The Bernoulli Operator This Is a Set of Working Notes
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The Transfer Operator Approach to ``Quantum Chaos
Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions Front page The transfer operator approach to “quantum chaos” Classical mechanics and the Laplace-Beltrami operator on PSL (Z) H 2 \ Tobias M¨uhlenbruch Joint work with D. Mayer and F. Str¨omberg Institut f¨ur Mathematik TU Clausthal [email protected] 22 January 2009, Fernuniversit¨at in Hagen Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions Outline of the presentation 1 Modular surface 2 Spectral theory 3 Geodesics and coding 4 Transfer operator 5 Connection to spectral theory 6 Conclusions Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions PSL Z The full modular group 2( ) Let PSL2(Z) be the full modular group PSL (Z)= S, T (ST )3 =1 = SL (Z) mod 1 2 h | i 2 {± } 0 1 1 1 1 0 with S = − , T = and 1 = . 1 0 0 1 0 1 a b az+b if z C, M¨obius transformations: z = cz+d ∈ c d a if z = . c ∞ Three orbits: the upper halfplane H = x + iy; y > 0 , the 1 { } projective real line PR and the lower half plane. z , z are PSL (Z)-equivalent if M PSL (Z) with 1 2 2 ∃ ∈ 2 M z1 = z2. The full modular group PSL2(Z) is generated by 1 translation T : z z + 1 and inversion S : z − . 7→ 7→ z (Closed) fundamental domain = z H; z 1, Re(z) 1 . F ∈ | |≥ | |≤ 2 Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions PSL Z H The modular surface 2( )\ The upper half plane H = x + iy; y > 0 can be viewed as a hyperbolic plane with constant{ negative} curvature 1. -
Arxiv:1910.08491V3 [Math.ST] 3 Nov 2020
Weakly stationary stochastic processes valued in a separable Hilbert space: Gramian-Cram´er representations and applications Amaury Durand ∗† Fran¸cois Roueff ∗ September 14, 2021 Abstract The spectral theory for weakly stationary processes valued in a separable Hilbert space has known renewed interest in the past decade. However, the recent literature on this topic is often based on restrictive assumptions or lacks important insights. In this paper, we follow earlier approaches which fully exploit the normal Hilbert module property of the space of Hilbert- valued random variables. This approach clarifies and completes the isomorphic relationship between the modular spectral domain to the modular time domain provided by the Gramian- Cram´er representation. We also discuss the general Bochner theorem and provide useful results on the composition and inversion of lag-invariant linear filters. Finally, we derive the Cram´er-Karhunen-Lo`eve decomposition and harmonic functional principal component analysis without relying on simplifying assumptions. 1 Introduction Functional data analysis has become an active field of research in the recent decades due to technological advances which makes it possible to store longitudinal data at very high frequency (see e.g. [22, 31]), or complex data e.g. in medical imaging [18, Chapter 9], [15], linguistics [28] or biophysics [27]. In these frameworks, the data is seen as valued in an infinite dimensional separable Hilbert space thus isomorphic to, and often taken to be, the function space L2(0, 1) of Lebesgue-square-integrable functions on [0, 1]. In this setting, a 2 functional time series refers to a bi-sequences (Xt)t∈Z of L (0, 1)-valued random variables and the assumption of finite second moment means that each random variable Xt belongs to the L2 Bochner space L2(Ω, F, L2(0, 1), P) of measurable mappings V : Ω → L2(0, 1) such that E 2 kV kL2(0,1) < ∞ , where k·kL2(0,1) here denotes the norm endowing the Hilbert space L2h(0, 1). -
An Introduction to Fractal Uncertainty Principle
An introduction to fractal uncertainty principle Semyon Dyatlov (MIT / UC Berkeley) The goal of this minicourse is to give a brief introduction to fractal uncertainty principle and its applications to transfer operators for Schottky groups Part 1: Schottky groups, transfer operators, and resonances Schottky groups of SL ( 2 IR ) on the action , Using ' 70 d z 3- H = fz E l Im its and on boundary H = IRV { a } by Mobius transformations : f- (Ibd ) ⇒ 8. z = aztbcztd Schottky groups provide interesting nonlinear dynamics on fractal limit sets and appear in many important applications To define a Schottky group, we fix: of r disks • 2. a collection nonintersecting IR in ¢ with centers in := . AIR Q . , Dzr Ij Dj := and 1 . • denote A { ,2r3 . Ifa Er Atr if I = , f - r if rsaE2r a , that • 8 . such , . Kr fix maps . , ' )= ti racial D; Da , E- Pc SLC 2,112) • The Schottky group free ti . is the group generated by . .fr Example of a Schottky group Here is a picture for the case of 4 disks: IC = = Cil K ( ) Dz , 8. ( Dj ) Dz , IDE - =D ID ) - Ds ( IC ID:) BCCI : , Oy , 83 ⇒ A Schottky quotients ' of Ta k i n g the quotient CH , ) action of P we a the , get by surface co - . convex compact hyperbolic µ ' - M - T l H IIF plane # Three-funneled surface in G - funnel infinite end Words and nested intervals the = 2r encodes Recall I 91 . } that , . P of the I := at r generators group , n : • Words of length " = W . an far / Vj , ajt, taj } ' = : . ⇒ = . ⑨ . - A , An OT A , Ah i • Group elements : : = E T - . -
The Theory of Filtrations of Subalgebras, Standardness and Independence
The theory of filtrations of subalgebras, standardness and independence Anatoly M. Vershik∗ 24.01.2017 In memory of my friend Kolya K. (1933{2014) \Independence is the best quality, the best word in all languages." J. Brodsky. From a personal letter. Abstract The survey is devoted to the combinatorial and metric theory of fil- trations, i. e., decreasing sequences of σ-algebras in measure spaces or decreasing sequences of subalgebras of certain algebras. One of the key notions, that of standardness, plays the role of a generalization of the no- tion of the independence of a sequence of random variables. We discuss the possibility of obtaining a classification of filtrations, their invariants, and various links to problems in algebra, dynamics, and combinatorics. Bibliography: 101 titles. Contents 1 Introduction 3 1.1 A simple example and a difficult question . .3 1.2 Three languages of measure theory . .5 1.3 Where do filtrations appear? . .6 1.4 Finite and infinite in classification problems; standardness as a generalization of independence . .9 arXiv:1705.06619v1 [math.DS] 18 May 2017 1.5 A summary of the paper . 12 2 The definition of filtrations in measure spaces 13 2.1 Measure theory: a Lebesgue space, basic facts . 13 2.2 Measurable partitions, filtrations . 14 2.3 Classification of measurable partitions . 16 ∗St. Petersburg Department of Steklov Mathematical Institute; St. Petersburg State Uni- versity Institute for Information Transmission Problems. E-mail: [email protected]. Par- tially supported by the Russian Science Foundation (grant No. 14-11-00581). 1 2.4 Classification of finite filtrations . 17 2.5 Filtrations we consider and how one can define them . -
Reproducing Kernel Hilbert Space Compactification of Unitary Evolution
Reproducing kernel Hilbert space compactification of unitary evolution groups Suddhasattwa Dasa, Dimitrios Giannakisa,∗, Joanna Slawinskab aCourant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA bFinnish Center for Artificial Intelligence, Department of Computer Science, University of Helsinki, Helsinki, Finland Abstract A framework for coherent pattern extraction and prediction of observables of measure-preserving, ergodic dynamical systems with both atomic and continuous spectral components is developed. This framework is based on an approximation of the generator of the system by a compact operator Wτ on a reproducing kernel Hilbert space (RKHS). A key element of this approach is that Wτ is skew-adjoint (unlike regularization approaches based on the addition of diffusion), and thus can be characterized by a unique projection- valued measure, discrete by compactness, and an associated orthonormal basis of eigenfunctions. These eigenfunctions can be ordered in terms of a Dirichlet energy on the RKHS, and provide a notion of coherent observables under the dynamics akin to the Koopman eigenfunctions associated with the atomic part of the spectrum. In addition, the regularized generator has a well-defined Borel functional calculus allowing tWτ the construction of a unitary evolution group fe gt2R on the RKHS, which approximates the unitary Koopman evolution group of the original system. We establish convergence results for the spectrum and Borel functional calculus of the regularized generator to those of the original system in the limit τ ! 0+. Convergence results are also established for a data-driven formulation, where these operators are approximated using finite-rank operators obtained from observed time series. An advantage of working in spaces of observables with an RKHS structure is that one can perform pointwise evaluation and interpolation through bounded linear operators, which is not possible in Lp spaces. -
The Entropy Function for Non Polynomial Problems and Its Applications for Turing Machines
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 30 January 2020 doi:10.20944/preprints202001.0360.v1 The entropy function for non polynomial problems and its applications for turing machines. Matheus Santana Harvard Extension School [email protected] Abstract We present a general process for the halting problem, valid regardless of the time and space computational complexity of the decision problem. It can be interpreted as the maximization of entropy for the utility function of a given Shannon-Kolmogorov-Bernoulli process. Applications to non-polynomials prob- lems are given. The new interpretation of information rate proposed in this work is a method that models the solution space boundaries of any decision problem (and non polynomial problems in general) as a communication channel by means of Information Theory. We described a sort method that order objects using the intrinsic information content distribution for the elements of a constrained solution space - modeled as messages transmitted through any communication systems. The limits of the search space are defined by the Kolmogorov-Chaitin complexity of the sequences encoded as Shannon-Bernoulli strings. We conclude with a discussion about the implications for general decision problems in Turing machines. Keywords: Computational Complexity, Information Theory, Machine Learning, Computational Statistics, Kolmogorov-Chaitin Complexity; Kelly criterion 1. Introduction Consider a Shanon-Bernoulli process P defined as a sequence of independent binary random variables X1, X2, X3 . Xn. For each element the value can be Preprint submitted to Journal of Information and Computation January 29, 2020 © 2020 by the author(s). Distributed under a Creative Commons CC BY license. -
Asymptotic Spectral Gap and Weyl Law for Ruelle Resonances of Open Partially Expanding Maps Jean-François Arnoldi, Frédéric Faure, Tobias Weich
Asymptotic spectral gap and Weyl law for Ruelle resonances of open partially expanding maps Jean-François Arnoldi, Frédéric Faure, Tobias Weich To cite this version: Jean-François Arnoldi, Frédéric Faure, Tobias Weich. Asymptotic spectral gap and Weyl law for Ruelle resonances of open partially expanding maps. Ergodic Theory and Dynamical Systems, Cambridge University Press (CUP), 2017, 37 (1), pp.1-58. 10.1017/etds.2015.34. hal-00787781v2 HAL Id: hal-00787781 https://hal.archives-ouvertes.fr/hal-00787781v2 Submitted on 4 Jun 2013 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. Asymptotic spectral gap and Weyl law for Ruelle resonances of open partially expanding maps Jean Francois Arnoldi∗, Frédéric Faure†, Tobias Weich‡ 01-06-2013 Abstract We consider a simple model of an open partially expanding map. Its trapped set in phase space is a fractal set. We first show that there is a well defined K discrete spectrum of Ruelle resonances which describes the asymptotisc of correlation functions for large time and which is parametrized by the Fourier component ν on the neutral direction of the dynamics. We introduce a specific hypothesis on the dynamics that we call “minimal captivity”. -
A Structural Approach to Solving the 6Th Hilbert Problem Udc 519.21
Teor Imovr.taMatem.Statist. Theor. Probability and Math. Statist. Vip. 71, 2004 No. 71, 2005, Pages 165–179 S 0094-9000(05)00656-3 Article electronically published on December 30, 2005 A STRUCTURAL APPROACH TO SOLVING THE 6TH HILBERT PROBLEM UDC 519.21 YU. I. PETUNIN AND D. A. KLYUSHIN Abstract. The paper deals with an approach to solving the 6th Hilbert problem based on interpreting the field of random events as a partially ordered set endowed with a natural order of random events obtained by formalization and modification of the frequency definition of probability. It is shown that the field of events forms an atomic generated, complete, and completely distributive Boolean algebra. The probability distribution of the field of events generated by random variables is studied. It is proved that the probability distribution generated by random variables is not a measure but only a finitely additive function of events in the case of continuous random variables (both rational- and real-valued). Introduction In 1900 in his lecture [1] David Hilbert formulated the problem of axiomatization of probability theory. Here is the corresponding quote from his lecture: “The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which mathematics plays an important part; in the first rank are the theory of probabilities and mechanics. As to the axioms of the theory of probabilities, it seems to me desirable that their logical investigation should be accompanied by a rigorous and satisfactory development of the method of mean values in mathematical physics, and in particular in the kinetic theory of gases.” As is well known [2], there is no generally accepted solution of this problem so far. -
Spectral Structure of Transfer Operators for Expanding Circle Maps
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Queen Mary Research Online SPECTRAL STRUCTURE OF TRANSFER OPERATORS FOR EXPANDING CIRCLE MAPS OSCAR F. BANDTLOW, WOLFRAM JUST, AND JULIA SLIPANTSCHUK Abstract. We explicitly determine the spectrum of transfer operators (acting on spaces of holomorphic functions) associated to analytic expanding circle maps arising from finite Blaschke products. This is achieved by deriving a convenient natural representation of the respective adjoint operators. 1. Introduction One of the major strands of modern ergodic theory is to exploit the rich links between dynamical systems theory and functional analysis, making the powerful tools of the latter available for the benefit of understanding complex dynamical be- haviour. In classical ergodic theory, composition operators occur naturally as basic objects for formulating concepts such as ergodicity or mixing [28]. These operators, known in this context as Koopman operators, are the formal adjoints of transfer op- erators, the spectral data of which provide insight into fine statistical properties of the underlying dynamical systems, such as rates of mixing (see, for example, [2, 7]). There exists a large body of literature on spectral estimates for transfer operators (and hence mixing properties) for various one- and higher-dimensional systems with different degree of expansivity or hyperbolicity (see [1] for an overview). However, to the best of our knowledge, no sufficiently rich class of dynamical systems to- gether with a machinery allowing for the explicit analytic determination of the entire spectrum is known. This is the gap the current contribution aims to fill. In the literature, the term `composition operator' is mostly used to refer to compositions with analytic functions mapping a disk into itself, a setting in which operator-theoretic properties such as boundedness, compactness, and most impor- tantly explicit spectral information are well-established (good references are [23] or the encyclopedia on the subject [9]). -
The Ergodic Theorem
The Ergodic Theorem 1 Introduction Ergodic theory is a branch of mathematics which uses measure theory to study the long term be- haviour of dynamic systems. The central object of consideration is known as a measure-preserving system, a type of dynamic system where the evolution of the system preserves a measure. Definition 1: Let (X; M; µ) be a finite measure space and let T map X to itself. We say that T is a measure-preserving transformation if for every A 2 M, we have µ(A) = µ(T −1(A)). The quadruple (X; M; µ, T ) is called a measure-preserving system (m.p.s.). Measure-preserving systems arise in a variety of contexts, such as probability theory, informa- tion theory, and of course in the study of dynamical systems. However, ergodic theory originated from statistical mechanics. In this setting, T represents the evolution of the system through time. Given a measurable function f : X ! R, the series of values f(x); f(T x); f(T 2x)::: are the values of a physical observable at certain time intervals. Of importance in statistical mechanics is the long-term average of these observables: N−1 1 X f (x) = f(T kx) N N k=0 The Ergodic Theorem (also known as the Pointwise or Birkhoff Ergodic Theorem) is central to the study of averages such as fN in the limit as N ! 1. In this paper we aim to prove the theorem, and then discuss a few of its applications. Before we can state the theorem, we need another definition. -
Harmonic Number Identities Via Polynomials with R-Lah Coefficients
Comptes Rendus Mathématique Levent Kargın and Mümün Can Harmonic number identities via polynomials with r-Lah coeYcients Volume 358, issue 5 (2020), p. 535-550. <https://doi.org/10.5802/crmath.53> © Académie des sciences, Paris and the authors, 2020. Some rights reserved. This article is licensed under the Creative Commons Attribution 4.0 International License. http://creativecommons.org/licenses/by/4.0/ Les Comptes Rendus. Mathématique sont membres du Centre Mersenne pour l’édition scientifique ouverte www.centre-mersenne.org Comptes Rendus Mathématique 2020, 358, nO 5, p. 535-550 https://doi.org/10.5802/crmath.53 Number Theory / Théorie des nombres Harmonic number identities via polynomials with r-Lah coeYcients Identités sur les nombres harmonique via des polynômes à coeYcients r-Lah , a a Levent Kargın¤ and Mümün Can a Department of Mathematics, Akdeniz University, Antalya, Turkey. E-mails: [email protected], [email protected]. Abstract. In this paper, polynomials whose coeYcients involve r -Lah numbers are used to evaluate several summation formulae involving binomial coeYcients, Stirling numbers, harmonic or hyperharmonic num- bers. Moreover, skew-hyperharmonic number is introduced and its basic properties are investigated. Résumé. Dans cet article, des polynômes à coeYcients faisant intervenir les nombres r -Lah sont utilisés pour établir plusieurs formules de sommation en fonction des coeYcients binomiaux, des nombres de Stirling et des nombres harmoniques ou hyper-harmoniques. De plus, nous introduisons le nombre asymétrique- hyper-harmonique et nous étudions ses propriétés de base. 2020 Mathematics Subject Classification. 11B75, 11B68, 47E05, 11B73, 11B83. Manuscript received 5th February 2020, revised 18th April 2020, accepted 19th April 2020. -
Covering Systems, Kubert Identities and Difference Equations
Mathematica Slovaca Štefan Porubský Covering systems, Kubert identities and difference equations Mathematica Slovaca, Vol. 50 (2000), No. 4, 381--413 Persistent URL: http://dml.cz/dmlcz/136785 Terms of use: © Mathematical Institute of the Slovak Academy of Sciences, 2000 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz Mathematlca Slovaca ©2000 Mathematical Institute Math. Slovaca, 50 (2000), No. 4, 381-413 Slovák Academy of Sciences With the deepest admiration to my esteemed friend Kdlmdn Gyory on the occasion of his sixties COVERING SYSTEMS, KUBERT IDENTITIES AND DIFFERENCE EQUATIONS STEFAN PORUBSKY (Communicated by Stanislav Jakubec ) ABSTRACT. A. S. Fraenkel proved that the following identities involving Ber noulli polynomials m • ч -»n(0) for all n > 0 are true if and only if the system of arithmetic congruences {a^ (mod 6^) : 1 < i < m} is an exact cover of Z. Generalizations of this result involving other functions and more general covering systems have been successively found by A. S. Fraenkel, J. Beebee, Z.-W. Sun and the author. Z.-W. Sun proved an alge braic characterization of functions capable of identities of this type, and indepen dently J. Beebee observed a connection of these results to the Raabe multiplica tion formula for Bernoulli polynomials and with the so-called Kubert identities.