SPHERICAL HARMONICS and INTEGRAL GEOMETRY on PROJECTIVE SPACES by Eric L
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Groups and Geometry 2018 Auckland University Projective Planes
Groups and Geometry 2018 Auckland University Projective planes, Laguerre planes and generalized quadrangles that admit large groups of automorphisms G¨unterSteinke School of Mathematics and Statistics University of Canterbury 24 January 2018 G¨unterSteinke Projective planes, Laguerre planes, generalized quadrangles GaG2018 1 / 26 (School of Mathematics and Statistics University of Canterbury) Projective and affine planes Definition A projective plane P = (P; L) consists of a set P of points and a set L of lines (where lines are subsets of P) such that the following three axioms are satisfied: (J) Two distinct points can be joined by a unique line. (I) Two distinct lines intersect in precisely one point. (R) There are at least four points no three of which are on a line. Removing a line and all of its points from a projective plane yields an affine plane. Conversely, each affine plane extends to a unique projective plane by adjoining in each line with an `ideal' point and adding a new line of all ideal points. G¨unterSteinke Projective planes, Laguerre planes, generalized quadrangles GaG2018 2 / 26 (School of Mathematics and Statistics University of Canterbury) Models of projective planes Desarguesian projective planes are obtained from a 3-dimensional vector space V over a skewfield F. Points are the 1-dimensional vector subspaces of V and lines are the 2-dimensional vector subspaces of V . In case F is a field one obtains the Pappian projective plane over F. There are many non-Desarguesian projective planes. One of the earliest and very important class is obtain by the (generalized) Moulton planes. -
Quaternion Zernike Spherical Polynomials
MATHEMATICS OF COMPUTATION Volume 84, Number 293, May 2015, Pages 1317–1337 S 0025-5718(2014)02888-3 Article electronically published on August 29, 2014 QUATERNION ZERNIKE SPHERICAL POLYNOMIALS J. MORAIS AND I. CAC¸ AO˜ Abstract. Over the past few years considerable attention has been given to the role played by the Zernike polynomials (ZPs) in many different fields of geometrical optics, optical engineering, and astronomy. The ZPs and their applications to corneal surface modeling played a key role in this develop- ment. These polynomials are a complete set of orthogonal functions over the unit circle and are commonly used to describe balanced aberrations. In the present paper we introduce the Zernike spherical polynomials within quater- nionic analysis ((R)QZSPs), which refine and extend the Zernike moments (defined through their polynomial counterparts). In particular, the underlying functions are of three real variables and take on either values in the reduced and full quaternions (identified, respectively, with R3 and R4). (R)QZSPs are orthonormal in the unit ball. The representation of these functions in terms of spherical monogenics over the unit sphere are explicitly given, from which several recurrence formulae for fast computer implementations can be derived. A summary of their fundamental properties and a further second or- der homogeneous differential equation are also discussed. As an application, we provide the reader with plot simulations that demonstrate the effectiveness of our approach. (R)QZSPs are new in literature and have some consequences that are now under investigation. 1. Introduction 1.1. The Zernike spherical polynomials. The complex Zernike polynomials (ZPs) have long been successfully used in many different fields of optics. -
Laplacian Eigenmodes for the Three Sphere
Laplacian eigenmodes for the three sphere M. Lachi`eze-Rey Service d’Astrophysique, C. E. Saclay 91191 Gif sur Yvette cedex, France October 10, 2018 Abstract The vector space k of the eigenfunctions of the Laplacian on the 3 V three sphere S , corresponding to the same eigenvalue λk = k (k + 2 − k 2), has dimension (k + 1) . After recalling the standard bases for , V we introduce a new basis B3, constructed from the reductions to S3 of peculiar homogeneous harmonic polynomia involving null vectors. We give the transformation laws between this basis and the usual hyper-spherical harmonics. Thanks to the quaternionic representations of S3 and SO(4), we are able to write explicitely the transformation properties of B3, and thus of any eigenmode, under an arbitrary rotation of SO(4). This offers the possibility to select those functions of k which remain invariant under V a chosen rotation of SO(4). When the rotation is an holonomy transfor- mation of a spherical space S3/Γ, this gives a method to calculates the eigenmodes of S3/Γ, which remains an open probleme in general. We illustrate our method by (re-)deriving the eigenmodes of lens and prism space. In a forthcoming paper, we present the derivation for dodecahedral space. 1 Introduction 3 The eigenvalues of the Laplacian ∆ of S are of the form λk = k (k + 2), + − k where k IN . For a given value of k, they span the eigenspace of ∈ 2 2 V dimension (k +1) . This vector space constitutes the (k +1) dimensional irreductible representation of SO(4), the isometry group of S3. -
Generalisations of the Fundamental Theorem of Projective
GENERALIZATIONS OF THE FUNDAMENTAL THEOREM OF PROJECTIVE GEOMETRY A thesis submitted for the degree of Doctor of Philosophy By Rupert McCallum Supervisor: Professor Michael Cowling School of Mathematics, The University of New South Wales. March 2009 Acknowledgements I would particularly like to thank my supervisor, Professor Michael Cowling, for conceiving of the research project and providing much valuable feedback and guidance, and providing help with writing the Extended Abstract. I would like to thank the University of New South Wales and the School of Mathematics and Statistics for their financial support. I would like to thank Dr Adam Harris for giving me helpful feedback on drafts of the thesis, and particularly my uncle Professor William McCallum for providing me with detailed comments on many preliminary drafts. I would like to thank Professor Michael Eastwood for making an important contribution to the research project and discussing some of the issues with me. I would like to thank Dr Jason Jeffers and Professor Alan Beardon for helping me with research about the history of the topic. I would like to thank Dr Henri Jimbo for reading over an early draft of the thesis and providing useful suggestions for taking the research further. I would like to thank Dr David Ullrich for drawing my attention to the theorem that if A is a subset of a locally compact Abelian group of positive Haar measure, then A+A has nonempty interior, which was crucial for the results of Chapter 8. I am very grateful to my parents for all the support and encouragement they gave me during the writing of this thesis. -
A User's Guide to Spherical Harmonics
A User's Guide to Spherical Harmonics Martin J. Mohlenkamp∗ Version: October 18, 2016 This pamphlet is intended for the scientist who is considering using Spherical Harmonics for some appli- cation. It is designed to introduce the Spherical Harmonics from a theoretical perspective and then discuss those practical issues necessary for their use in applications. I expect great variability in the backgrounds of the readers. In order to be informative to those who know less, without insulting those who know more, I have adopted the strategy \State even basic facts, but briefly." My dissertation work was a fast transform for Spherical Harmonics [6]. After its publication, I started receiving questions about Spherical Harmonics in general, rather than about my work specifically. This pamphlet aims to answer such general questions. If you find mistakes, or feel that important material is unclear or missing, please inform me. 1 A Theory of Spherical Harmonics In this section we give a development of Spherical Harmonics. There are other developments from other perspectives. The one chosen here has the benefit of being very concrete. 1.1 Mathematical Preliminaries We define the L2 inner product of two functions to be Z hf; gi = f(s)¯g(s)ds (1) R 2π whereg ¯ denotes complex conjugation and the integral is over the space of interest, for example 0 dθ on the R 2π R π 2 p 2 circle or 0 0 sin φdφdθ on the sphere. We define the L norm by jjfjj = hf; fi. The space L consists of all functions such that jjfjj < 1. -
Arxiv:1009.4249V12 [Math.MG] 17 Oct 2012 H Dniyof Identity the a K Ulig Fsmsml Leri Ruswihetbihsti.W O P Now We This
A LOCAL-TO-GLOBAL RESULT FOR TOPOLOGICAL SPHERICAL BUILDINGS RUPERT McCALLUM Abstract. Suppose that ∆, ∆′ are two buildings each arising from a semisimple algebraic group over a field, a topological field in the former case, and that for both the buildings the Coxeter diagram has no isolated nodes. We give conditions under which a partially defined injective chamber map, whose domain is the subcomplex of ∆ generated by a nonempty open set of chambers, and whose codomain is ∆′, is guaranteed to extend to a unique injective chamber map. Related to this result is a local version of the Borel-Tits theorem on abstract homomorphisms of simple algebraic groups. Keywords: topological geometry, local homomorphism, topological building, Borel-Tits the- orem Mathematicsl Subject Classification 2000: 51H10 1. Introduction Throughout the history of Lie theory there has been a notion of “local isomorphism”. Indeed, the original notion of Lie groups considered by Sophus Lie in [9] was an essentially local one. Suppose that G is an algebraic group defined over a Hausdorff topological field k. One may consider the notion of a local k-isogeny from the group G to a group G′ defined over k of the same dimension. This is a mapping defined on a nonempty open neighbourhood of the identity of G(k) in the strong k-topology, (see Definition 1.10), which “locally” acts as a k-isogeny, in the sense that it is a local k-homomorphism and its range is Zariski dense. As far as I know this notion has not been investigated systematically. In this paper we shall produce a local version of the Borel-Tits theorem which hints at the possibility that it may be fruitful to investigate this notion. -
Spherical-Harmonic Tensors Jm
Spherical-harmonic tensors Francisco Gonzalez Ledesma1 and Matthew Mewes2 1Department of Physics, Florida State University, Tallahassee, Florida 32306, USA 2Physics Department, California Polytechnic State University, San Luis Obispo, California 93407, USA The connection between spherical harmonics and symmetric tensors is explored. For each spher- ical harmonic, a corresponding traceless symmetric tensor is constructed. These tensors are then extended to include nonzero traces, providing an orthonormal angular-momentum eigenbasis for symmetric tensors of any rank. The relationship between the spherical-harmonic tensors and spin- weighted spherical harmonics is derived. The results facilitate the spherical-harmonic expansion of a large class of tensor-valued functions. Several simple illustrative examples are discussed, and the formalism is used to derive the leading-order effects of violations of Lorentz invariance in Newtonian gravity. I. INTRODUCTION Spherical harmonics Yjm provide an orthonormal basis for scalar functions on the 2-sphere and have numerous applications in physics and related fields. While they are commonly written in terms of the spherical-coordinate polar angle θ and azimuthal angle φ, spherical harmonics can be expressed in terms of cartesian coordinates, which is convenient in certain applications. The cartesian versions involve rank-j symmetric trace-free tensors jm [1–5]. These form a basis for traceless tensors and provide a link between functions on the sphere and symmetricY traceless tensors in three dimensions. This work builds on the above understanding in several ways. We first develop a new method for calculating the scalar spherical harmonics Yjm in terms of components of the direction unit vector n = sin θ cos φ ex + sin θ sin φ ey + cos θ ez . -
Electrodynamic Spherical Harmonic: Defi- Nition and Properties
Electrodynamic spherical harmonic A V Novitsky Department of Theoretical Physics, Belarusian State University, Nezavisimosti Avenue 4, 220050 Minsk, Belarus [email protected] Abstract Electrodynamic spherical harmonic is a second rank tensor in three- dimensional space. It allows to separate the radial and angle variables in vector solutions of Maxwell’s equations. Using the orthonormaliza- tion for electrodynamic spherical harmonic, a boundary problem on a sphere can be easily solved. 1 Introduction In this paper we introduce new function — electrodynamic spherical har- monic. It is represented as a second rank tensor in three-dimensional space. But the function differs from tensor spherical harmonic [1–5]. By its defini- tion the electrodynamic spherical function possesses a number of properties of usual scalar and vector harmonics and includes them as component parts. Why electrodynamic spherical harmonic? The word “electrodynamic” im- arXiv:0803.4093v1 [math-ph] 28 Mar 2008 plies that the function is applied for solution of vector field problems. We use it in electrodynamics, however one can apply the function for description of spin fields in quantum field theory. Electrodynamic spherical harmonic is not a simple designation of the well- known functions. It satisfies the Maxwell equations and describes the angular dependence of vector fields. The introduced function separates the variables (radial coordinate and angles) in the fields. Moreover, the notation of the fields in terms of electrodynamic spherical harmonic noticeably simplifies the solution of a boundary problem on spherical interface. Just application of 1 the ortonormalization condition allows to find the coefficients of spherical function expansion (e.g. scattering field amplitudes). -
Computation of Real-Valued Basis Functions Which Transform As Irreducible Representations of the Polyhedral Groups
COMPUTATION OF REAL-VALUED BASIS FUNCTIONS WHICH TRANSFORM AS IRREDUCIBLE REPRESENTATIONS OF THE POLYHEDRAL GROUPS NAN XU∗ AND PETER C. DOERSCHUKy Abstract. Basis functions which are invariant under the operations of a rotational point group G are able to describe any 3-D object which exhibits the rotational point group symmetry. However, in order to characterize the spatial statistics of an ensemble of objects in which each object is different but the statistics exhibit the symmetry, a complete set of basis functions is required. In particular, for each irreducible representation (irrep) of G, it is necessary to include basis functions that transform according to that irrep. This complete set of basis functions is a basis for square- integrable functions on the surface of the sphere in 3-D. Because the objects are real-valued, it is convenient to have real-valued basis functions. In this paper the existence of such real-valued bases is proven and an algorithm for their computation is provided for the icosahedral I and the octahedral O symmetries. Furthermore, it is proven that such a real-valued basis does not exist for the tetrahedral T symmetry because some irreps of T are essentially complex. The importance of these basis functions to computations in single-particle cryo electron microscopy is described. Key words. polyhedral groups, real-valued matrix irreducible representations, numerical com- putation of similarity transformations, basis functions that transform as a row of an irreducible representation of a rotation group, Classical groups (02.20.Hj), Special functions (02.30.Gp). AMS subject classifications. 2604, 2004, 57S17, 57S25 1. -
Spherical Harmonics and Homogeneous Har- Monic Polynomials
SPHERICAL HARMONICS AND HOMOGENEOUS HAR- MONIC POLYNOMIALS 1. The spherical Laplacean. Denote by S ½ R3 the unit sphere. For a function f(!) de¯ned on S, let f~ denote its extension to an open neighborhood N of S, constant along normals to S (i.e., constant along rays from the origin). We say f 2 C2(S) if f~ is a C2 function in N , and for such functions de¯ne a di®erential operator ¢S by: ¢Sf := ¢f;~ where ¢ on the right-hand side is the usual Laplace operator in R3. With a little work (omitted here) one may derive the expression for ¢ in polar coordinates (r; !) in R2 (r > 0;! 2 S): 2 1 ¢u = u + u + ¢ u: rr r r r2 S (Here ¢Su(r; !) is the operator ¢S acting on the function u(r; :) in S, for each ¯xed r.) A homogeneous polynomial of degree n ¸ 0 in three variables (x; y; z) is a linear combination of `monomials of degree n': d1 d2 d3 x y z ; di ¸ 0; d1 + d2 + d3 = n: This de¯nes a vector space (over R) denoted Pn. A simple combinatorial argument (involving balls and separators, as most of them do), seen in class, yields the dimension: 1 d := dim(P ) = (n + 1)(n + 2): n n 2 Writing a polynomial p 2 Pn in polar coordinates, we necessarily have: n p(r; !) = r f(!); f = pjS; where f is the restriction of p to S. This is an injective linear map p 7! f, but the functions on S so obtained are rather special (a dn-dimensional subspace of the in¯nite-dimensional space C(S) of continuous functions-let's call it Pn(S)) We are interested in the subspace Hn ½ Pn of homogeneous harmonic polynomials of degree n (¢p = 0). -
Arxiv:1805.12125V2 [Quant-Ph] 27 Sep 2018 Calculations in Quantum Mechanics? Even Quantum Experts Will find Something New in the Way We Compute the Spherical Harmonics
Condensed MatTER Physics, 2018, Vol. 21, No 3, 33002: 1–12 DOI: 10.5488/CMP.21.33002 HTtp://www.icmp.lviv.ua/journal Calculating SPHERICAL HARMONICS WITHOUT DERIVATIVES M. WEITZMAN1, J.K. Freericks2 1 3515 E. Rochelle Ave., Las Vegas, NV 89121, USA 2 Department OF Physics, GeorGETOWN University, 37th AND O Sts. NW, Washington, DC 20057, USA Received May 30, 2018 The DERIVATION OF SPHERICAL HARMONICS IS THE SAME IN NEARLY EVERY QUANTUM MECHANICS TEXTBOOK AND classroom. It IS FOUND TO BE DIffiCULT TO follow, HARD TO understand, AND CHALLENGING TO REPRODUCE BY MOST students. In THIS work, WE SHOW HOW ONE CAN DETERMINE SPHERICAL HARMONICS IN A MORE NATURAL WAY BASED ON OPERATORS AND A POWERFUL IDENTITY CALLED THE EXPONENTIAL DISENTANGLING OPERATOR IDENTITY (known IN QUANTUM optics, BUT LITTLE USED elsewhere). This NEW STRATEGY FOLLOWS NATURALLY AFTER ONE HAS INTRODUCED DirAC notation, COMPUTED THE ANGULAR MOMENTUM ALGEBRa, AND DETERMINED THE ACTION OF THE ANGULAR MOMENTUM RAISING AND LOWERING 2 OPERATORS ON THE SIMULTANEOUS ANGULAR MOMENTUM EIGENSTATES (under Lˆ AND Lˆ z). Key words: ANGULAR momentum, SPHERICAL harmonics, OPERATOR METHODS PACS: 01.40.gb, 02.20.Qs, 03.65.Fd, 31.15.-p 1. Introduction One of us (JKF), first met Prof. Stasyuk in the early 2000s when visiting Ukraine for a conference and then subsequently became involved in a long-standing collaboration with Andrij Shvaika, who is also at the Institute for Condensed Matter Physics in Lviv. It is always a pleasure to meet Prof. Stasyuk on subsequent trips to Lviv as his friendly smile and enthusiasm for physics is contagious. -
Geometrie Galois Actions 1
London Mathematical Society Lecture Note Series. 242 Geometrie Galois Actions 1. Around Grothendieck's Esquisse d'un Programme Edited by Leila Schneps CNRS and Pierre Locha!< CNRS gCAMBRIDGE V UNIVERSITY PRESS 32 Alexandre Grothendieck Esquisse d'un Programme 33 par le diviseur à croisements normaux de Mumford-Deligne. Une autre dif de voisinage tubulaire d'un sous-topos - et il est étonnant d'ailleurs que ce ficulté, moins sérieuse sans doute, c'est que le soi-disant "espace" modulaire travail (pour autant que je sache) n'ait toujours pas été fait, c'est-à-di:e est en fait une multiplicité - techniquement, cela s'exprime surtout par la né que personne (depuis plus de vingt ans qu'il existe un contexte de t?polo~le cessité de remplacer l'ensemble d'indices 1 pour les strates par une catégorie étale) ne semble en avoir eu besoin; un signe sûrement que la compre.hensl~n (essentiellement finie) cl 'indices, en l'occurrence celle d~s "graphes MDli 1 qui de la structure topologique des schémas n'a pas tellement progresse depms uparamètrent" les "structures combinatoires" possibles d'une courbe stable le travail d'Artin-Mazur... de type (g, v). Ceci dit, je puis affirmer que la théorie de dévissage générale, Une fois accompli le double travail de dégrossissage (plus ou moins heuris spécialement développée SO~S la pression du besoin de cette cause, s'est tique) autour de la notion de dévissage d'un espace ou d'un topos stratifié, révélée en effet un guide précieux, conduisant à une compréhension progres qui a été une étape cruciale dans ma compréhension des multiplicités modu sive, d'une cohérence sans failles, de certains aspects essentiels de la tour laires il est d'ailleurs apparu que pour les besoins de ces dernières, on peut de Teichmüller (c'est à dire, essentiellement de la "structure à l'infini" des sans doute court-circuiter au moins une bonne partie de cette théorie par groupes de Teichmüller ordinaires).