On a superadditivity property of Gram’s determinant

Summary Some superadditivity properties are given for Gram determinants involving inner products on a linear space.     

Brouwer and cardinalities

This note is a somewhat personal account of a paper that L.E.J. Brouwer published in 1908 and that dealt with the possible cardinalities of subsets of the continuum. That paper is of interest because it represents the first time that Brouwer presented his ideas on foundations in an international forum.I found Brouwer’s notions and arguments at times hard to grasp if not occasionally perplexing. I hope that this note contributes to a further discussion of the definitions and reasonings as presented in Brouwer’s paper.     

A structure theorem for small sumsets in nonabelian groups

Let G$G$ be an arbitrary finite group and let S$S$ and T$T$ be two subsets such that |S|≥2$\left|S\right|\ge 2$, |T|≥2$\left|T\right|\ge 2$, and |TS|≤|T|+|S|−1≤|G|−2$\left|TS\right|\le \left|T\right|+\left|S\right|-1\le \left|G\right|-2$. We show that if |S|≤|G|−4|G|^1/2$\left|S\right|\le \left|G\right|-4{\left|G\right|}^{1/2}$ then either S$S$ is a geometric progression or there exists a non-trivial subgroup H$H$ such that either |HS|≤|S|+|H|−1$\left|HS\right|\le \left|S\right|+\left|H\right|-1$ or |SH|≤|S|+|H|−1$\left|SH\right|\le \left|S\right|+\left|H\right|-1$. This extends to the nonabelian case classical results for abelian groups. When we remove the hypothesis |S|≤|G|−4|G|^1/2$\left|S\right|\le \left|G\right|-4{\left|G\right|}^{1/2}$ we show the existence of counterexamples to the above characterization whose structure is described precisely.     

Iterated sumsets and setpartitions

The Ramanujan Journal - Let $$G\cong {\mathbb {Z}}/m_1{\mathbb {Z}}\times \cdots \times {\mathbb {Z}}/m_r{\mathbb {Z}}$$ be a finite abelian group with $$m_1\mid \cdots \mid m_r=\exp (G)$$ . The...     

Classification theorems for sumsets modulo a prime

Let Zp be the finite field of prime order p and A be a subsequence of Zp. We prove several classification results about the following questions:(1) When can one represent zero as a sum of some elements of A?(2) When can one represent every element of Zp as a sum of some elements of A?(3) When can one represent every element of Zp as a sum of l elements of A?     

Projections, entropy and sumsets

In this paper we shall generalize Shearer??s entropy inequality and its recent extensions by Madiman and Tetali, and shall apply projection inequalities to deduce extensions of some of the inequalities concerning sums of sets of integers proved recently by Gyarmati, Matolcsi and Ruzsa. We shall also discuss projection and entropy inequalities and their connections.     

On submultiplicativity of spectral radius and transitivity of semigroups

It is shown that a transitive, closed, homogeneous semigroup of linear transformations on a finite-dimensional space either has zero divisors or is simultaneously similar to a group consisting of scalar multiples of unitary transformations. The proof begins with the result that for each closed homogeneous semigroup with no zero divisors there is a such that the spectral radius satisfies for all and in the semigroup.

It is also shown that the spectral radius is not -submultiplicative on any transitive semigroup of compact operators.

     

Diameter of sets and measure of sumsets

For bounded sets A, B of reals we show that wherea=(A),b=(B) andD is the diameter ofB. For large values ofa this yields (A+B)a+D.Supported by Hungarian National Foundation for Scientific Research, Grant No. 1901.     

A test for superadditivity of the mean value function of a non-homogeneous Poisson process

Let N(t) be a non-homogeneous Poisson process with mean value function Λ(t) and rate of occurrence λ(t). We propose a conditional test of the hypothesis that the process is homogeneous, versus alternatives for which the mean value function is superadditive. Specific models leading to superadditivity are presented, and the superadditive test is compared, on the basis of consistency and asymptotic relative efficiency, with the Cox-Lewis test, the latter being directed to alternatives where λ(t) is increasing.     

Possible cardinalities of irredundant bases for finite closure structures

We indicate certain connections between the rank and cardinality of a finite closure structure, and the relative sizes of its irredundant bases. A class of examples is described which shows that in general our theorem can not be strengthened.     

Hadamard product submultiplicativity of certain induced norms

In this paper we ask which norms on M_d induced by an absolute vector norm are sub-multiplicative with respect to the Hadamard product. We provide a simple necessary condition for submultiplicativity. We demonstrate that each norm on M_d induced by an l_p norm Hadamard submultiplicative and that the norms induced by certain polyhedral norms are Hadamard submultiplicative. We also consider some related inequalities.     

Hadamard product submultiplicativity of certain induced norms

In this paper we ask which norms on M_d induced by an absolute vector norm are sub-multiplicative with respect to the Hadamard product. We provide a simple necessary condition for submultiplicativity. We demonstrate that each norm on M_d induced by an l_p norm Hadamard submultiplicative and that the norms induced by certain polyhedral norms are Hadamard submultiplicative. We also consider some related inequalities.     

Restricted sumsets and a conjecture of Lev

LetA, B, S be finite subsets of an abelian groupG. Suppose that the restricted sumsetC={α+b: α ∈A, b ∈B, and α − b ∉S} is nonempty and somec∈C can be written asa+b witha∈A andb∈B in at mostm ways. We show that ifG is torsion-free or elementary abelian, then |C|≥|A|+|B|−|S|−m. We also prove that |C|≥|A|+|B|−2|S|−m if the torsion subgroup ofG is cyclic. In the caseS={0} this provides an advance on a conjecture of Lev. This author is responsible for communications, and supported by the National Science Fund for Distinguished Young Scholars (No. 10425103) and the Key Program of NSF (No. 10331020) in China.     

Cone superadditivity of discrete convex functions

A function f is said to be cone superadditive if there exists a partition of R ⁿ into a family of polyhedral convex cones such that f(z?+?x) + f(z?+?y) ≤ f(z) + f(z?+?x?+?y) holds whenever x and y belong to the same cone in the family. This concept is useful in nonlinear integer programming in that, if the objective function is cone superadditive, the global minimality can be characterized by local optimality criterion involving Hilbert bases. This paper shows cone superadditivity of L-convex and M-convex functions with respect to conic partitions that are independent of particular functions. L-convex and M-convex functions in discrete variables (integer vectors) as well as in continuous variables (real vectors) are considered.     

On the structure of the sumsets

Let A be a set of nonnegative integers. For h≥2, denote by hA the set of all the integers representable by a sum of h elements from A. In this paper, we prove that, if k≥3, and A={a₀,a₁,…,a_k−1} is a finite set of integers such that 0=a₀<a₁<?<a_k−1 and (a₁,…,a_k−1)=1, then there exist integers c and d and sets C⊆[0,c−2] and D⊆[0,d−2] such that hA=C∪[c,ha_k−1−d]∪(ha_k−1−D) for all . The result is optimal. This improves Nathanson’s result: h≥max{1,(k−2)(a_k−1−1)a_k−1}.