CONTRIBUTIONS TO ABSTRACT ANALYTIC NUMBER THEORY

Abstract

This paper reports on some recent contributions to the theory of multiplicative arithmetic semigroups, which have been initiated by John Knopfmacher's work on abstract analytic number theory. They concern weighted inversion theorems of the Wiener type, mean-value theorems for multiplicative functions, and Ramanu-jan expansions.     

Probabilistic number theory in additive arithmetic semigroups II

We prove two quantitative mean-value theorems of completely multiplicative functions on additive arithmetic semigroups. On the basis of the two theorems, a central limit theorem of additive functions on additive arithmetic semigroups is proved with a best possible error estimate. This generalizes the vital results of Halász and Elliott in classical probabilistic number theory to function fields. Received October 26, 1998; in final form April 5, 2000 / Published online October 11, 2000     

Probabilistic Number Theory in Additive Arithmetic Semigroups, III

We extend the investigation of quantitative mean-value theorems of completely multiplicative functions on additive arithmetic semigroups given in our previous paper. Then the new and old quantitative mean-value theorems are applied to the investigation of local distribution of values of a special additive function *(a). The result is unexpected from the point of view of classical number theory. This reveals the fact that the essential divergence of the theory of additive arithmetic semigroups from classical number theory is not related to the existence of a zero of the zeta function Z(y) at y = –q ^–1.     

Mean-Value Theorems for Multiplicative Functions on Additive Arithmetic Semigroups via Halász’s Method

 We extend the mean-value theorems for multiplicative functions f on additive arithmetic semigroups, which satisfy the condition ∣f(a)∣≤1, to a wider class of multiplicative functions f for which ∣f(a)∣ is bounded in some average sense, via Halász’s method in classical probabilistic number theory.     

Mean-Value Theorems for Multiplicative Functions on Additive Arithmetic Semigroups via Halász’s Method

 We extend the mean-value theorems for multiplicative functions f on additive arithmetic semigroups, which satisfy the condition ∣f(a)∣≤1, to a wider class of multiplicative functions f for which ∣f(a)∣ is bounded in some average sense, via Halász’s method in classical probabilistic number theory. Received October 18, 2001; in final form April 11, 2002     

The arithmetic extensions of a numerical semigroup

Abstract

In this paper we introduce the notion of extension of a numerical semigroup. We provide a characterization of the numerical semigroups whose extensions are all arithmetic and we give an algorithm for the computation of the whole set of arithmetic extension of a given numerical semigroup. As by-product, new explicit formulas for the Frobenius number and the genus of proportionally modular semigroups are obtained.     

Logarithmic density and measures on semigroups

Davenport and Erdős [3] proved that every setA of integers with the property thata ∈A impliesan ∈A for alln (multiplicative ideal) has a logarithmic density. I generalized [5] this result to sets with the property that if for some numbersa, b, n we havea ∈ A, b ∈ A andan ∈ A, then necessarilybn ∈ A, which I call quasi-ideals. Here a new proof of this theorem is given, applying a result on convolution of measures on discretes semigroups. This leads to further generalizations, including an improvement of a result of Warlimont [8] on ideals in abstract arithmetic semigroups. Supported by Hungarian National Foundation for Scientific Research, Grant No. 1901     

不可约特征标次数为等差数的有限可解群

设G为有限群,cd(G)表示G的所有复不可约特征标次数的集合.本文研究了不可约特征标次数为等差数的有限可解群,得到两个结果:如果cd(G)={1,1+d,1+2d,…,1+kd},则k≤2或cd(G)={1,2,3,4};如果cd(G)={1,a,a+d,a+2d,…,a+kd},|cd(G)|≥4,(a,d)=1,则cd(G)={1,2,2^e+1,2e+1,2^(e+1)},并给出了d>1时群的结构.     

Unbounded functional calculus for bounded groups with applications

In this paper, we develop the unbounded extension of the Hille–Phillips functional calculus for generators of bounded groups. Mathematical applications include the generalised Lévy–Khintchine formula for subordinate semigroups, the analyticity of semigroups generated by fractional powers of group generators, where the power is not an odd integer, and a shifted abstract Grünwald formula. We also give an application of the theory to subsurface hydrology, modeling solute transport on a regional scale using fractional dispersion along flow lines. M. Kovács is partially supported by postdoctoral grant No. 623-2005-5078 of the Swedish Research Council (VR).     

On modeling with multiplicative differential equations

This work is aimed to show that various problems from different fields can be modeled more efficiently using multiplicative calculus, in place of Newtonian calculus. Since multiplicative calculus is still in its infancy, some effort is put to explain its basic principles such as exponential arithmetic, multiplicative calculus, and multiplicative differential equations. Examples from finance, actuarial science, economics, and social sciences are presented with solutions using multiplicative calculus concepts. Based on the encouraging results obtained it is recommended that further research into this field be vested to exploit the applicability of multiplicative calculus in different fields as well as the development of multiplicative calculus concepts.     

On the correspondence between arithmetic theories and propositional proof systems – a survey

The purpose of this paper is to survey the correspondence between bounded arithmetic and propositional proof systems. In addition, it also contains some new results which have appeared as an extended abstract in the proceedings of the conference TAMC 2008 [11]. Bounded arithmetic is closely related to propositional proof systems; this relation has found many fruitful applications. The aim of this paper is to explain and develop the general correspondence between propositional proof systems and arithmetic theories, as introduced by Krají?ek and Pudlák [46]. Instead of focusing on the relation between particular proof systems and theories, we favour a general axiomatic approach to this correspondence. In the course of the development we particularly highlight the role played by logical closure properties of propositional proof systems, thereby obtaining a characterization of extensions of EF in terms of a simple combination of these closure properties (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiplicative Perturbations of Local C-Regularized Semigroups

In this paper, we investigate the multiplicative perturbations of local C-regularized semigroups on Banach spaces and establish some new multiplicative perturbation results which are generalizations of many existing theorems. Moreover, we give applications of the abstract results to three concrete problems including the mixed initial-boundary value problem for backwards heat equations with finite rank feedbacks.     

Abstract Cauchy Problems for Quasi-Linear Evolution Equations in the Sense of Hadamard

This paper is devoted to the well-posedness of abstract Cauchyproblems for quasi-linear evolution equations. The notion ofHadamard well-posedness is considered, and a new type of stabilitycondition is introduced from the viewpoint of the theory offinite difference approximations. The result obtained here generalizesnot only some results on abstract Cauchy problems closely relatedwith the theory of integrated semigroups or regularized semigroupsbut also the Kato theorem on quasi-linear evolution equations.An application to some quasi-linear partial differential equationof weakly hyperbolic type is also given. 2000 Mathematics SubjectClassification 34G20, 47J25 (primary), 47D60, 47D62 (secondary).     

Sequence encoding without induction

We show that the universally axiomatized, induction‐free theory $\mathsf {PA}^-$ is a sequential theory in the sense of Pudlák's 5 , in contrast to the closely related Robinson's arithmetic.     

Poisson Distribution in the Polynomial Semigroup

We consider the asymptotic behavior of the distributions of arithmetic functions in polynomial semigroups.__________Published in Lietuvos Matematikos Rinkinys, Vol. 44, No. 4, pp. 429–442, October–December, 2004.     

区间数的标准表示及其四则运算法则与泛灰数的内在联系

在回顾泛灰数四则运算法则基础上,给出了区间数的标准表示,论述了标准区间数的四则运算法则与泛灰数的内在联系及其应用前景.     

Arithmetic Processes in Semigroups

We consider the asymptotic behavior of the distributions of stochastic processes defined by multiplicative functions in an arithmetic semigroup.     

Equidistribution of numerical semigroup gaps modulo m

For a positive integer m, a finite set of integers is said to be equidistributed modulo m if the set contains an equal number of elements in each congruence class modulo m. In this paper, we consider the problem of determining when the set of gaps of a numerical semigroup S is equidistributed modulo m. Of particular interest is the case when the nonzero elements of an Apéry set of S form an arithmetic sequence. We explicitly describe such numerical semigroups S and determine conditions for which the sets of gaps of these numerical semigroups are equidistributed modulo m.     

Systems of proportionally modular Diophantine inequalities

The set of integer solutions to the inequality ax mod b≤c x is a numerical semigroup. We study numerical semigroups that are intersections of these numerical semigroups. Recently it has been shown that this class of numerical semigroups coincides with the class of numerical semigroups having a Toms decomposition. The first author was (partially) supported by the Centro de Matemática da Universidade do Porto (CMUP), financed by FCT (Portugal) through the programmes POCTI and POSI, with national and European Community structural funds. The last three authors are supported by the project MTM2004-01446 and FEDER funds. The authors would like to thank the referee for her/his comments and suggestions.