IA Scholar Query: Fourier meets möbius: fast subset convolution.
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Internet Archive Scholar query results feedeninfo@archive.orgTue, 11 Oct 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Linear algebra and group theory
https://scholar.archive.org/work/ctqxr7ssrfgmbpf3oz6e4wyf4y
This is an introduction to linear algebra and group theory. We first review the linear algebra basics, namely the determinant, the diagonalization procedure, and more, and with the determinant being constructed as it should, as a signed volume. We discuss then the basic applications of linear algebra to questions in analysis. Then we get into the study of the closed groups of unitary matrices G⊂ U_N, with some basic algebraic theory, and with a number of probability computations, in the finite group case. In the general case, where G⊂ U_N is compact, we explain how the Weingarten integration formula works, and we present some basic N→∞ applications.Teo Banicawork_ctqxr7ssrfgmbpf3oz6e4wyf4yTue, 11 Oct 2022 00:00:00 GMTAspects of Entanglement Entropy in Algebraic Quantum Field Theory
https://scholar.archive.org/work/355ctjbbm5fijdw3shkppveck4
In this thesis, we study aspects of entanglement theory of quantum field theories from an algebraic point of view. The main motivation is to gain insights about the general structure of the entanglement in QFT, towards a definition of an entropic version of QFT. In the opposite direction, we are also interested in exploring any consequence of the entanglement in algebraic QFT. This may help us to reveal unknown features of QFT, with the final aim of finding a dynamical principle which allows us to construct non-trivial and rigorous models of QFT. The algebraic approach is the natural framework to define and study entanglement in QFT, and hence, to pose the above inquiries. After a self-contained review of algebraic QFT and quantum information theory in operator algebras, we focus on our results. We compute, in a mathematically rigorous way, exact solutions of entanglement measures and modular Hamiltonians for specific QFT models, using algebraic tools from modular theory of von Neumann algebras. These calculations show explicitly non-local features of modular Hamiltonians and help us to solve ambiguities that arise in other non-rigorous computations. We also study aspects of entanglement entropy in theories having superselection sectors coming from global symmetries. We follow the algebraic perspective of Doplicher, Haag, and Roberts. In this way, we find an entropic order parameter that "measures" the size of the symmetry group, which is made out of a difference of two mutual informations. Moreover, we identify the main operators that take account of such a difference, and we obtain a new quantum information quantity, the entropic certainty relation, involving algebras containing such operators. This certainty relation keeps an intrinsic connection with subfactor theory of von Neumann algebras.Diego Pontellowork_355ctjbbm5fijdw3shkppveck4Fri, 26 Aug 2022 00:00:00 GMTAbstracts of the 8th International Conference on Speech Motor Control Groningen, August 2022
https://scholar.archive.org/work/zf42rnoharbcrcnxx5xdduvwty
This Supplement of 'Tijdschrift voor Stem-, Spraak- en Taalpathologie' (Journal of Voice, Speech and Language Pathology) contains the abstracts of the eighth edition of the International Conference on Speech Motor Control, held in Groningen, The Netherlands, August 24 - 27, 2022. With this eighth conference, a well-established Nijmegen (5 editions) - Groningen (6th & 7th edition) tradition continues. This conference, like the ones before, highlights new trends and state-of-the-art approaches in theoretical and applied research in the area of normal and disordered speech motor control. The five years since the previous conference in 2017 have yielded not only further insights in genetic, neural, physiological and developmental aspects of speech production, stuttering and other speech motor conditions, but have also advanced theoretical modelling. Combined with ongoing studies of populations that are increasingly better characterized genetically and neurobiologically, this quantitative boost of interdisciplinary results is now leading to a qualitative turning point in which large data sets are analyzed with powerful artificial intelligence and machine learning algorithms. The implementation of theories into computational models allows for the explicit testing of multifactorial interactions, thereby going beyond the more traditional single-factor experiments. Machine learning and the data sharing required to make this feasible are a special topic of the 2022 conference.Redactie SSTPwork_zf42rnoharbcrcnxx5xdduvwtyThu, 25 Aug 2022 00:00:00 GMTOn Gaussian multiplicative chaos and conformal field theory
https://scholar.archive.org/work/bhwea4esx5ci7euvz3qishl4ii
This thesis is concerned with conformally invariant stochastic processes in two dimensions and their applications to conformal field theory (CFT). The main probabilistic objects are the Gaussian free field (GFF) and the random geometries associated to it. Especially, we are interested in Gaussian multiplicative chaos (GMC), Schramm-Loewner evolution (SLE) and Liouville CFT, which can be understood as theories of random surfaces. From the point of view of physics, the idea of a "summing over surfaces" can be traced back to Polyakov's work on bosonic string theory. Indeed, the starting point of string theory is to replace a point particle by a one dimensional manifold (a string), so that one must replace the worldline by a worldsheet, i.e. an embedding of a surface into space-time. The path integral that Polyakov wrote down features a random conformal factor that should be described by the quantisation of the Liouville action. Therefore, this probability measure should describe random fluctuations around the uniform metric. Polyakov also suggested that the resulting quantum field theory should exhibit conformal invariance. This means that the Hilbert space of the theory should carry a projective unitary representation of the group of local conformal transformations, i.e. a unitary representation of the Virasoro algebra. Since it is an infinite dimensional Lie algebra, this is a huge constraint to put on a system and this led Belavin, Polyakov & Zamolodchikov to give an axiomatic framework for CFT based on the representation theory of the Virasoro algebra. Here, the game is somehow reversed: one tries to exhibit and classify all theories fitting in this framework. In particular, it is not even clear in the first place that such algebraic structures exist. In this context, Liouville theory is a success story in the interaction of algebra, geometry and probability. On the one hand, the algebraic point of view was successful in finding a theory fitting in the BPZ framework. On the other hand, it was unclear that t [...]Guillaume Baverez, Apollo-University Of Cambridge Repository, Jason Millerwork_bhwea4esx5ci7euvz3qishl4iiTue, 05 Apr 2022 00:00:00 GMTThis Week's Finds in Mathematical Physics (1-50)
https://scholar.archive.org/work/7dqichdfjraltdavtghiqngis4
These are the first 50 issues of This Week's Finds of Mathematical Physics, from January 19, 1993 to March 12, 1995. These issues focus on quantum gravity, topological quantum field theory, knot theory, and applications of n-categories to these subjects. However, there are also digressions into Lie algebras, elliptic curves, linear logic and other subjects. They were typeset in 2020 by Tim Hosgood. If you see typos or other problems please report them. (I already know the cover page looks weird).John C. Baezwork_7dqichdfjraltdavtghiqngis4Mon, 28 Feb 2022 00:00:00 GMTSome Fundamental Theorems in Mathematics
https://scholar.archive.org/work/6lqit72adje3zlo54s5zpgviem
An expository hitchhikers guide to some theorems in mathematics.Oliver Knillwork_6lqit72adje3zlo54s5zpgviemFri, 04 Feb 2022 00:00:00 GMTA friendly introduction to Fourier analysis on polytopes
https://scholar.archive.org/work/euftfnc5wjgihea4bts6etterm
This book is an introduction to the nascent field of Fourier analysis on polytopes, and cones. There is a rapidly growing number of applications of these methods, so it is appropriate to invite students, as well as professionals, to the field. Of the many applications of these techniques, we have chosen to focus on the following topics: (a) formulations for the Fourier transform of a polytope (b) Minkowski and Siegel's theorems in the geometry of numbers (c) tilings and multi-tilings of Euclidean space by translations of a polytope (d) Computing discrete volumes of polytopes, which are combinatorial approximations to the continuous volume (e) Optimizing sphere packings, and their packing density (f) Iterating the divergence theorem to give new formulations for the Fourier transform of a polytope, with applications (g) Shannon sampling, in several variables We assume familiarity with Linear Algebra, with some Calculus and infinite series. Throughout, we introduce the topics gently, by giving many examples and exercises, so that this book is ideally suited for a course, or for self-study.Sinai Robinswork_euftfnc5wjgihea4bts6ettermFri, 31 Dec 2021 00:00:00 GMT