operator which commutes with this integral operator. The second method is based on a Gaussian quadrature method applied to the finite Airy?s transform. The asymptotic behavior of the eigenfunctions is studied by the use of a WKB method. Some numerical examples are given to illustrate results of this work.     

An analogue of Beurling’s theorem for the Jacobi transform

In this paper, we prove Beurling＇s theorem for the Jacobi transform, from which we derive some other versions of uncertainty principles.     

On necessary conditions for global asymptotic stability of equilibrium for the Liénard equation

In [1], necessary and sufficient conditions for the global asymptotic stability of the trivial solution of the Liénard equation ?+f(x)?+g(x)=0, g(0)=0, were obtained under the condition $x\int_0^x {f(s)ds \geqslant 0.} $ In [1], the following problem was also posed: To determine whether condition (A) is a necessary condition for the global asymptotic stability of the trivial solution of the Liénard equation. The present paper answers this question, and the answer is negative, i.e., condition (A) is not a necessary condition.     

The Fourier–Stieltjes transform of Minkowskiʼs ?(x) function and an affirmative answer to Salemʼs problem

By using structural and asymptotic properties of the Kontorovich–Lebedev transform associated with Minkowski?s question mark function, we give an affirmative answer to the question posed by R. Salem (Trans. Amer. Math. Soc. 53 (3) (1943) 439) whether its Fourier–Stieltjes transform vanishes at infinity.     

Moritaʼs duality for split reductive groups

In this paper, we extend the work in [Z. Qi, C. Yang, Morita?s theory for the symplectic groups, Int. J. Number Theory 7 (2011) 2115–2137 [7]] to split reductive groups. We construct and study the holomorphic discrete series representations and the principal series representations of a split reductive group G over a p-adic field F as well as a duality between certain sub-representations of these two representations.     

Hardy’s and Miyachi’s Theorems for the First Hankel–Clifford Transform

Ukrainian Mathematical Journal - We present an analog of Hardy’s and Miyachi’s theorems for the first Hankel–Clifford transform.     

Riemann–Hilbert problem for the modified Landau–Lifshitz equation with nonzero boundary conditions

Theoretical and Mathematical Physics - We study a matrix Riemann–Hilbert (RH) problem for the modified Landau–Lifshitz (mLL) equation with nonzero boundary conditions at infinity. In...     

A note on Chowʼs entropy functional for the Gauss curvature flow

Based on the entropy formula for the Gauss curvature flow introduced by Bennett Chow, we define an entropy functional that is monotone along the unnormalized flow and whose critical point is a shrinking self-similar solution.     

Order of Selbergʼs and Ruelleʼs zeta functions for compact even-dimensional locally symmetric spaces

We prove that Selberg?s and Ruelle?s zeta functions considered by U. Bunke and M. Olbrich can be represented as quotients of two entire functions of order not larger than the dimension of the underlaying compact, even-dimensional, locally symmetric space.     

A proof of Perkoʼs conjectures for the Bogdanov–Takens system

The Bogdanov–Takens system has at most one limit cycle and, in the parameter space, it exists between a Hopf and a saddle-loop bifurcation curves. The aim of this paper is to prove the Perko?s conjectures about some analytic properties of the saddle-loop bifurcation curve. Moreover, we provide sharp piecewise algebraic upper and lower bounds for this curve.     

Non-degenerate necessary optimality conditions for the optimal control problem with equality-type state constraints

In this work, an optimal control problem with state constraints of equality type is considered. Novelty of the problem formulation is justified. Under various regularity assumptions imposed on the optimal trajectory, a non-degenerate Pontryagin Maximum Principle is proven. As a consequence of the maximum principle, the Euler–Lagrange and Legendre conditions for a variational problem with equality and inequality state constraints are obtained. As an application, the equation of the geodesic curve for a complex domain is derived. In control theory, the Maximum Principle suggests the global maximum condition, also known as the Weierstrass–Pontryagin maximum condition, due to which the optimal control function, at each instant of time, turns out to be a solution to a global finite-dimensional optimization problem.     

Bergeʼs maximum theorem for noncompact image sets

This note generalizes Berge?s maximum theorem to noncompact image sets. It also clarifies the results from Feinberg, Kasyanov and Zadoianchuk (2013) [7] on the extension to noncompact image sets of another Berge?s theorem, that states semi-continuity of value functions. Here we explain that the notion of a K$\mathbb{K}$-inf-compact function introduced there is applicable to metrizable topological spaces and to more general compactly generated topological spaces. For Hausdorff topological spaces we introduce the notion of a KN$\mathbb{K}\mathbb{N}$-inf-compact function (N$\mathbb{N}$ stands for “nets” in K$\mathbb{K}$-inf-compactness), which coincides with K$\mathbb{K}$-inf-compactness for compactly generated and, in particular, for metrizable topological spaces.