Oscillation and non-oscillation for second-order linear difference equations

We prove new oscillation and non-oscillation theorems for the second-order linear difference equation Δ²x_n−1 + p_nx_n = 0, where is a real sequence with p_n 0. These results are extensions of earlier results of Zhang and Zhou [Comput. Math. Appl. 39 (2000) 1–7].     

The zero-multiplicity of third-order linear recurrences associated to the Tribonacci sequence

In this paper, we consider third-order linear recurrences _{un}n≥0${\left\{u_n\right\}}_{n\ge 0}$ satisfying the recurrence relation _un+3=_un+2+_un+1+_un$u_{n+3}=u_{n+2}+u_{n+1}+u_n$ for all n≥0$n\ge 0$ and investigate the multiplicity of its zeros. We prove that _{un}n≥0${\left\{u_n\right\}}_{n\ge 0}$ has zero-multiplicity at most 2, except for nonzero multiples of shifts of the Tribonacci sequence which has zero-multiplicity 4 when the indices are extended to all the integers.     

Common values of a class of linear recurrences

Let $\left(a_n\right),\left(b_n\right)$ be linear recursive sequences of integers with characteristic polynomials $A\left(X\right),B\left(X\right)\in \mathbb{Z}\left[X\right]$ respectively. Assume that $A\left(X\right)$ has a dominating and simple real root $\alpha$, while $B\left(X\right)$ has a pair of conjugate complex dominating and simple roots $\beta ,\stackrel{?}{\beta }$. Assume further that $\alpha ,\beta ,\alpha /\beta$ and $\stackrel{?}{\beta }/\beta$ are not roots of unity and $\delta =log\left|\beta \right|/log\left|\alpha \right|\in \mathbb{Q}$. Then there are effectively computable constants $c_0,c_1>0$ such that the inequality $|a_n?b_m|>{\left|a_n\right|}^{1?\left(c_0{log}^{2}n\right)/n}$holds for all $n,m\in {\mathbb{Z}}_{\ge 0}^2$ with $max\{n,m\}>c_1$. We present $c_0$ explicitly.     

A second-order fluid queue with subordinator input and Markov-modulated linear output

We consider an infinite capacity second-order fluid queue with subordinator input and Markovmodulated linear release rate. The fluid queue level is described by a generalized Langevin stochastic differential equation （SDE）. Applying infinitesimal generator, we obtain the stationary distribution that satisfies an integro-differential equation. We derive the solution of the SDE and study the transient level＇s convergence in distribution. When the coefficients of the SDE are constants, we deduce the system transient property.     

On global meromorphic solutions of second-order linear differential equations with meromorphic coefficients

The main purpose of this article is to study the existence theories of global meromorphic solutions for some second-order linear differential equations with meromorphic coefficients, which perfect the solution theory of such equations.     

Diophantine representations of linear recurrences. I

Direct constructions of Diophantine representations of linear recurrent sequences are discussed. These constructions generalize already known results for second-order recurrences. Some connections of this problem with the theory of units in rings of algebraic integers are shown. It is proved that the required representations exist only for second-, third-, and fourth-order sequences. In the two last-mentioned cases certain additional restrictions on their coefficients must be imposed. Bibliography:14 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 52–60.     

Prime divisors of second-order linear recurrences. I

The truth of the Riemann hypothesis for Dedekind zeta functions over certain Kummer fields is shown to imply that the density of prime divisors of an infinite set of second-order linear recurrences is greater than 0. A rough estimate, suggested by R. R. Laxton, for calculating the density is also shown to be a consequence of the Riemann hypothesis holding over certain Kummer fields.     

Remarks on oscillation of second-order linear difference equation

In this remark, we shall show that the Lemma 1 and parts of the proof of Theorem 1 in the paper “Oscillation criteria for second-order linear difference equations” (Applied Mathematics and Computation 145 (2003) 591-601) are incorrect.     

Factorizations of tropical and sign polynomials

In this text, we study factorizations of polynomials over the tropical hyperfield and the sign hyperfield, which we call tropical polynomials and sign polynomials, respectively. We classify all irreducible polynomials in either case. We show that tropical polynomials factor uniquely into irreducible factors, but that unique factorization fails for sign polynomials. We describe division algorithms for tropical and sign polynomials by linear terms that correspond to roots of the polynomials.     

Index theory, nontrivial solutions, and asymptotically linear second-order Hamiltonian systems

In this paper, we consider the existence and multiplicity of solutions of second-order Hamiltonian systems. We propose a generalized asymptotically linear condition on the gradient of Hamiltonian function, classify the linear Hamiltonian systems, prove the monotonicity of the index function, and obtain some new conditions on the existence and multiplicity for generalized asymptotically linear Hamiltonian systems by global analysis methods such as the Leray-Schauder degree theory, the Morse theory, the Ljusternik-Schnirelman theory, etc.     

On the Relation Between the Determinant and the Permanent on Symmetric Matrices

Let H n ( F ) be the space of n -square symmetric matrices over the field F . We generalize the main result of [M.H. Lim (1979). A note on the relation between the determinant and the permanent. Linear and Multilinear Algebra , 7 , 145-147], proving that the determinant is not convertible into the permanent on H n ( F ), provided that n ? 3, F has at least n elements and the characteristic of F is not 2. The case n = 2 is also studied.     

Explicit bounds for multidimensional linear recurrences with restricted coefficients

This note employs path counting techniques to extend recent results on bounds for odd order linear recurrences to higher dimensions. The results imply optimal zero-free polydisks for multivariable power series with 0, 1 coefficients. Among the applications is a result that states that the optimal zero-free polydisk has radius approximately 1/(v+1), for large dimensions v.     

Perturbations of Triangular Matrix Factorizations

We derive exact perturbation expressions for the LU , LDU , LDL T and Cholesky factorizations, which are valid for all matrix perturbations that keep nonsingularity and factorability. These results are then used to give upper bounds that improve the known bounds in most of the cases.     

On the Relation Between the Determinant and the Permanent on Symmetric Matrices

Let H n ( F ) be the space of n -square symmetric matrices over the field F . We generalize the main result of [M.H. Lim (1979). A note on the relation between the determinant and the permanent. Linear and Multilinear Algebra , 7 , 145-147], proving that the determinant is not convertible into the permanent on H n ( F ), provided that n ≥3, F has at least n elements and the characteristic of F is not 2. The case n = 2 is also studied.     

A linear second-order ODE with only polynomial solutions

A linear second-order ODE featuring an arbitrary number of free parameters is identified, all solutions of which are polynomials.     

Local second-order approximations and applications in optimization

A new general abstract scheme for local second-order approximations and second-order generalized directional derivatives is presented. Applications to optimization are provided.     

Summations of linear recurrent sequences

We give an extension of Sister Celine’s method of proving hypergeometric sum identities that allows it to handle a larger variety of input summands. In particular, we extend the summand to powers of a C-finite sequence times a hypergeometric term. We then apply this to several problems. Some of these applications give new results, and some reprove already known results in an automated way.     

Factorizations and construction of linear phase paraunitary filter banks and higher multiplicity wavelets

It is known that paraunitary matrices can be factorized into shift products of orthogonal matrices or linear factors. When the number of rows of such a matrix (i.e. the number of channels of a paraunitary filter bank) is even, the symmetry constraints corresponding to the linear phase property of the filter bank can be expressed as restrictions on factors — except the very first one, all must be centrosymmetric. For odd numbers of rows the situation is more complicated. It turns out that paraunitary matrices comprising an even number of square blocks do not exist and quadratic centrosymmetric factors have to be used in the 0-shift product factorization. The centrosymmetric linear and quadratic factors can be easily obtained from partitions of centrosymmetric orthogonal matrices. Their parameterizations are also described.The characterizations of paraunitary matrices obtained from these factorizations are complete; the question of the number of free parameters is discussed. Furthermore, the proposed factorizations also allow us to derive lattice structures for linear phase paraunitary filter banks and, since the basic regularity conditions can be incorporated as a constraint on the first factor, they can be used also for the construction of symmetric higher multiplicity wavelets.and Cooperative Research Centre for Sensor Signal and Information ProcessingThe author is an Overseas Postgraduate Research Scholar supported by the Australian Government.     

Oscillation of second-order linear delay differential equations

The aim of this paper is to derive sufficient conditions for the linear delay differential equation (r(t)y′(t))′ + p(t)y(τ(t)) = 0 to be oscillatory by using a generalization of the Lagrange mean-value theorem, the Riccati differential inequality and the Sturm comparison theorem.      

Closed-form solutions of general second order linear recurrences and applications

Two closed-form solutions of a general second order linear recurrence with variable coefficients are established. The first form is combinatorial in nature and is derived through the use of a set which counts the number of elements which are two units apart. The second form is closely related to continued fractions and is derived through the use of continued fraction-like relation. Several applications including a number of verifications of conjectures produced from the Ramanujan Machine are worked out, which illustrate its versatility in this respect.