On Some Types of Rigid Sets in Banach Spaces

One of the properties characterizing Euclidean spaces says - roughly speaking- that their unit sphere has nice invariant properties. More precisely, a finite dimensional normed space has an Euclidean norm if and only if the group of isometries acts transitively on its unit sphere (the norm is “transitive”); such property of the sphere is also called “rigidity”. More recently, another notion of “rigidity” for compact sets, connected with “isometric sequences”, received some attention. Infinite rigid sets are diametral; moreover, under suitable assumptions on the space, they are also contained in the boundary of a sphere. These notions are connected with many problems, in different areas. Here we discuss and compare these two notions of rigid set, trying to indicate new relations among them and with some other properties of sets. Several examples complete the paper.     

Diversity in inside factorial monoids

In a recent paper (Diversity in Monoids, Czech. Math. J. 62 (2012), 795–809), the last two authors introduced and developed the monoid invariant “diversity” and related properties “homogeneity” and “strong homogeneity”. We investigate these properties within the context of inside factorial monoids, in which the diversity of an element counts the number of its different almost primary components. Inside factorial monoids are characterized via diversity and strong homogeneity. A new invariant complementary to diversity, height, is introduced. These two invariants are connected with the well-known invariant of elasticity.     

Large Cardinals and Ramifiability for Directed Sets

The notion of “ramifiability” (or “tree‐property”), usually applied to cardinals, can be extended to directed sets and is put in relation here with familiar “large cardinal” properties.     

关于自相似集的一个维数定理

本文对严格自相似集，提出了一个比“开集”条件更弱的“可解”条件，并且证明：在可解条件下，自相似集的Ｈａｕｓｄｏｒｆｆ维数及Ｂｏｕｌｉｇａｎｄ维数与其相似维数一致．     

Functions and Sets of Smooth Substructure: Relationships and Examples

The past decade has seen the introduction of a number of classes of nonsmooth functions possessing smooth substructure, e.g., “amenable functions”, “partly smooth functions”, and “g ° F decomposable functions”. Along with these classes a number of structural properties have been proposed, e.g., “identifiable surfaces”, “fast tracks”, and “primal-dual gradient structures”. In this paper we examine the relationships between these various classes of functions and their smooth substructures. In the convex case we show that the definitions of identifiable surfaces, fast tracks, and partly smooth functions are equivalent. In the nonconvex case we discuss when a primal-dual gradient structure or g ° F decomposition implies the function is partly smooth, and vice versa. We further provide examples to show these classes are not equal.     

Extending Independent Sets to Bases and the Axiom of Choice

We show that the both assertions “in every vector space B over a finite element field every subspace V ? B has a complementary subspace S” and “for every family ?? of disjoint odd sized sets there exists a subfamily ?={F_j:j ?ω} with a choice function” together imply the axiom of choice AC. We also show that AC is equivalent to the statement “in every vector space over ? every generating set includes a basis”.     

On Sets of Generators of Fuchsian Groups

We give applications of the discontinuity function of a discrete group for Fuchsian groups which act on the unit disk and which are finitely generated. We obtain two sets of theorems: One set corresponds to the Euclidean metric. The second set corresponds to the hyperbolic metric. These theorems state inequalities that involve combinatorial quantities (such as counting functions of the elements of the group with a given bounded length, or the order of growth of the group) and geometric quantities (such as distances of the images of a point under a fixed set of generators to the unit circle, or hyperbolic areas of certain disks). This paper is a sequel to the paper “The Discontinuity Function of Discrete Groups and Radius of Schlichtness” by the author, that recently appeared in this journal.     

Homology Groups of Cubical Sets with Connections

Applied Categorical Structures - Toward defining commutative cubes in all dimensions, Brown and Spencer introduced the notion of “connection” as a new kind of degeneracy. In this paper,...     

A Characterization of Ordered Sets and Lattices via Betweenness Relations

For a set X with at least 3 elements, we establish a canonical one to one correspondence between all betweenness relations satisfying certain axioms and all pairs of inverse orderings “<” and “>” defined on X for which the corresponding Hasse diagram is connected and all maximal chains contain at least 3 elements. For an ordering “<”, the corresponding betweenness relation B is given by $$B=\{(x,y,z)\in X^3\mid x<y<z {\rm \ or\ }z<y<x\}.$$ Moreover, by adding one more axiom, we obtain also a one to one correspondence between all pairs of dual lattices and all betweenness relations.     

Grasping Infinity by Finite Sets

We show that the existence of an infinite set can be reduced to the existence of finite sets “as big as we will”, provided that a multivalued extension of the relation of equipotence is admitted. In accordance, we modelize the notion of infinite set by a fuzzy subset representing the class of (finite) wide sets.     

Semipolar Sets and Intrinsic Hausdorff Measure

Potential Analysis - Given a “Green function” G on a locally compact space X with countable base, a Borel set A in X is called G-semipolar, if there is no measure ν ≠ 0...     

Configurations of singular fibres on rational elliptic surfaces in characteristic two

Antimatter domains are defined to be the integral domains which do not have any atoms. It is proved that each integral domain can be em-bedded as a subring of some antimatter domain which is not a field. Any fragmented domain is an antimatter domain, but the converse fails in each positive Krull dimension. A detailed study is made of the passage of the“an-timatter”property between the partners within an overring extension. Special attention is given to characterizing antimatter domains in classes of valuation domains, pseudo-valuation domains, and various types of pullbacks.     

Many-Sheeted Versions of the Pólya–Bernstein and Borel Theorems for Entire Functions of Order ρ ≠ 1 and Their Applications

The Puiseux series generated by the power function z = w^1/ρ, where ρ > 0,ρ ≠ 1, is considered. A version of the Pólya–Bernstein theorem for an entire function of order ρ ≠ 1 and normal type is proposed and applied to describe the domain of analytic continuation of this series. The domain of summability of a “regular” Puiseux series is found (this is a many-sheeted “Borel polygon”); in the case ρ = 1, the “one-sheeted” result of Borel is substantially extended. These results make it possible to describe domains of analytic continuation of the Puiseux expansions of popular many-sheeted functions (such as inverses of rational functions).     

On a Class of Recursively Enumerable Sets

We define a class of so-called ∑(n)-sets as a natural closure of recursively enumerable sets W_n under the relation “∈” and study its properties.     

On the Existence of Smooth Solutions for Fully Nonlinear Parabolic Equations with Measurable “Coefficients” without Convexity Assumptions

We show that for any uniformly parabolic fully nonlinear second-order equation with bounded measurable “coefficients” and bounded “free” term in any cylindrical smooth domain with smooth boundary data one can find an approximating equation which has a unique continuous solution with the first derivatives bounded and the second spacial derivatives locally bounded. The approximating equation is constructed in such a way that it modifies the original one only for large values of the unknown function and its spacial derivatives.     

Closing Aubry Sets I

Given a Tonelli Hamiltonian H:T*M → ? of class C^k, with k ≥ 2, we prove the following results: (1) Assume there exist a recurrent point of the projected Aubry set and a critical viscosity subsolution u such that u is a C¹ critical solution in an open neighborhood of the positive orbit of . Suppose further that u is “C² at ”. Then there exists a C^k potential V : M → ?, small in C²‐topology, for which the Aubry set of the new Hamiltonian H + V is either an equilibrium point or a periodic orbit. (2) If M is two dimensional, (1) holds replacing “C¹ critical solution + C² at ” by “CM³ critical subsolution”. These results can be considered as a first step through the attempt of proving the Mañé's conjecture in C²‐topology. In a second paper [27], we will generalize (2) to arbitrary dimension. Moreover, such an extension will need the introduction of some new techniques, which will allow us to prove in [27] the Mañé's density conjecture in C¹‐topology. Our proofs are based on a combination of techniques coming from finite‐dimensional control theory and Hamilton‐Jacobi theory, together with some of the ideas that were used to prove C¹‐closing lemmas for dynamical systems.© 2014 Wiley Periodicals, Inc.     

Sets of periods of dynamical systems

In this article, we present a coherent, though not exhaustive, account of some well-known and some recent results of many mathematicians (including our own) on the following question: Given a “nice” class of dynamical systems, which subsets of ? arise as the sets of periods of members of that class? While stating and explaining some elegant answers, proofs have been outlined or indicated occasionally.     

A Poisson random field model for intermittent phenomena with application to laminar-turbulent transition and material microstructure

The alternation of a physical system between two phases or states is referred to as intermittency. Examples of intermittent phenomena abound in applications and include the transition from laminar to turbulent flow over a flight vehicle and the presence of imperfections within material microstructure. It is shown that intermittent phenomena of this type can be modeled by two-state random fields with piecewise constant samples; we refer to the states of the random field as “off” and “on” or, equivalently, 0 and 1. These random fields can be calibrated to the available information, which consists of: (1) the marginal probability that the state of the system is “on”; and (2) the average number of fluctuations between states that occur within a bounded region. The proposed model is defined by a sequence of pulses of prescribed shape and unit magnitude, located at random (Poisson) points within a bounded domain. Properties of the model are discussed, and simple algorithms to generate samples of the random field are provided. Various applications are considered, including voids within material microstructure and the random vibration of a flight vehicle subjected to a transition from laminar to turbulent flow over its surface.     

Dual subgradient algorithms for large-scale nonsmooth learning problems

“Classical” First Order (FO) algorithms of convex optimization, such as Mirror Descent algorithm or Nesterov’s optimal algorithm of smooth convex optimization, are well known to have optimal (theoretical) complexity estimates which do not depend on the problem dimension. However, to attain the optimality, the domain of the problem should admit a “good proximal setup”. The latter essentially means that (1) the problem domain should satisfy certain geometric conditions of “favorable geometry”, and (2) the practical use of these methods is conditioned by our ability to compute at a moderate cost proximal transformation at each iteration. More often than not these two conditions are satisfied in optimization problems arising in computational learning, what explains why proximal type FO methods recently became methods of choice when solving various learning problems. Yet, they meet their limits in several important problems such as multi-task learning with large number of tasks, where the problem domain does not exhibit favorable geometry, and learning and matrix completion problems with nuclear norm constraint, when the numerical cost of computing proximal transformation becomes prohibitive in large-scale problems. We propose a novel approach to solving nonsmooth optimization problems arising in learning applications where Fenchel-type representation of the objective function is available. The approach is based on applying FO algorithms to the dual problem and using the accuracy certificates supplied by the method to recover the primal solution. While suboptimal in terms of accuracy guaranties, the proposed approach does not rely upon “good proximal setup” for the primal problem but requires the problem domain to admit a Linear Optimization oracle—the ability to efficiently maximize a linear form on the domain of the primal problem.