Many algorithms for computing the reliability of linear or circular consecutive-k-out-of-n:F systems appeared in this Transactions. The best complexity estimate obtained for solving this problem is O(k log(n/k)) operations in the case of i.i.d. components. Using fast algorithms for computing a selected term of a linear recurrence with constant coefficients, we provide an algorithm having arithm...

Oscillation properties of solutions of the forced second order linear difference equation ∆(rk∆xk) + ckxk+1 = hk are investigated. The authors show that if the forcing term h does not oscillate, in some sense, too rapidly, then the oscillation of the unforced equation implies oscillation of the forced equation. Some results illustrating this statement and extensions to the more general half-lin...

We solve a 1985 challenge problem posed by Lagarias [5] by determining, under GRH, the density of the set of prime numbers that occur as divisor of some term of the sequence {xn}∞ n=1 defined by the linear recurrence xn+1 = xn + xn−1 and the initial values x0 = 3 and x1 = 1. This is the first example of a ‘non-torsion’ second order recurrent sequence with irreducible recurrence relation for whi...

An introduction into the algebraic theory of several types of linear systems is given. In particular, linear ordinary and partial differential and difference equations are covered. Special emphasis is given to the formulation of formally wellposed initial value problem for treating solvability questions for general, i. e. also underand overdetermined, systems. A general framework for analysing ...

Parameters of linear systems are subject to time changes. That is why in order to construct such systems it is desirable to know if they are not only stable but also able to estimate the distance of the system from the boundary of the stability region in the parameter space. Therefore, it makes sense to investigate the geometry of the subset of stable polynomials in the space of characteristic ...

This paper investigates the number of spanning subgraphs of the product of an arbitrary graph G with the path graphs Pn on n vertices that meet certain properties: connectivity, acyclicity, Hamiltonicity, and restrictions on degree. A general method is presented for constructing a recurrence equation R(n) for the graphs G Pn, giving the number of spanning subgraphs that satisfy a given combinat...

We summarize some recent results on partial linear functional systems. By associating a finite-dimensional linear functional system to a Laurent-Ore module, Picard-Vessiot extensions are generalized from linear ordinary differential (difference) equations to finite-dimensional linear functional systems. A generalized Beke’s method is also presented for factoring Laurent-Ore modules and it will ...

Background • Legendre Sequence For a prime p > 2 let (s n) be the Legendre sequence defined as s n = 1, n p = −1, 0, otherwise, n ≥ 0, where. p denotes the Legendre symbol. • Sidelnikov Sequence Let q be an odd prime power, g a primitive element of F q , and let η denote the quadratic character of F Background • Legendre Sequence For a prime p > 2 let (s n) be the Legendre sequence defined as s...

We consider two algorithms which can be used for proving positivity of sequences that are defined by a linear recurrence equation with polynomial coefficients (P-finite sequences). Both algorithms have in common that while they do succeed on a great many examples, there is no guarantee for them to terminate, and they do in fact not terminate for every input. For some restricted classes of P-fin...

We derive sharp upper bounds for the size of the intersection of certain linear recurrence sequences. As a consequence of these, we partially resolve a conjecture of Yuan on simultaneous Pellian equations, under the condition that one of the parameters involved is suitably large.