On the reducible quintic complete base polynomials

Let B(x)=xm+bm−1xm−1+?+b₀∈Z[x]. If every element in Z[x]/(B(x)Z[x]) has a polynomial representative with coefficients in S={0,1,2,…,|b₀|−1} then B(x) is called a complete base polynomial. We prove that if B(x) is a completely reducible quintic polynomial with five distinct integer roots less than −1, then B is a complete base polynomial. This is the best possible result regarding the completely reducible polynomials so far. Meanwhile, we provide a Mathematica program for determining whether an input polynomial B(x) is a complete base polynomial or not. The program enables us to experiment with various polynomial examples, to decide if the potential result points in the desired direction and to formulate credible conjectures.     

Normalized trigonometric splines of Lagrange type

B _?-splines, which are a nonpolynomial generalization of the well-known B-splines, are investigated. B _?-splines arise from approximation relations regarded as a system of linear algebraical equations, from which both polynomial and nonpolynomial splines are derived. Third-order normalized trigonometric splines of Lagrange type (zero height) determined by the generating vector function ?(t) = (1, sin t, cos t, sin2 t) ^T are constructed. These splines are twice continuously differentiable and have minimal compact support. A system of functionals biorthogonal to B _?-splines is defined. The solution of the interpolation problem generated by the resulting biorthogonal system in the space of B _?-splines is found.     

Krylov type subspace methods for matrix polynomials

We consider solving eigenvalue problems or model reduction problems for a quadratic matrix polynomial Iλ² − Aλ − B with large and sparse A and B. We propose new Arnoldi and Lanczos type processes which operate on the same space as A and B live and construct projections of A and B to produce a quadratic matrix polynomial with the coefficient matrices of much smaller size, which is used to approximate the original problem. We shall apply the new processes to solve eigenvalue problems and model reductions of a second order linear input-output system and discuss convergence properties. Our new processes are also extendable to cover a general matrix polynomial of any degree.     

Compactly supported positive definite radial functions

We provide criteria for positive definiteness of radial functions with compact support. Based on these criteria we will produce a series of positive definite and compactly supported radial functions, which will be very useful in applications. The simplest ones arecut-off polynomials, which consist of a single polynomial piece on [0, 1] and vanish on [1, ∞). More precisely, for any given dimensionn and prescribedC ^k smoothness, there is a function inC ^k (? ⁿ ), which is a positive definite radial function with compact support and is a cut-off polynomial as a function of Euclidean distance. Another example is derived from odd-degreeB-splines.     

Generalized base representations

Let B∈Z[x] be a polynomial with b=B(0). Let S be a complete residue class modulo b containing 0. We attempt to classify the polynomials B and residue classes S so that for every polynomial P∈Z[x] there exists a polynomial Q with coefficients in S such that .     

Homogeneous multiplicative polynomial laws are determinants

Let k be a commutative ring. Let R,B be k-algebras with B commutative. Let p:R→B be a homogeneous multiplicative polynomial law of degree n. We show that p is obtained by left and right composing a determinant with some homomorphisms of k-algebras.     

The combinatorial interpretation of the Jacobi identity from Lie algebras

Many enumeration problems concerning sequences emerge as special cases of the combinatorial interpretation of the identity tr log(I ? B) = log det(I ? B), where B is a matrix over the ring of formal power series with zero constant term. The identity is obtained from the well-known formula of Jacobi, namely, det(e^A) = e^tr(A). The problems include the derangement problem, the Simon Newcomb problem, the Smirnov problem, the Ménage problem, and their various generalizations. A conjecture is obtained for the enumeration of sequences with no increasing p runs.     

Hypercyclic subspaces in omega

We show that any countable family of operators of the form P(B), where P is a non-constant polynomial and B is the backward shift operator on ω, the countably infinite product of lines, has a common hypercyclic subspace.     

Matchings in benzene chains

An explicit recurrence relation is derived for the matching polynomial of the general benzene chain B_n. A table is given for chains of length up to 6. Explicit formulae are then obtained for the first five coefficients of the matching polynomial of B_n. Finally, results are deduced for the number of perfect matchings and the number of matchings with two nodes.     

On the Bézoutian for polynomial matrices

The Bézoutian B of two polynomial matrices can be described as a solution of a linear matrix equation. This fact yields a new proof of the Barnett factorization of B.     

The cyclicity of a cubic system with nonradical Bautin ideal

We present a method for investigating the cyclicity of an elementary focus or center of a polynomial system of differential equations by means of complexification of the system and application of algorithms of computational algebra, showing an approach to treating the case that the Bautin ideal B of focus quantities is not a radical ideal (more precisely, when the ideal BK is not radical, where BK is the ideal generated by the shortest initial string of focus quantities that, like the Bautin ideal, determines the center variety). We illustrate the method with a family of cubic systems.     

Exact solution to a Lindley-type equation on a bounded support

We derive the limiting waiting-time distribution F_W of a model described by the Lindley-type equation W=max{0,B-A-W}, where B has a polynomial distribution. This exact solution is applied to derive approximations of F_W when B is generally distributed on a finite support. We provide error bounds for these approximations.     

Max Flow and Min Cut with bounded-length paths: complexity, algorithms, and approximation

We consider the “flow on paths” versions of Max Flow and Min Cut when we restrict to paths having at most B arcs, and for versions where we allow fractional solutions or require integral solutions. We show that the continuous versions are polynomial even if B is part of the input, but that the integral versions are polynomial only when B ≤ 3. However, when B ≤ 3 we show how to solve the problems using ordinary Max Flow/Min Cut. We also give tight bounds on the integrality gaps between the integral and continuous objective values for both problems, and between the continuous objective values for the bounded-length paths version and the version allowing all paths. We give a primal–dual approximation algorithm for both problems whose approximation ratio attains the integrality gap, thereby showing that it is the best possible primal–dual approximation algorithm.     

A new partial regularity proof for solutions of nonlinear elliptic systems

We prove partial regularity for weak solutionsu of the fully nonlinear elliptic system divA (x, u, Du) +B (x, u, Du)=0, whereA andB have natural polynomial growth, and whereB satisfies a two-sided or a one-sided condition. We do not employ a reverse Hölder inequality.     

Strong shift equivalence and shear adjacency of nonnegative square integer matrices

For two square matrices A, B of possibly different sizes with nonnegative integer entries, write A ≈₁ B if A = RS and B = SR for some two nonnegative integer matrices R,S. The transitive closure of this relation is called strong shift equivalence and is important in symbolic dynamics, where it has been shown by R.F. Williams to characterize the isomorphism of two topological Markov chains with transition matrices A and B. One invariant is the characteristic polynomial up to factors of λ. However, no procedure for deciding strong shift equivalence is known, even for 2×2 matrices A, B. In fact, for n × n matrices with n > 2, no nontrivial sufficient condition is known. This paper presents such a sufficient condition: that A and B are in the same component of a directed graph whose vertices are all n × n nonnegative integer matrices sharing a fixed characteristic polynomial and whose edges correspond to certain elementary similarities. For n > 2 this result gives confirmation of strong shift equivalences that previously could not be verified; for n = 2, previous results are strengthened and the structure of the directed graph is determined.     

Essential and inessential elements of a standard basis

In this paper we introduce the concept of inessential element of a standard basis B(I), where I is any homogeneous ideal of a polynomial ring. An inessential element is, roughly speaking, a form of B(I) whose omission produces an ideal having the same saturation as I; it becomes useless in any dehomogenization of I with respect to a linear form. We study the properties of B(I) linked to the presence of inessential elements and give some examples.     

Estimation of parameters in ARUMA models

Let X(n) be a time series satisfying the following ARUMA(p, d, q) models:U (B) A (B)X (n)=C (B) W (n)where U(B)=1+u(1)B+…+u(d) B~d is a polynomial with all roots on the unit circle, A(B)=1+a(1)B+…+a(p)Bp is a polynomial with all roots outside the unit circle, C(B)=1+c(1) B+…+c(q)Bq is a polynomial which is relatively prime with the polynomial U(B)A(B), B is thebackshift operator such that BX(n)=X(n-1), and (W (n), F(n), n≥1) is a sequence of martingaledifferences satisfying the following conditions:lim E (W (n)~2|F(n-1))=σ~2 a.s.n→∞sup E |W(n)|γ<∞ for some γ>2.n≥1The purpose of this paper is to provide consistent estimates of the parameters p, d, q, u(j) (j=1,2,…,d), and a(k) (k=1, 2.…, p).     

Nonexistence of Isochronous Centers in Planar Polynomial Hamiltonian Systems of Degree Four

In this paper we study centers of planar polynomial Hamiltonian systems and we are interested in the isochronous ones. We prove that every center of a polynomial Hamiltonian system of degree four (that is, with its homogeneous part of degree four not identically zero) is nonisochronous. The proof uses the geometric properties of the period annulus and it requires the study of the Hamiltonian systems associated to a Hamiltonian function of the form H(x, y)=A(x)+B(x) y+C(x) y²+D(x) y³.     

The representation type of Hecke algebras of type B

This paper determines the representation type of the Iwahori-Hecke algebras of type B when q≠±1. In particular, we show that a single parameter non-semisimple Iwahori-Hecke algebra of type B has finite representation type if and only if q is a simple root of the Poincaré polynomial, confirming a conjecture of Uno's (J. Algebra 149 (1992) 287).