Three Lectures on Algebraic Microlocal Analysis
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Dimension Formulas for the Hyperfunction Solutions to Holonomic D-Modules
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector ARTICLE IN PRESS Advances in Mathematics 180 (2003) 134–145 http://www.elsevier.com/locate/aim Dimension formulas for the hyperfunction solutions to holonomic D-modules Kiyoshi Takeuchi Institute of Mathematics, University of Tsukuba, 1-1-1, Tennodai, Tsukuba, Ibaraki 305-8571, Japan Received 15 July 2002; accepted 19 November 2002 Communicated by Takahiro Kawai Dedicated to Professor Mitsuo Morimoto on the Occasion of his 60th Birthday Abstract In this paper, we will prove an explicit dimension formula for the hyperfunction solutions to a class of holonomic D-modules. This dimension formula can be considered as a higher dimensional analogue of a beautiful theorem on ordinary differential equations due to Kashiwara (Master’s Thesis, University of Tokyo, 1970) and Komatsu (J. Fac. Sci. Univ. Tokyo Math. Sect. IA 18 (1971) 379). In the course of the proof, we will make use of a recent innovation of Schmid–Vilonen (Invent. Math. 124 (1996) 451) in the representation theory (in the theory of index theorems for constructible sheaves). r 2003 Elsevier Science (USA). All rights reserved. MSC: 32C38; 35A27 Keywords: D-module; Holonomic system; Index theorem 1. Introduction The aim of this paper is to prove a useful formula to compute the exact dimensions of the solutions to holonomic D-modules. The study of the dimension formula for the generalized function solutions to ordinary differential equations started only recently. In the 1970s, a beautiful theorem was obtained by Kashiwara [7] and Komatsu [12] using the hyperfunction theory. -
Algebraic Analysis of Rotation Data
11 : 2 2020 Algebraic Statistics ALGEBRAIC ANALYSIS OF ROTATION DATA MICHAEL F. ADAMER, ANDRÁS C. LORINCZ˝ , ANNA-LAURA SATTELBERGER AND BERND STURMFELS msp Algebraic Statistics Vol. 11, No. 2, 2020 https://doi.org/10.2140/astat.2020.11.189 msp ALGEBRAIC ANALYSIS OF ROTATION DATA MICHAEL F. ADAMER, ANDRÁS C. LORINCZ˝ , ANNA-LAURA SATTELBERGER AND BERND STURMFELS We develop algebraic tools for statistical inference from samples of rotation matrices. This rests on the theory of D-modules in algebraic analysis. Noncommutative Gröbner bases are used to design numerical algorithms for maximum likelihood estimation, building on the holonomic gradient method of Sei, Shibata, Takemura, Ohara, and Takayama. We study the Fisher model for sampling from rotation matrices, and we apply our algorithms to data from the applied sciences. On the theoretical side, we generalize the underlying equivariant D-modules from SO.3/ to arbitrary Lie groups. For compact groups, our D-ideals encode the normalizing constant of the Fisher model. 1. Introduction Many of the multivariate functions that arise in statistical inference are holonomic. Being holonomic roughly means that the function is annihilated by a system of linear partial differential operators with polynomial coefficients whose solution space is finite-dimensional. Such a system of PDEs can be written as a left ideal in the Weyl algebra, or D-ideal, for short. This representation allows for the application of algebraic geometry and algebraic analysis, including the use of computational tools, such as Gröbner bases in the Weyl algebra[28; 30]. This important connection between statistics and algebraic analysis was first observed by a group of scholars in Japan, and it led to their development of the holonomic gradient method (HGM) and the holonomic gradient descent (HGD). -
Mikio Sato, a Visionary of Mathematics, Volume 54, Number 2
Mikio Sato, a Visionary of Mathematics Pierre Schapira Like singularities in mathematics and physics, ideas People were essentially looking for existence propagate, and the speed of propagation depends theorems for linear partial differential equations highly on the energy put into promoting them. (PDE), and most of the proofs were reduced to Mikio Sato did not spend a great deal of time or finding “the right functional space”, to prove some energy popularizing his ideas. We can hope that a priori estimate and apply the Hahn-Banach his receipt of the 2002/2003 Wolf Prize1 will help theorem. make better known his deep work, which is perhaps It was in this environment that Mikio Sato too original to be immediately accepted. Sato does defined hyperfunctions in 1959–1960 as boundary not write a lot, does not communicate easily, and values of holomorphic functions, a discovery that attends very few meetings. But he invented a new allowed him to obtain a position at Tokyo Univer- way of doing analysis, “Algebraic Analysis”, and sity thanks to the clever patronage of Shokichi created a school, “the Kyoto school”. Iyanaga, an exceptionally open-minded person 2 Born in 1928 , Sato became known in mathe- and a great friend of French culture. Next, Sato matics only in 1959–1960 with his theory of spent two years in the USA, in Princeton, where he hyperfunctions. Indeed, his studies had been unsuccessfully tried to convince André Weil of the seriously disrupted by the war, particularly by relevance of his cohomological approach to the bombing of Tokyo. After his family home analysis. -
An Elementary Theory of Hyperfunctions and Microfunctions
PARTIAL DIFFERENTIAL EQUATIONS BANACH CENTER PUBLICATIONS, VOLUME 27 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1992 AN ELEMENTARY THEORY OF HYPERFUNCTIONS AND MICROFUNCTIONS HIKOSABURO KOMATSU Department of Mathematics, Faculty of Science, University of Tokyo Tokyo, 113 Japan 1. Sato’s definition of hyperfunctions. The hyperfunctions are a class of generalized functions introduced by M. Sato [34], [35], [36] in 1958–60, only ten years later than Schwartz’ distributions [40]. As we will see, hyperfunctions are natural and useful, but unfortunately they are not so commonly used as distributions. One reason seems to be that the mere definition of hyperfunctions needs a lot of preparations. In the one-dimensional case his definition is elementary. Let Ω be an open set in R. Then the space (Ω) of hyperfunctions on Ω is defined to be the quotient space B (1.1) (Ω)= (V Ω)/ (V ) , B O \ O where V is an open set in C containing Ω as a closed set, and (V Ω) (resp. (V )) is the space of all holomorphic functions on V Ω (resp. VO). The\ hyper- functionO f(x) represented by F (z) (V Ω) is written\ ∈ O \ (1.2) f(x)= F (x + i0) F (x i0) − − and has the intuitive meaning of the difference of the “boundary values”of F (z) on Ω from above and below. V Ω Fig. 1 [233] 234 H. KOMATSU The main properties of hyperfunctions are the following: (1) (Ω), Ω R, form a sheaf over R. B ⊂ Ω Namely, for all pairs Ω1 Ω of open sets the restriction mappings ̺Ω1 : (Ω) (Ω ) are defined, and⊂ for any open covering Ω = Ω they satisfy the B →B 1 α following conditions: S (S.1) If f (Ω) satisfies f = 0 for all α, then f = 0; ∈B |Ωα (S.2) If fα (Ωα) satisfy fα Ωα∩Ωβ = fβ Ωα∩Ωβ for all Ωα Ωβ = , then there is∈ Ban f (Ω) such| that f = f| . -
D-Modules, Perverse Sheaves, and Representation Theory
Progress in Mathematics Volume 236 Series Editors Hyman Bass Joseph Oesterle´ Alan Weinstein Ryoshi Hotta Kiyoshi Takeuchi Toshiyuki Tanisaki D-Modules, Perverse Sheaves, and Representation Theory Translated by Kiyoshi Takeuchi Birkhauser¨ Boston • Basel • Berlin Ryoshi Hotta Kiyoshi Takeuchi Professor Emeritus of Tohoku University School of Mathematics Shirako 2-25-1-1106 Tsukuba University Wako 351-0101 Tenoudai 1-1-1 Japan Tsukuba 305-8571 [email protected] Japan [email protected] Toshiyuki Tanisaki Department of Mathematics Graduate School of Science Osaka City University 3-3-138 Sugimoto Sumiyoshi-ku Osaka 558-8585 Japan [email protected] Mathematics Subject Classification (2000): Primary: 32C38, 20G05; Secondary: 32S35, 32S60, 17B10 Library of Congress Control Number: 2004059581 ISBN-13: 978-0-8176-4363-8 e-ISBN-13: 978-0-8176-4523-6 Printed on acid-free paper. c 2008, English Edition Birkhauser¨ Boston c 1995, Japanese Edition, Springer-Verlag Tokyo, D Kagun to Daisugun (D-Modules and Algebraic Groups) by R. Hotta and T. Tanisaki. All rights reserved. This work may not be translated or copied in whole or in part without the writ- ten permission of the publisher (Birkhauser¨ Boston, c/o Springer Science Business Media LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter de- veloped is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. -
27 Oct 2008 Masaki Kashiwara and Algebraic Analysis
Masaki Kashiwara and Algebraic Analysis Pierre Schapira Abstract This paper is based on a talk given in honor of Masaki Kashiwara’s 60th birthday, Kyoto, June 27, 2007. It is a brief overview of his main contributions in the domain of microlocal and algebraic analysis. It is a great honor to present here some aspects of the work of Masaki Kashiwara. Recall that Masaki’s work covers many fields of mathematics, algebraic and microlocal analysis of course, but also representation theory, Hodge the- ory, integrable systems, quantum groups and so on. Also recall that Masaki had many collaborators, among whom Daniel Barlet, Jean-Luc Brylinski, Et- surio Date, Ryoji Hotta, Michio Jimbo, Seok-Jin Kang, Takahiro Kawai, Tet- suji Miwa, Kiyosato Okamoto, Toshio Oshima, Mikio Sato, myself, Toshiyuki Tanisaki and Mich`ele Vergne. In each of the domain he approached, Masaki has given essential contri- butions and made important discoveries, such as, for example, the existence of crystal bases in quantum groups. But in this talk, I will restrict myself to describe some part of his work related to microlocal and algebraic analysis. The story begins long ago, in the early sixties, when Mikio Sato created a new branch of mathematics now called “Algebraic Analysis”. In 1959/60, arXiv:0810.4875v1 [math.HO] 27 Oct 2008 M. Sato published two papers on hyperfunction theory [24] and then devel- oped his vision of analysis and linear partial differential equations in a series of lectures at Tokyo University (see [1]). If M is a real analytic manifold and X is a complexification of M, hyperfunctions on M are cohomology classes supported by M of the sheaf X of holomorphic functions on X. -
Algebraic Analysis of Differential Equations
T. Aoki· H. Majima- Y. Takei· N. Tose (Eds.) Algebraic Analysis of Differential Equations from Microlocal Analysis to Exponential Asymptotics Festschrift in Honor of Takahiro Kawai ~ Springer Editors Takashi Aoki Department of Mathematics Kinki University Higashi-Osaka 577-8502, Japan e-mail: [email protected] Hideyuki Majima Department of Mathematics Ochanomizu University Tokyo 112-8610, Japan e-mail: [email protected] Yoshitsugu Takei Research Institute for Mathematical Sciences Kyoto University Kyoto 606-8502, Japan e-mail: [email protected] Nobuyuki Tose Faculty of Economics Keio University Yokohama 223-8521, Japan e-mail: [email protected] Library of Congress Control Number: 2007939560 ISBN 978-4-431-73239-6 Springer Tokyo Berlin Heidelberg New York Springer is a part of Springer Science+Business Media springer.com c Springer 2008 This work is subject to copyright. All rights are reserved, whether the whole or a part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Camera-ready copy prepared from the authors’ LATEXfiles. Printed and bound by Shinano Co. Ltd., Japan. SPIN: 12081080 Printed on acid-free paper Dedicated to Professor Takahiro Kawai on the Occasion of His Sixtieth Birthday Preface This is a collection of articles on algebraic analysis of differential equations and related topics. -
An Introduction to Constructive Algebraic Analysis and Its Applications Alban Quadrat
An introduction to constructive algebraic analysis and its applications Alban Quadrat To cite this version: Alban Quadrat. An introduction to constructive algebraic analysis and its applications. [Research Report] RR-7354, INRIA. 2010, pp.237. inria-00506104 HAL Id: inria-00506104 https://hal.inria.fr/inria-00506104 Submitted on 27 Jul 2010 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE Une introduction à l’analyse algébrique constructive et à ses applications Alban Quadrat N° 7354 July 2010 Algorithms, Certification, and Cryptography apport de recherche ISSN 0249-6399 ISRN INRIA/RR--7354--FR+ENG Une introduction à l’analyse algébrique constructive et à ses applications Alban Quadrat ∗ Theme : Algorithms, Certification, and Cryptography Équipe-Projet AT-SOP Rapport de recherche n° 7354 — July 2010 — 237 pages Résumé : Ce texte est une extension des notes de cours que j’ai préparés pour les Journées Nationales de Calcul Formel qui ont eu lieu au CIRM, Luminy (France) du 3 au 7 Mai 2010. Le but principal de ce cours était d’introduire la communauté française du calcul formel à l’analyse algébrique constructive, et particulièrement à la théorie des D-modules algébriques, à ses applications à la théorie mathématique des systèmes et à ses implantations dans des logiciels de calcul formel tels que Maple ou GAP4. -
Introduction to Hyperfunctions and Their Integral Transforms
[chapter] [chapter] [chapter] 1 2 Introduction to Hyperfunctions and Their Integral Transforms Urs E. Graf 2009 ii Contents Preface ix 1 Introduction to Hyperfunctions 1 1.1 Generalized Functions . 1 1.2 The Concept of a Hyperfunction . 3 1.3 Properties of Hyperfunctions . 13 1.3.1 Linear Substitution . 13 1.3.2 Hyperfunctions of the Type f(φ(x)) . 15 1.3.3 Differentiation . 18 1.3.4 The Shift Operator as a Differential Operator . 25 1.3.5 Parity, Complex Conjugate and Realness . 25 1.3.6 The Equation φ(x)f(x) = h(x)............... 28 1.4 Finite Part Hyperfunctions . 33 1.5 Integrals . 37 1.5.1 Integrals with respect to the Independent Variable . 37 1.5.2 Integrals with respect to a Parameter . 43 1.6 More Familiar Hyperfunctions . 44 1.6.1 Unit-Step, Delta Impulses, Sign, Characteristic Hyper- functions . 44 1.6.2 Integral Powers . 45 1.6.3 Non-integral Powers . 49 1.6.4 Logarithms . 51 1.6.5 Upper and Lower Hyperfunctions . 55 α 1.6.6 The Normalized Power x+=Γ(α + 1) . 58 1.6.7 Hyperfunctions Concentrated at One Point . 61 2 Analytic Properties 63 2.1 Sequences, Series, Limits . 63 2.2 Cauchy-type Integrals . 71 2.3 Projections of Functions . 75 2.3.1 Functions satisfying the H¨olderCondition . 77 2.3.2 Projection Theorems . 78 2.3.3 Convergence Factors . 87 2.3.4 Homologous and Standard Hyperfunctions . 88 2.4 Projections of Hyperfunctions . 91 2.4.1 Holomorphic and Meromorphic Hyperfunctions . 91 2.4.2 Standard Defining Functions . -
An Integral Formula for the Powered Sum of the Independent, Identically
An integral formula for the powered sum of the independent, identically and normally distributed random variables Tamio Koyama Abstract The distribution of the sum of r-th power of standard normal random variables is a generalization of the chi-squared distribution. In this paper, we represent the probability density function of the random variable by an one-dimensional absolutely convergent integral with the characteristic function. Our integral formula is expected to be applied for evaluation of the density function. Our integral formula is based on the inversion formula, and we utilize a summation method. We also discuss on our formula in the view point of hyperfunctions. 1 Introduction Let Xk k N be a sequence of the independent, identically distributed random variables, and suppose the { } ∈ distribution of each variable Xk is the standard normal distribution. Fix a positive integer r and n. In this paper, we discuss the sum of the r-th power of standard normal random variables. This distribution function can be written in the form of n dimensional integral: n 1 1 n P r 2 F (c) := Xk <c = n/2 exp xk dx1 . dxn (1) n r (2π) −2 ! Pk=1 xk<c ! Xk=1 Z kX=1 In the case where r = 2, this integral equals to the cumulative distribution function of the chi-squared n r statistics. The sum of the r-th power k=1 Xk is an generalization of the chi-squared statistics. The n 2 weighted sum k=1 wkXk , where wk is a positive real number, of the squared normal variables is another generalization of the chi-squared statistics.P Numerical analysis of the cumulative distribution function of the weighted sumP is discussed in [1] and [7]. -
Nonlinear Generalized Functions: Their Origin, Some Developments and Recent Advances
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Cadernos Espinosanos (E-Journal) S~aoPaulo Journal of Mathematical Sciences 7, 2 (2013), 201{239 Nonlinear Generalized Functions: their origin, some developments and recent advances Jean F. Colombeau Institut Fourier, Universit´ede Grenoble (retired) Abstract. We expose some simple facts at the interplay between math- ematics and the real world, putting in evidence mathematical objects " nonlinear generalized functions" that are needed to model the real world, which appear to have been generally neglected up to now by mathematicians. Then we describe how a "nonlinear theory of gen- eralized functions" was obtained inside the Leopoldo Nachbin group of infinite dimensional holomorphy between 1980 and 1985 **. This new theory permits to multiply arbitrary distributions and contains the above mathematical objects, which shows that the features of this theory are natural and unavoidable for a mathematical description of the real world. Finally we present direct applications of the theory such as existence-uniqueness for systems of PDEs without classical so- lutions and calculations of shock waves for systems in non-divergence form done between 1985 and 1995 ***, for which we give three examples of different nature (elasticity, cosmology, multifluid flows). * work done under support of FAPESP, processo 2011/12532-1, and thanks to the hospitality of the IME-USP. ** various supports from Fapesp, Finep and Cnpq between 1978 and 1984. *** support of "Direction des Recherches, Etudes´ et Techniques", Minist`ere de la D´efense,France, between 1985 and 1995. [email protected] 1. Mathematics and the real world. -
Elements of Computer-Algebraic Analysis 2 G
Operator algebras, partial classification Operator algebras, partial classification More general framework: G-algebras More general framework: G-algebras Recommended literature (textbooks) 1 J. C. McConnell, J. C. Robson, ”Noncommutative Noetherian Rings”, Graduate Studies in Mathematics, 30, AMS (2001) Elements of Computer-Algebraic Analysis 2 G. R. Krause and T. H. Lenagan, ”Growth of Algebras and Gelfand-Kirillov Dimension”, Graduate Studies in Mathematics, 22, AMS (2000) Viktor Levandovskyy 3 S. Saito, B. Sturmfels and N. Takayama, “Gr¨obner Lehrstuhl D f¨urMathematik, RWTH Aachen, Germany Deformations of Hypergeometric Differential Equations”, Springer, 2000 4 J. Bueso, J. G´omez–Torrecillas and A. Verschoren, 22.07. Tutorial at ISSAC 2012, Grenoble, France ”Algorithmic methods in non-commutative algebra. Applications to quantum groups”, Kluwer , 2003 5 H. Kredel, ”Solvable polynomial rings”, Shaker Verlag, 1993 6 H. Li, ”Noncommutative Gr¨obnerbases and filtered-graded transfer”, Springer, 2002 VL Elements of CAAN VL Elements of CAAN Operator algebras, partial classification Operator algebras, partial classification More general framework: G-algebras More general framework: G-algebras Recommended literature (textbooks and PhD theses) Software D-modules and algebraic analysis: 1 V. Ufnarovski, ”Combinatorial and Asymptotic Methods of Algebra”, Springer, Encyclopedia of Mathematical Sciences 57 kan/sm1 by N. Takayama et al. (1995) D-modules package in Macaulay2 by A. Leykin and H. Tsai 2 F. Chyzak, ”Fonctions holonomes en calcul formel”, Risa/Asir by M. Noro et al. PhD. Thesis, INRIA, 1998 OreModules package suite for Maple by D. Robertz, 3 V. Levandovskyy, ”Non-commutative Computer Algebra for A. Quadrat et al. polynomial algebras: Gr¨obner bases, applications and Singular:Plural with a D-module suite; by V.