LECTURE III. BiHamiltonian chains and it projections. Maciej B laszak. Poznań University, Poland


 Margaret Willis
 1 years ago
 Views:
Transcription
1 LECTURE III BiHamiltonian chains and it projections Maciej B laszak Poznań University, Poland Maciej B laszak (Poznań University, Poland) LECTURE III 1 / 18
2 BiHamiltonian chains Let (M, Π) be a Poisson manifold with Π of constant rank (almost everywhere), with Poisson algebra given by the bracket {F 1, F 2 } Π = Π(dF 1, df 2 ) = df 1, ΠdF 2. Remark. In local Darboux coordinates (q 1,..., q n, p 1,..., p n, c 1,..., c m ) Π takes the form 0 I 0 Π = I where coordinates c i are Casimir functions. Definition A linear combination Π λ = π 1 λπ 0 (λ R) of two Poisson tensors Π 0, Π 1 is called a Poisson pencil if Π λ is Poisson for any value of λ. Then Π 0 and Π 1 are called compatible. Maciej B laszak (Poznań University, Poland) LECTURE III 2 / 18
3 BiHamiltonian chains Given a Π λ we can often construct a sequence of vector fields X i that have two Hamiltonian representations (bihamiltonian chains) X i = Π 1 dh i = Π 0 dh i+1, where H i C (M) are called Hamiltonians of a chain. Consider a bipoisson manifold (M, Π 0, Π 1 ) of dim M = 2n + m, where Π 0, Π 1 is a pair of compatible Poisson tensors of rank 2n. We further assume that Π λ admits m Casimir functions which are polynomial in λ H (k) (λ) = n k H (k) i λ n k i, k = 1,..., m i=0 so, that n n m = n and H (k) i are functionally independent. Maciej B laszak (Poznań University, Poland) LECTURE III 3 / 18
4 BiHamiltonian chains The collection of n bihamiltonian vector fields Π λ dh (k) (λ) = 0 Π 0 dh (k) 0 = 0 Π 0 dh (k) 1 = X (k) 1 = Π 1 dh (k) 0. Π 0 dh n (k) k = X n (k) k = Π 1 dh (k) n k 1 0 = Π 1 dh n (k) k (3.1) Lemma where k = 1,..., m, define bihamiltonian systems of Gel fandzakharevich type. Notice that each chain starts from a Casimir of Π 0 and terminates with a Casimir of Π 1. All H (k) i pairwise commute wit respect to Π 0 and Π 1. Maciej B laszak (Poznań University, Poland) LECTURE III 4 / 18
5 BiHamiltonian chains GZ bihamiltonian chain Liouville integrable system Having GZ bihamiltonian system the crucial toward construction of separation coordinates is the projection of second Poisson tensor Π 1 onto the symplectic foliation of the first Poisson tensor Π 0. Maciej B laszak (Poznań University, Poland) LECTURE III 5 / 18
6 Poisson projections onto submanifolds Consider manifold M of dim M = m and foliation S consisting of leaves S ν parametrized by ν R r, so r is codimension of every leave. Let Z be a distribution transversal to S, i.e. T x M = T x S ν Z x where S ν is a leave that passes through x. It defines a decomposition ( ) TM X = X + X, X x T xs ν, (X ) x Z x and induces splitting of dual space T x M = T x S ν Z x, where T x S ν annihilates Z x and Z x annihilates T x S ν. Maciej B laszak (Poznań University, Poland) LECTURE III 6 / 18
7 Poisson projections onto submanifolds Thus, any oneform α on M has a unique decomposition T ( ) M α = α + α, α x T x S ν, (α ) x Zx. Definition A function F C (M) is called Zinvariant if L Z F = Z (F ) = 0 for any Z Z. Obviously df T S. Definition The Poisson tensor Π is said to be Zinvariant if L Z {F, G } Π = 0 for any pair of Zinvariant functions F, G and any Z Z. Maciej B laszak (Poznań University, Poland) LECTURE III 7 / 18
8 Poisson projections onto submanifolds For any Poisson tensor Π on M define the following bivector Π D Π D  deformation of Π. Π D (α, β) := Π(α, β ), α, β T M (3.2) Lemma The image of Π D is tangent to the foliation S. Indeed, α, Π D β = α, Πβ = 0 for β Z = Z ker Π D. So, Π D can be naturally restricted to any leave S ν : π = Π D S. Theorem If Π is Zinvariant, then Π D is Poisson and so π. Maciej B laszak (Poznań University, Poland) LECTURE III 8 / 18
9 Poisson projections onto submanifolds Proof. From Zinvariant of Π : L Z df, ΠdG = 0, so d df, ΠdG T S = ( d df, ΠdG ) = d df, ΠdG, hence, the Jacobi identity {{F, G } ΠD + c.p. = ( d df, ΠdG ), ΠdH + c.p. = d df, ΠdG, Π is fulfilled. Observation. Annihilator Z of TS is defined as soon as S is determined. Definition Distribution D = Π(Z ), associated with the foliation S, is called Dirac distribution. Maciej B laszak (Poznań University, Poland) LECTURE III 9 / 18
10 Poisson projections onto submanifolds Two limited cases are possible: 1 TM = D TS Dirac case 2 D TS tangent case. Let S be parametrized by ϕ i (x) C (M) : S ν = {x M : ϕ i (x) = ν i, i = 1,..., r}, so {d ϕ i } basis in Z. Denote by {Z i } a basis dual to {d ϕ i }, so Z i (ϕ j ) = δ ij. Then, X = X r i=1 X (ϕ i )Z i, α = α r i=1 α(z i )d ϕ i. Obviously X (ϕ i ) = 0 and α (Z i ) = 0. Maciej B laszak (Poznań University, Poland) LECTURE III 10 / 18
11 Poisson projections onto submanifolds Then, from Π D (α, β) = Π(α, β ) we get r Π D = Π X i Z i + 1 i=1 2 where X i = Πd ϕ i, ϕ ij = {ϕ i, ϕ j } Π. r ϕ ij Z i Z j (3.3) i,j=1 In the Dirac case we have a canonical choice of Z = D, as all X i are transversal to S and are linearly independent (det(ϕ ij ) = 0). Moreover Π is naturally Zinvariant since L Xi Π = 0. ϕ i (x) are then second class constraints in Dirac terminology. The basis {Z i } dual to {d ϕ i } can be expressed by X i : Z i = r ( ϕ 1 ) ji X j, Z j (ϕ i ) = δ ij. j=1 Maciej B laszak (Poznań University, Poland) LECTURE III 11 / 18
12 Poisson projections onto submanifolds Π D (3.3) attains the form Π D = Π 1 2 r ( ϕ 1 ) ij X j X i. i,j=1 This Poisson tensor defines the following bracket on C (M) : {F, G } ΠD = {F, G } Π r i,j=1 known as a Dirac bracket. In tangent case all X i are tangent to S, i.e. X i (ϕ j ) = Π(d ϕ i, d ϕ j ) = 0, so Π D = Π {F, ϕ i } Π ( ϕ 1 ) ij {ϕ j, G } Π r i=1 X i Z i and Z i must be find separately from case to case. Maciej B laszak (Poznań University, Poland) LECTURE III 12 / 18
13 From bihamiltonian to quasibihamiltonian chains Our foliation is symplectic foliation of Π 0, so Z = Ker Π 0 = Sp{dc i }. For GZbiHamiltonian chains Π 1 (dc i, dc j ) = 0, so we are in tangent case when project Π 1 onto S. Assume we found Z transversal to S and such that Π 1 is Z  invariant. Then Π D is Poisson with Ker Π 0 Ker Π D, so both tensors can be restricted to S. Let π 0 = Π 0 S, π 1 = Π D S. Maciej B laszak (Poznań University, Poland) LECTURE III 13 / 18
14 From bihamiltonian to quasibihamiltonian chains Theorem BiHamiltonian chains (3.1) restricted to S take the form of quasibihamiltonian chains ( ) π 1 dh (k) i = π 0 dh (k) m i+1 α (k) ij dh (j) 1 j=1, (3.4) where ( ) α (k) ij = Z j (H (k) i ), S h(k) i = (H (k) i ) S. It follows from the fact that Π 1 = Π D + m k=1 X (k) 1 Z k. Maciej B laszak (Poznań University, Poland) LECTURE III 14 / 18
15 From bihamiltonian to quasibihamiltonian chains Lemma {H (k) i, H (r) j } ΠD = 0. Theorem Necessary and sufficient condition for compatibility of Π 0 and Π D (hence π 0 and π 1 ) is L Zi Π 0 = 0. Z i are symmetries of Π 0 (so Π 0 is Z  invariant), hence Z is integrable distribution and [Z i, Z j ] = 0. Maciej B laszak (Poznań University, Poland) LECTURE III 15 / 18
16 Nmanifolds Definition ωn manofold is a bipoisson manifold (π 0, π 1 ) with two compatible Poisson tensors, where at least one (say π 0 ) is nondegenerated. So M is endowed with a symplectic form ω = ω 0 = π 1 and the (1, 1) tensor field N = π 1 ω 0. From compatibility of π 0 and π 1 follows that Nijenhuis torsion of N vanishes T (N)(X, Y ) = [NX, NY ] N[NX, y] N[X, NY ] + N 2 [X, Y ] = 0 and hence, the second twoform is closed. L NX N = NL X N ω 1 = ω 0 π 1 ω 0 Maciej B laszak (Poznań University, Poland) LECTURE III 16 / 18
17 Nmanifolds We further restrict to generic case, when N has at every point n distinct eigenvalues which are functionally independent. It means that π 1 is also nondegenerated and ω 1 is symplectic. Notice that Moreover, π 1 = Nπ 0, ω 1 = N ω 0, N = ω 0 π 1. {f, g} πi = ω i (X f, X g ), X h = π 0 dh, i = 1, 2. Let us come back to our quasibihamiltonian chain (3.4). The distribution tangent to the foliation defined by (h (1) 1,..., h(m) n m ) is bilagrangian: ω 0 (X h (k) i, X h (r) j ) = ω 1 (X h (k) i, X (r) h ) = 0. j Maciej B laszak (Poznań University, Poland) LECTURE III 17 / 18
18 Nmanifolds Moreover, quasibihamiltonian equations can be put into one matrix equation N m dh i = F ij dh j N dh = Fdh j=1 where (h 1,..., h n ) = (h (1) 1,..., h(m) n m ) and F control matrix (recursion matrix). Maciej B laszak (Poznań University, Poland) LECTURE III 18 / 18
DEFORMATION OF DIRAC STRUCTURES ALONG ISOTROPIC SUBBUNDLES. and MARCO ZAMBON
Vol. 65 (2010) REPORTS ON MATHEMATICAL PHYSICS No. 2 DEFORMATION OF DIRAC STRUCTURES ALONG ISOTROPIC SUBBUNDLES IVÁN CALVO Laboratorio Nacional de Fusión, Asociación EURATOMCIEMAT, E28040 Madrid, Spain
More informationMATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).
MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationTensors on a vector space
APPENDIX B Tensors on a vector space In this Appendix, we gather mathematical definitions and results pertaining to tensors. The purpose is mostly to introduce the modern, geometrical view on tensors,
More informationRemarks on Lagrangian singularities, caustics, minimum distance lines
Remarks on Lagrangian singularities, caustics, minimum distance lines Department of Mathematics and Statistics Queen s University CRM, Barcelona, Spain June 2014 CRM CRM, Barcelona, SpainJune 2014 CRM
More informationQuantum Mathematics. 1. Introduction... 3
Quantum Mathematics Table of Contents by Peter J. Olver University of Minnesota 1. Introduction.................... 3 2. Hamiltonian Mechanics............... 3 Hamiltonian Systems................ 3 Poisson
More informationMathematical Physics, Lecture 9
Mathematical Physics, Lecture 9 Hoshang Heydari Fysikum April 25, 2012 Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 1 / 42 Table of contents 1 Differentiable manifolds 2 Differential
More informationQuantum Mechanics and Representation Theory
Quantum Mechanics and Representation Theory Peter Woit Columbia University Texas Tech, November 21 2013 Peter Woit (Columbia University) Quantum Mechanics and Representation Theory November 2013 1 / 30
More informationOrthogonal Diagonalization of Symmetric Matrices
MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding
More informationCOBORDISM IN ALGEBRA AND TOPOLOGY
COBORDISM IN ALGEBRA AND TOPOLOGY ANDREW RANICKI (Edinburgh) http://www.maths.ed.ac.uk/ aar Dedicated to Robert Switzer and Desmond Sheiham Göttingen, 13th May, 2005 1 Cobordism There is a cobordism equivalence
More informationThe cover SU(2) SO(3) and related topics
The cover SU(2) SO(3) and related topics Iordan Ganev December 2011 Abstract The subgroup U of unit quaternions is isomorphic to SU(2) and is a double cover of SO(3). This allows a simple computation of
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22 RAVI VAKIL CONTENTS 1. Discrete valuation rings: Dimension 1 Noetherian regular local rings 1 Last day, we discussed the Zariski tangent space, and saw that it
More information1 Sets and Set Notation.
LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most
More informationExplicit Integration of some Integrable Systems of Classical Mechanics
Explicit Integration of some Integrable Systems of Classical Mechanics Inna Basak Gancheva Department de Matemàtica Aplicada I, Universitat Politecnica de Catalunya, Advisor: Yuri Fedorov November 2011
More informationMATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets.
MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in R n. Definition. Let V be a vector space. A function α
More informationLecture 18  Clifford Algebras and Spin groups
Lecture 18  Clifford Algebras and Spin groups April 5, 2013 Reference: Lawson and Michelsohn, Spin Geometry. 1 Universal Property If V is a vector space over R or C, let q be any quadratic form, meaning
More informationA GENERAL FRAMEWORK FOR NONHOLONOMIC MECHANICS: NONHOLONOMIC SYSTEMS ON LIE AFFGEBROIDS
A GENERAL FRAMEWORK FOR NONHOLONOMIC MECHANICS: NONHOLONOMIC SYSTEMS ON LIE AFFGEBROIDS DAVID IGLESIAS, JUAN C. MARRERO, D. MARTÍN DE DIEGO, AND DIANA SOSA Abstract. This paper presents a geometric description
More informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More informationMODULES OVER A PID KEITH CONRAD
MODULES OVER A PID KEITH CONRAD Every vector space over a field K that has a finite spanning set has a finite basis: it is isomorphic to K n for some n 0. When we replace the scalar field K with a commutative
More informationCARTAN S GENERALIZATION OF LIE S THIRD THEOREM
CARTAN S GENERALIZATION OF LIE S THIRD THEOREM ROBERT L. BRYANT MATHEMATICAL SCIENCES RESEARCH INSTITUTE JUNE 13, 2011 CRM, MONTREAL In many ways, this talk (and much of the work it reports on) owes its
More informationWhat is complete integrability in quantum mechanics
INTERNATIONAL SOLVAY INSTITUTES FOR PHYSICS AND CHEMISTRY Proceedings of the Symposium Henri Poincaré (Brussels, 89 October 2004) What is complete integrability in quantum mechanics L. D. Faddeev St.
More informationFiber bundles and nonabelian cohomology
Fiber bundles and nonabelian cohomology Nicolas Addington April 22, 2007 Abstract The transition maps of a fiber bundle are often said to satisfy the cocycle condition. If we take this terminology seriously
More informationUniversity of Lille I PC first year list of exercises n 7. Review
University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients
More informationSingular fibers of stable maps and signatures of 4 manifolds
359 399 359 arxiv version: fonts, pagination and layout may vary from GT published version Singular fibers of stable maps and signatures of 4 manifolds OSAMU SAEKI TAKAHIRO YAMAMOTO We show that for a
More informationUNIVERSAL PROLONGATION OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS ON FILTERED MANIFOLDS. Katharina Neusser
ARCHIVUM MATHEMATICUM BRNO) Tomus 45 29), 289 3 UNIVERSAL PROLONGATION OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS ON FILTERED MANIFOLDS Katharina Neusser Dedicated to Peter W. Michor at the occasion of his
More informationRINGS WITH A POLYNOMIAL IDENTITY
RINGS WITH A POLYNOMIAL IDENTITY IRVING KAPLANSKY 1. Introduction. In connection with his investigation of projective planes, M. Hall [2, Theorem 6.2]* proved the following theorem: a division ring D in
More informationLecture 1: Schur s Unitary Triangularization Theorem
Lecture 1: Schur s Unitary Triangularization Theorem This lecture introduces the notion of unitary equivalence and presents Schur s theorem and some of its consequences It roughly corresponds to Sections
More informationFiber Bundles and Connections. Norbert Poncin
Fiber Bundles and Connections Norbert Poncin 2012 1 N. Poncin, Fiber bundles and connections 2 Contents 1 Introduction 4 2 Fiber bundles 5 2.1 Definition and first remarks........................ 5 2.2
More informationBILINEAR FORMS KEITH CONRAD
BILINEAR FORMS KEITH CONRAD The geometry of R n is controlled algebraically by the dot product. We will abstract the dot product on R n to a bilinear form on a vector space and study algebraic and geometric
More informationFiber sums of genus 2 Lefschetz fibrations
Proceedings of 9 th Gökova GeometryTopology Conference, pp, 1 10 Fiber sums of genus 2 Lefschetz fibrations Denis Auroux Abstract. Using the recent results of Siebert and Tian about the holomorphicity
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationThe determinant of a skewsymmetric matrix is a square. This can be seen in small cases by direct calculation: 0 a. 12 a. a 13 a 24 a 14 a 23 a 14
4 Symplectic groups In this and the next two sections, we begin the study of the groups preserving reflexive sesquilinear forms or quadratic forms. We begin with the symplectic groups, associated with
More informationPOLYTOPES WITH MASS LINEAR FUNCTIONS, PART I
POLYTOPES WITH MASS LINEAR FUNCTIONS, PART I DUSA MCDUFF AND SUSAN TOLMAN Abstract. We analyze mass linear functions on simple polytopes, where a mass linear function is an affine function on whose value
More informationChapter 6. Orthogonality
6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be
More informationA new viewpoint on geometry of a lightlike hypersurface in a semieuclidean space
A new viewpoint on geometry of a lightlike hypersurface in a semieuclidean space Aurel Bejancu, Angel Ferrández Pascual Lucas Saitama Math J 16 (1998), 31 38 (Partially supported by DGICYT grant PB970784
More informationFIBER PRODUCTS AND ZARISKI SHEAVES
FIBER PRODUCTS AND ZARISKI SHEAVES BRIAN OSSERMAN 1. Fiber products and Zariski sheaves We recall the definition of a fiber product: Definition 1.1. Let C be a category, and X, Y, Z objects of C. Fix also
More informationMomentum Maps, Dual Pairs and Reduction in Deformation Quantization
Momentum Maps, Dual Pairs and Reduction in Deformation Quantization Henrique Bursztyn January, 2001 Abstract This paper is a brief survey of momentum maps, dual pairs and reduction in deformation quantization.
More informationINVARIANT METRICS WITH NONNEGATIVE CURVATURE ON COMPACT LIE GROUPS
INVARIANT METRICS WITH NONNEGATIVE CURVATURE ON COMPACT LIE GROUPS NATHAN BROWN, RACHEL FINCK, MATTHEW SPENCER, KRISTOPHER TAPP, AND ZHONGTAO WU Abstract. We classify the leftinvariant metrics with nonnegative
More informationCLASSIFICATIONS OF STAR PRODUCTS AND DEFORMATIONS OF POISSON BRACKETS
POISSON GEOMETRY BANACH CENTER PUBLICATIONS, VOLUME 51 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 000 CLASSIFICATIONS OF STAR PRODUCTS AND DEFORMATIONS OF POISSON BRACKETS PHILIPP E BO
More informationA QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS
A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors
More informationRow Ideals and Fibers of Morphisms
Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion
More informationExtrinsic geometric flows
On joint work with Vladimir Rovenski from Haifa Paweł Walczak Uniwersytet Łódzki CRM, Bellaterra, July 16, 2010 Setting Throughout this talk: (M, F, g 0 ) is a (compact, complete, any) foliated, Riemannian
More informationAlmost Quaternionic Structures on Quaternionic Kaehler Manifolds. F. Özdemir
Almost Quaternionic Structures on Quaternionic Kaehler Manifolds F. Özdemir Department of Mathematics, Faculty of Arts and Sciences Istanbul Technical University, 34469 MaslakIstanbul, TURKEY fozdemir@itu.edu.tr
More informationIRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible
More informationLecture 2. Observables
Lecture 2 Observables 13 14 LECTURE 2. OBSERVABLES 2.1 Observing observables We have seen at the end of the previous lecture that each dynamical variable is associated to a linear operator Ô, and its expectation
More informationModélisation et résolutions numérique et symbolique
Modélisation et résolutions numérique et symbolique via les logiciels Maple et Matlab Jeremy Berthomieu Mohab Safey El Din Stef Graillat Mohab.Safey@lip6.fr Outline Previous course: partial review of what
More informationsome algebra prelim solutions
some algebra prelim solutions David Morawski August 19, 2012 Problem (Spring 2008, #5). Show that f(x) = x p x + a is irreducible over F p whenever a F p is not zero. Proof. First, note that f(x) has no
More informationControllability and Observability
Controllability and Observability Controllability and observability represent two major concepts of modern control system theory These concepts were introduced by R Kalman in 1960 They can be roughly defined
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More information16.3 Fredholm Operators
Lectures 16 and 17 16.3 Fredholm Operators A nice way to think about compact operators is to show that set of compact operators is the closure of the set of finite rank operator in operator norm. In this
More informationMICROLOCAL ANALYSIS OF THE BOCHNERMARTINELLI INTEGRAL
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 00029939(XX)00000 MICROLOCAL ANALYSIS OF THE BOCHNERMARTINELLI INTEGRAL NIKOLAI TARKHANOV AND NIKOLAI VASILEVSKI
More informationMATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.
MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all ndimensional column
More informationContinuous Groups, Lie Groups, and Lie Algebras
Chapter 7 Continuous Groups, Lie Groups, and Lie Algebras Zeno was concerned with three problems... These are the problem of the infinitesimal, the infinite, and continuity... Bertrand Russell The groups
More informationThe Characteristic Polynomial
Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem
More informationMultiplicity. Chapter 6
Chapter 6 Multiplicity The fundamental theorem of algebra says that any polynomial of degree n 0 has exactly n roots in the complex numbers if we count with multiplicity. The zeros of a polynomial are
More information8.1 Examples, definitions, and basic properties
8 De Rham cohomology Last updated: May 21, 211. 8.1 Examples, definitions, and basic properties A kform ω Ω k (M) is closed if dω =. It is exact if there is a (k 1)form σ Ω k 1 (M) such that dσ = ω.
More informationFinite dimensional C algebras
Finite dimensional C algebras S. Sundar September 14, 2012 Throughout H, K stand for finite dimensional Hilbert spaces. 1 Spectral theorem for selfadjoint opertors Let A B(H) and let {ξ 1, ξ 2,, ξ n
More informationClassification of Cartan matrices
Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms
More informationClassification and reduction of polynomial Hamiltonians with three degrees of freedom
Monografías de la Real Academia de Ciencias de Zaragoza. 22: 101 109, (2003). Classification and reduction of polynomial Hamiltonians with three degrees of freedom S. Gutiérrez, J. Palacián and P. Yanguas
More informationit is easy to see that α = a
21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore
More informationWeek 5 Integral Polyhedra
Week 5 Integral Polyhedra We have seen some examples 1 of linear programming formulation that are integral, meaning that every basic feasible solution is an integral vector. This week we develop a theory
More informationSection 6.1  Inner Products and Norms
Section 6.1  Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,
More informationComputing divisors and common multiples of quasilinear ordinary differential equations
Computing divisors and common multiples of quasilinear ordinary differential equations Dima Grigoriev CNRS, Mathématiques, Université de Lille Villeneuve d Ascq, 59655, France Dmitry.Grigoryev@math.univlille1.fr
More informationSets of Fibre Homotopy Classes and Twisted Order Parameter Spaces
Communications in Mathematical Physics ManuscriptNr. (will be inserted by hand later) Sets of Fibre Homotopy Classes and Twisted Order Parameter Spaces Stefan BechtluftSachs, Marco Hien Naturwissenschaftliche
More informationOn the representability of the biuniform matroid
On the representability of the biuniform matroid Simeon Ball, Carles Padró, Zsuzsa Weiner and Chaoping Xing August 3, 2012 Abstract Every biuniform matroid is representable over all sufficiently large
More informationThe Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationFACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z
FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization
More informationResearch Article On the Rank of Elliptic Curves in Elementary Cubic Extensions
Numbers Volume 2015, Article ID 501629, 4 pages http://dx.doi.org/10.1155/2015/501629 Research Article On the Rank of Elliptic Curves in Elementary Cubic Extensions Rintaro Kozuma College of International
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More informationFrom Factorial Designs to Hilbert Schemes
From Factorial Designs to Hilbert Schemes Lorenzo Robbiano Università di Genova Dipartimento di Matematica Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015
More informationStructure of the Root Spaces for Simple Lie Algebras
Structure of the Root Spaces for Simple Lie Algebras I. Introduction A Cartan subalgebra, H, of a Lie algebra, G, is a subalgebra, H G, such that a. H is nilpotent, i.e., there is some n such that (H)
More informationON UNIQUE FACTORIZATION DOMAINS
ON UNIQUE FACTORIZATION DOMAINS JIM COYKENDALL AND WILLIAM W. SMITH Abstract. In this paper we attempt to generalize the notion of unique factorization domain in the spirit of halffactorial domain. It
More informationLet H and J be as in the above lemma. The result of the lemma shows that the integral
Let and be as in the above lemma. The result of the lemma shows that the integral ( f(x, y)dy) dx is well defined; we denote it by f(x, y)dydx. By symmetry, also the integral ( f(x, y)dx) dy is well defined;
More informationAlgébroïdes de Lie et mécanique lagrangienne
Yvette KosmannSchwarzbach Centre de Mathématiques Laurent Schwartz, École Polytechnique Université de Bourgogne 27 janvier 2010 Résumé Algébroïdes de Lie et algébroïdes de Courant dans le formalisme lagrangien
More informationDATA ANALYSIS II. Matrix Algorithms
DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More informationFUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES
FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied
More information88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a
88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small
More informationCODIMENSION ONE SYMPLECTIC FOLIATIONS AND REGULAR POISSON STRUCTURES
CODIMENSION ONE SYMPLECTIC FOLIATIONS AND REGULAR POISSON STRUCTURES VICTOR GUILLEMIN, EVA MIRANDA, AND ANA R. PIRES Abstract. In this short note we give a complete characterization of a certain class
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationSARD PROPERTY FOR THE ENDPOINT MAP ON SOME CARNOT GROUPS
SARD PROPERTY FOR THE ENDPOINT MAP ON SOME CARNOT GROUPS ENRICO LE DONNE, RICHARD MONTGOMERY, ALESSANDRO OTTAZZI, PIERRE PANSU, AND DAVIDE VITTONE Abstract. In CarnotCarathéodory or subriemannian geometry,
More informationSurface bundles over S 1, the Thurston norm, and the Whitehead link
Surface bundles over S 1, the Thurston norm, and the Whitehead link Michael Landry August 16, 2014 The Thurston norm is a powerful tool for studying the ways a 3manifold can fiber over the circle. In
More informationQuantum Field Theory and Representation Theory
Quantum Field Theory and Representation Theory Peter Woit woit@math.columbia.edu Department of Mathematics Columbia University Quantum Field Theory and Representation Theory p.1 Outline of the talk Quantum
More informationINTERACTION OF TWO CHARGES IN A UNIFORM MAGNETIC FIELD: II. SPATIAL PROBLEM
INTERACTION OF TWO CHARGES IN A UNIFORM MAGNETIC FIELD: II. SPATIAL PROBLEM D. PINHEIRO AND R. S. MACKAY Dedicated to the memory of John Greene. Abstract. The interaction of two charges moving in R 3 in
More informationLecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs
CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More informationInvariant Metrics with Nonnegative Curvature on Compact Lie Groups
Canad. Math. Bull. Vol. 50 (1), 2007 pp. 24 34 Invariant Metrics with Nonnegative Curvature on Compact Lie Groups Nathan Brown, Rachel Finck, Matthew Spencer, Kristopher Tapp and Zhongtao Wu Abstract.
More informationIdeal Class Group and Units
Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals
More informationMath 497C Sep 9, Curves and Surfaces Fall 2004, PSU
Math 497C Sep 9, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 2 15 sometries of the Euclidean Space Let M 1 and M 2 be a pair of metric space and d 1 and d 2 be their respective metrics We say
More information2 Complex Functions and the CauchyRiemann Equations
2 Complex Functions and the CauchyRiemann Equations 2.1 Complex functions In onevariable calculus, we study functions f(x) of a real variable x. Likewise, in complex analysis, we study functions f(z)
More informationGROUP ALGEBRAS. ANDREI YAFAEV
GROUP ALGEBRAS. ANDREI YAFAEV We will associate a certain algebra to a finite group and prove that it is semisimple. Then we will apply Wedderburn s theory to its study. Definition 0.1. Let G be a finite
More information10.3 POWER METHOD FOR APPROXIMATING EIGENVALUES
58 CHAPTER NUMERICAL METHODS. POWER METHOD FOR APPROXIMATING EIGENVALUES In Chapter 7 you saw that the eigenvalues of an n n matrix A are obtained by solving its characteristic equation n c nn c nn...
More informationCONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation
Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in
More informationA CONSTRUCTION OF THE UNIVERSAL COVER AS A FIBER BUNDLE
A CONSTRUCTION OF THE UNIVERSAL COVER AS A FIBER BUNDLE DANIEL A. RAMRAS In these notes we present a construction of the universal cover of a path connected, locally path connected, and semilocally simply
More informationFINITE DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROSSHEN S UNIMODULARITY CONJECTURE AARON TIKUISIS
FINITE DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROSSHEN S UNIMODULARITY CONJECTURE AARON TIKUISIS Abstract. It is shown that, for any field F R, any ordered vector space structure
More informationInner product. Definition of inner product
Math 20F Linear Algebra Lecture 25 1 Inner product Review: Definition of inner product. Slide 1 Norm and distance. Orthogonal vectors. Orthogonal complement. Orthogonal basis. Definition of inner product
More informationChapter 13: Basic ring theory
Chapter 3: Basic ring theory Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 3: Basic ring
More information