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1.
We investigate the dynamical properties of a skew product transformation Tφ on [0,1)×G defined by Tφ(x,g)=(Tx,gφ(x)) where T is the β-transformation for β?2 and φ(x) is a compact group G-valued step function with a finite number of discontinuities. We give several sufficient conditions for ergodicity and strong mixing of Tφ. As an application, we describe a class of step functions which satisfy the Central Limit Theorem for the β-transformations. As another application, we also consider a class of skew product transformations Tβ,a,w on [0,1)×[0,1) which maps where a,wR and give necessary and sufficient conditions for ergodicity and strong mixing.  相似文献   

2.
Given a real number β>1, a permutation π of length n is realized by the β-shift if there is some x∈[0,1] such that the relative order of the sequence x,f(x),…,fn−1(x), where f(x) is the fractional part of βx, is the same as that of the entries of π. Widely studied from such diverse fields as number theory and automata theory, β-shifts are prototypical examples of one-dimensional chaotic dynamical systems. When β is an integer, permutations realized by shifts were studied in Elizalde (2009) [5]. In this paper we generalize some of the results to arbitrary β-shifts. We describe a method to compute, for any given permutation π, the smallest β such that π is realized by the β-shift. We also give a way to determine the length of the shortest forbidden (i.e., not realized) pattern of an arbitrary β-shift.  相似文献   

3.
An α=(α1,…,αk)(0?αi?1) section of a family {K1,…,Kk} of convex bodies in Rd is a transversal halfspace H+ for which Vold(KiH+)=αi⋅Vold(Ki) for every 1?i?k. Our main result is that for any well-separated family of strictly convex sets, the space of α-sections is diffeomorphic to Sdk.  相似文献   

4.
For a given prime p, by studying p  -dissection identities for Ramanujan?s theta functions ψ(q)ψ(q) and f(−q)f(q), we derive infinite families of congruences modulo 2 for some ?  -regular partition functions, where ?=2,4,5,8,13,16?=2,4,5,8,13,16.  相似文献   

5.
6.
Let κQnt be the category of κ-quantales, quantales closed under κ-joins in which the monoid identity is the largest element. (κ is an infinite regular cardinal.) Although the lack of lattice completeness in this setting would seem to mitigate against the techniques which lend themselves so readily to the calculation of frame quotients, we show how to easily compute κQnt quotients by applying generalizations of the frame techniques to suitable extensions of this category.The second major tool in the analysis is the free κ-quantale over a λ-quantale, κ?λ. Surprisingly, these can be characterized intrinsically, and the generating sub-κ-quantale can even be identified. The result that the λ-free κ-quantales coincide with the λ-coherent κ-quantales directly generalizes Madden?s corresponding result for κ-frames.These tools permit a direct and intuitive construction of κQnt colimits. We provide two applications: an intrinsic characterization of κQnt colimits, and of free (over sets) κ-quantales. The latter is a direct generalization of Whitman?s condition for distributive lattices.  相似文献   

7.
8.
This work is concerned with dynamical systems in presence of symmetries and reversing symmetries. We describe a construction process of subspaces that are invariant by linear Γ-reversible-equivariant mappings, where Γ is the compact Lie group of all the symmetries and reversing symmetries of such systems. These subspaces are the σ-isotypic components, first introduced by Lamb and Roberts in (1999) [10] and that correspond to the isotypic components for purely equivariant systems. In addition, by representation theory methods derived from the topological structure of the group Γ, two algebraic formulae are established for the computation of the σ-index of a closed subgroup of Γ. The results obtained here are to be applied to general reversible-equivariant systems, but are of particular interest for the more subtle of the two possible cases, namely the non-self-dual case. Some examples are presented.  相似文献   

9.
Given an arbitrarily weak notion of left-〈f〉f-porosity and an arbitrarily strong notion of right-〈g〉g-porosity, we construct an example of closed subset of RR which is not σ  -left-〈f〉f-porous and is right-〈g〉g-porous. We also briefly summarize the relations between three different definitions of porosity controlled by a function; we then observe that our construction gives the example for any combination of these definitions of left-porosity and right-porosity.  相似文献   

10.
11.
We prove the following theorems:1. There exists an -covering with the property s 0.2. Under cov there exists X such that is not an -covering orX \ B is not an -covering].3. Also we characterize the property of being an -covering.  相似文献   

12.
In this paper we consider spaces of sequences which are valued in a topological space E and study generalized backward shifts associated to certain selfmappings of E. We characterize their universality in terms of dynamical properties of the underlying selfmappings. Applications to hypercyclicity theory are given. In particular, Rolewicz's theorem on hypercyclicity of scalar multiples of the classical backward shift is extended.  相似文献   

13.
Assume that XR?Q, and each clopen-valued lower semicontinuous multivalued map has a continuous selection . Our main result is that in this case, X is a σ-space. We also derive a partial converse implication, and present a reformulation of the Scheepers Conjecture in the language of continuous selections.  相似文献   

14.
We consider which ordinals, with the order topology, can be Stone-?ech remainders of which spaces of the form ψ(κ,M), where ω?κ is a cardinal number and Mω[κ] is a maximal almost disjoint family of countable subsets of κ (MADF). The cardinality of the continuum, denoted c, and its successor cardinal, c+, play important roles. We show that if κ>c+, then no ψ(κ,M) has any ordinal as a Stone-?ech remainder. If κ?c then for every ordinal δ<κ+ there exists Mδω[κ], a MADF, such that βψ(κ,Mδ)?ψ(κ,Mδ) is homeomorphic to δ+1. For κ=c+, βψ(κ,Mδ)?ψ(κ,Mδ) is homeomorphic to δ+1 if and only if c+?δ<c+ω.  相似文献   

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16.

Text

In this article we derive some new identities concerning π, algebraic radicals and some special occurrences of the Gauss hypergeometric function 2F1 in the analytic continuation. All of them have been derived by tackling some elliptic or hyperelliptic known integral, and looking for another representation of it by means of hypergeometric functions like those of Gauss, Appell or Lauricella. In any case we have focused on integrand functions having at least one couple of complex-conjugate roots. Founding upon a special hyperelliptic reduction formula due to Hermite (1876) [6], π is obtained as a ratio of a complete elliptic integral and the four-variable Lauricella function. Furthermore, starting with a certain binomial integral, we succeed in providing as a ratio of a linear combination of complete elliptic integrals of the first and second kinds to the Appell hypergeometric function of two complex-conjugate arguments. Each of the formulae we found theoretically has been satisfactorily tested by means of Mathematica®.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=rQqtVtAf-RQ.  相似文献   

17.
Let S be an infinite discrete semigroup with a point s such that stt for any tS. We prove that every closed left ideal of βS, where φ is a filter with a countable basis, is determined by a function on S which is slowly oscillating in the direction φ.  相似文献   

18.
19.
In this paper by a spectrum of mappings we mean a morphism of spectra of spaces. However, using the notion of a mapping of mappings, we give the definition of a spectrum of mappings similar to that of a spectrum of spaces. In this case, the formulations of the given results are also similar to the formulations of the corresponding results concerning the spectra of spaces.For the spectra of mappings we define the notion of a τ-spectrum of mappings factorizing in a special sense and prove a version of the Spectral Theorem for such spectra. Furthermore, to a given indexed collection F of mapping we associate a τ-spectrum factorizing in the above special sense whose mappings are Containing Mappings for F constructed in Iliadis (2005) [4]. These associated τ-spectra and the corresponding version of the Spectral Theorem imply that for a given indexed collection F of mappings any so-called “natural” τ-spectrum for F factorizing in the special sense contains a cofinal and τ-closed subspectrum whose mappings are Containing Mapping for F. Thus, Containing Mappigs for F appear here without any concrete construction. The associated τ-spectra are used also in order to define and characterize the so-called second-type saturated classes of mappings (which are “saturated” by universal elements).  相似文献   

20.
According to Mack a space is countably paracompact if and only if its product with [0,1] is δ-normal, i.e. any two disjoint closed sets, one of which is a regular Gδ-set, can be separated. In studying monotone versions of countable paracompactness, one is naturally led to consider various monotone versions of δ-normality. Such properties are the subject of this paper. We look at how these properties relate to each other and prove a number of results about them, in particular, we provide a factorization of monotone normality in terms of monotone δ-normality and a weak property that holds in monotonically normal spaces and in first countable Tychonoff spaces. We also discuss the productivity of these properties with a compact metrizable space.  相似文献   

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