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     

A Robertson-type uncertainty principle and quantum Fisher information

Let A₁,…,A_N be complex self-adjoint matrices and let ρ be a density matrix. The Robertson uncertainty principle

     

A restricted sum formula among multiple zeta values

For positive integers α₁,α₂,…,αr with αr?2, the multiple zeta value or r-fold Euler sum is defined as

     

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

     

A combinatorial identity with application to Catalan numbers

By a very simple argument, we prove that if l,m,n∈{0,1,2,…} then

     

On semigroup presentations and Adian graphs

Let P be a semigroup presentation of the form

〈a₁,…,a_n∣w₁=a₁,…,w_n=a_n〉.     

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

     

Maps preserving the spectrum of generalized Jordan product of operators

Let A₁,A₂ be standard operator algebras on complex Banach spaces X₁,X₂, respectively. For k?2, let (i₁,…,i_m) be a sequence with terms chosen from {1,…,k}, and define the generalized Jordan product

     

Invariant Banach limits and applications

Let ?_∞ be the space of all bounded sequences x=(x₁,x₂,…) with the norm

     

Von Neumann’s inequality for noncommuting contractions

Let T₁,…,T_d be linear contractions on a complex Hilbert space and p a complex polynomial in d variables which is a sum of d single variable polynomials. We show that the operator norm of p(T₁,…,T_d) is bounded by

     

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)     

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:

     

A new extension of the Erd?s-Heilbronn conjecture

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

     

On the spectral radius of positive operators on Banach sequence spaces

Let K₁,…,K_n be (infinite) non-negative matrices that define operators on a Banach sequence space. Given a function f:[0,∞)×…×[0,∞)→[0,∞) of n variables, we define a non-negative matrix and consider the inequality

     

On a modification of a problem of Bialostocki, Erd?s, and Lefmann

For positive integers m and r, one can easily show there exist integers N such that for every map Δ:{1,2,…,N}→{1,2,…,r} there exist 2m integers

x₁<?<x_m<y₁<?<y_m,     

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,     

Global stability of some symmetric difference equations

Suppose r∈(0,1],m∈N and 1?k₁<k₂<?<k_2m+1, and let S_2m+1={1,2,…,2m+1}. We show that every positive solution to the difference equation

     

Markov-Nikolskii type inequalities for exponential sums on finite intervals

Let Λn:={λ₀<λ₁<?<λn} be a set of real numbers. The collection of all linear combinations of eλ₀t,eλ₁t,…,eλnt over R will be denoted by

E(Λn):=span{eλ₀t,eλ₁t,…,eλnt}.