The Hidden Subgroup Problem in Affine Groups: Basis Selection in Fourier Sampling
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Affine Reflection Group Codes
Affine Reflection Group Codes Terasan Niyomsataya1, Ali Miri1,2 and Monica Nevins2 School of Information Technology and Engineering (SITE)1 Department of Mathematics and Statistics2 University of Ottawa, Ottawa, Canada K1N 6N5 email: {tniyomsa,samiri}@site.uottawa.ca, [email protected] Abstract This paper presents a construction of Slepian group codes from affine reflection groups. The solution to the initial vector and nearest distance problem is presented for all irreducible affine reflection groups of rank n ≥ 2, for varying stabilizer subgroups. Moreover, we use a detailed analysis of the geometry of affine reflection groups to produce an efficient decoding algorithm which is equivalent to the maximum-likelihood decoder. Its complexity depends only on the dimension of the vector space containing the codewords, and not on the number of codewords. We give several examples of the decoding algorithm, both to demonstrate its correctness and to show how, in small rank cases, it may be further streamlined by exploiting additional symmetries of the group. 1 1 Introduction Slepian [11] introduced group codes whose codewords represent a finite set of signals combining coding and modulation, for the Gaussian channel. A thorough survey of group codes can be found in [8]. The codewords lie on a sphere in n−dimensional Euclidean space Rn with equal nearest-neighbour distances. This gives congruent maximum-likelihood (ML) decoding regions, and hence equal error probability, for all codewords. Given a group G with a representation (action) on Rn, that is, an 1Keywords: Group codes, initial vector problem, decoding schemes, affine reflection groups 1 orthogonal n × n matrix Og for each g ∈ G, a group code generated from G is given by the set of all cg = Ogx0 (1) n for all g ∈ G where x0 = (x1, . -
Feature Matching and Heat Flow in Centro-Affine Geometry
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 093, 22 pages Feature Matching and Heat Flow in Centro-Affine Geometry Peter J. OLVER y, Changzheng QU z and Yun YANG x y School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA E-mail: [email protected] URL: http://www.math.umn.edu/~olver/ z School of Mathematics and Statistics, Ningbo University, Ningbo 315211, P.R. China E-mail: [email protected] x Department of Mathematics, Northeastern University, Shenyang, 110819, P.R. China E-mail: [email protected] Received April 02, 2020, in final form September 14, 2020; Published online September 29, 2020 https://doi.org/10.3842/SIGMA.2020.093 Abstract. In this paper, we study the differential invariants and the invariant heat flow in centro-affine geometry, proving that the latter is equivalent to the inviscid Burgers' equa- tion. Furthermore, we apply the centro-affine invariants to develop an invariant algorithm to match features of objects appearing in images. We show that the resulting algorithm com- pares favorably with the widely applied scale-invariant feature transform (SIFT), speeded up robust features (SURF), and affine-SIFT (ASIFT) methods. Key words: centro-affine geometry; equivariant moving frames; heat flow; inviscid Burgers' equation; differential invariant; edge matching 2020 Mathematics Subject Classification: 53A15; 53A55 1 Introduction The main objective in this paper is to study differential invariants and invariant curve flows { in particular the heat flow { in centro-affine geometry. In addition, we will present some basic applications to feature matching in camera images of three-dimensional objects, comparing our method with other popular algorithms. -
The Affine Group of a Lie Group
THE AFFINE GROUP OF A LIE GROUP JOSEPH A. WOLF1 1. If G is a Lie group, then the group Aut(G) of all continuous auto- morphisms of G has a natural Lie group structure. This gives the semi- direct product A(G) = G-Aut(G) the structure of a Lie group. When G is a vector group R", A(G) is the ordinary affine group A(re). Follow- ing L. Auslander [l ] we will refer to A(G) as the affine group of G, and regard it as a transformation group on G by (g, a): h-^g-a(h) where g, hEG and aGAut(G) ; in the case of a vector group, this is the usual action on A(») on R". If B is a compact subgroup of A(n), then it is well known that B has a fixed point on R", i.e., that there is a point xGR" such that b(x)=x for every bEB. For A(ra) is contained in the general linear group GL(« + 1, R) in the usual fashion, and B (being compact) must be conjugate to a subgroup of the orthogonal group 0(w + l). This conjugation can be done leaving fixed the (« + 1, w + 1)-place matrix entries, and is thus possible by an element of k(n). This done, the translation-parts of elements of B must be zero, proving the assertion. L. Auslander [l] has extended this theorem to compact abelian subgroups of A(G) when G is connected, simply connected and nil- potent. We will give a further extension. -
Fully Graphical Treatment of the Quantum Algorithm for the Hidden Subgroup Problem
FULLY GRAPHICAL TREATMENT OF THE QUANTUM ALGORITHM FOR THE HIDDEN SUBGROUP PROBLEM STEFANO GOGIOSO AND ALEKS KISSINGER Quantum Group, University of Oxford, UK iCIS, Radboud University. Nijmegen, Netherlands Abstract. The abelian Hidden Subgroup Problem (HSP) is extremely general, and many problems with known quantum exponential speed-up (such as integers factorisation, the discrete logarithm and Simon's problem) can be seen as specific instances of it. The traditional presentation of the quantum protocol for the abelian HSP is low-level, and relies heavily on the the interplay between classical group theory and complex vector spaces. Instead, we give a high-level diagrammatic presentation which showcases the quantum structures truly at play. Specifically, we provide the first fully diagrammatic proof of correctness for the abelian HSP protocol, showing that strongly complementary observables are the key ingredient to its success. Being fully diagrammatic, our proof extends beyond the traditional case of finite-dimensional quantum theory: for example, we can use it to show that Simon's problem can be efficiently solved in real quantum theory, and to obtain a protocol that solves the HSP for certain infinite abelian groups. 1. Introduction The advent of quantum computing promises to solve a number number of problems which have until now proven intractable for classical computers. Amongst these, one of the most famous is Shor's algorithm [Sho95, EJ96]: it allows for an efficient solution of the integer factorisation problem and the discrete logarithm problem, the hardness of which underlies many of the cryptographic algorithms which we currently entrust with our digital security (such as RSA and DHKE). -
Finite Groups with a Nilpotent Maximal Subgroup
FINITE GROUPS WITH A NILPOTENT MAXIMAL SUBGROUP ZVONIMIR JANKO (received 8 January 1964) Let G be a finite group all of whose proper subgroups are nilpotent. Then by a theorem of Schmidt-Iwasawa the group G is soluble. But what can we say about a finite group G if only one maximal subgroup is nilpotent? Let G be a finite group with a nilpotent maximal subgroup M. Then' the following results are known: THEOREM OF J. G. THOMPSON [4]. If M has odd order, then G is soluble. THEOREM OF DESKINS [1]. // M has class <Z 2, then G is soluble. THEOREM OF JANKO [3]. // the 2-Sylow subgroup M2 of M is abelian, then G is soluble. Now we can give a very simple proof of the following THEOREM. Let G be a finite group with a nilpotent maximal subgroup M. If a 2-Sylow subgroup M3 of M has class ^ 2, then G is soluble. This result is the best possible because the simple group LF(2, 17) has a 2-Sylow subgroup of class 3 which is a maximal subgroup. The theorem was announced without proof in [3]. The proof of the theorem will be independent of the theorem of Deskins. We shall give at first some definitions. Definition 1. A finite group G is called ^-nilpotent if it has a normal Sylow ^-complement. Definition 2. Let Z(GV) be the centre of a />-Sylow subgroup Gv of a finite group G. Then the group G is called ^-normal if Z(GV) is the centre of every p-Sylovr subgroup of G in which it is contained. -
View This Volume's Front and Back Matter
http://dx.doi.org/10.1090/gsm/092 Finite Group Theory This page intentionally left blank Finit e Grou p Theor y I. Martin Isaacs Graduate Studies in Mathematics Volume 92 f//s ^ -w* American Mathematical Society ^ Providence, Rhode Island Editorial Board David Cox (Chair) Steven G. Krantz Rafe Mazzeo Martin Scharlemann 2000 Mathematics Subject Classification. Primary 20B15, 20B20, 20D06, 20D10, 20D15, 20D20, 20D25, 20D35, 20D45, 20E22, 20E36. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-92 Library of Congress Cataloging-in-Publication Data Isaacs, I. Martin, 1940- Finite group theory / I. Martin Isaacs. p. cm. — (Graduate studies in mathematics ; v. 92) Includes index. ISBN 978-0-8218-4344-4 (alk. paper) 1. Finite groups. 2. Group theory. I. Title. QA177.I835 2008 512'.23—dc22 2008011388 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2008 by the American Mathematical Society. All rights reserved. Reprinted with corrections by the American Mathematical Society, 2011. -
Affine Dilatations
2.6. AFFINE GROUPS 75 2.6 Affine Groups We now take a quick look at the bijective affine maps. GivenanaffinespaceE, the set of affine bijections f: E → E is clearly a group, called the affine group of E, and denoted by GA(E). Recall that the group of bijective linear maps of the vector −→ −→ −→ space E is denoted by GL( E ). Then, the map f → f −→ defines a group homomorphism L:GA(E) → GL( E ). The kernel of this map is the set of translations on E. The subset of all linear maps of the form λ id−→, where −→ E λ ∈ R −{0}, is a subgroup of GL( E ), and is denoted as R∗id−→. E 76 CHAPTER 2. BASICS OF AFFINE GEOMETRY The subgroup DIL(E)=L−1(R∗id−→)ofGA(E) is par- E ticularly interesting. It turns out that it is the disjoint union of the translations and of the dilatations of ratio λ =1. The elements of DIL(E) are called affine dilatations (or dilations). Given any point a ∈ E, and any scalar λ ∈ R,adilata- tion (or central dilatation, or magnification, or ho- mothety) of center a and ratio λ,isamapHa,λ defined such that Ha,λ(x)=a + λax, for every x ∈ E. Observe that Ha,λ(a)=a, and when λ = 0 and x = a, Ha,λ(x)isonthelinedefinedbya and x, and is obtained by “scaling” ax by λ.Whenλ =1,Ha,1 is the identity. 2.6. AFFINE GROUPS 77 −−→ Note that H = λ id−→.Whenλ = 0, it is clear that a,λ E Ha,λ is an affine bijection. -
Homographic Wavelet Analysis in Identification of Characteristic Image Features
Optica Applicata, VoL X X X , No. 2 —3, 2000 Homographic wavelet analysis in identification of characteristic image features T adeusz Niedziela, Artur Stankiewicz Institute of Applied Physics, Military University of Technology, ul. Kaliskiego 2, 00-908 Warszawa, Poland. Mirosław Świętochowski Department of Physics, Warsaw University, ul. Hoża 69, 00-681 Warszawa, Poland. Wavelet transformations connected with subgroups SL(2, C\ performed as homographic transfor mations of a plane have been applied to identification of characteristic features of two-dimensional images. It has been proven that wavelet transformations exist for symmetry groups S17( 1,1) and SL(2.R). 1. Introduction In the present work, the problem of an analysis, processing and recognition of a two-dimensional image has been studied by means of wavelet analysis connected with subgroups of GL(2, C) group, acting in a plane by homographies hA(z) = - -- - -j , CZ + fl where A -(::)eG L (2, C). The existence of wavelet reversible transformations has been proven for individual subgroups in the group of homographic transfor mations. The kind of wavelet analysis most often used in technical applications is connected with affine subgroup h(z) = az + b, a. case for A = ^ of the symmetry of plane £ ~ S2 maintaining points at infinity [1], [2]. Adoption of the wider symmetry group means rejection of the invariability of certain image features, which is reasonable if the problem has a certain symmetry or lacks affine symmetry. The application of wavelet analysis connected with a wider symmetry group is by no means the loss of information. On the contrary, the information is duplicated for additional symmetries or coded by other means. -
Discrete Groups of Affine Isometries
Journal of Lie Theory Volume 9 (1999) 321{349 C 1999 Heldermann Verlag Discrete groups of affine isometries Herbert Abels Communicated by E. B. Vinberg Abstract. Given a finite set S of isometries of an affine Euclidean space. We ask when the group Γ generated by S is discrete. This includes as a special case the question when the group generated by a finite set of rotations is finite. This latter question is answered in an appendix. In the main part of the paper the general case is reduced to this special case. The result is phrased as a series of tests: Γ is discrete iff S passes all the tests. The testing procedure is algorithmic. Suppose we are given a finite set S of isometries of some affine Euclidean space. We ask when the group Γ generated by S is discrete. This includes as a special case the question when the group generated by a finite set of rotations is finite. We deal with this special case in an appendix. In the main part of the paper we reduce the general question to this special case. The main result is phrased as a series of tests. Γ is discrete iff S passes all the tests. The method is based on Bieberbach's theorems characterizing discrete groups of affine isometries. Bieberbach's second theorem says that a group Γ of isometries of an affine Euclidean space E is discrete (if and) only if there is a Γ{invariant affine subspace F of E such that the restriction homomorphism r from Γ to the group of isometries of F has finite kernel and a crystallographic group as image. -
On Properties of Translation Groups in the Affine General Linear
On properties of translation groups in the affine general linear group with applications to cryptography⋆ Marco Calderinia, Roberto Civinob, Massimiliano Salac aDepartment of Informatics, University of Bergen, Norway bDISIM, University of l’Aquila, Italy cDepartment of Mathematics, University of Trento, Italy Abstract The affine general linear group acting on a vector space over a prime field is a well-understood mathematical object. Its elementary abelian regular subgroups have recently drawn attention in applied mathematics thanks to their use in cryptography as a way to hide or detect weaknesses inside block ciphers. This paper is focused on building a convenient representation of their elements which suits better the purposes of the cryptanalyst. Sev- eral combinatorial counting formulas and a classification of their conjugacy classes are given as well. Keywords: Translation group, affine group, block ciphers, cryptanalysis. 1. Introduction The group of the translations of a vector space over a prime field is an elementary abelian regular subgroup of the corresponding symmetric group, and its normaliser, the affine general linear group, is a well-understood math- ematical object. Regular subgroups of the affine group and their connections with algebraic structures, such as radical rings [16] and braces [19], have already been studied in several works [18, 24, 27, 28]. More recently, ele- mentary abelian regular groups have been used in cryptography to define new operations on the message space of a block cipher and to implement statistical and group theoretical attacks [13, 15, 20]. All these objects are arXiv:1702.00581v4 [math.GR] 20 Nov 2020 well-known to be conjugated to the translation group, but this fact does not provide a simple description and representation of their elements which ⋆This research was partially funded by the Italian Ministry of Education, Universities and Research (MIUR), with the project PRIN 2015TW9LSR “Group theory and applica- tions”. -
Euclidean Group - Wikipedia, the Free Encyclopedia Page 1 of 6
Euclidean group - Wikipedia, the free encyclopedia Page 1 of 6 Euclidean group From Wikipedia, the free encyclopedia In mathematics, the Euclidean group E(n), sometimes called ISO( n) or similar, is the symmetry group of n-dimensional Euclidean space. Its elements, the isometries associated with the Euclidean metric, are called Euclidean moves . These groups are among the oldest and most studied, at least in the cases of dimension 2 and 3 — implicitly, long before the concept of group was known. Contents 1 Overview 1.1 Dimensionality 1.2 Direct and indirect isometries 1.3 Relation to the affine group 2 Detailed discussion 2.1 Subgroup structure, matrix and vector representation 2.2 Subgroups 2.3 Overview of isometries in up to three dimensions 2.4 Commuting isometries 2.5 Conjugacy classes 3 See also Overview Dimensionality The number of degrees of freedom for E(n) is n(n + 1)/2, which gives 3 in case n = 2, and 6 for n = 3. Of these, n can be attributed to available translational symmetry, and the remaining n(n − 1)/2 to rotational symmetry. Direct and indirect isometries There is a subgroup E+(n) of the direct isometries , i.e., isometries preserving orientation, also called rigid motions ; they are the rigid body moves. These include the translations, and the rotations, which together generate E+(n). E+(n) is also called a special Euclidean group , and denoted SE (n). The others are the indirect isometries . The subgroup E+(n) is of index 2. In other words, the indirect isometries form a single coset of E+(n). -
Fusion Systems
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 53, Number 4, October 2016, Pages 555–615 http://dx.doi.org/10.1090/bull/1538 Article electronically published on June 29, 2016 FUSION SYSTEMS MICHAEL ASCHBACHER AND BOB OLIVER Abstract. This is a survey article on the theory of fusion systems, a rela- tively new area of mathematics with connections to local finite group theory, algebraic topology, and modular representation theory. We first describe the general theory and then look separately at these connections. Contents Introduction 555 1. Basic properties of fusion systems 558 2. A local theory of fusion systems 573 3. Linking systems, transporter systems, and localities 584 4. Interactions with homotopy theory 590 5. Interactions with representation theory 597 6. Generalizations 606 7. Open questions and problems 608 About the authors 610 References 610 Introduction Let G be a finite group, p aprime,andS aSylowp-subgroup of G. Subsets X and Y of S are said to be fused in G if they are conjugate in G; that is, there exists g ∈ G such that gX := gXg−1 = Y . The use of the word “fusion” seems to be due to Richard Brauer, but the notion is much older, going back at least to Burnside in the late nineteenth century. The study of fusion in finite groups played an important role in the classification of the finite simple groups. Indeed fusion even leaked into the media in the movie It’s My Turn, where Jill Clayburgh plays a finite group theorist on the trail of a new sporadic group, talking to her graduate student (played by Daniel Stern) about the 2-fusion in her sought-after group.