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# Topic: Isospectral flow

 Flow Equations for Hamiltonians Volker Bach, Mainz that flow equations were also developed by the mathematicians Roger W. Brockett, Harvard and by Moody T. Chu, NCSU, and Kenneth R. Driessel, Univ. of Wyoming. The most important of these come from the fact that this equation is equivalent to a certain gradient flow on the space of orthogonal matrices. The flow approach has potential applications ranging from new development of numerical algorithms to the theoretical solution of open problems. www.tphys.uni-heidelberg.de /~statphys/flowmat.html   (444 words)

 AMCA: A centrosymmetric isospectral flow that preserves a matrix structure and its convergence by Oscar Rojo   (Site not responding. Last check: 2007-10-21) AMCA: A centrosymmetric isospectral flow that preserves a matrix structure and its convergence by Oscar Rojo A centrosymmetric isospectral flow that preserves a matrix structure and its convergence We introduce a isospectral flow evolving in the space of centrosymmetric matrices. at.yorku.ca /c/a/o/w/97.htm   (95 words)

 Dynamical Systems Group meeting flows) and relate the estimates of the global attractors to the physically asserted bounds on the number of degrees of freedom. The evolution of hydraulic conductivity and flow patterns, controlled by simultaneous precipitation and dissolution in porous rocks, was examined in a series of laboratory experiments. Because the dissolution of calcium carbonate is a mass-transfer limited process, higher flow rates cause a more rapid dissolution of the porous medium; in such cases, with dissolution dominating, highly conductive flow wormholes were observed to develop. www.wisdom.weizmann.ac.il /~gannar/Seminar/pre_lectures.html   (3797 words)

 Science Fair Projects - Isospectral In mathematics, two linear operators are called isospectral if they have the same spectrum. In the case of operators on infinite-dimensional spaces, the spectrum need not consist solely of isolated eigenvalues; the rest of this article will assume for clarity that we are talking about operators on finite-dimensional vector spaces. In that case, for complex square matrices, the relation of being isospectral for two diagonalizable matrices is just similarity. www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Isospectral_flow   (374 words)

 Stony Brook Math Calendar The first example of isospectral but non-isometric Riemannian manifolds was a pair of 16-dimensional flat tori, given by Milnor in 1964. We shall also describe a pair of isospectral platycosms' (flat 3-manifolds), and sketch the proof that this pair is unique (up to scale). Ricci Flow and the Geometrization of 3-Manifolds III www.math.sunysb.edu /~calendar/scott.php?LocationID=&Date=2003-03-01   (5495 words)

 Scientific Activities: The Minerva Center for Nonlinear Physics of Complex Systems Coil-stretch transition in polymer conformation in a random flow was identified and characterized. Similarly to flows of polymer solutions, solutions of vesicles are expected to show a random flow of the elastic turbulence type, the phenomenon that is looking for. Various other problems of interest, such as isospectrality (which relates to the question- "Can one hear the shape of a graph?"), quantum irreversibility (dephasing) and nodal structures of wave functions on graphs are also studied. www.weizmann.ac.il /acadaff/Scientific_Activities/current/Nonlinear_Physics_Center.html   (1678 words)

 BibTeX bibliography /home/ftp/pub/narep/u-manchester-mccm.bib Thus, admissible sets of data concerning systems of eigenvalues and eigenvectors are examined and procedures for generating associated (isospectral) families of systems are developed. In this paper it is assumed that one vibrating system is specified and the objective is to generate isospectral families of systems, i.e. Numerical results are presented for the Stokes equations arising in incompressible flow modelling and a variable diffusion equation that arises in modelling potential flow. www.maths.man.ac.uk /~nareports   (6794 words)

 [No title] Recall that the {\it stable} and {\it unstable distributions} participate in the definition of an Anosov flow \cite{An}. In the case of a smooth Anosov flow, these distributions are absolutely continuous \cite{An} and H\"older-continuous \cite{AS} but in general they are not smooth. In the case of an $n$-dimensional Anosov manifold $M$, the stable and unstable distributions are defined on the manifold $\Omega M$ of unit tangent vectors. www.ma.utexas.edu /mp_arc/html/papers/98-412   (4479 words)

 NSF-CBMS Regional Conference in the Mathematical Sciences: "Advances in Inverse Spectral Geometry" Abstract Examples include, among others, isospectral deformations arising from the representation theoretic techniques of Lecture 3, new examples of isospectral deformations constructed by R. Gornet by different methods, and isospectral nilmanifolds with different local geometry (to be discussed in lecture 8). We exhibit a pair of flat bordered surfaces which are isospectral for the Neumann boundary conditions, one of which is orientable while the other is nonorientable. The surfaces are constructed using the orbifold version of Sunada's theorem, and Neumann isospectrality is verified explicitly by transplantation of eigenfunctions. www.math.ttu.edu /current/cbms1996/abstract.html   (1059 words)

 Conference on Inverse Spectral Geometry-Abstracts of Presentations The generator, B, of the Lax-Phillips semigroup has spectrum given in terms of the eigenvalues of the Laplacian and the poles of the scattering matrix. The first is that an isospectral family of Riemannian orbifolds, satisfying a lower Ricci curvature bound, contains orbifolds with points of only finitely many possible isotropy types. Wide range of new examples for isospectral metrics with different local geometries are constructed. www.ms.uky.edu /~isgconf/abstracts.html   (1278 words)

 Paper availability and download   (Site not responding. Last check: 2007-10-21) is an isospectral flow if the eigenvalues and the dimensions of the associated eigenspaces are the same for all parameter values \sigma. This paper presents the most general form for isospectral flows of linear dynamic systems of orders p=2,3,4, and the forms for isospectral flows for even higher order systems are evident from the patterns emerging. Based on the definition of a class of coordinate transformations called structure-preserving transformations, the concept of isospectrality and the associated flows is seen to extend to cases where m does not equal n. www.aer.bris.ac.uk /contact/academic/friswell/PDF_Files/J102.html   (197 words)

 vita2 We show that the map defines a gradient-like flow which deforms a given curve to a conformal circle. Given a stochastic flow on Euclidean space generated as the solution of a stochastic differential equation, we study the domain valued process obtained by allowing the flow to act on the space of smoothly bounded domains with compact closure in Euclidean space. Given a pair of planar isospectral, nonisometric polygons constructed as a quotient of the plane by a finite group, we construct an associated pair of planar isospectral, nonisometric weighted graphs. proteus.network.ncf.edu /~mcdonald/papers/papers.htm   (1575 words)

 Advances in Computation: Theory and Practice, Book Series, Volume 4, Abstracts In this paper a technique based on isospectral flows is introduced to compute the eigenvalues of displacement structured matrices. The flow starting from a matrix with a given displacement structure is enforced to preserve that structure, by adding a suitable constraint to the classical formulation. In particular, existence and uniqueness of the flow is proven, under the assumption that the starting matrix is sufficiently close to the kernel of the appropriate displacement operator. www.cs.okstate.edu /~actp/volumes/vol04abs.html   (1865 words)

 Weekly Calendar   (Site not responding. Last check: 2007-10-21) Time and brain permitting it would be nice to see a glimpse of the "spectral curve" and the associated algebraic integrable system for these flows and observe the relationship with "finite type" solutions for a discrete version of the KdV system. In particular, it seems that a new approach is needed to explore this relationship in the presence of large isometry groups. After reviewing the isospectral problem and techniques for constructing isospectral manifolds, we will discuss progress the speaker and Alejandro Uribe have made in understanding whether isospectral manifolds constructed via the so-called torus method have the same length spectrum. www.math.umass.edu /~cal/oldcal.html   (286 words)

 Southeast Geometry Seminar Although some results exist about isospectral graphs and spectral asymptotics, this seems like a rich enough set of examples to warrant further exploration. Abstract: The lecture explains how the geometry of necks in a hypersurface can be controlled by a priori estimates for the curvature and then gives an explicit surgery construction. Finally it is shown how the surgery can be used to extend mean curvature flow beyond singularities for hypersurfaces with the sum of the two lowest principal curvature positive everywhere. www.math.uab.edu /sgs/sgs5   (456 words)

 Fields Institute - Numerical and Computational Challenges in Science and Engineering - Lectures Matrices that appear in physical problems are not arbitrary: they have specified patterns of zero and non-zero terms, and specified patterns of signs. Totally positive, and the related descriptors strictly totally positive, oscillatory, and sign-oscillatory are related to important examples of such sign patterns. isospectral families of matrices for various vibrating systems. www.fields.utoronto.ca /programs/scientific/01-02/numerical/lectures/gladwell_jan.html   (99 words)

 Citebase - Isospectral Flow and Liouville-Arnold Integration in Loop Algebras A number of examples of Hamiltonian systems that are integrable by classical means are cast within the framework of isospectral flows in loop algebras. Integrable Hamiltonians are obtained by restriction of elements of the ring of spectral invariants to the image of these moment maps. The isospectral property follows from the Adler-Kostant-Symes theorem, and gives rise to invariant spectral curves. citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:hep-th/9306127   (278 words)

 Flow Equations for Hamiltonians   (Site not responding. Last check: 2007-10-21) The method of flow equations is a non-perturbative method that can be used to diagonalize, block-diagonalize or renormalize a given Hamiltonian. The flow equation desribes the flow of a Hamiltonian H as a function of the flow-parameter l and has the form dH(l)/dl = [η(l),H(l)], where the generator η(l) of the unitary transformation has typically the form η(l)=[H(l),X]. Meanwhile (2002) we have learned from Volker Bach, that such a method called double bracket flow and isospectral flow, resp., has also been developed by the mathematicians Roger W. Brockett, Moody T. Chu, and Kenneth R. Driessel. www.tphys.uni-heidelberg.de /~statphys/floweq.html   (341 words)

 [No title]   (Site not responding. Last check: 2007-10-21) The research involves the study of manifolds without conjugate points, sharp isoperimetric inequalities for manifolds of various dimensions, degenerate systems of partial differential equations (especially equations of prescribed curvature, the geometry of isospectral manifolds, certain classes of minimal submanifolds, and certain applications of geometry to physiology. The investigators on this award are involved in developing a sophisticated mathematical model which predicts the behavior of fragments in solvent over time. The model predicts the evolution of the stone/solvent interface and is a modification of the curvature flow in differential geometry studied by many researchers. www.cs.utexas.edu /users/yguan/NSFAbstracts/Abstracts/MPS/DMS.MPS.a9203362.txt   (163 words)

 Some reports by Kenneth R. Driessel   (Site not responding. Last check: 2007-10-21) On matrix structures invariant under Toda-like isospectral flows: sign-scaled algebras' (with D. Ashlock and I. Hentzel), September, 1997. On the geometry of some isospectral surfaces' presented June 5, 1998 at the ILAS meeting at the University of Wisconsin at Madison. Matrix structures invariant under Toda-like isospectral flows: sign-scaled algebras' presented June 7, 1997 at the ILAS meeting at the University of Manitoba. members.hpnx.com /driessel/reports.html   (421 words)

 [No title]   (Site not responding. Last check: 2007-10-21) The isospectral problem and Korteweg-deVriess flow arising from complex-valued initial data was studied in the 1980's, marking the starting point for this research. This approach promises to be a powerful tool for tackling some of the open problems in the field, such as practical computation of band edges (by reduction to a linear algebraic eigenvalue problem) and the classification of isospectral manifolds of elliptic finite- gap potentials. Moreover it sheds light on an unexpected relationship between spectral properties in the algebraic as well as in the functional analytic sense of differential operators and global analytic properties of solutions of the associated differential equations. www.cs.utexas.edu /users/yguan/NSFAbstracts/Abstracts/MPS/DMS.MPS.a9401816.txt   (326 words)

 Applied Mathematics - Staff Publications H.P.W. Gottlieb, 2006, 'Radially isospectral annular membranes', IMA Journal of Applied Mathematics, vol. H.P.W. Gottlieb, 2005, 'Transformations between Isospectral Membranes Yield Conformal Maps', IMA Journal of Applied Mathematics, 70, 748-752. Peter R. Johnston, 2004, 'Higher degrees and honours bachelor degrees in mathematics and statistics completed in Australia in 2003', Australian Mathematical Society Gazette, 31 (5), 314-319. www.cit.gu.edu.au /maths/staffpubs.html   (1281 words)

 Search Results for Conformal The Ruth Lyttle Satter Prize is awarded to Sun-Yung Alice Chang for her deep contributions to the study of partial differential equations on Riemannian manifolds and in particular for her work on extremal problems in spectral geometry and the compactness of isospectral metrics within a fixed conformal class on a compact 3-manifold. We have also applied this approach in conformal geometry to the isospectral compactness problem on 3-manifolds when the metrics are restricted in any given conformal class. After considering the problem of conformal mapping on the half-plane of finite polygonal regions bounded by straight lines and circular arcs she applied these ideas to the physical problem of the two-dimensional seepage flow of ground water in an earth dam of a particular shape. www-groups.dcs.st-and.ac.uk /history/Search/historysearch.cgi?SUGGESTION=Conformal&CONTEXT=1   (2229 words)

 Isospectral flows and linear programming   (Site not responding. Last check: 2007-10-21) Brockett has studied the isospectral flow H ' = [H, [H, N]], with [A, B] = AB - BA, on spaces of real symmetric matrices. The flow diagonalises real symmetric matrices and can be used to solve linear programming problems with compact convex constraints. We show that the flow converges exponentially fast to the optimal solution of the programming problem and we give explicit estimates for the time needed by the flow to approach an anziamj.austms.org.au /V34/part4/Helmke.html   (126 words)

 Introduction It is defined as the isospectral deformation of a one-dimensional almost periodic Schrödinger operator It can be integrated by conjugating the flow to a linear flow on the Jacobi variety of a Riemann surface defined by the isospectrally deformed Jacobi matrix. Solving the Einstein equations in the almost periodic case and a better understanding of the geodesic flow in an almost periodic metric are problems that have not yet been addressed. www.math.harvard.edu /~knill/oldinterests/complexity/node1.html   (876 words)

 On the Discretization of Double-Bracket Flows - Iserles (ResearchIndex)   (Site not responding. Last check: 2007-10-21) We show that the solution of the isospectral flow Y = [[Y; N ]; Y ], Y (0) = Y0 2 Sym(n), can be represented in the form Y (t) = e \Omega\Gamma Y0e, where the Taylor expansion of\Omega can be constructed explicitly, term-by-term, identifying individual expansion terms with certain rooted trees with bicolour leaves. On The Optimality Of Double-Bracket Flows - Bloch, Iserles (2003) Arieh Iserles, On the discretization of double-bracket flows, Found. citeseer.ist.psu.edu /iserles01discretization.html   (309 words)

 Birkhoff Billiards They form a limiting case of the geodesic flow and illustrate theorems in topology, geometry or ergodic theory. which still features Kac's problem "can one hear the shape of a smooth drum" (until now, known isospectral counterexamples to Kac's question are not smooth). Billiards are used to investigate the transition from quantum mechanics (the Dirichlet problem, asymptotics of eigenvalues and eigenfunctions) to classical mechanics (the geodesic flow). www.dynamical-systems.org /billiard/info.html   (594 words)

 >Applications of Orthogonal Integration Techniques by Erik Van Vleck, Colorado School of Mines Of course, for stability reasons, explicit computation of Y has to be avoided, and one seeks to compute the Q and R factors directly. Applications include approximation of Lyapunov exponents, continuous orthonormalization technique for solving two-point boundary value problems, and eigenvalue calculation (isospectral flow). The University of Minnesota is an equal opportunity educator and employer. www.ima.umn.edu /dynsys/wkshp_abstracts/vanvleck1.html   (87 words)

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