910 research outputs found

    Classification of multiplicity free quasi-Hamiltonian manifolds

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    A quasi-Hamiltonian manifold is called multiplicity free if all of its symplectic reductions are 0-dimensional. In this paper, we classify compact, multiplicity free, twisted quasi-Hamiltonian manifolds for simply connected, compact Lie groups. Thereby, we recover old and find new examples of these structures.Comment: v1: 35 pages, this is a complete revision of arxiv:1612.03843. Since some omitted parts have already been cited, I opted for a new submission under a new title. v2: 39 pages, revised according to the advice of a very helpful refere

    On Ramsey-type properties of the distance in nonseparable spheres

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    Given an uncountable subset Y\mathcal Y of a nonseparable Banach space, is there an uncountable Z⊆Y\mathcal Z\subseteq \mathcal Y such that the distances between any two distinct points of Z\mathcal Z are more or less the same? If an uncountable subset Y\mathcal Y of a nonseparable Banach space does not admit an uncountable Z⊆Y\mathcal Z\subseteq \mathcal Y, where any two points are distant by more than r>0r>0, is it because Y\mathcal Y is the countable union of sets of diameters not bigger than rr? We investigate connections between the set-theoretic phenomena involved and the geometric properties of uncountable subsets of nonseparable Banach spaces of densities up to 2ω2^\omega related to uncountable (1+)(1+)-separated sets, equilateral sets or Auerbach systems. The results include geometric dichotomies for a wide range of classes of Banach spaces, some in ZFC, some under the assumption of OCA+MA and some under a hypothesis on the descriptive complexity of the space as well as constructions (in ZFC or under CH) of Banach spaces where the geometry of the unit sphere displays anti-Ramsey properties. This complements classical theorems for separable spheres and the recent results of H\'ajek, Kania, Russo for densities above 2ω2^\omega as well as offers a synthesis of possible phenomena and categorization of examples for uncountable densities up to 2ω2^\omega obtained previousy by the author and Guzm\'an, Hru\v{s}\'ak, Ryduchowski and Wark. It remains open if the dichotomies may consistently hold for all Banach spaces of the first uncountable density or if the strong anti-Ramsey properties of the distance on the unit sphere of a Banach space can be obtained in ZFC

    Global universal approximation of functional input maps on weighted spaces

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    We introduce so-called functional input neural networks defined on a possibly infinite dimensional weighted space with values also in a possibly infinite dimensional output space. To this end, we use an additive family as hidden layer maps and a non-linear activation function applied to each hidden layer. Relying on Stone-Weierstrass theorems on weighted spaces, we can prove a global universal approximation result for generalizations of continuous functions going beyond the usual approximation on compact sets. This then applies in particular to approximation of (non-anticipative) path space functionals via functional input neural networks. As a further application of the weighted Stone-Weierstrass theorem we prove a global universal approximation result for linear functions of the signature. We also introduce the viewpoint of Gaussian process regression in this setting and show that the reproducing kernel Hilbert space of the signature kernels are Cameron-Martin spaces of certain Gaussian processes. This paves the way towards uncertainty quantification for signature kernel regression.Comment: 57 pages, 4 figure

    C∗C^{\ast}-algebraic approach to the principal symbol. III

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    We treat the notion of principal symbol mapping on a compact smooth manifold as a ∗\ast-homomorphism of C∗C^{\ast}-algebras. Principal symbol mapping is built from the ground, without referring to the pseudodifferential calculus on the manifold. Our concrete approach allows us to extend Connes Trace Theorem for compact Riemannian manifolds

    Generalised subbundles and distributions: A comprehensive review

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    Distributions, i.e., subsets of tangent bundles formed by piecing together subspaces of tangent spaces, are commonly encountered in the theory and application of differential geometry. Indeed, the theory of distributions is a fundamental part of mechanics and control theory. The theory of distributions is presented in a systematic way, and self-contained proofs are given of some of the major results. Parts of the theory are presented in the context of generalised subbundles of vector bundles. Special emphasis is placed on understanding the r\^ole of sheaves and understanding the distinctions between the smooth or finitely differentiable cases and the real analytic case. The Orbit Theorem and applications, including Frobenius's Theorem and theorems on the equivalence of families of vector fields, are considered in detail. Examples illustrate the phenomenon that can occur with generalised subbundles and distributions

    Aspects Topologiques des Représentations en Analyse Calculable

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    Computable analysis provides a formalization of algorithmic computations over infinite mathematical objects. The central notion of this theory is the symbolic representation of objects, which determines the computation power of the machine, and has a direct impact on the difficulty to solve any given problem. The friction between the discrete nature of computations and the continuous nature of mathematical objects is captured by topology, which expresses the idea of finite approximations of infinite objects.We thoroughly study the multiple interactions between computations and topology, analysing the information that can be algorithmically extracted from a representation. In particular, we focus on the comparison between two representations of a single family of objects, on the precise relationship between algorithmic and topological complexity of problems, and on the relationship between finite and infinite representations.L’analyse calculable permet de formaliser le traitement algorithmique d’objets mathématiques infinis. La théorie repose sur une représentation symbolique des objets, dont le choix détermine les capacités de calcul de la machine, notamment sa difficulté à résoudre chaque problème donné. La friction entre le caractère discret du calcul et la nature continue des objets est capturée par la topologie, qui exprime l’idée d’approximation finie d’objets infinis.Nous étudions en profondeur les multiples interactions entre calcul et topologie, cherchant à analyser l’information qui peut être extraite algorithmiquement d’une représentation. Je me penche plus particulièrement sur la comparaison entre deux représentations d’une même famille d’objets, sur les liens détaillés entre complexité algorithmique et topologique des problèmes, ainsi que sur les relations entre représentations finies et infinies

    Compactifications of pseudofinite and pseudo-amenable groups

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    We first give simplified and corrected accounts of some results in \cite{PiRCP} on compactifications of pseudofinite groups. For instance, we use a classical theorem of Turing \cite{Turing} to give a simplified proof that any definable compactification of a pseudofinite group has an abelian connected component. We then discuss the relationship between Turing's work, the Jordan-Schur Theorem, and a (relatively) more recent result of Kazhdan \cite{Kazh} on approximate homomorphisms, and we use this to widen our scope from finite groups to amenable groups. In particular, we develop a suitable continuous logic framework for dealing with definable homomorphisms from pseudo-amenable groups to compact Lie groups. Together with the stabilizer theorems of \cite{HruAG,MOS}, we obtain a uniform (but non-quantitative) analogue of Bogolyubov's Lemma for sets of positive measure in discrete amenable groups. We conclude with brief remarks on the case of amenable topological groups.Comment: 23 page

    Graph of groups decompositions of graph braid groups

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    We provide an explicit construction that allows one to easily decompose a graph braid group as a graph of groups. This allows us to compute the braid groups of a wide range of graphs, as well as providing two general criteria for a graph braid group to split as a non-trivial free product, answering two questions of Genevois. We also use this to distinguish certain right-angled Artin groups and graph braid groups. Additionally, we provide an explicit example of a graph braid group that is relatively hyperbolic, but is not hyperbolic relative to braid groups of proper subgraphs. This answers another question of Genevois in the negative.Comment: Version accepted for publication. Several minor corrections and clarifications have been made and new classes of examples have been added. Code is also available, implementing algorithms in this paper to compute free splittings and presentations of graph braid groups; see https://github.com/danberlyne/graph-braid-splitter and https://github.com/danberlyne/graph-braid-presente

    Computability and Tiling Problems

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    In this thesis we will present and discuss various results pertaining to tiling problems and mathematical logic, specifically computability theory. We focus on Wang prototiles, as defined in [32]. We begin by studying Domino Problems, and do not restrict ourselves to the usual problems concerning finite sets of prototiles. We first consider two domino problems: whether a given set of prototiles SS has total planar tilings, which we denote TILETILE, or whether it has infinite connected but not necessarily total tilings, WTILEWTILE (short for `weakly tile'). We show that both TILE≡mILL≡mWTILETILE \equiv_m ILL \equiv_m WTILE, and thereby both TILETILE and WTILEWTILE are Σ11\Sigma^1_1-complete. We also show that the opposite problems, ¬TILE\neg TILE and SNTSNT (short for `Strongly Not Tile') are such that ¬TILE≡mWELL≡mSNT\neg TILE \equiv_m WELL \equiv_m SNT and so both ¬TILE\neg TILE and SNTSNT are both Π11\Pi^1_1-complete. Next we give some consideration to the problem of whether a given (infinite) set of prototiles is periodic or aperiodic. We study the sets PTilePTile of periodic tilings, and ATileATile of aperiodic tilings. We then show that both of these sets are complete for the class of problems of the form (Σ11∧Π11)(\Sigma^1_1 \wedge \Pi^1_1). We also present results for finite versions of these tiling problems. We then move on to consider the Weihrauch reducibility for a general total tiling principle CTCT as well as weaker principles of tiling, and show that there exist Weihrauch equivalences to closed choice on Baire space, CωωC_{\omega^\omega}. We also show that all Domino Problems that tile some infinite connected region are Weihrauch reducible to CωωC_{\omega^\omega}. Finally, we give a prototile set of 15 prototiles that can encode any Elementary Cellular Automaton (ECA). We make use of an unusual tile set, based on hexagons and lozenges that we have not see in the literature before, in order to achieve this.Comment: PhD thesis. 179 pages, 13 figure

    Sequential parametrized topological complexity of sphere bundles

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    Autonomous motion of a system (robot) is controlled by a motion planning algorithm. A sequential parametrized motion planning algorithm \cite{FP22} works under variable external conditions and generates continuous motions of the system to attain the prescribed sequence of states at prescribed moments of time. Topological complexity of such algorithms characterises their structure and discontinuities. Information about states of the system consistent with states of the external conditions is described by a fibration p:E→Bp: E\to B where the base BB parametrises the external conditions and each fibre p−1(b)p^{-1}(b) is the configuration space of the system constrained by external conditions b∈Bb\in B; more detail on this approach is given below. Our main goal in this paper is to study the sequential topological complexity of sphere bundles ξ˙:E˙→B\dot \xi: \dot E\to B; in other words we study {\it \lq\lq parametrized families of spheres\rq\rq} and sequential parametrized motion planning algorithms for such bundles. We use the Euler and Stiefel - Whitney characteristic classes to obtain lower bounds on the topological complexity. We illustrate our results by many explicit examples. Some related results for the special case r=2r=2 were described earlier in \cite{FW23}
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