Search results for "Countable set"
showing 10 items of 56 documents
Quantum, stochastic, and pseudo stochastic languages with few states
2014
Stochastic languages are the languages recognized by probabilistic finite automata (PFAs) with cutpoint over the field of real numbers. More general computational models over the same field such as generalized finite automata (GFAs) and quantum finite automata (QFAs) define the same class. In 1963, Rabin proved the set of stochastic languages to be uncountable presenting a single 2-state PFA over the binary alphabet recognizing uncountably many languages depending on the cutpoint. In this paper, we show the same result for unary stochastic languages. Namely, we exhibit a 2-state unary GFA, a 2-state unary QFA, and a family of 3-state unary PFAs recognizing uncountably many languages; all th…
Generalized countable iterated function systems
2011
One of the most common and most general way to generate fractals is by using iterated function systems which consists of a finite or infinitely many maps. Generalized countable iterated function systems (GCIFS) are a generalization of countable iterated function systems by considering contractions from X ? X into X instead of contractions on the metric space X to itself, where (X, d) is a compact metric space. If all contractions of a GCIFS are Lipschitz with respect to a parameter and the supremum of the Lipschitz constants is finite, then the associated attractor depends continuously on the respective parameter.
Sets versus trial sequences, Hausdorff versus von Mises: “Pure” mathematics prevails in the foundations of probability around 1920
2010
Abstract The paper discusses the tension which occurred between the notions of set (with measure) and (trial-) sequence (or—to a certain degree—between nondenumerable and denumerable sets) when used in the foundations of probability theory around 1920. The main mathematical point was the logical need for measures in order to describe general nondiscrete distributions, which had been tentatively introduced before (1919) based on von Mises’s notion of the “Kollektiv.” In the background there was a tension between the standpoints of pure mathematics and “real world probability” (in the words of J.L. Doob) at the time. The discussion and publication in English translation (in Appendix ) of two …
Bases and quasi-reflexivity in Fréchet spaces
2005
A Frechet space E is quasi-reflexive if, either dim(E″/E) < ∞, or E″[β(E″,E′)]/E is isomorphic to ω. A Frechet space E is totally quasi-reflexive if every separated quotient is quasi-reflexive. In this paper we show, using Schauder bases, that E is totally quasi-reflexive if and only if it is isomorphic to a closed subspace of a countable product of quasi-reflexive Banach spaces. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
Topological Linearly Ordered Spaces Determined by Pervin's Quasi-uniformity
1994
: In this paper we study those uniform linearly ordered spaces that are determined by Pervin's quasi-uniformity. In particular, we show that the space of all countable ordinals is determined by a unique quasi-uniformity.
Spiking patterns emerging from wave instabilities in a one-dimensional neural lattice.
2003
The dynamics of a one-dimensional lattice (chain) of electrically coupled neurons modeled by the FitzHugh-Nagumo excitable system with modified nonlinearity is investigated. We have found that for certain conditions the lattice exhibits a countable set of pulselike wave solutions. The analysis of homoclinic and heteroclinic bifurcations is given. Corresponding bifurcation sets have the shapes of spirals twisting to the same center. The appearance of chaotic spiking patterns emerging from wave instabilities is discussed.
Existentially closed locally cofinite groups
1992
Let be a class of finite groups. Then a c-group shall be a topological group which has a fundamental system of open neighbourhoods of the identity consisting of normal subgroups with -factor groups and trivial intersection. In this note we study groups which are existentially closed (e.c.) with respect to the class Lc of all direct limits of c-groups (where satisfies certain closure properties). We show that the so-called locally closed normal subgroups of an e.c. Lc-group are totally ordered via inclusion. Moreover it turns out that every ∀2-sentence, which is true for countable e.c. L-groups, also holds for e.c. Lc-groups. This allows it to transfer many known properties from e.c. L-group…
Affine Surfaces With a Huge Group of Automorphisms
2013
We describe a family of rational affine surfaces S with huge groups of automorphisms in the following sense: the normal subgroup Aut(S)alg of Aut(S) generated by all algebraic subgroups of Aut(S) is not generated by any countable family of such subgroups, and the quotient Aut(S)/Aut(S)alg cointains a free group over an uncountable set of generators.
Strictly convex corners scatter
2017
We prove the absence of non-scattering energies for potentials in the plane having a corner of angle smaller than $\pi$. This extends the earlier result of Bl{\aa}sten, P\"aiv\"arinta and Sylvester who considered rectangular corners. In three dimensions, we prove a similar result for any potential with a circular conic corner whose opening angle is outside a countable subset of $(0,\pi)$.
On Compacta K for Which C(K) Has Some Good Renorming Properties
2019
By renorming it is usually meant obtaining equivalent norms in a Banach space with better properties, like being local uniformly rotund (LUR) or Kadets. In these notes we are concerned with C(K) spaces and pointwise lower semicontinuous Kadets or LUR renormings on them. If a C(K) space admits some of such equivalent norms then this space, endowed with the pointwise topology, has a countable cover by sets of small local norm-diameter (SLD). This property may be considered as the topological baseline for the existence of a pointwise lower semicontinuous Kadets, or even LUR renorming, since in many concrete cases it is the first step to obtain such a norm. In these notes we survey some methods…