Kerdock Codes Determine Unitary 2-Designs
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Arxiv:1702.00823V1 [Stat.OT] 2 Feb 2017
Nonparametric Spherical Regression Using Diffeomorphic Mappings M. Rosenthala, W. Wub, E. Klassen,c, Anuj Srivastavab aNaval Surface Warfare Center, Panama City Division - X23, 110 Vernon Avenue, Panama City, FL 32407-7001 bDepartment of Statistics, Florida State University, Tallahassee, FL 32306 cDepartment of Mathematics, Florida State University, Tallahassee, FL 32306 Abstract Spherical regression explores relationships between variables on spherical domains. We develop a nonparametric model that uses a diffeomorphic map from a sphere to itself. The restriction of this mapping to diffeomorphisms is natural in several settings. The model is estimated in a penalized maximum-likelihood framework using gradient-based optimization. Towards that goal, we specify a first-order roughness penalty using the Jacobian of diffeomorphisms. We compare the prediction performance of the proposed model with state-of-the-art methods using simulated and real data involving cloud deformations, wind directions, and vector-cardiograms. This model is found to outperform others in capturing relationships between spherical variables. Keywords: Nonlinear; Nonparametric; Riemannian Geometry; Spherical Regression. 1. Introduction Spherical data arises naturally in a variety of settings. For instance, a random vector with unit norm constraint is naturally studied as a point on a unit sphere. The statistical analysis of such random variables was pioneered by Mardia and colleagues (1972; 2000), in the context of directional data. Common application areas where such data originates include geology, gaming, meteorology, computer vision, and bioinformatics. Examples from geographical domains include plate tectonics (McKenzie, 1957; Chang, 1986), animal migrations, and tracking of weather for- mations. As mobile devices become increasingly advanced and prevalent, an abundance of new spherical data is being collected in the form of geographical coordinates. -
Limits of Geometries
Limits of Geometries Daryl Cooper, Jeffrey Danciger, and Anna Wienhard August 31, 2018 Abstract A geometric transition is a continuous path of geometric structures that changes type, mean- ing that the model geometry, i.e. the homogeneous space on which the structures are modeled, abruptly changes. In order to rigorously study transitions, one must define a notion of geometric limit at the level of homogeneous spaces, describing the basic process by which one homogeneous geometry may transform into another. We develop a general framework to describe transitions in the context that both geometries involved are represented as sub-geometries of a larger ambi- ent geometry. Specializing to the setting of real projective geometry, we classify the geometric limits of any sub-geometry whose structure group is a symmetric subgroup of the projective general linear group. As an application, we classify all limits of three-dimensional hyperbolic geometry inside of projective geometry, finding Euclidean, Nil, and Sol geometry among the 2 limits. We prove, however, that the other Thurston geometries, in particular H × R and SL^2 R, do not embed in any limit of hyperbolic geometry in this sense. 1 Introduction Following Felix Klein's Erlangen Program, a geometry is given by a pair (Y; H) of a Lie group H acting transitively by diffeomorphisms on a manifold Y . Given a manifold of the same dimension as Y , a geometric structure modeled on (Y; H) is a system of local coordinates in Y with transition maps in H. The study of deformation spaces of geometric structures on manifolds is a very rich mathematical subject, with a long history going back to Klein and Ehresmann, and more recently Thurston. -
Workshop on Applied Matrix Positivity International Centre for Mathematical Sciences 19Th – 23Rd July 2021
Workshop on Applied Matrix Positivity International Centre for Mathematical Sciences 19th { 23rd July 2021 Timetable All times are in BST (GMT +1). 0900 1000 1100 1600 1700 1800 Monday Berg Bhatia Khare Charina Schilling Skalski Tuesday K¨ostler Franz Fagnola Apanasovich Emery Cuevas Wednesday Jain Choudhury Vishwakarma Pascoe Pereira - Thursday Sharma Vashisht Mishra - St¨ockler Knese Titles and abstracts 1. New classes of multivariate covariance functions Tatiyana Apanasovich (George Washington University, USA) The class which is refereed to as the Cauchy family allows for the simultaneous modeling of the long memory dependence and correlation at short and intermediate lags. We introduce a valid parametric family of cross-covariance functions for multivariate spatial random fields where each component has a covariance function from a Cauchy family. We present the conditions on the parameter space that result in valid models with varying degrees of complexity. Practical implementations, including reparameterizations to reflect the conditions on the parameter space will be discussed. We show results of various Monte Carlo simulation experiments to explore the performances of our approach in terms of estimation and cokriging. The application of the proposed multivariate Cauchy model is illustrated on a dataset from the field of Satellite Oceanography. 1 2. A unified view of covariance functions through Gelfand pairs Christian Berg (University of Copenhagen) In Geostatistics one examines measurements depending on the location on the earth and on time. This leads to Random Fields of stochastic variables Z(ξ; u) indexed by (ξ; u) 2 2 belonging to S × R, where S { the 2-dimensional unit sphere { is a model for the earth, and R is a model for time. -
Doubling Algorithms with Permuted Lagrangian Graph Bases
DOUBLING ALGORITHMS WITH PERMUTED LAGRANGIAN GRAPH BASES VOLKER MEHRMANN∗ AND FEDERICO POLONI† Abstract. We derive a new representation of Lagrangian subspaces in the form h I i Im ΠT , X where Π is a symplectic matrix which is the product of a permutation matrix and a real orthogonal diagonal matrix, and X satisfies 1 if i = j, |Xij | ≤ √ 2 if i 6= j. This representation allows to limit element growth in the context of doubling algorithms for the computation of Lagrangian subspaces and the solution of Riccati equations. It is shown that a simple doubling algorithm using this representation can reach full machine accuracy on a wide range of problems, obtaining invariant subspaces of the same quality as those computed by the state-of-the-art algorithms based on orthogonal transformations. The same idea carries over to representations of arbitrary subspaces and can be used for other types of structured pencils. Key words. Lagrangian subspace, optimal control, structure-preserving doubling algorithm, symplectic matrix, Hamiltonian matrix, matrix pencil, graph subspace AMS subject classifications. 65F30, 49N10 1. Introduction. A Lagrangian subspace U is an n-dimensional subspace of C2n such that u∗Jv = 0 for each u, v ∈ U. Here u∗ denotes the conjugate transpose of u, and the transpose of u in the real case, and we set 0 I J = . −I 0 The computation of Lagrangian invariant subspaces of Hamiltonian matrices of the form FG H = H −F ∗ with H = H∗,G = G∗, satisfying (HJ)∗ = HJ, (as well as symplectic matrices S, satisfying S∗JS = J), is an important task in many optimal control problems [16, 25, 31, 36]. -
Generating Sequences of PSL(2,P) Which Will Eventually Lead Us to Study How This Group Behaves with Respect to the Replacement Property
Generating Sequences of PSL(2,p) Benjamin Nachmana aCornell University, 310 Malott Hall, Ithaca, NY 14853 USA Abstract Julius Whiston and Jan Saxl [14] showed that the size of an irredundant generating set of the group G = PSL(2,p) is at most four and computed the size m(G) of a maximal set for many primes. We will extend this result to a larger class of primes, with a surprising result that when p 1 mod 10, m(G) = 3 except for the special case p = 7. In addition, we6≡ will ± determine which orders of elements in irredundant generating sets of PSL(2,p) with lengths less than or equal to four are possible in most cases. We also give some remarks about the behavior of PSL(2,p) with respect to the replacement property for groups. Keywords: generating sequences, general position, projective linear group 1. Introduction The two dimensional projective linear group over a finite field with p ele- ments, PSL(2,p) has been extensively studied since Galois, who constructed them and showed their simplicity for p > 3 [15]. One of their nice prop- erties is due to a theorem by E. Dickson, which shows that there are only a small number of possibilities for the isomorphism types of maximal sub- arXiv:1210.2073v2 [math.GR] 4 Aug 2014 groups. There are two possibilities: ones that exist for all p and ones that exist for exceptional primes. A more recent proof of Dickson’s Theorem can be found in [10] and a complete proof due to Dickson is in [3]. -
Structure-Preserving Flows of Symplectic Matrix Pairs
STRUCTURE-PRESERVING FLOWS OF SYMPLECTIC MATRIX PAIRS YUEH-CHENG KUO∗, WEN-WEI LINy , AND SHIH-FENG SHIEHz Abstract. We construct a nonlinear differential equation of matrix pairs (M(t); L(t)) that are invariant (Structure-Preserving Property) in the class of symplectic matrix pairs X12 0 IX11 (M; L) = S2; S1 X = [Xij ]1≤i;j≤2 is Hermitian ; X22 I 0 X21 where S1 and S2 are two fixed symplectic matrices. Furthermore, its solution also preserves de- flating subspaces on the whole orbit (Eigenvector-Preserving Property). Such a flow is called a structure-preserving flow and is governed by a Riccati differential equation (RDE) of the form > > W_ (t) = [−W (t);I]H [I;W (t) ] , W (0) = W0, for some suitable Hamiltonian matrix H . We then utilize the Grassmann manifolds to extend the domain of the structure-preserving flow to the whole R except some isolated points. On the other hand, the Structure-Preserving Doubling Algorithm (SDA) is an efficient numerical method for solving algebraic Riccati equations and nonlinear matrix equations. In conjunction with the structure-preserving flow, we consider two special classes of sym- plectic pairs: S1 = S2 = I2n and S1 = J , S2 = −I2n as well as the associated algorithms SDA-1 k−1 and SDA-2. It is shown that at t = 2 ; k 2 Z this flow passes through the iterates generated by SDA-1 and SDA-2, respectively. Therefore, the SDA and its corresponding structure-preserving flow have identical asymptotic behaviors. Taking advantage of the special structure and properties of the Hamiltonian matrix, we apply a symplectically similarity transformation to reduce H to a Hamiltonian Jordan canonical form J. -
2-Arcs of Maximal Size in the Affine and the Projective Hjelmslev Plane Over Z25
Advances in Mathematics of Communications doi:10.3934/amc.2011.5.287 Volume 5, No. 2, 2011, 287{301 2-ARCS OF MAXIMAL SIZE IN THE AFFINE AND THE PROJECTIVE HJELMSLEV PLANE OVER Z25 Michael Kiermaier, Matthias Koch and Sascha Kurz Institut f¨urMathematik Universit¨atBayreuth D-95440 Bayreuth, Germany (Communicated by Veerle Fack) Abstract. It is shown that the maximal size of a 2-arc in the projective Hjelmslev plane over Z25 is 21, and the (21; 2)-arc is unique up to isomorphism. Furthermore, all maximal (20; 2)-arcs in the affine Hjelmslev plane over Z25 are classified up to isomorphism. 1. Introduction It is well known that a Desarguesian projective plane of order q admits a 2-arc of size q + 2 if and only if q is even. These 2-arcs are called hyperovals. The biggest 2-arcs in the Desarguesian projective planes of odd order q have size q + 1 and are called ovals. For a projective Hjelmslev plane over a chain ring R of composition length 2 and size q2, the situation is somewhat similar: In PHG(2;R) there exists a hyperoval { that is a 2-arc of size q2 + q + 1 { if and only if R is a Galois ring of size q2 with q even, see [7,6]. For the case q odd and R not a Galois ring it was recently shown 2 that the maximum size m2(R) of a 2-arc is q , see [4]. In the remaining cases, the situation is less clear. For even q and R not a Galois ring, it is known that the maximum possible size of a 2-arc is lower bounded by q2 + 2 [4] and upper bounded by q2 + q [6]. -
Linear Transformations on Matrices
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 198, 1974 LINEAR TRANSFORMATIONSON MATRICES BY D. I. DJOKOVICO) ABSTRACT. The real orthogonal group 0(n), the unitary group U(n) and the symplec- tic group Sp (n) are embedded in a standard way in the real vector space of n x n real, complex and quaternionic matrices, respectively. Let F be a nonsingular real linear transformation of the ambient space of matrices such that F(G) C G where G is one of the groups mentioned above. Then we show that either F(x) ■ ao(x)b or F(x) = ao(x')b where a, b e G are fixed, x* is the transpose conjugate of the matrix x and o is an automorphism of reals, complexes and quaternions, respectively. 1. Introduction. In his survey paper [4], M. Marcus has stated seven conjec- tures. In this paper we shall study the last two of these conjectures. Let D be one of the following real division algebras R (the reals), C (the complex numbers) or H (the quaternions). As usual we assume that R C C C H. We denote by M„(D) the real algebra of all n X n matrices over D. If £ G D we denote by Ï its conjugate in D. Let A be the automorphism group of the real algebra D. If D = R then A is trivial. If D = C then A is cyclic of order 2 with conjugation as the only nontrivial automorphism. If D = H then the conjugation is not an isomorphism but an anti-isomorphism of H. -
The Sturm-Liouville Problem and the Polar Representation Theorem
THE STURM-LIOUVILLE PROBLEM AND THE POLAR REPRESENTATION THEOREM JORGE REZENDE Grupo de Física-Matemática da Universidade de Lisboa Av. Prof. Gama Pinto 2, 1649-003 Lisboa, PORTUGAL and Departamento de Matemática, Faculdade de Ciências da Universidade de Lisboa e-mail: [email protected] Dedicated to the memory of Professor Ruy Luís Gomes Abstract: The polar representation theorem for the n-dimensional time-dependent linear Hamiltonian system Q˙ = BQ + CP , P˙ = −AQ − B∗P , with continuous coefficients, states that, given two isotropic solu- tions (Q1,P1) and (Q2,P2), with the identity matrix as Wronskian, the formula Q2 = r cos ϕ, Q1 = r sin ϕ, holds, where r and ϕ are continuous matrices, det r =6 0 and ϕ is symmetric. In this article we use the monotonicity properties of the matrix ϕ arXiv:1006.5718v1 [math.CA] 29 Jun 2010 eigenvalues in order to obtain results on the Sturm-Liouville prob- lem. AMS Subj. Classification: 34B24, 34C10, 34A30 Key words: Sturm-Liouville theory, Hamiltonian systems, polar representation. 1. Introduction Let n = 1, 2,.... In this article, (., .) denotes the natural inner 1 product in Rn. For x ∈ Rn one writes x2 =(x, x), |x| =(x, x) 2 . If ∗ M is a real matrix, we shall denote M its transpose. Mjk denotes the matrix entry located in row j and column k. In is the identity n × n matrix. Mjk can be a matrix. For example, M can have the four blocks M11, M12, M21, M22. In a case like this one, if M12 = M21 =0, we write M = diag (M11, M22). -
DECOMPOSITION of SYMPLECTIC MATRICES INTO PRODUCTS of SYMPLECTIC UNIPOTENT MATRICES of INDEX 2∗ 1. Introduction. Decomposition
Electronic Journal of Linear Algebra, ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 35, pp. 497-502, November 2019. DECOMPOSITION OF SYMPLECTIC MATRICES INTO PRODUCTS OF SYMPLECTIC UNIPOTENT MATRICES OF INDEX 2∗ XIN HOUy , ZHENGYI XIAOz , YAJING HAOz , AND QI YUANz Abstract. In this article, it is proved that every symplectic matrix can be decomposed into a product of three symplectic 0 I unipotent matrices of index 2, i.e., every complex matrix A satisfying AT JA = J with J = n is a product of three −In 0 T 2 matrices Bi satisfying Bi JBi = J and (Bi − I) = 0 (i = 1; 2; 3). Key words. Symplectic matrices, Product of unipotent matrices, Symplectic Jordan Canonical Form. AMS subject classifications. 15A23, 20H20. 1. Introduction. Decomposition of elements in a matrix group into products of matrices with a special nature (such as unipotent matrices, involutions and so on) is a popular topic studied by many scholars. In the n × n matrix k ring Mn(F ) over a field F , a unipotent matrix of index k refers to a matrix A satisfying (A − In) = 0. Fong and Sourour in [4] proved that every matrix in the group SLn(C) (the special linear group over complex field C) is a product of three unipotent matrices (without limitation on the index). Wang and Wu in [6] gave a further result that every matrix in the group SLn(C) is a product of four unipotent matrices of index 2. In particular, decompositions of symplectic matrices have drawn considerable attention from numerous 0 I scholars (see [1, 3]). -
An Elementary Proof That Symplectic Matrices Have Determinant One
Advances in Dynamical Systems and Applications (ADSA). ISSN 0973-5321 Volume 12, Number 1 (2017), pp. 15–20 © Research India Publications https://dx.doi.org/10.37622/ADSA/12.1.2017.15-20 An elementary proof that symplectic matrices have determinant one Donsub Rim University of Washington, Department of Applied Mathematics, Seattle, WA, 98195, USA. E-mail: [email protected] Abstract We give one more proof of the fact that symplectic matrices over real and complex fields have determinant one. While this has already been proved many times, there has been lasting interest in finding in an elementary proof [2, 5]. The result is restricted to the real and complex case due to its reliance on field-dependent spec- tral theory, however in this setting we obtain a proof which is more elementary in the sense that it is direct and requires only well-known facts. Finally, an explicit formula for the determinant of conjugate symplectic matrices in terms of its square subblocks is given. AMS subject classification: 15A15, 37J10. Keywords: Symplectic Matrix, Determinants, Canonical transformations. 1. Introduction × A symplectic matrix is a matrix A ∈ K2N 2N over a field K that is defined by the property T OIN N×N A JA = J where J := ,IN := (identity in K .) (1.1) −IN O We are concerned with the problem of showing that det(A) = 1. In this paper we focus on the special case when K = R or C. It is straightforward to show that such a matrix A has det(A) =±1. However, there is an apparent lack of an entirely elementary proof verifying that det(A) is indeed equal to +1 [2, 5], although various proofs have appeared in the past. -
Exponential Polar Factorization of the Fundamental Matrix of Linear Differential Systems
Exponential polar factorization of the fundamental matrix of linear differential systems Ana Arnal∗ Fernando Casasy Institut de Matem`atiquesi Aplicacions de Castell´o(IMAC) and Departament de Matem`atiques,Universitat Jaume I, E-12071 Castell´on,Spain. Abstract We propose a new constructive procedure to factorize the fundamental real matrix of a linear system of differential equations as the product of the exponentials of a symmetric and a skew-symmetric matrix. Both matrices are explicitly constructed as series whose terms are computed recursively. The procedure is shown to converge for sufficiently small times. In this way, explicit exponential representations for the factors in the analytic polar decomposition are found. An additional advantage of the algorithm proposed here is that, if the exact solution evolves in a certain Lie group, then it provides approximations that also belong to the same Lie group, thus preserving important qualitative properties. Keywords: Exponential factorization, polar decomposition MSC (2000): 15A23, 34A45, 65L99 1 Introduction Given the non-autonomous system of linear ordinary differential equations dU = A(t) U; U(0) = I (1) dt with A(t) a real analytic N × N matrix, the Magnus expansion allows one to represent the fundamental matrix U(t) locally as U(t) = exp(Ω(t)); Ω(0) = O; (2) where the exponent Ω(t) is given by an infinite series 1 X Ω(t) = Ωm(t) (3) m=1 ∗Corresponding author. Email: [email protected] yEmail: [email protected] 1 whose terms are linear combinations of integrals and nested commutators in- volving the matrix A at different times [13].