Some simple conditions implying topological transitivity for interval maps

Summary. We provide some sufficient conditions for topological transitivity of piecewise monotonic maps on [0,1]. Our theorems provide shorter and elementary proofs for some known recent results.     

Some reflections on mathematicians’ views of quantization

We start with a short presentation of the difference in attitude between mathematicians and physicists even in their treatment of physical reality, and look at the paradigm of quantization as an illustration. In particular, we stress the differences in motivation and realization between the Berezin and deformation quantization approaches, exposing briefly Berezin’s view of quantization as a functor. We continue with a schematic overview of deformation quantization and of its developments in contrast with the latter and discuss related issues, in particular, the spectrality question. We end by a very short survey of two main avatars of deformation quantization, quantum groups and quantum spaces (especially noncommutative geometry) presented in that perspective. Bibliography: 74 titles. This paper is dedicated to the memory of “Alik” Berezin Published in Zapiski Nauchnykh Seminarov POMI, Vol. 331, 2006, pp. 199–220.     

On the transitivity property

In this short note general transitivity properties of fuzzy binary relations are discussed. It is shown that under some weak conditions any fuzzy binary relation can be regarded as a transitive one with respect to a properly chosen t-norm. A representation theorem for such relations is formulated.     

Some comments on preference order dynamic programming models

A simple deterministic dynamic programming model is used as a general framework for the analysis of stochastic versions of three classical optimization problems: knapsack, traveling salesperson, and assembly line balancing problems. It is shown that this model can provide an alternative to the preference order models proposed for these problems. Counterexample to the optimality of the preference order models are presented.     

Investing in the stock market: Statistical pooling of individual preference judgments

Literature concerning the quality of individual and face-to-face group judgments has generally concluded that both groups and statistically pooled individuals outperform randomly chosen or average individuals. This paper extends previous research by comparing statistically pooled individual judgments of both individuals and face-to-face groups in a stock selection task. In general, decisions that would have resulted from statistically pooled judgments were better (as assessed by future stock value) than those that would have resulted from individual or face-to-face group judgments. In choosing among pooling methods, majority rule is often thought to be a very compelling criterion. However, majority rule can produce intransitive group preferences. Methods that use some procedure to resolve the intransitivities of majority rule did not perform well relative to other non-majority rule based methods. Another class of pooling methods, termed equity methods, used in conjunction with ordinal judgments, are recommended based on simplicity, performance, and fairness criteria. The results are discussed in terms of the nature of the task.     

Scale transitivity in the AHP

Transitivity is important in multicriteria decision-making. The analytic hierarchy process (AHP), as one of the widely used decision analysis tools, is criticized since it suffers from scale intransitivity. This paper first reviews and compares different scales from different aspects, then discusses the transitivity of AHP scales and derives a scale based on the transitivity, so it is naturally transitive. Besides, two approaches are provided to determine the scale parameter for the derived transitive scale. In order to deal with the transitivity problem, the AHP provides a consistency index for testing pairwise comparison consistency. So, finally, this paper proposes a consistency measure to reflect the judgmental inconsistency.     

Some personal views on the current state and the future of locational analysis

In this paper a group of participants of the 12th European Summer Institute which took place in Tenerife, Spain in June 1995 present their views on the state of the art and the future trends in Locational Analysis. The issues discussed include modeling aspects in discrete Location Theory, the influence of the distance function, the relation between discrete, network and continuous location, heuristic techniques, the state of technology and undesirable facility location. Some general questions are stated regarding the applicability of location models, promising research directions and the way technology affects the development of solution techniques.     

A study on the transitivity of probabilistic and fuzzy relations

Given a set of alternatives we consider a fuzzy relation and a probabilistic relation defined on such a set. We investigate the relation between the T-transitivity of the fuzzy relation and the cycle-transitivity of the associated probabilistic relation. We provide a general result, valid for any t-norm and we later provide explicit expressions for important particular cases. We also apply the results obtained to explore the transitivity satisfied by the probabilistic relation defined on a set of random variables. We focus on uniform continuous random variables.     

On the transitivity of regular differential forms

IfR/k resp.S/R are genetically smooth algebras of finite type, equidimensional of relative dimensiond resp.n, then the canonical isomorphism of meromorphic differential forms induces a homomorphism _R/k ^d _S/R ⁿ _S/k ^d+n of regular differential forms, which is an isomorphism ifR/k andS/R are Cohen-Macaulay or ifR/k is Gorenstein.     

Topological transitivity of cylindrical cascades

The existence is proved of a topologically transitive (t.t.) homeomorphism U of the space W = × Z of the formU (, z)=(T,, z+f ()) ( , z Z), where is a complete separable metric space, T is a t.t. homeomorphism of onto itself, Z is a separable banach space, andf is a continuous map: z. For the special case W = S¹×R, T=+ ( is incommensurable with 2) the existence is proved of t.t. homeomorphisms (1) of two types: 1) with zero measure of the set of transitive points, 2) with zero measure of the set of intransitive points. An example is presented of a continuous functionf: S¹R for which the corresponding homeomorphism (1) is t.t. for all incommensurable with 2.Translated from Matematicheskie Zametki, Vol. 14, No. 3, pp. 441–452, September, 1973.The author thanks D. V. Anosov for advice and interest in the work.     

A note on shadowing with chain transitivity

Let f : X → X be a continuous map of a compact metric space X. The map f induces in a natural way a map f_M on the space M(X) of probability measures on X, and a transformation f_K on the space K(X) of closed subsets of X. In this paper, we show that if (X, f) is a chain transitive system with shadowing property, then exactly one of the following two statements holds:

(a)

fⁿ and (f_K)ⁿ are syndetically sensitive for all n ? 1.

(b)

fⁿ and (f_K)ⁿ are equicontinuous for all n ? 1.

In particular, we show that for a continuous map f : X → X of a compact metric space X with infinite elements, if f is a chain transitive map with the shadowing property, then fⁿ and (f_K)ⁿ are syndetically sensitive for all n ? 1. Also, we show that if f_M (resp. f_K) is chain transitive and syndetically sensitive, and f_M (resp. f_K) has the shadowing property, then f is sensitive.In addition, we introduce the notion of ergodical sensitivity and present a sufficient condition for a chain transitive system (X, f) (resp. (M(X), f_M)) to be ergodically sensitive. As an application, we show that for a L-hyperbolic homeomorphism f of a compact metric space X, if f has the AASP, then fⁿ is syndetically sensitive and multi-sensitive for all n ? 1.     

Full transitivity of Abelian groups

We study fully transitive Abelian groups and their direct sums and direct products. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 3, pp. 107–140, 2007.     

The transitivity of induced maps

For a metric continuum X, we consider the hyperspaces X² and C(X) of the closed and nonempty subsets of X and of subcontinua of X, respectively, both with the Hausdorff metric. For a given map we investigate the transitivity of the induced maps and . Among other results, we show that if X is a dendrite or a continuum of type λ and is a map, then C(f) is not transitive. However, if X is the Hilbert cube, then there exists a transitive map such that f² and C(f) are transitive.     

Neighbour transitivity on codes in Hamming graphs

We consider a code to be a subset of the vertex set of a Hamming graph. In this setting a neighbour of the code is a vertex which differs in exactly one entry from some codeword. This paper examines codes with the property that some group of automorphisms acts transitively on the set of neighbours of the code. We call these codes neighbour transitive. We obtain sufficient conditions for a neighbour transitive group to fix the code setwise. Moreover, we construct an infinite family of neighbour transitive codes, with minimum distance δ = 4, where this is not the case. That is to say, knowledge of even the complete set of code neighbours does not determine the code.