Survival of contact processes on the hierarchical group

We consider contact processes on the hierarchical group, where sites infect other sites at a rate depending on their hierarchical distance, and sites become healthy with a constant recovery rate. If the infection rates decay too fast as a function of the hierarchical distance, then we show that the critical recovery rate is zero. On the other hand, we derive sufficient conditions on the speed of decay of the infection rates for the process to exhibit a nontrivial phase transition between extinction and survival. For our sufficient conditions, we use a coupling argument that compares contact processes on the hierarchical group with freedom two with contact processes on a renormalized lattice. An interesting novelty in this renormalization argument is the use of a result due to Rogers and Pitman on Markov functionals.     

Optimization in Control and Learning in Coupled Map Lattice Systems

In this paper, we analyze various control algorithms that have been proposed for controlling spatiotemporal chaos in a globally coupled map lattice (CML) system. We reformulate the choice of feedback parameters in such systems as a constrained optimization problem and provide numerical and experimental results on the choice of optimal parameters for controlling the mean global Lyapunov exponent of a lattice. Finally, we propose a scheme to use this optimization technique to solve a learning problem in which such a CML system can be used to emulate the dynamics of an epileptic brain. This work was supported by NIH-NIBIB and CRDF grants.     

On Persistence and Stability of Some Spatially Inhomogeneous Systems

Local and global stability and persistence of some coupled map lattices (CMLs) and partial differential equations are studied. A logistic CML with noninteger time step and delay is introduced. The persistence results for reaction-diffusion equations are extendable to the telegraph reaction-diffusion equation for a sufficiently small delay parameter. The stability and persistence results are applied to ecology, physics, economics, and immunology.     

Dynamics of renormalization group in the lower half-plane of the coupling constants of the fermionic hierarchical model

We consider four-component fermionic (Grassmann-valued) field on the hierarchical lattice. The Gaussian part of the Hamiltonian in the model is invariant under the block-spin renormalization group transformation with given degree of normalization factor (renormalization group parameter). The non-Gaussian part of the Hamiltonian is given by the sum of the selfinteraction forms of the 2-nd and 4-th order. The action of the renormalization group transformation in this model is reduced to the rational map in the plane of coupling constants. We investigate the global dynamics of this map in the case when the coupling constant of the 4-th order form is less than zero (lower half-plane) and the renormalization group parameter belongs to the interval [1, 3/2).     

On Julia sets concerning phase transitions

The sets of the points corresponding to the phase transitions of the Potts model on the diamond hierarchical lattice for antiferromagnetic coupling are studied. These sets are the Julia sets of a family of rational mappings. It is shown that they may be disconnected sets. Furthermore, the topological structures of these sets are described completely.     

Continuum limit in the fermionic hierarchical model

We discuss the problem of rigorously constructing the continuum limit in the fermionic hierarchical model. The continuum limit constructed as the limit of fields on the refined hierarchical lattices is a field on a p-adic continuum. We investigate the problem of reconstructing the coupling constants of the continuum model from the coupling constants of the discretized model. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 1, pp. 40–50, January, 1999.     

Bursting synchronization in networks with long-range coupling mediated by a diffusing chemical substance

Many networks of physical and biological interest are characterized by a long-range coupling mediated by a chemical which diffuses through a medium in which oscillators are embedded. We considered a one-dimensional model for this effect for which the diffusion is fast enough so as to be implemented through a coupling whose intensity decays exponentially with the lattice distance. In particular, we analyzed the bursting synchronization of neurons described by two timescales (spiking and bursting activity), and coupled through such a long-range interaction network. One of the advantages of the model is that one can pass from a local (Laplacian) type of coupling to a global (all-to-all) one by varying a single parameter in the interaction term. We characterized bursting synchronization using an order parameter which undergoes a transition as the coupling parameters are changed through a critical value. We also investigated the role of an external time-periodic signal on the bursting synchronization properties of the network. We show potential applications in the control of pathological rhythms in biological neural networks.     

Critical exponents in fermionic hierarchical model

We consider the problem of algebraic computation of the critical exponent ν in the 2N-component fermionic Dyson model on a hierarchical lattice without the use of perturbation theory. Analyzing the results in a particular case when N = 2, we conclude that an algebraic approach in this model gives the same expression for ν as the approach of functional integration via Feynman diagrams in the p-adic ϕ ⁴-model.     

Bayesian Inference in Hidden Markov Random Fields for Binary Data Defined on Large Lattices

Hidden Markov random fields represent a complex hierarchical model, where the hidden latent process is an undirected graphical structure. Performing inference for such models is difficult primarily because the likelihood of the hidden states is often unavailable. The main contribution of this article is to present approximate methods to calculate the likelihood for large lattices based on exact methods for smaller lattices. We introduce approximate likelihood methods by relaxing some of the dependencies in the latent model, and also by extending tractable approximations to the likelihood, the so-called pseudolikelihood approximations, for a large lattice partitioned into smaller sublattices. Results are presented based on simulated data as well as inference for the temporal-spatial structure of the interaction between up- and down-regulated states within the mitochondrial chromosome of the Plasmodium falciparum organism. Supplemental material for this article is available online.     

时空Chaos研究中的CML模型

通过对有关差分格式的稳定性分析，我们提出了一类新的格点耦合映射（ＣＭＬ）模型。数值试验表明：我们提出的ＣＭＬ模型是一类有效的研究时空复杂性的模型，特别是对于强耦合系统。     

Nonparametric Bayesian topic modelling with the hierarchical Pitman–Yor processes

The Dirichlet process and its extension, the Pitman–Yor process, are stochastic processes that take probability distributions as a parameter. These processes can be stacked up to form a hierarchical nonparametric Bayesian model. In this article, we present efficient methods for the use of these processes in this hierarchical context, and apply them to latent variable models for text analytics. In particular, we propose a general framework for designing these Bayesian models, which are called topic models in the computer science community. We then propose a specific nonparametric Bayesian topic model for modelling text from social media. We focus on tweets (posts on Twitter) in this article due to their ease of access. We find that our nonparametric model performs better than existing parametric models in both goodness of fit and real world applications.     

A Network Optimization Model for Budget Planning in Multi-Objective Hierarchical Systems

This paper presents a planning/budgeting scheme for hierarchical systems. A multi-objective network optimization model for multilayer budget allocation is suggested. The network presents the hierarchical structure of the system. The budget allocations are the flows in the network. Each component in the system (arc in the network) has lower and upper bounds. The model maximizes the additive utility function of the system, expressed as a weighted summation over the preferences of the system's components in the various levels. The preferences are evaluated by using a multigoal approach, utilizing the Analytical Hierarchy Process (AHP). Finally, the model is conceptually compared with other known budgeting procedures and models, such as ZBB, PBBS and cost benefit analyses.     

Robust estimation in inverse problems via quantile coupling

In this article we consider a sequence of hierarchical space model of inverse problems.The underlying function is estimated from indirect observations over a variety of error distributions including those that are heavy-tailed and may not even possess variances or means.The main contribution of this paper is that we establish some oracle inequalities for the inverse problems by using quantile coupling technique that gives a tight bound for the quantile coupling between an arbitrary sample p-quantile and a normal variable,and an automatic selection principle for the nonrandom filters.This leads to the data-driven choice of weights.We also give an algorithm for its implementation.The quantile coupling inequality developed in this paper is of independent interest,because it includes the median coupling inequality in literature as a special case.     

The phase transitions of the planar random-cluster and Potts models with $$q \ge 1$$ are sharp

We prove that random-cluster models with \(q \ge 1\) on a variety of planar lattices have a sharp phase transition, that is that there exists some parameter \(p_c\) below which the model exhibits exponential decay and above which there exists a.s. an infinite cluster. The result may be extended to the Potts model via the Edwards–Sokal coupling. Our method is based on sharp threshold techniques and certain symmetries of the lattice; in particular it makes no use of self-duality. Part of the argument is not restricted to planar models and may be of some interest for the understanding of random-cluster and Potts models in higher dimensions. Due to its nature, this strategy could be useful in studying other planar models satisfying the FKG lattice condition and some additional differential inequalities.     

CONTACT PROCESS ON HEXAGONAL LATTICE

In this article, we discuss several properties of the basic contact process on hexagonal lattice H, showing that it behaves quite similar to the process on d-dimensional lattice Zd in many aspects. Firstly, we construct a coupling between the contact process on hexagonal lattice and the oriented percolation, and prove an equivalent finite space-time condition for the survival of the process. Secondly, we show the complete convergence theorem and the polynomial growth hold for the contact process on hexagonal lattice. Finally, we prove exponential bounds in the supercritical case and exponential decay rates in the subcritical case of the process.     

The lattice of integer partitions and its infinite extension

In this paper, we use a simple discrete dynamical model to study integer partitions and their lattice. The set of reachable configurations of the model, with the order induced by the transition rule defined on it, is the lattice of all partitions of a positive integer, equipped with a dominance ordering. We first explain how this lattice can be constructed by an algorithm in linear time with respect to its size by showing that it has a self-similar structure. Then, we define a natural extension of the model to infinity, which we compare with the Young lattice. Using a self-similar tree, we obtain an encoding of the obtained lattice which makes it possible to enumerate easily and efficiently all the partitions of a given integer. This approach also gives a recursive formula for the number of partitions of an integer, and some informations on special sets of partitions, such as length bounded partitions.     

The heterogeneous multiscale method as a framework for coupling the finite element method and molecular dynamics

Hierarchical two-scale methods are computationally very powerful as there is no direct coupling between the macro- and microscale. Such schemes develop first a microscale model under macroscopic constraints, then the macroscopic constitutive laws are found by averaging over the microscale. The heterogeneous multiscale method (HMM) is a general top-down approach for the design of multiscale algorithms. While this method is mainly used for concurrent coupling schemes in the literature, the proposed methodology also applies to a hierarchical coupling. This contribution discusses a hierarchical two-scale setting based on the heterogeneous multi-scale method for quasi-static problems: the macroscale is treated by continuum mechanics and the finite element method and the microscale is treated by statistical mechanics and molecular dynamics. Our investigation focuses on an optimised coupling of solvers on the macro- and microscale which yields a significant decrease in computational time with no associated loss in accuracy. In particular, the number of time steps used for the molecular dynamics simulation is adjusted at each iteration of the macroscopic solver. A numerical example demonstrates the performance of the model. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Renormalization-Group Transformation in a 2n-Component Fermionic Hierarchical Model

We study the 2N-component fermionic model on a hierarchical lattice and give explicit formulas for the renormalization-group transformation in the space of coefficients that determine a Grassmann-valued density of the free measure. We evaluate the inverse renormalization-group transformation. The de.nition of the renormalization-group fixed points reduces to a solution of a system of algebraic equations. We investigate solutions of this system for N = 1, 2, 3. For α = 1, we prove an analogue of the central limit theorem for fermionic 2N-component fields. We discover an interesting relation between renormalization-group transformations in bosonic and fermionic hierarchical models and show that one of these transformations is obtained from the other by replacing N with -N. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 2, pp. 251–266, February, 2006.     

The Use of a Simple Multiple-Criteria Model to Assist in Selection from a Shortlist

The use of a simple multiple-criteria model to assist in decision making is described. The model, a hierarchical additive weighted value-function, was used as a part of a decision-making process to select, from a shortlist of three, the company with which to place a contract for the development of a computerized financial management system. The multiple-criteria model and its use are described in detail. To conclude, there is a discussion on the contribution of the model to the decision-making process as perceived by the decision-making group.