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Register a new userFinitely Supported Sets Containing Infinite Uniformly Supported Subsets
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Publication date
2019
fields
Informatics Engineering
and research's language is
English
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The theory of finitely supported algebraic structures represents a reformulation of Zermelo-Fraenkel set theory in which every construction is finitely supported according to the action of a group of permutations of some basic elements named atoms. In this paper we study the properties of finitely supported sets that contain infinite uniformly supported subsets, as well as the properties of finitely supported sets that do not contain infinite uniformly supported subsets. For classical atomic sets, we study whether they contain or not infinite uniformly supported subsets.
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Given a compactly supported probability measure on a Riemannian manifold, we study the asymptotic speed at which it can be approximated (in Wasserstein distance of any exponent p) by finitely supported measure. This question has been studied under the names of ``quantization of distributions and, when p=1, ``location problem. When p=2, it is linked with Centroidal Voronoi Tessellations.
We study the orthogonal polynomials associated with the equilibrium measure, in logarithmic potential theory, living on the attractor of an Iterated Function System. We construct sequences of discrete measures, that converge weakly to the equilibrium measure, and we compute their Jacobi matrices via standard procedures, suitably enhanced for the scope. Numerical estimates of the convergence rate to the limit Jacobi matrix are provided, that show stability and efficiency of the whole procedure. As a secondary result, we also compute Jacobi matrices of equilibrium measures on finite sets of intervals, and of balanced measures of Iterated Function Systems. These algorithms can reach large orders: we study the asymptotic behavior of the orthogonal polynomials and we show that they can be used to efficiently compute Greens functions and conformal mappings of interest in constructive function theory.
Altenbernd, Thomas and Wohrle have considered in [ATW02] acceptance of languages of infinite two-dimensional words (infinite pictures) by finite tiling systems, with the usual acceptance conditions, such as the Buchi and Muller ones, firstly used for infinite words. Many classical decision problems are studied in formal language theory and in automata theory and arise now naturally about recognizable languages of infinite pictures. We first review in this paper some recent results of [Fin09b] where we gave the exact degree of numerous undecidable problems for Buchi-recognizable languages of infinite pictures, which are actually located at the first or at the second level of the analytical hierarchy, and highly undecidable. Then we prove here some more (high) undecidability results. We first show that it is $Pi_2^1$-complete to determine whether a given Buchi-recognizable languages of infinite pictures is unambiguous. Then we investigate cardinality problems. Using recent results of [FL09], we prove that it is $D_2(Sigma_1^1)$-complete to determine whether a given Buchi-recognizable language of infinite pictures is countably infinite, and that it is $Sigma_1^1$-complete to determine whether a given Buchi-recognizable language of infinite pictures is uncountable. Next we consider complements of recognizable languages of infinite pictures. Using some results of Set Theory, we show that the cardinality of the complement of a Buchi-recognizable language of infinite pictures may depend on the model of the axiomatic system ZFC. We prove that the problem to determine whether the complement of a given Buchi-recognizable language of infinite pictures is countable (respectively, uncountable) is in the class $Sigma_3^1 setminus (Pi_2^1 cup Sigma_2^1)$ (respectively, in the class $Pi_3^1 setminus (Pi_2^1 cup Sigma_2^1)$).
In set theory without the Axiom of Choice (AC), we observe new relations of the following statements with weak choice principles. 1. Every locally finite connected graph has a maximal independent set. 2. Every locally countable connected graph has a maximal independent set. 3. If in a partially ordered set all antichains are finite and all chains have size $aleph_{alpha}$, then the set has size $aleph_{alpha}$ if $aleph_{alpha}$ is regular. 4. Every partially ordered set has a cofinal well-founded subset. 5. If $G=(V_{G},E_{G})$ is a connected locally finite chordal graph, then there is an ordering $<$ of $V_{G}$ such that ${w < v : {w,v} in E_{G}}$ is a clique for each $vin V_{G}$.
Insights to the mechanism of CdSe nanoparticle attachment to carbon nanotubes following the hot injection method are discussed. It was observed that the presence of water improves the nanotube coverage while Cl containing media are responsible for the shape transformation of the nanoparticles and further attachment to the carbon lattice. The experiments also show that the mechanism taking place involves the right balance of several factors, namely, low passivated nanoparticle surface, particles with well-defined crystallographic facets, and interaction with an organics-free sp2 carbon lattice. Furthermore, this procedure can be extended to cover graphene by quantum dots.
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