Functionals and the Functional Derivative
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Math 346 Lecture #3 6.3 the General Fréchet Derivative
Math 346 Lecture #3 6.3 The General Fr´echet Derivative We now extend the notion of the Fr´echet derivative to the Banach space setting. Throughout let (X; k · kX ) and (Y; k · kY ) be Banach spaces, and U an open set in X. 6.3.1 The Fr´echet Derivative Definition 6.3.1. A function f : U ! Y is Fr´echet differentiable at x 2 U if there exists A 2 B(X; Y ) such that kf(x + h) − f(x) − A(h)k lim Y = 0; h!0 khkX and we write Df(x) = A. We say f is Fr´echet differentiable on U if f is Fr´echet differ- entiable at each x 2 U. We often refer to Fr´echet differentiable simply as differentiable. Note. When f is Fr´echet differentiable at x 2 U, the derivative Df(x) is unique. This follows from Proposition 6.2.10 (uniqueness of the derivative for finite dimensional X and Y ) whose proof carries over without change to the general Banach space setting (see Remark 6.3.9). Remark 6.3.2. The existence of the Fr´echet derivative does not change when the norm on X is replace by a topologically equivalent one and/or the norm on Y is replaced by a topologically equivalent one. Example 6.3.3. Any L 2 B(X; Y ) is Fr´echet differentiable with DL(x)(v) = L(v) for all x 2 X and all v 2 X. The proof of this is exactly the same as that given in Example 6.2.5 for finite-dimensional Banach spaces. -
A Short Introduction to the Quantum Formalism[S]
A short introduction to the quantum formalism[s] François David Institut de Physique Théorique CNRS, URA 2306, F-91191 Gif-sur-Yvette, France CEA, IPhT, F-91191 Gif-sur-Yvette, France [email protected] These notes are an elaboration on: (i) a short course that I gave at the IPhT-Saclay in May- June 2012; (ii) a previous letter [Dav11] on reversibility in quantum mechanics. They present an introductory, but hopefully coherent, view of the main formalizations of quantum mechanics, of their interrelations and of their common physical underpinnings: causality, reversibility and locality/separability. The approaches covered are mainly: (ii) the canonical formalism; (ii) the algebraic formalism; (iii) the quantum logic formulation. Other subjects: quantum information approaches, quantum correlations, contextuality and non-locality issues, quantum measurements, interpretations and alternate theories, quantum gravity, are only very briefly and superficially discussed. Most of the material is not new, but is presented in an original, homogeneous and hopefully not technical or abstract way. I try to define simply all the mathematical concepts used and to justify them physically. These notes should be accessible to young physicists (graduate level) with a good knowledge of the standard formalism of quantum mechanics, and some interest for theoretical physics (and mathematics). These notes do not cover the historical and philosophical aspects of quantum physics. arXiv:1211.5627v1 [math-ph] 24 Nov 2012 Preprint IPhT t12/042 ii CONTENTS Contents 1 Introduction 1-1 1.1 Motivation . 1-1 1.2 Organization . 1-2 1.3 What this course is not! . 1-3 1.4 Acknowledgements . 1-3 2 Reminders 2-1 2.1 Classical mechanics . -
Functional Integration on Paracompact Manifods Pierre Grange, E
Functional Integration on Paracompact Manifods Pierre Grange, E. Werner To cite this version: Pierre Grange, E. Werner. Functional Integration on Paracompact Manifods: Functional Integra- tion on Manifold. Theoretical and Mathematical Physics, Consultants bureau, 2018, pp.1-29. hal- 01942764 HAL Id: hal-01942764 https://hal.archives-ouvertes.fr/hal-01942764 Submitted on 3 Dec 2018 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. Functional Integration on Paracompact Manifolds Pierre Grangé Laboratoire Univers et Particules, Université Montpellier II, CNRS/IN2P3, Place E. Bataillon F-34095 Montpellier Cedex 05, France E-mail: [email protected] Ernst Werner Institut fu¨r Theoretische Physik, Universita¨t Regensburg, Universita¨tstrasse 31, D-93053 Regensburg, Germany E-mail: [email protected] ........................................................................ Abstract. In 1948 Feynman introduced functional integration. Long ago the problematic aspect of measures in the space of fields was overcome with the introduction of volume elements in Probability Space, leading to stochastic formulations. More recently Cartier and DeWitt-Morette (CDWM) focused on the definition of a proper integration measure and established a rigorous mathematical formulation of functional integration. CDWM’s central observation relates to the distributional nature of fields, for it leads to the identification of distribution functionals with Schwartz space test functions as density measures. -
FROM CLASSICAL MECHANICS to QUANTUM FIELD THEORY, a TUTORIAL Copyright © 2020 by World Scientific Publishing Co
FROM CLASSICAL MECHANICS TO QUANTUM FIELD THEORY A TUTORIAL 11556_9789811210488_TP.indd 1 29/11/19 2:30 PM This page intentionally left blank FROM CLASSICAL MECHANICS TO QUANTUM FIELD THEORY A TUTORIAL Manuel Asorey Universidad de Zaragoza, Spain Elisa Ercolessi University of Bologna & INFN-Sezione di Bologna, Italy Valter Moretti University of Trento & INFN-TIFPA, Italy World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO 11556_9789811210488_TP.indd 2 29/11/19 2:30 PM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. FROM CLASSICAL MECHANICS TO QUANTUM FIELD THEORY, A TUTORIAL Copyright © 2020 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 978-981-121-048-8 For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/11556#t=suppl Desk Editor: Nur Syarfeena Binte Mohd Fauzi Typeset by Stallion Press Email: [email protected] Printed in Singapore Syarfeena - 11556 - From Classical Mechanics.indd 1 02-12-19 3:03:23 PM January 3, 2020 9:10 From Classical Mechanics to Quantum Field Theory 9in x 6in b3742-main page v Preface This book grew out of the mini courses delivered at the Fall Workshop on Geometry and Physics, in Granada, Zaragoza and Madrid. -
Differentiability Problems in Banach Spaces
Derivatives Differentiability problems in Banach spaces For vector valued functions there are two main version of derivatives: Gâteaux (or weak) derivatives and Fréchet (or strong) derivatives. For a function f from a Banach space X 1 David Preiss into a Banach space Y the Gâteaux derivative at a point x0 2 X is by definition a bounded linear operator T : X −! Y so that Expanded notes of a talk based on a nearly finished research monograph “Fréchet differentiability of Lipschitz functions and porous sets in Banach for every u 2 X, spaces” f (x + tu) − f (x ) written jointly with lim 0 0 = Tu t!0 t Joram Lindenstrauss2 and Jaroslav Tišer3 The operator T is called the Fréchet derivative of f at x if it is a with some results coming from a joint work with 0 4 5 Gâteaux derivative of f at x0 and the limit above holds uniformly Giovanni Alberti and Marianna Csörnyei in u in the unit ball (or unit sphere) in X. So T is the Fréchet derivative of f at x0 if 1Warwick 2Jerusalem 3Prague 4Pisa 5London f (x0 + u) = f (x0) + Tu + o(kuk) as kuk ! 0: 1 / 24 2 / 24 Existence of derivatives Sharpness of Lebesgue’s result The first continuous nowhere differentiable f : R ! R was constructed by Bolzano about 1820 (unpublished), who Lebesgue’s result is sharp in the sense that for every A ⊂ of however did not give a full proof. Around 1850, Riemann R measure zero there is a Lipschitz (and monotone) function mentioned such an example, which was later found slightly f : ! which fails to have a derivative at any point of A. -
Well-Defined Lagrangian Flows for Absolutely Continuous Curves of Probabilities on the Line
Graduate Theses, Dissertations, and Problem Reports 2016 Well-defined Lagrangian flows for absolutely continuous curves of probabilities on the line Mohamed H. Amsaad Follow this and additional works at: https://researchrepository.wvu.edu/etd Recommended Citation Amsaad, Mohamed H., "Well-defined Lagrangian flows for absolutely continuous curves of probabilities on the line" (2016). Graduate Theses, Dissertations, and Problem Reports. 5098. https://researchrepository.wvu.edu/etd/5098 This Dissertation is protected by copyright and/or related rights. It has been brought to you by the The Research Repository @ WVU with permission from the rights-holder(s). You are free to use this Dissertation in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you must obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/ or on the work itself. This Dissertation has been accepted for inclusion in WVU Graduate Theses, Dissertations, and Problem Reports collection by an authorized administrator of The Research Repository @ WVU. For more information, please contact [email protected]. Well-defined Lagrangian flows for absolutely continuous curves of probabilities on the line Mohamed H. Amsaad Dissertation submitted to the Eberly College of Arts and Sciences at West Virginia University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Adrian Tudorascu, Ph.D., Chair Harumi Hattori, Ph.D. Harry Gingold, Ph.D. Tudor Stanescu, Ph.D. Charis Tsikkou, Ph.D. Department Of Mathematics Morgantown, West Virginia 2016 Keywords: Continuity Equation, Lagrangian Flow, Optimal Transport, Wasserstein metric, Wasserstein space. -
Value Functions and Optimality Conditions for Nonconvex
Value Functions and Optimality Conditions for Nonconvex Variational Problems with an Infinite Horizon in Banach Spaces∗ H´el`ene Frankowska† CNRS Institut de Math´ematiques de Jussieu – Paris Rive Gauche Sorbonne Universit´e, Campus Pierre et Marie Curie Case 247, 4 Place Jussieu, 75252 Paris, France e-mail: [email protected] Nobusumi Sagara‡ Department of Economics, Hosei University 4342, Aihara, Machida, Tokyo, 194–0298, Japan e-mail: [email protected] February 11, 2020 arXiv:1801.00400v3 [math.OC] 10 Feb 2020 ∗The authors are grateful to the two anonymous referees for their helpful comments and suggestions on the earlier version of this manuscript. †The research of this author was supported by the Air Force Office of Scientific Re- search, USA under award number FA9550-18-1-0254. It also partially benefited from the FJMH Program PGMO and from the support to this program from EDF-THALES- ORANGE-CRITEO under grant PGMO 2018-0047H. ‡The research of this author benefited from the support of JSPS KAKENHI Grant Number JP18K01518 from the Ministry of Education, Culture, Sports, Science and Tech- nology, Japan. Abstract We investigate the value function of an infinite horizon variational problem in the infinite-dimensional setting. Firstly, we provide an upper estimate of its Dini–Hadamard subdifferential in terms of the Clarke subdifferential of the Lipschitz continuous integrand and the Clarke normal cone to the graph of the set-valued mapping describing dynamics. Secondly, we derive a necessary condition for optimality in the form of an adjoint inclusion that grasps a connection between the Euler–Lagrange condition and the maximum principle. -
On Fréchet Differentiability of Lipschitz Maps
Annals of Mathematics, 157 (2003), 257–288 On Fr´echet differentiability of Lipschitz maps between Banach spaces By Joram Lindenstrauss and David Preiss Abstract Awell-known open question is whether every countable collection of Lipschitz functions on a Banach space X with separable dual has a common point ofFr´echet differentiability. We show that the answer is positive for some infinite-dimensional X. Previously, even for collections consisting of two functions this has been known for finite-dimensional X only (although for one function the answer is known to be affirmative in full generality). Our aims are achieved by introducing a new class of null sets in Banach spaces (called Γ-null sets), whose definition involves both the notions of category and mea- sure, and showing that the required differentiability holds almost everywhere with respect to it. We even obtain existence of Fr´echet derivatives of Lipschitz functions between certain infinite-dimensional Banach spaces; no such results have been known previously. Our main result states that a Lipschitz map between separable Banach spaces is Fr´echet differentiable Γ-almost everywhere provided that it is reg- ularly Gˆateaux differentiable Γ-almost everywhere and the Gˆateaux deriva- tives stay within a norm separable space of operators. It is easy to see that Lipschitz maps of X to spaces with the Radon-Nikod´ym property are Gˆateaux differentiable Γ-almost everywhere. Moreover, Gˆateaux differentiability im- plies regular Gˆateaux differentiability with exception of another kind of neg- ligible sets, so-called σ-porous sets. The answer to the question is therefore positive in every space in which every σ-porous set is Γ-null. -
Gateaux Differentiability Revisited Malek Abbasi, Alexander Kruger, Michel Théra
Gateaux differentiability revisited Malek Abbasi, Alexander Kruger, Michel Théra To cite this version: Malek Abbasi, Alexander Kruger, Michel Théra. Gateaux differentiability revisited. 2020. hal- 02963967 HAL Id: hal-02963967 https://hal.archives-ouvertes.fr/hal-02963967 Preprint submitted on 12 Oct 2020 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. Noname manuscript No. (will be inserted by the editor) Gateaux differentiability revisited Malek Abbasi · Alexander Y. Kruger · Michel Thera´ Received: date / Accepted: date Abstract We revisit some basic concepts and ideas of the classical differential calculus and convex analysis extending them to a broader frame. We reformulate and generalize the notion of Gateaux differentiability and propose new notions of generalized derivative and general- ized subdifferential in an arbitrary topological vector space. Meaningful examples preserving the key properties of the original notion of derivative are provided. Keywords Gateaux differentiability · Moreau–Rockafellar subdifferential · Convex function · Directional derivative. Mathematics Subject Classification (2000) 49J52 · 49J53 · 90C30 1 Introduction Gateaux derivatives are widely used in the calculus of variations, optimization and physics. According to Laurent Mazliak [7,8], Gateaux differentiability first appeared in Gateaux’s notes [3,4] under the name variation premiere` . -
Functional Derivative
Appendix Functional Derivative Appendix A: Calculus of Variation Consider a mapping from a vector space of functions to real number. Such mapping is called functional. One of the most representative examples of such a mapping is the action functional of analytical mechanics, A ,which is defined by Zt2 A½xtðÞ Lxt½ðÞ; vtðÞdt ðA:1Þ t1 where t is the independent variable, xtðÞis the trajectory of a particle, vtðÞ¼dx=dt is the velocity of the particle, and L is the Lagrangian which is the difference of kinetic energy and the potential: m L ¼ v2 À uxðÞ ðA:2Þ 2 Calculus of variation is the mathematical theory to find the extremal function xtðÞwhich maximizes or minimizes the functional such as Eq. (A.1). We are interested in the set of functions which satisfy the fixed boundary con- ditions such as xtðÞ¼1 x1 and xtðÞ¼2 x2. An arbitrary element of the function space may be defined from xtðÞby xtðÞ¼xtðÞþegðÞt ðA:3Þ where ε is a real number and gðÞt is any function satisfying gðÞ¼t1 gðÞ¼t2 0. Of course, it is obvious that xtðÞ¼1 x1 and xtðÞ¼2 x2. Varying ε and gðÞt generates any function of the vector space. Substitution of Eq. (A.3) to Eq. (A.1) gives Zt2 dx dg aðÞe A½¼xtðÞþegðÞt L xtðÞþegðÞt ; þ e dt ðA:4Þ dt dt t1 © Springer Science+Business Media Dordrecht 2016 601 K.S. Cho, Viscoelasticity of Polymers, Springer Series in Materials Science 241, DOI 10.1007/978-94-017-7564-9 602 Appendix: Functional Derivative From Eq. -
ECE 821 Optimal Control and Variational Methods Lecture Notes
ECE 821 Optimal Control and Variational Methods Lecture Notes Prof. Dan Cobb Contents 1 Introduction 3 2 Finite-Dimensional Optimization 5 2.1 Background ........................................ 5 2.1.1 EuclideanSpaces ................................. 5 2.1.2 Norms....................................... 6 2.1.3 MatrixNorms................................... 8 2.2 Unconstrained Optimization in Rn ........................... 9 2.2.1 Extrema...................................... 9 2.2.2 Jacobians ..................................... 9 2.2.3 CriticalPoints................................... 10 2.2.4 Hessians...................................... 11 2.2.5 Definite Matrices . 11 2.2.6 Continuity and Continuous Differentiability . 14 2.2.7 Second Derivative Conditions . 15 2.3 Constrained Optimization in Rn ............................. 16 2.3.1 Constrained Extrema . 16 2.3.2 OpenSets..................................... 17 2.3.3 Strict Inequality Constraints . 18 2.3.4 Equality Constraints and Lagrange Multipliers . 19 2.3.5 Second Derivative Conditions . 21 2.3.6 Non-Strict Inequality Constraints . 23 2.3.7 Mixed Constraints . 27 3 Calculus of Variations 27 3.1 Background ........................................ 27 3.1.1 VectorSpaces ................................... 27 3.1.2 Norms....................................... 28 3.1.3 Functionals .................................... 30 3.2 Unconstrained Optimization in X ............................ 32 3.2.1 Extrema...................................... 32 3.2.2 Differentiation of Functionals . -
35 Nopw It Is Time to Face the Menace
35 XIV. CORRELATIONS AND SUSCEPTIBILITY A. Correlations - saddle point approximation: the easy way out Nopw it is time to face the menace - path integrals. We were tithering on the top of this ravine for long enough. We need to take the plunge. The meaning of the path integral is very simple - tofind the partition function of a system locallyfluctuating, we need to consider the all the possible patterns offlucutations, and the only way to do this is through a path integral. But is the path integral so bad? Well, if you believe that anyfluctuation is still quite costly, then the path integral, like the integral over the exponent of a quadratic function, is pretty much determined by where the argument of the exponent is a minimum, plus quadraticfluctuations. These belief is translated into equations via the saddle point approximation. Wefind the minimum of the free energy functional, and then expand upto second order in the exponent. 1 1 F [m(x),h] = ddx γ( m (x))2 +a(T T )m (x)2 + um (x)4 hm + ( m (x))2 ∂2F (219) L ∇ min − c min 2 min − min 2 − min � � � I write here deliberately the vague partial symbol, in fact, this should be the functional derivative. Does this look familiar? This should bring you back to the good old classical mechanics days. Where the Lagrangian is the only thing you wanted to know, and the Euler Lagrange equations were the only guidance necessary. Non of this non-commuting business, or partition function annoyance. Well, these times are back for a short time! In order tofind this, we do a variation: m(x) =m min +δm (220) Upon assignment wefind: u F [m(x),h] = ddx γ( (m +δm)) 2 +a(T T )(m +δm) 2 + (m +δm) 4 h(m +δm) (221) L ∇ min − c min 2 min − min � � � Let’s expand this to second order: d 2 2 d x γ (mmin) + δm 2γ m+γ( δm) → ∇ 2 ∇ · ∇ ∇ 2 +a (mmin) +δm 2am+aδm � � ∇ u 4 ·3 2 2 (222) + 2 mmin +2umminδm +3umminδm hm hδm] − min − Collecting terms in power ofδm, thefirst term is justF L[mmin].