Sets in Abelian groups with distinct sums of pairs

A subset S={s₁,…,sk} of an Abelian group G is called an St-set of size k if all sums of t different elements in S are distinct. Let s(G) denote the cardinality of the largest S₂-set in G. Let v(k) denote the order of the smallest Abelian group for which s(G)?k. In this article, bounds for s(G) are developed and v(k) is determined for k?15 by computing s(G) for Abelian groups of order up to 183 using exhaustive backtrack search with isomorph rejection.     

Difference Sets and Recursion Theory

There is a recursive set of natural numbers which is the difference set of some recursively enumerable set but which is not the difference set of any recursive set.     

Sets with more differences than sums

We show that a random set of integers with density 0 has almost always more differences than sums.      

Sets of integers with missing differences

This paper deals with the problem of finding the maximal density μ(M) of sets of integers in which the differences given by a set M do not occur (M-sets). Some general estimates are given, μ(M) is compared to other set functions, and expressions for μ(M) are given for most members of the families {1, j, k} and {1, 2, j, k}     

Constructions of Partial Difference Sets and Relative DifferenceSets Using Galois Rings

We use Galois rings to construct partial difference sets and relative difference sets in non-elementary abelianp-groups. As an example, we also use Galois ringG R(4, 2) to construct a (96,20,4) difference set in Z₄ × Z₄ × Z₆.Dedicated to Hanfried Lenz on the occasion of his 80th birthday     

A Note on Certain 2-Groups with Hadamard Difference Sets

Nontrivial difference sets in 2-groups are part of the family of Hadamarddifference sets. An abelian group of order 2^2d+2 has a difference setif and only if the exponent of the group is less than or equal to2 ^d+2. We provide an exponent bound for a more general type of 2-groupwhich has a Hadamard difference set. A recent construction due to Davis and Iiamsshows that we can attain this bound in at least half of the cases.     

拟正则双序集用矩形双序集的余扩张

称双序集E为双序集F用矩形双序集的余扩张,若存在满双序集态射θ：E→F,使对每个α∈F,αθ－1是E的矩形双序子集．本文讨论了拟正则双序集的这种余扩张的性质,给出了它们的结构．作为应用,证明了拟正则的硬双序集实为正则双序集．     

Rings characterized by direct sums of cs modules

It is shown that a ring for which every CS right module is ∑CS is right artinian. As a consequence, it is also shown that over a ring R every direct sum of CS right R-modules is CS iff R is right artinian and the composition length of every uniform right R-module is at most 2.     

New trigonometric sums by sampling theorem

We use a sampling theorem associated with second-order discrete eigenvalue problems to derive some trigonometric identities extending the results of Byrne and Smith [G.J. Byrne, S.J. Smith, Some integer-valued trigonometric sums, Proc. Edinburg Math. Soc. 40 (1997) 393-401]. We derive both integral and non-integral valued trigonometric sums. We give illustrative examples involving representations of the trigonometric sums and in an integral-valued polynomial in (2n+1) of degree 2m, .     

Large Sets and Overlarge Sets of Triangle-Decomposition

There are six types of triangles:undirected triangle,cyclic triangle,transitive triangle,mixed-1triangle,mixed-2 triangle and mixed-3 triangle.The triangle-decompositions for the six types of triangles havealready been solved.For the first three types of triangles,their large sets have already been solved,and theiroverlarge sets have been investigated.In this paper,we establish the spectrum of LT_i(v,λ),OLT_i(v)(i=1,2),and give the existence of LT_3(v,λ)and OLT_3(v,λ)with λ even.     

Sum rules for Jacobi matrices and divergent Lieb-Thirring sums

Let E_j be the eigenvalues outside [-2,2] of a Jacobi matrix with a_n-1∈?² and b_n→0, and μ^′ the density of the a.c. part of the spectral measure for the vector δ₁. We show that if b_n∉?⁴, b_n+1-b_n∈?², then

     

Combinatorial sums and finite differences

We present a new approach to evaluating combinatorial sums by using finite differences. Let and be sequences with the property that Δb_k=a_k for k?0. Let , and let . We derive expressions for g_n in terms of h_n and for h_n in terms of g_n. We then extend our approach to handle binomial sums of the form , , and , as well as sums involving unsigned and signed Stirling numbers of the first kind, and . For each type of sum we illustrate our methods by deriving an expression for the power sum, with a_k=k^m, and the harmonic number sum, with a_k=H_k=1+1/2+?+1/k. Then we generalize our approach to a class of numbers satisfying a particular type of recurrence relation. This class includes the binomial coefficients and the unsigned Stirling numbers of the first kind.     

On Pseudocomplemented and Stone Ordered Sets

The aim of this paper is to characterize both the pseudocomplemented and Stone ordered sets in a manner similar to that used previously for Boolean and distributive ordered sets. The sublattice G(A) of the Dedekind–Mac Neille completion DM(A) of an ordered set A generated by A is said to be the characteristic lattice of A. We will show that there are distributive pseudocomplemented ordered sets whose characteristic lattices are not pseudocomplemented. We can define a stronger notion of pseudocomplementedness by demanding that both A and G(A) be pseudocomplemented. It turns out that the two concepts are the same for finite and Stone ordered sets.     

Strongly Amorphous Sets and Dual Dedekind Infinity

1. If A is strongly amorphous (i.e., all relations on A are definable), then its power set P(A) is dually Dedekind infinite, i. e., every function from P(A) onto P(A) is injective. 2. The class of “inexhaustible” sets is not closed under supersets unless AC holds.     

Grasping Infinity by Finite Sets

We show that the existence of an infinite set can be reduced to the existence of finite sets “as big as we will”, provided that a multivalued extension of the relation of equipotence is admitted. In accordance, we modelize the notion of infinite set by a fuzzy subset representing the class of (finite) wide sets.     

Minimal Kakeya Sets

Blokhuis and Mazzocca (A. Blokhuis and F. Mazzocca, The finite field Kakeya problem (English summary). Building bridges. Bolyai Soc Math Stud 19 (2008) 205–218) provide a strong answer to the finite field analog of the classical Kakeya problem, which asks for the minimum size of a point set in an affine plane π that contains a line in every direction. In this article, we consider the related problem of minimal Kakeya sets, namely Kakeya sets containing no smaller Kakeya sets, and provide an interesting infinite family of minimal Kakeya sets that are not of extremal size.     

Similar Transposed Sets of Polynomials

This paper studies the effectiveness of another kind of transposed sets of polynomials of one complex variable in closed regions, open discs, at the origin and for all entire functions. In addition, an upper bound for the order of these sets in this case is obtained.     

Differences of Computably Enumerable Sets

We consider the ower semilattice 𝒟 of differences of c.e. sets under inclusion. It is shown that 𝒟 is not distributive as a semilattice, and that the c.e. sets form a definable subclass.     

闭模糊集构成凸模糊集的充要条件

本文通过引入弱拟凸模糊集的概念，针对欧几里空间上的闭模糊集，给出了它构成凸模糊集的一个充要条件，从而丰富了凸模糊集的理论及其应用。     

Pluripolar Hulls and Complete Pluripolar Sets

We study the pluripolar hull of a complex subvariety in the complement of a closed complete pluripolar set. A result on propagation of pluripolar hulls is also given.