C~n中单位球面上的不变g-函数及BMOA

设B是C~n中的单位球,S是C~n中的单位球面。■(f)是定义在S上的不变g-函数．设f∈ BMOA,若存在正测度集ES,使■(f)＜ ∞在E上成立,则■(f)＜ ∞在S上几乎处处成立,同时■(f)∈BMO(S)且存在常数C,使得‖■(f)‖≤C‖f‖.     

李超代数■(2)到奇Hamilton超代数HO的低维上同调

本文在特征p> 3的域上,采用将典型李超代数■(2)嵌入到奇Hamilton超代数HO的零阶化分支的方法,使得在伴随表示的意义下,HO成为■(2)-模.通过对HO进行子模分解和权空间分解,得到了■(2)到HO的低维上同调.     

带自同构箭图在有限域上的表示

设F_q是一个含有q个元素的有限域且κ=F_q是它的代数闭包.设Q是一个带有自同构σ的箭图且■(Q,σ;q)是对应的F_q-代数,本文主要利用箭图Q在域κ上的F-稳定表示研究■(Q,σ;q)-模,并且给出不可分解■(Q,σ;q)-模同构类个数的若干多项式.特别地,本文将引入σ-绝对不可分解■(Q,σ;q)-模的概念.当Q是仿射箭图时,给出具有给定维数向量的σ-绝对不可分解■(Q,σ;q)-模同构类个数的公式.     

曲线■(m,n>0)为双曲线的证明

<正>在人民教育出版社出版的?普通高中教科书数学(B版)?必修第一册"均值不等式及其应用"的内容中,给出曲线方程■(图1),此曲线方程可以推广到一般形式为:■(m>0,n>0),该曲线是双曲线,但是由于方程形式与双曲线标准方程■(a>0,b>0)差异较大,     

E-紧框架小波来自MRA的特征刻画

设E=■或■,■(x)∈L~2(R~2)且■_(jk)(x)=2■(E~jx-k),其中j∈Z,k∈Z~2.若{■_(jk)|jJ∈Z,k∈Z~2}构成L~2(R~2)的紧框架,则称■(x)为E-紧框架小波.本文给出E-紧框架小波是MRA E-紧框架小波的一个充要条件,即E紧框架小波■来自多尺度分析当且仅当线性空间F_■(ξ)的维数为0或1,其中F_■(ξ)=■(ξ)|j■1},■_j(ξ)={■((E~T)~j(ξ+2kπ))}_(k∈EZ~2,j■1。     

自相似集的球密度性质

本文讨论利用上、下球密度计算自相似集的Hausdorff中心测度与填充测度的问题.设E是满足强分离条件的自相似集,s为其Hausdorff维数,μ为定义于E上的自相似测度,则有如下结论:(1)如果存在x₀∈E,使得x₀关于μ的上球密度■(μ,x₀)=■,则对μ-几乎所有x∈E,有■(μ,x)≥■;(2)如果存在y₀∈E,使得y₀关于μ的下球密度■(μ,y₀)=■,则对μ-几乎所有y∈E,有■(μ,g)≤■.运用这一结论,对自相似集的测度计算问题进行了讨论.     

一类具复杂偏差变元的中立型微分方程的周期解

本文研究了一类具复杂偏差变元的中立型泛函微分方程■(t)=θ■(t-r) α(t)f(x(t)) β(t)g(x(x(t))) p(t)的周期解的存在性,得到了周期解存在的充分条件,并给出了所得结论的几个简单应用．     

超Schr?dinger代数■(1/1)的上同调

本文具体计算了系数在超Schr?dinger代数■(1/1)的平凡模和有限维不可约模中的第一阶上同调群与第二阶上同调群,并给出了系数在通用包络代数U(■(1/1))中■(1/1)的第一阶与第二阶上同调群的维数是无限维的.     

模糊可测函数

本文首先定义了模糊σ-域,算子～:■(X)→■(X)和σ:■(X)→■(X),证明了算子～与算子σ是可交的。然后利用集值分析知识讨论了模糊函数的逆函数,模糊函数的可测性以及它的可积性,导出了模糊函数可测、可积的充分必要条件。     

Bergman空间上Cowen-Douglas算子的换位

令H~∞(D)表示单位圆盘D上的有界解析函数全体构成的代数,对于■∈H~∞(D),在某种条件下,证明T=M■是一个指标为n的Cowen-Douglaus算子,并且给出了■(T)/rad■(T)可交换的一个充分条件.当n=1时,刻画了T的换位.     

On the average of the number of imaginary quadratic fields with a given class number

Let \(\mathcal {F}(h)\) be the number of imaginary quadratic fields with class number h. In this note, we improve the error term in Soundararajan’s asymptotic formula for the average of \(\mathcal {F}(h)\). Our argument leads to a similar refinement of the asymptotic for the average of \(\mathcal {F}(h)\) over odd h, which was recently obtained by Holmin, Jones, Kurlberg, McLeman and Petersen.     

关于实二次域类数的上下界

设Ｄ是无平方因子正整数，Δ，ｈ，ε分别是实二次的判别式、类数和基本单位数，本文运用初等方法证明了：（ｉ）在假定广义Ｒｉｅｍａｎｎ猜想成立的条件下，当Δ＞１０￣９时，（ｉｉ）当Ｄ＝ｐ，ｐ是奇素数时，     

也谈实二次域类数的可除性

设ｄ无平方因子，ｈ（ｄ）是二次域Ｑ（ｄ）的类数，本文证明了：若１＋４ｋ２ｎ＝ｄａ２，ａ，ｋ＞１，ｎ＞２为正整数，且ａ＜０．９ｋ３５ｎ或ｎ的奇素因子ｐ和ｋ的素因子ｑ均适合（ｐ，ｑ－１）＝１，则除（ａ，ｄ，ｋ，ｎ）＝（５，４１，２，４）以外，ｈ（ｄ）≡０（ｍｏｄｎ）．同时，我们猜测：上述结果中的条件（ｐ，ｑ－１）＝１是不必要的．     

关于Q(4k~(2n)+1)的理想类群的循环子群

设ｄ,ａ,ｋ,ｎ是适合４ｋ２ｎ＋１＝ｄａ２,ｋ＞１,ｎ＞２,ｄ无平方因子的正整数;又设Ｃ（Ｋ）和ｈ（Ｋ）分别是实二次域Ｋ的理想类群和类数．本文证明了：当ａ＜０．５ｋ０．５６ｎ时,则ｈ（ｋ）＝０（ｍｏｄｎ）和Ｃ（Ｋ）必有ｎ阶循环子群．     

丢番图方程与实二次域类数的可除性

设ｄ无平方因子，ｈ（ｄ）是二次域的类数。本文证明了：在方程Ｕ￣２－ｄＶ￣２＝４，（Ｕ，Ｖ）＝１有整数解时，丢番图方程４ｘ￣（２ｎ）－ｄｙ￣２＝－１，ｎ＞２无｜ｙ｜＞１的整数解；如果正整数ａ，ｋ，ｎ满足，ｋ＞１，ｎ＞２且而是Ｐｅｌｌ方程ｘ￣２－ｄｙ￣２＝－１的基本解，则ｈ（ｄ）≡０（ｍｏｄｎ）。     

On the divisor function in short intervals

Let d(n) denote the number of positive divisors of the natural number n. The aim of this paper is to investigate the validity of the asymptotic formula

$\begin{array}{lll}\sum \limits_{x < n \leq x+h(x)}d(n)\sim h(x)\log x\end{array}$

for \({x \to + \infty,}\) assuming a hypothetical estimate on the mean

$\begin{array}{lll} \int \limits_X^{X+Y}(\Delta(x+h(x))-\Delta (x))^2\,{d}x, \end{array}$

which is a weakened form of a conjecture of M. Jutila.

     

Assessing the impact of the environmental contamination on the transmission of Ebola virus disease (EVD)

This paper deals with the following biological question: how influential is the environmental contamination on the transmission of EVD? Based on the works in (Bibby et al., Environ Sci Technol Lett 2:2–6, 2015; Leroy et al., Nature 438: 575–576, 2005; World Health Organization. Unprecedented number of medical staff infected with Ebola), we design a new mathematical model to address this question by assessing the effect of the Ebola virus contaminated environment on the dynamical transmission of EVD. The formulated model captures two infection pathways through both direct human-to-human transmission and indirect human-to-environment-to-human transmission by incorporating the environment as a transition and/or reservoir of Ebola viruses. We compute the basic reproduction number \({\mathcal {R}}^{env}_0\) for the model with environmental contamination and prove that the disease-free equilibrium is globally asymptotically stable (GAS) whenever \({\mathcal {R}}^{env}_0 \le 1\). When \({\mathcal {R}}^{env}_0 > 1\), we show that the said model has a unique endemic equilibrium which is GAS. Similar results hold for the free environmental contamination sub-model (without the incorporation of the indirect transmission). More precisely, for the latter model, calculate the corresponding basic reproduction number \({\mathcal {R}}^{h}_0\) and establish the GAS of the disease-free and endemic equilibria, whenever \({\mathcal {R}}^{h}_0 \le 1\) and \({\mathcal {R}}^{h}_0 > 1\), respectively. At the endemic level, we show that the number of infected individuals for the full model with the environmental contamination is greater than the corresponding number for the free environmental contamination sub-model. In conjunction with the inequality \({\mathcal {R}}^{h}_0 < {\mathcal {R}}^{env}_0\), our finding suggests a negative answer to the biological question under investigation, i.e. the contaminated environment plays a detrimental role on the transmission dynamics of EVD by increasing the endemic level and/or the severity of the outbreak. Therefore, it is natural to implement a control strategy which aim at reducing the severity of the disease by providing adequate hygienic living conditions, educate populations at risk to follow rigorously those basic hygienic conditions as well as ask them avoid contact with suspected contaminated objects. Further, we perform numerical simulations to support the theory.     

Multiplicity of virtual levels at the lower edge of the continuous spectrum of a two-particle Hamiltonian on a lattice

We consider a system of two arbitrary quantum particles on a three-dimensional lattice with special dispersion functions (describing site-to-site particle transport), where the particles interact by a chosen attraction potential. We study how the number of eigenvalues of the family of the operators h(k) depends on the particle interaction energy and the total quasimomentum \(k \in \mathbb{T}^3\) (where \(\mathbb{T}^3\) is a three-dimensional torus). Depending on the particle interaction energy, we obtain conditions under which the left edge of the continuous spectrum is simultaneously a multiple virtual level and an eigenvalue of the operator h(0).     

A realization theorem for sets of lengths

Text

By a result of G. Freiman and A. Geroldinger [G. Freiman, A. Geroldinger, An addition theorem and its arithmetical application, J. Number Theory 85 (1) (2000) 59-73] it is known that the set of lengths of factorizations of an algebraic integer (in the ring of integers of an algebraic number field), or more generally of an element of a Krull monoid with finite class group, has a certain structure: it is an almost arithmetical multiprogression for whose difference and bound only finitely many values are possible, and these depend just on the class group. We establish a sort of converse to this result, showing that for each choice of finitely many differences and of a bound there exists some number field such that each almost arithmetical multiprogression with one of these difference and that bound is up to shift the set of lengths of an algebraic integer of that number field. Moreover, we give an explicit sufficient condition on the class group of the number field for this to happen.

Video

For a video summary of this paper, please visit http://www.youtube.com/watch?v=c61xM-5D6Do.     

A note on the normality of unramified,abelian extensions of quadratic extensions

Let F, K and L be algebraic number fields such that , [KF]=2 and [LK]=n. It is a simple consequence of the class field theory that, if L is an abelian, unramified extension of K and (n,h)=1, where h is the class number of F, then L is normal over F. The purpose of this note is to point out the necessity of the condition (n,h)=1 by constructing for any field F with even class number a tower of fields with [KF]=2, [LK]=2 where L is unramified over K, but L is not normal over F.