Control System Analysis on Symmetric Cones
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Part I. Origin of the Species Jordan Algebras Were Conceived and Grew to Maturity in the Landscape of Physics
1 Part I. Origin of the Species Jordan algebras were conceived and grew to maturity in the landscape of physics. They were born in 1933 in a paper \Uber VerallgemeinerungsmÄoglichkeiten des Formalismus der Quantenmechanik" by the physicist Pascual Jordan; just one year later, with the help of John von Neumann and Eugene Wigner in the paper \On an algebraic generalization of the quantum mechanical formalism," they reached adulthood. Jordan algebras arose from the search for an \exceptional" setting for quantum mechanics. In the usual interpretation of quantum mechanics (the \Copenhagen model"), the physical observables are represented by Hermitian matrices (or operators on Hilbert space), those which are self-adjoint x¤ = x: The basic operations on matrices or operators are multiplication by a complex scalar ¸x, addition x + y, multipli- cation xy of matrices (composition of operators), and forming the complex conjugate transpose matrix (adjoint operator) x¤. This formalism is open to the objection that the operations are not \observable," not intrinsic to the physically meaningful part of the system: the scalar multiple ¸x is not again hermitian unless the scalar ¸ is real, the product xy is not observable unless x and y commute (or, as the physicists say, x and y are \simultaneously observable"), and the adjoint is invisible (it is the identity map on the observables, though nontrivial on matrices or operators in general). In 1932 the physicist Pascual Jordan proposed a program to discover a new algebraic setting for quantum mechanics, which would be freed from dependence on an invisible all-determining metaphysical matrix structure, yet would enjoy all the same algebraic bene¯ts as the highly successful Copenhagen model. -
Geometry of the Shilov Boundary of a Bounded Symmetric Domain Jean-Louis Clerc
Geometry of the Shilov Boundary of a Bounded Symmetric Domain Jean-Louis Clerc To cite this version: Jean-Louis Clerc. Geometry of the Shilov Boundary of a Bounded Symmetric Domain. journal of geometry and symmetry in physics, 2009, Vol. 13, pp. 25-74. hal-00381665 HAL Id: hal-00381665 https://hal.archives-ouvertes.fr/hal-00381665 Submitted on 6 May 2009 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. Geometry of the Shilov Boundary of a Bounded Symmetric Domain Jean-Louis Clerc today Abstract In the first part, the theory of bounded symmetric domains is pre- sented along two main approaches : as special cases of Riemannian symmetric spaces of the noncompact type on one hand, as unit balls in positive Hermitian Jordan triple systems on the other hand. In the second part, an invariant for triples in the Shilov boundary of such a domain is constructed. It generalizes an invariant constructed by E. Cartan for the unit sphere in C2 and also the triple Maslov index on the Lagrangian manifold. 1 Introduction The present paper is an outgrowth of the cycle of conferences delivred by the author at the Tenth International Conference on Geometry, Integrability and Quantization, held in Varna in June 2008. -
Spectral Cones in Euclidean Jordan Algebras
Spectral cones in Euclidean Jordan algebras Juyoung Jeong Department of Mathematics and Statistics University of Maryland, Baltimore County Baltimore, Maryland 21250, USA [email protected] and M. Seetharama Gowda Department of Mathematics and Statistics University of Maryland, Baltimore County Baltimore, Maryland 21250, USA [email protected] August 2, 2016 Abstract A spectral cone in a Euclidean Jordan algebra V of rank n is of the form K = λ−1(Q); where Q is a permutation invariant convex cone in Rn and λ : V!Rn is the eigenvalue map (which takes x to λ(x), the vector of eigenvalues of x with entries written in the decreasing order). In this paper, we describe some properties of spectral cones. We show, for example, that spectral cones are invariant under automorphisms of V, that the dual of a spectral cone is a spectral cone when V is simple or carries the canonical inner product, and characterize the pointedness/solidness of a spectral cone. We also show that for any spectral cone K in V, dim(K) 2 f0; 1; m − 1; mg, where dim(K) denotes the dimension of K and m is the dimension of V. Key Words: Euclidean Jordan algebra, spectral cone, automorphism, dual cone, dimension AMS Subject Classification: 15A18, 15A51, 15A57, 17C20, 17C30, 52A20 1 1 Introduction Let V be a Euclidean Jordan algebra of rank n and λ : V!Rn denote the eigenvalue map (which takes x to λ(x), the vector of eigenvalues of x with entries written in the decreasing order). A set E in V is said to be a spectral set [1] if there exists a permutation invariant set Q in Rn such that E = λ−1(Q): A function F : V!R is said to be a spectral function [1] if there is a permutation invariant function f : Rn !R such that F = f ◦ λ. -
Arxiv:1106.4415V1 [Math.DG] 22 Jun 2011 R,Rno Udai Form
JORDAN STRUCTURES IN MATHEMATICS AND PHYSICS Radu IORDANESCU˘ 1 Institute of Mathematics of the Romanian Academy P.O.Box 1-764 014700 Bucharest, Romania E-mail: [email protected] FOREWORD The aim of this paper is to offer an overview of the most important applications of Jordan structures inside mathematics and also to physics, up- dated references being included. For a more detailed treatment of this topic see - especially - the recent book Iord˘anescu [364w], where sugestions for further developments are given through many open problems, comments and remarks pointed out throughout the text. Nowadays, mathematics becomes more and more nonassociative (see 1 § below), and my prediction is that in few years nonassociativity will govern mathematics and applied sciences. MSC 2010: 16T25, 17B60, 17C40, 17C50, 17C65, 17C90, 17D92, 35Q51, 35Q53, 44A12, 51A35, 51C05, 53C35, 81T05, 81T30, 92D10. Keywords: Jordan algebra, Jordan triple system, Jordan pair, JB-, ∗ ∗ ∗ arXiv:1106.4415v1 [math.DG] 22 Jun 2011 JB -, JBW-, JBW -, JH -algebra, Ricatti equation, Riemann space, symmet- ric space, R-space, octonion plane, projective plane, Barbilian space, Tzitzeica equation, quantum group, B¨acklund-Darboux transformation, Hopf algebra, Yang-Baxter equation, KP equation, Sato Grassmann manifold, genetic alge- bra, random quadratic form. 1The author was partially supported from the contract PN-II-ID-PCE 1188 517/2009. 2 CONTENTS 1. Jordan structures ................................. ....................2 § 2. Algebraic varieties (or manifolds) defined by Jordan pairs ............11 § 3. Jordan structures in analysis ....................... ..................19 § 4. Jordan structures in differential geometry . ...............39 § 5. Jordan algebras in ring geometries . ................59 § 6. Jordan algebras in mathematical biology and mathematical statistics .66 § 7. -
[Math.OC] 2 Apr 2001
UNIVERSITY OF CAMBRIDGE Numerical Analysis Reports SELF–SCALED BARRIERS FOR IRREDUCIBLE SYMMETRIC CONES Raphael Hauser and Yongdo Lim arXiv:math/0104020v1 [math.OC] 2 Apr 2001 DAMTP 2001/NA04 April 2001 Department of Applied Mathematics and Theoretical Physics Silver Street Cambridge England CB3 9EW SELF–SCALED BARRIERS FOR IRREDUCIBLE SYMMETRIC CONES RAPHAEL A. HAUSER, YONGDO LIM April 2, 2001 Abstract. Self–scaled barrier functions are fundamental objects in the theory of interior–point methods for linear optimization over symmetric cones, of which linear and semidefinite program- ming are special cases. We are classifying all self–scaled barriers over irreducible symmetric cones and show that these functions are merely homothetic transformations of the universal barrier func- tion. Together with a decomposition theorem for self–scaled barriers this concludes the algebraic classification theory of these functions. After introducing the reader to the concepts relevant to the problem and tracing the history of the subject, we start by deriving our result from first principles in the important special case of semidefinite programming. We then generalise these arguments to irreducible symmetric cones by invoking results from the theory of Euclidean Jordan algebras. Key Words Semidefinite programming, self–scaled barrier functions, interior–point methods, symmetric cones, Euclidean Jordan algebras. AMS 1991 Subject Classification Primary 90C25, 52A41, 90C60. Secondary 90C05, 90C20. Contact Information and Credits Raphael Hauser, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, England. [email protected]. Research supported in part by the Norwegian Research Council through project No. 127582/410 “Synode II”, by the Engineering and Physical Sciences Research Council of the UK under grant No. -
Arxiv:1207.3214V1 [Math.MG] 13 Jul 2012 Ftosubspaces Two of Esc E.Tepouto W Oe Sjs H Sa Produc Usual the Just Is Cones Two of Product Cone the a Sets
SYMMETRIC CONES, THE HILBERT AND THOMPSON METRICS BOSCHÉ AURÉLIEN Abstract. Symmetric cones can be endowed with at least two in- teresting non Riemannian metrics: the Hilbert and the Thompson metrics. It is trivial that the linear maps preserving the cone are isometries for those two metrics. Oddly enough those are not the only isometries in general. We give here a full description of the isometry groups for both the Hilbert and the Thompson metrics using essentially the theory of euclidean Jordan algebras. Those results were already proved for the symmetric cone of complexe positive hermitian matrices by L. Molnár in [7]. In this paper how- ever we do not make any assumption on the symmetric cone under scrutiny (it could be reducible and contain exceptional factors). 1. Preliminaries A cone is a subset C of some euclidean space Rn that is invariant by positive scalar exterior multiplication. A convex cone is a cone that is also a convex subset of Rn. A cone C is proper (resp. open) if its closure contains no complete line (resp. if it’s interior is not empty). In this paper we deal exclusively with open proper cones and C will always be such a set. The product of two cones is just the usual product of sets. A cone C ∈ Rn is reducible if Rn splits orthogonally as the sum of two subspaces A and B each containing a cone CA and CB such that C is the product of CA and CB , i.e. the set of all sums of elements of CA n and CB. -
A Characterization of Symmetric Cones by an Order-Reversing Property of the Pseudoinverse Maps
A CHARACTERIZATION OF SYMMETRIC CONES BY AN ORDER-REVERSING PROPERTY OF THE PSEUDOINVERSE MAPS CHIFUNE KAI Abstract. When a homogeneous convex cone is given, a natural partial order is introduced in the ambient vector space. We shall show that a homogeneous convex cone is a symmetric cone if and only if Vinberg’s ¤-map and its inverse reverse the order. Actually our theorem is formulated in terms of the family of pseudoinverse maps including the ¤-map, and states that the above order-reversing property is typical of the ¤-map of a symmetric cone which coincides with the inverse map of the Jordan algebra associated with the symmetric cone. 1. Introduction Let Ω be an open convex cone in a finite-dimensional real vector space V which is regular, that is, Ω\(¡Ω) = f0g, where Ω stands for the closure of Ω. For x; y 2 V , we write x ºΩ y if x ¡ y 2 Ω. Clearly, this defines a partial order in V . In the special case that Ω is the one-dimensional symmetric cone R>0, the order ºΩ is the usual one and is reversed by taking inverse numbers in R>0. In general, let a symmetric cone Ω ½ V be given. Then V has a structure of the Jordan algebra associated with Ω. In this case, the Jordan algebra inverse map is an involution on Ω and reverses the order ºΩ. This order-reversing property helps our geometric understanding of the Jordan algebra inverse map. In this paper, we shall investigate this order-reversing property in a more general setting and give a characterization of symmetric cones. -
Spectral Functions and Smoothing Techniques on Jordan Algebras
Universite¶ catholique de Louvain Faculte¶ des Sciences appliquees¶ Departement¶ d'Ingenierie¶ mathematique¶ Center for Operation Research and Econometrics Center for Systems Engineering and Applied Mechanics Spectral Functions and Smoothing Techniques on Jordan Algebras Michel Baes Thesis submitted in partial ful¯llment of the requirements for the degree of Docteur en Sciences Appliqu¶ees Dissertation Committee: Fran»coisGlineur Universit¶ecatholique de Louvain Yurii Nesterov (advisor) Universit¶ecatholique de Louvain Cornelius Roos Technische Universiteit Delft Jean-Pierre Tignol Universit¶ecatholique de Louvain Paul Van Dooren (advisor) Universit¶ecatholique de Louvain Jean-Philippe Vial Universit¶ede Gen`eve Vincent Wertz (chair) Universit¶ecatholique de Louvain ii Contents List of notation vii 1 Introduction and preliminaries 1 1.1 Comparing algorithms .............................. 2 1.2 Linear Programming ............................... 3 1.3 Convex Programming .............................. 5 1.4 Self-scaled Optimization, and formally real Jordan algebras ......... 10 1.5 A closer look at interior-point methods ..................... 12 1.5.1 Newton's Algorithm: solving unconstrained problems ........ 12 1.5.2 Barrier methods: dealing with constraints ............... 13 1.5.3 Choosing an appropriate barrier .................... 13 1.5.4 Path-following interior-point methods for Linear Programming ... 17 1.5.5 Path-following interior-point methods for Self-Scaled Programming . 18 1.6 Smoothing techniques .............................. 20 1.7 Eigenvalues in Jordan algebra make it work: more applications ....... 22 1.7.1 A concavity result ............................ 22 1.7.2 Augmented barriers in Jordan algebras ................. 23 1.8 Overview of the thesis and research summary ................. 25 2 Jordan algebras 27 2.1 The birth of Jordan algebras .......................... 28 2.2 Algebras and Jordan algebras ......................... -
Commutation Principles in Euclidean Jordan Algebras and Normal Decomposition Systems∗
SIAM J. OPTIM. c 2017 Society for Industrial and Applied Mathematics Vol. 27, No. 3, pp. 1390{1402 COMMUTATION PRINCIPLES IN EUCLIDEAN JORDAN ALGEBRAS AND NORMAL DECOMPOSITION SYSTEMS∗ M. SEETHARAMA GOWDAy AND JUYOUNG JEONGy Abstract. The commutation principle of Ram´ırez, Seeger, and Sossa [SIAM J. Optim., 23 (2013), pp. 687{694] proved in the setting of Euclidean Jordan algebras says that when the sum of a Fr´echet differentiable function Θ and a spectral function F is minimized (maximized) over a spectral set Ω, any local minimizer (respectively, maximizer) a operator commutes with the Fr´echet derivative Θ0(a). In this paper, we extend this result to sets and functions which are (just) invariant under algebra automorphisms. We also consider a similar principle in the setting of normal decomposition systems. Key words. Euclidean Jordan algebra, (weakly) spectral sets/functions, automorphisms, com- mutation principle, normal decomposition system, variational inequality problem, cone complemen- tarity problem AMS subject classifications. 17C20, 17C30, 52A41, 90C25 DOI. 10.1137/16M1071006 1. Introduction. Let V be a Euclidean Jordan algebra of rank n [3] and λ : V!Rn denote the eigenvalue map (which takes x to λ(x), the vector of eigenvalues of x with entries written in the decreasing order). A set Ω in V is said to be a spectral set [1] if it is of the form Ω = λ−1(Q), where Q is a permutation invariant set in Rn. A function F : V!R is said to be a spectral function [1] if it is of the form F = f ◦λ, where f : Rn !R is a permutation invariant function. -
Maximal Invariants Over Symmetric Cones
Maximal Invariants Over Symmetric Cones Emanuel Ben-David Stanford University August 2010 Abstract In this paper we consider some hypothesis tests within a family of Wishart distributions, where both the sample space and the parameter space are symmetric cones. For such testing problems, we first derive the joint density of the ordered eigenvalues of the generalized Wishart distribution and propose a test statistic analog to that of classical multivariate statistics for test- ing homoscedasticity of covariance matrix. In this generalization of Bartlett’s test for equality of variances to hypotheses of real, complex, quaternion, Lorentz and octonion types of covari- ance structures. 1 Introduction Consider a statistical model consisting of a sample space X and unknown probability measures Pθ, with θ in the parameter space Θ. In this setting, the statistical inference about the model is often concentrated on parameter estimation and hypothesis testing. In the latter, we typically test the hypothesis H0 : θ ∈ Θ0 ⊂ Θ vs. the hypothesis H : θ ∈ Θ\Θ0. Systematically, this means to find a suitable test statistic t from X to a measurable space Y, and the distribution of tPθ, the transformed measure with respect to Pθ under t. For a Gaussian model, where the sample space is Rn, the probability measures are multivariate normal distribution Nn(0, Σ) and the parameter space is PDn(R), the cone of n × n positive definite matrices over R, some classical examples of such hypotheses are the sphericity hypothesis: 2 2 H0 : Σ= σ In vs. H : Σ , σ In, (1.1) arXiv:1201.0514v1 [math.ST] 2 Jan 2012 for some σ> 0; the complex structure hypothesis A −B H0 : Σ= ∈ PDk(C) vs. -
Optimization Over Symmetric Cones Under Uncertainty
OPTIMIZATION OVER SYMMETRIC CONES UNDER UNCERTAINTY By BAHA' M. ALZALG A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY WASHINGTON STATE UNIVERSITY Department of Mathematics DECEMBER 2011 To the Faculty of Washington State University: The members of the Committee appointed to examine the dissertation of BAHA' M. ALZALG find it satisfactory and recommend that it be accepted. K. A. Ariyawansa, Professor, Chair Robert Mifflin, Professor David S. Watkins, Professor ii For all the people iii ACKNOWLEDGEMENTS My greatest appreciation and my most sincere \Thank You!" go to my advisor Pro- fessor Ari Ariyawansa for his guidance, advice, and help during the preparation of this dissertation. I am also grateful for his offer for me to work as his research assistant and for giving me the opportunity to write papers and visit several conferences and workshops in North America and overseas. I wish also to express my appreciation and gratitude to Professor Robert Mifflin and Professor David S. Watkins for taking the time to serve as committee members and for various ways in which they helped me during all stages of doctoral studies. I want to thank all faculty, staff and graduate students in the Department of Mathe- matics at Washington State University. I would especially like to thank from the faculty Associate Professor Bala Krishnamoorthy, from the staff Kris Johnson, and from the students Pietro Paparella for their kind help. Finally, no words can express my gratitude to my parents and my grandmother for love and prayers. I also owe a special gratitude to my brothers and sisters, and some relatives in Jordan for their support and encouragement. -
Gradings on Simple Exceptional Jordan Systems and Structurable Algebras
2017 25 Diego Aranda Orna Gradings on simple exceptional Jordan systems and structurable algebras Departamento Matemáticas Director/es Elduque Palomo, Alberto Carlos Reconocimiento – NoComercial – © Universidad de Zaragoza SinObraDerivada (by-nc-nd): No se permite un uso comercial de la obra Servicio de Publicaciones original ni la generación de obras derivadas. ISSN 2254-7606 Tesis Doctoral Autor Director/es UNIVERSIDAD DE ZARAGOZA Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es Departamento Director/es Reconocimiento – NoComercial – © Universidad de Zaragoza SinObraDerivada (by-nc-nd): No se permite un uso comercial de la obra Servicio de Publicaciones original ni la generación de obras derivadas. ISSN 2254-7606 Tesis Doctoral Autor Director/es UNIVERSIDAD DE ZARAGOZA Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es Departamento Director/es Reconocimiento – NoComercial – © Universidad de Zaragoza SinObraDerivada (by-nc-nd): No se permite un uso comercial de la obra Servicio de Publicaciones original ni la generación de obras derivadas. ISSN 2254-7606 Tesis Doctoral GRADINGS ON SIMPLE EXCEPTIONAL JORDAN SYSTEMS AND STRUCTURABLE ALGEBRAS Autor Diego Aranda Orna Director/es Elduque Palomo, Alberto Carlos UNIVERSIDAD DE ZARAGOZA Matemáticas 2017 Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es Departamento Director/es Reconocimiento – NoComercial – © Universidad de Zaragoza SinObraDerivada (by-nc-nd): No se permite un uso comercial de la obra Servicio de Publicaciones original ni la generación de obras derivadas. ISSN 2254-7606 Tesis Doctoral Autor Director/es UNIVERSIDAD DE ZARAGOZA Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es DOCTORAL THESIS Gradings on simple exceptional Jordan systems and structurable algebras Author Diego Aranda Orna Supervisor Alberto Elduque UNIVERSIDAD DE ZARAGOZA Departamento de Matem´aticas 2016 Acknowledgements I am very grateful to Dr.