二向受力不等的平面薄膜自由振动问题解

本文中求解了双向受力不等的矩形、圆形、椭圆形平面薄膜的自振频率与振型,还给出了任意外形边界的平面薄膜的近似解.矩形薄膜,先经过坐标变换将方程变换成常见的薄膜振动方程,因此很容易求得解.圆形薄膜.先将坐标作与上述同样的变换,再把它变换成椭圆坐标,将方程化为马丢(Mathieu)方程,这样利用马丢函数的性质,不难求得其解.椭圆形薄膜解法与圆形薄膜相似.文末还给出了例题.     

On the two-dimensional Knapsack Problem

We address the two-dimensional Knapsack Problem (2KP), aimed at packing a maximum-profit subset of rectangles selected from a given set into another rectangle. We consider the natural relaxation of 2KP given by the one-dimensional KP with item weights equal to the rectangle areas, proving the worst-case performance of the associated upper bound, and present and compare computationally four exact algorithms based on the above relaxation, showing their effectiveness.     

Lagrange四边形单位分解有限元法的最优误差分析

用构造最优局部逼近空间的方法对Lagrange型四边形单位分解有限元法进行了最优误差分析.单位分解取Lagrange型四边形上的标准双线性基函数,构造了一个特殊的局部多项式逼近空间,给出了具有2阶再生性的Lagrange型四边形单位分解有限元插值格式,从而得到了高于局部逼近阶的最优插值误差.     

Rectangular tileability and complementary tileability are undecidable

Does a given set of polyominoes tile some rectangle? We show that this problem is undecidable. In a different direction, we also consider tiling a cofinite subset of the plane. The tileability is undecidable for many variants of this problem. However, we present an algorithm for testing whether the complement of a finite region is tileable by a set of rectangles.     

Dyadic Hermite interpolation on a rectangular mesh

Given f and ∇f at the vertices of a rectangular mesh, we build an interpolating function f by a subdivision algorithm. The construction on each elementary rectangle is independent of any disjoint rectangle. From the Hermite data associated with the vertices of a rectangle R, the function f is defined on a dense subset of R. Sufficient conditions are found in order to extend f to a C ¹ function. Moreover, infinite products and generalized radii of matrices are used to study the convergence to a C ¹ function. This convergence depends on the five parameters introduced in the algorithm. This revised version was published online in June 2006 with corrections to the Cover Date.     

Some weak fragments of Martin’s axiom related to the rectangle refining property

We introduce the anti-rectangle refining property for forcing notions and investigate fragments of Martin’s axiom for ℵ₁ dense sets related to the anti-rectangle refining property, which is close to some fragment of Martin’s axiom for ℵ₁ dense sets related to the rectangle refining property, and prove that they are really weaker fragments. T. Yorioka was partially supported by Grant-in-aids for Scientific Research No.16340022 and No.18840022.     

Finding squares and rectangles in sets of points

A number of problems concerning sets of points in the plane are studied, e.g. establishing whether it contains a subset of size 4, which are the vertices of a square or rectangle. Both the problems of finding axis-parallel squares and rectangles, and arbitrarily oriented squares and rectangles are studied. Efficient algorithms are obtained for all of them. Furthermore, we investigate the generalizations tod-dimensional space, where the problem is to find hyperrectangles and hypercubes. Also, upper and lower bounds are given for combinatorial problems on the maxium number of subsets of size 4, of which the points are the vertices of a square or rectangle. Then we state an equivalence between the problem of finding rectangles, and the problem of findingK _{2, 2} subgraphs in bipartite graphs. Thus we immediately have an efficient algorithm for this graph problem.This work was partially supported by the ESPRIT Basic Research Action No. 3075 (project ALCOM). Work of the second author was also supported by the Dutch Organisation for Scientific Research (N.W.O.).     

Deleting Vertices and Interlacing Laplacian Eigenvalues

The authors obtain an interlacing relation between the Laplacian spectra of a graph G and its subgraph G - U, which is obtained from G by deleting all the vertices in the vertex subset U together with their incident edges. Also, some applications of this interlacing property are explored and this interlacing property is extended to the edge weighted graphs.     

Gerechte designs with rectangular regions

A gerechte framework is a partition of an n × n array into n regions of n cells each. A realization of a gerechte framework is a latin square of order n with the property that when its cells are partitioned by the framework, each region contains exactly one copy of each symbol. A gerechte design is a gerechte framework together with a realization. We investigate gerechte frameworks where each region is a rectangle. It seems plausible that all such frameworks have realizations, and we present some progress toward answering this question. In particular, we show that for all positive integers s and t, any gerechte framework where each region is either an s × t rectangle or a t × s rectangle is realizable. © 2011 Wiley Periodicals, Inc. J Combin Designs 20:112‐123, 2012     

On Vector-Valued Function Spaces with Helly's Property

If a vector-valued function space with a Hausdorff locally convex topology has a property such that every closed strongly bounded subset is compact, then we name this property Helly's property. In this paper, we show a class of vector-valued function spaces with Helly's property and consider convegence of vector measures and best approximations in function spaces in this class.     

On Some Variants of Connectedness

Does the closure of a subset M of a topological space X possess property P, whenever X and the boundary of M have this property? We shall consider connectivity properties P with positive and negative answers to this question.     

空间（Lμ^p）^m的弱Banach-Saks性质

证明了由m个Lμp空间产生的Banach向量空间（Lμp）m的弱Banach-Saks性质,其中m是自然数, 1 p 〈＋∞.当m= 1时,这就是著名的Banach-Saks-Szlenk定理.运用该性质,还给出了定义在向量空间Rm的一个凸集上的非负连续凸函数与取值在空间（Lpμ）m的一个弱紧子集中的向量值函数的复合函数的积分不等式.当这些向量值函数属于由m个Lμ∞空间产生的积空间（Lμ∞）m的一个弱＊紧子集时,类似的积分不等式还是成立的.     

Unimodality and Lie Superalgebras

It is well-known how the representation theory of the Lie algebra sl(2, ?) can be used to prove that certain sequences of integers are unimodal and that certain posets have the Sperner property. Here an analogous theory is developed for the Lie superalgebra osp(1,2). We obtain new classes of unimodal sequences (described in terms of cycle index polynomials) and a new class of posets (the “super analogue” of the lattice L(m,n) of Young diagrams contained in an m × n rectangle) which have the Sperner property.     

Carleman estimates and unique continuation property for the Kadomtsev–Petviashvili equations

We study the unique continuation property for the generalized Kadomtsev–Petviashvili (KP) equations and its regularized version. We use Carleman estimates to prove that if the solution of the KP equations vanishes in an open subset, then this solution is identically equal to zero in the horizontal component of the open subset.     

Fixed point property for nonexpansive mappings versus that for nonexpansive semigroups

The main result of this paper is that a closed convex subset of a Banach space has the fixed point property for nonexpansive mappings if and only if it has the fixed point property for nonexpansive semigroups.     

Multidimensional bony attractors

In this paper we study attractors of skew products, for which the following dichotomy is ascertained. These attractors either are not asymptotically stable or possess the following two surprising properties. The intersection of the attractor with some invariant submanifold does not coincide with the attractor of the restriction of the skew product to this submanifold but contains this restriction as a proper subset. Moreover, this intersection is thick on the submanifold, that is, both the intersection and its complement have positive relative measure. Such an intersection is called a bone, and the attractor itself is said to be bony. These attractors are studied in the space of skew products. They have the important property that, on some open subset of the space of skew products, the set of maps with such attractors is, in a certain sense, prevalent, i.e., ??big.?? It seems plausible that attractors with such properties also form a prevalent subset in an open subset of the space of diffeomorphisms.     

A random weak ergodic property of infinite products of operators in metric spaces

ABSTRACT

We study the random weak ergodic property of infinite products of mappings acting on complete metric spaces. Our results describe an aspect of the asymptotic behaviour of random infinite products of such mappings. More precisely, we show that in appropriate spaces of sequences of operators there exists a subset, which is a countable intersection of open and everywhere dense sets, such that each sequence belonging to this subset has the random weak ergodic property. Then we show that several known results in the literature can be deduced from our general result.     

全函数空间和置换空间的一些性质

In this paper, an equivalent relation among the reflexivity, weak sequential completeness and bounded completeness in full function space is given. Some results on weakly sequential compactness of subset and the property （u） in substitution spaces are obtained.     

Counting dissections into integral squares

A squared rectangle is a rectangle dissected into squares. Similarly a rectangled rectangle is a rectangle dissected into rectangles. The classic paper ‘The dissection of rectangles into squares’ of Brooks, Smith, Stone and Tutte described a beautiful connection between squared rectangles and harmonic functions. In this paper we count dissections of a rectangle into a set of integral squares or a set of integral rectangles. Here, some squares and rectangles may have the same size. We introduce a method involving a recurrence relation of large sized matrices to enumerate squared and rectangled rectangles of a given sized rectangle and propose the asymptotic behavior of their growth rates.