The Experts below are selected from a list of 6642 Experts worldwide ranked by ideXlab platform
Nevio Carpanese - One of the best experts on this subject based on the ideXlab platform.
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on the geometry of Symplectic pencils arising from discrete time Matrix equations
Systems & Control Letters, 2002Co-Authors: Nevio CarpaneseAbstract:Abstract In this paper general Symplectic Matrix pencils are considered disregarding the particular Matrix equations from which they arise. A parameterization of the Lagrangian deflating subspaces is given with the only assumption of regularity of the Matrix pencil.
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the parameterization of lagrangian subspaces of Symplectic Matrix pencils
European Control Conference, 2001Co-Authors: Nevio CarpaneseAbstract:Geometric methods for the solution of Matrix equations are recognized to be important in establishing effective numerical algorithms. In this paper general Symplectic Matrix pencils are considered disregarding the particular Matrix equations from which they arise. The problem of parameterization of the set of deflating Lagrangian subspaces of the pencil is solved.
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On the Geometry of Symplectic Matrix Pencils
IFAC Proceedings Volumes, 2001Co-Authors: Nevio CarpaneseAbstract:Abstract The role played by Matrix pencils to characterize solutions of Matrix equations is given as the motivation to seek the parameterization of deflating subspaces of Symplectic Matrix pencils. This parameterization is given with the only assumption of regularity of the pencil. The comparison and extension of the results known for the Hamiltonian matrices is evidenced.
Harold Widom - One of the best experts on this subject based on the ideXlab platform.
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Correlation Functions, Cluster Functions, and Spacing Distributions for Random Matrices
Journal of Statistical Physics, 1998Co-Authors: Craig A. Tracy, Harold WidomAbstract:The usual formulas for the correlation functions in orthogonal and Symplectic Matrix models express them as quaternion determinants. From this representation one can deduce formulas for spacing probabilities in terms of Fredholm determinants of Matrix-valued kernels. The derivations of the various formulas are somewhat involved. In this article we present a direct approach which leads immediately to scalar kernels for the unitary ensembles and Matrix kernels for the orthogonal and Symplectic ensembles, and the representations of the correlation functions, cluster functions, and spacing distributions in terms of them.
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On orthogonal and Symplectic Matrix ensembles
Communications in Mathematical Physics, 1996Co-Authors: Craig A. Tracy, Harold WidomAbstract:The focus of this paper is on the probability,Eβ(O;J), that a setJ consisting of a finite union of intervals contains no eigenvalues for the finiteN Gaussian Orthogonal (β=1) and Gaussian Symplectic (β=4) Ensembles and their respective scaling limits both in the bulk and at the edge of the spectrum. We show how these probabilities can be expressed in terms of quantities arising in the corresponding unitary (β=2) ensembles. Our most explicit new results concern the distribution of the largest eigenvalue in each of these ensembles. In the edge scaling limit we show that these largest eigenvalue distributions are given in terms of a particular Painleve II function.
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on orthogonal and Symplectic Matrix ensembles
arXiv: Exactly Solvable and Integrable Systems, 1995Co-Authors: Craig A. Tracy, Harold WidomAbstract:The focus of this paper is on the probability, $E_\beta(0;J)$, that a set $J$ consisting of a finite union of intervals contains no eigenvalues for the finite $N$ Gaussian Orthogonal ($\beta=1$) and Gaussian Symplectic ($\beta=4$) Ensembles and their respective scaling limits both in the bulk and at the edge of the spectrum. We show how these probabilities can be expressed in terms of quantities arising in the corresponding unitary ($\beta=2$) ensembles. Our most explicit new results concern the distribution of the largest eigenvalue in each of these ensembles. In the edge scaling limit we show that these largest eigenvalue distributions are given in terms of a particular Painlev\'e II function.
F W Williams - One of the best experts on this subject based on the ideXlab platform.
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physical interpretation of the Symplectic orthogonality of the eigensolutions of a hamiltonian or Symplectic Matrix
Computers & Structures, 1993Co-Authors: Zhong Wanxie, F W WilliamsAbstract:Abstract The physical interpretation of the adjoint Symplectic orthogonality between the eigenvectors of a Hamiltonian Matrix, or of a Symplectic Matrix, is shown in this note to be that it corresponds to the well-known Betti reciprocal theorem.
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Wave propagation in substructural chain-type structures excited by harmonic forces
International Journal of Mechanical Sciences, 1993Co-Authors: Min Zhou, Wanxie Zhong, F W WilliamsAbstract:Abstract A general method for vibration response and wave propagation in a substructural chain-type structure is presented in this paper. The analysis is based on the finite element and Symplectic Matrix methods. The dynamic stiffness matrices of a typical substructure and an irregular substructure of the substructural chain-type system are obtained by finite element discretization. The Symplectic Matrix of a single typical substructure is derived from the condensed dynamic stiffness Matrix. After determining the eigenvalues and the associated eigenvectors of the Symplectic Matrix, the state vector at any station of the substructural chain is expanded in terms of the eigenvectors. From the compatibility conditions between the irregular substructure and the typical substructures, the governing dynamic equations for wave propagation are derived. A numerical example for a periodically simply supported infinite uniform Timoshenko beam is given.
Andrew Waldron - One of the best experts on this subject based on the ideXlab platform.
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Duality of Orthogonal and Symplectic Matrix Integrals and Quaternionic Feynman Graphs
Communications in Mathematical Physics, 2003Co-Authors: Motohico Mulase, Andrew WaldronAbstract:We present an asymptotic expansion for quaternionic self-adjoint Matrix integrals. The Feynman diagrams appearing in the expansion are ordinary ribbon graphs and their non-orientable counterparts. The result exhibits a striking duality between quaternionic self-adjoint and real symmetric Matrix integrals. The asymptotic expansions of these integrals are given in terms of summations over topologies of compact surfaces, both orientable and non-orientable, for all genera and an arbitrary positive number of marked points on them. We show that the Gaussian Orthogonal Ensemble (GOE) and Gaussian Symplectic Ensemble (GSE) have exactly the same graphical expansion term by term (when appropriately normalized),except that the contributions from non-orientable surfaces with odd Euler characteristic carry the opposite sign. As an application, we give a new topological proof of the known duality for correlations of characteristic polynomials. Indeed, we show that this duality is equivalent to Poincare duality of graphs drawn on a compact surface. Another application of our graphical expansion formula is a simple and simultaneous (re)derivation of the Central Limit Theorem for GOE, GUE (Gaussian Unitary Ensemble) and GSE: The three cases have exactly the same graphical limiting formula except for an overall constant that represents the type of the ensemble.Comment: 39 pages, AMS LaTeX, 49 .eps figures, references update
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duality of orthogonal and Symplectic Matrix integrals and quaternionic feynman graphs
Communications in Mathematical Physics, 2003Co-Authors: Motohico Mulase, Andrew WaldronAbstract:We present an asymptotic expansion for quaternionic self-adjoint Matrix integrals. The Feynman diagrams appearing in the expansion are ordinary ribbon graphs and their non-orientable counterparts. We show that the 2N×2N Gaussian Orthogonal Ensemble (GOE) and N×N Gaussian Symplectic Ensemble (GSE) have exactly the same expansion term by term, except that the contributions from graphs on a non-orientable surface with odd Euler characteristic carry the opposite sign. As an application, we give a new topological proof of the known duality for correlations of characteristic polynomials, demonstrating that this duality is equivalent to Poincare duality of graphs drawn on a compact surface. Another consequence of our graphical expansion formula is a simple and simultaneous (re)derivation of the Central Limit Theorem for GOE, GUE (Gaussian Unitary Ensemble) and GSE: The three cases have exactly the same graphical limiting formula except for an overall constant that represents the type of the ensemble.
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duality of orthogonal and Symplectic Matrix integrals and quaternionic feynman graphs
arXiv: Mathematical Physics, 2002Co-Authors: Motohico Mulase, Andrew WaldronAbstract:We present an asymptotic expansion for quaternionic self-adjoint Matrix integrals. The Feynman diagrams appearing in the expansion are ordinary ribbon graphs and their non-orientable counterparts. The result exhibits a striking duality between quaternionic self-adjoint and real symmetric Matrix integrals. The asymptotic expansions of these integrals are given in terms of summations over topologies of compact surfaces, both orientable and non-orientable, for all genera and an arbitrary positive number of marked points on them. We show that the Gaussian Orthogonal Ensemble (GOE) and Gaussian Symplectic Ensemble (GSE) have exactly the same graphical expansion term by term (when appropriately normalized),except that the contributions from non-orientable surfaces with odd Euler characteristic carry the opposite sign. As an application, we give a new topological proof of the known duality for correlations of characteristic polynomials. Indeed, we show that this duality is equivalent to Poincare duality of graphs drawn on a compact surface. Another application of our graphical expansion formula is a simple and simultaneous (re)derivation of the Central Limit Theorem for GOE, GUE (Gaussian Unitary Ensemble) and GSE: The three cases have exactly the same graphical limiting formula except for an overall constant that represents the type of the ensemble.
Aiqing Zhu - One of the best experts on this subject based on the ideXlab platform.
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unit triangular factorization of the Matrix Symplectic group
SIAM Journal on Matrix Analysis and Applications, 2020Co-Authors: Pengzhan Jin, Yifa Tang, Aiqing ZhuAbstract:In this work, we prove that any Symplectic Matrix can be factored into no more than 9 unit triangular Symplectic matrices. This structure-preserving factorization of the Symplectic matrices immedia...
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Unit triangular factorization of the Matrix Symplectic group
arXiv: Symplectic Geometry, 2019Co-Authors: Pengzhan Jin, Yifa Tang, Aiqing ZhuAbstract:In this work, we prove that any Symplectic Matrix can be factored into no more than 9 unit triangular Symplectic matrices. This structured preserving factorization of the Symplectic matrices immediately reveals several important inferences, such as, (\romannumeral1) the determinant of Symplectic Matrix is one, (\romannumeral2) the Matrix Symplectic group is path connected, (\romannumeral3) all the unit triangular Symplectic matrices forms a set of generators of the Matrix Symplectic group, (\romannumeral4) the $2d$-by-$2d$ Matrix Symplectic group is a smooth manifold of dimension $2d^{2}+d$. Furthermore, this factorization yields effective methods for the unconstrained parametrization of the Matrix Symplectic group as well as its structured subsets. The unconstrained parametrization enables us to apply faster and more efficient unconstrained optimization algorithms to the problem with Symplectic constraints.
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unit triangular factorization of the Matrix Symplectic group
arXiv: Symplectic Geometry, 2019Co-Authors: Pengzhan Jin, Yifa Tang, Aiqing ZhuAbstract:In this work, we prove that any Symplectic Matrix can be factored into no more than 9 unit triangular Symplectic matrices. This structure-preserving factorization of the Symplectic matrices immediately reveals two well-known features that, (i) the determinant of any Symplectic Matrix is one, (ii) the Matrix Symplectic group is path connected, as well as a new feature that (iii) all the unit triangular Symplectic matrices form a set of generators of the Matrix Symplectic group. Furthermore, this factorization yields effective methods for the unconstrained parametrization of the Matrix Symplectic group as well as its structured subsets. The unconstrained parametrization enables us to apply faster and more efficient unconstrained optimization algorithms to the problems with Symplectic constraints under certain circumstances.
