270 results on '"Patras, Frédéric"'
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252. Cumulant–Cumulant Relations in Free Probability Theory from Magnus' Expansion.
- Author
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Celestino, Adrian, Ebrahimi-Fard, Kurusch, Patras, Frédéric, and Perales, Daniel
- Subjects
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PROBABILITY theory , *CUMULANTS , *BOOLEAN functions , *FUNCTIONALS - Abstract
Relations between moments and cumulants play a central role in both classical and non-commutative probability theory. The latter allows for several distinct families of cumulants corresponding to different types of independences: free, Boolean and monotone. Relations among those cumulants have been studied recently. In this work, we focus on the problem of expressing with a closed formula multivariate monotone cumulants in terms of free and Boolean cumulants. In the process, we introduce various constructions and statistics on non-crossing partitions. Our approach is based on a pre-Lie algebra structure on cumulant functionals. Relations among cumulants are described in terms of the pre-Lie Magnus expansion combined with results on the continuous Baker–Campbell–Hausdorff formula due to A. Murua. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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253. Molien generating functions and integrity bases for the action of the SO(3) and O(3) groups on a set of vectors.
- Author
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Dhont, Guillaume, Cassam-Chenaï, Patrick, and Patras, Frédéric
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GENERATING functions , *QUADRUPOLE moments , *MOLECULAR magnetic moments , *QUANTUM chemistry , *POTENTIAL energy - Abstract
The construction of integrity bases for invariant and covariant polynomials built from a set of three dimensional vectors under the SO (3) and O (3) symmetries is presented. This paper is a follow-up to our previous work that dealt with a set of two dimensional vectors under the action of the SO (2) and O (2) groups (Dhont and Zhilinskií in J Phys A Math Theor 46:455202, 2013). The expressions of the Molien generating functions as one rational function are a useful guide to build integrity bases for the rings of invariants and the free modules of covariants. The structure of the non-free modules of covariants is more complex. In this case, we write the Molien generating function as a sum of rational functions and show that its symbolic interpretation leads to the concept of generalized integrity basis. The integrity bases and generalized integrity bases for O (3) are deduced from the SO (3) ones. The results are useful in quantum chemistry to describe the potential energy or multipole moment hypersurfaces of molecules. In particular, the generalized integrity bases that are required for the description of the electric and magnetic quadrupole moment hypersurfaces of tetratomic molecules are given for the first time. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
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254. A theory of pictures for quasi-posets.
- Author
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Foissy, Loïc, Malvenuto, Claudia, and Patras, Frédéric
- Subjects
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COMBINATORICS , *MATHEMATICS , *ADDITIVE combinatorics , *SYMMETRIC functions - Abstract
The theory of pictures between posets is known to encode much of the combinatorics of symmetric group representations and related topics such as Young diagrams and tableaux. Many reasons, combinatorial (e.g. since semi-standard tableaux can be viewed as double quasi-posets) and topological (quasi-posets identify with finite topologies) lead to extend the theory to quasi-posets. This is the object of the present article. [ABSTRACT FROM AUTHOR]
- Published
- 2017
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255. Infinitesimal and B∞-algebras, finite spaces, and quasi-symmetric functions.
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Foissy, Loïc, Malvenuto, Claudia, and Patras, Frédéric
- Subjects
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SYMMETRIC functions , *TOPOLOGICAL spaces , *SET theory , *MATHEMATICAL symmetry , *HOPF algebras - Abstract
Finite topological spaces are in bijective correspondence with preorders on finite sets. We undertake their study using combinatorial tools that have been developed to investigate general discrete structures. A particular emphasis will be put on recent topological and combinatorial Hopf algebra techniques. We will show that the linear span of finite spaces carries generalized Hopf algebraic structures that are closely connected with familiar constructions and structures in topology (such as the one of cogroups in the category of associative algebras that has appeared e.g. in the study of loop spaces of suspensions). The most striking results that we obtain are certainly that the linear span of finite spaces carries the structure of the enveloping algebra of a B ∞ -algebra, and that there are natural (Hopf algebraic) morphisms between finite spaces and quasi-symmetric functions. In the process, we introduce the notion of Schur–Weyl categories in order to describe rigidity theorems for cogroups in the category of associative algebras and related structures, as well as to account for the existence of natural operations (graded permutations) on them. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
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256. A duality formula for Feynman–Kac path particle models.
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Del Moral, Pierre, Kohn, Robert, and Patras, Frédéric
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DUALITY theory (Mathematics) , *FEYNMAN integrals , *MATHEMATICAL models , *MARKOV processes , *TAYLOR'S series - Abstract
This Note and its extended version [10] present a new duality formula between genetic type genealogical tree based particle models and Feynman–Kac measures on path spaces. Among others, this formula allows us to design reversible Gibbs–Glauber Markov chains for Feynman–Kac integration on path spaces. Our approach yields new Taylor series expansions of the particle Gibbs–Glauber semigroup around its equilibrium measure w.r.t. the size of the particle system, generalizing the recent work of Andrieu, Doucet, and Holenstein [1] . We analyze the rate of convergence to equilibrium in terms of the ratio of the length of the trajectories to the number of particles. The analysis relies on a tree-based functional and combinatorial representation of a class of Feynman–Kac particle models with a frozen ancestral line. We illustrate the impact of these results in the context of Quantum and Diffusion Monte Carlo methods. [ABSTRACT FROM AUTHOR]
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- 2015
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257. Husserl between Formalism and Intuitionism
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Dodd, James, Bernet, R., editor, Taminiaux, J., editor, IJsseling, S., editor, Leonardy, H., editor, Lories, D., editor, Melle, U., editor, Bernasconi, R., editor, Carr, D., editor, Casey, E.S., editor, Cobb-Stevens, R., editor, Courtine, J.F., editor, Dastur, F., editor, Düsing, K., editor, Hart, J., editor, Held, K., editor, Kaehler, K.E., editor, Lohmar, D., editor, McKenna, W.R., editor, Mohanty, J.N., editor, Orth, E.W., editor, Ricœur, P., editor, Sini, C., editor, Sokolowski, R., editor, Waldenfels, B., editor, Boi, Luciano, editor, Kerszberg, Pierre, editor, and Patras, Frédéric, editor
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- 2007
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258. Mettre les structures en mouvement: La phénoménologie et la dynamique de l'intuition conceptuelle. Sur la pertinence phénoménologique de la théorie des catégories
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Benoist, Jocelyn, Bernet, R., editor, Taminiaux, J., editor, IJsseling, S., editor, Leonardy, H., editor, Lories, D., editor, Melle, U., editor, Bernasconi, R., editor, Carr, D., editor, Casey, E.S., editor, Cobb-Stevens, R., editor, Courtine, J.F., editor, Dastur, F., editor, Düsing, K., editor, Hart, J., editor, Held, K., editor, Kaehler, K.E., editor, Lohmar, D., editor, McKenna, W.R., editor, Mohanty, J.N., editor, Orth, E.W., editor, Ricœur, P., editor, Sini, C., editor, Sokolowski, R., editor, Waldenfels, B., editor, Boi, Luciano, editor, Kerszberg, Pierre, editor, and Patras, Frédéric, editor
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- 2007
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259. The Two-Sidedness and the Rationalistic Ideal of Formal Logic: Husserl and Gödel
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Cassou-Noguès, Pierre, Bernet, R., editor, Taminiaux, J., editor, IJsseling, S., editor, Leonardy, H., editor, Lories, D., editor, Melle, U., editor, Bernasconi, R., editor, Carr, D., editor, Casey, E.S., editor, Cobb-Stevens, R., editor, Courtine, J.F., editor, Dastur, F., editor, Düsing, K., editor, Hart, J., editor, Held, K., editor, Kaehler, K.E., editor, Lohmar, D., editor, McKenna, W.R., editor, Mohanty, J.N., editor, Orth, E.W., editor, Ricœur, P., editor, Sini, C., editor, Sokolowski, R., editor, Waldenfels, B., editor, Boi, Luciano, editor, Kerszberg, Pierre, editor, and Patras, Frédéric, editor
- Published
- 2007
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260. Mathematical Concepts and Physical Objects
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Longo, Giuseppe, Bernet, R., editor, Taminiaux, J., editor, IJsseling, S., editor, Leonardy, H., editor, Lories, D., editor, Melle, U., editor, Bernasconi, R., editor, Carr, D., editor, Casey, E.S., editor, Cobb-Stevens, R., editor, Courtine, J.F., editor, Dastur, F., editor, Düsing, K., editor, Hart, J., editor, Held, K., editor, Kaehler, K.E., editor, Lohmar, D., editor, McKenna, W.R., editor, Mohanty, J.N., editor, Orth, E.W., editor, Ricœur, P., editor, Sini, C., editor, Sokolowski, R., editor, Waldenfels, B., editor, Boi, Luciano, editor, Kerszberg, Pierre, editor, and Patras, Frédéric, editor
- Published
- 2007
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261. Understanding Quantum Mechanics with Bohr and Husserl
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Lurçat, François, Bernet, R., editor, Taminiaux, J., editor, IJsseling, S., editor, Leonardy, H., editor, Lories, D., editor, Melle, U., editor, Bernasconi, R., editor, Carr, D., editor, Casey, E.S., editor, Cobb-Stevens, R., editor, Courtine, J.F., editor, Dastur, F., editor, Düsing, K., editor, Hart, J., editor, Held, K., editor, Kaehler, K.E., editor, Lohmar, D., editor, McKenna, W.R., editor, Mohanty, J.N., editor, Orth, E.W., editor, Ricœur, P., editor, Sini, C., editor, Sokolowski, R., editor, Waldenfels, B., editor, Boi, Luciano, editor, Kerszberg, Pierre, editor, and Patras, Frédéric, editor
- Published
- 2007
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262. Perseverance and Adjustment: On Weyl's Phenomenological Philosophy of Nature
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Kerszberg, Pierre, Bernet, R., editor, Taminiaux, J., editor, IJsseling, S., editor, Leonardy, H., editor, Lories, D., editor, Melle, U., editor, Bernasconi, R., editor, Carr, D., editor, Casey, E.S., editor, Cobb-Stevens, R., editor, Courtine, J.F., editor, Dastur, F., editor, Düsing, K., editor, Hart, J., editor, Held, K., editor, Kaehler, K.E., editor, Lohmar, D., editor, McKenna, W.R., editor, Mohanty, J.N., editor, Orth, E.W., editor, Ricœur, P., editor, Sini, C., editor, Sokolowski, R., editor, Waldenfels, B., editor, Boi, Luciano, editor, Kerszberg, Pierre, editor, and Patras, Frédéric, editor
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- 2007
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263. On Naturalizing Free
- Author
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Rinofner-Kreidl, Sonja, Bernet, R., editor, Taminiaux, J., editor, IJsseling, S., editor, Leonardy, H., editor, Lories, D., editor, Melle, U., editor, Bernasconi, R., editor, Carr, D., editor, Casey, E.S., editor, Cobb-Stevens, R., editor, Courtine, J.F., editor, Dastur, F., editor, Düsing, K., editor, Hart, J., editor, Held, K., editor, Kaehler, K.E., editor, Lohmar, D., editor, McKenna, W.R., editor, Mohanty, J.N., editor, Orth, E.W., editor, Ricœur, P., editor, Sini, C., editor, Sokolowski, R., editor, Waldenfels, B., editor, Boi, Luciano, editor, Kerszberg, Pierre, editor, and Patras, Frédéric, editor
- Published
- 2007
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264. Space and Movement. On Husserl's Geometry of the Visual Field
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Giorello, Giulio, Sinigaglia, Corrado, Bernet, R., editor, Taminiaux, J., editor, IJsseling, S., editor, Leonardy, H., editor, Lories, D., editor, Melle, U., editor, Bernasconi, R., editor, Carr, D., editor, Casey, E.S., editor, Cobb-Stevens, R., editor, Courtine, J.F., editor, Dastur, F., editor, Düsing, K., editor, Hart, J., editor, Held, K., editor, Kaehler, K.E., editor, Lohmar, D., editor, McKenna, W.R., editor, Mohanty, J.N., editor, Orth, E.W., editor, Ricœur, P., editor, Sini, C., editor, Sokolowski, R., editor, Waldenfels, B., editor, Boi, Luciano, editor, Kerszberg, Pierre, editor, and Patras, Frédéric, editor
- Published
- 2007
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265. On the Relationship between Parts and Wholes in Husserl's Phenomenology
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Casari, Ettore, Bernet, R., editor, Taminiaux, J., editor, IJsseling, S., editor, Leonardy, H., editor, Lories, D., editor, Melle, U., editor, Bernasconi, R., editor, Carr, D., editor, Casey, E.S., editor, Cobb-Stevens, R., editor, Courtine, J.F., editor, Dastur, F., editor, Düsing, K., editor, Hart, J., editor, Held, K., editor, Kaehler, K.E., editor, Lohmar, D., editor, McKenna, W.R., editor, Mohanty, J.N., editor, Orth, E.W., editor, Ricœur, P., editor, Sini, C., editor, Sokolowski, R., editor, Waldenfels, B., editor, Boi, Luciano, editor, Kerszberg, Pierre, editor, and Patras, Frédéric, editor
- Published
- 2007
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266. Husserl and the Phenomenology of Attention
- Author
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Bégout, Bruce, Bernet, R., editor, Taminiaux, J., editor, IJsseling, S., editor, Leonardy, H., editor, Lories, D., editor, Melle, U., editor, Bernasconi, R., editor, Carr, D., editor, Casey, E.S., editor, Cobb-Stevens, R., editor, Courtine, J.F., editor, Dastur, F., editor, Düsing, K., editor, Hart, J., editor, Held, K., editor, Kaehler, K.E., editor, Lohmar, D., editor, McKenna, W.R., editor, Mohanty, J.N., editor, Orth, E.W., editor, Ricœur, P., editor, Sini, C., editor, Sokolowski, R., editor, Waldenfels, B., editor, Boi, Luciano, editor, Kerszberg, Pierre, editor, and Patras, Frédéric, editor
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- 2007
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267. Phénoménologie et méréologie de la perception spatiale, de Husserl aux théoriciens de la Gestalt
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Boi, Luciano, Bernet, R., editor, Taminiaux, J., editor, IJsseling, S., editor, Leonardy, H., editor, Lories, D., editor, Melle, U., editor, Bernasconi, R., editor, Carr, D., editor, Casey, E.S., editor, Cobb-Stevens, R., editor, Courtine, J.F., editor, Dastur, F., editor, Düsing, K., editor, Hart, J., editor, Held, K., editor, Kaehler, K.E., editor, Lohmar, D., editor, McKenna, W.R., editor, Mohanty, J.N., editor, Orth, E.W., editor, Ricœur, P., editor, Sini, C., editor, Sokolowski, R., editor, Waldenfels, B., editor, Boi, Luciano, editor, Kerszberg, Pierre, editor, and Patras, Frédéric, editor
- Published
- 2007
- Full Text
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268. Cumulants in Non-commutative:Probability via Hopf Algebras
- Author
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Rodriguez, Adrian de Jesus Celestino, Ebrahimi-Fard, Kurusch, and Patras, Frédéric
- Subjects
Mathematics and natural science: 400::Mathematics: 410 [VDP] - Abstract
The notion of cumulants plays a significant role in the combinatorial study of noncommutative probability theory. In this thesis, we study several problems associated with the notions of cumulants for free, Boolean and monotone independence via a unified Hopf algebraic framework for non-commutative probability. In this framework, developed by Ebrahimi-Fard and Patras [EFP15, EFP16, EFP18], the relations between moments and the different brands of cumulants are described by three different exponentials that relate the group of characters and the Lie algebra of infinitesimal characters on a particular word Hopf algebra H. The first question that we address is to show how the algebraic central limit theorems for free, Boolean and monotone independence enter into the shuffle-algebraic picture for non-commutative probability. We analyze how the different notions of additive convolutions are described in the shuffle framework and prove combinatorial formulas for the powers of the half-shuffle products. Next, we focus on extending the shuffle framework in order to consider the extension of non-commutative probability called infinitesimal non-commutative probability. We show how infinitesimal cumulants also correspond to infinitesimal characters on H and describe the shuffle-algebraic equations for the infinitesimal moment-cumulant relations. We also prove that the combinatorial relations between infinitesimal cumulants follow via the extended shuffle-algebraic framework. Afterwards, we concentrate on the problem of writing the multivariate monotone cumulants of random variables in terms of their moments. The starting point to obtain this formula is to compute a logarithm of a certain character on H. Then by investigating a connection of H with a Hopf algebra of decorated rooted trees, we compute the logarithm instead in the Hopf algebra of trees, which yields a new combinatorial formula from moments to monotone cumulants in terms of Schröder trees. Thereafter, we turn to the problem, left open in [AHLV15], of expressing multivariate monotone cumulants in terms of the free and Boolean cumulants. Our approach relies on a pre-Lie algebraic relation between the three infinitesimal characters associated to the free, Boolean and monotone cumulants. This relation is known as the pre-Lie Magnus expansion and is defined in terms of iterations of a pre-Lie product. Our problem is then transformed into finding a combinatorial formula in terms of non-crossing partitions of the iterated pre-Lie products and identifying and describing the coefficients that govern the transition from free and Boolean to monotone cumulants. The coefficients obtained in the solution of the previous problem hint at a deeper relation between non-commutative probability, combinatorics of rooted trees and free pre-Lie algebras. Our last question is to have a more systematic understanding of the previous relation. In the process, we develop a concrete and effective method of computations on a specific class of pre-Lie algebras, where iterations of the pre-Lie product can be computed in terms of forest formulas for iterated coproducts. We illustrate our method by retrieving the formulas between monotone and free (Boolean) cumulants, as well as the computation of the pre-Lie Magnus expansion on the generator of the free pre-Lie algebra of rooted trees.
- Published
- 2022
269. Cumulants, free cumulants and half-shuffles.
- Author
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Ebrahimi-Fard K and Patras F
- Abstract
Free cumulants were introduced as the proper analogue of classical cumulants in the theory of free probability. There is a mix of similarities and differences, when one considers the two families of cumulants. Whereas the combinatorics of classical cumulants is well expressed in terms of set partitions, that of free cumulants is described and often introduced in terms of non-crossing set partitions. The formal series approach to classical and free cumulants also largely differs. The purpose of this study is to put forward a different approach to these phenomena. Namely, we show that cumulants, whether classical or free, can be understood in terms of the algebra and combinatorics underlying commutative as well as non-commutative (half-)shuffles and (half-) unshuffles. As a corollary, cumulants and free cumulants can be characterized through linear fixed point equations. We study the exponential solutions of these linear fixed point equations, which display well the commutative, respectively non-commutative, character of classical and free cumulants.
- Published
- 2015
- Full Text
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270. [Building mathematics in imagination].
- Author
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Patras F
- Abstract
The extraordinary quantitative achievements of contemporary science often hide their qualitative dimension. In mathematics, the understanding of fundamental theoretical phenomena we have got today goes much beyond that achieved in previous periods. This also holds when it comes to the theorisation of mathematical practice.Philosophically, these changes remain largely to be properly analyzed. The present article will address this issue from the point of view of Bachelard's epistemology.
- Published
- 2015
- Full Text
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