On the log-convexity of combinatorial sequences

We give new sufficient conditions for a sequence of polynomials to have only real zeros based on the method of interlacing zeros. As applications we derive several well-known facts, including the reality of zeros of orthogonal polynomials, matching polynomials, Narayana polynomials and Eulerian polynomials. We also settle certain conjectures of Stahl on genus polynomials by proving them for certain classes of graphs, while showing that they are false in general.     

The uncapacitated facility location problem with demand-dependent setup and service costs and customer-choice allocation

We consider a generalization of the uncapacitated facility location problem, where the setup cost for a facility and the price charged for service may depend on the number of customers patronizing the facility. Customers are represented by the nodes of the transportation network, and facilities can be located only at nodes; a customer selects a facility to patronize so as to minimize his (her) expenses (price for service + the part of transportation costs paid by the customer). We assume that transportation costs are paid partially by the service company and partially by customers. The objective is to choose locations for facilities and balanced prices so as to either minimize the expenses of the service company (the sum of the total setup cost and the total part of transportation costs paid by the company), or to maximize the total profit. A polynomial-time dynamic programming algorithm for the problem on a tree network is developed.     

Lie,Jordan and proper codimensions of associative algebras

We study the exponential rate of growth of the sequence of proper, Lie and Jordan codimensions of an associative algebra. We show that for any finite dimensional associative algebra, the exponential rates of growth can be explicitly computed and are strictly related to the PI-exponent of the algebra. The first author was partially supported by MIUR of Italy. The second author was partially supported by RFBR grant No 06-01-00485 and SSC-5666.2006.1     

Real rank versus nonnegative rank

We consider the set of m×n nonnegative real matrices and define the nonnegative rank of a matrix A to be the minimum k such that A=BC where B is m×k and C is k×n. Given that the real rank of A is j for some j, we give bounds on the nonnegative rank of A and A².     

A generalization of the Motzkin–Straus theorem to hypergraphs

In 1965, Motzkin and Straus established a remarkable connection between the global maxima of the Lagrangian of a graph G over the standard simplex and the clique number of G. In this paper, we provide a generalization of the Motzkin–Straus theorem to k-uniform hypergraphs (k-graphs). Specifically, given a k-graph G, we exhibit a family of (parameterized) homogeneous polynomials whose local (global) minimizers are shown to be in one-to-one correspondence with maximal (maximum) cliques of G.     

SOME RESULTS CONCERNING GROWTH OF POLYNOMIALS

In this paper, we establish inequalities for polynomials with restricted zeros, which in particular yields interesting generalizations of some Zygmund type inequalities for polynomial.     

N-Lie代数的态像结构

研究一类有限维的可解n-Lie代数,提出了n-Lie代数的态像、态像结构和函子的概念,并对其性质进行了研究.     

Multivariate Markov polynomial inequalities and Chebyshev nodes

This article considers the extension of V.A. Markov's theorem for polynomial derivatives to polynomials with unit bound on the closed unit ball of any real normed linear space. We show that this extension is equivalent to an inequality for certain directional derivatives of polynomials in two variables that have unit bound on the Chebyshev nodes. We obtain a sharpening of the Markov inequality for polynomials whose values at specific points have absolute value less than one. We also obtain an interpolation formula for polynomials in two variables where the interpolation points are Chebyshev nodes.     

Nonparametric nonlinear regression using polynomial and neural approximators: a numerical comparison

The solution of nonparametric regression problems is addressed via polynomial approximators and one-hidden-layer feedforward neural approximators. Such families of approximating functions are compared as to both complexity and experimental performances in finding a nonparametric mapping that interpolates a finite set of samples according to the empirical risk minimization approach. The theoretical background that is necessary to interpret the numerical results is presented. Two simulation case studies are analyzed to fully understand the practical issues that may arise in solving such problems. The issues depend on both the approximation capabilities of the approximating functions and the effectiveness of the methodologies that are available to select the tuning parameters, i.e., the coefficients of the polynomials and the weights of the neural networks. The simulation results show that the neural approximators perform better than the polynomial ones with the same number of parameters. However, this superiority can be jeopardized by the presence of local minima, which affects the neural networks but does not regard the polynomial approach.     

Implementation of a restarted Krylov subspace method for the evaluation of matrix functions

A new implementation of restarted Krylov subspace methods for evaluating f(A)b for a function f, a matrix A and a vector b is proposed. In contrast to an implementation proposed previously, it requires constant work and constant storage space per restart cycle. The convergence behavior of this scheme is discussed and a new stopping criterion based on an error indicator is given. The performance of the implementation is illustrated for three parabolic initial value problems, requiring the evaluation of exp(A)b.