The inverse problem of geometric and golden means of positive definite matrices

In this paper we prove that the inverse mean problem of geometric and golden means of positive definite matrices

一个素变量丢番图问题

设k和r是满足k≥3及r≥Ψ(k)+1的正整数,这里当3≤k≤4时,Ψ(k)=2~(k-1);而当k≥5时,Ψ(k)=1/2k(k+1).假定δ和ε是给定的足够小的正数,λ_1,λ_2,…,λ_(r+1)是不全同号且两两之比不全为有理数的非零实数.对于任意实数η与0σ2~(1-2k)/r-1,证明了:存在一个正数序列X→+∞,使得不等式|λ_1p_1~k+λ_2p_2~k+···+λ_rp_r~k+λ_(r+1)p_(r+1)+η|(max(1≤j≤r+1)p_j)~(-σ)有》■X~(■-(2~(1-2k))/(r-1)+ε组素数解(p_1,p_2,…,p_(r+1)),这里(δX)~(1/k)≤p_j≤X~(1/k)(1≤j≤r)及δX≤p_(r+1)≤X.这改进了之前的结果.     

自正则化和Davis大数律和重对数律的精确渐近性

本文证明了自正则化Davis大数律和重对数律的精确渐近性, 即 {\heiti\bf 定理1}\hy 设$\ep X=0$, 且$\ep X^2I_{(|X|\leq x)}$在无穷远处是缓变函数, 则$\lim_{\varepsilon\searrow0}\varepsilon^2\tsm_{n\geq3}\frac{1}{n\log n}\pr\Big(\Big|\frac{S_n}{V_n}\Big|\geq\varepsilon\sqrt{\log\log n}\Big)=1.${\heiti\bf 定理2}\hy 设$\ep X=0$, 且$\ep X^2I_{(|X|\leq x)}$在无穷远处是缓变函数, 则对本文证明了目正则化Davis大数律和重对数律的精确渐近性,即定理1设EX=0,且EX~2I_(|x|≤x)在无穷远处是缓变函数,则■定理2设EX=0,且EX~2I_(|x|≤x)在无穷远处是缓变函数,则对0≤δ≤1,有■其中N为标准正态随机变量．     

Exact Distribution of the Local Score for Markovian Sequences

Let be a sequence of letters taken in a finite alphabet Θ. Let be a scoring function and the corresponding score sequence where X _i = s(A _i ). The local score is defined as follows: . We provide the exact distribution of the local score in random sequences in several models. We will first consider a Markov model on the score sequence , and then on the letter sequence . The exact P-value of the local score obtained with both models are compared thanks to several datasets. They are also compared with previous results using the independent model.     

A uniform version of a theorem by Dvir and Moran

Acta Mathematica Hungarica - Let $$0\leq d\leq n$$ be integers and $$\mathcal{F}$$ a family with $${\rm VCdim} ( \mathcal{F} \Delta \mathcal{F} )\leq d$$ . Analogously to the celebrated...     

广义Macaulay-Northcott模与Tor-群

Let （S,≤） be a strictly totally ordered monoid which is also artinian, and R a right noetherian ring. Assume that M is a finitely generated right R-module and N is a left Rmodule. Denote by [[MS,≤]] and [NS,≤] the module of generalized power series over M, and the generalized Macaulay-Northcott module over N, respectively. Then we show that there exists an isomorphism of Abelian groups：Tori[[ RS,≤]]（[[MS,≤]],[NS,≤]）≌ s∈S ToriR （M,N）.     

三维液晶关于压力水平梯度的相关正则性准则

This paper is devoted to investigating regularity criteria for the 3-D nematic liquid crystal flows in terms of horizontal derivative components of the pressure and gradient of the orientation field. More precisely, we mainly proved that the strong solution(u, d)can be extended beyond T, provided that the horizontal derivative components of the pressure■ and gradient of the orientation field satisfy■ and■.     

Norm Inequalities for Commutators of Self-adjoint Operators

Let A, B, and X be bounded linear operators on a complex separable Hilbert space. It is shown that if A and B are self-adjoint with and for some real numbers a ₁, a ₂, b ₁, and b ₂, then for every unitarily invariant norm|||·|||,

An Extension of the Admissibility-Type Conditions for the Exponential Dichotomy of C 0-Semigroups

In the present paper we obtain a sufficient condition for the exponential dichotomy of a strongly continuous, one-parameter semigroup , in terms of the admissibility of the pair . It is already known the equivalence between the -admissibility condition and and the hyperbolicity of a C ₀-semigroup , when we assume a priori that the kernel of the dichotomic projector (denoted here by X ₂) is T(t)-invariant and is an invertible operator. We succeed to prove in this paper that the admissibility of the pair still implies the existence of an exponential dichotomy for a C ₀-semigroup even in the general case where the kernel of the dichotomic projector, X ₂, is not assumed to be T(t)-invariant.      

Size of graphs and digraphs with given diameter and connectivity constraints

Acta Mathematica Hungarica - We determine the maximum size of a graph of given order, diameter, and edge-connectivity $$\lambda$$ for $$2\leq \lambda \leq 7$$ . This completes the determination of...     

Small bound for birational automorphism groups of algebraic varieties (with an Appendix by Yujiro Kawamata)

We give an effective upper bound of |Bir(X)| for the birational automorphism group of an irregular n-fold (with n = 3) of general type in terms of the volume V = V(X) under an “albanese smoothness and simplicity” condition. To be precise, . An optimum linear bound is obtained for those threefolds with non-maximal albanese dimension. For all n ≥ 3, a bound is obtained when alb _X is generically finite, alb(X) is smooth and Alb(X) is simple. The author is supported by an Academic Research Fund of NUS.     

具$p$-Laplacian 算子的多点边值问题迭代解的存在性

利用单调迭代技巧和推广的Mawhin定理得到下述带有p-Laplacian算子的多点边值问题迭代解的存在性,{(Фp(u'))' f(t,u, Tu)=0, 0(≤)t(≤)1,u(0)=q-1∑i=1γiu(δi),u(1)=m-1∑i=1ηiu(ξi),其中Фp(s)=|s|p-2s,p＞1;0＜δi＜1,γi＞0,1(≤)i(≤)q-1;0＜ξi＜1,ηi(≥)0,1(≤)i(≤)m-1且q-1∑i=1γi＜1,m-1∑i=1ηi(≤)1;Tu(t)=∫t0k(t,s)u(s)ds,k(t,s)∈C(I×I,R ).     

关于Banach空间一阶非线性脉冲积分-微分方程初值问题解存在性的注记

设m(t)∈C[Jk,R ](k=1,2,…,m),且满足不等式m(t)<(L1 L2t)∫tn(s)ds L3t∫a m(s)ds ∑o0满足KaLs(eδ(L1 aL2)-1)相似文献   

Aizenman's Theorem for Orthogonal Polynomials on the Unit Circle

For suitable classes of random Verblunsky coefficients, including independent, identically distributed, rotationally invariant ones, we prove that if $$ \bbE \biggl( \int\f{d\theta}{2\pi} \biggl|\biggl( \f{\calC + e^{i\theta}}{\calC-e^{i\theta}} \biggr)_{k\ell}\biggr|^p \biggr) \leq C_1 e^{-\kappa_1 \abs{k-\ell}} $$ for some $\kappa_1 < 0$ and $p < 1$, then for suitable $C_2$ and $\kappa_2 >0$, $$ \bbE \Bigl( \sup_n \abs{(\calC^n)_{k\ell}}\Bigr) \leq C_2 e^{-\kappa_2 \abs{k-\ell}}. $$ Here $\calC$ is the CMV matrix.     

Gradient Estimates for p-Laplacian Lichnerowicz Equation on Noncompact Metric Measure Space

Abstract In this paper,the authors obtain the gradient estimates for positive solutions to the weighted p-Laplacian Lichnerowicz equation■on noncompact smooth metric measure space,where c is a nonnegative constant,and p,σ(1p≤2,σ≤p-1)are real constants.Moreover,by the gradient estimate,they can get the corresponding Liouville theorem and Harnack inequality.     

Asymptotic Expansions of the Cubic Spline Collocation Solution for Second-Order Ordinary Differential Equations

In this paper, we consider the following problem $$\begin{cases}u''(x)=f(x,u(x),u'(x)), a\leq x\leq b,\\u(a)=u(b)=0,\end{cases}$$ and obtain the theorem.     

OM AN INVERSE THEOREM OF APPROXIMATION

The author gives some disagreement to the following result, which is published in [1]. Let ${L_{n}(f)}$ be mass-concerntative,$\phi\rightarrow 0(n\rightarrow \infty), 0<\alpha\leq2$ and $$C^{-1}\leq \phi_{n+1}/\phi_{n}\leq C (n=1,2,\ldots)$$ for some constrant $C>0$. Then for any $f\in C[-2a,2a]$, $$\parallel L_{n}(f)-f\parallel_{C[ a,a]}= O(\phi^{\alpha}_{n})$$ inplies $f \in Lip^{*}\alpha$, where $$Lip*\alpha={f\in C[-2a,2a]|\omega_{2}(f,\delta)_{[-2a,2a]}=O(\delta^{\alpha})}.$$ Then some similar results on $C_{2\pi$ are given, and further some results on $C[-2a,2a]$ are established by adding some proper conditions.     

On a second conjecture of Stolarsky: the sum of digits of polynomial values

Let q, r ≥ 2 be integers, and denote by s _q the sum-of-digits function in base q. In 1978, K.B. Stolarsky conjectured that $$\lim_{N \to \infty} \frac{1}{N} \sum_{n \leq N} \frac{s_2(n^r)}{s_2(n)} \leq r.$$ In this paper we prove this conjecture. We show that for polynomials ${P_1(X), P_2(X) \in \mathbb{Z}[X]}$ of degrees r ₁, r ₂ ≥ 1 and integers q ₁, q ₂ ≥ 2, we have $$\lim_{N \to \infty} \frac{1}{N} \sum_{n \leq N}\frac{s_{q_1}(P_1(n))}{s_{q_2}(P_2(n))} = \frac{r_1 (q_1 - 1) {\rm log}q_2}{r_2(q_2 - 1) {\rm log} q_1}.$$ We also present a variant of the problem to polynomial values of prime numbers.     

An Integral Representation Result for the Γ-Limit of Functionals with Non-standard Growth Conditions in the Case of Elasticity

An integral representation result on regular functions is proved for the o -limit of a sequence of integral functionals defined in the vectorial case and modelled on elasticity theory functional Z z f (( x , e ( u )) dx where convex lagrangians satisfy a non-standard estimate $$ -c_{0} + c_{1} | \xi|^{\alpha }\leq f ( (x,\xi ) \leq c_{0} + c_{2} | \xi|^{\beta },\quad 1 \lt \alpha \leq \beta \lt \frac {n\alpha }{n-\alpha },\enskip c_{0}\geq 0,\enskip c_{1},c_{2} \gt 0. $$ When the limit integrand does not show Lavrent'ev phenomenon the representation result is also true on the whole space W 1, f ( z ; R n ).     

Inequalities of the Hermite–Hadamard Type Involving Numerical Differentiation Formulas

Results in Mathematics - We observe that the Hermite–Hadamard inequality written in the form $$f\left(\frac{x+y}{2}\right)\leq\frac{F(y)-F(x)}{y-x}\leq\frac{f(x)+f(y)}{2}$$ may be viewed as...