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 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 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 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 summation formulas related to Riemann Zeta function from hypergeometric series

Applying Gauss and Watson’s famous hypergeometric summation theorems, the authors establish two pattern infinite summation formulas involving generalized harmonic numbers related to Riemann Zeta function.     

Infinite type homeomorphisms of the circle and convergence of Fourier series

We consider the problem of convergence of Fourier series when we make a change of variable. Under a certain reasonable hypothesis, we give a necessary and sufficient condition for a homeomorphism of the circle to transform absolutely convergent Fourier series into uniformly convergent Fourier series.

     

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.     

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.     

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 series identities derived from the very well-poised $$\Omega $$ Ω -sum

The Ramanujan Journal - For the very well-poised $$\Omega $$ -series, a universal iteration pattern is established that yields numerous infinite series identities including several important ones...     

Three‐Phase Barker Arrays

A 3‐phase Barker array is a matrix of third roots of unity for which all out‐of‐phase aperiodic autocorrelations have magnitude 0 or 1. The only known truly two‐dimensional 3‐phase Barker arrays have size 2 × 2 or 3 × 3. We use a mixture of combinatorial arguments and algebraic number theory to establish severe restrictions on the size of a 3‐phase Barker array when at least one of its dimensions is divisible by 3. In particular, there exists a double‐exponentially growing arithmetic function T such that no 3‐phase Barker array of size with exists for all . For example, , , and . When both dimensions are divisible by 3, the existence problem is settled completely: if a 3‐phase Barker array of size exists, then .     

Proof of the Barker array conjecture

Using only elementary methods, we prove Alquaddoomi and Scholtz's conjecture of 1989, that no Barker array having exists except when .

     

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