Related papers: A problem on completeness of exponentials

Many mathematical statements have the following form. If something is true for all finite subsets of an infinite set $I$, then it is true for all of $I$. This paper describes some old and new results on infinite sets of linear and…

The $\lambda$-exponential family generalizes the standard exponential family via a generalized convex duality motivated by optimal transport. It is the constant-curvature analogue of the exponential family from the information-geometric…

We prove that the residual girth of any finitely generated linear group is at most exponential. This means that the smallest finite quotient in which the $n$-ball injects has at most exponential size. If the group is also not virtually…

We show that the essentially algebraic theory of generalized algebraic theories, regarded as a category with finite limits, has a universal exponentiable arrow in the sense that any exponentiable arrow in any category with finite limits is…

In this paper, we study the existence of the density associated to the exponential functional of the L\'evy process $\xi$, \[ I_{\ee_q}:=\int_0^{\ee_q} e^{\xi_s} \, \mathrm{d}s, \] where $\ee_q$ is an independent exponential r.v. with…

We examine convergent representations for the sum of Bessel functions \[\sum_{n=1}^\infty \frac{J_\mu(na) J_\nu(nb)}{n^{\alpha}}\] for $\mu$, $\nu\geq0$ and positive values of $a$ and $b$. Such representations enable easy computation of the…

We show that for entire maps of the form $z \mapsto \lambda \exp(z)$ such that the orbit of zero is bounded and such that Lebesgue almost every point is transitive, no absolutely continuous invariant probability measure can exist. This…

We use the combinatorial properties of central sets to prove a result about the existence of exponential monochromatic patterns, in the style of Hindman's Finite Sums Theorem. More precisely, we prove that for every finite coloring of the…

Given integers $\ell > m >0$, we define monic polynomials $X_n$, $Y_n$, and $Z_n$ with the property that $\mu$ is a zero of $X_n$ if and only if the triple $(\mu,\mu+m,\mu+\ell)$ satisfies $x^n + y^n = z^n$. It is shown that the…

We show that there is a measure $\mu$, defined on the hyperbolic plane and with polynomial growth, such that the centered maximal operator associated to $\mu$ does not satisfy weak type $(1,1)$ bounds.

We study behavior of a measure on $[0,\infty)$ by considering its Laplace transform. If it is possible to extend the Laplace transform to a complex half-plane containing the imaginary axis, then the exponential decay of the tail of the…

We give several general theorems concerning positive definite solutions of Riemann-Hilbert problems on the real line. Furthermore, as an example, we apply our theory to the characteristic function of a class of L\'{e}vy processes and we…

Let $n$ be a positive integer, and let $R$ be a (possibly infinite dimensional) finitely presented algebra over a computable field of characteristic zero. We describe an algorithm for deciding (in principle) whether $R$ has at most finitely…

For a transcendental entire function $f$, the property that there exists $r>0$ such that $m^n(r)\to\infty$ as $n\to\infty$, where $m(r)=\min \{|f(z)|:|z|=r\}$, is related to conjectures of Eremenko and of Baker, for both of which order…

We study the existence/nonexistence of positive solution to the problem of the type: \begin{equation}\tag{$P_{\lambda}$} \begin{cases} \Delta^2u-\mu a(x)u=f(u)+\lambda b(x)\quad\textrm{in $\Omega$,}\\ u>0 \quad\textrm{in $\Omega$,}\\…

Let $M$ be a $d$-dimensional closed Riemannian manifold, let $f\in\mathrm{Diff}^{1+\beta}(M)$, and denote by $m$ the Riemannian volume form of $M$. We prove that if $m\circ f^{-n}\xrightarrow[n\to\infty]{}\mu$ exponentially fast, then $\mu$…

Let $f$ be a $C^{1+\alpha}$ diffeomorphism of a compact manifold $M$ preserving a smooth measure $\mu$. We show that if $f:(M,\mu)\to (M,\mu)$ is exponentially mixing then it is Bernoulli.

We show the existence of infinitely many positive solutions, defined on the real line, for the nonlinear scalar ODE \[ \ddot u + (a^+(t) - \mu a^-(t)) u^3 = 0, \] where $a$ is a periodic, sign-changing function, and the parameter $\mu>0$ is…

We consider a solution to a parametric family of the Cauchy problems for $m$th-order linear differential equations with constant coefficients. Parameters of the family are the coefficients of the differential equation and the initial values…

Motzkin posed the problem of finding the maximal density $\mu(M)$ of sets of integers in which the differences given by a set $M$ do not occur. The problem is already settled when $|M|\leq 2$ and $M$ is a finite arithmetic progression. In…