Interaction information and Markov chain

The problem of multivariate information analysis is considered. First, the interaction information in each dimension is defined analogously according to McGill [4] and then applied to Markov chains. The property of interaction information zero deeply relates to a certain class of weakly dependent random variables. For homogeneous, recurrent Markov chains with m states, m ≥ n ≥3, the zero criterion of n-dimensional interaction information is achieved only by (n ? 2)-dependent Markov chains, which are generated by some nilpotent matrices. Further for Gaussian Markov chains, it gives the decomposition rule of the variables into mutually correlated subchains.     

Extremal double hexagonal chains with respect to k-matchings and k-independent sets

“Double hexagonal chains” can be considered as benzenoids constructed by successive fusions of successive naphthalenes along a zig-zag sequence of triples of edges as appear on opposite sides of each naphthalene unit. In this paper, we discuss the numbers of k-matchings and k-independent sets of double hexagonal chains, as well as Hosoya indices and Merrifield-Simmons indices, and obtain some extremal results: among all the double hexagonal chains with the same number of naphthalene units, (a) the double linear hexagonal chain has minimal k-matching number and maximal k-independent set number and (b) the double zig-zag hexagonal chain has maximal k-matching number and minimal k-independent set number, which are extensions to hexagonal chains [L. Zhang and F. Zhang, Extremal hexagonal chains concerning k-matchings and k-independent sets, J. Math. Chem. 27 (2000) 319-329].     

Controlling multiparticle system on the line. II. Periodic case

As in [A. Sarychev, Controlling multiparticle system on the line. I, J. Differential Equations 246 (12) (2009) 4772-4790] we consider classical system of interacting particles P₁,…,Pn on the line with only neighboring particles involved in interaction. On the contrast to [A. Sarychev, Controlling multiparticle system on the line. I, J. Differential Equations 246 (12) (2009) 4772-4790] now periodic boundary conditions are imposed onto the system, i.e. P₁ and Pn are considered neighboring. Periodic Toda lattice would be a typical example. We study possibility to control periodic multiparticle systems by means of forces applied to just few of its particles; mainly we study system controlled by single force. The free dynamics of multiparticle systems in periodic and nonperiodic case differ substantially. We see that also the controlled periodic multiparticle system does not mimic its nonperiodic counterpart.Main result established is global controllability by means of single controlling force of the multiparticle system with a generic potential of interaction. We study the nongeneric potentials for which controllability and accessibility properties may lack. Results are formulated and proven in Sections 2, 3.     

Dynamical analysis of a delay model of phytoplankton-zooplankton interaction

The interaction of toxic-phytoplankton-zooplankton systems and their dynamical behavior will be considered in this paper based upon nonlinear ordinary differential equation model system. We induced a discrete time delay to the both of the consume response function and distribution of toxic substance term to describe the delay in the conversion of nutrient consumed to species and the fact that the time required for the phytoplankton species to mature before they can produce toxic substances. We generalized the model in [1] and explicit results are derived for globally asymptotically stability of the boundary equilibrium. Using numerical simulation method, we determine there is a parameter range for the delay parameter τ where more complicated dynamics occurs, and this appears to be a new result. Significant outcomes of our numerical findings and their interpretations from ecological point of view are provided in this paper.     

The relation between invariant integrals of the linear isotropic theory of elasticity and integrals defined by the duality principle

Invariant integrals of the linear isotropic theory of elasticity, determined by a certain specified elastic field, are considered, and also invariant integrals generated by the interaction of the specified field with an arbitrary secondary field. For all types of invariant integral, generated by the interaction of the specified elastic field and an arbitrary secondary elastic field, transformations of the secondary fields are found for which the invariant integrals considered turn out to be equal to the RG-integrals, determined by the duality principle, of the specified elastic field and the transformed secondary elastic field. The invariant J-, L- and M-integrals themselves are also expressed in terms of the RG-integrals of the specified elastic field and its corresponding transformation.     

A new analytical method for the linearization of dynamic equation on measure chains

In this paper, by introducing the concept of topological equivalence on measure chain, we investigate the relationship between the linear system xΔ=A(t)x and the nonlinear system xΔ=A(t)x+f(t,x). Some sufficient conditions are obtained to guarantee the existence of a equivalent function H(t,x) sending the (c,d)-quasibounded solutions of nonlinear system xΔ=A(t)x+f(t,x) onto those of linear system xΔ=A(t)x. Our results generalize the Palmer's linearization theorem in [K.J. Palmer, A generalization of Hartman's linearization theorem, J. Math. Anal. Appl. 41 (1973) 753-758] to dynamic equation measure chains. In the present paper, we give a new analytical method to study the topological equivalence problem on measure chains. As we will see, due to the completely different method to investigate the topological equivalence problem, we have a considerably different result from that in the pioneering work of Hilger [S. Hilger, Generalized theorem of Hartman-Grobman on measure chains, J. Aust. Math. Soc. Ser. A 60 (2) (1996) 157-191]. Moreover, we prove that equivalent function H(t,x) is also ω-periodic when the systems are ω-periodic. Hilger [S. Hilger, Generalized theorem of Hartman-Grobman on measure chains, J. Aust. Math. Soc. Ser. A 60 (2) (1996) 157-191] never considered this important property of the equivalent function H(t,x).     

Algorithm for a general discrete k-out-of-n: G system subject to several types of failure with an indefinite number of repairpersons

A discrete k-out-of-n: G system with multi-state components is modelled by means of block-structured Markov chains. An indefinite number of repairpersons are assumed and PH distributions for the lifetime of the units and for the repair time are considered. The units can undergo two types of failures, repairable or non-repairable. The repairability of the failure can depend on the time elapsed up to failure. The system is modelled and the stationary distribution is built by using matrix analytic methods. Several performance measures of interest, such as the conditional probability of failure for the units and for the system, are built into the transient and stationary regimes. Rewards are included in the model. All results are shown in a matrix algorithmic form and are implemented computationally with Matlab. A numerical example of an optimization problem shows the versatility of the model.     

Some stochastic properties of “semi-magic” and “magic” Markov chains

Gustafson and Styan (Gustafson and Styan, Superstochastic matrices and Magic Markov chains, Linear Algebra Appl. 430 (2009) 2705-2715) examined the mathematical properties of superstochastic matrices, the transition matrices of “magic” Markov chains formed from scaled “magic squares”. This paper explores the main stochastic properties of such chains as well as “semi-magic” chains (with doubly-stochastic transition matrices). Stationary distribution, generalized inverses of Markovian kernels, mean first passage times, variances of the first passage times and expected times to mixing are considered. Some general results are developed, some observations from the chains generated by MATLAB are discussed, some conjectures are presented and some special cases, involving three and four states, are explored in detail.     

Backward uniqueness of the s.c. semigroup arising in parabolic-hyperbolic fluid-structure interaction

A 2-d or 3-d fluid-structure interaction model in its linear form is considered, for which semigroup well-posedness (with explicit generator) was recently established in [G. Avalos, R. Triggiani, The coupled PDE-system arising in fluid-structure interaction. Part I: Explicit semigroup generator and its spectral properties, in: Fluids and Waves, in: Contemp. Math., vol. 440, Amer. Math. Soc., 2007, pp. 15-55; G. Avalos, R. Triggiani, The coupled PDE-system arising in fluid-structure interaction. Part II: Uniform stabilization with boundary dissipation at the interface, Discrete Contin. Dyn. Syst., in press]. This is a system which couples at the interface the linear version of the Navier-Stokes equations with the equations of linear elasticity (wave-like). In this paper, we establish a backward uniqueness theorem for such a parabolic-hyperbolic coupled PDE system. If {eAt}_t?0 is the (contraction) s.c. semigroup describing its evolution on the finite energy space H, then eATy₀=0 for some T>0 and y₀∈H, implies y₀=0. This property has implications in establishing unique continuation and controllability properties, as in the case of thermoelastic equations [M. Eller, I. Lasiecka, R. Triggiani, Simultaneous exact/approximate boundary controllability of thermoelastic plates with variable coefficient, in: Marcel Dekker Lect. Notes Pure Appl. Math., vol. 216, February 2001, pp. 109-230, invited paper for the special volume entitled Shape Optimization and Optimal Designs, J. Cagnol, J.P. Zolesio (Eds). (Preliminary version is in invited paper in: A.V. Balakrishnan (Ed.), Semigroup of Operators and Applications, Birkhäuser, 2000, pp. 335-351.); M. Eller, I. Lasiecka, R. Triggiani, Simultaneous exact/approximate boundary controllability of thermoelastic plates with variable thermal coefficient and moment control, J. Math. Anal. Appl. 251 (2000) 452-478; M. Eller, I. Lasiecka, R. Triggiani, Simultaneous exact/approximate boundary controllability of thermoelastic plates with variable thermal coefficient and clamped controls, Discrete Contin. Dyn. Syst. 7 (2) (2001) 283-301].     

Synchronization phenomena in the system consisting of m coupled Berger plates

The dynamical system arising in the study of nonlinear oscillations of a number of coupled Berger plates is considered. The dependence of the long-time behavior of the trajectories of the system on the properties of the coupling operator is studied. It is shown that the global attractor of the dynamical system is continuous with respect to the coupling parameter γ expressing the intensity of plate interaction. When γ→∞ it converges upper semicontinuously to the attractor of the system generated by the projection of the vector field of the coupled system on the kernel of the coupling operator. For the particular case of 3-diagonal coupling operator the synchronization phenomenon at the level of attractors is stated for large values of γ as well as the absence of synchronization for γ small. The case of cluster synchronization is also considered.     

Strongly ergodic Markov chains and rates of convergence using spectral conditions

For finite Markov chains the eigenvalues of P can be used to characterize the chain and also determine the geometric rate at which Pⁿ converges to Q in case P is ergodic. For infinite Markov chains the spectrum of P plays the analogous role. It follows from Theorem 3.1 that 6Pⁿ?Q6?Cβⁿ if and only if P is strongly ergodic. The best possible rate for β is the spectral radius of P?Q which in this case is the same as sup{|λ|: λ ? σ (P), λ ≠;1}. The question of when this best rate equals δ(P) is considered for both discrete and continous time chains. Two characterizations of strong ergodicity are given using spectral properties of P? Q (Theorem 3.5) and spectral properties of a submatrix of P (Theorem 3.16).     

Convolution products of probability measures on completely simple semigroups

Convolution products of probability measures are considered in the context of completely simple semigroups. Given a sequence of measures (μn)⊂Prob(S), where S is a finite completely simple semigroup, results are proven which (1) relate the supports of the measures in the sequence to the supports of the tail limit measures, and (2) determine necessary and sufficient conditions for convergence of the convolution products in the case of rectangular groups. An example showing how the theory can be used to analyze the convergence behavior of non-homogeneous Markov chains is included.     

Algorithms for finding steady state probabilities for some special classes of finite state Markov chains

Efficient algorithms for finding steady state probabilities are presented and compared with the Gaussian elimination method for two special classes of finite state Markov chains. One class has block matrix steps and a possible jump of up to k block steps, and the other is a generalization of the class considered by Shanthikumar and Sargent where each element is a matrix.     

An algorithmic approach for a special class of Markov chains

In this paper, we develop an algorithmic method for the evaluation of the steady state probability vector of a special class of finite state Markov chains. For the class of Markov chains considered here, it is assumed that the matrix associated with the set of linear equations for the steady state probabilities possess a special structure, such that it can be rearranged and decomposed as a sum of two matrices, one lower triangular with nonzero diagonal elements, and the other an upper triangular matrix with only very few nonzero columns. Almost all Markov chain models of queueing systems with finite source and/or finite capacity and first-come-first-served or head of the line nonpreemptive priority service discipline belongs to this special class.     

Oscillations of one-dimensional linear systems with mobile weights under the action of longitudinal impacts : PMM vol. 42, no. 6, 1978, pp. 1141–1145

Longitudinal osciallations of a one-dimensional system which can be represented by a rod interacting with various kinds of inertial mobile media, are considered. It is assumed that the media do not react with each other, can only move along the rod and, that there is no internal interaction between the elements of the media. The model can be used to study the oscillations of sufficiently long chains of rigid bodies to which other mobile bodies are attached by means of deformable elements, oscillations of one-dimensional systems of rigid bodies with cavities partially filled with fluid, etc. A transitional mode of motion in similar systems was studied in [1].     

An algorithm for computing Jordan chains and inverting analytic matrix functions

We present an efficient algorithm for obtaining a canonical system of Jordan chains for an n × n regular analytic matrix function A(λ) that is singular at the origin. For any analytic vector function b(λ), we show that each term in the Laurent expansion of A(λ)⁻¹b(λ) may be obtained from the previous terms by solving an (n + d) × (n+d) linear system, where d is the order of the zero of det A(λ) at λ = 0. The matrix representing this linear system contains A(0) as a principal submatrix, which can be useful if A(0) is sparse. The last several iterations can be eliminated if left Jordan chains are computed in addition to right Jordan chains. The performance of the algorithm in floating point and exact (rational) arithmetic is reported for several test cases. The method is shown to be forward stable in floating point arithmetic.     

Existence and stability of periodic solutions for chains of connected oscillators

We consider a nonlinear system of difference equations. This system corresponds to chains of N symmetrically connected oscillators with sufficiently general type of connection, which includes, among others, local and global connection. We prove a theorem on the existence and stability of space-time periodic solutions of such systems for sufficiently small values of the parameter of connection ?.     

Characterization of the inverse of a particular circulant matrix by means of a continued fraction

The matrix A of the constraining system of a particular discrete optimization problem, which has been posed in [3] for modeling a particular scheduling program and for which an optimal solution has been found in [1], is considered and an explicit form of its inverse is given by means of a continued fraction. The knowledge of the inverse enables one to obtain an explicit form of the set of optimal solutions of the problem.Lastly, the connection between A^-1 and the definition of equi-assignment of binary vectors [5] is analysed.     

Retailer order commitments improve distribution supply chain performance

Manufacturing, retail, and healthcare organizations are improving their distribution supply chains by implementing order commitment – an agreement under which some delivery quantities are fixed a priori while some others are freely chosen by the recipients. We show that these strategies are effective whenever the supplier’s cost is at least 29.3% of the total system cost. This result is robust to supply chain operating structure, and savings increase as the supplier’s share of the total system cost increases.     

Permanence of a discrete nonlinear N-species cooperation system with time delays and feedback controls

A discrete nonlinear N-species cooperation system with time delays and feedback controls is considered in this paper. Sufficient conditions which ensure the permanence of the system are obtained. It is shown that these conditions are weaker than those of Chen [F.D. Chen, Permanence of a discrete N-species cooperation system with time delays and feedback controls, Appl. Math. Comput. 186(2007) 23-29], that is, our investigation shows that the additional condition in Chen’s paper is not necessary.