On properties of topological pressure

We consider a parametric family of continuous maps of a compact metric space which continuously depend on a parameter ranging over a metric space. The topological pressure of maps in any such family is studied as a function of the parameter from the viewpoint of the Baire classification of functions.     

A unified framework for the discretization of nonlinear operator equations

A general theory for the discretization of non-linear operator equations is presented. A given operator with certain continuity and compactness properties is approximated by a sequence of operators acting in different spaces, usually finite dimensional. Connection maps, such as restriction and interpolation, relate the spaces. The abstract convergence theory is formulated in terms of metric spaces. Specializations and applications to differential and integral equations involve normed linear spaces. The case with the same setting for the original and approximate problems was treated in [1]. For typical problems, both types of discretization methods are available. They are related by means of the connection maps.     

Spherical shells as obstructions for the extension of holomorphic mappings

In this paper we study the extension properties of holomorphic and meromorphic maps into complex manifolds that carry a pluriclosed Hermitian metric. For example, any compact, complex surface admits such a metric. We prove that the only obstructions for the Hartogs-type extendability of holomorphic maps are spherical shells and rational curves.     

H?lder metric regularity of set-valued maps

This paper is devoted to metric regularity of set-valued maps from a complete metric space to a Banach space. In particular we extend a known characterization of the regularity modulus to maps defined on reflexive spaces. The higher order metric regularity, i.e. an extension of metric regularity to H?lder context, is also investigated using high order variations of set-valued maps and results of similar nature are obtained for conical metric regularity.     

Study on chaos induced by turbulent maps in noncompact sets

This paper is concerned with chaos induced by strictly turbulent maps in noncompact sets of complete metric spaces. Two criteria of chaos for such types of maps are established, and then a criterion of chaos, characterized by snap-back repellers in complete metric spaces, is obtained. All the maps presented in this paper are proved to be chaotic either in the sense of both Li–Yorke and Wiggins or in the sense of both Li–Yorke and Devaney. The results weaken the assumptions in some existing criteria of chaos. Several illustrative examples are provided with computer simulation.     

Fixed point theorems for a broad class of multimaps

Multimaps from a metric space into the collection of all its nonempty closed subsets are considered and fixed point theorems for such maps have been obtained under new contractive conditions not involving the extended Hausdorff metric.     

A combinatorial approach to cartograms

A homeomorphism from ² to itself distorts metric quantities, such as distance and area. We describe an algorithm that constructs homeomorphisms with prescribed area distortion. Such homeomorphisms can be used to generate cartograms, which are geographic maps purposely distorted so their area distributions reflects a variable different from area, as for example population density. The algorithm generates the homeomorphism through a sequence of local piecewise linear homeomorphic changes. Sample results produced by the preliminary implementation of the method are included.     

度量空间的序列覆盖边界紧映象(英文)

本文主要讨论了度量空间的序列覆盖边界紧映象.用序列商、序列覆盖或1-序列覆盖的纤维边界紧或有限来刻画具有sn网或弱基的空间.主要结果如下:(1)度量空间上的序列覆盖边界紧映射是1-序列覆盖映射;(2)空间X是度量空间的序列商边界紧映象当且仅当X是snf-第一可数空间;(3)空间X是度量空间的序列覆盖边界紧S映象当且仅当X有点可数sn-网.     

Connectedness in Metric Frames

A new metric diameter for a locally connected metric frame is constructed which is finer than the original one. Amongst other results, this construction allows a proof that the category of uniformly locally connected metric frames and uniform frame maps is reflective in the category of locally connected metric frames and uniform frame maps. Mathematics Subject Classification (2000) 54A05.     

Implicit multifunction theorems with positively homogeneous maps

We extend Robinson’s and Ledyaev and Zhu’s implicit multifunction theorems using the language of generalized derivatives with positively homogeneous maps, allowing us to obtain results that more closely resemble the classical (single-valued) implicit function theorem. We highlight that using linear openness instead of metric regularity gives simpler proofs and stronger results. As part of our analysis, we study perturbations of generalized linear openness and metric regularity. Finally, we discuss how our methods may also be adapted to study generalized calmness.     

Integrably asymptotic affine homeomorphisms of the circle and Teichmüller spaces

A quasisymmetric homeomorphism of the unit circle S1 is called integrably asymptotic affine if it admits a quasiconformal extension into the unit disk so that its complex dilatation is square integrable in the Poincar metric on the unit disk. Let QS*(S1) be the space of such maps. Here we give some characterizations and properties of maps in QS(S1). We also show that QS*(S1)/M(O)b(S1) is the completion of Diff(S1)/M(O)b(S1) in the Weil-Petersson metric.     

Parsimonious representation of nonlinear dynamical systems through manifold learning: A chemotaxis case study

Nonlinear manifold learning algorithms, such as diffusion maps, have been fruitfully applied in recent years to the analysis of large and complex data sets. However, such algorithms still encounter challenges when faced with real data. One such challenge is the existence of “repeated eigendirections,” which obscures the detection of the true dimensionality of the underlying manifold and arises when several embedding coordinates parametrize the same direction in the intrinsic geometry of the data set. We propose an algorithm, based on local linear regression, to automatically detect coordinates corresponding to repeated eigendirections. We construct a more parsimonious embedding using only the eigenvectors corresponding to unique eigendirections, and we show that this reduced diffusion maps embedding induces a metric which is equivalent to the standard diffusion distance. We first demonstrate the utility and flexibility of our approach on synthetic data sets. We then apply our algorithm to data collected from a stochastic model of cellular chemotaxis, where our approach for factoring out repeated eigendirections allows us to detect changes in dynamical behavior and the underlying intrinsic system dimensionality directly from data.     

Set-valued dynamic systems and random iterations of set-valued weaker contractions in uniform spaces

Set-valued weaker contractions in uniform, locally convex and metric spaces are defined and dynamic systems of such weaker contractions are studied. Conditions guaranteeing the convergence of generalized sequences of random iterations and iterations and the existence and uniqueness of endpoints of set-valued weaker contractions are established. Our definitions and results are new for set-valued maps in uniform, locally convex and metric spaces and even for single-valued maps.     

On the complexity of expansive measures

We prove that the existence of positively expansive measures for continuous maps on compact metric spaces implies the existence of e 0 and a sequence of(m, e)-separated sets whose cardinalities go to infinite as m →∞. We then prove that maps exhibiting such a constant e and the positively expansive maps share some properties.     

Grain rotations and distortions in the asymptotic variance of vacancy of the Boolean model

We consider the asymptotic variance of vacancy (AVV) in the high intensity small-grain Boolean model. Subjecting the grains to rotations or, more generally, linear distortions gives rise to a function which maps distortion distributions to the AVV of the corresponding Boolean model. We mainly study continuity properties of this function, where we use the L¹ Wasserstein metric on distortion distributions. An important role in the formulation and derivation of our results is played by notions of symmetry commonly used in multivariate analysis and stochastic simulation, such as conjugation-invariance and group models.     

Semigroups of locally injective maps and transfer operators

We consider semigroups of continuous, surjective, locally injective maps of a compact metric space, and whether such semigroups admit a transfer operator.     

On Uniform Lipschitz-Connectedness in Metric Spaces

We show that the category of uniformly Lipschitz-connected metric spaces and Lipschitz maps is coreflective in the category of Lipschitz-connected metric spaces and Lipschitz maps.     

Stable maps of Polish spaces

We define the notions of stable and transquotient maps and study the relation between these classes of maps. The class of stable maps contains all closed and open maps and their compositions. The transquotient maps preserve the property of being a Polish space, and every stable map between separable metric spaces is transquotient.

In particular, a composition of closed and open maps (the intermediary spaces may not be metric) preserves the property of being a Polish space. This generalizes the results of Sierpinski and Vainstein for open and closed maps.

     

Covering maps of spaces of compact subsets

Mappings generated by covering maps of metric spaces are considered in the spaces of compact subsets of metric spaces. Sufficient conditions for the covering property of maps of spaces of compact sets and sufficient conditions for the existence of coincidence sets of maps of metric spaces are obtained.