Infinite groups with large balls of torsion elements and small entropy

We exhibit infinite, solvable, virtually abelian groups with a fixed number of generators, having arbitrarily large balls consisting of torsion elements. We also provide a sequence of 3-generator non-virtually nilpotent polycyclic groups of algebraic entropy tending to zero. All these examples are obtained by taking appropriate quotients of finitely presented groups mapping onto the first Grigorchuk group. Received: 3 August 2005     

Infinite products in a Banach algebra

A theory is developed for infinite products in a noncommutative Banach algebra. Sufficient conditions for the convergence of such a product are given. Conditions for analyticity of the product are also given in the case when the factors depend on a complex parameter.     

无穷级数与无穷乘积

通过构造新的级数以研究原来级数通项的极限性质,从而得到其敛散性.该方法在精细判别和无穷乘积研究有重要有用.     

Infinite paths in a Lorentz lattice gas model

We consider infinite paths in an illumination problem on the lattice ℤ², where at each vertex, there is either a two-sided mirror (with probability p≥ 0) or no mirror (with probability 1 −p). The mirrors are independently oriented NE-SW or NW-SE with equal probability. We consider beams of light which are shone from the origin and deflected by the mirrors. The beam of light is either periodic or unbounded. The novel feature of this analysis is that we concentrate on the measure on the space of paths. In particular, under the assumption that the set of unbounded paths has positive measure, we are able to establish a useful ergodic property of the measure. We use this to prove results about the number and geometry of infinite light beams. Extensions to higher dimensions are considered. Received: 14 November 1996 / Revised version: 1 September 1998     

Infinite positive-definite quadratic programming in a Hilbert space

This note generalizes the results of Benson, Smith, Schochetman, and Bean (Ref. 1) regarding the minimization of a positive-definite functional over the countable intersection of closed convex sets in a Hilbert space. A finite approximating subproblem for the general case is shown to have the same strong convergence properties of the earlier work without any of the specialized structures imposed therein. In particular, the current development does not rely on any properties ofl ² and does not require the Hilbert space to be separable.     

Infinite flocks of a quadratic cone

In this article, we construct some infinite analogues of the Fisher flocks. We also consider maximal partial flocks which may be constructed in several ways. In particular, we show that any derivation of a flock inPG(3,K), forK a full field, produces either a flock or a maximal partial flock. We provide constructions which produce maximal partial flocks or maximal partial spreads.In memoriam, Giuseppe TalliniThis article was written while the authors were visiting the Ricco Institute during June of 1994 and was partially supported by Glasgow-Caledonian University. Also, the authors are indebted to Christian Boltanski,D — S, in the writing of this article.The authors also are indebted to the referee for helpful suggestions with regard to this article.     

Expanded mixed FEM with lowest order RT elements for nonlinear and nonlocal parabolic problems

In this paper, an expanded mixed finite element method with lowest order Raviart Thomas elements is developed and analyzed for a class of nonlinear and nonlocal parabolic problems. After obtaining some regularity results for the exact solution, a priori error estimates for the semidiscrete problem are established. Based on a linearized backward Euler method, a complete discrete scheme is proposed and a variant of Brouwer’s fixed point theorem is used to derive an existence of a fully discrete solution. Further, a priori error estimates for the fully discrete scheme are established. Finally, numerical experiments are conducted to confirm our theoretical findings.     

Banach空间中含有无穷多个跳跃点的一阶脉冲积分-微分方程的无穷边值问题

通过建立一个新的比较引理,应用上下解方法和单调迭代技术,研究了Banach空间中含有无穷多个跳跃点的一阶脉冲积分-微分方程无穷边值问题在任意闭区间上最小解和最大解的存在性.     

Infinite Faces of a Perfect Voronoi Polyhedron

Infinite faces of perfect Voronoi polyhedra are studied. The result substantially varies depending on whether the perfect Voronoi polyhedron is considered as the closed or nonclosed convex hull of the set of Voronoi points (see Theorem 1 in Sec. 5 and Theorem 2 in Sec. 7).     

Infinite antichains in semilattices

In this paper we consider infinite antichains and the semilattices that they generate, mainly in the context of continuous semilattices. Conditions are first considered that lead to the antichain generating a copy of a countable product of the two-element semilattice. Then a special semilattice, called , is defined, its basic properties developed, and it is shown in our main result that a continuous semilattice with an infinite antichain converging to a larger element contains a semilattice copy of . The paper closes with a consideration of countable antichains that converge to a lower element or a parallel element and the kinds of semilattices generated in this context.The first two authors gratefully acknowledge the partial support of the National Science Foundation and the hospitality of Oxford University during the preparation of this paper.     

Infinite matroids in graphs

It has recently been shown that infinite matroids can be axiomatized in a way that is very similar to finite matroids and permits duality. This was previously thought impossible, since finitary infinite matroids must have non-finitary duals.In this paper we illustrate the new theory by exhibiting its implications for the cycle and bond matroids of infinite graphs. We also describe their algebraic cycle matroids, those whose circuits are the finite cycles and double rays, and determine their duals. Finally, we give a sufficient condition for a matroid to be representable in a sense adapted to infinite matroids. Which graphic matroids are representable in this sense remains an open question.     

Infinite Circulant Digraphs and Random Infinite Circulant Digraphs

In this paper, we completely determine the connectivity of every infinite circulant digraphs and prove that almost all infinite circulant digraphs are infinitely strongly connected and therefore have both one- and two-way infinite Hamiltonian paths. Received February 4, 1998, Accepted May 16, 2002     

Infinite friezes

We provide a characterization of infinite frieze patterns of positive integers via triangulations of an infinite strip in the plane. In the periodic case, these triangulations may be considered as triangulations of annuli. We also give a geometric interpretation of all entries of infinite friezes via matching numbers.     

一类无穷矩阵变换的刻划

对于一类经典的矢值序列空间,引入了一类重要子集,它包括了该序列空间的全部全有界集和许多非全有界集.利用Antosik-Mikusinski基本矩阵定理和该子集族,获得了一个无穷矩阵收敛定理,进而给出了一类经典无穷矩阵变换的刻划.     

一类无穷级数的求和

应用微分方程的理论和线性代数中的克莱姆法则,范德蒙行列式,得到一类无穷级数∑x<'mn>/(mn)!的求和公式,其中m是任意给定的正整数.     

除环上无限方阵的对角化

讨论了除环上无限方阵的对角化问题,证明了除环上行列有限方阵等价于一类特殊对角矩阵Dn的两个等价条件。