相关论文: Schoenberg's Problem on Positive Definite Function…

We study expansions in non-integer negative base -{\beta} introduced by Ito and Sadahiro. Using countable automata associated with (-{\beta})-expansions, we characterize the case where the (-{\beta})-shift is a system of finite type. We…

In this paper, we study the following class of weighted Choquard equations \begin{align*} -\Delta u =\lambda u + \Bigg(\displaystyle\int\limits_\Omega \frac{Q(|y|)F(u(y))}{|x-y|^\mu}dy\Bigg) Q(|x|)f(u) ~~\textrm{in}~~ \Omega~~ \text{and}~~…

For $\ba \in \R_{\geq 0}^{n}$, the Tesler polytope $\tes_{n}(\ba)$ is the set of upper triangular matrices with non-negative entries whose hook sum vector is $\ba$. Motivated by a conjecture of Morales', we study the questions of whether…

We prove that for $q\in (0,1)$, the partial theta function $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ has no zeros in the closed domain $\{ \{ |x|\leq 3\} \cap \{${\rm Re}$x\leq 0\} \cap \{ |${\rm Im}$x|\leq 3/\sqrt{2}\} \} \subset…

In this paper we deal with the cubic Schr\"odinger system $ -\Delta u_i = \sum_{j=1}^n \beta_{ij}u_j^2 u_i$, $u_1,\dots,u_n \geq 0$ in $\mathbb{R}^N (N\leq 3)$, where $\beta=(\beta_{i,j})_{ij}$ is a symmetric matrix with real coefficients…

We study local and global properties of positive solutions of $-{\Delta}u=u^p]{\left |{\nabla u}\right |}^q$ in a domain ${\Omega}$ of ${\mathbb R}^N$, in the range $1\<p+q$, $p\geq 0$, $0\leq q\< 2$. We first prove a local Harnack…

We study the positive-definiteness of a family of $L^2(\mathbf{R})$ integral operators with kernel $K_{t, a}(x, y) = (1 + (x - y)^2 + a(x^2 + y^2)^t)^{-1}$, with $t > 0$ and $a > 0$. When $0 < t \le 1$, the known theory of positive-definite…

This paper deals with the following fractional Schr$ \ddot{\textrm{o}}$dinger equations with Choquard-type nonlinearities \begin{equation*} \left\{\begin{array}{r@{\ \ }c@{\ \ }ll} (-\Delta)^{\frac{\alpha}{2}}u + u - C_{n,-\beta}…

This article deals with the existence and non-existence of positive solutions for the eigenvalue problem driven by nonhomogeneous fractional $p\& q$ Laplacian operator with indefinite weights $$\left(-\Delta_p\right)^{\alpha}u +…

We present a rational extension of Newton diagram for the positivity of ${}_1F_2$ generalized hypergeometric functions. As an application, we give upper and lower bounds for the transcendental roots $\beta(\alpha)$ of \begin{align*}…

We present a characterization for the continuous, isotropic and positive definite kernels on a product of spheres along the lines of a classical result of I. J. Schoenberg on positive definiteness on a single sphere. We also discuss a few…

Let $q\in(1,2)$. A $q$-expansion of a number $x$ in $[0,\frac{1}{q-1}]$ is a sequence $(\delta_i)_{i=1}^\infty\in\{0,1\}^{\mathbb{N}}$ satisfying $$ x=\sum_{i=1}^\infty\frac{\delta_i}{q^i}.$$ Let $\mathcal{B}_{\aleph_0}$ denote the set of…

In a recent paper [15], Hilbert space operators $T$ with the property that each sequence of the form $\{\|T^n h\|^2\}_{n=0}^{\infty}$ is conditionally positive definite in a semigroup sense were introduced. In the present paper, this line…

For almost all tuples $(x_1,\dots,x_n)$ of complex numbers, a strong version of Schanuel's Conjecture is true: the $2n$ numbers $x_1,\dots,x_n, {\mathrm e}^{x_1},\dots, {\mathrm e}^{x_n}$ are algebraically independent. Similar statements…

This note proves the following inequality: if $n=3k$ for some positive integer $k$, then for any $n$ positive definite matrices $A_1,A_2,\cdots,A_n$, \begin{equation} \frac{1}{n^3}\Big\|\sum_{j_1,j_2,j_3=1}^{n}A_{j_1}A_{j_2}A_{j_3}\Big\|…

In this paper we study the expansions of real numbers in positive and negative real base as introduced by R\'enyi, and Ito & Sadahiro, respectively. In particular, we compare the sets $\mathbb{Z}_\beta^+$ and $\mathbb{Z}_{-\beta}$ of…

In this paper, we found new q-binomial formula for Q-commutative operators. Expansion coefficients in this formula are given by q-binomial coefficients with two bases (q,Q), determined by Q-commutative q-Pascal triangle. Our formula…

We review the indefinite sublinear elliptic equation $-\Delta u=a(x)u^{q}$ in a smooth bounded domain $\Omega\subset\mathbb{R}^{N}$, with Dirichlet or Neumann homogeneous boundary conditions. Here $0<q<1$ and $a$ is continuous and changes…

We present a quite curious generalization of multi-step Fibonacci numbers. For any positive rational $q$, we enumerate binary words of length $n$ whose maximal factors of the form $0^a1^b$ satisfy $a = 0$ or $aq > b$. When $q$ is an integer…

A classical theorem of Herglotz states that a function $n\mapsto r(n)$ from $\mathbb Z$ into $\mathbb C^{s\times s}$ is positive definite if and only there exists a $\mathbb C^{s\times s}$-valued positive measure $d\mu$ on $[0,2\pi]$ such…