Convex integer maximization via Graver bases

We present a new algebraic algorithmic scheme to solve convex integer maximization problems of the following form, where c is a convex function on R^d and w₁x,…,w_dx are linear forms on Rⁿ,

max{c(w₁x,…,w_dx):Ax=b,x∈Nⁿ}.     

A generalization of a classical zero-sum problem

In this note, we supply the details of the proof of the fact that if a₁,…,a_n+Ω(n) are integers, then there exists a subset M⊂{1,…,n+Ω(n)} of cardinality n such that the equation

     

The zeros of a quadratic form at square-free points

Let F(x₁,…,xn) be a nonsingular indefinite quadratic form, n=3 or 4. For n=4, suppose the determinant of F is a square. Results are obtained on the number of solutions of

F(x₁,…,xn)=0     

Sets whose sumset avoids a thin sequence

Let {a₁,a₂,a₃,…} be an unbounded sequence of positive integers with an+1/an approaching α as n→∞, and let β>max(α,2). We show that for all sufficiently large x?0, if A⊂[0,x] is a set of nonnegative integers containing 0 and satisfying

     

OMC for scalar multiples of unitaries

We establish the following case of the Determinantal Conjecture of Marcus [M. Marcus, Derivations, Plücker relations and the numerical range, Indiana Univ. Math. J. 22 (1973) 1137-1149] and de Oliveira [G.N. de Oliveira, Research problem: Normal matrices, Linear and Multilinear Algebra 12 (1982) 153-154]. Let A and B be unitary n × n matrices with prescribed eigenvalues a₁, … , a_n and b₁, … , b_n, respectively. Then for any scalars t and s

     

Fractal interpolation on the Sierpinski Gasket

We prove for the Sierpinski Gasket (SG) an analogue of the fractal interpolation theorem of Barnsley. Let V₀={p₁,p₂,p₃} be the set of vertices of SG and the three contractions of the plane, of which the SG is the attractor. Fix a number n and consider the iterations uw=uw₁uw₂?uwn for any sequence w=(w₁,w₂,…,wn)∈n{1,2,3}. The union of the images of V₀ under these iterations is the set of nth stage vertices Vn of SG. Let F:Vn→R be any function. Given any numbers αw(w∈n{1,2,3}) with 0<|αw|<1, there exists a unique continuous extension of F, such that

f(uw(x))=αwf(x)+hw(x)     

A note on the continuous mixing set

The continuous mixing set is , where w₁,…,w_n>0 and f₁,…,f_n∈ℜ. Let m=|{w₁,…,w_n}|. We show that when w₁|?|w_n, optimization over S can be performed in time O(n^m+1), and in time O(nlogn) when w₁=?=w_n=1.     

Representation of group elements as subsequence sums

Let G be a finite (additive written) abelian group of order n. Let w₁,…,w_n be integers coprime to n such that w₁+w₂+?+w_n≡0 (mod n). Let I be a set of cardinality 2n-1 and let ξ={x_i:i∈I} be a sequence of elements of G. Suppose that for every subgroup H of G and every a∈G, ξ contains at most terms in a+H.Then, for every y∈G, there is a subsequence {y₁,…,y_n} of ξ such that y=w₁y₁+?+w_ny_n.Our result implies some known generalizations of the Erd?s-Ginzburg-Ziv Theorem.     

The structure of max-min hyperplanes

In this article, continuing [12,13], further contributions to the theory of max-min convex geometry are given. The max-min semiring is the set endowed with the operations ⊕=max,⊗=min in . A max-min hyperplane (briefly, a hyperplane) is the set of all points satisfying an equation of the form

a₁⊗x₁⊕…⊕a_n⊗x_n⊕a_n+1=b₁⊗x₁⊕…⊕b_n⊗x_n⊕b_n+1,     

Degree complete graphs

Ryser [Combinatorial Mathematics, Carus Mathematical Monograph, vol. 14, Wiley, New York, 1963] introduced a partially ordered relation ‘?’ on the nonnegative integral vectors. It is clear that if S=(s₁,s₂,…,s_n) is an out-degree vector of an orientation of a graph G with vertices 1,2,…,n, then

(Π)     

Operator algebras associated with unitary commutation relations

We define nonselfadjoint operator algebras with generators Le₁,…,Len,Lf₁,…,Lfm subject to the unitary commutation relations of the form

     

The hilbert boundary value problem for generalized analytic functions in clifford analysis

Let ?_0,n be the real Clifford algebra generated by e₁, e₂,…, e_n satisfying e_ie_j+e_je_i=−2δ_ij, i, j=1,2,…, n. e₀ is the unit element. Let Ω be an open set. A function f is called left generalized analytic in Ω if f satisfies the equation

equation(0.1)

Lf=0,$Lf=0,$

where

L=q₀e₀∂x₀+ q₁e₁∂x₁+…+qnen_∂xn,$L=q_0{e}_0\partial x_0+\hspace{0.17em}{q}_1{e}_1\partial x_1+\dots +q_n{e}_n{\partial }_{x_n},$

qi <0, i=0,1,…, n. In this article, we first give the kernel function for the generalized analytic function. Further, the Hilbert boundary value problem for generalized analytic functions in ?ⁿ⁺¹₊ will be investigated.     

Noncommutative convexity arises from linear matrix inequalities

This paper concerns polynomials in g noncommutative variables x=(x₁,…,xg), inverses of such polynomials, and more generally noncommutative “rational expressions” with real coefficients which are formally symmetric and “analytic near 0.” The focus is on rational expressions r=r(x) which are “matrix convex” near 0; i.e., those rational expressions r for which there is an ?>0 such that if X=(X₁,…,Xg) is a g-tuple of n×n symmetric matrices satisfying

     

On the minimum number of completely 3-scrambling permutations

A family P={π₁,…,π_q} of permutations of [n]={1,…,n} is completely k-scrambling [Spencer, Acta Math Hungar 72; Füredi, Random Struct Algor 96] if for any distinct k points x₁,…,x_k∈[n], permutations π_i's in P produce all k! possible orders on π_i(x₁),…,π_i(x_k). Let N^*(n,k) be the minimum size of such a family. This paper focuses on the case k=3. By a simple explicit construction, we show the following upper bound, which we express together with the lower bound due to Füredi for comparison.

     

A new extension of the Erd?s-Heilbronn conjecture

Let A₁,…,An be finite subsets of a field F, and let

     

Construction of nonnegative symmetric matrices with given spectrum

Let σ = (λ₁, … , λ_n) be the spectrum of a nonnegative symmetric matrix A with the Perron eigenvalue λ₁, a diagonal entry c and let τ = (μ₁, … , μ_m) be the spectrum of a nonnegative symmetric matrix B with the Perron eigenvalue μ₁. We show how to construct a nonnegative symmetric matrix C with the spectrum

(λ₁+max{0,μ₁-c},λ₂,…,λ_n,μ₂,…,μ_m).     

Numerical ranges of nilpotent operators

For any operator A on a Hilbert space, let W(A), w(A) and w₀(A) denote its numerical range, numerical radius and the distance from the origin to the boundary of its numerical range, respectively. We prove that if Aⁿ=0, then w(A)?(n-1)w₀(A), and, moreover, if A attains its numerical radius, then the following are equivalent: (1) w(A)=(n-1)w₀(A), (2) A is unitarily equivalent to an operator of the form aA_n⊕A^′, where a is a scalar satisfying |a|=2w₀(A), A_n is the n-by-n matrix

     

On various restricted sumsets

For finite subsets A₁,…,A_n of a field, their sumset is given by . In this paper, we study various restricted sumsets of A₁,…,A_n with restrictions of the following forms:

     

The Eulerian distribution on involutions is indeed unimodal

Let In,k (respectively Jn,k) be the number of involutions (respectively fixed-point free involutions) of {1,…,n} with k descents. Motivated by Brenti's conjecture which states that the sequence In,0,In,1,…,In,n−1 is log-concave, we prove that the two sequences In,k and J2n,k are unimodal in k, for all n. Furthermore, we conjecture that there are nonnegative integers an,k such that