633 research outputs found
Clustering and Arnoux-Rauzy words
We characterize the clustering of a word under the Burrows-Wheeler transform
in terms of the resolution of a bounded number of bispecial factors belonging
to the language generated by all its powers. We use this criterion to compute,
in every given Arnoux-Rauzy language on three letters, an explicit bound
such that each word of length at least is not clustering; this bound is
sharp for a set of Arnoux-Rauzy languages including the Tribonacci one. In the
other direction, we characterize all standard Arnoux-Rauzy clustering words,
and all perfectly clustering Arnoux-Rauzy words. We extend some results to
episturmian languages, characterizing those which produce infinitely many
clustering words, and to larger alphabets
Primitive Automata that are Synchronizing
A deterministic finite (semi)automaton is primitive if its transition monoid
(semigroup) acting on the set of states has no non-trivial congruences. It is
synchronizing if it contains a constant map (transformation). In analogy to
synchronizing groups, we study the possibility of characterizing automata that
are synchronizing if primitive. We prove that the implication holds for several
classes of automata. In particular, we show it for automata whose every letter
induce either a permutation or a semiconstant transformation (an idempotent
with one point of contraction) unless all letters are of the first type. We
propose and discuss two conjectures about possible more general
characterizations.Comment: Note: The weak variant of our conjecture in a stronger form has been
recently solved by Mikhail Volkov arXiv:2306.13317, together with several new
results concerning our proble
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Forward Limit Sets of Semigroups of Substitutions and Arithmetic Progressions in Automatic Sequences
This thesis deals with symbolic sequences generated by semigroups of substitutions acting on finite alphabets.
First, we investigate the underlying structure of certain automatic sequences by studying the maximum length A(d) of the monochromatic arithmetic progressions of difference d appearing in these sequences. For example, for the Thue-Morse sequence and a class of generalised Thue-Morse sequences, we give exact values of A(d) or upper bounds on it, for certain differences d. For aperiodic, primitive, bijective substitutions and spin substitutions, which are generalisations of the Thue-Morse and Rudin-Shapiro substitutions, respectively, we study the asymptotic growth rate of A(d). In particular, we prove that there exists a subsequence (d_n) of differences along which A(d_n) grows at least polynomially in d_n. Explicit upper and lower bounds for the growth exponent can be derived from a finite group associated to the substitution considered.
Next, we introduce the forward limit set Λ of a semigroup S generated by a family of substitutions of a finite alphabet, which typically coincides with the set of all possible s-adic limits of that family. We provide several alternative characterisations of the forward limit set. For instance, we prove that Λ is the unique maximal closed and strongly S-invariant subset of the space of all infinite words, and we prove that it is the closure of the image under S of the set of all fixed points of S. It is usually difficult to compute a forward limit set explicitly; however, we show that, provided certain assumptions hold, Λ is uncountable, and we supply upper bounds on its size in terms of logarithmic Hausdorff dimension
Lie algebra actions on module categories for truncated shifted Yangians
We develop a theory of parabolic induction and restriction functors relating
modules over Coulomb branch algebras, in the sense of
Braverman-Finkelberg-Nakajima. Our functors generalize Bezrukavnikov-Etingof's
induction and restriction functors for Cherednik algebras, but their definition
uses different tools.
After this general definition, we focus on quiver gauge theories attached to
a quiver . The induction and restriction functors allow us to define a
categorical action of the corresponding symmetric Kac-Moody algebra
on category for these Coulomb branch
algebras. When is of Dynkin type, the Coulomb branch algebras are
truncated shifted Yangians and quantize generalized affine Grassmannian slices.
Thus, we regard our action as a categorification of the geometric Satake
correspondence.
To establish this categorical action, we define a new class of "flavoured"
KLRW algebras, which are similar to the diagrammatic algebras originally
constructed by the second author for the purpose of tensor product
categorification. We prove an equivalence between the category of
Gelfand-Tsetlin modules over a Coulomb branch algebra and the modules over a
flavoured KLRW algebra. This equivalence relates the categorical action by
induction and restriction functors to the usual categorical action on modules
over a KLRW algebra.Comment: 66 pages, version 2: many corrections, improved treatment of GK
dimension, 71 page
Automorphisms of a Generalized Quadrangle of Order 6
In this thesis, we study the symmetries of the putative generalized quadrangle of order 6. Although it is unknown whether such a quadrangle Q can exist, we show that if it does, that Q cannot be transitive on either points or lines. We first cover the background necessary for studying this problem. Namely, the theory of groups and group actions, the theory of generalized quadrangles, and automorphisms of GQs. We then prove that a generalized quadrangle Q of order 6 cannot have a point- or line-transitive automorphism group, and we also prove that if a group G acts faithfully on Q such that 259 | |G|, then G is not solvable. Along the way, we develop techniques for studying composite order automorphisms of a generalized quadrangle. Specifically, we deal with automorphisms of order pk and pq, where p and q are prime
From telescopes to frames and simple groups
We introduce the notion of a telescope of groups. Very roughly a telescope is
a directed system of groups that contains various commuting images of some
fixed group . Telescopes are inspired from the theory of groups acting on
rooted trees. Imitating known constructions of branch groups, we obtain a
number of examples of -telescopes and discuss several applications. We give
examples of -generated infinite amenable simple groups. We show that every
finitely generated residually finite (amenable) group embeds into a finitely
generated (amenable) LEF simple group. We construct -generated frames in
products of finite simple groups and show that there are Grothendieck pairs
consisting of amenable groups and groups with property . We give
examples of automorphisms of finitely generated, residually finite, amenable
groups that are not inner, but become inner in the profinite completion. We
describe non-elementary amenable examples of finitely generated, residually
finite groups all of whose finitely generated subnormal subgroups are direct
factors.Comment: 41 pages, comments welcom
Quantum reservoir computing in finite dimensions
Most existing results in the analysis of quantum reservoir computing (QRC)
systems with classical inputs have been obtained using the density matrix
formalism. This paper shows that alternative representations can provide better
insights when dealing with design and assessment questions. More explicitly,
system isomorphisms are established that unify the density matrix approach to
QRC with the representation in the space of observables using Bloch vectors
associated with Gell-Mann bases. It is shown that these vector representations
yield state-affine systems (SAS) previously introduced in the classical
reservoir computing literature and for which numerous theoretical results have
been established. This connection is used to show that various statements in
relation to the fading memory (FMP) and the echo state (ESP) properties are
independent of the representation, and also to shed some light on fundamental
questions in QRC theory in finite dimensions. In particular, a necessary and
sufficient condition for the ESP and FMP to hold is formulated using standard
hypotheses, and contractive quantum channels that have exclusively trivial
semi-infinite solutions are characterized in terms of the existence of
input-independent fixed points.Comment: 19 pages, 4 figure
Classification of lagrangian planes in Kummer-type hyperk\"ahler manifolds
We generalise a result from [Bak15] on K3 type hyperk\"ahler manifolds
proving that a line in a lagrangian plane on a hyperk\"ahler manifold of
Kummer type has Beauville-Bogomolov-Fujiki square and order 2
in the discriminant group of . Vice versa, an extremal
primitive ray of the Mori cone verifying these conditions is in fact the class
of a line in some lagrangian plane. In doing so, we show, on moduli spaces of
Bridgeland stable objects on an abelian surface, that lagrangian planes on the
fibre of the Albanese map correspond to sublattices of the Mukai lattice
verifying some numerical condition
Special cubic zeros and the dual variety
Let be a diagonal cubic form over in variables. From the
dual variety in the delta method of Duke--Friedlander--Iwaniec and Heath-Brown,
we unconditionally extract a weighted count of certain special integral zeros
of in regions of diameter . Heath-Brown did the same in
variables, but our analysis differs and captures some novel features. We also
put forth an axiomatic framework for more general .Comment: 30 pages; added references; changed title; conceptual improvement
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