Number of infections suffered by a focal individual in a two‐strain SIS model with partial cross‐immunity

We study a stochastic model for the spread of two pathogen strains—termed type 1 and type 2—among a homogeneously mixing community consisting of a finite number of individuals. In the model, we assume partial cross‐immunity, exogenous streams of infection, and that the degree of severity of a newly infective individual depends on who this infective individual was infected by. The aim is to characterize the joint probability distribution of the numbers M₁ and M₂ of type‐1 and type‐2 infections suffered by a focal individual during an outbreak of the disease. We present iterative procedures for computing the probability mass function of (M₁,M₂) under the assumption that the initial state of the focal individual is known, and a numerical study of the model is performed to investigate the influence of certain key parameters on the spread of resistant bacteria in hospitals.     

Convergence in measure with respect to a matrix-valued measure and some matrix completion problems

For a matrix-valued measure M we introduce a notion of convergence in measure M, which generalizes the notion of convergence in measure with respect to a scalar measure and takes into account the matrix structure of M. Let S be a subset of the set of matrices of given size. It is easy to see that the set of S-valued measurable functions is closed under convergence in measure with respect to a matrix-valued measure if and only if S is a ρ-closed set, i.e. if and only if SP is closed for any orthoprojector P. We discuss the behaviour of ρ-closed sets under operations of linear algebra and the ρ-closedness of particular classes of matrices.     

On P-coherent endomorphism rings

A ring is called right P-coherent if every principal right ideal is finitely presented. Let M _R be a right R-module. We study the P-coherence of the endomorphism ring S of M _R . It is shown that S is a right P-coherent ring if and only if every endomorphism of M _R has a pseudokernel in add M _R ; S is a left P-coherent ring if and only if every endomorphism of M _R has a pseudocokernel in add M _R . Some applications are given.     

Divisibility among power GCD matrices and among power LCM matrices on three coprime divisor chains

Let h be a positive integer and S?=?{x ₁,?…?,?x _h } be a set of h distinct positive integers. We say that the set S is a divisor chain if x _σ(1) ∣?…?∣ x _σ(h) for a permutation σ of {1,?…?,?h}. If the set S can be partitioned as S?=?S ₁?∪?S ₂?∪?S ₃, where S ₁, S ₂ and S ₃ are divisor chains and each element of S _i is coprime to each element of S _j for all 1?≤?i?<?j?≤?3, then we say that the set S consists of three coprime divisor chains. The matrix having the ath power (x _i , x _j ) ^a of the greatest common divisor of x _i and x _j as its i, j-entry is called the ath power greatest common divison (GCD) matrix on S, denoted by (S ^?a ). The ath power least common multiple (LCM) matrix [S ^?a ] can be defined similarly. In this article, let a and b be positive integers and let S consist of three coprime divisor chains with 1?∈?S. We show that if a?∣?b, then the ath power GCD matrix (S ^?a ) (resp., the ath power LCM matrix [S ^?a ]) divides the bth power GCD matrix (S ^?b ) (resp., the bth power LCM matrix [S ^?b ]) in the ring M _h (Z) of h?×?h matrices over integers. We also show that the ath power GCD matrix (S ^?a ) divides the bth power LCM matrix [S ^?b ] in the ring M _h (Z) if a?∣?b. However, if a???b, then such factorizations are not true. Our results extend Hong's and Tan's theorems and also provide further evidences to the conjectures of Hong raised in 2008.     

Minimal entropy and collapsing with curvature bounded from below

We show that if a closed manifold M admits an ℱ-structure (not necessarily polarized, possibly of rank zero) then its minimal entropy vanishes. In particular, this is the case if M admits a non-trivial S ¹-action. As a corollary we obtain that the simplicial volume of a manifold admitting an ℱ-structure is zero.?We also show that if M admits an ℱ-structure then it collapses with curvature bounded from below. This in turn implies that M collapses with bounded scalar curvature or, equivalently, its Yamabe invariant is non-negative.?We show that ℱ-structures of rank zero appear rather frequently: every compact complex elliptic surface admits one as well as any simply connected closed 5-manifold.?We use these results to study the minimal entropy problem. We show the following two theorems: suppose that M is a closed manifold obtained by taking connected sums of copies of S ⁴, ℂP ², ²,S ²×S ²and the K3 surface. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S ⁴,ℂP ²,S ²×S ²,ℂP ²#  ² or ℂP ²# ℂP ². Finally, suppose that M is a closed simply connected 5-manifold. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S ⁵,S ³×S ², then on trivial S ³-bundle over S ² or the Wu-manifold SU(3)/SO(3). Oblatum 13-III-2002 & 12-VIII-2002?Published online: 8 November 2002 G.P. Paternain was partially supported by CIMAT, Guanajuato, México.?J. Petean is supported by grant 37558-E of CONACYT.     

Geometric Convergence Rates for Time-Sampled Markov Chains

We consider time-sampled Markov chain kernels, of the form P = _n n P ⁿ . We prove bounds on the total variation distance to stationarity of such chains. We are motivated by the analysis of near-periodic MCMC algorithms.     

Factorization of Markov Chains

Existence of following factorization is proved:

Strong Stability of Queues with Multiple Vacation of the Server

Abstract

The main purpose of this article is to use the strong stability method to approximate the characteristics of the M/G/1//N queue with server vacation by those of the classical M/G/1//N queue, when the rate of the vacations is sufficiently small. This last queue is simpler and more exploitable in practice. For this, we proof the stability conditions and next obtain quantitative stability estimates with an exact computation of constants. From these theoretical results, we can develop an algorithm in order to check the conditions of approximation. These results of approximation have a great practical and economic interest in reliability systems and maintenance optimization policy, when we consider elements with constant failure rate.     

Smoothness equivalence properties of univariate subdivision schemes and their projection analogues

We study the following modification of a linear subdivision scheme S: let M be a surface embedded in Euclidean space, and P a smooth projection mapping onto M. Then the P-projection analogue of S is defined as T := P ◦ S. As it turns out, the smoothness of the scheme T is always at least as high as the smoothness of the underlying scheme S or the smoothness of P minus 1, whichever is lower. To prove this we use the method of proximity as introduced by Wallner et al. (Constr Approx 24(3):289–318, 2006; Comput Aided Geom Design 22(7):593–622, 2005). While smoothness equivalence results are already available for interpolatory schemes S, this is the first result that confirms smoothness equivalence properties of arbitrary order for general non-interpolatory schemes.     

Singular perturbed Markov chains and exact behaviors of simulated annealing processes

We study asymptotic properties of discrete and continuous time generalized simulated annealing processesX(·) by considering a class of singular perturbed Markov chains which are closely related to the large deviation of perturbed diffusion processes. Convergence ofX(t) in probability to a setS ₀ of desired states, e.g., the set of global minima, and in distribution to a probability concentrated onS ₀ are studied. The corresponding two critical constants denoted byd and withd are given explicitly. When the cooling schedule is of the formc/logt, X(t) converges weakly forc>0. Whether the weak limit depends onX(0) or concentrates onS ₀ is determined by the relation betweenc, d, and . Whenc>, the expression for the rate of convergence for each state is also derived.     

A divide and conquer approach to computing the mean first passage matrix for Markov chains via Perron complement reductions

Let M_T be the mean first passage matrix for an n‐state ergodic Markov chain with a transition matrix T. We partition T as a 2×2 block matrix and show how to reconstruct M_T efficiently by using the blocks of T and the mean first passage matrices associated with the non‐overlapping Perron complements of T. We present a schematic diagram showing how this method for computing M_T can be implemented in parallel. We analyse the asymptotic number of multiplication operations necessary to compute M_T by our method and show that, for large size problems, the number of multiplications is reduced by about 1/8, even if the algorithm is implemented in serial. We present five examples of moderate sizes (of orders 20–200) and give the reduction in the total number of flops (as opposed to multiplications) in the computation of M_T. The examples show that when the diagonal blocks in the partitioning of T are of equal size, the reduction in the number of flops can be much better than 1/8. Copyright © 2001 John Wiley & Sons, Ltd.     

Random invariant measures for Markov chains,and independent particle systems

Summary Let P be the transition operator for a discrete time Markov chain on a space S. The object of the paper is to study the class of random measures on S which have the property that MP=M in distribution. These will be called random invariant measures for P. In particular, it is shown that MP=M in distribution implies MP=M a.s. for various classes of chains, including aperiodic Harris recurrent chains and aperiodic irreducible random walks. Some of this is done by exploiting the relationship between random invariant measures and entrance laws. These results are then applied to study the invariant probability measures for particle systems in which particles move independently in discrete time according to P. Finally, it is conjectured that every Markov chain which has a random invariant measure also has a deterministic invariant measure.Research supported in part by N.S.F. Grant No. MCS 77-02121     

Additive Maps Preserving Moore–Penrose Inverses of Matrices on Symmetric Matrix Spaces

Suppose F is a field of characteristic not 2. Let M_n F and S_n F be the n × n full matrix space and symmetric matrix space over F, respectively. All additive maps from S_n F to S_n F (respectively, M_n F) preserving Moore–Penrose inverses of matrices are characterized. We first characterize all additive Moore–Penrose inverse preserving maps from S_n F to M_n F, and thereby, all additive Moore–Penrose inverse preserving maps from S_n F to itself are characterized by restricting the range of these additive maps into the symmetric matrix space.     

Sign patterns for eigenmatrices of nonnegative matrices

For a square (0,?1,??1) sign pattern matrix S, denote the qualitative class of S by Q(S). In this article, we investigate the relationship between sign patterns and matrices that diagonalize an irreducible nonnegative matrix. We explicitly describe the sign patterns S such that every matrix in Q(S) diagonalizes some irreducible nonnegative matrix. Further, we characterize the sign patterns S such that some member of Q(S) diagonalizes an irreducible nonnegative matrix. Finally, we provide necessary and sufficient conditions for a multiset of real numbers to be realized as the spectrum of an irreducible nonnegative matrix M that is diagonalized by a matrix in the qualitative class of some S ² NS sign pattern.     

Nonnegative rank-preserving operators

Analogues of characterizations of rank-preserving operators on field-valued matrices are determined for matrices witheentries in certain structures $\text{S}$ contained in the nonnegative reals. For example, if $\text{S}$ is the set of nonnegative members of a real unique factorization domain (e.g. the nonnegative reals or the nonnegative integers), M is the set of m×n matrices with entries in $\text{S}$, and min(m,n)?4, then a “linear” operator on M preserves the “rank” of each matrix in M if and only if it preserves the ranks of those matrices in M of ranks 1, 2, and 4. Notions of rank and linearity are defined analogously to the field-valued concepts. Other characterizations of rank-preserving operators for matrices over these and other structures $\text{S}$ are also given.     

On a class of semicommutative modules

Let R be a ring with identity, M a right R-module and S = End _R (M). In this note, we introduce S-semicommutative, S-Baer, S-q.-Baer and S-p.q.-Baer modules. We study the relations between these classes of modules. Also we prove if M is an S-semicommutative module, then M is an S-p.q.-Baer module if and only if M[x] is an S[x]-p.q.-Baer module, M is an S-Baer module if and only if M[x] is an S[x]-Baer module, M is an S-q.-Baer module if and only if M[x] is an S[x]-q.-Baer module.     

A simple proof for the matrix-geometric theorem

For block-partitioned matrices of the GI/M/1 type, it has been shown by M. F. Neuts that the stationary probability vector, when it exists, has a matrix-geometric form. We present here a new proof, which we believe to be the simplest available today.     

Findings about the BMMPP for modeling dependent and simultaneous data in reliability and queueing systems

The batch Markov‐modulated Poisson process (BMMPP) is a subclass of the versatile batch Markovian arrival process (BMAP), which has been widely used for the modeling of dependent and correlated simultaneous events (as arrivals, failures, or risk events). Both theoretical and applied aspects are examined in this paper. On one hand, the identifiability of the stationary BMMPP_m(K ) is proven, where K is the maximum batch size and m is the number of states of the underlying Markov chain. This is a powerful result for inferential issues. On the other hand, some novelties related to the correlation and autocorrelation structures are provided.