On Regularizations of the Delta Distribution
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Geometric Integration Theory Contents
Steven G. Krantz Harold R. Parks Geometric Integration Theory Contents Preface v 1 Basics 1 1.1 Smooth Functions . 1 1.2Measures.............................. 6 1.2.1 Lebesgue Measure . 11 1.3Integration............................. 14 1.3.1 Measurable Functions . 14 1.3.2 The Integral . 17 1.3.3 Lebesgue Spaces . 23 1.3.4 Product Measures and the Fubini–Tonelli Theorem . 25 1.4 The Exterior Algebra . 27 1.5 The Hausdorff Distance and Steiner Symmetrization . 30 1.6 Borel and Suslin Sets . 41 2 Carath´eodory’s Construction and Lower-Dimensional Mea- sures 53 2.1 The Basic Definition . 53 2.1.1 Hausdorff Measure and Spherical Measure . 55 2.1.2 A Measure Based on Parallelepipeds . 57 2.1.3 Projections and Convexity . 57 2.1.4 Other Geometric Measures . 59 2.1.5 Summary . 61 2.2 The Densities of a Measure . 64 2.3 A One-Dimensional Example . 66 2.4 Carath´eodory’s Construction and Mappings . 67 2.5 The Concept of Hausdorff Dimension . 70 2.6 Some Cantor Set Examples . 73 i ii CONTENTS 2.6.1 Basic Examples . 73 2.6.2 Some Generalized Cantor Sets . 76 2.6.3 Cantor Sets in Higher Dimensions . 78 3 Invariant Measures and the Construction of Haar Measure 81 3.1 The Fundamental Theorem . 82 3.2 Haar Measure for the Orthogonal Group and the Grassmanian 90 3.2.1 Remarks on the Manifold Structure of G(N,M).... 94 4 Covering Theorems and the Differentiation of Integrals 97 4.1 Wiener’s Covering Lemma and its Variants . -
Numerical Solution of Ordinary Differential Equations
NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS Kendall Atkinson, Weimin Han, David Stewart University of Iowa Iowa City, Iowa A JOHN WILEY & SONS, INC., PUBLICATION Copyright c 2009 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created ore extended by sales representatives or written sales materials. The advice and strategies contained herin may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. -
On the Use of Discrete Fourier Transform for Solving Biperiodic Boundary Value Problem of Biharmonic Equation in the Unit Rectangle
On the Use of Discrete Fourier Transform for Solving Biperiodic Boundary Value Problem of Biharmonic Equation in the Unit Rectangle Agah D. Garnadi 31 December 2017 Department of Mathematics, Faculty of Mathematics and Natural Sciences, Bogor Agricultural University Jl. Meranti, Kampus IPB Darmaga, Bogor, 16680 Indonesia Abstract This note is addressed to solving biperiodic boundary value prob- lem of biharmonic equation in the unit rectangle. First, we describe the necessary tools, which is discrete Fourier transform for one di- mensional periodic sequence, and then extended the results to 2- dimensional biperiodic sequence. Next, we use the discrete Fourier transform 2-dimensional biperiodic sequence to solve discretization of the biperiodic boundary value problem of Biharmonic Equation. MSC: 15A09; 35K35; 41A15; 41A29; 94A12. Key words: Biharmonic equation, biperiodic boundary value problem, discrete Fourier transform, Partial differential equations, Interpola- tion. 1 Introduction This note is an extension of a work by Henrici [2] on fast solver for Poisson equation to biperiodic boundary value problem of biharmonic equation. This 1 note is organized as follows, in the first part we address the tools needed to solve the problem, i.e. introducing discrete Fourier transform (DFT) follows Henrici's work [2]. The second part, we apply the tools already described in the first part to solve discretization of biperiodic boundary value problem of biharmonic equation in the unit rectangle. The need to solve discretization of biharmonic equation is motivated by its occurence in some applications such as elasticity and fluid flows [1, 4], and recently in image inpainting [3]. Note that the work of Henrici [2] on fast solver for biperiodic Poisson equation can be seen as solving harmonic inpainting. -
Introduction to Sobolev Spaces
Introduction to Sobolev Spaces Lecture Notes MM692 2018-2 Joa Weber UNICAMP December 23, 2018 Contents 1 Introduction1 1.1 Notation and conventions......................2 2 Lp-spaces5 2.1 Borel and Lebesgue measure space on Rn .............5 2.2 Definition...............................8 2.3 Basic properties............................ 11 3 Convolution 13 3.1 Convolution of functions....................... 13 3.2 Convolution of equivalence classes................. 15 3.3 Local Mollification.......................... 16 3.3.1 Locally integrable functions................. 16 3.3.2 Continuous functions..................... 17 3.4 Applications.............................. 18 4 Sobolev spaces 19 4.1 Weak derivatives of locally integrable functions.......... 19 1 4.1.1 The mother of all Sobolev spaces Lloc ........... 19 4.1.2 Examples........................... 20 4.1.3 ACL characterization.................... 21 4.1.4 Weak and partial derivatives................ 22 4.1.5 Approximation characterization............... 23 4.1.6 Bounded weakly differentiable means Lipschitz...... 24 4.1.7 Leibniz or product rule................... 24 4.1.8 Chain rule and change of coordinates............ 25 4.1.9 Equivalence classes of locally integrable functions..... 27 4.2 Definition and basic properties................... 27 4.2.1 The Sobolev spaces W k;p .................. 27 4.2.2 Difference quotient characterization of W 1;p ........ 29 k;p 4.2.3 The compact support Sobolev spaces W0 ........ 30 k;p 4.2.4 The local Sobolev spaces Wloc ............... 30 4.2.5 How the spaces relate.................... 31 4.2.6 Basic properties { products and coordinate change.... 31 i ii CONTENTS 5 Approximation and extension 33 5.1 Approximation............................ 33 5.1.1 Local approximation { any domain............. 33 5.1.2 Global approximation on bounded domains....... -
Proving the Regularity of the Reduced Boundary of Perimeter Minimizing Sets with the De Giorgi Lemma
PROVING THE REGULARITY OF THE REDUCED BOUNDARY OF PERIMETER MINIMIZING SETS WITH THE DE GIORGI LEMMA Presented by Antonio Farah In partial fulfillment of the requirements for graduation with the Dean's Scholars Honors Degree in Mathematics Honors, Department of Mathematics Dr. Stefania Patrizi, Supervising Professor Dr. David Rusin, Honors Advisor, Department of Mathematics THE UNIVERSITY OF TEXAS AT AUSTIN May 2020 Acknowledgements I deeply appreciate the continued guidance and support of Dr. Stefania Patrizi, who supervised the research and preparation of this Honors Thesis. Dr. Patrizi kindly dedicated numerous sessions to this project, motivating and expanding my learning. I also greatly appreciate the support of Dr. Irene Gamba and of Dr. Francesco Maggi on the project, who generously helped with their reading and support. Further, I am grateful to Dr. David Rusin, who also provided valuable help on this project in his role of Honors Advisor in the Department of Mathematics. Moreover, I would like to express my gratitude to the Department of Mathematics' Faculty and Administration, to the College of Natural Sciences' Administration, and to the Dean's Scholars Honors Program of The University of Texas at Austin for my educational experience. 1 Abstract The Plateau problem consists of finding the set that minimizes its perimeter among all sets of a certain volume. Such set is known as a minimal set, or perimeter minimizing set. The problem was considered intractable until the 1960's, when the development of geometric measure theory by researchers such as Fleming, Federer, and De Giorgi provided the necessary tools to find minimal sets. -
TERMINOLOGY UNIT- I Finite Element Method (FEM)
TERMINOLOGY UNIT- I The Finite Element Method (FEM) is a numerical technique to find Finite Element approximate solutions of partial differential equations. It was originated from Method (FEM) the need of solving complex elasticity and structural analysis problems in Civil, Mechanical and Aerospace engineering. Any continuum/domain can be divided into a number of pieces with very Finite small dimensions. These small pieces of finite dimension are called Finite Elements Elements. Field The field variables, displacements (strains) & stresses or stress resultants must Conditions satisfy the governing condition which can be mathematically expressed. A set of independent functions satisfying the boundary conditions is chosen Functional and a linear combination of a finite number of them is taken to approximately Approximation specify the field variable at any point. A structure can have infinite number of displacements. Approximation with a Degrees reasonable level of accuracy can be achieved by assuming a limited number of of Freedom displacements. This finite number of displacements is the number of degrees of freedom of the structure. The formulation for structural analysis is generally based on the three Numerical fundamental relations: equilibrium, constitutive and compatibility. There are Methods two major approaches to the analysis: Analytical and Numerical. Analytical approach which leads to closed-form solutions is effective in case of simple geometry, boundary conditions, loadings and material properties. Analytical However, in reality, such simple cases may not arise. As a result, various approach numerical methods are evolved for solving such problems which are complex in nature. For numerical approach, the solutions will be approximate when any of these Numerical relations are only approximately satisfied. -
Lecture 7: the Complex Fourier Transform and the Discrete Fourier Transform (DFT)
Lecture 7: The Complex Fourier Transform and the Discrete Fourier Transform (DFT) c Christopher S. Bretherton Winter 2014 7.1 Fourier analysis and filtering Many data analysis problems involve characterizing data sampled on a regular grid of points, e. g. a time series sampled at some rate, a 2D image made of regularly spaced pixels, or a 3D velocity field from a numerical simulation of fluid turbulence on a regular grid. Often, such problems involve characterizing, detecting, separating or ma- nipulating variability on different scales, e. g. finding a weak systematic signal amidst noise in a time series, edge sharpening in an image, or quantifying the energy of motion across the different sizes of turbulent eddies. Fourier analysis using the Discrete Fourier Transform (DFT) is a fun- damental tool for such problems. It transforms the gridded data into a linear combination of oscillations of different wavelengths. This partitions it into scales which can be separately analyzed and manipulated. The computational utility of Fourier methods rests on the Fast Fourier Transform (FFT) algorithm, developed in the 1960s by Cooley and Tukey, which allows efficient calculation of discrete Fourier coefficients of a periodic function sampled on a regular grid of 2p points (or 2p3q5r with slightly reduced efficiency). 7.2 Example of the FFT Using Matlab, take the FFT of the HW1 wave height time series (length 24 × 60=25325) and plot the result (Fig. 1): load hw1 dat; zhat = fft(z); plot(abs(zhat),’x’) A few elements (the second and last, and to a lesser extent the third and the second from last) have magnitudes that stand above the noise. -
Polarization Fields and Phase Space Densities in Storage Rings: Stroboscopic Averaging and the Ergodic Theorem
Physica D 234 (2007) 131–149 www.elsevier.com/locate/physd Polarization fields and phase space densities in storage rings: Stroboscopic averaging and the ergodic theorem✩ James A. Ellison∗, Klaus Heinemann Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, United States Received 1 May 2007; received in revised form 6 July 2007; accepted 9 July 2007 Available online 14 July 2007 Communicated by C.K.R.T. Jones Abstract A class of orbital motions with volume preserving flows and with vector fields periodic in the “time” parameter θ is defined. Spin motion coupled to the orbital dynamics is then defined, resulting in a class of spin–orbit motions which are important for storage rings. Phase space densities and polarization fields are introduced. It is important, in the context of storage rings, to understand the behavior of periodic polarization fields and phase space densities. Due to the 2π time periodicity of the spin–orbit equations of motion the polarization field, taken at a sequence of increasing time values θ,θ 2π,θ 4π,... , gives a sequence of polarization fields, called the stroboscopic sequence. We show, by using the + + Birkhoff ergodic theorem, that under very general conditions the Cesaro` averages of that sequence converge almost everywhere on phase space to a polarization field which is 2π-periodic in time. This fulfills the main aim of this paper in that it demonstrates that the tracking algorithm for stroboscopic averaging, encoded in the program SPRINT and used in the study of spin motion in storage rings, is mathematically well-founded. -
Discretization of Integro-Differential Equations
Discretization of Integro-Differential Equations Modeling Dynamic Fractional Order Viscoelasticity K. Adolfsson1, M. Enelund1,S.Larsson2,andM.Racheva3 1 Dept. of Appl. Mech., Chalmers Univ. of Technology, SE–412 96 G¨oteborg, Sweden 2 Dept. of Mathematics, Chalmers Univ. of Technology, SE–412 96 G¨oteborg, Sweden 3 Dept. of Mathematics, Technical University of Gabrovo, 5300 Gabrovo, Bulgaria Abstract. We study a dynamic model for viscoelastic materials based on a constitutive equation of fractional order. This results in an integro- differential equation with a weakly singular convolution kernel. We dis- cretize in the spatial variable by a standard Galerkin finite element method. We prove stability and regularity estimates which show how the convolution term introduces dissipation into the equation of motion. These are then used to prove a priori error estimates. A numerical ex- periment is included. 1 Introduction Fractional order operators (integrals and derivatives) have proved to be very suit- able for modeling memory effects of various materials and systems of technical interest. In particular, they are very useful when modeling viscoelastic materials, see, e.g., [3]. Numerical methods for quasistatic viscoelasticity problems have been studied, e.g., in [2] and [8]. The drawback of the fractional order viscoelastic models is that the whole strain history must be saved and included in each time step. The most commonly used algorithms for this integration are based on the Lubich convolution quadrature [5] for fractional order operators. In [1, 2], we develop an efficient numerical algorithm based on sparse numerical quadrature earlier studied in [6]. While our earlier work focused on discretization in time for the quasistatic case, we now study space discretization for the fully dynamic equations of mo- tion, which take the form of an integro-differential equation with a weakly singu- lar convolution kernel. -
Approximations in Numerical Analysis, Cont'd
Jim Lambers MAT 460/560 Fall Semester 2009-10 Lecture 7 Notes These notes correspond to Section 1.3 in the text. Approximations in Numerical Analysis, cont'd Convergence Many algorithms in numerical analysis are iterative methods that produce a sequence fαng of ap- proximate solutions which, ideally, converges to a limit α that is the exact solution as n approaches 1. Because we can only perform a finite number of iterations, we cannot obtain the exact solution, and we have introduced computational error. If our iterative method is properly designed, then this computational error will approach zero as n approaches 1. However, it is important that we obtain a sufficiently accurate approximate solution using as few computations as possible. Therefore, it is not practical to simply perform enough iterations so that the computational error is determined to be sufficiently small, because it is possible that another method may yield comparable accuracy with less computational effort. The total computational effort of an iterative method depends on both the effort per iteration and the number of iterations performed. Therefore, in order to determine the amount of compu- tation that is needed to attain a given accuracy, we must be able to measure the error in αn as a function of n. The more rapidly this function approaches zero as n approaches 1, the more rapidly the sequence of approximations fαng converges to the exact solution α, and as a result, fewer iterations are needed to achieve a desired accuracy. We now introduce some terminology that will aid in the discussion of the convergence behavior of iterative methods. -
L P and Sobolev Spaces
NOTES ON Lp AND SOBOLEV SPACES STEVE SHKOLLER 1. Lp spaces 1.1. Definitions and basic properties. Definition 1.1. Let 0 < p < 1 and let (X; M; µ) denote a measure space. If f : X ! R is a measurable function, then we define 1 Z p p kfkLp(X) := jfj dx and kfkL1(X) := ess supx2X jf(x)j : X Note that kfkLp(X) may take the value 1. Definition 1.2. The space Lp(X) is the set p L (X) = ff : X ! R j kfkLp(X) < 1g : The space Lp(X) satisfies the following vector space properties: (1) For each α 2 R, if f 2 Lp(X) then αf 2 Lp(X); (2) If f; g 2 Lp(X), then jf + gjp ≤ 2p−1(jfjp + jgjp) ; so that f + g 2 Lp(X). (3) The triangle inequality is valid if p ≥ 1. The most interesting cases are p = 1; 2; 1, while all of the Lp arise often in nonlinear estimates. Definition 1.3. The space lp, called \little Lp", will be useful when we introduce Sobolev spaces on the torus and the Fourier series. For 1 ≤ p < 1, we set ( 1 ) p 1 X p l = fxngn=1 j jxnj < 1 : n=1 1.2. Basic inequalities. Lemma 1.4. For λ 2 (0; 1), xλ ≤ (1 − λ) + λx. Proof. Set f(x) = (1 − λ) + λx − xλ; hence, f 0(x) = λ − λxλ−1 = 0 if and only if λ(1 − xλ−1) = 0 so that x = 1 is the critical point of f. In particular, the minimum occurs at x = 1 with value f(1) = 0 ≤ (1 − λ) + λx − xλ : Lemma 1.5. -
There Is Only One Fourier Transform Jens V
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 25 December 2017 doi:10.20944/preprints201712.0173.v1 There is only one Fourier Transform Jens V. Fischer * Microwaves and Radar Institute, German Aerospace Center (DLR), Germany Abstract Four Fourier transforms are usually defined, the Integral Fourier transform, the Discrete-Time Fourier transform (DTFT), the Discrete Fourier transform (DFT) and the Integral Fourier transform for periodic functions. However, starting from their definitions, we show that all four Fourier transforms can be reduced to actually only one Fourier transform, the Fourier transform in the distributional sense. Keywords Integral Fourier Transform, Discrete-Time Fourier Transform (DTFT), Discrete Fourier Transform (DFT), Integral Fourier Transform for periodic functions, Fourier series, Poisson Summation Formula, periodization trick, interpolation trick Introduction Two Important Operations The fact that “there is only one Fourier transform” actually There are two operations which are being strongly related to needs no proof. It is commonly known and often discussed in four different variants of the Fourier transform, i.e., the literature, for example in [1-2]. discretization and periodization. For any real-valued T > 0 and δ(t) being the Dirac impulse, let However frequently asked questions are: (1) What is the difference between calculating the Fourier series and the (1) Fourier transform of a continuous function? (2) What is the difference between Discrete-Time Fourier Transform (DTFT) be the function that results from a discretization of f(t) and let and Discrete Fourier Transform (DFT)? (3) When do we need which Fourier transform? and many others. The default answer (2) today to all these questions is to invite the reader to have a look at the so-called Fourier-Poisson cube, e.g.