A Hyperplane Inequality for Measures of Convex Bodies in ? n , n≤4

Let 2≤n≤4. We show that for an arbitrary measure μ with even continuous density in ℝ ⁿ and any origin-symmetric convex body K in ℝ ⁿ ,

The evaluation of a quartic integral via Schwinger, Schur and Bessel

We provide additional methods for the evaluation of the integral

Growth rate for beta-expansions

Let β > 1 and let m > β be an integer. Each x ? I_b:=[0,\fracm-1b-1]{x\in I_\beta:=[0,\frac{m-1}{\beta-1}]} can be represented in the form

Distance near the origin between elements of a strongly continuous semigroup

Set

Multiplicative type functional equations arising from characterization problems

We give the general and the so-called density function solutions of equation

On expansions of numbers in alternating <Emphasis Type="Italic">s</Emphasis>-adic series and Ostrogradskii series of the first and second kind

We present expansions of real numbers in alternating s-adic series (1 < s ∈ N), in particular, s-adic Ostrogradskii series of the first and second kind. We study the “geometry” of this representation of numbers and solve metric and probability problems, including the problem of structure and metric-topological and fractal properties of the distribution of the random variable

Quotient mean,its invariance with respect to a quasi-arithmetic mean-type mapping,and some applications

Under some conditions on the functions f and g defined in a real interval I the function

Spectral properties of higher order anharmonic oscillators

We discuss spectral properties of the selfadjoint operator - \fracd²dt² + ( \fract^{k + 1}k + 1 - a )² \begin{gathered} - \frac{{{d^2}}}{{d{t^2}}} + {\left( {\frac{{{t^{k + 1}}}}{{k + 1}} - \alpha } \right)^2} \hfill \\ \hfill \\ \end{gathered} in L ²(ℝ) for odd integers k. We prove that the minimum over α of the ground state energy of this operator is attained at a unique point which tends to zero as k tends to infinity. We also show that the minimum is nondegenerate. These questions arise naturally in the spectral analysis of Schr?dinger operators with magnetic field. Bibliography: 13 titles. Illustrations: 2 figures.     

Regularizing and decay rate estimates for solutions to the Cauchy problem of the Debye–Hückel system

In this paper we establish some regularizing and decay rate estimates for mild solutions of the Debye–Hückel system. We prove that if the initial data belong to the critical Lebesgue space L^\fracn2(\mathbbRⁿ){L^{\frac{n}{2}}(\mathbb{R}^{n})} , then the L ^q -norm ( \fracn2 ￡ q ￡ ￥{\frac{n}{2} \leq q \leq \infty}) of the βth order spatial derivative of mild solutions are majorized by K₁(K₂|b|)^|b|t^{-\frac|b|2-1+\fracn2q}{K_{1}(K_{2}|\beta|)^{|\beta|}t^{-\frac{|\beta|}{2}-1+\frac{n}{2q}}} for some constants K ₁ and K ₂. These estimates particularly imply that mild solutions are analytic in the space variable, and provide decay estimates in the time variable for higher-order derivatives of mild solutions. We also prove that similar estimates also hold for mild solutions whose initial data belong to the critical homogeneous Besov space [(B)\dot]^-2+\fracnp_p,￥(\mathbbRⁿ){\dot{B}^{-2+\frac{n}{p}}_{p,\infty}(\mathbb{R}^n)} ( \fracn2 < p < n{\frac{n}{2} < p < n}).     

Numerical verification of condition for approximately midconvex functions

Let X be a normed space and V be a convex subset of X. Let a\colon \mathbbR₊ ? \mathbbR₊{\alpha \colon \mathbb{R}_+ \to \mathbb{R}_+}. A function f \colon V ? \mathbbR{f \colon V \to \mathbb{R}} is called α-midconvex if

Large time asymptotics of the doubly nonlinear equation in the non-displacement convexity regime

We study the long-time asymptotics of the doubly nonlinear diffusion equation ${\rho_t={\rm div}(|\nabla\rho^m |^{p-2} \nabla\left(\rho^m\right))}We study the long-time asymptotics of the doubly nonlinear diffusion equation r_t=div(|?r^m |^p-2 ?(r^m)){\rho_t={\rm div}(|\nabla\rho^m |^{p-2} \nabla\left(\rho^m\right))} in \mathbbRⁿ{\mathbb{R}^n}, in the range \fracn-pn(p-1) < m < \fracn-p+1n(p-1){\frac{n-p}{n(p-1)} < m < \frac{n-p+1}{n(p-1)}} and 1 < p < ∞ where the mass of the solution is conserved, but the associated energy functional is not displacement convex. Using a linearisation of the equation, we prove an L ¹-algebraic decay of the non-negative solution to a Barenblatt-type solution, and we estimate its rate of convergence. We then derive the nonlinear stability of the solution by means of some comparison method between the nonlinear equation and its linearisation. Our results cover the exponent interval \frac2nn+1 < p < \frac2n+1n+1{\frac{2n}{n+1} < p < \frac{2n+1}{n+1}} where a rate of convergence towards self-similarity was still unknown for the p-Laplacian equation.     

Singular oscillatory integrals on {\mathbb{R}^n}

Let ${\mathcal{P}_{d,n}}Let P_d,n{\mathcal{P}_{d,n}} denote the space of all real polynomials of degree at most d on \mathbbRⁿ{\mathbb{R}^n} . We prove a new estimate for the logarithmic measure of the sublevel set of a polynomial P ? P_d,1{P\in \mathcal{P}_{d,1}} . Using this estimate, we prove that

supP ? Pd,n| p.v.ò\mathbbRneiP(x)\fracW(x/|x|)|x|ndx| ￡ c log d (||W||L logL(Sn-1)+1),\mathop{\rm sup}\limits_ {P \in \mathcal{P}_{d,n}}\left| p.v.\int_{\mathbb{R}^{n}}{e^{iP(x)}}{\frac{\Omega(x/|x|)}{|x|^n}dx}\right | \leq c\,{\rm log}\,d\,(||\Omega||_L \log L(S^{n-1})+1),      13. On the log-concavity of Hilbert series of Veronese subrings and Ehrhart series      Matthias Beck  Alan Stapledon 《Mathematische Zeitschrift》2010,264(1):195-207 For every positive integer n, consider the linear operator U n on polynomials of degree at most d with integer coefficients defined as follows: if we write ${\frac{h(t)}{(1 - t)^{d + 1}}=\sum_{m \geq 0} g(m) \, t^{m}}For every positive integer n, consider the linear operator U n on polynomials of degree at most d with integer coefficients defined as follows: if we write \frach(t)(1 - t)d + 1=?m 3 0 g(m)  tm{\frac{h(t)}{(1 - t)^{d + 1}}=\sum_{m \geq 0} g(m) \, t^{m}} , for some polynomial g(m) with rational coefficients, then \fracUnh(t)(1- t)d+1 = ?m 3 0g(nm)  tm{\frac{{\rm{U}}_{n}h(t)}{(1- t)^{d+1}} = \sum_{m \geq 0}g(nm) \, t^{m}} . We show that there exists a positive integer n d , depending only on d, such that if h(t) is a polynomial of degree at most d with nonnegative integer coefficients and h(0) = 1, then for n ≥ n d , U n h(t) has simple, real, negative roots and positive, strictly log concave and strictly unimodal coefficients. Applications are given to Ehrhart δ-polynomials and unimodular triangulations of dilations of lattice polytopes, as well as Hilbert series of Veronese subrings of Cohen–Macauley graded rings.      14. Uniform estimates for order statistics and Orlicz functions      Yehoram Gordon  Alexander E. Litvak  Carsten Schütt  Elisabeth Werner 《Positivity》2012,16(1):1-28 We establish uniform estimates for order statistics: Given a sequence of independent identically distributed random variables ξ 1, … , ξ n and a vector of scalars x = (x 1, … , x n ), and 1 ≤ k ≤ n, we provide estimates for \mathbb E   k-min1 ￡ i ￡ n |xixi|{\mathbb E \, \, k-{\rm min}_{1\leq i\leq n} |x_{i}\xi _{i}|} and \mathbb E k-max1 ￡ i ￡ n|xixi|{\mathbb E\,k-{\rm max}_{1\leq i\leq n}|x_{i}\xi_{i}|} in terms of the values k and the Orlicz norm ||yx||M{\|y_x\|_M} of the vector y x  = (1/x 1, … , 1/x n ). Here M(t) is the appropriate Orlicz function associated with the distribution function of the random variable |ξ 1|, G(t) = \mathbb P ({ |x1| ￡ t}){G(t) =\mathbb P \left(\left\{ |\xi_1| \leq t\right\}\right)}. For example, if ξ 1 is the standard N(0, 1) Gaussian random variable, then G(t) = ?{\tfrac2p}ò0t e-\fracs22ds {G(t)= \sqrt{\tfrac{2}{\pi}}\int_{0}^t e^{-\frac{s^{2}}{2}}ds }  and M(s)=?{\tfrac2p}ò0se-\frac12t2dt{M(s)=\sqrt{\tfrac{2}{\pi}}\int_{0}^{s}e^{-\frac{1}{2t^{2}}}dt}. We would like to emphasize that our estimates do not depend on the length n of the sequence.      15. Green–Samoilenko operator in the theory of invariant sets of nonlinear differential equations      M. O. Perestyuk  V. Yu. Slyusarchuk 《Ukrainian Mathematical Journal》2009,61(7):1123-1136 We establish conditions for the existence of an invariant set of the system of differential equations \fracdj dt = a( j ),    \fracdxdt = P( j )x + F( j, x ), \frac{{d{\rm{\varphi}} }}{{dt}} = a\left( {\rm{\varphi}} \right),\quad \frac{{dx}}{{dt}} = P\left( {\rm{\varphi}} \right)x + F\left( {{\rm{\varphi}}, x} \right),      16. Roots of Ehrhart polynomials arising from graphs      Tetsushi Matsui  Akihiro Higashitani  Yuuki Nagazawa  Hidefumi Ohsugi  Takayuki Hibi 《Journal of Algebraic Combinatorics》2011,34(4):721-749 Several polytopes arise from finite graphs. For edge and symmetric edge polytopes, in particular, exhaustive computation of the Ehrhart polynomials not merely supports the conjecture of Beck et al. that all roots α of Ehrhart polynomials of polytopes of dimension D satisfy −D≤Re(α)≤D−1, but also reveals some interesting phenomena for each type of polytope. Here we present two new conjectures: (1) the roots of the Ehrhart polynomial of an edge polytope for a complete multipartite graph of order d lie in the circle |z+\fracd4| ￡ \fracd4|z+\frac{d}{4}| \le \frac{d}{4} or are negative integers, and (2) a Gorenstein Fano polytope of dimension D has the roots of its Ehrhart polynomial in the narrower strip -\fracD2 ￡ Re(a) ￡ \fracD2-1-\frac{D}{2} \leq \mathrm{Re}(\alpha) \leq \frac{D}{2}-1. Some rigorous results to support them are obtained as well as for the original conjecture. The root distribution of Ehrhart polynomials of each type of polytope is plotted in figures.      17. Global existence and uniqueness for the magnetic Hartree equation      Pei Cao 《Journal of Evolution Equations》2011,11(4):811-825 In this paper, we consider lliut=Hu+\frac1|x|*|u|2u,    (x,t) ? \mathbbRN×\mathbbR.\begin{array}{ll}iu_{t}=Hu+\frac{1}{|x|}*|u|^{2}u,\quad (x,t)\in \mathbb{R}^{N}\times\mathbb{R}.\end{array}      18. Spectral decomposition and Gelfand’s theorem      A. Driouich  O. El-Mennaoui  M. Jazar 《Semigroup Forum》2010,80(3):391-404 We consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set $\sigma(A)=\{i\lambda_{k};k\in\mathbb{\mathbb{Z}}^{*}\}We consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set s(A)={ilk;k ? \mathbb\mathbbZ*}\sigma(A)=\{i\lambda_{k};k\in\mathbb{\mathbb{Z}}^{*}\} is discrete and satisfies ?\frac1|lk|ldkn < ￥\sum \frac{1}{|\lambda_{k}|^{\ell}\delta_{k}^{n}}<\infty , where ℓ is a nonnegative integer and dk=min(\frac|lk+1-lk|2,\frac|lk-1-lk|2)\delta _{k}=\min(\frac{|\lambda_{k+1}-\lambda _{k}|}{2},\frac{|\lambda _{k-1}-\lambda _{k}|}{2}) . In this case, Theorem 3, we show by using Gelfand’s Theorem that there exists a family of projectors (Pk)k ? \mathbb\mathbbZ*(P_{k})_{k\in\mathbb{\mathbb{Z}}^{*}} such that, for any x∈D(A n+ℓ ), the decomposition ∑P k x=x holds.      19. Criteria of exponential decay for eigenfunctions of some classes of integral operators      V. M. Kaplitsky 《Journal of Mathematical Sciences》2010,165(4):455-462 We study sufficient conditions for exponential decay at infinity for eigenfunctions of a class of integral equations in unbounded domains in ℝ n . We consider integral operators K whose kernels have the form k( x,y ) = c( x,y )\frace - a| x - y || x - y |b , ( x,y ) ? W×W, k\left( {x,y} \right) = c\left( {x,y} \right)\frac{{{e^{ - \alpha \left| {x - y} \right|}}}}{{{{\left| {x - y} \right|}^\beta }}},\,\left( {x,y} \right) \in \Omega \times \Omega,      20. Generalized uncertainty inequalities      Alessio Martini 《Mathematische Zeitschrift》2010,265(4):831-848 The Heisenberg–Pauli–Weyl (HPW) uncertainty inequality on \mathbbRn{\mathbb{R}^n} says that || f ||2 ￡ Ca,b|| |x|a f||2\fracba+b|| (-D)b/2f||2\fracaa+b.\| f \|_2 \leq C_{\alpha,\beta}\| |x|^\alpha f\|_2^\frac{\beta}{\alpha+\beta}\| (-\Delta)^{\beta/2}f\|_2^\frac{\alpha}{\alpha+\beta}.      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

On the log-concavity of Hilbert series of Veronese subrings and Ehrhart series

For every positive integer n, consider the linear operator U _n on polynomials of degree at most d with integer coefficients defined as follows: if we write ${\frac{h(t)}{(1 - t)^{d + 1}}=\sum_{m \geq 0} g(m) \, t^{m}}For every positive integer n, consider the linear operator U _n on polynomials of degree at most d with integer coefficients defined as follows: if we write \frach(t)(1 - t)^{d + 1}=?_{m 3 0} g(m)  t^m{\frac{h(t)}{(1 - t)^{d + 1}}=\sum_{m \geq 0} g(m) \, t^{m}} , for some polynomial g(m) with rational coefficients, then \fracU_nh(t)(1- t)^d+1 = ?_{m 3 0}g(nm)  t^m{\frac{{\rm{U}}_{n}h(t)}{(1- t)^{d+1}} = \sum_{m \geq 0}g(nm) \, t^{m}} . We show that there exists a positive integer n _d , depending only on d, such that if h(t) is a polynomial of degree at most d with nonnegative integer coefficients and h(0) = 1, then for n ≥ n _d , U _n h(t) has simple, real, negative roots and positive, strictly log concave and strictly unimodal coefficients. Applications are given to Ehrhart δ-polynomials and unimodular triangulations of dilations of lattice polytopes, as well as Hilbert series of Veronese subrings of Cohen–Macauley graded rings.     

Uniform estimates for order statistics and Orlicz functions

We establish uniform estimates for order statistics: Given a sequence of independent identically distributed random variables ξ ₁, … , ξ _n and a vector of scalars x = (x ₁, … , x _n ), and 1 ≤ k ≤ n, we provide estimates for \mathbb E   k-min_{1 ￡ i ￡ n} |x_ix_i|{\mathbb E \, \, k-{\rm min}_{1\leq i\leq n} |x_{i}\xi _{i}|} and \mathbb E k-max_{1 ￡ i ￡ n}|x_ix_i|{\mathbb E\,k-{\rm max}_{1\leq i\leq n}|x_{i}\xi_{i}|} in terms of the values k and the Orlicz norm ||y_x||_M{\|y_x\|_M} of the vector y _x  = (1/x ₁, … , 1/x _n ). Here M(t) is the appropriate Orlicz function associated with the distribution function of the random variable |ξ ₁|, G(t) = \mathbb P ({ |x₁| ￡ t}){G(t) =\mathbb P \left(\left\{ |\xi_1| \leq t\right\}\right)}. For example, if ξ ₁ is the standard N(0, 1) Gaussian random variable, then G(t) = ?{\tfrac2p}ò₀^t e^-\fracs22ds {G(t)= \sqrt{\tfrac{2}{\pi}}\int_{0}^t e^{-\frac{s^{2}}{2}}ds }  and M(s)=?{\tfrac2p}ò₀^se^-\frac12t2dt{M(s)=\sqrt{\tfrac{2}{\pi}}\int_{0}^{s}e^{-\frac{1}{2t^{2}}}dt}. We would like to emphasize that our estimates do not depend on the length n of the sequence.     

Green–Samoilenko operator in the theory of invariant sets of nonlinear differential equations

We establish conditions for the existence of an invariant set of the system of differential equations

Roots of Ehrhart polynomials arising from graphs

Several polytopes arise from finite graphs. For edge and symmetric edge polytopes, in particular, exhaustive computation of the Ehrhart polynomials not merely supports the conjecture of Beck et al. that all roots α of Ehrhart polynomials of polytopes of dimension D satisfy −D≤Re(α)≤D−1, but also reveals some interesting phenomena for each type of polytope. Here we present two new conjectures: (1) the roots of the Ehrhart polynomial of an edge polytope for a complete multipartite graph of order d lie in the circle |z+\fracd4| ￡ \fracd4|z+\frac{d}{4}| \le \frac{d}{4} or are negative integers, and (2) a Gorenstein Fano polytope of dimension D has the roots of its Ehrhart polynomial in the narrower strip -\fracD2 ￡ Re(a) ￡ \fracD2-1-\frac{D}{2} \leq \mathrm{Re}(\alpha) \leq \frac{D}{2}-1. Some rigorous results to support them are obtained as well as for the original conjecture. The root distribution of Ehrhart polynomials of each type of polytope is plotted in figures.     

Global existence and uniqueness for the magnetic Hartree equation

In this paper, we consider

Spectral decomposition and Gelfand’s theorem

We consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set $\sigma(A)=\{i\lambda_{k};k\in\mathbb{\mathbb{Z}}^{*}\}We consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set s(A)={il_k;k ? \mathbb\mathbbZ^*}\sigma(A)=\{i\lambda_{k};k\in\mathbb{\mathbb{Z}}^{*}\} is discrete and satisfies ?\frac1|l_k|^ld_kⁿ < ￥\sum \frac{1}{|\lambda_{k}|^{\ell}\delta_{k}^{n}}<\infty , where ℓ is a nonnegative integer and d_k=min(\frac|l_k+1-l_k|2,\frac|l_k-1-l_k|2)\delta _{k}=\min(\frac{|\lambda_{k+1}-\lambda _{k}|}{2},\frac{|\lambda _{k-1}-\lambda _{k}|}{2}) . In this case, Theorem 3, we show by using Gelfand’s Theorem that there exists a family of projectors (P_k)_{k ? \mathbb\mathbbZ^*}(P_{k})_{k\in\mathbb{\mathbb{Z}}^{*}} such that, for any x∈D(A ^n+ℓ ), the decomposition ∑P _k x=x holds.     

Criteria of exponential decay for eigenfunctions of some classes of integral operators

We study sufficient conditions for exponential decay at infinity for eigenfunctions of a class of integral equations in unbounded domains in ℝ ⁿ . We consider integral operators K whose kernels have the form

Generalized uncertainty inequalities

The Heisenberg–Pauli–Weyl (HPW) uncertainty inequality on \mathbbRⁿ{\mathbb{R}^n} says that