Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userConvexity properties of the difference over the real axis between the Steklov zeta functions of a smooth planar domain with $2pi$ perimeter and of the unit disk
88
0
0.0
(
0
)
Authors
Alexandre Jollivet
Ask ChatGPT about the research
No Arabic abstract
We consider the zeta function $zeta_Omega$ for the Dirichlet-to-Neumann operator of a simply connected planar domain $Omega$ bounded by a smooth closed curve of perimeter $2pi$. We prove that $zeta_Omega(0)ge zeta_{mathbb{D}}(0)$ with equality if and only if $Omega$ is a disk where $mathbb{D}$ denotes the closed unit disk. We also provide an elementary proof that for a fixed real $s$ satisfying $sle-1$ the estimate $zeta_Omega(s)ge zeta_{mathbb{D}}(s)$ holds with equality if and only if $Omega$ is a disk. We then bring examples of domains $Omega$ close to the unit disk where this estimate fails to be extended to the interval $(0,2)$. Other computations related to previous works are also detailed in the remaining part of the text.
rate research
Read More
We consider the Steklov zeta function $zeta$ $Omega$ of a smooth bounded simply connected planar domain $Omega$ $subset$ R 2 of perimeter 2$pi$. We provide a first variation formula for $zeta$ $Omega$ under a smooth deformation of the domain. On the base of the formula, we prove that, for every s $in$ (--1, 0) $cup$ (0, 1), the difference $zeta$ $Omega$ (s) -- 2$zeta$ R (s) is non-negative and is equal to zero if and only if $Omega$ is a round disk ($zeta$ R is the classical Riemann zeta function). Our approach gives also an alternative proof of the inequality $zeta$ $Omega$ (s) -- 2$zeta$ R (s) $ge$ 0 for s $in$ (--$infty$, --1] $cup$ (1, $infty$); the latter fact was proved in our previous paper [2018] in a different way. We also provide an alternative proof of the equality $zeta$ $Omega$ (0) = 2$zeta$ R (0) obtained by Edward and Wu [1991].
We consider the one dimensional focusing (cubic) Nonlinear Schrodinger equation (NLS) in the semiclassical limit with exponentially decaying complex-valued initial data, whose phase is multiplied by a real parameter. We prove smooth dependence of the asymptotic solution on the parameter. Numerical results supporting our estimates of important quantities are presented.
Formulas relating Poincare-Steklov operators for Schroedinger equations related by Darboux-Moutard transformations are derived. They can be used for testing algorithms of reconstruction of the potential from measurements at the boundary.
In this Ph.D. dissertation (2018, Emory University) we prove theorems at the intersection of the additive and multiplicative branches of number theory, bringing together ideas from partition theory, $q$-series, algebra, modular forms and analytic number theory. We present a natural multiplicative theory of integer partitions (which are usually considered in terms of addition), and explore new classes of partition-theoretic zeta functions and Dirichlet series -- as well as Eulerian $q$-hypergeometric series -- enjoying many interesting relations. We find a number of theorems of classical number theory and analysis arise as particular cases of extremely general combinatorial structure laws. Among our applications, we prove explicit formulas for the coefficients of the $q$-bracket of Bloch-Okounkov, a partition-theoretic operator from statistical physics related to quasi-modular forms; we prove partition formulas for arithmetic densities of certain subsets of the integers, giving $q$-series formulas to evaluate the Riemann zeta function; we study $q$-hypergeometric series related to quantum modular forms and the strange function of Kontsevich; and we show how Ramanujans odd-order mock theta functions (and, more generally, the universal mock theta function $g_3$ of Gordon-McIntosh) arise from the reciprocal of the Jacobi triple product via the $q$-bracket operator, connecting also to unimodal sequences in combinatorics and quantum modular-like phenomena.
It is shown that a strong solution of the Camassa-Holm equation, initially decaying exponentially together with its spacial derivative, must be identically equal to zero if it also decays exponentially at a later time. In particular, a strong solution of the Cauchy problem with compact initial profile can not be compactly supported at any later time unless it is the zero solution.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
