633 research outputs found

    Clustering and Arnoux-Rauzy words

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    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 KK such that each word of length at least KK 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

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    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

    Lie algebra actions on module categories for truncated shifted Yangians

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    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 Γ\Gamma. The induction and restriction functors allow us to define a categorical action of the corresponding symmetric Kac-Moody algebra gΓ\mathfrak{g}_{\Gamma} on category O \mathcal O for these Coulomb branch algebras. When Γ \Gamma 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

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    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

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    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 BB. 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 BB-telescopes and discuss several applications. We give examples of 22-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 22-generated frames in products of finite simple groups and show that there are Grothendieck pairs consisting of amenable groups and groups with property (Ï„)(\tau). 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

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    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

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    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 XX of Kummer type has Beauville-Bogomolov-Fujiki square −n+12-\frac{n+1}{2} and order 2 in the discriminant group of H2(X,Z)H^2(X,\mathbb{Z}). 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

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    Let FF be a diagonal cubic form over Z\mathbb{Z} in 66 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 FF in regions of diameter X→∞X \to \infty. Heath-Brown did the same in 44 variables, but our analysis differs and captures some novel features. We also put forth an axiomatic framework for more general FF.Comment: 30 pages; added references; changed title; conceptual improvement
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