Nonlinear Generalized Functions: Their Origin, Some Developments and Recent Advances
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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). -
GENERALIZED FUNCTIONS LECTURES Contents 1. the Space
GENERALIZED FUNCTIONS LECTURES Contents 1. The space of generalized functions on Rn 2 1.1. Motivation 2 1.2. Basic definitions 3 1.3. Derivatives of generalized functions 5 1.4. The support of generalized functions 6 1.5. Products and convolutions of generalized functions 7 1.6. Generalized functions on Rn 8 1.7. Generalized functions and differential operators 9 1.8. Regularization of generalized functions 9 n 2. Topological properties of Cc∞(R ) 11 2.1. Normed spaces 11 2.2. Topological vector spaces 12 2.3. Defining completeness 14 2.4. Fréchet spaces 15 2.5. Sequence spaces 17 2.6. Direct limits of Fréchet spaces 18 2.7. Topologies on the space of distributions 19 n 3. Geometric properties of C−∞(R ) 21 3.1. Sheaf of distributions 21 3.2. Filtration on spaces of distributions 23 3.3. Functions and distributions on a Cartesian product 24 4. p-adic numbers and `-spaces 26 4.1. Defining p-adic numbers 26 4.2. Misc. -not sure what to do with them (add to an appendix about p-adic numbers?) 29 4.3. p-adic expansions 31 4.4. Inverse limits 32 4.5. Haar measure and local fields 32 4.6. Some basic properties of `-spaces. 33 4.7. Distributions on `-spaces 35 4.8. Distributions supported on a subspace 35 5. Vector valued distributions 37 Date: October 7, 2018. 1 GENERALIZED FUNCTIONS LECTURES 2 5.1. Smooth measures 37 5.2. Generalized functions versus distributions 38 5.3. Some linear algebra 39 5.4. Generalized functions supported on a subspace 41 6. -
Schwartz Functions on Nash Manifolds and Applications to Representation Theory
SCHWARTZ FUNCTIONS ON NASH MANIFOLDS AND APPLICATIONS TO REPRESENTATION THEORY AVRAHAM AIZENBUD Joint with Dmitry Gourevitch and Eitan Sayag arXiv:0704.2891 [math.AG], arxiv:0709.1273 [math.RT] Let us start with the following motivating example. Consider the circle S1, let N ⊂ S1 be the north pole and denote U := S1 n N. Note that U is diffeomorphic to R via the stereographic projection. Consider the space D(S1) of distributions on S1, that is the space of continuous linear functionals on the 1 1 1 Fr´echet space C (S ). Consider the subspace DS1 (N) ⊂ D(S ) consisting of all distributions supported 1 at N. Then the quotient D(S )=DS1 (N) will not be the space of distributions on U. However, it will be the space S∗(U) of Schwartz distributions on U, that is continuous functionals on the Fr´echet space S(U) of Schwartz functions on U. In this case, S(U) can be identified with S(R) via the stereographic projection. The space of Schwartz functions on R is defined to be the space of all infinitely differentiable functions that rapidly decay at infinity together with all their derivatives, i.e. xnf (k) is bounded for any n; k. In this talk we extend the notions of Schwartz functions and Schwartz distributions to a larger geometric realm. As we can see, the definition is of algebraic nature. Hence it would not be reasonable to try to extend it to arbitrary smooth manifolds. However, it is reasonable to extend this notion to smooth algebraic varieties. -
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. -
Fundamental Theorems in Mathematics
SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635]. -
Mathematical Problems on Generalized Functions and the Can–
Mathematical problems on generalized functions and the canonical Hamiltonian formalism J.F.Colombeau [email protected] Abstract. This text is addressed to mathematicians who are interested in generalized functions and unbounded operators on a Hilbert space. We expose in detail (in a “formal way”- as done by Heisenberg and Pauli - i.e. without mathematical definitions and then, of course, without mathematical rigour) the Heisenberg-Pauli calculations on the simplest model close to physics. The problem for mathematicians is to give a mathematical sense to these calculations, which is possible without any knowledge in physics, since they mimick exactly usual calculations on C ∞ functions and on bounded operators, and can be considered at a purely mathematical level, ignoring physics in a first step. The mathematical tools to be used are nonlinear generalized functions, unbounded operators on a Hilbert space and computer calculations. This text is the improved written version of a talk at the congress on linear and nonlinear generalized functions “Gf 07” held in Bedlewo-Poznan, Poland, 2-8 September 2007. 1-Introduction . The Heisenberg-Pauli calculations (1929) [We,p20] are a set of 3 or 4 pages calculations (in a simple yet fully representative model) that are formally quite easy and mimick calculations on C ∞ functions. They are explained in detail at the beginning of this text and in the appendices. The H-P calculations [We, p20,21, p293-336] are a basic formulation in Quantum Field Theory: “canonical Hamiltonian formalism”, see [We, p292] for their relevance. The canonical Hamiltonian formalism is considered as mainly equivalent to the more recent “(Feynman) path integral formalism”: see [We, p376,377] for the connections between the 2 formalisms that complement each other. -
Introduction to the Fourier Transform
Chapter 4 Introduction to the Fourier transform In this chapter we introduce the Fourier transform and review some of its basic properties. The Fourier transform is the \swiss army knife" of mathematical analysis; it is a powerful general purpose tool with many useful special features. In particular the theory of the Fourier transform is largely independent of the dimension: the theory of the Fourier trans- form for functions of one variable is formally the same as the theory for functions of 2, 3 or n variables. This is in marked contrast to the Radon, or X-ray transforms. For simplicity we begin with a discussion of the basic concepts for functions of a single variable, though in some de¯nitions, where there is no additional di±culty, we treat the general case from the outset. 4.1 The complex exponential function. See: 2.2, A.4.3 . The building block for the Fourier transform is the complex exponential function, eix: The basic facts about the exponential function can be found in section A.4.3. Recall that the polar coordinates (r; θ) correspond to the point with rectangular coordinates (r cos θ; r sin θ): As a complex number this is r(cos θ + i sin θ) = reiθ: Multiplication of complex numbers is very easy using the polar representation. If z = reiθ and w = ½eiÁ then zw = reiθ½eiÁ = r½ei(θ+Á): A positive number r has a real logarithm, s = log r; so that a complex number can also be expressed in the form z = es+iθ: The logarithm of z is therefore de¯ned to be the complex number Im z log z = s + iθ = log z + i tan¡1 : j j Re z µ ¶ 83 84 CHAPTER 4. -
Tempered Generalized Functions Algebra, Hermite Expansions and Itoˆ Formula
TEMPERED GENERALIZED FUNCTIONS ALGEBRA, HERMITE EXPANSIONS AND ITOˆ FORMULA PEDRO CATUOGNO AND CHRISTIAN OLIVERA Abstract. The space of tempered distributions S0 can be realized as a se- quence spaces by means of the Hermite representation theorems (see [2]). In this work we introduce and study a new tempered generalized functions algebra H, in this algebra the tempered distributions are embedding via its Hermite ex- pansion. We study the Fourier transform, point value of generalized tempered functions and the relation of the product of generalized tempered functions with the Hermite product of tempered distributions (see [6]). Furthermore, we give a generalized Itˆoformula for elements of H and finally we show some applications to stochastic analysis. 1. Introduction The differential algebras of generalized functions of Colombeau type were developed in connection with non linear problems. These algebras are a good frame to solve differential equations with rough initial date or discontinuous coefficients (see [3], [8] and [13]). Recently there are a great interest in develop a stochastic calculus in algebras of generalized functions (see for instance [1], [4], [11], [12] and [14]), in order to solve stochastic differential equations with rough data. A Colombeau algebra G on an open subset Ω of Rm is a differential algebra containing D0(Ω) as a linear subspace and C∞(Ω) as a faithful subalgebra. The embedding of D0(Ω) into G is done via convolution with a mollifier, in the simplified version the embedding depends on the particular mollifier. The algebra of tempered generalized functions was introduced by J. F. Colombeau in [5] in order to develop a theory of Fourier transform in algebras of generalized functions (see [8] and [7] for applications and references). -
Applications of Fourier Transforms to Generalized Functions
Applications of Fourier Transforms to Generalized Functions WITPRESS WIT Press publishes leading books in Science and Technology. Visit our website for the current list of titles. www.witpress.com WITeLibrary Home of the Transactions of the Wessex Institute, the WIT electronic-library provides the international scientific community with immediate and permanent access to individual papers presented at WIT conferences. Visit the WIT eLibrary athttp://library.witpress.com Applications of Fourier Transforms to Generalized Functions M. Rahman Halifax, Nova Scotia, Canada M. Rahman Halifax, Nova Scotia, Canada Published by WIT Press Ashurst Lodge, Ashurst, Southampton, SO40 7AA, UK Tel: 44 (0) 238 029 3223; Fax: 44 (0) 238 029 2853 E-Mail: [email protected] http://www.witpress.com For USA, Canada and Mexico WIT Press 25 Bridge Street, Billerica, MA 01821, USA Tel: 978 667 5841; Fax: 978 667 7582 E-Mail: [email protected] http://www.witpress.com British Library Cataloguing-in-Publication Data A Catalogue record for this book is available from the British Library ISBN: 978-1-84564-564-9 Library of Congress Catalog Card Number: 2010942839 The texts of the papers in this volume were set individually by the authors or under their supervision. No responsibility is assumed by the Publisher, the Editors and Authors for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. The Publisher does not necessarily endorse the ideas held, or views expressed by the Editors or Authors of the material contained in its publications. -
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. -
Generalized Functions: Homework 1
Generalized Functions: Homework 1 Exercise 1. Prove that there exists a function f Cc∞(R) which isn’t the zero function. ∈ Solution: Consider the function: 1 e x x > 0 g = − 0 x 0 ( ≤ 1 We claim that g C∞(R). Indeed, all of the derivatives of e− x are vanishing when x 0: ∈ n → d 1 1 lim e− x = lim q(x) e− x = 0 x 0 n x 0 → dx → ∙ where q(x) is a rational function of x. The function f = g(x) g(1 x) is smooth, as a multiplication of two smooth functions, has compact support∙ − (the interval 1 1 [0, 1]), and is non-zero (f 2 = e ). Exercise 2. Find a sequence of functions fn n Z (when fn Cc∞ (R) n) that weakly converges to Dirac Delta function.{ } ∈ ∈ ∀ Solution: Consider the sequence of function: ˜ 1 1 fn = g x g + x C∞(R) n − ∙ n ∈ c where g(x) Cc∞(R) is the function defined in the first question. Notice that ∈ 1 1 the support of fn is the interval n , n . Normalizing these functions so that the integral will be equal to 1 we get:− 1 ∞ − f = d(n)f˜ , d(n) = f˜ (x)dx n n n Z −∞ For every test function F (x) C∞(R), we can write F (x) = F (0) + x G(x), ∈ c ∙ where G(x) C∞(R). Therefore we have: ∈ c 1/n ∞ F (x) f (x)dx = F (x) f (x)dx ∙ n ∙ n Z 1Z/n −∞ − 1/n 1/n = F (0) f (x)dx + xG(x) f (x)dx ∙ n ∙ n 1Z/n 1Z/n − − 1/n = F (0) + xG(x) f (x)dx ∙ n 1Z/n − 1 1/n ∞ ∞ So F (x) fn(x)dx F (x) δ(x)dx = 1/n xG(x) fn(x)dx. -
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.