Frustrated Magnetic Systems and the q-Oscillator

We analyze the spherical model with frustration induced by an external gauge field. The case of the infinite-dimensional model has recently been reduced to a problem of q-deformed oscillators with q parametrizing the amount of frustration. We find a complete analytic solution of the model by using a convenient representation of the q-oscillator algebra, the q-Hermite polynomials. The low-temperature phase does not exhibit a glassy behavior. With respect to the usual unfrustrated spherical model, the effect of frustration is only quantitative. A glassy low-temperature phase is expected for the more complicated XY model whose study is in progress. Bibliography: 15 titles.     

Thermoviscoplasticity of Glassy Polymers in the Logarithmic Strain Space Based on the Free Volume Theory

In this contribution a new constitutive model of finite thermo-visco-plastic behavior of amorphous glassy polymers and details of its numerical implementation are outlined. In contrast to existing kinematical approaches to finite plasticity of glassy polymers, the formulation applies a plastic metric theory based on an additive split of Lagrangian Hencky-type strains into elastic and plastic parts [1, 3]. The characteristic strain hardening of the model is derived from a polymer network model, the thermo-visco-plastic flow rule in the logarithmic strain space uses structures of the free volume flow theory [4]. The integration of this micromechanically motivated approach in a three-dimensional computational model is the key novel aspect of this work. An important aspect of this work is the model validation based on experimental findings, whereas the excellent performance of the proposed formulation is demonstrated by means of numerical examples. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global optima results for the Kauffman NK model

The Kauffman NK model has been used in theoretical biology, physics and business organizations to model complex systems with interacting components. Recent NK model results have focused on local optima. This paper analyzes global optima of the NK model. The resulting global optimization problem is transformed into a stochastic network model that is closely related to two well-studied problems in operations research. This leads to applicable strategies for explicit computation of bounds on the global optima particularly with K either small or close to N. A general lower bound, which is sharp for K = 0, is obtained for the expected value of the global optimum of the NK model. A detailed analysis is provided for the expectation and variance of the global optimum when K = N−1. The lower and upper bounds on the expectation obtained for this case show that there is a wide gap between the values of the local and the global optima. They also indicate that the complexity catastrophe that occurs with the local optima does not arise for the global optima.     

Bayesian Inference for the RC(m) Association Model

Describing the structure in a two-way contingency table in terms of an RC(m) association model, we are concerned with the computation of posterior distributions of the model parameters using prior distributions which take into account the nonlinear restrictions of the model. We are further involved with the determination of the order of association m, based on Bayesian arguments. Using projection methods, a prior distribution over the parameters of the simpler RC(m) model is induced from a prior of the parameters of the saturated model. The fit of the assumed RC(m) model is evaluated using the posterior distribution of its distance from the full model. Our methods are illustrated with a popular dataset.     

Exact Analysis of the State-Dependent Polling Model

We consider a polling model in which a number of queues are served, in cyclic order, by a single server. Each queue has its own distinct Poisson arrival stream, service time, and switchover time (the server's travel time from that queue to the next) distribution. A setup time is incurred if the polled queue has one or more customers present. This is the polling model with State-Dependent service (the SD model). The SD model is inherently complex; hence, it has often been approximated by the much simpler model with State-Independent service (the SI model) in which the server always sets up for a service at the polled queue, regardless of whether it has customers or not. We provide an exact analysis of the SD model and obtain the probability generating function of the joint queue length distribution at a polling epoch, from which the moments of the waiting times at the various queues are obtained. A number of numerical examples are presented, to reveal conditions under which the SD model could perform worse than the corresponding SI model or, alternately, conditions under which the SD model performs better than a corresponding model in which all setup times are zero. We also present expressions for a variant of the SD model, namely, the SD model with a patient server.     

A function space model approach to the rational evaluation subgroups

Let f : U → X be a map from a connected nilpotent space U to a connected rational space X. The evaluation subgroup G _*(U, X; f), which is a generalization of the Gottlieb group of X, is investigated. The key device for the study is an explicit Sullivan model for the connected component containing f of the function space of maps from U to X, which is derived from the general theory of such a model due to Brown and Szczarba (Trans Am Math Soc 349, 4931–4951, 1997). In particular, we show that non Gottlieb elements are detected by analyzing a Sullivan model for the map f and by looking at non-triviality of higher order Whitehead products in the homotopy group of X. The Gottlieb triviality of a fibration in the sense of Lupton and Smith (The evaluation subgroup of a fibre inclusion, 2006) is also discussed from the function space model point of view. Moreover, we proceed to consideration of the evaluation subgroup of the fundamental group of a nilpotent space. In consequence, the first Gottlieb group of the total space of each S ¹-bundle over the n-dimensional torus is determined explicitly in the non-rational case.      

From the Ehrenfest model to time-fractional stochastic processes

The Ehrenfest model is considered as a good example of a Markov chain. I prove in this paper that the time-fractional diffusion process with drift towards the origin, is a natural generalization of the modified Ehrenfest model. The corresponding equation of evolution is a linear partial pseudo-differential equation with fractional derivatives in time, the orders lying between 0 and 1. I focus on finding a precise explicit analytical solution to this equation depending on the interval of the time. The stationary solution of this model is also analytically and numerically calculated. Then I prove that the difference between the discrete approximate solution at time t_n, n≥0, and the stationary solution obeys a power law with exponent between 0 and 1. The reversibility property is discussed for the Ehrenfest model and its fractional version with a new observation.     

Testing the diameter of graphs

We propose a general model for testing graph properties, which extends and simplifies the bounded degree model of Goldreich and Ron [Property Testing in Bounded Degree Graphs, Proc. 31st Annual ACM Symposium on the Theory of Computing, 1997, pp. 406–415.] In this model, we present a family of algorithms that test whether the diameter of a graph is bounded by a given parameter D, or is ?‐far from any graph with diameter at most β(D). The function β(D) ranges between D+4 and 4D+2, depending on the algorithm. All our algorithms run in time polynomial in 1/?. © 2002 Wiley Periodicals, Inc. Random Struct. Alg. 20: 165–183, 2002     

A probabilistic analysis of the set covering problem

A probabilistic analysis of the minimum cardinality set covering problem (SCP) is developed, considering a stochastic model of the (SCP), withn variables andm constraints, in which the entries of the corresponding (m, n) incidence matrix are independent Bernoulli distributed random variables, each with constant probabilityp of success. The behaviour of the optimal solution of the (SCP) is then investigated as bothm andn grow asymptotically large, assuming either an incremental model for the evolution of the matrix (for each size, the matrixA is obtained bordering a matrix of smaller size by new columns and rows) or an independent one (for each size, an entirely new set of entries forA are considered). Two functions ofm are identified, which represent a lower and an upper bound onn in order the (SCP) to be a.e. feasible and not trivial. Then, forn lying within these bounds, an asymptotic formula for the optimum value of the (SCP) is derived and shown to hold a.e.The performance of two simple randomized algorithms is then analyzed. It is shown that one of them produces a solution value whose ratio to the optimum value asymptotically approaches 1 a.e. in the incremental model, but not in the independent one, in which case the ratio is proved to be tightly bounded by 2 a.e. Thus, in order to improve the above result, a second randomized algorithm is proposed, for which it is proved that the ratio between the approximate solution value and the optimum approaches 1 a.e. also in the independent model.     

Weak covering and the tree property

Suppose that there is no transitive model of ZFC + there is a strong cardinal, and let K denote the core model. It is shown that if has the tree property then and is weakly compact in K. Received: 11 June 1997     

Duality in inhomogeneous random graphs,and the cut metric

The classical random graph model G(n, c/n) satisfies a “duality principle”, in that removing the giant component from a supercritical instance of the model leaves (essentially) a subcritical instance. Such principles have been proved for various models; they are useful since it is often much easier to study the subcritical model than to directly study small components in the supercritical model. Here we prove a duality principle of this type for a very general class of random graphs with independence between the edges, defined by convergence of the matrices of edge probabilities in the cut metric. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 399–411, 2011     

Experimental evidence for the occurrence of E 8 in nature and the radii of the Gosset circles

A recent experimental discovery involving the spin structure of electrons in a cold one-dimensional magnet points to a validation of a (1989) Zamolodchikov model involving the exceptional Lie group E ₈. The model predicts 8 particles and predicts the ratio of their masses. The conjectures have now been validated experimentally, at least for the first five masses. The Zamolodchikov model was extended in 1990 to a Kateev–Zamolodchikov model involving E ₆ and E ₇ as well. In a seemingly unrelated matter, the vertices of the 8-dimensional Gosset polytope identifies with the 240 roots of E ₈. Under the famous two-dimensional (Peter McMullen) projection of the polytope, the images of the vertices are arranged in eight concentric circles, hereafter referred to as the Gosset circles. The McMullen projection generalizes to any complex simple Lie algebra (in particular not restricted to A-D-E types) whose rank is greater than 1. The Gosset circles generalize as well, using orbits of the Coxeter element on roots. Applying results in Kostant (Am J Math 81:973–1032, 1959), I found some time ago a very easily defined operator A on a Cartan subalgebra, the ratio of whose eigenvalues is exactly the ratio of squares of the radii r _i of the generalized Gosset circles. The two matters considered above relate to one another in that the ratio of the masses in the E ₆, E ₇, E ₈ Kateev–Zamolodchikov models are exactly equal to the ratios of the radii of the corresponding generalized Gosset circles.     

A primal‐dual method for the Meyer model of cartoon and texture decomposition

In this paper, we study the original Meyer model of cartoon and texture decomposition in image processing. The model, which is a minimization problem, contains an l₁‐based TV‐norm and an l_∞‐based G‐norm. The main idea of this paper is to use the dual formulation to represent both TV‐norm and G‐norm. The resulting minimization problem of the Meyer model can be given as a minimax problem. A first‐order primal‐dual algorithm can be developed to compute the saddle point of the minimax problem. The convergence of the proposed algorithm is theoretically shown. Numerical results are presented to show that the original Meyer model can decompose better cartoon and texture components than the other testing methods.     

On the moving Preisach model

It is shown that the moving model, which is a variant of the Preisach model for hysteresis, possesses the wiping out property and is continuous in C[0, T] under natural assumptions.     

Routing Problems on the Mesh of Buses

The mesh of buses (MBUSs) is a parallel computation model which consists ofn×nprocessors,nrow buses, andncolumn buses, but no local connections between neighboring processors. Annlower bound for the permutation routing on this model is shown. The proof does not depend on common predetermined assumptions such as “if a packet has to move horizontally then it has to ride on a horizontal bus at least once.” As for upper bounds, a 1.5nalgorithm is shown.     

Modelling the dynamics of stochastic local search on <Emphasis Type="Italic">k</Emphasis>-<Emphasis Type="SmallCaps">sat</Emphasis>

A new analytical tool is presented to provide a better understanding of the search space of k-sat. This tool, termed the local value distribution , describes the probability of finding assignments of any value q′ in the neighbourhood of assignments of value q. The local value distribution is then used to define a Markov model to model the dynamics of a corresponding stochastic local search algorithm for k-sat. The model is evaluated by comparing the predicted algorithm dynamics to experimental results. In most cases the fit of the model to the experimental results is very good, but limitations are also recognised.     

Asymptotic relations betweenL- andM-estimators in the linear model

We obtain Bahadur-type representations for one-stepL-estimators,M- and one-stepM-estimators in the linear model. The order of the remainder terms in these representations depends on the smooth-ness of the weight function forL-estimators and on the smoothness of the ψ-function forM- and one-stepM-estimators. We use the representations to investigate the asymptotic relations between these estimators. In particular, we show that asymptotically equivalentL- andM-estimators of the slope parameter exist even when the underlying distribution is asymmetric. It is important to consider the asymmetric case for both practical and robustness reasons: first, there is no compelling argument which precludes asymmetric distributions from arising in practice, and, secondly, even if a symmetric model can be posited, it is important to allow for the possibility of mild (and therefore difficult to detect) departures from the symmetric model.     

On one method of finding the equilibrium state

An algorithm for solving a linear “production-exchange” model is described. The model reduces to a parametric model with strictly concave utility functions. Some properties of the parametric model are studied. Since the excessive demand map satisfies the discovered preference condition, it is possible to apply the second form of Chebyshev centers. For a sufficiently small parameter, the algorithm converges to the equilibrium state of the initial linear model. Bibliography:11 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 80, 1996, pp. 59–67.     

Efficient and accurate spectral method for the time-fractional dual-phase-lag heat transfer model and its parameter estimation

In the current paper, a heat transfer model is suggested based on a time-fractional dual-phase-lag (DPL) model. We discuss the model in two parts, the direct problem and the inverse problem. Firstly, for solving it, the finite difference/Legendre spectral method is constructed. In the temporal direction, we employ the weighted and shifted Grünwald approximation, which can achieve second order convergence. In the spatial direction, the Legendre spectral method is used, it can obtain spectral accuracy. The stability and convergence are theoretically analyzed. For the inverse problem, the Bayesian method is used to construct an algorithm to estimate the four parameters for the model, namely, the time-fractional order α, the time-fractional order β, the delay time τ_T, and the relaxation time τ_q. Next, numerical experiments are provided to demonstrate the effectiveness of our scheme, with the values of τ_q and τ_T for processed meat employed. We also make a comparison with another method. The data obtained for the direct problem are used in the parameter estimation. The paper provides an accurate and efficient numerical method for the time-fractional DPL model.     

On the discrete velocity models of the Boltzmann equation in several dimensions

Summary The existence, local in time, of the solution of the discrete velocity model of the Boltzmann equation, is discussed. Then for the 2r-plane regular model a theorem which gives the global existence and the asymptotic behaviour of the solution is proved.