On Some Inequalities for Gaussian Measures 815
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Sharp Finiteness Principles for Lipschitz Selections: Long Version
Sharp finiteness principles for Lipschitz selections: long version By Charles Fefferman Department of Mathematics, Princeton University, Fine Hall Washington Road, Princeton, NJ 08544, USA e-mail: [email protected] and Pavel Shvartsman Department of Mathematics, Technion - Israel Institute of Technology, 32000 Haifa, Israel e-mail: [email protected] Abstract Let (M; ρ) be a metric space and let Y be a Banach space. Given a positive integer m, let F be a set-valued mapping from M into the family of all compact convex subsets of Y of dimension at most m. In this paper we prove a finiteness principle for the existence of a Lipschitz selection of F with the sharp value of the finiteness number. Contents 1. Introduction. 2 1.1. Main definitions and main results. 2 1.2. Main ideas of our approach. 3 2. Nagata condition and Whitney partitions on metric spaces. 6 arXiv:1708.00811v2 [math.FA] 21 Oct 2017 2.1. Metric trees and Nagata condition. 6 2.2. Whitney Partitions. 8 2.3. Patching Lemma. 12 3. Basic Convex Sets, Labels and Bases. 17 3.1. Main properties of Basic Convex Sets. 17 3.2. Statement of the Finiteness Theorem for bounded Nagata Dimension. 20 Math Subject Classification 46E35 Key Words and Phrases Set-valued mapping, Lipschitz selection, metric tree, Helly’s theorem, Nagata dimension, Whitney partition, Steiner-type point. This research was supported by Grant No 2014055 from the United States-Israel Binational Science Foundation (BSF). The first author was also supported in part by NSF grant DMS-1265524 and AFOSR grant FA9550-12-1-0425. -
Gaussian Measures in Traditional and Not So Traditional Settings
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 33, Number 2, April 1996 GAUSSIAN MEASURES IN TRADITIONAL AND NOT SO TRADITIONAL SETTINGS DANIEL W. STROOCK Abstract. This article is intended to provide non-specialists with an intro- duction to integration theory on pathspace. 0: Introduction § Let Q1 denote the set of rational numbers q [0, 1], and, for a Q1, define the ∈ ∈ translation map τa on Q1 by addition modulo 1: a + b if a + b 1 τab = ≤ for b Q1. a + b 1ifa+b>1 ∈ − Is there a probability measure λQ1 on Q1 which is translation invariant in the sense that 1 λQ1 (Γ) = λQ1 τa− Γ for every a Q1? To answer∈ this question, it is necessary to be precise about what it is that one is seeking. Indeed, the answer will depend on the level of ambition at which the question is asked. For example, if denotes the algebra of subsets Γ Q1 generated by intervals A ⊆ [a, b] 1 q Q1 : a q b Q ≡ ∈ ≤ ≤ for a b from Q1, then it is easy to check that there is a unique, finitely additive way to≤ define λ on .Infact, Q1 A λ 1 [a, b] 1 = b a, for a b from Q1. Q Q − ≤ On the other hand, if one is more ambitious and asks for a countably additive λQ1 on the σ-algebra generated by (equivalently, all subsets of Q1), then one is asking A for too much. In fact, since λQ1 ( q ) = 0 for every q Q1, countable additivity forces { } ∈ λ 1 (Q1)= λ 1 q =0, Q Q { } q 1 X∈Q which is obviously irreconcilable with λQ1 (Q1) = 1. -
Infinite Dimension
Tel Aviv University, 2006 Gaussian random vectors 10 2 Basic notions: infinite dimension 2a Random elements of measurable spaces: formu- lations ......................... 10 2b Random elements of measurable spaces: proofs, andmorefacts .................... 14 2c Randomvectors ................... 17 2d Gaussian random vectors . 20 2e Gaussian conditioning . 24 foreword Up to isomorphism, we have for each n only one n-dimensional Gaussian measure (like a Euclidean space). What about only one infinite-dimensional Gaussian measure (like a Hilbert space)? Probability in infinite-dimensional spaces is usually overloaded with topo- logical technicalities; spaces are required to be Hilbert or Banach or locally convex (or not), separable (or not), complete (or not) etc. A fuss! We need measurability, not continuity; a σ-field, not a topology. This is the way to a single infinite-dimensional Gaussian measure (up to isomorphism). Naturally, the infinite-dimensional case is more technical than finite- dimensional. The crucial technique is compactness. Back to topology? Not quite so. The art is, to borrow compactness without paying dearly to topol- ogy. Look, how to do it. 2a Random elements of measurable spaces: formulations Before turning to Gaussian measures on infinite-dimensional spaces we need some more general theory. First, recall that ∗ a measurable space (in other words, Borel space) is a set S endowed with a σ-field B; ∗ B ⊂ 2S is a σ-field (in other words, σ-algebra, or tribe) if it is closed under finite and countable operations (union, intersection and comple- ment); ∗ a collection closed under finite operations is called a (Boolean) algebra (of sets); Tel Aviv University, 2006 Gaussian random vectors 11 ∗ the σ-field σ(C) generated by a given set C ⊂ 2S is the least σ-field that contains C; the algebra α(C) generated by C is defined similarly; ∗ a measurable map from (S1, B1) to (S2, B2) is f : S1 → S2 such that −1 f (B) ∈ B1 for all B ∈ B2. -
Proved for Real Hilbert Spaces. Time Derivatives of Observables and Applications
AN ABSTRACT OF THE THESIS OF BERNARD W. BANKSfor the degree DOCTOR OF PHILOSOPHY (Name) (Degree) in MATHEMATICS presented on (Major Department) (Date) Title: TIME DERIVATIVES OF OBSERVABLES AND APPLICATIONS Redacted for Privacy Abstract approved: Stuart Newberger LetA andH be self -adjoint operators on a Hilbert space. Conditions for the differentiability with respect totof -itH -itH <Ae cp e 9>are given, and under these conditionsit is shown that the derivative is<i[HA-AH]e-itHcp,e-itHyo>. These resultsare then used to prove Ehrenfest's theorem and to provide results on the behavior of the mean of position as a function of time. Finally, Stone's theorem on unitary groups is formulated and proved for real Hilbert spaces. Time Derivatives of Observables and Applications by Bernard W. Banks A THESIS submitted to Oregon State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy June 1975 APPROVED: Redacted for Privacy Associate Professor of Mathematics in charge of major Redacted for Privacy Chai an of Department of Mathematics Redacted for Privacy Dean of Graduate School Date thesis is presented March 4, 1975 Typed by Clover Redfern for Bernard W. Banks ACKNOWLEDGMENTS I would like to take this opportunity to thank those people who have, in one way or another, contributed to these pages. My special thanks go to Dr. Stuart Newberger who, as my advisor, provided me with an inexhaustible supply of wise counsel. I am most greatful for the manner he brought to our many conversa- tions making them into a mutual exchange between two enthusiasta I must also thank my parents for their support during the earlier years of my education.Their contributions to these pages are not easily descerned, but they are there never the less. -
Locally Symmetric Submanifolds Lift to Spectral Manifolds Aris Daniilidis, Jérôme Malick, Hristo Sendov
Locally symmetric submanifolds lift to spectral manifolds Aris Daniilidis, Jérôme Malick, Hristo Sendov To cite this version: Aris Daniilidis, Jérôme Malick, Hristo Sendov. Locally symmetric submanifolds lift to spectral mani- folds. 2012. hal-00762984 HAL Id: hal-00762984 https://hal.archives-ouvertes.fr/hal-00762984 Submitted on 17 Dec 2012 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. Locally symmetric submanifolds lift to spectral manifolds Aris DANIILIDIS, Jer´ ome^ MALICK, Hristo SENDOV December 17, 2012 Abstract. In this work we prove that every locally symmetric smooth submanifold M of Rn gives rise to a naturally defined smooth submanifold of the space of n × n symmetric matrices, called spectral manifold, consisting of all matrices whose ordered vector of eigenvalues belongs to M. We also present an explicit formula for the dimension of the spectral manifold in terms of the dimension and the intrinsic properties of M. Key words. Locally symmetric set, spectral manifold, permutation, symmetric matrix, eigenvalue. AMS Subject Classification. Primary 15A18, 53B25 Secondary 47A45, 05A05. Contents 1 Introduction 2 2 Preliminaries on permutations 4 2.1 Permutations and partitions . .4 2.2 Stratification induced by the permutation group . -
Probability Measures on Infinite-Dimensional Stiefel Manifolds
JOURNAL OF GEOMETRIC MECHANICS doi:10.3934/jgm.2017012 ©American Institute of Mathematical Sciences Volume 9, Number 3, September 2017 pp. 291–316 PROBABILITY MEASURES ON INFINITE-DIMENSIONAL STIEFEL MANIFOLDS Eleonora Bardelli‡ Microsoft Deutschland GmbH, Germany Andrea Carlo Giuseppe Mennucci∗ Scuola Normale Superiore Piazza dei Cavalieri 7, 56126, Pisa, Italia Abstract. An interest in infinite-dimensional manifolds has recently appeared in Shape Theory. An example is the Stiefel manifold, that has been proposed as a model for the space of immersed curves in the plane. It may be useful to define probabilities on such manifolds. Suppose that H is an infinite-dimensional separable Hilbert space. Let S ⊂ H be the sphere, p ∈ S. Let µ be the push forward of a Gaussian measure γ from TpS onto S using the exponential map. Let v ∈ TpS be a Cameron–Martin vector for γ; let R be a rotation of S in the direction v, and ν = R#µ be the rotated measure. Then µ, ν are mutually singular. This is counterintuitive, since the translation of a Gaussian measure in a Cameron– Martin direction produces equivalent measures. Let γ be a Gaussian measure on H; then there exists a smooth closed manifold M ⊂ H such that the projection of H to the nearest point on M is not well defined for points in a set of positive γ measure. Instead it is possible to project a Gaussian measure to a Stiefel manifold to define a probability. 1. Introduction. Probability theory has been widely studied for almost four cen- turies. Large corpuses have been written on the theoretical aspects. -
13 the Minkowski Bound, Finiteness Results
18.785 Number theory I Fall 2015 Lecture #13 10/27/2015 13 The Minkowski bound, finiteness results 13.1 Lattices in real vector spaces In Lecture 6 we defined the notion of an A-lattice in a finite dimensional K-vector space V as a finitely generated A-submodule of V that spans V as a K-vector space, where K is the fraction field of the domain A. In our usual AKLB setup, A is a Dedekind domain, L is a finite separable extension of K, and the integral closure B of A in L is an A-lattice in the K-vector space V = L. When B is a free A-module, its rank is equal to the dimension of L as a K-vector space and it has an A-module basis that is also a K-basis for L. We now want to specialize to the case A = Z, and rather than taking K = Q, we will instead use the archimedean completion R of Q. Since Z is a PID, every finitely generated Z-module in an R-vector space V is a free Z-module (since it is necessarily torsion free). We will restrict our attention to free Z-modules with rank equal to the dimension of V (sometimes called a full lattice). Definition 13.1. Let V be a real vector space of dimension n. A (full) lattice in V is a free Z-module of the form Λ := e1Z + ··· + enZ, where (e1; : : : ; en) is a basis for V . n Any real vector space V of dimension n is isomorphic to R . -
On Abstract Wiener Measure*
Balram S. Rajput Nagoya Math. J. Vol. 46 (1972), 155-160 ON ABSTRACT WIENER MEASURE* BALRAM S. RAJPUT 1. Introduction. In a recent paper, Sato [6] has shown that for every Gaussian measure μ on a real separable or reflexive Banach space {X, || ||) there exists a separable closed sub-space I of I such that μ{X) = 1 and μ% == μjX is the σ-extension of the canonical Gaussian cylinder measure μ^ of a real separable Hubert space <%f such that the norm ||-||^ = || \\/X is contiunous on g? and g? is dense in X. The main purpose of this note is to prove that ll llx is measurable (and not merely continuous) on Jgf. From this and the Sato's result mentioned above, it follows that a Gaussian measure μ on a real separable or reflexive Banach space X has a restriction μx on a closed separable sub-space X of X, which is an abstract Wiener measure. Gross [5] has shown that every measurable norm on a real separ- able Hubert space <%f is admissible and continuous on Sίf We show con- versely that any continuous admissible norm on J£? is measurable. This result follows as a corollary to our main result mentioned above. We are indebted to Professor Sato and a referee who informed us about reference [2], where, among others, more general results than those included here are proved. Finally we thank Professor Feldman who supplied us with a pre-print of [2].* We will assume that the reader is familiar with the notions of Gaussian measures and Gaussian cylinder measures on Banach spaces. -
Change of Variables in Infinite-Dimensional Space
Change of variables in infinite-dimensional space Denis Bell Denis Bell Change of variables in infinite-dimensional space Change of variables formula in R Let φ be a continuously differentiable function on [a; b] and f an integrable function. Then Z φ(b) Z b f (y)dy = f (φ(x))φ0(x)dx: φ(a) a Denis Bell Change of variables in infinite-dimensional space Higher dimensional version Let φ : B 7! φ(B) be a diffeomorphism between open subsets of Rn. Then the change of variables theorem reads Z Z f (y)dy = (f ◦ φ)(x)Jφ(x)dx: φ(B) B where @φj Jφ = det @xi is the Jacobian of the transformation φ. Denis Bell Change of variables in infinite-dimensional space Set f ≡ 1 in the change of variables formula. Then Z Z dy = Jφ(x)dx φ(B) B i.e. Z λ(φ(B)) = Jφ(x)dx: B where λ denotes the Lebesgue measure (volume). Denis Bell Change of variables in infinite-dimensional space Version for Gaussian measure in Rn − 1 jjxjj2 Take f (x) = e 2 in the change of variables formula and write φ(x) = x + K(x). Then we obtain Z Z − 1 jjyjj2 <K(x);x>− 1 jK(x)j2 − 1 jjxjj2 e 2 dy = e 2 Jφ(x)e 2 dx: φ(B) B − 1 jxj2 So, denoting by γ the Gaussian measure dγ = e 2 dx, we have Z <K(x);x>− 1 jjKx)jj2 γ(φ(B)) = e 2 Jφ(x)dγ: B Denis Bell Change of variables in infinite-dimensional space Infinite-dimesnsions There is no analogue of the Lebesgue measure in an infinite-dimensional vector space. -
Graduate Research Statement
Research Statement Irina Holmes October 2013 My current research interests lie in infinite-dimensional analysis and geometry, probability and statistics, and machine learning. The main focus of my graduate studies has been the development of the Gaussian Radon transform for Banach spaces, an infinite-dimensional generalization of the classical Radon transform. This transform and some of its properties are discussed in Sections 2 and 3.1 below. Most recently, I have been studying applications of the Gaussian Radon transform to machine learning, an aspect discussed in Section 4. 1 Background The Radon transform was first developed by Johann Radon in 1917. For a function f : Rn ! R the Radon transform is the function Rf defined on the set of all hyperplanes P in n given by: R Z Rf(P ) def= f dx; P where, for every P , integration is with respect to Lebesgue measure on P . If we think of the hyperplane P as a \ray" shooting through the support of f, the in- tegral of f over P can be viewed as a way to measure the changes in the \density" of P f as the ray passes through it. In other words, Rf may be used to reconstruct the f(x, y) density of an n-dimensional object from its (n − 1)-dimensional cross-sections in differ- ent directions. Through this line of think- ing, the Radon transform became the math- ematical background for medical CT scans, tomography and other image reconstruction applications. Besides the intrinsic mathematical and Figure 1: The Radon Transform theoretical value, a practical motivation for our work is the ability to obtain information about a function defined on an infinite-dimensional space from its conditional expectations. -
The Symmetric Theory of Sets and Classes with a Stronger Symmetry Condition Has a Model
The symmetric theory of sets and classes with a stronger symmetry condition has a model M. Randall Holmes complete version more or less (alpha release): 6/5/2020 1 Introduction This paper continues a recent paper of the author in which a theory of sets and classes was defined with a criterion for sethood of classes which caused the universe of sets to satisfy Quine's New Foundations. In this paper, we describe a similar theory of sets and classes with a stronger (more restrictive) criterion based on symmetry determining sethood of classes, under which the universe of sets satisfies a fragment of NF which we describe, and which has a model which we describe. 2 The theory of sets and classes The predicative theory of sets and classes is a first-order theory with equality and membership as primitive predicates. We state axioms and basic definitions. definition of class: All objects of the theory are called classes. axiom of extensionality: We assert (8xy:x = y $ (8z : z 2 x $ z 2 y)) as an axiom. Classes with the same elements are equal. definition of set: We define set(x), read \x is a set" as (9y : x 2 y): elements are sets. axiom scheme of class comprehension: For each formula φ in which A does not appear, we provide the universal closure of (9A :(8x : set(x) ! (x 2 A $ φ))) as an axiom. definition of set builder notation: We define fx 2 V : φg as the unique A such that (8x : set(x) ! (x 2 A $ φ)). There is at least one such A by comprehension and at most one by extensionality. -
Spectral Properties of Structured Kronecker Products and Their Applications
Spectral Properties of Structured Kronecker Products and Their Applications by Nargiz Kalantarova A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Combinatorics and Optimization Waterloo, Ontario, Canada, 2019 c Nargiz Kalantarova 2019 Examining Committee Membership The following served on the Examining Committee for this thesis. The decision of the Examining Committee is by majority vote. External Examiner: Maryam Fazel Associate Professor, Department of Electrical Engineering, University of Washington Supervisor: Levent Tun¸cel Professor, Deptartment of Combinatorics and Optimization, University of Waterloo Internal Members: Chris Godsil Professor, Department of Combinatorics and Optimization, University of Waterloo Henry Wolkowicz Professor, Department of Combinatorics and Optimization, University of Waterloo Internal-External Member: Yaoliang Yu Assistant Professor, Cheriton School of Computer Science, University of Waterloo ii I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii Statement of contributions The bulk of this thesis was authored by me alone. Material in Chapter5 is related to the manuscript [75] that is a joint work with my supervisor Levent Tun¸cel. iv Abstract We study certain spectral properties of some fundamental matrix functions of pairs of sym- metric matrices. Our study includes eigenvalue inequalities and various interlacing proper- ties of eigenvalues. We also discuss the role of interlacing in inverse eigenvalue problems for structured matrices. Interlacing is the main ingredient of many fundamental eigenvalue inequalities.