Lecture Notes 9: Measure and Probability Part A: Measure and Integration
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Measure-Theoretic Probability I
Measure-Theoretic Probability I Steven P.Lalley Winter 2017 1 1 Measure Theory 1.1 Why Measure Theory? There are two different views – not necessarily exclusive – on what “probability” means: the subjectivist view and the frequentist view. To the subjectivist, probability is a system of laws that should govern a rational person’s behavior in situations where a bet must be placed (not necessarily just in a casino, but in situations where a decision must be made about how to proceed when only imperfect information about the outcome of the decision is available, for instance, should I allow Dr. Scissorhands to replace my arthritic knee by a plastic joint?). To the frequentist, the laws of probability describe the long- run relative frequencies of different events in “experiments” that can be repeated under roughly identical conditions, for instance, rolling a pair of dice. For the frequentist inter- pretation, it is imperative that probability spaces be large enough to allow a description of an experiment, like dice-rolling, that is repeated infinitely many times, and that the mathematical laws should permit easy handling of limits, so that one can make sense of things like “the probability that the long-run fraction of dice rolls where the two dice sum to 7 is 1/6”. But even for the subjectivist, the laws of probability should allow for description of situations where there might be a continuum of possible outcomes, or pos- sible actions to be taken. Once one is reconciled to the need for such flexibility, it soon becomes apparent that measure theory (the theory of countably additive, as opposed to merely finitely additive measures) is the only way to go. -
Chapter 5: Derivatives and Integration
Chapter 5: Derivatives and Integration Chandrajit Bajaj and Andrew Gillette October 22, 2010 Contents 1 Curvature Computations 2 2 Numerical and Symbolic Integration 2 2.1 Cubature Formulae Notation . .2 2.2 Constructing Cubature Formulae . .3 2.2.1 Interpolatory Cubature Formulae . .3 2.2.2 Ideal Theory . .4 2.2.3 Bounds . .5 2.2.4 The characterization of minimal formulae and the reproducing kernel . .5 2.3 Gauss Formula on an A-patch . .6 2.3.1 Gauss Points and Weights . .6 2.3.2 Error Analysis . .6 2.4 T-method . .7 2.5 S-method . .8 3 Generalized Born Electrostatics 9 3.1 Geometric model . 11 3.1.1 Gaussian surface . 11 3.1.2 Triangular mesh . 11 3.1.3 Algebraic spline molecular surface (ASMS) . 12 3.2 Fast solvation energy computation . 13 3.2.1 Method . 13 3.2.2 Fast summation . 15 3.2.3 Error analysis . 16 3.3 Fast solvation force computation . 20 3.4 Results . 23 3.4.1 Conclusion . 25 3.5 NFFT . 26 3.6 NFFTT ........................................... 29 3.7 Continuity of f ....................................... 32 4 Poisson Boltzmann Electrostatics 33 4.1 Motivation . 34 4.2 The Poisson-Boltzmann Equation . 35 4.2.1 Boundary Integral Formulation . 37 4.2.2 Discretization by the Collocation Method . 37 4.3 BEM for Molecular Surfaces . 38 4.3.1 Construction of the Molecular Surface . 40 1 4.3.2 Surface Parametrization . 40 4.3.3 Selection of Basis Functions . 41 4.3.4 Quadrature . 41 4.4 Polarization Energy Computation . 42 4.5 Implementation Details . -
Measure Theory and Probability
Measure theory and probability Alexander Grigoryan University of Bielefeld Lecture Notes, October 2007 - February 2008 Contents 1 Construction of measures 3 1.1Introductionandexamples........................... 3 1.2 σ-additive measures ............................... 5 1.3 An example of using probability theory . .................. 7 1.4Extensionofmeasurefromsemi-ringtoaring................ 8 1.5 Extension of measure to a σ-algebra...................... 11 1.5.1 σ-rings and σ-algebras......................... 11 1.5.2 Outermeasure............................. 13 1.5.3 Symmetric difference.......................... 14 1.5.4 Measurable sets . ............................ 16 1.6 σ-finitemeasures................................ 20 1.7Nullsets..................................... 23 1.8 Lebesgue measure in Rn ............................ 25 1.8.1 Productmeasure............................ 25 1.8.2 Construction of measure in Rn. .................... 26 1.9 Probability spaces ................................ 28 1.10 Independence . ................................. 29 2 Integration 38 2.1 Measurable functions.............................. 38 2.2Sequencesofmeasurablefunctions....................... 42 2.3 The Lebesgue integral for finitemeasures................... 47 2.3.1 Simplefunctions............................ 47 2.3.2 Positivemeasurablefunctions..................... 49 2.3.3 Integrablefunctions........................... 52 2.4Integrationoversubsets............................ 56 2.5 The Lebesgue integral for σ-finitemeasure................. -
An Approximate Logic for Measures
AN APPROXIMATE LOGIC FOR MEASURES ISAAC GOLDBRING AND HENRY TOWSNER Abstract. We present a logical framework for formalizing connections be- tween finitary combinatorics and measure theory or ergodic theory that have appeared in various places throughout the literature. We develop the basic syntax and semantics of this logic and give applications, showing that the method can express the classic Furstenberg correspondence as well as provide short proofs of the Szemer´edi Regularity Lemma and the hypergraph removal lemma. We also derive some connections between the model-theoretic notion of stability and the Gowers uniformity norms from combinatorics. 1. Introduction Since the 1970’s, diagonalization arguments (and their generalization to ul- trafilters) have been used to connect results in finitary combinatorics with results in measure theory or ergodic theory [14]. Although it has long been known that this connection can be described with first-order logic, a recent striking paper by Hrushovski [22] demonstrated that modern developments in model theory can be used to give new substantive results in this area as well. Papers on various aspects of this interaction have been published from a va- riety of perspectives, with almost as wide a variety of terminology [30, 10, 9, 1, 32, 13, 4]. Our goal in this paper is to present an assortment of these techniques in a common framework. We hope this will make the entire area more accessible. We are typically concerned with the situation where we wish to prove a state- ment 1) about sufficiently large finite structures which 2) concerns the finite analogs of measure-theoretic notions such as density or integrals. -
Dynamic Ham-Sandwich Cuts in the Plane∗
Dynamic Ham-Sandwich Cuts in the Plane∗ Timothy G. Abbott† Michael A. Burr‡ Timothy M. Chan§ Erik D. Demaine† Martin L. Demaine† John Hugg¶ Daniel Kanek Stefan Langerman∗∗ Jelani Nelson† Eynat Rafalin†† Kathryn Seyboth¶ Vincent Yeung† Abstract We design efficient data structures for dynamically maintaining a ham-sandwich cut of two point sets in the plane subject to insertions and deletions of points in either set. A ham- sandwich cut is a line that simultaneously bisects the cardinality of both point sets. For general point sets, our first data structure supports each operation in O(n1/3+ε) amortized time and O(n4/3+ε) space. Our second data structure performs faster when each point set decomposes into a small number k of subsets in convex position: it supports insertions and deletions in O(log n) time and ham-sandwich queries in O(k log4 n) time. In addition, if each point set has convex peeling depth k, then we can maintain the decomposition automatically using O(k log n) time per insertion and deletion. Alternatively, we can view each convex point set as a convex polygon, and we show how to find a ham-sandwich cut that bisects the total areas or total perimeters of these polygons in O(k log4 n) time plus the O((kb) polylog(kb)) time required to approximate the root of a polynomial of degree O(k) up to b bits of precision. We also show how to maintain a partition of the plane by two lines into four regions each containing a quarter of the total point count, area, or perimeter in polylogarithmic time. -
Universität Karlsruhe (TH) Institut Für Baustatik a Mixed Shell Formulation Accounting for Thickness Strains and Finite Strain
Universit¨at Karlsruhe (TH) Institut fur¨ Baustatik A mixed shell formulation accounting for thickness strains and finite strain 3d-material models S. Klinkel, F. Gruttmann, W. Wagner Mitteilung 4(2006) BAUSTATIK Universit¨at Karlsruhe (TH) Institut fur¨ Baustatik A mixed shell formulation accounting for thickness strains and finite strain 3d-material models S. Klinkel, F. Gruttmann, W. Wagner Mitteilung 4(2006) BAUSTATIK Prof. Dr.–Ing. W. Wagner Telefon: (0721) 608–2280 Institut fur¨ Baustatik Telefax: (0721) 608–6015 c Universit¨at Karlsruhe E–mail: [email protected] Postfach 6980 Internet: http://www.bs.uni-karlsruhe.de 76128 Karlsruhe A mixed shell formulation accounting for thickness strains and finite strain 3d-material models Sven Klinkel∗, Friedrich Gruttmann+, Werner Wagner∗ ∗ Institut f¨ur Baustatik, Universit¨at Karlsruhe (TH), Kaiserstr. 12, 76131 Karlsruhe, Germany + Institut f¨ur Werkstoffe und Mechanik im Bauwesen, Technische Universit¨at Darmstadt, Petersenstr. 12, 64287 Darmstadt, Germany Contents 1 Introduction 2 2 Variational formulation of the shell equations 4 3 Finite Element Equations 6 3.1 Interpolation of the initial and current reference surface ............. 6 3.2 Interpolation of the stress resultants ........................ 9 3.3 Interpolation of the shell strains .......................... 9 3.4 Linearized variational formulation ......................... 11 4 Examples 13 4.1 Membrane and bending patch test ......................... 14 4.2 Channel section cantilever with plasticity ..................... 14 4.3 Stretching of a rubber sheet ............................ 16 4.4 Square plate ..................................... 19 4.5 Conical shell ..................................... 21 5 Conclusions 23 A Finite element matrices B and kσ 24 B Non-linear constitutive equations 25 B.1 Hyper-elastic material ............................... 25 B.2 Finite strain J2-plasticity model ......................... -
Functional RG Flow Equation: Regularization and Coarse-Graining
Functional RG flow equation: regularization and coarse-graining in phase space G. P. Vacca∗ and L. Zambelli† Dip. di Fisica, Universit`adegli Studi di Bologna INFN Sez. di Bologna via Irnerio 46, I-40126 Bologna, Italy Abstract Starting from the basic path integral in phase space we reconsider the functional approach to the RG flow of the one particle irreducible effective average action. On employing a balanced coarse-graining procedure for the canonical variables we obtain a functional integral with a non trivial measure which leads to a modified flow equation. We first address quantum mechanics for boson and fermion degrees of freedom and we then extend the construction to quantum field theories. For this modified flow equation we discuss the reconstruction of the bare action and the implications on the computation of the vacuum energy density. 1 Introduction Renormalization analysis in quantum (and statistical) field theory is one of the main tools at our disposal to investigate non trivial theories. Both perturbative and non perturbative techniques have been developed. Some analytical results can be obtained starting from a functional formulation, where the generator of the correlation functions is written in terms of a functional (path) integral. The Wilsonian idea of renormalization [1], which started from the analysis of Kadanoff’s blocking and scaling of spin systems (more generally coarse-graining), can also be conve- niently formulated analytically in a functional integral formulation. The idea of a step by step integration of the quantum fluctuations typically belonging to a momentum shell, followed by rescaling, can be implemented in a smooth way [2, 3] and leads to a differential arXiv:1103.2219v2 [hep-th] 27 May 2011 equation for Wilson’s effective action as a function of the scale parameter. -
21 TOPOLOGICAL METHODS in DISCRETE GEOMETRY Rade T
21 TOPOLOGICAL METHODS IN DISCRETE GEOMETRY Rade T. Zivaljevi´cˇ INTRODUCTION A problem is solved or some other goal achieved by “topological methods” if in our arguments we appeal to the “form,” the “shape,” the “global” rather than “local” structure of the object or configuration space associated with the phenomenon we are interested in. This configuration space is typically a manifold or a simplicial complex. The global properties of the configuration space are usually expressed in terms of its homology and homotopy groups, which capture the idea of the higher (dis)connectivity of a geometric object and to some extent provide “an analysis properly geometric or linear that expresses location directly as algebra expresses magnitude.”1 Thesis: Any global effect that depends on the object as a whole and that cannot be localized is of homological nature, and should be amenable to topological methods. HOW IS TOPOLOGY APPLIED IN DISCRETE GEOMETRIC PROBLEMS? In this chapter we put some emphasis on the role of equivariant topological methods in solving combinatorial or discrete geometric problems that have proven to be of relevance for computational geometry and topological combinatorics and with some impact on computational mathematics in general. The versatile configuration space/test map scheme (CS/TM) was developed in numerous research papers over the years and formally codified in [Ziv98].ˇ Its essential features are summarized as follows: CS/TM-1: The problem is rephrased in topological terms. The problem should give us a clue how to define a “natural” configuration space X and how to rephrase the question in terms of zeros or coincidences of the associated test maps. -
1. Σ-Algebras Definition 1. Let X Be Any Set and Let F Be a Collection Of
1. σ-algebras Definition 1. Let X be any set and let F be a collection of subsets of X. We say that F is a σ-algebra (on X), if it satisfies the following. (1) X 2 F. (2) If A 2 F, then Ac 2 F. 1 (3) If A1;A2; · · · 2 F, a countable collection, then [n=1An 2 F. In the above situation, the elements of F are called measurable sets and X (or more precisely (X; F)) is called a measurable space. Example 1. For any set X, P(X), the set of all subsets of X is a σ-algebra and so is F = f;;Xg. Theorem 1. Let X be any set and let G be any collection of subsets of X. Then there exists a σ-algebra, containing G and smallest with respect to inclusion. Proof. Let S be the collection of all σ-algebras containing G. This set is non-empty, since P(X) 2 S. Then F = \A2SA can easily be checked to have all the properties asserted in the theorem. Remark 1. In the above situation, F is called the σ-algebra generated by G. Definition 2. Let X be a topological space (for example, a metric space) and let B be the σ-algebra generated by the set of all open sets of X. The elements of B are called Borel sets and (X; B), a Borel measurable space. Definition 3. Let (X; F) be a measurable space and let Y be any topo- logical space and let f : X ! Y be any function. -
Measure and Integration
Chapter 5 Measure and integration In calculus you have learned how to calculate the size of different kinds of sets: the length of a curve, the area of a region or a surface, the volume or mass of a solid. In probability theory and statistics you have learned how to compute the size of other kinds of sets: the probability that certain events happen or do not happen. In this chapter we shall develop a general theory for the size of sets, a theory that covers all the examples above and many more. Just as the concept of a metric space gave us a general setting for discussing the notion of distance, the concept of a measure space will provide us with a general setting for discussing the notion of size. In calculus we use integration to calculate the size of sets. In this chap- ter we turn the situation around: We first develop a theory of size and then use it to define integrals of a new and more general kind. As we shall sometimes wish to compare the two theories, we shall refer to integration as taught in calculus as Riemann-integration in honor of the German math- ematician Bernhard Riemann (1826-1866) and the new theory developed here as Lebesgue integration in honor of the French mathematician Henri Lebesgue (1875-1941). Let us begin by taking a look at what we might wish for in a theory of size. Assume what we want to measure the size of subsets of a set X (if 2 you need something concrete to concentrate on, you may let X = R and 2 3 think of the area of subsets of R , or let X = R and think of the volume of 3 subsets of R ). -
Jhep09(2008)065
Published by Institute of Physics Publishing for SISSA Received: May 6, 2008 Revised: August 14, 2008 Accepted: August 26, 2008 Published: September 11, 2008 From loops to trees by-passing Feynman’s theorem JHEP09(2008)065 Stefano Catani INFN, Sezione di Firenze and Dipartimento di Fisica, Universit`adi Firenze, I-50019 Sesto Fiorentino, Florence, Italy E-mail: [email protected] Tanju Gleisberg Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309, U.S.A. E-mail: [email protected] Frank Krauss Institute for Particle Physics Phenomenology, Durham University, Durham DH1 3LE, U.K. E-mail: [email protected] Germ´an Rodrigo Instituto de F´ısica Corpuscular, CSIC-Universitat de Val`encia, Apartado de Correos 22085, E-46071 Valencia, Spain E-mail: [email protected] Jan-Christopher Winter Fermi National Accelerator Laboratory, Batavia, IL 60510, U.S.A. E-mail: [email protected] Abstract: We derive a duality relation between one-loop integrals and phase-space inte- grals emerging from them through single cuts. The duality relation is realized by a modifica- tion of the customary +i0 prescription of the Feynman propagators. The new prescription regularizing the propagators, which we write in a Lorentz covariant form, compensates for the absence of multiple-cut contributions that appear in the Feynman Tree Theorem. The duality relation can be applied to generic one-loop quantities in any relativistic, local and unitary field theories. We discuss in detail the duality that relates one-loop and tree-level Green’s functions. We comment on applications to the analytical calculation of one-loop scattering amplitudes, and to the numerical evaluation of cross-sections at next-to-leading order. -
Annales De L'ihp, Section A
ANNALES DE L’I. H. P., SECTION A A. BASSETTO M. TOLLER Harmonic analysis on the one-sheet hyperboloid and multiperipheral inclusive distributions Annales de l’I. H. P., section A, tome 18, no 1 (1973), p. 1-38 <http://www.numdam.org/item?id=AIHPA_1973__18_1_1_0> © Gauthier-Villars, 1973, tous droits réservés. L’accès aux archives de la revue « Annales de l’I. H. P., section A » implique l’accord avec les conditions générales d’utilisation (http://www.numdam. org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Inst. Henri Poincaré Section A : Vol. XVIII, no 1, 1973, 1 Physique théorique. Harmonic analysis on the one-sheet hyperboloid and multiperipheral inclusive distributions A. BASSETTO and M. TOLLER C. E. R. N., Geneva ABSTRACT. - The harmonic analysis of the n particle inclusive distri- butions and the partial diagonalization of the ABFST multiperipheral integral equation at vanishing momentum transfer are treated rigorously on the basis of the harmonic analysis of distributions defined on the one-sheet hyperboloid in four dimensions. A complete and consistent treatment is given of the Radon and Fourier transforms on the hyper- boloid and of the diagonalization of invariant kernels. The final result is a special form of the 0 (3, 1) expansion of the inclusive distributions, which exhibits peculiar dynamical features, in particular fixed poles. at the nonsense points, which are essential in order to get the experi- mentally observed behaviour.