Ein verschärfter und verallgemeinerter Satz von Alexandrov

In this paper we prove the following theorem: Suppose that n≥3 and 1≤jn $$(\forall a,b) d(a,b) : = \sum\limits_{\nu = 1}^j { (a_\nu - b_\nu )^2 - \sum\limits_{\nu = j + 1}^n { (a_\nu - b_\nu )^2 .} }$$ If a function f:?ⁿ→?ⁿ satisfies the condition: (*) $$(\forall x,y \in \mathbb{R}^n ) d(f(x),f(y)) = 0 \Leftrightarrow d(x,y) = 0,$$ then f is affine. Moreover, f preserves distances up to a constant factor C≠0, i.e. d(f(x),f(y))=C·d(x,y) for every x,y. In contrast to Alexandrov's result [1] we do not assume that f is bijective, and we also do not assume that j=n?1. A very important part of our proof will be the discussion of a functional equation.     

Generalized sine equations,II

In this paper we solve the equations

The research and progress of the enumeration of lattice paths

The enumeration of lattice paths is an important counting model in enumerative combinatorics. Because it can provide powerful methods and technical support in the study of discrete structural objects in different disciplines, it has attracted much attention and is a hot research field. In this paper, we summarize two kinds of the lattice path counting models that are single lattice paths and family of nonintersecting lattice paths and their applications in terms of the change of dimensions, steps, constrained conditions, the positions of starting and end points, and so on. (1) The progress of classical lattice path such as Dyck lattice is introduced. (2) A method to study the enumeration of lattice paths problem by generating function is introduced. (3) Some methods of studying the enumeration of lattice paths problem by matrix are introduced. (4) The family of lattice paths problem and some counting methods are introduced. (5) Some applications of family of lattice paths in symmetric function theory are introduced, and a related open problem is proposed.     

ON A MULTILINEAR OSCILLATORY SINGULAR INTEGRAL OPERATOR （I）

ＯＮＡＭＵＬＴＩＬＩＮＥＡＲＯＳＣＩＬＬＡＴＯＲＹＳＩＮＧＵＬＡＲＩＮＴＥＧＲＡＬＯＰＥＲＡＴＯＲ（Ｉ）ＣＨＥＮＷＥＮＧＵＨＵＧＵＯＥＮＬＵＳＨＡＮＺＨＥＮＭａｎｕｓｃｒｉｐｔｒｅｃｅｉｖｅｄＯｃｔｏｂｅｒ１８，１９９４．ＲｅｖｉｓｅｄＤｅｃｅ...     

有向图的L(j,k)数的上界

给定正整数j≥k,有向图D的一个L(j,k)-标号是指从V(D)到非负整数集的一个函数f,使得当x在D中邻接到y时|f(x)-f(y)|≥j,当x在D中到y距离为二时|f(x)-f(y)|≥k.f的像元素称为标号.L(j,k)一标号问题就是确定(?)j,k-数(?)j,k(D),这个参数等于(?) max{f(x)|x∈V(D)},这里f取遍D的所有L(j,k)-标号.本文根据有向图的有向着色数及最长有向路的长度来研究(?)j,k-数,证明了:(1)对任何有向着色数为(?)(D)的有向图D,(?)j,k(D)≤((?)(D)-1)j;(2)对任何最长有向路的长度为l的有向图D,如果不含有向圈或者D中最长有向圈长度为l 1,则(?)j,k(D)≤lj.并且这两个界都是可达的.最后我们对l=3的有向图给出了3j-L(j,k)-labelling的一个有效算法.     

非线性边值问题的奇摄动

本文研究边值问题:εy"=f(x,y,y',ε,μ)(μ0(ε,μ)y(x,ε,μ)|(x=1-μ)=φ₁(ε,μ)其中ε,μ是两个正的小参数 在f_y’≤-k<0和其他适当的限制下,存在一个解且满足其中y₀,0(x)是退化问题 f(x,y,y',0,0)=0(01(0,0)的解,而y_i-j,j(x)(j=0,1,…,i;i=1,2,…m)能够从某些线性方程逐次求得.     

Буняковский不等式之推广及其对于积分方程与希尔伯特空间之应用

<正> 不等式■(1) 通常称为布湼可夫斯基不等式,或席瓦耳智不等式,在本文中,作者推广此不等式为这里我们用 det u_(ij)(i,j=1,2,…,n)表第i列j行之元为 u_(ij)之n列行列式,f_i,g_j(i,j=1,2,…,n)表任一希尔伯特空间之任意二组之元,(f_i,g_j)表f_i与g_j二元之内乘积.     

Multiplicative symmetry and related functional equations

Summary Let (G, *) be a commutative monoid. Following J. G. Dhombres, we shall say that a functionf: G G is multiplicative symmetric on (G, *) if it satisfies the functional equationf(x * f(y)) = f(y * f(x)) for allx, y inG. (1)Equivalently, iff: G G satisfies a functional equation of the following type:f(x * f(y)) = F(x, y) (x, y G), whereF: G × G G is a symmetric function (possibly depending onf), thenf is multiplicative symmetric on (G, *).In Section I, we recall the results obtained for various monoidsG by J. G. Dhombres and others concerning the functional equation (1) and some functional equations of the formf(x * f(y)) = F(x, y) (x, y G), (E) whereF: G × G G may depend onf. We complete these results, in particular in the case whereG is the field of complex numbers, and we generalize also some results by considering more general functionsF. In Section II, we consider some functional equations of the formf(x * f(y)) + f(y * f(x)) = 2F(x, y) (x, y K), where (K, +, ·) is a commutative field of characteristic zero, * is either + or · andF: K × K K is some symmetric function which has already been considered in Section I for the functional equation (E). We investigate here the following problem: which conditions guarantee that all solutionsf: K K of such equations are multiplicative symmetric either on (K, +) or on (K, ·)? Under such conditions, these equations are equivalent to some functional equations of the form (E) for which the solutions have been given in Section I. This is a partial answer to a question asked by J. G. Dhombres in 1973.     

A functional equation related to symmetry of operators

In this note we determine the unique solution to the functional equation \(f(x + y) ( x- y) = \left( f(x) -f (y) \right) (x + y)\). We require no additional assumptions on the function \(f\). Moreover we solve this functional equation if \(f\) is only defined on a finite interval. The interest in this type of functional equation is motivated by the study of symmetrizing measures for (the generator of) a Lévy-driven Ornstein–Uhlenbeck process.     

矩阵Kronecker积的推广

由x,y的p次多项式f(x,y)=∑pi,j=0aijxiyj给出f(x,y)的广义Kronecker积f(A,B)=∑pi,j=0aijAi Bj,得到f(A,B)的特征值的分布,推广了已知的一些结果.     

A periodicity theorem for the octahedron recurrence

The octahedron recurrence lives on a 3-dimensional lattice and is given by . In this paper, we investigate a variant of this recurrence which lives in a lattice contained in . Following Speyer, we give an explicit non-recursive formula for the values of this recurrence and use it to prove that it is periodic of period n+m. We then proceed to show various other hidden symmetries satisfied by this bounded octahedron recurrence. An earlier version of this work has circulated under the name “A coboundary category defined using the octahedron recurrence.”     

Dyadic Bivariate Fourier Multipliers for Multi-Wavelets in L2(R2)

The single 2 dilation orthogonal wavelet multipliers in one dimensional case and single A-dilation(where A is any expansive matrix with integer entries and|det A|=2) wavelet multipliers in high dimensional case were completely characterized by the Wutam Consortium(1998) and Z. Y. Li, et al.(2010). But there exist no more results on orthogonal multivariate wavelet matrix multipliers corresponding integer expansive dilation matrix with the absolute value of determinant not 2 in L~2(R~2). In this paper, we choose 2I2=(_0~2 _2~0)as the dilation matrix and consider the 2 I2-dilation orthogonal multivariate waveletΨ = {ψ_1, ψ_2, ψ_3},(which is called a dyadic bivariate wavelet) multipliers. We call the3 × 3 matrix-valued function A(s) = [ f_(i, j)(s)]_(3×3), where fi, jare measurable functions, a dyadic bivariate matrix Fourier wavelet multiplier if the inverse Fourier transform of A(s)( ψ_1(s), ψ_2(s), ψ_3(s)) ~T=( g_1(s), g_2(s), g_3(s))~ T is a dyadic bivariate wavelet whenever(ψ_1, ψ_2, ψ_3) is any dyadic bivariate wavelet. We give some conditions for dyadic matrix bivariate wavelet multipliers. The results extended that of Z. Y. Li and X. L.Shi(2011). As an application, we construct some useful dyadic bivariate wavelets by using dyadic Fourier matrix wavelet multipliers and use them to image denoising.     

一类超椭圆方程的整数解

设ｍ，ｎ∈Ｎ；ｍ≥２，ｎ≥２，ｍｎ≥６，ｆ（ｘ）＝ｘｍ＋ａ１ｘｍ－１＋…＋ａｍ∈Ｚ［ｘ］，Ｈ＝ｍａｘ（｜ａ１｜，…，｜ａｍ｜）．本文运用组合分析方法证明了：当ｍ≡０（ｍｏｄｎ），ａ１，…，ａｍ不全为零，而且其中第一个非零系数ａｓ与ｎ互素时，方程ｆ（ｘ）＝ｙｎ，ｘ，ｙ∈Ｚ，仅有有限多组解（ｘ，ｙ），而且这些解都满足｜ｘ｜＜（４ｍＨ）２ｍ／ｎ＋１以及｜ｙ｜＜（４ｍＨ）４ｍ２／ｎ２＋ｍ／ｎ＋１     

Counting Lattice Paths With Four Types of Steps

The main purpose of this paper is to derive generating functions for the numbers of lattice paths running from (0, 0) to any (n, k) in \({\mathbb{Z} \times \mathbb{N}}\) consisting of four types of steps: horizontal H = (1, 0), vertical V = (0, 1), diagonal D = (1, 1), and sloping L = (–1, 1). These paths generalize the well-known Delannoy paths which consist of steps H, V, and D. Several restrictions are considered. However, we mainly treat with those which will be needed to get the generating function for the numbers R(n, k) of these lattice paths whose points lie in the integer rectangle \({\{(x, y) \in \mathbb{N}^2 : 0 \leq x \leq n, 0 \leq y \leq k\}}\) . Recurrence relation, generating functions and explicit formulas are given. We show that most of considered numbers define Riordan arrays.     

一类向量四阶非线性微分方程边值问题的奇摄动

我们研究伴有边界摄动的向量边值问题:
ε2y(4)=f(x,y,y″,ε,μ)(μy(x,ε,μ)|_x=μ=A₁(ε,μ),y(x,ε,μ)|_x=1-μ=B₁(ε,μ)
y″(x,ε,μ)|_x=μ=A₂(ε,μ),y″(x,ε,μ)|_x=1-μ=B₂(ε,μ)
其中y,f,A_j和B_j(j=1,2)是n维向量函数和ε,μ是两个正的小参数.虽然纯量边值问题曾有人研究过,但这样的向量边值问题尚未被研究.在适当的假设下,利用微分不等式方法,我们找到向量边值问题的一个解和获得一致有效的渐近展开式.     

The Variety of Commutative BCI-Algebras is 2-Based

In this note, we first solve the following open problem in [5]: Can the variety of commutative BCI-algebras be defined by two identities? An algebra of type (2, 0) is a commutative BCI-algebra if and only if it satisfies $u\ast\left(((x \ast y) \ast (x \ast y))(z \ast y)\right) = uIn this note, we first solve the following open problem in [5]: Can the variety of commutative BCI-algebras be defined by two identities? An algebra of type (2, 0) is a commutative BCI-algebra if and only if it satisfies and (see Theorem 2 below).Next, we prove that I-variety [2] is also 2-based. Finally, we show that I-variety is a proper subvariety of the variety of commutative BCI-algebras.AMS Subject Classification (2000): 03G25, 06A10, 06D99     

Verallgemeinerungen eines Satzes von S. Piccard

The following result is due to S. Piccard ([12], S.30): “If A,B ?? are Baire sets of second category and if the function f: ?×?→? is defined by f(x,y):=x?y (x,y ε ?), then the interior of f(A×B) is non void”. In this note the two main results assure, that the theorem of S. Piccard remains valid, if (1) ? is replaced by topological spaces X,Y,Z, (2) f:X×Y→Z is a function, which satisfies a certain global (respectively local) solvability condition, (3) A ?X contains a Baire set of second category and (4) B ?Y is only of second category.     

交换环上幂映射的周期轨道分支的对称性

设Ｒ是个交换环，带离散拓扑，是由ｆ（ｒ）＝ｒ￣ｎ（任ｒ∈Ｒ）定义的幂映射．又设ｘ及ｙ均是ｆ的周期点，其周期分别是ｋ及ｌ．记称Ｗ＿ｘ为ｆ的含ｘ的周期轨道分支，本文证明：（１）Ｗ＿ｘ在ｆ之下具有循环对称性，即存在着周期为ｋ的周期映射ｈ＿ｘ：Ｗ＿ｘ→Ｗ＿ｘ使得ｆｈ＿ｘ＝ｈ＿ｘｆ｜Ｗ＿ｘ且ｈ＿ｘ（ｘ）＝ｆ（ｘ）；（２）当ｌ是ｋ的因数且存在ｕ∈Ｒ使得ｙ＝ｕｘ时，　存在着映射ξ＿ｕ：Ｗ＿ｘ→Ｗ＿ｙ满足；（ｉｉｉ）若还存在着ｖ∈Ｒ使得ｘ＋ｖｙ，且ｌ＝ｋ，则此ξ＿ｖ与ξ＿ｖ互逆．     

Generalizations Of A “folk-Theorem” On Simple Functional Equations In A Single Variable

It is known (“mathematical folklore”) that, to every function defined on [1,2], there exists a solution of f(2x) = 2f(x) on ]0,∞[ of which the given function is a restriction to [1,2]. With a little care in the definition on [1,2], with still a lot of arbitrariness left, the resulting solution will be continuous, even C^∞ on ]0,∞[ (a behaviour markedly different from that of the Cauchy equation f(x + y) = f(x) + f(y), which has f(x) = cx as only continuous solution on ]0,∞[, even though, with y = x, it degenerates into the above equation). If 0 is added to the domain and we choose the “arbitrary function” bounded on [1,2[, then the solution will even be continuous (from the right) at 0. However, if f is supposed to be differentiable at 0 (from the right), then f(x) = cx is the only solution on [0,∞[. p In this paper we present similar and further results concerning general, Cⁿ (n ≤ ∞), analytic, locally monotonie or γ-th order convex solutions of the somewhat more general equation f(kx) = k^γf(x) (k ≠ 1 a positive, γ a real constant), which seems to be of importance in meterology. Some of the results are not quite what one expects.     

The butterfly decomposition of plane trees

We introduce the notion of doubly rooted plane trees and give a decomposition of these trees, called the butterfly decomposition, which turns out to have many applications. From the butterfly decomposition we obtain a one-to-one correspondence between doubly rooted plane trees and free Dyck paths, which implies a simple derivation of a relation between the Catalan numbers and the central binomial coefficients. We also establish a one-to-one correspondence between leaf-colored doubly rooted plane trees and free Schröder paths. The classical Chung-Feller theorem as well as some generalizations and variations follow quickly from the butterfly decomposition. We next obtain two involutions on free Dyck paths and free Schröder paths, leading to parity results and combinatorial identities. We also use the butterfly decomposition to give a combinatorial treatment of Klazar's generating function for the number of chains in plane trees. Finally we study the total size of chains in plane trees with n edges and show that the average size of such chains tends asymptotically to (n+9)/6.