Narrow Results

The paper investigates two-dimensional recognizable languages that are defined by the so-called "dot systems" that are special subgroups of (ℤ/2ℤ)^ℤ2. The dot shapes that provide directional transitivity or mixing for the related language are investigated. It is shown that languages defined by parallelogram shapes fail to be transitive in the direction of a defining vector and hence fail to be mixing, while certain triangular shapes guarantee that the factor language of the associated dot system will be mixing. Dot systems belong to a class of two-dimensional shift spaces that have a factor language such that every admissable block can be extended to a configuration of the entire plane. For this class of shift spaces we introduce a finite graph (i.e., a finite state automaton) that recognizes two-dimensional local languages, then show that certain transitivity properties may be observed from the structure of the finite graph.

Let ${\mathrm{\text{AAut}}}_{\mathrm{\text{hol}}}(X)$ be the subgroup of the group ${\mathrm{\text{Aut}}}_{\mathrm{\text{hol}}}(X)$ of holomorphic automorphisms of a normal affine algebraic surface $X$ generated by elements of flows associated with complete algebraic vector fields. Our main result is a classification of all normal affine algebraic surfaces $X$ quasi-homogeneous under ${\mathrm{\text{AAut}}}_{\mathrm{\text{hol}}}(X)$ in terms of the dual graphs of the boundaries $\bar{X}\backslash X$ of their SNC-completions $\bar{X}$.

A typical approach to handle the complexity of multi-faceted decision-making problems is to use multi-attribute decision-making (MADM) methods based on pairwise comparisons. Fuzzy set theory has also been employed to cope with the uncertainty and vagueness involved in conducting the comparisons between components of a decision model. An important issue regarding the reliability of the output is the consistency of pairwise comparisons provided by the decision maker(s). Using the MADM method developed by Lu et al. (2007) as a foundation, this paper proposes an algorithm for evaluating the consistency level of pairwise comparison matrices, where linguistic data are used. A crisp numeric scale has been introduced to consider the priority of linguistic data and to avoid the complexity of handling fuzzy calculations in consistency evaluation of pairwise comparison matrices. As an advantage, the proposed method of consistency evaluation is capable of assessing the degree of inconsistency among the pairwise comparisons. Therefore, the acceptance or rejection of the pairwise comparisons can be determined based on the desired degree of tolerance in accepting inconsistent judgments. An application of a revised MADM method is then demonstrated in a case study involving flood mitigation project selection in Australia.

Let $X$ be a compact metric space without isolated points and let $f:X\to X$ be a continuous map. In this paper, if $(X,f)$ is a transitive dynamical system with a repelling periodic point, then $f$ is chaotic in the sense of Kato. In addition, if $f$ is weakly topologically mixing, then $f$ is chaotic in the strong sense of Kato.

The dynamic Parrondo's paradox, which establishes the possibility of combining two chaotic systems to produce a simple one (or vice versa), is analyzed. In particular, we find properties such that the paradox is impossible. Some additional problems are pointed out.

This paper is concerned with Devaney chaos in nonautonomous discrete systems. It is shown that in its definition, the two former conditions, i.e. transitivity and density of periodic points, in a set imply the last one, i.e. sensitivity, in the case that the set is unbounded, while a similar result holds under two additional conditions in the other case that the set is bounded. Some chaotic behavior is studied for a class of nonautonomous discrete systems, each of which is governed by a convergent sequence of continuous maps. In addition, the concepts of some pseudo-orbits and shadowing properties are introduced for nonautonomous discrete systems, and it is shown that some shadowing properties of the system and density of periodic points imply that the system is Devaney chaotic under the condition that the sequence of continuous maps is uniformly convergent in a compact metric space.

In this paper, we introduce the concepts of the large deviations theorem of weaker types, i.e. type I, type I${}^{\prime }$, type II, type II${}^{\prime }$, type III, and type III${}^{\prime }$, and present a systematic study of the ergodic and chaotic properties of dynamical systems satisfying the large deviations theorem of various types. Some characteristics of the ergodic measure are obtained and then applied to prove that every dynamical system satisfying the large deviations theorem of type I${}^{\prime }$ is ergodic, which is equivalent to the large deviations theorem of type II${}^{\prime }$ in this regard, and that every uniquely ergodic dynamical system restricted on its support satisfies the large deviations theorem. Moreover, we prove that every dynamical system satisfying the large deviations theorem of type III is an $E$-system.

In this paper, some characterizations are obtained on the transitivity, mildly mixing property, a-transitivity, equicontinuity, uniform rigidity and proximality of Zadeh’s extensions restricted on some invariant closed subsets of all upper semi-continuous fuzzy sets in the level-wise metric. In particular, it is proved that a dynamical system is weakly mixing (resp., mildly mixing, weakly mixing and a-transitive, equicontinuous, uniformly rigid) if and only if the corresponding Zadeh’s extension is transitive (resp., mildly mixing, a-transitive, equicontinuous, uniformly rigid).

This paper firstly proves that every dynamical system defined on a Hausdorff uniform space with topologically ergodic shadowing is topologically mixing, thus topologically chain mixing. Then, the following is proved: (1) every weakly mixing dynamical system defined on a second countable Baire–Hausdorff uniform space is chaotic in the sense of both Li–Yorke and Auslander–Yorke; (2) every point transitive dynamical system defined on a Hausdorff uniform space is either almost equicontinuous or sensitive.

This paper proves that some shadowing properties are sufficient conditions for being transitive or point-transitive for a nonautonomous discrete dynamical system. Moreover, considering weak mixing property and transitivity via Furstenberg family, this paper reveals the relationship for transitivity and mixing between $p$-periodic systems and their induced autonomous discrete dynamical systems.

In group decision making (GDM) processes, prior to the selection of the best alternative(s), it would be desirable that experts achieve a high degree of consensus or agreement between them. Due to the complexity of most decision making problems, individuals' preferences may not satisfy formal properties. ‘Consistency’ is one of such properties, and it is associated with the transitivity property. Obviously, when carrying out a rational decision making, consistent information, i.e. information which does not imply any kind of contradiction, is more appropriate than information containing some contradictions. Therefore, in a GDM process, consistency should also be sought after. In this paper we present a consensus model for GDM problems that proceeds from consistency to consensus. This model integrates a novel consistency reaching module based on consistency measures. In particular, the model generates advice on how experts should change their preferences in order to increase their consistency. Also, the consensus model is considered adaptive because the search for consensus is adapted to the level of agreement achieved at each consensus round.

Dealing with incomplete information is an important problem in decision making. In this paper, we present a short discussion on this topic and a new estimation method of missing values in an incomplete fuzzy preference relation which is based on the modelling of consistency of preferences via a representable uninorm.

One of the most important issues in the field of fuzzy preference modelling is the construction of a fuzzy strict preference relation and a fuzzy indifference relation from a fuzzy weak preference relation. Here, we focus on a particular class of fuzzy weak preference relations, the so-called fuzzy orderings. The definition of a fuzzy ordering involves a fuzzy equivalence relation and, in this paper, the latter will be considered as the corresponding fuzzy indifference relation. We search for fuzzy strict preference relations compatible with a given fuzzy ordering and its fuzzy indifference relation. In many situations, depending on the t-norm and t-conorm used, this quest results in a unique fuzzy strict preference relation. Our aim is to characterize these fuzzy strict preference relations and to study their transitivity.

In this paper we study interval-valued fuzzy relations. We consider preference relations, i.e. a triplet consisting of strict preference, indifference and incomparability which are defined with the use of a fuzzy negation. We analyze the preservation of the fuzzy negation based reciprocity property of interval-valued fuzzy relations by aggregation functions and by some basic interval-valued fuzzy relations. We use diverse representa-tions of aggregation functions. We also consider the connection between N-reciprocal relations and transitivity properties. We provide a numerical example where the final alternative is chosen with the use of generalized voting method, where admissible linear orders for intervals are applied.

This paper proposes a preference index for ranking the most applicable vague data, i.e. closed intervals and LR-fuzzy numbers. In this approach, first a preference degree is suggested to compare two closed intervals. Then, we extended the proposed criterion from the space of closed intervals into the the space of fuzzy numbers. Using some theorems and lemmas, it will be proved that the extended preference degrees will satisfy many common and desired properties based on the proposed axiomatic definitions introduced for similarity measures. Some numerical examples are also used to illustrate the proposed ranking methods. Moreover, the proposed methods are examined to compare other existing methods and the feasibility and effectiveness of the proposed methods will be cleared via some numerical comparisons. Comparative examples show that the proposed method overcomes certain unreasonable and shortcomings exhibited by some existing methods in some common situations.

The paper considers the two main game-theoretic models, such as coalition and cooperative. The authors are of the opinion that definitions and notions of cooperative games and coalition games are different, but both games are coopetitive games. Transitivity and superadditivity are presented as the main characteristic functions of coopetitive games. The individual and collective rationality were identified as unconditional requirements for the optimal distribution between players. Furthermore, the additional income added to the guaranteed amount occurs in the event of coopetition. Any substantial coopetitive game has an infinite number of transactions. The authors highlighted that the dominant transaction is the transaction that is better for all coalition numbers without exceptions and it can be reached by the coalition. In addition, the authors propose using Shapley system of axioms to identify coopetitive game results.

In this paper, we study $N$-expansive homeomorphisms on surfaces. We prove that when $f$ is a $2$-expansive homeomorphism defined on a compact boundaryless surface $M$ with non-wandering set $\mathrm{\Omega }(f)$ being the whole of $M$ then $f$ is expansive. This condition on the non-wandering set cannot be relaxed: we present an example of a $2$-expansive homeomorphisms on a surface with genus $2$ with wandering points that is not expansive.

This paper presents the first part of a general study of the structure of nine preference-based choice functions introduced by Barrett et al.² More precisely, we show that, as for crisp total pre-orders, first and last alternatives exist in a finite set of alternatives equipped with a strongly complete fuzzy pre-order. We use that result to characterize each of those crisp choice functions for crisp total pre-orders and strongly complete fuzzy pre-orders. We study, by means of those characterizations, the consistency of those preference-based choice functions when preferences are strongly complete fuzzy pre-orders (thereby crisp total pre-orders), that is, we check if each choice function satisfies or violates each of six consistency conditions introduced by Sen.¹¹

This paper is devoted to study some chaotic properties of iterated function systems (IFSs). Specially, a new notion named thick chaotic IFSs is introduced. The relationship between thick chaos and another properties of some notions in dynamical systems are studied.

For a graph $G=(V,E)$, a partition $\pi =(V_1,V_2,\dots ,V_k)$ of the vertex set $V$ is a transitive partition if $V_i$ dominates $V_j$ whenever $i<j$. The transitivity$\mathrm{\text{ Tr}}(G)$ of a graph $G$ is the maximum order of a transitive partition of $V$. For any positive integer $k$, we characterize the smallest tree with transitivity $k$ and obtain an algorithm to determine the transitivity of any tree of finite order.