On Waring's Problem for Cubes and Smooth Weyl Sums

Non-trivial estimates for fractional moments of smooth cubicWeyl sums are developed. Complemented by bounds for such sumsof use on both the major and minor arcs in a Hardy-Littlewooddissection, these estimates are applied to derive an upper boundfor the sth moment of the smooth cubic Weyl sum of the expectedorder of magnitude as soon as s> 7.691. Related argumentsdemonstrate that all large integers n are represented as thesum of eight cubes of natural numbers, all of whose prime divisorsare at most exp (c(log nlog log n)^1/2}, for a suitable positivenumber c. This conclusion improves a previous result of G. Harcosin which nine cubes are required. 1991 Mathematics Subject Classification:11P05, 11L15, 11P55.     

Sumsets contained in infinite sets of integers

If the positive integers are partitioned into a finite number of cells, then Hindman proved that there exists an infinite set B such that all finite, nonempty sums of distinct elements of B all belong to one cell of the partition. Erdös conjectured that if A is a set of integers with positive asymptotic density, then there exist infinite sets B and C such that B + C ? A. This conjecture is still unproved. This paper contains several results on sumsets contained in finite sets of integers. For example, if A is a set of integers of positive upper density, then for any n there exist sets B and F such that B has positive upper density, F has cardinality n, and B + F ? A.     

The Frobenius problem, sums of powers of integers, and recurrences for the Bernoulli numbers

In the Frobenius problem with two variables, one is given two positive integers a and b that are relative prime, and is concerned with the set of positive numbers NR that have no representation by the linear form ax+by in nonnegative integers x and y. We give a complete characterization of the set NR, and use it to establish a relation between the power sums over its elements and the power sums over the natural numbers. This relation is used to derive new recurrences for the Bernoulli numbers.     

A result on Waring–Goldbach problem for cubes

In this paper, we prove that, with at most O(N ^5/6+ε ) exceptions, all positive integers n ⩽ N can be written as sums of a cube and four cubes of primes.     

Extrapolating the mean-values of multiplicative functions

It is shown that certain commonly occurring conditions may be factored out of sums of multiplicative arithmetic functions.A function is arithmetic if it is defined on the positive integers. Those complex-valued arithmetic functions g which satisfy the relation g(ab) = g(a)g(b) for all coprime pairs of positive integers a, b are here called multiplicative. In this paper g will be a multiplicative function which satisfies |g(n)| ≤ 1 for all positive integers n.     

Gauss sums and multinomial coefficients

Consider a Gauss sum for a finite field of characteristic p, where p is an odd prime. When such a sum (or a product of such sums) is a p-adic integer we show how it can be realized as a p-adic limit of a sequence of multinomial coefficients. As an application we generalize some congruences of Hahn and Lee to exhibit p-adic limit formulae, in terms of multinomial coefficients, for certain algebraic integers in imaginary quadratic fields related to the splitting of rational primes. We also give an example illustrating how such congruences arise from a p-integral formal group law attached to the p-adic unit part of a product of Gauss sums.     

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.     

General power sums of integers that are,and are not,represented in the two-element Frobenius problem

A general method is presented for evaluating the sums of mth powers of the integers that can, and that cannot, be represented in the two-element Frobenius problem. Generating functions are introduced and used for that purpose. Explicit formulas for the desired sums are obtained and specific examples are discussed.     

On the generalization of the Euler polynomials with the real parameters

In this study, we give multiplication formula for generalized Euler polynomials of order α and obtain some explicit recursive formulas. The multiple alternating sums with positive real parameters a and b are evaluated in terms of both generalized Euler and generalized Bernoulli polynomials of order α. Finally we obtained some interesting special cases.     

Sums of the squares of terms of sequence {<Emphasis Type="Italic">u</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>}

In this paper, we consider generalized Fibonacci type second order linear recurrence {u _n }. We derive a generating matrix for both the sums of squares, ∑ _i=0 ⁿ u _i ² and the products of the form u _n u _n+2. We also derive explicit formulas for the sums and products by using matrix methods. Then we give a matrix method to generate the sums of product of two consecutive terms u _n u _n+1 as well as the product, u _n u _n+2. Further we give generating functions and combinatorial representations of the sums of squares of terms of {u _n } and the product, u _n u _n+2.     

Stability of sums of weighted nonnegative random variables

A stability result for sums of weighted nonnegative random variables is established and then it is utilized to obtain, among other things, a slight generalization of the Borel-Cantelli lemma and to show that the work of Jamison, Orey, and Pruitt (Z. Wahrsch. Verw. Gebiete4 (1965), 40–44) on almost sure convergence of weighted averages of independent random variables remains valid if the assumption of independence on the random variables is replaced by pairwise independence.     

Convolution identities and lacunary recurrences for Bernoulli numbers

We extend Euler's well-known quadratic recurrence relation for Bernoulli numbers, which can be written in symbolic notation as n(B₀+B₀)=−nBn−1−(n−1)Bn, to obtain explicit expressions for n(Bk+Bm) with arbitrary fixed integers k,m?0. The proof uses convolution identities for Stirling numbers of the second kind and for sums of powers of integers, both involving Bernoulli numbers. As consequences we obtain new types of quadratic recurrence relations, one of which gives B6k depending only on B2k,B2k+2,…,B4k.     

Latin squares with pairwise orthogonal conjugates

Let L¹ denote the set of integers n such that there exists an idempotent Latin square of order n with all of its conjugates distinct and pairwise orthogonal. It is known that L¹ contains all sufficiently large integers. That is, there is a smallest integer n_o such that L¹ contains all integers greater than n_o. However, no upper bound for n_o has been given and the term “sufficiently large” is unspecified. The main purpose of this paper is to establish a concrete upper bound for n_o. In particular it is shown that L¹ contain all integers n>5594, with the possible exception of n=6810.     

Hilbertian Jamison sequences and rigid dynamical systems

A strictly increasing sequence (nk)_k?0 of positive integers is said to be a Hilbertian Jamison sequence if for any bounded operator T on a separable Hilbert space such that supk?0‖Tnk‖<+∞, the set of eigenvalues of modulus 1 of T is at most countable. We first give a complete characterization of such sequences. We then turn to the study of rigidity sequences (nk)_k?0 for weakly mixing dynamical systems on measure spaces, and give various conditions, some of which are closely related to the Jamison condition, for a sequence to be a rigidity sequence. We obtain on our way a complete characterization of topological rigidity and uniform rigidity sequences for linear dynamical systems, and we construct in this framework examples of dynamical systems which are both weakly mixing in the measure-theoretic sense and uniformly rigid.     

Universally bad integers and the 2-adics

In his 1964 paper, de Bruijn (Math. Comp. 18 (1964) 537) called a pair (a,b) of positive odd integers good, if , where is the set of nonnegative integers whose 4-adic expansion has only 0's and 1's, otherwise he called the pair (a,b) bad. Using the 2-adic integers we obtain a characterization of all bad pairs. A positive odd integer u is universally bad if (ua,b) is bad for all pairs of positive odd integers a and b. De Bruijn showed that all positive integers of the form u=2^k+1 are universally bad. We apply our characterization of bad pairs to give another proof of this result of de Bruijn, and to show that all integers of the form u=φ_p^k(4) are universally bad, where p is prime and φ_n(x) is the nth cyclotomic polynomial. We consider a new class of integers we call de Bruijn universally bad integers and obtain a characterization of such positive integers. We apply this characterization to show that the universally bad integers u=φ_p^k(4) are in fact de Bruijn universally bad for all primes p>2. Furthermore, we show that the universally bad integers φ_2^k(4), and more generally, those of the form 4^k+1, are not de Bruijn universally bad.     

A new class of antimagic Cartesian product graphs

An antimagic labeling of a finite undirected simple graph with m edges and n vertices is a bijection from the set of edges to the integers 1,…,m such that all n-vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel [N. Hartsfield, G. Ringel, Pearls in Graph Theory, Academic Press, INC., Boston, 1990, pp. 108-109, Revised version, 1994] conjectured that every simple connected graph, except K₂, is antimagic. In this article, we prove that a new class of Cartesian product graphs are antimagic. In particular, by combining this result and the antimagicness result on toroidal grids (Cartesian products of two cycles) in [Tao-Ming Wang, Toroidal grids are anti-magic, in: Proc. 11th Annual International Computing and Combinatorics Conference COCOON’2005, in: LNCS, vol. 3595, Springer, 2005, pp. 671-679], all Cartesian products of two or more regular graphs of positive degree can be proved to be antimagic.     

On products of integers

The main objective of this paper is to investigate the relation between the number of integers in a given subset $\text{A}$ of the integers 1, 2,…, n and the number of integers that can be chosen from 1, 2,…, n so that their pairwise products all appear in $\text{A}$. Other related problems are also considered.     

Extremal problems for numerical series

In this paper, we consider extremal problems for numerical positive series. The terms of these series are pairwise products of the elements of two sequences, one of which is fixed and the other varies within a given set of sequences. We obtain exact solutions for a number of such problems. As one of the possible applications of the results obtained, we find solutions of some extremal problems related to best n-term approximations of periodic functions.     

Design and analysis of predictive sorting algorithms

The focus of this research is the class of sequential algorithms, called predictive sorting algorithms, for sorting a given set ofn elements using pairwise comparisons. The order in which these pairwise comparisons are made is defined by a fixed sequence of all unordered pairs of distinct integers {1, 2, ...,n} called a sort sequence. A predictive sorting algorithm associated with a sort sequence specifies pairwise comparisons of elements in the input set in the order defined by the sort sequence, except that the comparisons whose outcomes can be inferred from the preceding pairs of comparisons are not performed. In this paper predictive sorting algorithms are obtained, based on known sorting algorithms, and are shown to be required on the averageO(n logn) comparisons.     

Sets of permutations that generate the symmetric group pairwise

The paper contains proofs of the following results. For all sufficiently large odd integers n, there exists a set of ²n−1 permutations that pairwise generate the symmetric group Sn. There is no set of ²n−1+1 permutations having this property. For all sufficiently large integers n with n≡2mod4, there exists a set of ²n−2 even permutations that pairwise generate the alternating group An. There is no set of ²n−2+1 permutations having this property.