Arithmetic properties of certain functions in several variables

If T is an n × n matrix with nonnegative integral entries, we define a transformation T: Cⁿ → Cⁿ by w = Tz where

$\text{W}{}_1\text{=}\text{∏}\text{j=1}\text{n}\text{z}{}_{\mathrm{j}}{}^{\mathrm{t}}\text{ij}\text{(1?i?n).}$

We consider functions f(z) of n complex variables which satisfy a functional equation giving f(Tz) as a rational function of ¹f(z) and we obtain conditions under which such a function f(z) takes transcendental values at algebraic points.     

Arithmetic and other properties of certain Delphic semigroups. II

Summary This paper continues [2]. We show that the sets of infinitely divisible elements of the Delphic semigroups ⁺ (of positive renewal sequences) and P (of standard p-functions) are additively convex, and do a Choquet analysis in each case. We draw up the (M, m) diagram for members of , and deduce from it that the product topology on is metrizable. Finally we look at the arithmetic of , showing that the simples are residual in it, and partially identifying I ₀, the set of infinitely divisible elements without simple factors. Many examples are given.I am indebted to Professor D. G. Kendall for his constant help and encouragement in the course of the research leading to this paper and [2].     

Arithmetic and other properties of certain Delphic semigroups. I

Summary Some aspects of Delphic semigroups in general — in particular, the idea of an hereditary subsemigroup, which has many uses in connexion with Delphic semigroups — are first treated. After that, attention is directed to the arithmetic of ⁺, the semigroup of positive renewal sequences. In a Delphic semigroup the aboriginal elements are the simples and the members of I ₀: a class of simples of ⁺ is constructed and the simples are shown to be residual. I ₀ is explicitly identified, and this leads to a canonical factorization of ⁺. The properties of division in ⁺ are discussed.     

Local properties of sums of certain random series

Conditions are given which insure the uniform convergence of random series of the form , where P_n(x) is a trigonometric polynomial of degree n. Conditions are also given under which S(x) satisfies Hölder's conditions.Translated from Matematicheskie Zametki, Vol. 3, No. 3, pp. 261–270, March, 1968.In conclusion the author expresses his appreciation to M. I. Yadrenko for assistance with thework.     

Conservative extension of polyadic MV-algebras to polyadic pavelka algebras

In this paper we prove polyadic counterparts of the Hájek, Paris and Shepherdson's conservative extension theorems of Łukasiewicz predicate logic to rational Pavelka predicate logic. We also discuss the algebraic correspondents of the provability and truth degree for polyadic MV-algebras and prove a representation theorem similar to the one for polyadic Pavelka algebras.     

Arithmetic properties of coefficients of half-integral weight Maass–Poincaré series

Zagier [23] proved that the generating functions for the traces of level 1 singular moduli are weight 3/2 modular forms. He also obtained generalizations for “twisted traces”, and for traces of special non-holomorphic modular functions. Using properties of Kloosterman-Salié sums, and a well known reformulation of Salié sums in terms of orbits of CM points, we systematically show that such results hold for arbitrary weakly holomorphic and cuspidal half-integral weight Poincaré series in Kohnen’s Γ₀(4) plus-space. These results imply the aforementioned results of Zagier, and they provide exact formulas for such traces.     

Arithmetic of quaternions and Eisenstein series

In the paper one computes the Fourier coefficients of the Eisenstein series of the orthogonal group of signature (1, 4). The formulas show that the restriction of the Eisenstein series to the “imaginary” axis is a Dirichlet series, whose coefficients are the products of the L-series by the number of the representations of the given number as a sum of three squares. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 160, pp. 82–90, 1987.     

Simple polyadic groups

The main aim of this article is to establish a characterization of simple polyadic groups in terms of ordinary groups and their automorphisms. We give two different definitions of simpleness for polyadic groups, from the point of views of universal algebra, UAS (universal algebraic simpleness), and group theory, GTS (group theoretical simpleness). We obtain necessary and sufficient conditions for a polyadic group to be UAS or GTS.     

New properties of a certain method of summation of generalized hypergeometric series

In a recent paper (Appl. Math. Comput. 215:1622–1645, 2009), the authors proposed a method of summation of some slowly convergent series. The purpose of this note is to give more theoretical analysis for this transformation, including the convergence acceleration theorem in the case of summation of generalized hypergeometric series. Some new theoretical results and illustrative numerical examples are given.     

Arithmetic linear series with base conditions

In this note, we study the volume of arithmetic linear series with base conditions. As an application, we consider the problem of Zariski decompositions on arithmetic varieties.     

Arithmetic properties of polynomials

Periodica Mathematica Hungarica - First, we prove that the Diophantine system $$\begin{aligned} f(z)=f(x)+f(y)=f(u)-f(v)=f(p)f(q) \end{aligned}$$ has infinitely many integer solutions for...     

Arithmetic of semigroups of series in multiplicative systems

We study the arithmetic of a semigroup M_P\mathcal{M}_{\mathcal{P}} of functions with operation of multiplication representable in the form f(x) = ?_{n = 0}^￥ a_nc_n(x)    ( a_n 3 0,?_{n = 0}^￥ a_n = 1 ) f(x) = \sum\nolimits_{n = 0}^\infty {{a_n}{\chi_n}(x)\quad \left( {{a_n} \ge 0,\sum\nolimits_{n = 0}^\infty {{a_n} = 1} } \right)} , where { c_n }_{n = 0}^￥ \left\{ {{\chi_n}} \right\}_{n = 0}^\infty is a system of multiplicative functions that are generalizations of the classical Walsh functions. For the semigroup M_P\mathcal{M}_{\mathcal{P}}, analogs of the well-known Khinchin theorems related to the arithmetic of a semigroup of probability measures in R ⁿ are true. We describe the class I₀(M_P)I_0(\mathcal{M}_{\mathcal{P}}) of functions without indivisible or nondegenerate idempotent divisors and construct a class of indecomposable functions that is dense in M_P\mathcal{M}_{\mathcal{P}} in the topology of uniform convergence.     

Arithmetic properties of overpartition triples

Let p_3(n) be the number of overpartition triples of n. By elementary series manipulations,we establish some congruences for p_3(n) modulo small powers of 2, such as p_3(16 n + 14) ≡ 0(mod 32), p_3(8 n + 7) ≡ 0(mod 64).We also find many arithmetic properties for p_3(n) modulo 7, 9 and 11, involving the following infinite families of Ramanujan-type congruences: for any integers α≥ 1 and n ≥ 0, we have p_3 (3~(2α+1)(3n + 2))≡ 0(mod 9 · 2~4), p_3(4~(α-1)(56 n + 49)) ≡ 0(mod 7),p_3 (7~(2α+1)(7 n + 3))≡ p_3 (7~(2α+1)(7 n + 5))≡ p_3 (7~(2α+1)(7 n + 6))≡ 0(mod 7),and for r ∈ {1, 2, 3, 4, 5, 6},p_3(11 · 7~(4α-1)(7 n + r)≡ 0(mod 11).