Infinite products for power series

An algorithm is introduced and shown to lead to a unique infinite product representation for a given formal power series A(z) with A(0) = 1. The infinite product is , where all b_n ≠ 0, r_n , and r_n+1 > r_n. The degree of approximation by the polynomial (1 + b₁z^r₁) · · · (1 + b_nz^r_n) is also considered.     

Infinite series representations for complex numbers

We introduce two new algorithms that lead to finite or infinite series expansions for complex number in terms of ‘integral digits’ within the complex quadratic fields ? \(\left( {\sqrt { - m} } \right)\) , form=1, 2,…, 11. In particular, we derive complex number representations as sums of reciprocal of Gaussion integers and as sums of reciprocals of algebraic integers in ? \(\left( {\sqrt { - m} } \right)\) , form=2, 3, 7 and 11. In addition to convergence of the various algorithms we investigate the representation of ‘rationals’ relative to the fields ? \(\left( {\sqrt { - m} } \right)\) , form=1, 2, 3, 7 and 11.     

Infinite type power series subspaces of finite type power series spaces

Nuclear power series spaces of finite type, Λ₀(α), and infinite type, Λ_∞(α), are considered. Sufficient conditions are given on α for which there exists a β such that Λ_∞(β) is isomorphic to a subspace of Λ₀(α) and also for which there does not exist such a β. In certain cases it is possible to take β=α. The results in this paper are related to earlier results by S. Rolewicz and V. P. Zaharyuta.     

Infinite type power series subspaces of infinite type power series spaces

We consider the problem of finding all power series subspaces of a given infinite type power series space. A necessary condition is obtained which is similar to a property of complemented subspaces of nuclear Fréchet spaces. In some cases, a complete solution is obtained and this leads to new information on a conjecture of C. Bessaga. This research was partially supported by the National Science Foundation.     

Infinite series of combinatorially regular polyhedra in three-space

The paper discusses polyhedral realizations in ordinary Euclidean 3-space of Coxeter's regular skew polyhedra {4, p|4 ^p/2]–1} and their duals on an orientable surface of genus 2 ^p–3(p–4)+1. Our considerations are based on work of Coxeter, Ringel and McMullen et al., revealing that certain polyhedral manifolds discovered by the last three authors are in fact the polyhedra in question. We also describe Kepler-Poinsot-type polyhedra in 3-space obtained by projections from Coxeter's regular skew star-polyhedra in 4 dimensions.     

Infinite products of relations, set-valued series and uniform openness of multifunctions

Using the notations of convergent series of sets and convergent products of relations, general open mapping theorems are presented which encompass classical results of Banach, Ptak, Khanh, and others.     

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.     

Wieferich pairs and Barker sequences

We show that if a Barker sequence of length n > 13 exists, then either n = 189 260 468 001 034 441 522 766 781 604, or n > 2 · 10³⁰. This improves the lower bound on the length of a long Barker sequence by a factor of more than 10⁷. We also show that all but fewer than 1600 integers n ≤ 4 · 10²⁶ can be eliminated as the order of a circulant Hadamard matrix. These results are obtained by completing extensive searches for Wieferich prime pairs (q, p), which are defined by the relation q^p-1 o 1{q^{p-1} equiv1} mod p ², and analyzing their results in combination with a number of arithmetic restrictions on n.     

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.     

A note on the nonexistence of Barker sequences

A Barker sequence is a sequence with elements ±1 such that all out-of-phase aperiodic autocorrelation coefficients are 0, 1 or -1. It is known that if a Barker sequence of length s > 13 exists then s = 4N ² for some odd integer N 55, and it has long been conjectured that no such sequence exists. We review some previous attempts to improve the bound on N which, unfortunately, contain errors. We show that a recent theorem of Eliahou et al. [5] rules out all but six values of N less than 5000, the smallest of which is 689.     

模糊数无穷乘积

在定义了区间无穷乘积、模糊数无穷乘积及其收敛的基础上 ,利用 [3]中得到的关于模糊级数收敛的某些结论 ,得出模糊数无穷乘积与实数无穷乘积理论类似的结果 ,从而说明本文定义的模糊数无穷乘积确为实数无穷乘积的推广。     

无限正则p-群

对一类无限正则p-群进行了研究,得到了一个正则的局部幂零p-群G如果满足|G(Ω)1(G)|＜∞,那么G是幂零的且G是可除阿贝尔p-群被有限群的扩张.进而,还研究了一类无限的非正则p-群,但它的所有真商群或者真的无限子群是正则群.在假设这类群存在拟循环子群的情况下,在定理1.2和1.3给出了这类群的结构的刻画.     

Infinite Regress Arguments

Infinite regress arguments play an important role in many distinct philosophical debates. Yet, exactly how they are to be used to demonstrate anything is a matter of serious controversy. In this paper I take up this metaphilosophical debate, and demonstrate how infinite regress arguments can be used for two different purposes: either they can refute a universally quantified proposition (as the Paradox Theory says), or they can demonstrate that a solution never solves a given problem (as the Failure Theory says). In the meantime, I show that Black’s view on infinite regress arguments (1996, this journal) is incomplete, and how his criticism of Passmore can be countered.