Narrow Results

In this paper, we focus on studying the non-probability Gibbs measures for a Hard-Core (HC) model on a Cayley tree of order $k\ge 2$, where the set of integers $\mathbb{Z}$ is the set of spin values. It is well known that each Gibbs measure, whether it be a gradient or non-probability measure, of this model corresponds to a boundary law. A boundary law can be thought of as an infinite-dimensional vector function (with strictly positive coordinates) defined at the vertices of the Cayley tree, which satisfies a nonlinear functional equation. Furthermore, every normalizable boundary law corresponds to a Gibbs measure. However, a non-normalizable boundary law can define the gradient or non-probability Gibbs measures. In this paper, we investigate the conditions for uniqueness and non-uniqueness of translation-invariant and periodic non-probability Gibbs measures for the HC model on a Cayley tree of any order $k\ge 2$.

The purpose of this review paper is to present systematically all known results on Gibbs measures on Cayley trees (Bethe lattices). There are about 150 papers which contain mathematically rigorous results about Gibbs measures on Cayley trees. This review is mainly based on the recently published mathematical papers. The method used for the description of Gibbs measures on Cayley trees is the method of Markov random field theory and recurrent equations of this theory, but the modern theory of Gibbs measures on trees uses new tools such as group theory, information flows on trees, node-weighted random walks, contour methods on trees and nonlinear analysis. We discuss all the mentioned methods which were developed recently. Thus, the paper informs the reader about what is (mathematically) done in the theory of Gibbs measures on trees and about where the corresponding results were published. We only give proofs which were not published in literature. Moreover, we give several open problems.

In this paper, we give a systematic review of the theory of Gibbs measures of Potts model on Cayley trees (developed since 2013) and discuss many applications of the Potts model to real world situations: mainly biology, physics, and some examples of alloy behavior, cell sorting, financial engineering, flocking birds, flowing foams, image segmentation, medicine, sociology, etc.

Kittel’s 1D model represents a natural DNA with two strands as a (molecular) zipper, which may be separated as the temperature is varied. We define multidimensional version of this model on a Cayley tree and study the set of Gibbs measures. We reduce description of Gibbs measures to solving of a nonlinear functional equation, with unknown functions (called boundary laws) defined on vertices of the Cayley tree. Each boundary law defines a Gibbs measure. We give a general formula of free energy depending on the boundary law. Moreover, we find some concrete boundary laws and corresponding Gibbs measures. Explicit critical temperature for occurrence of a phase transition (non-uniqueness of Gibbs measures) is obtained.

In this work, we study the opinion dynamics of the Sznajd model with anticonformity on a fully-connected network. We consider four agents with two different configurations; three against one (3–1) and two against two (2–2). We consider two different individual behaviors, conformity and anticonformity, and observe the effect on the critical behavior of the model. We analyze the differences between the phase transitions that occur for both agent configurations. We find that both agent configurations have a different critical point. The critical point of the 3–1 agent is smaller than that of the 2–2 agent configuration. From the simulation and analytical result, we find that the critical point for the 3–1 occurs at $p_c=0.2$, and for the 2–2, at $p_c=0.5$. From the social viewpoint, the consensus process in a population is faster with a larger influencer in the same number of small group of the population. In addition, we find the critical exponents for both configurations are the same, that are $\beta \approx 0.5,\gamma \approx 1$ and $\nu \approx 2$. Our results suggest that both models are identical and in the mean-field Ising universality class.

In this paper, we consider the realization of configuration of limit cycles of piecewise linear systems on the plane. We show that any configuration of finite number of Jordan curves can be realized by a discontinuous piecewise linear system with two zones separated by a continuous curve.

In this paper, we present a framework for maintainable software design, based on a multilevel understanding of software function. This framework is the novel functional representation ZD, in which domain concepts (e.g., employee records) as well as computational concepts (e.g., stacks and queues) are represented in a reusable manner. We use ZD to support the construction of a library of reusable components for novel configurations. In order to achieve this, we provide mechanisms for supporting flexible configurations, address problems of determining the correctness of a combination of library components, and consider the computational complexity of finding combinations. These problems all stem from problems of interactions between components. Therefore, the structure of ZD is focused on representing and handling these interactions. We show how the approach based on ZD resolves certain well-known problems faced by other library-based component reuse architectures.

We propose a natural scheme to measure the joint separation of a cluster of objects in general geometric settings. In particular, here the measure is developed for finite sets of planes in ℝ³ in terms of extreme configurations of vectors on the planes of a given set. We prove geometric and graph-theoretic results about extreme configurations on arbitrary finite plane sets. We then specialize to the planes bounding a regular polyhedron in order to exploit the symmetries. However, even then results are non-trivial and surprising – extreme configurations on regular polyhedra may turn out to be highly irregular.

Giving a condition for the amenability of groups, Rosenblatt and Willis first introduced the concept of configuration. From the beginning of the theory, the question whether the concept of configuration equivalence coincides with the concept of group isomorphism was posed. We negatively answer this question by introducing two non-isomorphic, solvable and hence amenable groups which are configuration equivalent. Also, we will prove this conjecture, due to Rosenblatt and Willis, whether the configuration equivalent groups include the free non-Abelian group of the same rank or not. We show that two-sided equivalent groups have same class numbers.

Shell model calculations have been performed in $f_{5/2}p{g}_{9/2}$ model space using two different interactions viz. jj44bpn and jun45pn to explore nuclear structure in ${}^{63,65,67}$Ga. Calculated excitation energies are compared with previously reported experimental values and a good agreement has been observed. Transitions strengths are also calculated using two sets of effective charges for proton and neutron and are compared with nearby ${}^{62,64,66}$Zn isotopes. Occupation probabilities of protons and neutrons corresponding to individual orbitals (namely $f_{5/2}$, $p_{3/2}$, $p_{1/2}$ and $g_{9/2}$ orbital), and dominant particle configurations for individual spin states have been presented as well. Calculations suggest major role of intruder $1g_{9/2}$ orbital in constructing the wave functions of higher angular momentum states, whereas, the lower excited states are mainly dominated by contributions from $fp$ orbitals.

This research work investigates the simultaneous interaction of entrepreneurial orientation dimensions of the firm with the leader’s psychological factors for a better explanation of SMEs’ performance. The universal and contingency approach are reductionist and do not allow us to achieve the research objective. So, to study the simultaneous interaction between a great number of variables, we are mobilizing for this purpose the configurational approach and particularly, the perspective of alignment (fit) as “configuration”. Indeed, we are pursuing a quantitative methodological approach and conducting a field survey through a research questionnaire distributed to 100 industrial Tunisian SMEs.

The results of this research reveal a taxonomy of four configurations of industrial Tunisian SMEs significantly different from each other and which have different effects on performance. The configurations are named, the “Leader”, the “Creative”, the “Ambitious” and the “Conservative”.

Workflow modeling languages allow for the specification of executable business processes. They, however, typically do not provide any guidance for the adaptation of workflow models, i.e. they do not offer any methods or tools explaining and highlighting which adaptations of the models are feasible and which are not. Therefore, an approach to identify so-called configurable elements of a workflow modeling language and to add configuration opportunities to workflow models is presented in this paper. Configurable elements are the elements of a workflow model that can be modified such that the behavior represented by the model is restricted. More precisely, a configurable element can be either set to enabled, to blocked, or to hidden. To ensure that such configurations lead only to desirable models, our approach allows for imposing so-called requirements on the model's configuration. They have to be fulfilled by any configuration, and limit therefore the freedom of configuration choices. The identification of configurable elements within the workflow modeling language of YAWL and the derivation of the new "configurable YAWL" language provide a concrete example for a rather generic approach. A transformation of configured models into lawful YAWL models demonstrates its applicability.

We consider models with nearest-neighbor interactions and with the set [0, 1] of spin values, on a Cayley tree of order k ≥ 1. We show that periodic Gibbs measures are either translation-invariant or periodic with period two. We describe two-periodic Gibbs measures of the model. For k = 1 we show that there is no any periodic Gibbs measure. In case k ≥ 2 we get a sufficient condition on Hamiltonian of the model with uncountable set of spin values under which the model has no periodic Gibbs measure. We construct several models which have at least two periodic Gibbs measures.

In this paper, we consider a Hard-Core (HC) model with two spin values on Cayley trees. The conception of alternative Gibbs measure is introduced and translational invariance conditions for alternative Gibbs measures are found. Also, we show that the existence of alternative Gibbs measures which are not translation-invariant. In addition, we study free energy of the model.

For the solid-on-solid (SOS) model with an external field and with spin values from the set of all integers on a Cayley tree, each (gradient) Gibbs measure corresponds to a boundary law (an infinite-dimensional vector function defined on vertices of the Cayley tree) satisfying a nonlinear functional equation. Recently some translation-invariant and height-periodic (non-normalizable) solutions to the equation are found. Here, our aim is to find non-height-periodic and non-normalizable boundary laws for the SOS model. By such a solution one can construct a non-probability Gibbs measure. We find explicitly several non-normalizable boundary laws. Moreover, we reduce the problem to solving of a nonlinear, second-order difference equation. We give analytic and numerical analyses of the difference equation.

In this paper we investigate the configuration of switches suitable for high performance communication, and propose communication schemes which exploit the structural strengths of that configuration. In switch-based networks, communication performance heavily depends on the configuration of switches and communication schemes for the networks. The deadlock problem caused by wormhole routing is another crucial factor affecting communication performance. Thus, we first evaluate several candidate configurations in terms of deadlock avoidance, scalability, flexibility, cost, and network properties (bandwidth, diameter, and average distance), and verify that the incomplete fat tree is the most promising configuration. Next, we show how to implement a routing algorithm and a unicast-based multicast algorithm on the incomplete fat tree configuration. The routing algorithm always finds a shortest path and fully utilizes network resources without using a routing tables. The multicast algorithm is optimal in that it is contention-free and requires a minimum number of communication steps.

The Tarski number of a group $G$ is the minimal number of the pieces of paradoxical decompositions of that group. Using configurations along with a matrix combinatorial property, we construct paradoxical decompositions. We also compute an upper bound for the Tarski number of a given non-amenable group by counting the number of paths in a diagram associated to the group.

The concept of configuration was first introduced by Rosenblatt and Willis to give a characterization for the amenability of groups. Then Rejali and Yousofzadeh introduced the notion of two-sided configuration to study the normal subsets of a group. In [A, Abdollahi, A. Rejali and G. A. Willis, Group properties characterised by configurations, Illinois J. Math.48(3) (2004) 861–873.], the authors have asked that if two configuration equivalent groups are isomorphic? We show that if $G$ and $H$ have same two-sided configuration sets and $N$ is a normal subgroup of $G$ with polycyclic or FC quotient, then there is a normal subgroup $𝔑$ of $H$ such that $G/N\cong H/𝔑$. Also, we show that if $G$ and $H$ are two-sided equivalent groups, and if one of them is polycyclic or FC, then they are isomorphic.

This study examines the complex and evolutionary nature of supply network configuration. Taking a bottom-up approach, we examine how supply network configuration at the macro-level evolves as a result of individual retailers' dynamic choice of procurement strategies at the micro-level. Employing agent-based modeling, we focus on the effects of switch cost and distributors' ordering policy on the evolution of supply network configuration. Our results show that (1) supply networks tend to evolve into a set of separate supply chains when switch cost is high and into an integrated network when switch cost is low, (2) a responsive ordering policy adopted by distributors is more conducive to the integrated network configuration than a non-responsive policy, and (3) lack of coordination among retailers in their dynamic choice of procurement strategies hurts not only the overall system performance, but also retailers themselves. More importantly, our study demonstrates the capabilities of agent-based modeling as a methodology for researching complex supply network issues.

The contour argument was introduced by Peierls for two dimensional Ising model. Peierls benefited from the particular symmetries of the Ising model. For non-symmetric models the argument was developed by Pirogov and Sinai. It is very general and rather difficult. Intuitively clear that the Peierls argument does work for any symmetric model. But contours defined in Pirogov–Sinai theory do not work if one wants to use Peierls argument for more general symmetric models. We give a new definition of contour which allows relatively easier proof to the main result of the Pirogov–Sinai theory for symmetric models. Namely, our contours allow us to apply the classical Peierls argument (with contour removal operation).