910 research outputs found
Classification of multiplicity free quasi-Hamiltonian manifolds
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
Given an uncountable subset of a nonseparable Banach space, is
there an uncountable such that the distances
between any two distinct points of are more or less the same? If
an uncountable subset of a nonseparable Banach space does not
admit an uncountable , where any two points are
distant by more than , is it because is the countable union
of sets of diameters not bigger than ?
We investigate connections between the set-theoretic phenomena involved and
the geometric properties of uncountable subsets of nonseparable Banach spaces
of densities up to related to uncountable -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 as well as offers a synthesis of possible
phenomena and categorization of examples for uncountable densities up to
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
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
-algebraic approach to the principal symbol. III
We treat the notion of principal symbol mapping on a compact smooth manifold
as a -homomorphism of -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
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
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
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
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
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 has total planar tilings, which we denote , or whether
it has infinite connected but not necessarily total tilings, (short for
`weakly tile'). We show that both , and
thereby both and are -complete. We also show that
the opposite problems, and (short for `Strongly Not Tile')
are such that and so both
and are both -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 of periodic tilings, and of
aperiodic tilings. We then show that both of these sets are complete for the
class of problems of the form . 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
as well as weaker principles of tiling, and show that there exist Weihrauch
equivalences to closed choice on Baire space, . We also show
that all Domino Problems that tile some infinite connected region are Weihrauch
reducible to . 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
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 where
the base parametrises the external conditions and each fibre is
the configuration space of the system constrained by external conditions ; more detail on this approach is given below. Our main goal in this paper is
to study the sequential topological complexity of sphere bundles ; 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 were
described earlier in \cite{FW23}
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