56 research outputs found

    NIDUS IDEARUM. Scilogs, IV: vinculum vinculorum

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    Welcome into my scientific lab! My lab[oratory] is a virtual facility with non-controlled conditions in which I mostly perform scientific meditation and chats: a nest of ideas (nidus idearum, in Latin). I called the jottings herein scilogs (truncations of the words scientific, and gr. Λόγος – appealing rather to its original meanings ground , opinion , expectation ), combining the welly of both science and informal (via internet) talks (in English, French, and Romanian). * In this fourth book of scilogs collected from my nest of ideas, one may find new and old questions and solutions, referring mostly to topics on NEUTROSOPHY – email messages to research colleagues, or replies, notes about authors, articles, or books, so on. Feel free to budge in or just use the scilogs as open source for your own ideas

    Symétries de jauge non-commutative et diffusions pseudo-unitaires

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    This thesis is devoted to the study of two quite different questions, which are related by the tools that we used to study them.The first question is that of the definition of lattice gauge theories with a non-commutative structure group. Here, by non-commutative, we do not mean non-Abelian, but instead non-commutative in the general sense of non-commutative geometry.The second question is that of the behaviour of Brownian diffusions on non-compact matrix groups of a specific kind, namely groups of pseudo-orthogonal, pseudo-unitary or pseudo-symplectic matrices.In the first chapter, we investigate lattice and continuous quantum gauge theories on the Euclidean plane with a structure group that is replaced by a Zhang algebra. Zhang algebras are non-commutative analogues of groups and contain the class of Voiculescu's dual groups.We are interested in non-commutative analogues of random gauge fields, which we describe through the random holonomy that they induce.We propose a general definition of a holonomy field with Zhang gauge symmetry, and construct such a field starting from a quantum Lévy process on a Zhang algebra.As an application, we define higher dimensional generalizations of the so-called master field.In the second chapter, we study matricial approximations of higher dimensional master fields constructed in the previous chapter.These approximations (in non-commutative distribution) are obtained by extracting blocks of a Brownian unitary diffusion (with entries in the algebras of real, complex or quaternionic numbers) and letting the dimension of these blocks tend to infinity. We divide our study into two parts: in the first one, we extract square blocks while in the second one we allow rectangular blocks.In both cases, free probability theory appears as the natural framework in which the limiting distributions are most accurately described.In the last two chapters, we use tools introduced (Zhang algebras and coloured Brauer diagrams) in the first two ones to study Brownian motion on pseudo-unitary matrices in high dimensions.We prove convergence in non-commutative distribution of the pseudo-unitary Brownian motions we consider, to free with amalgamation semi-groups under the hypothesis of convergence of the normalized signature of the metric.In the split case, meaning that at least asymptotically the metric has as much negative directions as positive ones, the limiting distribution is that of a free L\'evy process, which is a solution of a free stochastic differential equation.We leave open the question of such a realization of the limiting distribution in the general case.In addition, we provide (intriguing) numerical evidences for the convergence of the spectral distribution of such random matrices and make two conjectures. At the end of the thesis, we prove asymptotic normality for the fluctuations.Cette thèse est consacrée à l'étude de deux questions très différentes, reliées par les outils que nous utilisons pour les étudier. La première question est celle de la définition des théories de jauge sur un réseau avec un groupe de structure non commutatif. Ici, non commutatif ne signifie pas non Abelian, mais plutôt non commutatif au sens général de la géométrie non commutative. La deuxième question est celle du comportement des diffusions Browniennes sur des groupes matriciels non compacts d'un type spécifique, à savoir des groupes de matrices pseudo-orthogonales, pseudo-unitaires ou pseudo-symplectiques.Dans le premier chapitre, nous étudions des théories de jauge quantiques sur un réseau et leur limite continue sur le plan euclidien ayant une algèbre de Zhang pour groupe de stucture. Les algèbres de Zhang sont des analogues non commutatifs des groupes et contiennent la classe des groupes duaux de Voiculescu. Nous nous intéressons donc aux analogues non commutatifs des champs de jauges quantiques, que nous décrivons par l'holonomie aléatoire qu'ils induisent. Nous proposons une définition générale d'un champ d'holonomies ayant une symétrie de jauge présentant la structure d'une algèbre de Zhang, et construisons un tel champ à partir d'un processus quantique de Lévy sur une algèbre de Zhang.Dans le deuxième chapitre, nous étudions les approximations matricielles des champs maîtres en dimensions supérieures construits dans le chapitre précédent. Ces approximations (en distribution non commutative) sont obtenues en extrayant des blocs d'une diffusion unitaire Brownienne (à coefficients dans les algèbres de nombres réels, complexes ou quaternioniques) et en laissant la dimension de ces blocs tendre vers l'infini. Nous divisons notre étude en deux parties : dans la première, nous extrayons des blocs carrés tandis que dans la seconde, nous autorisons des blocs rectangulaires.Dans les deux derniers chapitres, nous utilisons les outils introduits (algèbres de Zhang et diagrammes de Brauer colorés) dans les deux premiers pour étudier des diffusions sur des groupes de matrices pseudo-unitaires. Nous prouvons la convergence non commutative des mouvements Browniens pseudo-unitaires que nous considérons vers des semi-groupes libres avec amalgamation sous l'hypothèse de convergence de la signature normalisée de la métrique de l'espace sous-jacent.Dans le cas déployé, c'est-à-dire, qu'au moins asymptotiquement, la métrique a autant de directions négatives que de directions positives, la distribution limite est la distribution d'un processus de Lévy, solution d'une équation différentielle stochastique libre. Nous laissons ouverte la question d'une telle réalisation de la distribution limite dans le cas général.De plus, nous présentons des résultats numériques sur la convergence de la distribution spectrale de ces matrices aléatoires et faisons deux conjectures. Dans le dernier chapitre, nous prouvons la normalité asymptotique des fluctuations

    Multifraction reduction II: Conjectures for Artin-Tits groups

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    Multifraction reduction is a new approach to the word problem for Artin-Tits groups and, more generally, for the enveloping group of a monoid in which any two elements admit a greatest common divisor. This approach is based on a rewrite system ("reduction") that extends free group reduction. In this paper, we show that assuming that reduction satisfies a weak form of convergence called semi-convergence is sufficient for solving the word problem for the enveloping group, and we connect semi-convergence with other conditions involving reduction. We conjecture that these properties are valid for all Artin-Tits monoids, and provide partial results and numerical evidence supporting such conjectures.Comment: 41 pages , v2 : cross-references updated , v3 : exposition improved, typos corrected, final version due tu appear in Journal of Combinatorial Algebr

    Matricial approximations of higher dimensional master fields

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    We study matricial approximations of master fields constructed in [6]. These approximations (in non-commutative distribution) are obtained by extracting blocks of a Brownian unitary diffusion (with entries in R, C or K) and letting the dimension of these blocks to tend to infinity. We divide our study into two parts: in the first one, we extract square blocks while in the second one we allow rectangular blocks. In both cases, free probability theory appears as the natural framework in which the limiting distributions are most accurately described

    A Polynomial-Time Algorithm for the Lambek Calculus with Brackets of Bounded Order

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    Lambek calculus is a logical foundation of categorial grammar, a linguistic paradigm of grammar as logic and parsing as deduction. Pentus (2010) gave a polynomial-time algorithm for determining provability of bounded depth formulas in L*, the Lambek calculus with empty antecedents allowed. Pentus\u27 algorithm is based on tabularisation of proof nets. Lambek calculus with brackets is a conservative extension of Lambek calculus with bracket modalities, suitable for the modeling of syntactical domains. In this paper we give an algorithm for provability in Lb*, the Lambek calculus with brackets allowing empty antecedents. Our algorithm runs in polynomial time when both the formula depth and the bracket nesting depth are bounded. It combines a Pentus-style tabularisation of proof nets with an automata-theoretic treatment of bracketing
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