Solutions to arithmetic convolution equations

In the -algebra of arithmetic functions , endowed with the usual pointwise linear operations and the Dirichlet convolution, let denote the convolution power with factors . We investigate the solvability of polynomial equations of the form

with fixed coefficients . In some cases the solutions have specific properties and can be determined explicitly. We show that the property of the coefficients to belong to convergent Dirichlet series transfers to those solutions , whose values are simple zeros of the polynomial . We extend this to systems of convolution equations, which need not be of polynomial-type.

     

Averages of shifted convolution sums for arithmetic functions

Let f be a full-level cusp form for GLm(Z) with Fourier coefficients Af(cm-2,…, c1, n): Let λ(n) be either the von Mangoldt function Λ(n) or the k-th divisor function τk(n): We consider averages of shifted convolution sums of the type Σ|h|≤H |ΣX相似文献   

Factorization of convolution type vector integro-differential equations

The paper investigates a class of convolution type integro-differential equations for vector-functions. Some special factorization methods are used to prove the solvability in several functional spaces.     

Decompositions of the free additive convolution

We introduce and study a new type of convolution of probability measures, denoted μν and called the s-free additive convolution, which is defined by the subordination functions associated with the free additive convolution. We derive an alternating decomposition of μν for compactly supported μ and ν, using another convolution called orthogonal additive convolution. This decomposition leads to two types of ‘complete’ alternating decompositions of the free additive convolution μ?ν. More importantly, we develop an operatorial approach to the subordination property and introduce the associated notion of s-free independence. Moreover, we establish relations between convolutions associated with the main notions of noncommutative independence (free, monotone and boolean). Finally, our result leads to natural decompositions of the free product of rooted graphs.     

On arithmetical functions related to the Ramanujan sum

Let m and n be positive integers, and μ the M"bius function. And let S _f(m,n) be the function defined by , where f is an arithmetical function. We show that this function has many properties like the Ramanujan sum. Firstly we study the partial summation formula involving S _f(m,n) and taking f=μ, we obtain the Dirichlet series with the coefficients Sμ(m,n) and Sμ(m,n)d(m). Moreover we show a certain property which is analogous to the orthogonality relation of the Ramanujan sums. This revised version was published online in June 2006 with corrections to the Cover Date.     

Finite element approximation of diffusion equations with convolution terms

Approximation of solutions to diffusion equations with memory represented by convolution integral terms is considered. Such problems arise from modeling of flows in fissured media. Convergence of the method is proved and results of numerical experiments confirming the theoretical results are presented. The advantages of implementation of the algorithm in a multiprocessing environment are discussed.

     

On bounded solutions to convolution equations

Periodicity of bounded solutions for convolution equations on a separable abelian metric group is established, and related Liouville type theorems are obtained. A non-constant Borel and bounded harmonic function is constructed for an arbitrary convolution semigroup on any infinite-dimensional separable Hilbert space, generalizing a classical result by Goodman (1973).

     

The boundaries of the solutions of the linear Volterra integral equations with convolution kernel

Some boundaries about the solution of the linear Volterra integral equations of the second type with unit source term and positive monotonically increasing convolution kernel were obtained in Ling, 1978 and 1982. A method enabling the expansion of the boundary of the solution function of an equation in this type was developed in I. Özdemir and Ö. F. Temizer, 2002.

In this paper, by using the method in Özdemir and Temizer, it is shown that the boundary of the solution function of an equation in the same form can also be expanded under different conditions than those that they used.

     

On solvability of singular integral-differential equations with convolution

In this paper, we study a class of singular integral-different equations of convolution type with Cauchy kernel. By means of the classical boundary value theory, of the theory of Fourier analysis, and of the principle of analytic continuation, we transform the equations into the Riemann-Hilbert problems with discontinuous coefficients and obtain the general solutions and conditions of solvability in class $\{0\}$. Thus, the result in this paper generalizes the classical theory of integral equations and boundary value problems.     

On a generalized Hankel type convolution of generalized functions

The classical generalized Hankel type convolution are defined and extended to a class of generalized functions. Algebraic properties of the convolution are explained and the existence and significance of an identity element are discussed.     

Changing the variable in convolution of distributions

In this paper the author has extended the concept of changing the variables in distributions to the convolution of distributions. For an infinitely differentiable functionh(x), he has first defined the convolution of two distributions f(h(x)) and g(h(x)) and then proved some of its properties. Finally, he has applied his results to the fractional integral and fractional differential operators     

Intuitionistic weak arithmetic

We construct -framed Kripke models of i₁ and i₁ non of whose worlds satisfies xy(x=2yx=2y+1) and x,yzExp(x, y, z) respectively. This will enable us to show that i₁ does not prove ¬¬xy(x=2yx=2y+1) and i₁ does not prove ¬¬x, yzExp(x, y, z). Therefore, i₁¬¬lop and i₁¬¬i₁. We also prove that HAl₁ and present some remarks about i₂. Mathematics Subject Classification (2000):03F30, 03F55, 03H15.     

Nonradial Hörmander algebras of several variables and convolution operators

A characterization of the closed principal ideals in nonradial Hörmander algebras of holomorphic functions of several variables in terms of the behaviour of the generator is obtained. This result is applied to study the range of convolution operators and ultradifferential operators on spaces of quasianalytic functions of Beurling type. Contrary to what is known to happen in the case of non-quasianalytic functions, an ultradistribution on a space of quasianalytic functions is constructed such that the range of the operator does not contain the real analytic functions.

     

Inclusion relations and convolution properties of a certain class of analytic functions associated with the Ruscheweyh derivatives

Let A denote the class of functions f(z) with

f(0)=f^′(0)−1=0,     

Bivariate delta-evolution equations and convolution polynomials: Computing polynomial expansions of solutions

This paper describes an application of Rota and collaborator’s ideas, about the foundation on combinatorial theory, to the computing of solutions of some linear functional partial differential equations. We give a dynamical interpretation of the convolution families of polynomials. Concretely, we interpret them as entries in the matrix representation of the exponentials of certain contractive linear operators in the ring of formal power series. This is the starting point to get symbolic solutions for some functional-partial differential equations. We introduce the bivariate convolution product of convolution families to obtain symbolic solutions for natural extensions of functional-evolution equations related to delta-operators. We put some examples to show how these symbolic methods allow us to get closed formulas for solutions of genuine partial differential equations. We create an adequate framework to base theoretically some of the performed constructions and to get some existence and uniqueness results.     

Long arithmetic progressions in critical sets

Given a density 0<σ?1, we show for all sufficiently large primes p that if S⊆Z/pZ has the least number of three-term arithmetic progressions among all sets with at least σp elements, then S contains an arithmetic progression of length at least log^1/4+o(1)p.     

Separations of first and second order theories in bounded arithmetic

We prove that PTC_N(n) (the polynomial time closure of the nonstandard natural number n in the model N of S₂.) cannot be a model of U¹₂. This implies that there exists a first order sentence of bounded arithmetic which is provable in U¹₂ but does not hold in PTC_N(n).     

On the index of fractions with square-free denominators in arithmetic progressions

We prove asymptotic formulas for the first and second moments of the index of fractions with square-free denominators of order Q streaming in a given arithmetic progression as Q→∞. A. Zaharescu was supported by NSF grant number DMS-0456615. This research was also partially supported by the CERES Program 4-147/2004 of the Romanian Ministry of Education and Research.     

Minimal arithmetic thickness connecting discrete planes

While connected arithmetic discrete lines are entirely characterized, only partial results exist for the more general case of arithmetic discrete hyperplanes. In the present paper, we focus on the three-dimensional case, that is on arithmetic discrete planes. Thanks to arithmetic reductions on a vector , we provide algorithms either to determine whether a given arithmetic discrete plane with as normal vector is connected, or to compute the minimal thickness for which an arithmetic discrete plane with normal vector is connected.