Theories of rings with divisibility

We consider several different theories of commutative rings with a new predicate symbol | together with the additional axiom a|b ? ?c (a . c = b) and show that none of these has a model companion.     

Universal number partition problem with divisibility

We examine a version of the Universal Number Partition Problem with a divisibility property referred to as the Universal Shelf Packing Problem (USPP). We show that if a shelf length is a product of powers of two primes the USPP is always partitionable. In the case where the shelf length is a product of three distinct primes we propose an efficient scheme to determine when such a case is not partitionable. We also show that a shelf length that is a product of powers of four or more primes always has at least one partition failure. Our analysis uses elementary number theory, known results related to the linear Diophantine Frobenius problem, and a new result on Frobenius gaps.     

Diophantine approximation with mild divisibility constraints

Given an irrational number α and a sequence B of coprime positive integers with the sum of inverses convergent, we investigate the problem of finding small values of nα, with n B-free.     

Tests of divisibility

Some procedures are developed for testing divisibility by prime numbers composed of two or more digits. Accelerating the tests is also considered.     

Rings with divisibility on chains of ideals

We present a generalization of the ascending and descending chain condition on one-sided ideals by means of divisibility on chains. We say that a ring R satisfies ACC_d on right ideals if in every ascending chain of right ideals of R, each right ideal in the chain, except for a finite number of right ideals, is a left multiple of the following one; that is, each right ideal in the chain, except for a finite number, is divisible by the following one. We study these rings and prove some results about them. Dually, we say that a ring R satisfies DCC_d on right ideals if in every descending chain of right ideals of R, each right ideal in the chain, except for a finite number of right ideals, is divisible by the previous one. We study these conditions on rings, in general and in special cases.     

Infinite divisibility II

Khinchin's theorem on the infinite divisibility of the limit of an infinitesimal triangular array of distributions is extended to distributions on a broad class of groups.     

Quaternion algebras with derivations

We study derivations on quaternion algebras that stabilise quadratic subfields. Following the work of L. Juan and A. Magid [10], we provide an explicit construction of a differential splitting field for a given differential quaternion algebra. We also examine the presence and impact of those derivations on quaternion algebras that admit new constants.     

Taylor series and divisibility

LetA be an augmentedK-algebra; defineT:A →A ?k _kA byT(a)=1?a ?a?1,a ∈A. We prove, under some conditions, thatg is in the subalgebraK[f] ofA generated byf if and only ifT(g) is in the principal ideal generated byT(f) inA?k _kA. WhenA=K[[X]],T(f) is a multiple ofT(X) if and only iff belongs to the ringL obtained by localizingK[X] at (X).     

Mean matrices and infinite divisibility

We consider matrices M with entries m_ij = m(λ_i, λ_j) where λ₁, … ,λ_n are positive numbers and m is a binary mean dominated by the geometric mean, and matrices W with entries w_ij = 1/m (λ_i, λ_j) where m is a binary mean that dominates the geometric mean. We show that these matrices are infinitely divisible for several much-studied classes of means.     

Quaternion Fourier Transform on Quaternion Fields and Generalizations

We treat the quaternionic Fourier transform (QFT) applied to quaternion fields and investigate QFT properties useful for applications. Different forms of the QFT lead us to different Plancherel theorems. We relate the QFT computation for quaternion fields to the QFT of real signals. We research the general linear (GL) transformation behavior of the QFT with matrices, Clifford geometric algebra and with examples. We finally arrive at wide-ranging non-commutative multivector FT generalizations of the QFT. Examples given are new volume-time and spacetime algebra Fourier transformations. I thank my family and FTHD organizer S.L. Eriksson. Soli Deo Gloria     

四元数非中心x^2分布，t分布，F分布及性质

补充四元数线性变换下四元数正态分布的性质,给出四元数非中心X^2分布、t分布,F分布的定义,导出密度函数及其性质,并研究四元数正态分布条件下样本均值及方差的分布。     

Degree divisibility in character correspondences

Recent results of the author prove the existence of bijections of certain sets of irreducible characters which satisfy many nice properties. In particular, these correspondences imply a strengthened version of the McKay Conjecture for solvable finite groups which takes into account fields of values and Schur indices. In the present paper, we prove that these correspondences have also the additional property that the degree of the corresponding character divides the degree of the original character, and we say something about the quotient as well.     

Unique divisibility in Abelian groups

Acta Mathematica Hungarica -     

On the Quaternion Ball and the Quaternion Projective Space

Since the quaternion ball was used to study the AdS/CFT problems tor spinor fields, it is worthwhile to study further the geometry （in sense of Klein） and analysis on it and on its extended space （in the sense of Behnke-Thullen）, the quaternion projective space.     

Arithmetic of divisibility in finite models

We prove that the finite‐model version of arithmetic with the divisibility relation is undecidable (more precisely, it has Π⁰₁‐complete set of theorems). Additionally we prove FM‐representability theorem for this class of finite models. This means that a relation R on natural numbers can be described correctly on each input on almost all finite divisibility models if and only if R is of degree ≤ 0 ′. We obtain these results by interpreting addition and multiplication on initial segments of finite models with divisibility only. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Primitive divisors of elliptic divisibility sequences

Silverman proved the analogue of Zsigmondy's Theorem for elliptic divisibility sequences. For elliptic curves in global minimal form, it seems likely this result is true in a uniform manner. We present such a result for certain infinite families of curves and points. Our methods allow the first explicit examples of the elliptic Zsigmondy Theorem to be exhibited. As an application, we show that every term beyond the fourth of the Somos-4 sequence has a primitive divisor.