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

     

Interior estimates for solutions of a fourth order nonlinear partial differential equation

Let be a locally strongly convex hypersurface, given by the graph of a convex function xn+1=f(x₁,…,xn) defined in a convex domain Ω⊂Rn. M is called a α-extremal hypersurface, if f is a solution of

     

A technique to study the correlation measures of binary sequences

Let E^N=(e₁,e₂,…,e_N) be a binary sequence with e_i∈{+1,−1}. For 2≤k≤N, the correlation measure of order k of the sequence is defined by Mauduit and Sárközy as

     

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 generalized Roper-Suffridge extension operator for some biholomorphic mappings

Suppose f is a spirallike function of type β (or starlike function of order α) on the unit disk D in C. Let , where 1?p₁?2 (or 0<p₁?2), pj?1, j=2,…,n, are real numbers. In this paper, we prove that

     

A new extension of the Erd?s-Heilbronn conjecture

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

     

Invariant Banach limits and applications

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

     

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

     

Certain series attached to an even number of elliptic modular forms

For j=1,…,n let f_j(z) and g_j(z) be holomorphic modular forms for such that f_j(z)g_j(z) is a cusp form. We define a series

     

The asymptotic number of spanning trees in circulant graphs

Let T(G) be the number of spanning trees in graph G. In this note, we explore the asymptotics of T(G) when G is a circulant graph with given jumps.The circulant graph is the 2k-regular graph with n vertices labeled 0,1,2,…,n−1, where node i has the 2k neighbors i±s₁,i±s₂,…,i±s_k where all the operations are . We give a closed formula for the asymptotic limit as a function of s₁,s₂,…,s_k. We then extend this by permitting some of the jumps to be linear functions of n, i.e., letting s_i, d_i and e_i be arbitrary integers, and examining

     

The numerical range of a tridiagonal operator

Let r be a real number and A a tridiagonal operator defined by Aej=ej−1+rjej+1, j=1,2,…, where {e₁,e₂,…} is the standard orthonormal basis for ?²(N). Such tridiagonal operators arise in Rogers-Ramanujan identities. In this paper, we study the numerical ranges of these tridiagonal operators and finite-dimensional tridiagonal matrices. In particular, when r=−1, the numerical range of the finite-dimensional tridiagonal matrix is the convex hull of two explicit ellipses. Applying the result, we obtain that the numerical range of the tridiagonal operator is the square

     

On the prime power factorization of n!

In this paper, we prove two results. The first theorem uses a paper of Kim (J. Number Theory 74 (1999) 307) to show that for fixed primes p₁,…,p_k, and for fixed integers m₁,…,m_k, with , the numbers (e_p1(n),…,e_pk(n)) are uniformly distributed modulo (m₁,…,m_k), where e_p(n) is the order of the prime p in the factorization of n!. That implies one of Sander's conjectures from Sander (J. Number Theory 90 (2001) 316) for any set of odd primes. Berend (J. Number Theory 64 (1997) 13) asks to find the fastest growing function f(x) so that for large x and any given finite sequence , there exists n<x such that the congruences hold for all i?f(x). Here, p_i is the ith prime number. In our second result, we are able to show that f(x) can be taken to be at least , with some absolute constant c₁, provided that only the first odd prime numbers are involved.     

Solvability of singular second order m-point boundary value problems

Let be a function satisfying Carathéodory's conditions and (1−t)e(t)∈L¹(0,1). Let ξi∈(0,1), ai∈R, i=1,…,m−2, 0<ξ₁<ξ₂<?<ξm−2<1 be given. This paper is concerned with the problem of existence of a C¹[0,1) solution for the m-point boundary value problem

     

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

     

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