DOCSLIB.ORG

Notes on Integral Geometry Uga, Fall 2011

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Notes on Integral Geometry Uga, Fall 2011 NOTES ON INTEGRAL GEOMETRY UGA, FALL 2011 Contents 1. Introduction 1 1.1. Goals of the subject and the course 1 1.2. Buffon needle problem 2 1.3. Integral geometry of the euclidean plane 2 1.4. Relaxation of smoothness conditions. 6 2. Classical integral geometry 6 2.1. Steiner's formula 6 2.2. Statement of Hadwiger's theorem, and its integral geometric consequences 10 2.3. Support functions, the cosine transform and zonoids 14 2.4. Proof of Hadwiger's theorem 17 2.5. Even valuations: Klain functions, Crofton measures and the Alesker Fourier transform 19 2.6. The McMullen decomposition 20 2.7. Nijenhuis's theorem 20 2.8. The Alesker product, part I 23 2.9. Review of differential forms 24 2.10. Integration of differential forms and the normal cycle 28 2.11. Curvature measures and valuations. First variation and the kernel theorem 31 2.12. Kinematic formulas for invariant curvature measures 34 2.13. The transfer principle 37 2.14. Integral geometry of real space forms 40 3. Alesker theory for affine spaces 46 3.1. Alesker irreducibility and its consequences 46 3.2. Convolution 49 3.3. Constant coefficient valuations 52 n 4. Integral geometry of (C ;U(n)) 54 4.1. Hermitian intrinsic volumes and Tasaki valuations 54 4.2. On kU(n)(χ) 56 4.3. Algebraic structure of ValU(n) 57 4.4. Dictionaries 61 n 4.5. Global kinematic formulas in C 63 4.6. Two dualities 64 References 66 1. Introduction 1.1. Goals of the subject and the course. Geometric probability. Rota: \continuous combi- natorics" Geometry of singular spaces: language of valuations and curvature measures is natural. Date: December 5, 2011. 1 2 NOTES ON INTEGRAL GEOMETRY UGA, FALL 2011 To give an account of the basic vocabulary, which (as usual in geometry) includes a variety of complementary approaches. To describe and (partially) prove recent results about the integral geometry of the unitary group. 1.2. Buffon needle problem. Reference: [21] Imagine the euclidean plane ruled by a system of parallel lines one inch apart. Drop a needle of length ` ≤ one inch in a random position on the plane. What is the probability that the needle crosses one of the lines? First we reframe the question as: what is the expectation E(`) of the number of lines crossed by a needle of unrestricted length? If ` ≤ 1 then this is the same as the first question. Observe that E(`1 +`2) = E(`1)+E(`2): for we may think of the long needle as the concatenation of the two shorter ones. Although the two crossing events are not independent, dropping one of the needles at random also gives a random position for the other. In fact it is not even necessary that the two needles be colinear, and furthermore one may concatenate any number of needles and the same additivity will result. Finally, (by continuity and perhaps a leap of faith) the needle may even be curved. So the function E(`) is linear, and gives the expected number of crossings of any rectifiable curve of length `. Thus E(`) = c` for some constant c. To evaluate the constant c, take the curve to be a circle C of diameter 1. In this case C crosses exactly two lines regardless of its position. Thus 2 cπ = E(π) = 2 =) c = : π This argument is an instance of the \template method": once we know that an integral-geometric formula of a certain kind exists, in order to specify the constants one finds one or more convenient geometric objects for which everything is directly computable. Then the constants can be computed by working out the resulting linear equations. Ideally the templates can be chosen so that the system is diagonal. 1.3. Integral geometry of the euclidean plane. 2 2 1.3.1. Haar measures on Gr1(R ) and SO(2). It is convenient to identify R ' C. The euclidean 1 iθ group SO(2) is diffeomorphic to S × C by identifying (e ; z0) to the map iθ z 7! z0 + e z: iθ Thus z0 is the image of the origin and e is the (constant) derivative of the map. 3 Consider the the measure on SO(2) given by identifying it locally with R via the coordinates θ; x; y and taking Lebesgue measure there. Since iθ i i(θ+ ) iθ (e ; z0) · (e ; w0) = (e ; e w0 + z0) Fubini's theorem shows that this measure is invariant under the action of the group on itself: if E ⊂ SO(2) 3 g then m(gE) = m(E). The affine Grassmannian Gr1(C) is a 2-dimensional manifold. It can be viewed as a quotient of SO(2) as follows. Clearly SO(2) acts transitively on Gr1(C). The subgroup H := {±1g×R ⊂ SO(2) is the stabilizer of R ⊂ C. Hence the map g 7! gR gives an identification of the left coset space SO(2) =H ' Gr1(C). iθ −iθ The locally defined functions (e ; z0) 7! θ; =(e z0) on SO(2) are invariant, up to sign, under the right action of H. Therefore they are lifts of locally defined functions θ; p on Gr1. Geometrically, θ is the angle that the line makes with the x-axis. Exercise 1.1. Show that p = ± the distance from the line to the origin. Exercise 1.2. θ; p give local coordinates on Gr1. The resulting measure dpdθ is invariant under the action of SO(2). NOTES ON INTEGRAL GEOMETRY UGA, FALL 2011 3 1.3.2. Crofton's formula for the length of a rectifiable plane curve. Recall that a continuous curve 2 γ : [0; 1] ! R is rectifiable if N X jγj := sup jγ(ti) − γ(ti−1j < 1: P=ft0<···<tN g i=1 Here the sup is taken over all partitions P of [0; 1] and #(γ \ `) := #fs : γ(s) 2 `g. Lemma 1.3. Let f : R ! R. Then the set of all local extreme values of f is countable. Proof. A point x 2 R is a local max if there is an open interval (p; q) 3 x such that f(x) = maxp;q f. Clearly p; q may be taken to be rational. Thus the set of all local maximum values is contained in the countable set fsup f : p; q 2 Qg: (p;q) The set of local minimum values is likewise countable. Lemma 1.4. Let f : R ! R be continuous. Suppose that c is not a local extreme value of f, and that f(x0) = c. Let > 0 be given. Then for any sufficiently fine partition P of R there are points a1; a2 2 P, with jx − aij < , such that f(a1) < c < f(a2). 2 Theorem 1.5. Let γ ⊂ R be a rectifiable curve. Then Z jγj = 4 #(γ \ `(θ; p)) dθdp: Gr1 Proof. Given a partition P, let γP denote the corresponding piecewise linear path, with vertices at the points γ(t); t 2 P. On the other hand, one easily calculates that for a line segments σ Z #(σ \ `(θ; p)) dθdp = 4jσj Gr1 and therefore Z #(γP \ `(θ; p)) dθdp = 4jγP j: Gr1 Thus Z sup #(γP \ `(θ; p)) dθdp = 4jγj: (1) P Gr1 and it remains to show that the left hand side equals R #(γ \ `(θ; p)) dθdp. Gr1 Let P1; P2;::: be a sequence of successive refinements with mesh ! 0. By the intermediate value theorem, given any line ` #(` \ γPi ) ≤ #(` \ γPi+1 ) ≤ #(` \ γ): By (1) and the dominated convergence theorem, #(` \ γP ) ≤ N(`) := lim #(` \ γP ) < 1 i i!1 i for all i. Let u be a unit vector and consider the continuous function f; = u · γ. The level sets f −1(c) correspond to the intersections between γ and the lines ? to u. By Lemma 1.3, among these lines ` only countably many correspond to local extreme values c. By Lemma 1.4, if ` is one of the generic lines that do not, then for sufficiently large i #(` \ γPi ) = #(` \ γ): Thus the functions ` 7! #(γPi \ `) increase to ` 7! #(γ \ `) at a.e. line `, which with (1) and dominated convergence concludes the proof. 4 NOTES ON INTEGRAL GEOMETRY UGA, FALL 2011 1.3.3. Poincar´eformula. Theorem 1.6. Let β; γ be C1 curves in the plane. Then Z #(β \ gγ) dg = 4jβjjγj (2) SO(2) Proof. Consider the map F : S1 × β × γ ! SO(2); (eiθ; p; q) 7! (eiθ; β(s) − eiθγ(t)): Clearly F (p; q; eiθ)q = p, and the cardinality of F −1(g) is #(β \ gγ). Thus the change of variables formula gives Z Z #(β \ gγ) dg = jdet DF j (3) SO(2) S1×β×γ | though strictly speaking we should replace F in the right hand integral by its expression in local coordinates (θ ◦ F; x ◦ F; y ◦ F ). For simplicity of notation we abbreviate this as (θ; x; y). Let s; t be the respective arclength coordinates on β; γ and the angular parameter on the S1 factor in the domain. Then @θ = 1 @ @θ @θ = = 0 @x @y so the determinant in question is @x @x @s @t 0 iθ 0 0 iθ 0 0 iθ 0 det @y @y = <β =(e γ ) − =β <(e γ ) = hβ ; ie γ i: @s @t The integral with respect to θ of the absolute value of this expression is clearly independent of s; t, so by (3) the formula (2) holds for some value of the constant. To evaluate the constant we use the template method, taking β; γ to be circles of radius 1.
Load more

Recommended publications
  • Integral Geometry and Algebraic Structures for Tensor Valuations
    Integral Geometry and Algebraic Structures for Tensor Valuations Andreas Bernig and Daniel Hug Abstract In this survey, we consider various integral geometric formulas for tensor- valued valuations that have been obtained by different methods. Furthermore we explain in an informal way recently introduced algebraic structures on the space of translation invariant, smooth tensor valuations, including convolution, product, Poincare´ duality and Alesker-Fourier transform, and their relation to kinematic formulas for tensor valuations. In particular, we describe how the algebraic viewpoint leads to new intersectional kinematic formulas and substantially simplified Crofton formulas for translation invariant tensor valuations. We also highlight the connection to general integral geometric formulas for area measures. 1 Introduction An important part of integral geometry is devoted to the investigation of integrals (mean values) of the form Z j(K \ gL) m(dg); G n where K; L ⊂ R are sets from a suitable intersection stable class of sets, G is a group n acting on R and thus on its subsets, m is a Haar measure on G, and j is a functional with values in some vector space W. Common choices for W are the reals or the space of signed Radon measures. Instead of the intersection, Minkowski addition is another natural choice for a set operation which has been studied. The principle aim then is to express such integrals by means of basic geometric functionals of K Andreas Bernig Institut fur¨ Mathematik, Goethe-Universitat¨ Frankfurt, Robert-Mayer-Str. 10, 60054 Frankfurt, Germany, e-mail: [email protected], Daniel Hug Karlsruhe Institute of Technology (KIT), Department of Mathematics, D-76128 Karlsruhe, Germany, e-mail: [email protected] 59 60 Andreas Bernig and Daniel Hug and L.
    [Show full text]
  • Integral Geometry and Radon Transforms, by Sigurdur Helgason, Springer, New York, 2010, Xiv+301 Pp., ISBN 978-1-4419-6054-2, Hardcover
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY S 0273-0979(2012)01391-5 Article electronically published on December 6, 2012 Integral geometry and Radon transforms, by Sigurdur Helgason, Springer, New York, 2010, xiv+301 pp., ISBN 978-1-4419-6054-2, hardcover 1. Integral geometry The book under review is an excellent introduction to the group theoretical and analytic aspects of the field by one of its pioneers. Before reviewing the book, we will provide an overview of the field. Integral geometry draws together analysis, geometry, and numerical mathemat- ics. It has direct applications in PDEs, group representations, and the applied mathematical field of tomography. The fundamental problem in integral geometry is to determine properties of a function f in the plane or three-dimensional space or other manifolds from knowing the integrals of f over lines, planes, hyperplanes, spheres, or other submanifolds. The history of integral geometry starts in the early 1900s with Radon, Funk, Lorenz, and others. In 1917, Johann Radon [60] proved an inversion formula for whathasbecomeknownastheclassical Radon line transform: if is a line in the plane and f is an integrable function, then ż (1.1) RLfpq“ fpxqds, xP where ds is the arc length measure on the line . This is the mathematical model behind X-ray tomography, which we discuss in Section 5. One can extend this n to the hyperplane transform, RH , which integrates over all hyperplanes H in R . 3 RadonalsoprovedaninversionformulaforRH in R . (According to Bockwinkel [5], the inversion for RH had already been obtained by H. A. Lorenz before 1906.) In 1936, Cramer and Wold [10] proved injectivity of the hyperplane transform for probability measures.
    [Show full text]
  • Even If Google Sent You Here, What Follows Is NOT the Book RANDOM
    i Note: Even if Google sent you here, what follows is NOT the book RANDOM FIELDS AND GEOMETRY published with Springer in 2007, but rather a companion volume, still under production, that gives a simpler version of the theory of the first book as well as lots of applications. You can find the original Random Fields and Geometry on the Springer site. Meanwhile, enjoy what is available of the second volume, and keep in mind that we are happy to receive all comments, suggestions and, most of all, cor- rections. ii Applications of RANDOM FIELDS AND GEOMETRY Foundations and Case Studies Robert J. Adler Faculty of Industrial Engineering and Management and Faculty of Electrical Engineering Technion { Israel Institute of Technology Haifa, Israel e-mail: [email protected] Jonathan E. Taylor Department of Statistics Stanford University Stanford, California. e-mail: [email protected] Keith J. Worsley Department of Mathematics and Statistics McGill University Montr´eal,Canada and Department of Statistics University of Chicago Chicago, Illinois. e-mail: [email protected] May 20, 2015 0 Preface Robert J. Adler Jonathan E. Taylor1 Keith Worsley Haifa, Israel Stanford, California Montreal, Canada ie.technion.ac.il/Adler.phtml www-stat.stanford.edu/∼jtaylo www.math.mcgill.ca/keith 1Research supported in part by NSF grant DMS-0405970 Contents 0 Preface .................................................... iii 1 Introduction ............................................... 1 1.1 An Initial Example......................................1 1.2 Gaussian and Related Random Fields......................3 1.3 Shape and the Euler Characteristic........................4 1.4 The Large-Scale Structure of the Universe..................7 1.4.1 Survey Data......................................7 1.4.2 Designing and Testing a Model......................8 1.5 Brain Imaging.........................................
    [Show full text]
  • Integral Geometry and Radon Transforms
    TO ARTIE Preface This book deals with a special subject in the wide field of Geometric Anal- ysis. The subject has its origins in results by Funk [1913] and Radon [1917] determining, respectively, a symmetric function on the two-sphere S2 from its great circle integrals and an integrable function on R2 from its straight line integrals. (See References.) The first of these is related to a geometric theorem of Minkowski [1911] (see Ch. III, 1). While the above work of Funk and Radon§ lay dormant for a while, Fritz John revived the subject in important papers during the thirties and found significant applications to differential equations. More recent applications to X-ray technology and tomography have widened interest in the subject. This book originated with lectures given at MIT in the Fall of 1966, based mostly on my papers during 1959–1965 on the Radon transform and its generalizations. The viewpoint of these generalizations is the following. The set of points on S2 and the set of great circles on S2 are both acted on transitively by the group O(3). Similarly, the set of points in R2 and the set P2 of lines in R2 are both homogeneous spaces of the group M(2) of rigid motions of R2. This motivates our general Radon transform definition from [1965a] and [1966a], which forms the framework of Chapter II: Given two homogeneous spaces X = G/K and Ξ = G/H of the same group G, two elements x = gK and ξ = γH are said to be incident (denoted x#ξ) if gK γH = (as subsets of G).
    [Show full text]
  • Recent Developments in the Stereological Analysis of Particles*
    Ann. Inst. Statist. Math. Vol. 43, No. 3, 455-468 (1991) RECENT DEVELOPMENTS IN THE STEREOLOGICAL ANALYSIS OF PARTICLES* EVA B. VEDEL JENSEN Department of Theoretical Statistics, Institute of Mathematics, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark (Received November 14, 1989; revised March 26, 1991) Abstract. Recent developments in the stereological analysis of particles are reviewed. The trend has been towards methods which are applicable without specific assumptions about particle shape. Geometric samples of a local 3-d character are used. Stereological estimators of particle intensity, particle size distribution and particle interaction are presented and discussed. Key words and phrases: Intensity, K-function, mark distribution, marked point processes, measure decomposition, nucleator, second-order properties, stereology, stochastic geometry. 1. Introduction Stereology is the science of making statistical inference about spatial struc- tures from samples of a geometric nature. Such methods are used in the study of the different components of spatial materials such as metals, minerals, synthetic materials or biological tissues. The physical size of the components of interest may be of the order of #m and are then studied by some microscopic technique. Until recently, the geometric samples used in a stereological analysis have consisted of line or plane sections of the structure. The step from spatial structures to their sections involves a great loss of information and so traditional stereological methods commonly yield only "global" information of a statistical character. A further consequence of this loss of information is that, in order to give a spatial interpretation to size data collected on sections, the solutions of ill-posed problems are required: numerical solutions in which small deviations due to measurement error can lead to large discrepancies in the final solution (cf.
    [Show full text]
  • Alegbraic Integral Geometry
    ALGEBRAIC INTEGRAL GEOMETRY JOSEPH H.G. FU Contents 1. Introduction 1 1.1. About these notes 2 1.2. Acknowledgments 2 1.3. Notation and conventions 2 2. Classical integral geometry 3 2.1. Intrinsic volumes and Federer curvature measures 4 2.2. Other incarnations of the normal cycle 6 2.3. Crofton formulas 7 2.4. The classical kinematic formulas 8 2.5. The Weyl principle 11 3. Curvature measures and the normal cycle 13 3.1. Properties of the normal cycle 13 3.2. General curvature measures 14 3.3. Kinematic formulas for invariant curvature measures 15 3.4. The transfer principle 17 4. Integral geometry of euclidean spaces via Alesker theory 19 4.1. Survey of valuations on finite dimensional real vector spaces 19 4.2. Constant coefficient valuations 23 4.3. The ftaig for isotropic structures on euclidean spaces 26 n 4.4. The classical integral geometry of R 28 5. Valuations and integral geometry on isotropic manifolds 31 5.1. Brief definition of valuations on manifolds 31 5.2. First variation, the Rumin operator and the kernel theorem 33 5.3. The filtration and the transfer principle for valuations 36 5.4. The ftaig for compact isotropic spaces 38 5.5. Analytic continuation 38 5.6. Integral geometry of real space forms 43 6. Hermitian integral geometry 45 U(n) n 6.1. Algebra structure of Val (C ) 46 6.2. Hermitian intrinsic volumes and special cones 49 6.3. Tasaki valuations, and a mysterious duality 51 n 6.4. Determination of the kinematic operator of (C ;U(n)) 53 Date: February 9, 2013.
    [Show full text]
  • Some Classical Problems in Random Geometry Pierre Calka
    Some Classical Problems in Random Geometry Pierre Calka To cite this version: Pierre Calka. Some Classical Problems in Random Geometry. Stochastic geometry, 2237, pp.1- 43, 2019, Lecture Notes in Mathematics, 978-3-030-13546-1. 10.1007/978-3-030-13547-8_1. hal- 02377523 HAL Id: hal-02377523 https://hal.archives-ouvertes.fr/hal-02377523 Submitted on 23 Nov 2019 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. Some classical problems in random geometry Pierre Calka Abstract This chapter is intended as a first introduction to selected topics in random geometry. It aims at showing how classical questions from recreational mathematics can lead to the modern theory of a mathematical domain at the interface of prob- ability and geometry. Indeed, in each of the four sections, the starting point is a historical practical problem from geometric probability. We show that the solution of the problem, if any, and the underlying discussion are the gateway to the very rich and active domain of integral and stochastic geometry, which we describe at a basic level. In particular, we explain how to connect Buffon’s needle problem to integral geometry, Bertrand’s paradox to random tessellations, Sylvester’s four-point prob- lem to random polytopes and Jeffrey’s bicycle wheel problem to random coverings.
    [Show full text]
  • The Radon Transform Joshua Benjamin
    THE RADON TRANSFORM JOSHUA BENJAMIN III CLASS OF 2020 [email protected] Acknowledgments I would like to thank: Fabian Haiden for allowing me to create an exposition and for oering advice as to what topic I should explore. I would also like to thank Kevin Yang and Reuben Stern for aiding me in checking for errors and oering advice on what information was superuous. ii To My Haters iii THE RADON TRANSFORM JOSHUA BENJAMIN III FABIAN HAIDEN iv Contents 1 Introduction 1 2 e Radon Transform 2 2.1 Two Dimensions (R2)............................2 2.2 Higher Dimensions (Rn)..........................3 2.3 Dual Transform . .4 2.4 Gaussian distribution . .4 3 Properties 5 3.1 Homogeneity and scaling . .5 3.2 Linearity . .5 3.3 Transform of a linear transformation . .6 3.4 Shiing . .6 4 Inversion 7 4.1 Inversion formulas . .7 v 4.2 Unication . .8 5 Relation to the Fourier Transform 9 5.1 Fourier and Radon . .9 5.2 Another Inversion . 10 6 Application 11 vi Chapter 1 Introduction e reconstruction problem has long been of interest to mathematicians, and with the advancements in experimental science, applied elds also needed a solution to this is- sue. e reconstruction problem concerns distributions and proles. Given the pro- jected distribution (or prole), what can be known about the actual distribution? In 1917, Johann Radon of Austria presented a solution to the reconstruction problem (he was not aempting to solve this problem) with the Radon transform and its inversion formula. He developed this solution aer building on the work of Hermann Minkowski and Paul Funk while maintaining private conversation with Gustav Herglotz.
    [Show full text]
  • 12 DISCRETE ASPECTS of STOCHASTIC GEOMETRY Rolf Schneider
    12 DISCRETE ASPECTS OF STOCHASTIC GEOMETRY Rolf Schneider INTRODUCTION Stochastic geometry studies randomly generated geometric objects. The present chapter deals with some discrete aspects of stochastic geometry. We describe work that has been done on finite point sets, their convex hulls, infinite discrete point sets, arrangements of flats, and tessellations of space, under various assumptions of randomness. Typical results concern expectations of geometrically defined random variables, or probabilities of events defined by random geometric configurations. The selection of topics must necessarily be restrictive. We leave out the large number of special elementary geometric probability problems that can be solved explicitly by direct, though possibly intricate, analytic calculations. We pay special attention to either asymptotic results, where the number of points considered tends to infinity, or to inequalities, or to identities where the proofs involve more delicate geometric or combinatorial arguments. The close ties of discrete geometry with convexity are reflected: we consider convex hulls of random points, intersections of random halfspaces, and tessellations of space into convex sets. There are many topics that one might classify under `discrete aspects of stochas- tic geometry', such as optimization problems with random data, the average-case analysis of geometric algorithms, random geometric graphs, random coverings, per- colation, shape theory, and several others. All of these have to be excluded here. 12.1 RANDOM POINTS The setup is a finite number of random points or, alternatively, a point process, in a topological space Σ. Mostly, the space Σ is Rd (d ≥ 2), the d-dimensional Euclidean space, with scalar product h· ; ·i and norm k · k.
    [Show full text]
  • Integral Geometry and Tomography
    Integral Geometry and Tomography Proceedings of a Joint Summer Research Conference held June 24-30, 1989 .....•. .....·' ' .""·... Q ' ..• •' ..~ ... : ~ "· ":. \. ....... , ~ - . .. ... '......... - http://dx.doi.org/10.1090/conm/113 Titles in This Series Volume 1 Markov random fields and their 19 Proceedings of the Northwestern applications, Ross Kindermann and homotopy theory conference, Haynes J. Laurie Snell R. Miller and Stewart B. Priddy, Editors 2 Proceedings of the conference on 20 Low dimensional topology, Samuel J. integration, topology, and geometry in Lomonaco, Jr., Editor linear spaces, William H. Graves, Editor 21 Topological methods in nonlinear 3 The closed graph and P-closed functional analysis, S. P. Singh, graph properties in general topology, S. Thomeier, and B. Watson, Editors T. R. Hamlett and L. L. Herrington 22 Factorizations of b" ± 1, b = 2, 4 Problems of elastic stability and 3, 5, 6, 7, 10, 11, 12 up to high vibrations, Vadim Komkov, Editor powers, John Brillhart, D. H. Lehmer, 5 Rational constructions of modules J. L. Selfridge, Bryant Tuckerman, and for simple Lie algebras, George B. S. S. Wagstaff, Jr. Seligman 23 Chapter 9 of Ramanujan's second 6 Umbral calculus and Hopf algebras, notebook-Infinite series identities, Robert Morris, Editor transformations, and evaluations, Bruce C. Berndt and Padmini T. Joshi 7 Complex contour integral representation of cardinal spline 24 Central extensions, Galois groups, and functions, Walter Schempp ideal class groups of number fields, 8 Ordered fields and real algebraic A. Frohlich geometry, D. W. Dubois and T. Recio, 25 Value distribution theory and its Editors applications, Chung-Chun Yang, Editor 9 Papers in algebra, analysis and 26 Conference in modern analysis statistics, R.
    [Show full text]
  • Arxiv:Math/0606755V2 [Math.PR] 12 Oct 2006 Optto (Pasco)
    Average volume, curvatures, and Euler characteristic of random real algebraic varieties Peter B¨urgisser∗ October 25, 2018 Abstract We determine the expected curvature polynomial of random real projec- tive varieties given as the zero set of independent random polynomials with Gaussian distribution, whose distribution is invariant under the action of the orthogonal group. In particular, the expected Euler characteristic of such ran- dom real projective varieties is found. This considerably extends previously known results on the number of roots, the volume, and the Euler characteristic of the real solution set of random polynomial equations. Key words. Random polynomials, real zeros of random polynomial equations, Euler characteristic, volume of tubes, curvature polynomial, kinematic formula, or- thogonal invariance AMS subject classifications. 60D05, 14P25, 53C65, 60G60, 60G15 1 Introduction 1.1 Real zeros of random polynomials The study of the real zeros of random polyomials started with a paper by Bloch and P´olya [6] who investigated polynomials in one variable with independent random coefficients from the set 1, 0, 1 . The question was further investigated by Little- {− } wood and Offord [24, 25] who estimated the average number of roots with respect to arXiv:math/0606755v2 [math.PR] 12 Oct 2006 different probability distributions of the coefficients. The first asymptotically sharp estimate on the average number of real roots was obtained by Kac [19]. He showed that the expected number of real roots of a random degree d polynomial is asymp- 2 totically π ln d if the coefficients are independent and standard normal distributed. Erd¨os and Offord [12] obtained similar results for random coefficients in 1, 1 .
    [Show full text]
  • Differential Geometry, Lie Groups and Symmetric Spaces Over General Base Fields and Rings Wolfgang Bertram
    Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings Wolfgang Bertram To cite this version: Wolfgang Bertram. Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings. Memoirs of the American Mathematical Society, American Mathematical Society, 2008, 00 (00), pp.00. hal-00004190v2 HAL Id: hal-00004190 https://hal.archives-ouvertes.fr/hal-00004190v2 Submitted on 7 Dec 2006 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. Wolfgang Bertram (Institut Elie Cartan, Nancy) Differential Geometry over General Base Fields and Rings hal-00004190, version 2 - 7 Dec 2006 Version to appear in the Memoirs of the AMS. Last updated: november mmvi mmvii Contents Introduction ................................................... ................1 I. Basic notions 1. Differential calculus .................................................. 14 2. Manifolds ................................................... ......... 20 3. Tangent bundle and general fiber bundles ............................ 22 4. The Lie bracket of vector fields .......................................25 5. Lie groups and symmetric spaces: basic facts ......................... 30 II. Interpretation of tangent objects via scalar extensions 6. Scalar extensions. I: Tangent functor and dual numbers .............. 36 7. Scalar extensions. II: Higher order tangent functors ...................42 8. Scalar extensions. III: Jet functor and truncated polynomial rings .... 50 III.
    [Show full text]