Ｙ-数值半径的范数性质(英文)

Given an n×n complex matrix A and an n-dimensional complex vector y=(ν1 , ··· , νn ), the y-numerical radius of A is the nonnegative quantity ry(A)=max{n∑j=1ν*jAx︱:Axj︱: x*jxj=1,xj ∈Cn}.Here Cn is an n-dimensional linear space overthe complex field C. For y = (1, 0, ··· , 0) it reduces to the classical radius r(A) =max {|x*Ax|: x*x=1}.We show that ry is a generalized matrix norm if and only ifn∑j=1νj≠ 0.Next, we study some properties of the y-numerical radius of matrices andvectors with non-negative entries.     

Ruzsa’s constant on additive functions

A function f : N → R is called additive if f(mn)= f(m)+f(n)for all m, n with(m, n)= 1. Let μ(x)= max n≤x(f(n)f(n + 1))and ν(x)= max n≤x(f(n + 1)f(n)). In 1979, Ruzsa proved that there exists a constant c such that for any additive function f , μ(x)≤ cν(x 2 )+ c f , where c f is a constant depending only on f . Denote by R af the least such constant c. We call R af Ruzsa's constant on additive functions. In this paper, we prove that R af ≤ 20.     

On a certain sum-product estimate in fields of prime order

Let Fp be the finite field of p elements with p prime.If A is a subset of Fp and g is an element of F*p with order ν,then max{|A + g·A|,|A·A|} (ν/(ν + |A|2) )1/12|A|13/12.     

Sign changes of Fourier coefficients of Maass eigenforms

Let f be a Maass eigenform that is a new form of level N on Γ0(N),with Laplace eigenvalue 1/4 + νf2 . Then,all its Fourier coefficients {λf(n)}∞n=1 are real,and we may normalize so that λf (1) = 1. It is proved,in this paper,that the first sign change in the sequence {λf(n)}n∞=1 occurs at some n satisfying n《{(3+|νf|2)N }1/2-δ and (n,N)=1. This generalizes the previous results for holomorphic Hecke eigenforms.     

SUPPORT SIZES OF COMPLETELY DECOMPOSABLE TS （ν, 4）

In this article the spectrum of support sizes of a completely decomposable TS(ν,,4)for any positive integer ν≡1,3 (mod 6) is determined.     

MODERATE DEVIATIONS FOR THE BESSEL CLOCK

By the method of change measures,the moderate deviations for the Bessel clock t 0 ds X(ν) s is studied,where(X(ν) t,t≥ 0) is a squared Bessel process with index ν0.The rate function can be given explicitly.Furthermore,the functional moderate deviations for the Bessel clock are obtained.     

EXISTENCE OF PERIODIC SOLUTIONS OF SYSTEM OF DIFFERENCE EQUATIONS

This paper studies the nonautonomous nonlinear system of difference equationsΔx(n)=A(n)x(n)+f(n,x(n)),n∈Z,(*) where x(n)∈R~N,A(n)=(a_(ij)(n))N×N is an N×N matrix,with a-(ij)∈C(R,R) for i,j= 1,2,3,...,N,and f=(f_1,f_2,...,f_N)~T∈C(R×R~N,R~N),satisfying A(t+ω)=A(t),f(t+ω,z)=f(t,z) for any t∈R,(t,z)∈R×R~N andωis a positive integer.Sufficient conditions for the existence ofω-periodic solutions to equations (*) are obtained.     

Embeddings of Kirkman Systems KS(2,4,v)

A balanced incomplete block design(BIBD)B(κ,λ;ν)is a pair(X,A),whereX is a v-set of elements(called points)and A is a collection of k-subsets of X(called blocks)such that every unordered pair of points of X appears in exactlyλ blocks of A.A B(κ,1;ν)is called a Steiner system and denoted by S(2,κ,ν). A parallel class in a BIBD is a set of blocks that forms a partition of the     

On Rationality of Generating Function for the Number of Spanning Trees in Circulant Graphs

Let F(x)=∑∞n=1 Tsi,s2,...,sk(n)x^n be the generating function for the number,Ts1bs2,...,sk(n) of spanning trees in the circulant graph Cn(s1,S2,...,Sk).We show that F(x)is a rational function with integer coefficients satisfying the property F(x)=F(l/x).A similar result is also true for the circulant graphs C2n(s1,S2,....,Sk,n)of odd valency.We illustrate the obtained results by a series of examples.     

On Kantorovich-Stieltjes operators

Let ν be a finite Borel measure on[0,1]The Kantorovich-Stieltjes polynomials are de-fined byK_n ν=(n+1)N_(k,n)(nN),where N_(k,n)(x)=x~k(1-x)~(n-k)(x[0,1],k=1,2,…,n)are the basic Bernsteinpolynomials and I_(k,n):=[k/(n+1),(k+1)/(n+1)](k=0,1,…,n;nN).We prove that the maximaloperator of the sequence(K_n)is of weak type and the sequence of polynomials(K_n ν)con-verges a.e.on[0,1]to the Radon-Nikodym derivative of the absolutely continuous part of     

经典分形集测度上估的计算机搜索Ⅱ──对典型例子Sierpinski垫片计数技术和格点跟踪技术的剖析

相同生标题的文献山,通过编码技术导出了义献问给出的Sierninski垫片Hausdorff测度＊”（S的L方估值函数…X）．本文的第一个目的,是利用计数技术,推导出更好的上方估值函数“一及其可计算的过剩近似值。N（）的数学表达式其中l二0．lllZ…2n…,t二k·2－N二0．tltZ…iN,（二0f122··＜N是二进制小数,ZI—…二打．1二0,。三2满足！1＋…十认．1二！。l＝1,＊表示l的整数部分,unitx二1,若xZ地否则unitx二队尽管op00的上述表达式把人们从看图作业和手工计算的状态中解脱出来,但其计算复杂性却仍然是指数爆炸的,对大如10的N…     

THE FIRST BOUNDARY VALUE PROBLEM FOR A CLASS OF QUASILINEAR DEGENERATE ELLIPTIC EQUATIONS

In this paper, the first boundary value problem for quasilinear equation of the form △A（u,x）＋∑i=1^m δb^i（u,x）/δxi＋c（u,x）=0,Au（u,x） ≥0is studied. By using the compensated compactness theory, some results on the existence of weak solution are established. In addition, under certain condition the uniqueness of solution is proved.     

滞后线性控制系统的输出能控性

本文对具有滞后的线性泛函微分方程所确定的滞后线性控制系统ｘ（ｔ）＝Ａ（ｔ）ｘ（ｔ）十Ｂ（ｔ）ｘ（ｔ－１）＋Ｃ（ｔ）ｕ（ｔ）ｔ∈（０，Ｔ］ｘ（ｔ）＝Φ（ｔ）ｔ∈［１，０］ｙ（ｔ）＝Ｄ（ｔ）ｘ（ｔ）＋Ｅ（ｔ）ｕ（ｔ）ｔ∈［－１，Ｔ］给出一些关于欧氏空间完全输出能控性的重要结论。并且对自治的滞后线性控制系统做了进一步研究，得到令人满意的充要条件；特别是通过与相应的常微分方程所确定的控制系统的输出能控性相比较，得到比较简明的判则。     

一类抛物型方程系数反问题的分裂算法

考虑利用终端时刻的温度u(x,T)=Z_T(x)反演热传导方程u_t-a~2u_(xx) q(x)u=0,x∈(0,1)中的未知系数q(x)的反问题.通过引进变换v(x,t)=(u_t(x,t)/u(x,t))将此非线性不适定问题的求解分解为两步.首先利用输入数据迭代求解一个非线性的正问题(该过程独立于未知系数),得到其迭代解v~(k)(x,t).其次利用q(x)与v(x,t)的关系式求出q(x)的近似解.对提出的反演方法,证明了采用的变换的可行性,得到了原反问题与由变换后的非线性正问题反演q(x)的等价性并且证明了迭代解的收敛性,给出了收敛速度.数值结果表明了该方法的有效性.     

求解非线性对流-扩散问题的特征—差分法

近年来,已有不少文献讨论发展型方程的特征-有限元或特征-差分解法.在这一方向上,J.Douglas,Jr.及T.F Russell[1]是重要的工作,它讨论了一维对流-扩散方程C(x)?u/?t+b(x)?u/?x-?/?x(a(x)?u/?x)=f(x,t)的数值求解问题,建立了基于线性插值与基于二次插值的两种特征-差分格式,给出了误差分析,将前一种格式推广到     

关于双曲型方程解的爆破

在RNR+(N2)中考虑非线性双曲型方程 utt-DI(aij(x)Dju)=|u|p-1u.Kato1980年证明了当1<p [SX(]N+1[]N-1[SX)]时,Cauchy问题的解可能在有限时刻爆破.本文使用不同的方法估计, 把Kato的结果改进为1<p<[SX(]N+3[]N-1[SX)].     

四阶微分方程的迭代解

利用一个构造性的方法,在假设边值问题存在上解α和下解β,满足β≤α的前提下,给出了两个单调序列它们一致收敛于如下两类边值问题的极值解u(4)(x)-Mu″(x)=f(x,u(x),u'(x),u″(x),u″'(x)),0＜x＜1,u(0)=u'(1)=u″(0)=u″'(1)=0;u(4)(x)-Mu″(x)=g(x,u(x),u'(x),u″(x)),0＜x＜1,u(0)=u'(1)=u″(0)=u″'(1)=0.     

四阶方程两点边值问题Hermite有限元解的渐近展式与外推

1引言有限元解的渐近展式是提高微分方程数值解精度的重要工具,比如亏量校正和外推就是建立在有限元解的渐近展式的基础之上．许多作者对此进行了大量的研究(见[1]-[4]),特别是文[1],提出了在研究有限元解的渐近展式中十分有用的能量嵌入技巧．本文利用能量嵌入定理得到了四阶方程两点边值问题Hermite有限元解及其二阶平均导数的渐近展式,进一步我们还讨论了它们的Richardson外推公式．考虑四阶方程两点边值问题     

STABILITY OF THE RAREFACTION WAVE FOR THE GENERALIZED KDV-BURGERS EQUATION

This paper is concerned with the stability of the rarefaction wave for the generalized KdV-Burgers equationRoughly speaking, under the assumption that u_ < u+, the solution u(x,t) to Cauchy problem (1) satisfying sup \u(x,t) -uR(x/t)| -0 as t - , where uR(x/t) is the rarefac-tion wave of the non- viscous Burgers equation ut + f(u)x=0 with Riemann initial data     

关于线性偏微分方程的Cauchy问题的唯一性

<正> Cauchy问题的唯一性是偏微分方程的基本问题之一.经典的Cauchy-Kowalewski定理断言,解析方程或方程组的解析解是唯一的.1901年,Holmgren证明了,线性的解析方程或方程组的光滑解的唯一性.在取消关于系数的解析性的假设这个方向上的第一个结果是由Carleman在1939年给出的,他证明了两个自变量的相应结果,其中假设方程的主部的系数是实的,以及特征根是单重的,因而特征根的虚部如果不恒为零则总不为零.