On the existence of certain cyclic difference families and difference matrices

A theorem is proved to the effect that if there exists a BIB-schema with parameters (p^m–1,k, k–1), where k¦(p^m–1), p is prime, and m is a natural number, then there exists a BIB-schema (p^mn–1),k, k–1). A consequence is the existnece of a cyclic BIB-schema (p^mn–1, p^m–1, p^m–2) (p^m–1 is prime) that specifies each ordered pair of difference elements at any distance = 1, 2, ..., p^m–2 (cyclically) precisely once. Recursive theorems on the existence of difference matrices and (v, k, k)-difference families in the group Z_v of residue classes mod v are proved, along with a theorem on difference families in an additive abelian group.Translated from Matematicheskie Zametki, Vol. 52, No. 1, pp. 114–119, July, 1992.     

The generating elements of certain Volterra operators connected with third- and fourth-order differential operators

Sufficient conditions are established forf (x) to be the generating function for the Volterra operator which is inverse to the Cauchy operator:l [y]=y⁽ⁿ⁾+p₂(x)y^(n–2) + ... +pn(x)y, y(0)=y(0)=...=y^(n–1)(0)=0 (n=3, 4), when the coefficients pi(x) are not analytic.Translated from Matematicheskie Zametki, Vol. 3, No. 6, pp. 715–720, June, 1968.     

Mean-Value Theorems in Arithmetic Semigroups

This paper reports on recent progress in the theory of multiplicative arithmetic semigroups, which has been initiated by John Knopfmacher's work on abstract analytic number theory. In particular, it deals with abstract versions of the mean-value theorems of Delange, of Wirsing, and of Halász for multiplicative functions on arithmetic semigroups G with Axiom A . The Turán Kubilius inequality is transferred to G , and methods developed by Rényi, Daboussi and Indlekofer, Lucht and Reifenrath are utilized. As byproduct a new proof of the abstract prime number theorem is obtained. This revised version was published online in June 2006 with corrections to the Cover Date.     

Detailed Error Analysis for a Fractional Adams Method

We investigate a method for the numerical solution of the nonlinear fractional differential equation D _* y(t)=f(t,y(t)), equipped with initial conditions y ^(k)(0)=y ₀ ^(k), k=0,1,...,–1. Here may be an arbitrary positive real number, and the differential operator is the Caputo derivative. The numerical method can be seen as a generalization of the classical one-step Adams–Bashforth–Moulton scheme for first-order equations. We give a detailed error analysis for this algorithm. This includes, in particular, error bounds under various types of assumptions on the equation. Asymptotic expansions for the error are also mentioned briefly. The latter may be used in connection with Richardson's extrapolation principle to obtain modified versions of the algorithm that exhibit faster convergence behaviour.     

On the Construction of Cosine Operator Functions and Semigroups on Function Spaces with Generator a(x)(d2dx2)+b(x)(d/dx)+c(x); Theory

In this paper we develop a method to solve exactly partial differential equations of the type ( ⁿ /t ⁿ )f(x,t)=(a(x)( ⁿ /x ⁿ )+b(x) (/x+c(x))f(x,t); n=1,2, with several boundary conditions, where f·,t) lies in a function space. The most powerful tool here is the theory of cosine operator functions and their connection to (holomorphic) semigroups. The method is that generally we are able to unify and generalize many theorems concerning problems in the theories of holomorphic semigroups, cosine operator functions, and approximation theory, especially these dealing with approximation by projections. These applications will be found in [14].     

An O(n 3 L) primal interior point algorithm for convex quadratic programming

We present a primal interior point method for convex quadratic programming which is based upon a logarithmic barrier function approach. This approach generates a sequence of problems, each of which is approximately solved by taking a single Newton step. It is shown that the method requires iterations and O(n ^3.5 L) arithmetic operations. By using modified Newton steps the number of arithmetic operations required by the algorithm can be reduced to O(n ³ L).This research was supported in part by NSF Grant DMS-85-12277 and ONR Contract N-00014-87-K0214. It was presented at the Meeting on Mathematische Optimierung, Mathematisches Forschungsinstitut, Oberwolfach, West Germany, January 3–9, 1988.     

More Delphic theory and practice

Summary We find conditions on semigroups satisfying Kendall's [5] Delphic postulates A and B such that they then satisfy also postulate C (the central limit theorem). These conditions are of the type that the semigroup possess enough continuous homomorphisms (each into the additive reals or circle group) to separate its points. We show that the classical Delphic semigroups (of probability laws on the line, renewal sequences, and p-functions) satisfy our conditions, and thus get the classical results as consequences of the abstract theorems.We also find some curious spiral Delphic semigroups for counter-examples to do with the I ₀-problem.I am very grateful to Professor J. F. C. Kingman, for proposing, at my Ph. D. interview, the problems examined here.     

The boundary behavior of components of polyharmonic functions

We consider the following representation of polyharmonic functions on the unit ballD ^m : 450-1 where the _j are harmonic onD ^m . We study the relation between uniform boundary properties ofƒ (its smoothness and growth while approaching the boundary) and the same properties of the terms in this representation. The theorems proved in this paper generalize some results obtained by Dolzhenko in the theory of polyanalytic functions.Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 518–530, October, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01366.     

Zur numerischen Lösung des ersten biharmonischen Randwertproblems

Summary The system of equations resulting from a mixed finite element approximation of the first biharmonic boundary value problem is solved by various preconditioned Uzawa-type iterative methods. The preconditioning matrices are based on simple finite element approximations of the Laplace operator and some factorizations of the corresponding matrices. The most efficient variants of these iterative methods require asymptoticallyO(h ^–0,5In ^–1) iterations andO(h ^–p–0,5In ^–1) arithmetic operations only, where denotes the relative accuracy andh is a mesh-size parameter such that the number of unknowns grows asO(h ^–p ),h0.

     

Distribution of Values of Additive Functions with Respect to the Logarithmic Frequency

We study the weak convergence of distribution functions _x(n x: f _x (n) < u). Here _x denotes the logarithmic frequency and f _x , x 6, is a set of integer-valued strongly additive functions. The method of factorial moments is basic in the proofs.__________Translated from Lietuvos Matematikos Rinkinys, Vol. 44, No. 4, pp. 546–557, October–December, 2004.     

An implicit function theorem

Suppose thatF:DR ⁿ×R^mRⁿ, withF(x ⁰,y ⁰)=0. The classical implicit function theorem requires thatF is differentiable with respect tox and moreover that ₁ F(x ⁰,y ⁰) is nonsingular. We strengthen this theorem by removing the nonsingularity and differentiability requirements and by replacing them with a one-to-one condition onF as a function ofx.     

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     

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     

Linear Extensions of Additive Partial Orders

Addive partial orders arise naturally in theories of comparativeprobability and subset preferences. An additive partial order is a partialorder on the family of subsets of ann-element set that satisfies . This is reformulated as a subset P of {1,0,–1}ⁿ that excludes 0 and containsx+y whenever x,y P and x+y {1,0,–1}ⁿ. Additional conditions of positivity andcompleteness give rise to positive additive partial orders and additivelinear orders respectively. The paper investigates conditions under which anadditive partial order is included in, or extendable to, an additive linearorder. The additive dimension of an extendable additive partial order isdefined and computed for several classes of additive orders.     

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.     

A Graham-Sloane Type Construction for s-Surjective Matrices

We give a construction of (n–s)-surjective matrices with n columns over using Abelian groups and additive s-bases. In particular we show that the minimum number of rows ms _q(n,n–s) in such a matrix is at most s ^s q ^n–s for all q, n and s.     

Convergence of linear multistep methods for differential equations with discontinuities

Summary A new stability functional is introduced for analyzing the stability and consistency of linear multistep methods. Using it and the general theory of [1] we prove that a linear multistep method of design orderqp1 which satisfies the weak stability root condition, applied to the differential equationy (t)=f (t, y (t)) wheref is Lipschitz continuous in its second argument, will exhibit actual convergence of ordero(h ^p–1) ify has a (p–1)th derivativey ^(p–1) that is a Riemann integral and ordero(h ^p) ify ^(p–1) is the integral of a function of bounded variation. This result applies for a functiony taking on values in any real vector space, finite or infinite dimensional.This work was supported by Grant GJ-938 from the National Science Foundation     

Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces

In this paper, we achieve the general solution and the generalized Hyers–Ulam–Rassias stability of the following functional equation

f(x+ky)+f(x−ky)=k²f(x+y)+k²f(x−y)+2(1−k²)f(x)