Characteristic roots and vectors of a diifferentiable family of symmetric matrices

LetA(x) be a differentiable family of k × k symmetric matrices where x runs through a domain D in RⁿWe prove that if λ is a continuous function onDsuch that, for every x εD,λ(x) is a characteristic root of A(x) of constant multiplicity m, then λ is a differentiable function and there exists, locally, a differentiable family of ortho-normal bases for the eigenspace. The case n = 1 has been known in the standard treatises on the perturbation theory for linear operators.     

Convolution integral equations with Gegenbauer function kernel

In this paper we develop a concise and transparent approach for solving Mellin convolution equations where the convolutor is the product of an algebraic function and a Gegenbauer function. Our method is primarily based on

1. the use of fractional integral/differential operators;

2. a formula for Gegenbauer functions which is a fractional extension of the Rodrigues formula for Gegenbauer polynomials (see Theorem 3);

3. an intertwining relation concerning fractional integral/differential operators (see Theorem 1), which in the integer case reads (d/dx)²ⁿ⁺¹ = (x⁻¹ d/dx)ⁿx²ⁿ⁺¹(x⁻¹ d/dx)ⁿ⁺¹.

Thus we cover most of the known results on this type of integral equations and obtain considerable extensions. As a special illustration we present the Gegenbauer transform pair associated to the Radon transformation.     

用广义多项式逼近在正数轴上的函数

<正> 导言伯恩斯坦曾经证明:设 F(x)是偶的整函数,其泰勒系数不是负数,并且它的性(род,genus)大于零.如果 f(x)在(—∞,∞)上连续,并且适合     

Nonlinear system identification employing automatic differentiation

An optimization based state and parameter estimation method is presented where the required Jacobian matrix of the cost function is computed via automatic differentiation. Automatic differentiation evaluates the programming code of the cost function and provides exact values of the derivatives. In contrast to numerical differentiation it is not suffering from approximation errors and compared to symbolic differentiation it is more convenient to use, because no closed analytic expressions are required. Furthermore, we demonstrate how to generalize the parameter estimation scheme to delay differential equations, where estimating the delay time requires attention.     

Comparison theorems for the oscillation of higher order difference equations with deviating arguments

The difference equation Δy + δp(k)f (y (g (k))) = 0, where p(k) is positive, is classified into four cases according to is odd or even and δ is 1 or −1. In each case, we shall offer comparison theorems for the oscillation of the difference equation. Examples are also included to illustrate the importance of the results obtained.     

Krylov subspace methods for eigenvalues with special properties and their analysis for normal matrices

In this paper we propose a general approach by which eigenvalues with a special property of a given matrix A can be obtained. In this approach we first determine a scalar function ψ: C → C whose modulus is maximized by the eigenvalues that have the special property. Next, we compute the generalized power iterations uinj + 1 = ψ(A)u_j, j = 0, 1,…, where u₀ is an arbitrary initial vector. Finally, we apply known Krylov subspace methods, such as the Arnoldi and Lanczos methods, to the vector u_n for some sufficiently large n. We can also apply the simultaneous iteration method to the subspace span{x⁽ⁿ⁾₁,…,x⁽ⁿ⁾_k} with some sufficiently large n, where x^(j+1)_m = ψ(A)x^(j)_m, j = 0, 1,…, m = 1,…, k. In all cases the resulting Ritz pairs are approximations to the eigenpairs of A with the special property. We provide a rather thorough convergence analysis of the approach involving all three methods as n → ∞ for the case in which A is a normal matrix. We also discuss the connections and similarities of our approach with the existing methods and approaches in the literature.     

Betti numbers of perfect homogeneous ideals

In this paper we study the graded minimal free resolution of the ideal, I, of any arithmetically Cohen-Macaulay projective variety. First we determine the range of the shifts (twisting numbers) that can possibly occur in the resolution, in terms of the Hilbert function of I. Then we find conditions under which some of the twisting numbers do not occur. Finally, in some ‘good’ cases, all the Betti numbers are (recursively) computed, in terms of the Hilbert function of I or that of Extⁿ_R(R/I,R), where R is a polynomial ring over a field and n is the height of I in R.     

Weak convergence to the multiple Stratonovich integral

We have considered the problem of the weak convergence, as tends to zero, of the multiple integral processes

in the space , where fL²([0,T]ⁿ) is a given function, and {η(t)}_>0 is a family of stochastic processes with absolutely continuous paths that converges weakly to the Brownian motion. In view of the known results when n2 and f(t₁,…,t_n)=1_{t1<t2<<tn}, we cannot expect that these multiple integrals converge to the multiple Itô–Wiener integral of f, because the quadratic variations of the η are null. We have obtained the existence of the limit for any {η}, when f is given by a multimeasure, and under some conditions on {η} when f is a continuous function and when f(t₁,…,t_n)=f₁(t₁)f_n(t_n)1_{t1<t2<<tn}, with f_iL²([0,T]) for any i=1,…,n. In all these cases the limit process is the multiple Stratonovich integral of the function f.     

Wilson's functional equation for vector and matrix functions

We determine the general solution of the functional equation

where is a 2-divisible abelian group, is a vector-valued function and is a matrix-valued function. Using this result we solve the scalar equation

which contains as special cases, among others, the d'Alembert and Wilson equations and the parallelogram law.

     

｛0,1,3｝问题摄动形式所对应的测度

给定实数λ,α以及Ｒ上（以λ,α为参数）的压缩自相似映射Ｓ１（ｘ）＝λｘ, Ｓ２（ｘ）＝ λｘ＋ａ, Ｓ３（ｘ）＝ λｘ＋３,记满足测度方程ｖ＝（１／３）∑ｉ＝１ｖｏＳｉ－１的唯一概率测度为ｕλ,α本文得到：（１）当固定 λ∈Ａ Ｅ（１／３, ２／５）时,则在 Ｌｅｂｅｓｇｕｅ测度意义下,对于 ａ．ｅ．的 ａ∈（０,１）,测度 ｕλ,α绝对连续,且存在平方可积密度．（２）若λ－１是 Ｐ．Ｖ．数,且 α是λ的有理系数多项式,则测度ｕλ,α是奇异测度．     

On the extreme eigenvalues of hermitian (block) toeplitz matrices

We are concerned with the behavior of the minimum (maximum) eigenvalue λ₀⁽ⁿ⁾ (λ_n⁽ⁿ⁾) of an (n + 1) × (n + 1) Hermitian Toeplitz matrix T_n(ƒ) where ƒ is an integrable real-valued function. Kac, Murdoch, and Szegö, Widom, Parter, and R. H. Chan obtained that λ₀⁽ⁿ⁾ — min ƒ = O(1/n^2k) in the case where ƒ C^2k, at least locally, and ƒ — inf ƒ has a zero of order 2k. We obtain the same result under the second hypothesis alone. Moreover we develop a new tool in order to estimate the extreme eigenvalues of the mentioned matrices, proving that the rate of convergence of λ₀⁽ⁿ⁾ to inf ƒ depends only on the order ρ (not necessarily even or integer or finite) of the zero of ƒ — inf ƒ. With the help of this tool, we derive an absolute lower bound for the minimal eigenvalues of Toeplitz matrices generated by nonnegative L¹ functions and also an upper bound for the associated Euclidean condition numbers. Finally, these results are extended to the case of Hermitian block Toeplitz matrices with Toeplitz blocks generated by a bivariate integrable function ƒ.     

Bondage number of strong product of two paths

The bondage number b(G) of a graph G is the cardinality of a minimum set of edges whose removal from G results in a graph with a domination number greater than that of G. In this paper, we obtain the exact value of the bondage number of the strong product of two paths. That is, for any two positive integers m≥2 and n≥2, b(P_m?P_n) = 7 - r(m) - r(n) if (r(m), r(n)) = (1, 1) or (3, 3), 6 - r(m) - r(n) otherwise, where r(t) is a function of positive integer t, defined as r(t) = 1 if t ≡ 1 (mod 3), r(t) = 2 if t ≡ 2 (mod 3), and r(t) = 3 if t ≡ 0 (mod 3).     

當若阿-富理埃級數的係數

<正> 不能用蔡查羅(Cesaro)的平均法求它的和;這是哈戴和立篤耳武德老早指出過的,他們的解析是依靠着(?)函數的理論。 其後,鐵起馬虛用初等的方法作成如下的實例:     

线性可移算子与可调和的随机过程

<正> §1.引言T.Onoyama 利用了 Weyl-Stone-Titchmarsh 的特征函数(Eigenfunction)展开公式,对具有二阶矩的实的连续机过程(本文中所用的极限,系指在均方意义下的极限)求得了下列隨机函数方程     

U-统计量的精致渐近性

设{X_n．n≥1}是一非退化的i．i．d．随机变量序列,U_n是以二维Borel可测对称函数h(x,y)为核函数的U-统计量．记U_n=2/(n(n-1))Σ_≤i≤j≤nh(X_i,X_j)．本文分别在核函数h(x,y)只有4/3阶矩或4/3+δ,0＜δ≤1的情况下,对非常广泛的一类权函数(x)与边界函数b(x)得到了如下关于U-统计量U_n的精致渐近性:不仅使得已有的结果成为我们的特况,还大大降低了其中的矩条件．     

Some involutory similarities

Matrices A,B over an arbitrary field F, when given to be similar to each other, are shown to be involutorily similar (over F) to each other (i.e.B = CAC^-1for some C = C^-1over F) in the following cases: (1)B= aI - Afor some a ε F and (2) B = A^-1. Result (2) for the cases where char F ≠ 2 is essentially a 1966 result of Wonenburger.     

Stack words, standard Young tableaux, permutations with forbidden subsequences and planar maps

Stack words stem from studies on stack-sortable permutations and represent classical combinatorial objects such as standard Young tableaux, permutations with forbidden sequences and planar maps. We extend existing enumerative results on stack words and we also obtain new results. In particular, we make a correspondence between nonseparable 3×n rectangular standard Young tableaux (or stack words where elements satisfy a ‘Towers of Hanoi’ condition) and nonseparable cubic rooted planar maps with 2n vertices enumerated by 2ⁿ(3n)!/((2n+1)!(n+1)!). Moreover, these tableaux without two consecutive integers in the same row are in bijection with nonseparable rooted planar maps with n+1 edges enumerated by 2(3n)!/((2n+1)!(n+1)!).     

Kirchhoff方程单峰解的局部唯一性

该文主要证明了以下非线Kirchhoff问题的单峰解的局部唯一性-(∈^2a+∈b∫R^3|▽u|^2dx)△u+u=K(x)|u|^p-1u,u> 0,x∈R^3,其中∈>0任意小,a,b> 0,1相似文献   

开拓劳宾生和戈鲁辛的几个定理

<正> 1.设 p 次对称函数(?)在单位圆|z|<1中是正则的单叶的,此种函数的全体成一函数族 S_p.当p=1时,简讯 S_1为 S.设ω=f(z)∈S_p 映照|z|<1于 W 面上时,其像关于原点成星形,此种 f(z)成 S_p 之一子族S_p.设 f(z)∈S_p,     

The exact measures of a class of self-similar sets on the plane

Let S (C) R2 be the attractor of the iterated function system{f1,f2,f3}iterating on the unit equilateral triangle So,where fi(x)=λix bi,I=1,2,3,x=(x1,x2),b1=(0,0),62=(1-λ2,0),63=(1-λ3/2,√3/2(1-λ3)).This paper determines the exact Hausdorff measure,centred covering measure and packing measure of S under some conditions relating to the ontraction parameter.