DUALITY PHENOMENA and VOLUME INEQUALITIES in CONVEX GEOMETRY a Dissertation Submitted to Kent State University in Partial Fulfil
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Wiener Measures on Certain Banach Spaces Over Non-Archimedean Local fields Compositio Mathematica, Tome 93, No 1 (1994), P
COMPOSITIO MATHEMATICA TAKAKAZU SATOH Wiener measures on certain Banach spaces over non-archimedean local fields Compositio Mathematica, tome 93, no 1 (1994), p. 81-108 <http://www.numdam.org/item?id=CM_1994__93_1_81_0> © Foundation Compositio Mathematica, 1994, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 81 © 1994 Kluwer Academic Publishers. Printed in the Netherlands. Wiener measures on certain Banach spaces over non-Archimedean local fields TAKAKAZU SATOH Department of Mathematics, Faculty of Science, Saitama University, 255 Shimo-ookubo, Urawu, Saitama, 338, Japan Received 8 June 1993; accepted in final form 29 July 1993 Dedicated to Proffessor Hideo Shimizu on his 60th birthday 1. Introduction The integral representation is a fundamental tool to study analytic proper- ties of the zeta function. For example, the Euler p-factor of the Riemann zeta function is where dx is a Haar measure of Q, and p/( p - 1) is a normalizing constant so that measure of the unit group is 1. What is necessary to generalize such integrals to a completion of a ring finitely generated over Zp? For example, let T be an indeterminate. The two dimensional local field Qp«T» is an infinite dimensional Qp-vector space. -
Topics in Algorithmic, Enumerative and Geometric Combinatorics
Thesis for the Degree of Doctor of Philosophy Topics in algorithmic, enumerative and geometric combinatorics Ragnar Freij Division of Mathematics Department of Mathematical Sciences Chalmers University of Technology and University of Gothenburg G¨oteborg, Sweden 2012 Topics in algorithmic, enumerative and geometric combinatorics Ragnar Freij ISBN 978-91-7385-668-3 c Ragnar Freij, 2012. Doktorsavhandlingar vid Chalmers Tekniska H¨ogskola Ny serie Nr 3349 ISSN 0346-718X Department of Mathematical Sciences Chalmers University of Technology and University of Gothenburg SE-412 96 GOTEBORG,¨ Sweden Phone: +46 (0)31-772 10 00 [email protected] Printed in G¨oteborg, Sweden, 2012 Topics in algorithmic, enumerative and geometric com- binatorics Ragnar Freij ABSTRACT This thesis presents five papers, studying enumerative and extremal problems on combinatorial structures. The first paper studies Forman’s discrete Morse theory in the case where a group acts on the underlying complex. We generalize the notion of a Morse matching, and obtain a theory that can be used to simplify the description of the G-homotopy type of a simplicial complex. As an application, we determine the S2 × Sn−2-homotopy type of the complex of non-connected graphs on n nodes. In the introduction, connections are drawn between the first paper and the evasiveness conjecture for monotone graph properties. In the second paper, we investigate Hansen polytopes of split graphs. By applying a partitioning technique, the number of nonempty faces is counted, and in particular we confirm Kalai’s 3d-conjecture for such polytopes. Further- more, a characterization of exactly which Hansen polytopes are also Hanner polytopes is given. -
Mixtures of Gaussian Cylinder Set Measures and Abstract Wiener Spaces As Models for Detection Annales De L’I
ANNALES DE L’I. H. P., SECTION B A. F. GUALTIEROTTI Mixtures of gaussian cylinder set measures and abstract Wiener spaces as models for detection Annales de l’I. H. P., section B, tome 13, no 4 (1977), p. 333-356 <http://www.numdam.org/item?id=AIHPB_1977__13_4_333_0> © Gauthier-Villars, 1977, tous droits réservés. L’accès aux archives de la revue « Annales de l’I. H. P., section B » (http://www.elsevier.com/locate/anihpb) implique l’accord avec les condi- tions générales d’utilisation (http://www.numdam.org/conditions). Toute uti- lisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Inst. Henri Pincaré, Section B : Vol. XIII, nO 4, 1977, p. 333-356. Calcul des Probabilités et Statistique. Mixtures of Gaussian cylinder set measures and abstract Wiener spaces as models for detection A. F. GUALTIEROTTI Dept. Mathematiques, ~cole Polytechnique Fédérale, 1007 Lausanne, Suisse 0. INTRODUCTION AND ABSTRACT Let H be a real and separable Hilbert space. Consider on H a measurable family of weak covariance operators { Re, 0 > 0 } and, on !?+ : = ]0, oo [, a probability distribution F. With each Re is associated a weak Gaussian distribution Jlo. The weak distribution p is then defined by Jl can be studied, as we show below, within the theory of abstract Wiener spaces. The motivation for considering such measures comes from statistical communication theory. There, signals of finite energy are often modelled as random variables, normally distributed, with values in a L2-space. -
Plenary Lectures
Plenary lectures A. D. Alexandrov and the Birth of the Theory of Tight Surfaces Thomas Banchoff Brown University, Providence, Rhode Island 02912, USA e-mail: Thomas [email protected] Alexander Danilovich Alexandrov was born in 1912, and just over 25 years later in 1938 (the year of my birth), he proved a rigidity theorem for real analytic surfaces of type T with minimal total absolute curvature. He showed in particular that that two real analytic tori of Type T in three-space that are iso- metric must be congruent. In 1963, twenty-five years after that original paper, Louis Nirenberg wrote the first generalization of that result, for five times differentiable surfaces satisfying some technical hypothe- ses necessary for applying techniques of ordinary and partial differential equations. Prof. Shiing-Shen Chern, my thesis advisor at the University of California, Berkeley, gave me this paper to present to his graduate seminar and challenged me to find a proof without these technical hypotheses. I read Konvexe Polyeder by A. D. Alexandrov and the works of Pogoreloff on rigidity for convex surfaces using polyhedral methods and I decided to try to find a rigidity theorem for Type T surfaces using similar techniques. I found a condition equivalent to minimal total absolute curvature for smooth surfaces, that also applied to polyhedral surfaces, namely the Two-Piece Property or TPP. An object in three- space has the TPP if every plane separates the object into at most two connected pieces. I conjectured a rigidity result similar to those of Alexandroff and Nirenberg, namely that two polyhedral surfaces in three-space with the TPP that are isometric would have to be congruent. -
Minimal Volume Product of Three Dimensional Convex Bodies with Various Discrete Symmetries
Minimal volume product of three dimensional convex bodies with various discrete symmetries Hiroshi Iriyeh ∗ Masataka Shibata † August 12, 2021 Abstract We give the sharp lower bound of the volume product of three dimensional convex bodies which are invariant under a discrete subgroup of O(3) in several cases. We also characterize the convex bodies with the minimal volume product in each case. In particular, this provides a new partial result of the non-symmetric version of Mahler’s conjecture in the three dimensional case. 1 Introduction and main results 1.1 Mahler’s conjecture and its generalization Let K be a convex body in Rn, i.e, K is a compact convex set in Rn with nonempty interior. Then K ⊂ Rn is said to be centrally symmetric if it satisfies that K = −K. Denote by Kn the set of all convex bodies in Rn equipped with the Hausdorff metric and n n by K0 the set of all K ∈ K which are centrally symmetric. The interior of K ∈ Kn is denoted by int K. For a point z ∈ int K, the polar body of K with respect to z is defined by z n K = {y ∈ R ;(y − z) · (x − z) ≤ 1 for any x ∈ K} , where · denotes the standard inner product on Rn. Then an affine invariant P(K) := min |K| |Kz| (1) z∈int K is called volume product of K, where |K| denotes the n-dimensional volume of K in Rn. arXiv:2007.08736v2 [math.MG] 8 Oct 2020 It is known that for each K ∈ Kn the minimum of (1) is attained at the unique point z on K, which is called Santaló point of K (see, e.g., [13]). -
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. -
On K-Level Matroids: Geometry and Combinatorics Francesco Grande
On k-level matroids: geometry and combinatorics ⊕ ⊕ Dissertation zur Erlangung des akademisches Grades eines Doktors der Naturwissenschaften am Fachbereich Mathematik und Informatik der Freien Universität Berlin vorgelegt von Francesco Grande Berlin 2015 Advisor and first reviewer: Professor Dr. Raman Sanyal Second reviewer: Professor Rekha Thomas, Ph.D. Date of the defense: October 12, 2015 Summary The Theta rank of a finite point configuration V is the maximal degree nec- essary for a sum-of-squares representation of a non-negative linear function on V . This is an important invariant for polynomial optimization that is in general hard to determine. We study the Theta rank of point configu- rations via levelness, that is a discrete-geometric invariant, and completely classify the 2-level (equivalently Theta-1) configurations whose convex hull is a simple or a simplicial polytope. We consider configurations associated to the collection of bases of matroids and show that the class of matroids with bounded Theta rank or levelness is closed under taking minors. This allows us to find a characterization of matroids with bounded Theta rank or levelness in terms of forbidden minors. We give the complete (finite) list of excluded minors for Theta-1 matroids which generalize the well-known series-parallel graphs. Moreover, we char- acterize the class of Theta-1 matroids in terms of the degree of generation of the vanishing ideal and in terms of the psd rank for the associated matroid base polytope. We analyze in full detail Theta-1 matroids from a constructive perspective and discover that they are sort-closed, which allows us to determine a uni- modular triangulation of every matroid base polytope and to characterize its volume by means of permutations. -
Dynamics of Infinite-Dimensional Groups and Ramsey-Type Phenomena
\rtp-impa" i i 2005/5/12 page 1 i i Dynamics of infinite-dimensional groups and Ramsey-type phenomena Vladimir Pestov May 12, 2005 i i i i \rtp-impa" i i 2005/5/12 page 2 i i i i i i \rtp-impa" i i 2005/5/12 page i i i Contents Preface iii 0 Introduction 1 1 The Ramsey{Dvoretzky{Milman phenomenon 17 1.1 Finite oscillation stability . 17 1.2 First example: the sphere S1 . 27 1.2.1 Finite oscillation stability of S1 . 27 1.2.2 Concentration of measure on spheres . 32 1.3 Second example: finite Ramsey theorem . 41 1.4 Counter-example: ordered pairs . 47 2 The fixed point on compacta property 49 2.1 Extremely amenable groups . 49 2.2 Three main examples . 55 2.2.1 Example: the unitary group . 55 2.2.2 Example: the group Aut (Q; ) . 60 ≤ 2.2.3 Counter-example: the infinite symmetric group 62 2.3 Equivariant compactification . 63 2.4 Essential sets and the concentration property . 66 2.5 Veech theorem . 71 2.6 Big sets . 76 2.7 Invariant means . 81 3 L´evy groups 95 3.1 Unitary group with the strong topology . 95 i i i i i \rtp-impa" i i 2005/5/12 page ii i i ii CONTENTS 3.2 Group of measurable maps . 98 3.2.1 Construction and example . 98 3.2.2 Martingales . 101 3.2.3 Unitary groups of operator algebras . 110 3.3 Groups of transformations of measure spaces . 115 3.3.1 Maurey's theorem . -
The Singular Moving Points Are What Is Left of the Moving Points
BOOK REVIEWS 695 points. Those which are not singular, are regular. The regular moving points are the union of disjoint annuli; the singular moving points are what is left of the moving points. The description of the flow then is given in terms of the regular moving points, where it can be completely described in terms of simple "patches"-certain standard annular flows; the fixed points-where it is, of course, fixed; and the singular moving points-where life gets complicated. To develop the theory for singular moving points, the author first considers a very special case: the case of flows in a multiply-connected region of the plane where every orbit is aperiodic and has all its endpoints in the boundary. These he calls Kaplan-Mar kus flows, after the two who first began develop ment of the theory of such flows. Complete success in describing such flows has eluded the author and his coworkers except, to some extent, where the set of singular fixed points has only finitely many components. The later is, however, basic, and so for many cases, another patch in the quilt yields to description. After this, the author explores various ways to combine such flows with regular flows and develop further theory. Flows without stagnation points can be described rather completely. For flows with a finite number of stagnation points, or whose set of stagnation points has countable closure, a considerable amount is known, though the information is not as complete as for the no stagnation point case. In all cases where a description is possible, it is the set of regular moving points that supplies the main body of information. -
On Superpositional Filtrations
Munich Personal RePEc Archive On superpositional filtrations Strati, Francesco 20 August 2012 Online at https://mpra.ub.uni-muenchen.de/40879/ MPRA Paper No. 40879, posted 29 Aug 2012 04:19 UTC On superpositional filtrations Francesco Strati Department of Statistics, University of Messina Messina, Italy E-mail: [email protected] Abstract In this present work I shall define the basic notions of superpositional filtrations. Given a superposition integral I shall find a general measure theory by means of cylinder sets and then I shall define the properties of the sfiltration for a general process X. n §1. We denote by Sn ≡ S(R ) the Schwartz space endowed with a natural topology (Sτ ) which stems from the seminorm α β ||ϕ||α,β ≡ sup |x D ϕ(x)| < ∞. x∈Rn ′ ′ Rn We call this kind of space S-space, and it is a Fr´echet space [FrB]. We denote by Sn ≡S ( ) the space of tempered distributions. It is the topological dual of the topological vector space (Sn, Sτ ) [FrT]. A locally integrable function can be a tempered distribution if, for some constants a and C |f(x)|dx ≤ aGC , G → ∞ Z|x|≤G and thus |f(x)ϕ(x)|dx < ∞, ∀ϕ ∈S′. Rn Z Therefore we can say that Rn f(x)ϕ(x)dx is a tempered distribution [FrT]. ′ ′ We know that the superpositionR integral [FrB] runs through S -spaces, rather given a ∈ Sm Rn ′ ′ Rm ′ and a summable v ∈S( , S ), the superposition integral of v on Sm( ) is the element of Sn defined by lim (αk ◦ vˆ)= av = a ◦ vˆ k→+∞ Rm Z 2010 Mathematics Subject Classification: Primary 28C05, 28C15 ; Secondary 28A05. -
Incidence Graphs and Unneighborly Polytopes
Incidence Graphs and Unneighborly Polytopes vorgelegt von Diplom-Mathematiker Ronald Frank Wotzlaw aus G¨ottingen Von der Fakult¨at II – Mathematik und Naturwissenschaften der Technischen Universit¨at Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften – Dr. rer. nat. – genehmigte Dissertation Promotionsausschuss: Vorsitzender: Prof. Fredi Tr¨oltzsch Berichter: Prof. G¨unter M. Ziegler Prof. Gil Kalai Tag der wissenschaftlichen Aussprache: 6. Februar 2009 Berlin 2009 D 83 . for though it cannot hope to be useful or informative on all matters, it does at least make the reassuring claim, that where it is inaccurate it is at least definitively inaccurate. In cases of major discrepancy it’s always reality that’s got it wrong. The Restaurant at the End of the Universe Douglas Adams For Finja and Nora Preface This thesis is about polytopes and “their life in high dimensions.” Nearly all questions considered here are trivial or simple exercises when restricted to dimensions that we can “see.” More often than not, the interesting examples start to appear in dimensions d = 6, 7, 8 or even much higher. Nevertheless, I have tried to illustrate ideas with pictures throughout. The reader should be warned however that they rarely show the real nature of the mathematical problem at hand. Some of the results are generalized to objects that are more abstract than polytopes. I have taken a pragmatical point of view with respect to the level of generality that theorems are stated and proved: If a generalization was asked for in the existing literature, it is provided (if possible). The following is a list of the main results in this thesis in order of appearance: The connectivity of (k,ℓ)-incidence graphs is determined for the case • ℓ 2k + 1 in Chapter 3. -
Arxiv:1905.02109V1 [Math.FA] 6 May 2019 Hl N14,F On( John F
SCIENCE CHINA Mathematics CrossMark . ARTICLES . doi: Infinite dimensional Cauchy-Kowalevski and Holmgren type theorems Jiayang YU1 & Xu ZHANG 2,∗ 1 School of Mathematics, Sichuan University, Chengdu 610064, P. R. China; 2 School of Mathematics, Sichuan University, Chengdu 610064, P. R. China Email: [email protected], zhang [email protected] Abstract The aim of this paper is to show Cauchy-Kowalevski and Holmgren type theorems with infinite number of variables. We adopt von Koch and Hilbert’s definition of analyticity of functions as monomial expansions. Our Cauchy-Kowalevski type theorem is derived by modifying the classical method of majorants. Based on this result, by employing some tools from abstract Wiener spaces, we establish our Holmgren type theorem. Keywords Cauchy-Kowalevski theorem, Holmgren theorem, monomial expansions, abstract Wiener space, divergence theorem, method of majorants MSC(2010) Primary 35A10, 26E15, 46G20; Secondary 46G20, 46G05, 58B99. Citation: Jiayang Yu, Xu Zhang. Infinite dimensional Cauchy-Kowalevski and Holmgren type theorems. Sci China Math, 2019, 60, doi: 1 Introduction The classical Cauchy-Kowalevski theorem asserts the local existence and uniqueness of analytic solutions to quite general partial differential equations with analytic coefficients and initial data in the finite dimensional Euclidean space n (for any given n ). In 1842, A. L. Cauchy first proved this theorem arXiv:1905.02109v1 [math.FA] 6 May 2019 R N ∈ for the second order case; while in 1875, S. Kowalevski proved the general result. Both of them used the method of majorants. Because of its fundamental importance, there exists continued interest to generalize and/or improve this theorem (See [2,10,24,26–29,31–33,35] and the references cited therin).