The Ramsey numbers r(Pm, Kn)

This note evaluates the Ramsey numbers r(P_m,K_n), and discusses developments in 0 generalized Ramsey theory for graphs.     

On properties of discrete (r, q) and (s, T) inventory systems

We consider single-item (r, q) and (s, T) inventory systems with integer-valued demand processes. While most of the inventory literature studies continuous approximations of these models and establishes joint convexity properties of the policy parameters in the continuous space, we show that these properties no longer hold in the discrete space, in the sense of linear interpolation extension and L^?-convexity. This nonconvexity can lead to failure of optimization techniques based on local optimality to obtain the optimal inventory policies. It can also make certain comparative properties established previously using continuous variables invalid. We revise these properties in the discrete space.     

On the damped nonlinear evolution equation wtt = σ(w)xx − ywt

Initial boundary value problems for the damped nonlinear wave equation w_tt = σ(w)_xx ? yw_t arise in several areas of applied mathematics and, in particular, in studies of shearing flow in a nonlinear viscoelastic fluid; the problems of global existence and nonexistence of smooth solutions have been extensively studied in the strictly hyperbolic case σ′(δ) ? ε > 0, ?δ?R¹ as well as in the case where σ′(0) > 0 and the initial data are chosen so small that σ′(w) > 0 for as long as a smooth solution w(x, t) exists. In this paper the global nonexistence problem is studied for the cases σ′(0) = 0 and σ′(0) > 0 but σ′(δ) < 0 for ¦δ¦ sufficiently large and growth estimates which are valid on the maximal interval of existence of a sufficiently smooth solution are derived.     

Reliability of a consecutive (r, s)-out-of-(m, n):F lattice system with conditions on the number of failed components in the system

A consecutive(r, s)-out-of-(m, n):F lattice system which is defined as a two-dimensional version of a consecutive k-out-of-n:F system is used as a reliability evaluation model for a sensor system, an X-ray diagnostic system, a pattern search system, etc. This system consists of m × n components arranged like an (m, n) matrix and fails iff the system has an (r, s) submatrix that contains all failed components. In this paper we deal a combined model of a k-out-of-mn:F and a consecutive (r, s)-out-of-(m, n):F lattice system. Namely, the system has one more condition of system down, that is the total number of failed components, in addition to that of a consecutive (r, s)-out-of-(m, n):F lattice system. We present a method to obtain reliability of the system. The proposed method obtains the reliability by using a combinatorial equation that does not depend on the system size. Some numerical examples are presented to show the relationship between component reliability and system reliability.     

Polynomial algorithms for m × (m + 1) integer programs and m × (m + k) diophantine systems

Recent results of Kannan and Bachem (on computing the Smith Normal Form of a matrix) and Lenstra (on solving integer inequality systems) are used with classical results by Smith to obtain polynomial-time algorithms for solving m × (m + 1) equality constrained integer programs and m × (m + k) systems of diophantine equations for fixed k.     

Nonnegative (s, t)-potent matrices

This paper contains a study of matrices satisfying A^s = A^t for different positive integers s and t. Representations, similar to Flor's well-known characterization of a nonnegative idempotent matrix, are obtained for nonnegative matrices of this type.     

Group gradings on the superalgebras M(m,n), A(m,n) and P(n)

We classify gradings by arbitrary abelian groups on the classical simple Lie superalgebras $P(n)$, $n\ge 2$, and on the simple associative superalgebras $M(m,n)$, $m,n\ge 1$, over an algebraically closed field: fine gradings up to equivalence and G-gradings, for a fixed group G, up to isomorphism. As a corollary, we also classify up to isomorphism the G-gradings on the classical Lie superalgebra $A(m,n)$ that are induced from G-gradings on $M(m+1,n+1)$. In the case of Lie superalgebras, the characteristic is assumed to be 0.     

The Cauchy problem for ut=Δu+|∇u|

With q a positive real number, the nonlinear partial differential equation in the title of the paper arises in the study of the growth of surfaces. In that context it is known as the generalized deterministic KPZ equation. The paper is concerned with the initial-value problem for the equation under the assumption that the initial-data function is bounded and continuous. Results on the existence, uniqueness, and regularity of solutions are obtained.     

Regularizing effects for ut + Aϑ(u) = 0 in L1

Various initial-boundary value problems and Cauchy problems can be written in the form $\text{du}\text{dt}\text{+ A?(u) = 0}$, where ?:R→$\text{R}$ is nondecreasing and A is the linear generator of strongly continuous nonexpansive semigroup e^?tA in an L¹ space. For example, if A = ?Δ (subject, perhaps, to suitable boundary conditions) we obtain equations arising in flow in a porous medium or plasma physics (depending on the choice of ?) while if $\text{A =}\text{?}\text{?x}$ acting in L¹($\text{R}$) we have a scalar conservation law. In this paper we show that if M, m > 0 and m?′² ? ν??′' ? M?′², where ν ? {1,?1}, then (roughly speaking), the norm of $\text{t}\text{du}\text{dt}$ may be estimated in terms of the initial data u₀ in L¹. Such estimates give information about the regularity of solutions, asymptotic behaviour, etc., in applications. Side issues, such as the introduction of sufficiently regular approximate problems on which estimates can be made and the assignment of a precise meaning to the operator A?, are also dealt with. These considerations are of independent interest.     

The M/Ek/r machine interference model steady state equations and numerical solutions

This paper studies the machine interference problem in which the running times follow the negative exponential distribution, the repair times the Erlang distribution and the number of operatives is more than one. The steady state equations are derived and it is shown that unlike the case of the M/E_k/r ordinary queueing model, the solution cannot be taken in closed form. An efficient numerical procedure is developed instead, based on a decomposition principle. Tabulated results of the average number of machines running and the operative utilization for a range of the problem parameters are given, for the cases M/E₃/2 and M/E₃/3. A tentative conclusion for a closeness in performance between the models M/M/r and M/E_k/r is drawn.     

Surface Measures and Tightness of (r,p)-Capacities on Poisson Space

We prove tightness of (r,p)-Sobolev capacities on configuration spaces equipped with Poisson measure. By using this result we construct surface measures on configuration spaces in the spirit of the Malliavin calculus. A related Gauss-Ostrogradskii formula is obtained.     

On polynomial functions (mod m)

In this paper, we obtain a canonical representation for those polynomials (with integer coefficients) which vanish (mod m), a canonical representation for each polynomial function (mod m) and an expression for the number of polynomial functions (mod m). This number turns out to be (weakly) multiplicative in m.     

Weak solution of (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0))u′(0) = ku(0), r(L, u(L))u′ (L) = hu(L) and k, h are suitable elements of [0, ∞]

We consider weak solutions to the nonlinear boundary value problem (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0)) u′(0) = ku(0), r(L, u(L)) u′(L) = hu(L) and k, h are suitable elements of [0, ∞]. In addition to studying some new boundary conditions, we also relax the constraints on r(x, u) and (Fu)(x). r(x, u) > 0 may have a countable set of jump discontinuities in u and r(x, u)^?1?L_q((0, L) × (0, p)). F is an operator from a suitable set of functions to a subset of L_p(0, L) which have nonnegative values. F includes, among others, examples of the form (Fu)(x) = (1 ? H(x ? x₀)) u(x₀), (Fu)(x) = ∫_x^Lf(y, u(y)) dy where f(y, u) may have a countable set of jump discontinuities in u or F may be chosen so that (Fu)′(x) = ? g(x, u(x)) u′(x) ? q(x) u(x) ? f(x, u(x)) where q is a distributional derivative of an L₂(0, L) function.