Spatial No-Waiting Stations with Moving Customers

We analyze spatial MAP/G/∞-, spatial MAP/G/c/0¹ and spatial Cox/G/∞-stations with group arrivals over some Polish space X (with Borel σ-algebra X), including the aspect of customer motion in space. For models with MAP-input, characteristic differential equations are set up that describe the dynamics of phase dependent random functions Q _r;ij (u,t;S′), where Q _r;ij (u,t;S′) is the probability to observe, at time u≤t, the number r of those customers in some source set S′∈X, who will be in a destination set S∈X at time t. For Cox/G/∞-stations, i.e., infinite server stations with doubly stochastic input, the arrival intensities as well as service times may depend on some general stochastic process (J′ _t ) _t≥0 with countable state space. For that case we obtain explicit expressions for space–time distributions as well as stationary and non-stationary characteristics.     

Enumeration of sequences of given specification according to rises,falls and maxima

Letn = (a₁.a₂ … a_N) denote a sequence of integers a_i={1.2.…n}. A rise is a part a_i.a_i+1 with a_i <a_i+1: a fall is a pair with a_ia_i+1: a level is a pair with a_i = a_i+1. A maximum is a triple a_i-1.a_ia_i+1 with a_i-1?a_i.a_i?a_i+1. If e_i is the number of a_j?n witha_j = i, then [e₁…e_n] is called the specification of n. In addition, a conventional rise is counted to the left of a₁ and a conventional fall to the right of a_N: ifa₁?a₂, then a₁ is counted as a conventional maximum, similarly if a_N-1 ? a_N thena_N is a conventional maximum. Simon Newcomb's problem is to find the number of sequences n with given specification and r rises; the refined problem of determining the number of sequences of given specification with r rises and s falls has also been solved recently. The present paper is concerned with the problem of finding the number of sequences of given specification with r rises, s falls. λ levels and λ maxima. A generating function for this enumerant is obtained as the quotient of two continuants. In certain special cases this result simplifies considerably.     

Enumeration of rectangular arrays of zeros and ones

For r=1,2 the rectangular arrays of zeros and ones with r rows and n columns, with m_i zeros and r_i changes in the ith row, and with s_i vertical changes between the ith and (i+1)st rows, i=1,…,r-1, are enumerated. The number of arrays of zeros and ones with 3 rows and n columns, with r_i changes in the ith row, i=1,2,3, and with s_i vertical changes between the ith and (i+1)st rows, i=1,2, is also determined.     

Graphs determined by their generalized characteristic polynomials

For a given graph G with (0, 1)-adjacency matrix A_G, the generalized characteristic polynomial of G is defined to be ?_G=?_G(λ,t)=det(λI-(A_G-tD_G)), where I is the identity matrix and D_G is the diagonal degree matrix of G. In this paper, we are mainly concerned with the problem of characterizing a given graph G by its generalized characteristic polynomial ?_G. We show that graphs with the same generalized characteristic polynomials have the same degree sequence, based on which, a unified approach is proposed to show that some families of graphs are characterized by ?_G. We also provide a method for constructing graphs with the same generalized characteristic polynomial, by using GM-switching.     

On (0, 1)-matrices with prescribed row and column sum vectors

Given partitions R and S with the same weight, the Robinson-Schensted-Knuth correspondence establishes a bijection between the class A(R,S) of (0, 1)-matrices with row sum R and column sum S and pairs of Young tableaux of conjugate shapes λ and λ^∗, with S?λ?R^∗. An algorithm for constructing a matrix in A(R,S) whose insertion tableau has a prescribed shape λ, with S?λ?R^∗, is provided. We generalize some recent constructions due to R. Brualdi for the extremal cases λ=S and λ=R^∗.     

Characterization of the convergence of weighted averages of sequences and functions

The weighted averages of a sequence (c _k ), c _k ?? ?, with respect to the weights (p _k ), p _k ?? 0, with {fx135-1} are defined by {fx135-2} while the weighted average of a measurable function f: ?₊ ?? ? with respect to the weight function p(t) ?? 0 with {fx135-3}. Under mild assumptions on the weights, we give necessary and sufficient conditions under which the finite limit ?? _n ?? L as n ?? ?? or ??(t) ?? L as t ?? ?? exists, respectively. These characterizations may find applications in probability theory.     

On averaging principle for diffusion processes with null-recurrent fast component

An averaging principle is proved for diffusion processes of type (X_ε(t),Y_ε(t)) with null-recurrent fast component X_ε(t). In contrast with positive recurrent setting, the slow component Y_ε(t) alone cannot be approximated by diffusion processes. However, one can approximate the pair (X_ε(t),Y_ε(t)) by a Markov diffusion with coefficients averaged in some sense.     

On the optimal allocation of two active redundancies in a two-component series system

Components C₁ and C₂ form a series system. Suppose we can allocate the spare R₁ in parallel with C₁ and the spare R₂ in parallel with C₂, or otherwise, allocate R₁ with C₂ and R₂ with C₁. In this paper, we compare these two options using hazard rate ordering.     

n-almost prime submodules

Let R be a commutative ring with identity. A proper submodule N of an R-module M will be called prime [resp. n-almost prime], if for r ∈ R and a ∈ M with ra ∈ N [resp. ra ∈ N \ (N: M) ^n?1 N], either a ∈ N or r ∈ (N: M). In this note we will study the relations between prime, primary and n-almost prime submodules. Among other results it is proved that:

1. If N is an n-almost prime submodule of an R-module M, then N is prime or N = (N: M)N, in case M is finitely generated semisimple, or M is torsion-free with dim R = 1.
2. Every n-almost prime submodule of a torsion-free Noetherian module is primary.
3. Every n-almost prime submodule of a finitely generated torsion-free module over a Dedekind domain is prime.
4. There exists a finitely generated faithful R-module M such that every proper submodule of M is n-almost prime, if and only if R is Von Neumann regular or R is a local ring with the maximal ideal m such that m ² = 0.
5. If I is an n-almost prime ideal of R and F is a flat R-module with IF ≠ F, then IF is an n-almost prime submodule of F.

     

A note on random walk in random scenery

We consider a random walk in random scenery {Xn=η(S₀)+?+η(Sn),n∈N}, where a centered walk {Sn,n∈N} is independent of the scenery {η(x),x∈Zd}, consisting of symmetric i.i.d. with tail distribution P(η(x)>t)∼exp(−cαtα), with 1?α<d/2. We study the probability, when averaged over both randomness, that {Xn>ny} for y>0, and n large. In this note, we show that the large deviation estimate is of order exp(−ca(ny)), with a=α/(α+1).     

Mixed problems for the time-dependent telegraph equation: Continuous numerical solutions with a priori error bounds

This paper deals with the construction of continuous numerical solutions of mixed problems described by the time-dependent telegraph equation u_tt + c(t)u_t + b(t)u = a(t)u_xx, 0 < x < d, t > 0. Here a(t), b(t), and c(t) are positive functions with appropiate additional alternative hypotheses. First, using the separation of variables technique a theoretical series solution is obtained. Then, after truncation using one-step matrix methods and interpolating functions, a continuous numerical solution with a prefixed accuracy in a bounded subdomain is constructed.     

Well-posedness of a class of perturbed optimization problems in Banach spaces

Let X be a Banach space and Z a nonempty subset of X. Let J:Z→R be a lower semicontinuous function bounded from below and p?1. This paper is concerned with the perturbed optimization problem of finding z₀∈Z such that ‖x−z₀p^‖+J(z₀)=infz∈Z{‖x−zp^‖+J(z)}, which is denoted by minJ(x,Z). The notions of the J-strictly convex with respect to Z and of the Kadec with respect to Z are introduced and used in the present paper. It is proved that if X is a Kadec Banach space with respect to Z and Z is a closed relatively boundedly weakly compact subset, then the set of all x∈X for which every minimizing sequence of the problem minJ(x,Z) has a converging subsequence is a dense Gδ-subset of X?Z₀, where Z₀ is the set of all points z∈Z such that z is a solution of the problem minJ(z,Z). If additionally p>1 and X is J-strictly convex with respect to Z, then the set of all x∈X for which the problem minJ(x,Z) is well-posed is a dense Gδ-subset of X?Z₀.     

Some degenerate and quasilinear parabolic systems not in divergence form

This paper deals with positive solutions of degenerate and quasilinear parabolic systems not in divergence form: u_t=u^p(Δu+av), v_t=v^q(Δv+bu), with null Dirichlet boundary conditions and positive initial conditions, where p, q, a and b are all positive constants. The local existence and uniqueness of classical solution are proved. Moreover, it will be proved that all solutions exist globally if and only if ab?λ₁², where λ₁ is the first eigenvalue of −Δ in Ω with homogeneous Dirichlet boundary condition.     

Robustness of optimal designs for correlated random variables

Suppose that Y = (Y_i) is a normal random vector with mean Xb and covariance σ²I_n, where b is a p-dimensional vector (b_j), X = (X_ij) is an n × p matrix with X_ij ∈ {−1, 1}; this corresponds to a factorial design with −1, 1 representing low or high level respectively, or corresponds to a weighing design with −1, 1 representing an object j with weight b_j placed on the left and right of a chemical balance respectively. E-optimal designs Z are chosen that are robust in the sense that they remain E-optimal when the covariance of Y_i, Y_i′ is ρ > 0 for i ≠ i′. Within a smaller class of designs similar results are obtained with respect to a general class of optimality criteria which include the A- and D-criteria.     

The graphs G(n, k) of the Johnson Schemes are unique for n ≥ 20

Let G(n, k) denote the graph of the Johnson Scheme J(n, k), i.e., the graph whose vertices are all k-subsets of a fixed n-set, with two vertices adjacent if and only if their intersection is of size k ? 1. It is known that G(n, k) is a distance regular graph with diameter k. Much work has been devoted to the question of whether a distance regular graph with the parameters of G(n, k) must isomorphic to G(n, k). In this paper, this question is settled affirmatively for n ≥ 20. In fact the result is proved with weaker conditions.     

On Turan hypergraphs

Let α(H) be the stability number of a hypergraph H = (X, $\text{E}$). T(n, k, α) is the smallest q such that there exists a k-uniform hypergraph H with n vertices, q edges and with α(H) ? α. A k-uniform hypergraph H, with n vertices, T(n, k, α) edges and α(H) ?α is a Turan hypergraph. The value of T(n, 2, α) is given by a theorem of Turan. In this paper new lower bounds to T(n, k, α) are obtained and it is proved that an infinity of affine spaces are Turan hypergraphs.     

Time-varying minimum cost flow problems

In this paper, we study a minimum cost flow problem on a time-varying network. Let N(V,A,l,b,c_r,c_w) be a network with an arc set A and a vertex set V. Each a∈A is associated with three integer parameters: a positive transit time b(a,t), an arbitrary transit cost c_r(a,t), and a positive capacity limit l(a,t). Each x∈V is associated with two integer parameters: a waiting cost c_w(x,t) and a vertex capacity l(x,t). All these parameters are functions of the discrete time t=0,1,2,… The objective is to find an optimal schedule to send a flow from the origin (the source vertex) to its destination (the sink vertex) with the minimum cost, subject to the constraint that the flow must arrive at the destination before a deadline T. Three versions of the problem are examined, which are classified depending on whether waiting at the intermediate vertices of the network is strictly prohibited, arbitrarily allowed, or bounded. Three algorithms with pseudopolynomial time complexity are proposed, which can find optimal solutions to the three versions of the problem, respectively.     

The n-lie property of the Jacobian as a condition for complete integrability

We prove that an associative commutative algebra U with derivations D ₁, ..., D _n ε DerU is an n-Lie algebra with respect to the n-multiplication D ₁ ^ ? ^ D _n if the system {D ₁, ..., D _n } is in involution. In the case of pairwise commuting derivations this fact was established by V. T. Filippov. One more formulation of the Frobenius condition for complete integrability is obtained in terms of n-Lie multiplications. A differential system {D ₁, ..., D _n } of rank n on a manifold M ^m is in involution if and only if the space of smooth functions on M is an n-Lie algebra with respect to the Jacobian Det(D _i u _j ).     

Lipschitz lower sigma-exponents of linear differential systems

For the lower sigma-exponent of the linear differential system ? = A(t)x, x ∈ R ⁿ , t ≥ 0, defined by the formula Δ_σ(A) ≡ inf_λ[Q]≤-σ λ ₁(A + Q), σ > 0, on the basis of the lower characteristic exponents λ ₁(A+Q) of perturbed linear systems with Lyapunov exponents λ[Q] ≤ ?σ < 0 of perturbations Q, we prove the following general form as a function of the parameter σ > 0. For any nondecreasing bounded function f(σ) of the parameter σ ∈ (0,+∞) that coincides with a constant on some infinite interval (σ ₀,+∞), σ ₀ ≥ 0, and satisfies the Lipschitz condition on the complementary interval (0, σ ₀], we prove the existence of a linear system with coefficient matrix A _f (t) bounded on the half-line [0,+∞) whose lower sigma-exponent Δ_σ(A _f ) coincides with the function f(σ) on the entire interval (0,+∞).     

On the maximal eccentric connectivity indices of graphs

For a connected simple graph G, the eccentricity ec(v) of a vertex v in G is the distance from v to a vertex farthest from v, and d(v) denotes the degree of a vertex v. The eccentric connectivity index of G, denoted by ξc(G), is defined as v∈V(G)d(v)ec(v). In this paper, we will determine the graphs with maximal eccentric connectivity index among the connected graphs with n vertices and m edges(n ≤ m ≤ n + 4), and propose a conjecture on the graphs with maximal eccentric connectivity index among the connected graphs with n vertices and m edges(m ≥ n + 5).