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Although histogram methods have been extremely effective for analyzing data from Monte Carlo simulations, they do have certain limitations, including the range over which they are valid and the difficulties of combining data from independent simulations. In this paper, we describe a complementary approach to extracting information from Monte Carlo simulations that uses the matrix of transition probabilities. Combining the Transition Matrix with an N-fold way simulation technique produces an extremely flexible and efficient approach to rather general Monte Carlo simulations.

In this paper, we first analytically calculate the eigenvalues of the transition matrix of a structure with very complex architecture and their multiplicities. We call this structure polymer network. Based on the eigenvalues obtained in the iterative manner, we then calculate the eigentime identity. We highlight two scaling behaviors (logarithmic and linear) for this quantity, strongly depending on the value of the weight factor. Finally, by making use of the obtained eigenvalues, we determine the weighted counting of spanning trees.

In quantum information and quantum computation, a bipartite system provides a basic few-body framework for investigating significant properties of thermodynamics and statistical mechanics. A Hamiltonian model for a bipartite system is introduced to analyze the important role of interaction between bipartite subsystems in quantum non-equilibrium thermodynamics. We illustrate discrimination between such quantum thermodynamics and classical few-body non-equilibrium thermodynamics. By proposing a detailed balance condition of the bipartite system, we generally investigate the properties of the entropy and heat of our model, as well as the relation between them.

This study describes the spatial disorder of one-dimensional Cellular Neural Networks (CNN) with a biased term by applying the iteration map method. Under certain parameters, the map is one-dimensional and the spatial entropy of stable stationary solutions can be obtained explicitly as a staircase function.

This work investigates the complexity of one-dimensional cellular neural network mosaic patterns with spatially variant templates on finite and infinite lattices. Various boundary conditions are considered for finite lattices and the exact number of mosaic patterns is computed precisely. The entropy of mosaic patterns with periodic templates can also be calculated for infinite lattices. Furthermore, we show the abundance of mosaic patterns with respect to template periods and, which differ greatly from cases with spatially invariant templates.

We study a one-dimensional Cellular Neural Network with an output function which is nonflat at infinity. Spatial chaotic regions are completely characterized. Moreover, each of their exact corresponding entropy is obtained via the method of transition matrices. We also study the bifurcation phenomenon of mosaic patterns with bifurcation parameters z and β. Here z is a source (or bias) term and β is the interaction weight between the neighboring cells. In particular, we find that by injecting the source term, i.e. z ≠ 0, a lot of new chaotic patterns emerge with a smaller interaction weight β. However, as β increases to a certain range, most of previously observed chaotic patterns disappear, while other new chaotic patterns emerge.

We describe the fine structure of the global attractor of the Cahn–Hilliard equation on two-dimensional square domains. This is accomplished by combining recent numerical results on the set of equilibrium solutions due to [Maier-Paape & Miller, 2002] with algebraic Conley index techniques. Using the information on the set of equilibria as assumption, we build Morse decompositions and connection matrices. The latter imply existence of heteroclinic connections between the equilibria inside the attractor. While path-following of the parameter range of Cahn–Hilliard, we find more and more complicated dynamical behavior. One of our main results describes the fine structure of the attractor for mean mass zero with four stable cosine structured equilibria and eight other stable equilibria that have a quarter circle nodal line. Besides that, we also study the attractor in symmetry fixed point spaces where we, for example, find nonunique connection matrices and saddle–saddle connections of Morse sets.

A directed network such as the WWW can be represented by a transition matrix. Comparing this matrix to a Frobenius–Perron matrix of a chaotic piecewise-linear one-dimensional map whose domain can be divided into Markov subintervals, we are able to relate the network structure itself to chaotic dynamics. Just like various large deviation properties of local expansion rates (finite-time Lyapunov exponents) related to chaotic dynamics, we can also discuss those properties of network structure.

For an irreducible transition matrix A of size m × m, which is not a permutation, a map f : X → X is said to be strictly A-coupled-expanding if there are nonempty sets V₁,…, V_m ⊂ X such that the distance between any two of them is positive and f(V_i) ⊃ V_j holds whenever a_ij = 1. This paper presents two theorems that give sufficient conditions for a strictly A-coupled-expanding map to be chaotic on part of its domain in the sense of, respectively, Auslander and Yorke and Devaney. These results improve on the work of Zhang and Shi [2010]. An example is provided to illustrate that the class of maps the new theorems apply to is significantly wider.

In the case of one-dimensional cellular automaton (CA), a hybrid CA (HCA) is the member whose evolution of the cells is dependent on nonunique global functions. The HCAs exhibit a wide range of traveling and stationary localizations in their evolution. We focus on HCA with memory (HCAM) because they produce a host of gliders and complicated glider collisions by introducing the hybrid mechanism. In particular, we undertake an exhaustive search of gliders and describe their collisions using quantitative approach in HCAM$(43,74)$. By introducing the symbol vector space and exploiting the mathematical definition of HCAM, we present an analytical method of complex asymptotic dynamics of the gliders.

The Hausdorff dimension of Julia sets of expanding maps can be computed by the eigenvalue algorithm. In this work, an implementation of this algorithm for quadratic polynomial, that allows the calculation of the Hausdorff dimension of Julia sets for complex parameters, is done. In particular, the parameters in a neighborhood of the parabolic parameter $c=1/4$ are analyzed and a small oscillation in Hausdorff dimension is shown.

We investigate the structure of the characteristic polynomial det(xI - T) of a transition matrix T that is associated to a train track representative of a pseudo-Anosov map [F] acting on a surface. As a result we obtain three new polynomial invariants of [F], one of them being the product of the other two, and all three being divisors of det(xI - T). The degrees of the new polynomials are invariants of [F] and we give simple formulas for computing them by a counting argument from an invariant train-track. We give examples of genus 2 pseudo-Anosov maps having the same dilatation, and use our invariants to distinguish them.

Several methods have been developed in order to solve electrical circuits consisting of resistors and an ideal voltage source. A correspondence with random walks avoids difficulties caused by choosing directions of currents and signs in potential differences. Starting from the random-walk method, we introduce a reduced transition matrix of the associated Markov chain whose dominant eigenvector alone determines the electric potentials at all nodes of the circuit and the equivalent resistance between the nodes connected to the terminals of the voltage source. Various means to find the eigenvector are developed from its definition. A few example circuits are solved in order to show the usefulness of the present approach.

Based on 2001, 2005, 2011, 2015 Landsat /TM images and ArcGIS spatial analysis techniques, the dynamic changes of Linpan landscape in the second ring of Chengdu city were analyzed by using transfer contribution rate. The results demonstrated that with the rapid urbanization, Linpan and farmland mainly converted to urban construction land. The ecological environment and quantity of Linpan was damaged severely. The ecological planning strategies in different level of optimizing Linpan landscape were put forward. The research provides an important basis for protection and development of rural landscape of western Sichuan plain.