443 research outputs found
Krein systems
In the present paper we extend results of M.G. Krein associated to the
spectral problem for Krein systems to systems with matrix valued accelerants
with a possible jump discontinuity at the origin. Explicit formulas for the
accelerant are given in terms of the matrizant of the system in question.
Recent developments in the theory of continuous analogs of the resultant
operator play an essential role
Strings in five-dimensional anti-de Sitter space with a symmetry
The equation of motion of an extended object in spacetime reduces to an
ordinary differential equation in the presence of symmetry. By properly
defining of the symmetry with notion of cohomogeneity, we discuss the method
for classifying all these extended objects. We carry out the classification for
the strings in the five-dimensional anti-de Sitter space by the effective use
of the local isomorphism between \SO(4,2) and \SU(2,2). We present a
general method for solving the trajectory of the Nambu-Goto string and apply to
a case obtained by the classification, thereby find a new solution which has
properties unique to odd-dimensional anti-de Sitter spaces. The geometry of the
solution is analized and found to be a timelike helicoid-like surface
The symmetric-Toeplitz linear system problem in parallel
[EN] Many algorithms exist that exploit the special structure of
Toeplitz matrices for solving linear systems. Nevertheless, these algorithms
are difficult to parallelize due to its lower computational cost and
the great dependency of the operations involved that produces a great
communication cost. The foundation of the parallel algorithm presented
in this paper consists of transforming the Toeplitz matrix into a another
structured matrix called Cauchy¿like. The particular properties of
Cauchy¿like matrices are exploited in order to obtain two levels of parallelism
that makes possible to highly reduce the execution time. The
experimental results were obtained in a cluster of PC¿s.Supported by Spanish MCYT and FEDER under Grant TIC 2003-08238-C02-02Alonso-Jordá, P.; Vidal Maciá, AM. (2005). The symmetric-Toeplitz linear system problem in parallel. Computational Science -- ICCS 2005,Pt 1, Proceedings. 3514:220-228. https://doi.org/10.1007/11428831_28S2202283514Sweet, D.R.: The use of linear-time systolic algorithms for the solution of toeplitz problems. k Technical Report JCU-CS-91/1, Department of Computer Science, James Cook University, Tue, 23 April 1996 15, 17, 55 GMT (1991)Evans, D.J., Oka, G.: Parallel solution of symmetric positive definite Toeplitz systems. Parallel Algorithms and Applications 12, 297–303 (1998)Gohberg, I., Koltracht, I., Averbuch, A., Shoham, B.: Timing analysis of a parallel algorithm for Toeplitz matrices on a MIMD parallel machine. Parallel Computing 17, 563–577 (1991)Gallivan, K., Thirumalai, S., Dooren, P.V.: On solving block toeplitz systems using a block schur algorithm. In: Proceedings of the 23rd International Conference on Parallel Processing, Boca Raton, FL, USA, vol. 3, pp. 274–281. CRC Press, Boca Raton (1994)Thirumalai, S.: High performance algorithms to solve Toeplitz and block Toeplitz systems. Ph.d. th., Grad. College of the U. of Illinois at Urbana–Champaign (1996)Alonso, P., Badía, J.M., Vidal, A.M.: Parallel algorithms for the solution of toeplitz systems of linear equations. In: Wyrzykowski, R., Dongarra, J., Paprzycki, M., Waśniewski, J. (eds.) PPAM 2004. LNCS, vol. 3019, pp. 969–976. Springer, Heidelberg (2004)Anderson, E., et al.: LAPACK Users’ Guide. SIAM, Philadelphia (1995)Blackford, L., et al.: ScaLAPACK Users’ Guide. SIAM, Philadelphia (1997)Alonso, P., Badía, J.M., González, A., Vidal, A.M.: Parallel design of multichannel inverse filters for audio reproduction. In: Parallel and Distributed Computing and Systems, IASTED, Marina del Rey, CA, USA, vol. II, pp. 719–724 (2003)Loan, C.V.: Computational Frameworks for the Fast Fourier Transform. SIAM Press, Philadelphia (1992)Heinig, G.: Inversion of generalized Cauchy matrices and other classes of structured matrices. Linear Algebra and Signal Proc., IMA, Math. Appl. 69, 95–114 (1994)Gohberg, I., Kailath, T., Olshevsky, V.: Fast Gaussian elimination with partial pivoting for matrices with displacement structure. Mathematics of Computation 64, 1557–1576 (1995)Alonso, P., Vidal, A.M.: An efficient and stable parallel solution for symmetric toeplitz linear systems. TR DSIC-II/2005, DSIC–Univ. Polit. Valencia (2005)Kailath, T., Sayed, A.H.: Displacement structure: Theory and applications. SIAM Review 37, 297–386 (1995
Minimal symmetric Darlington synthesis
We consider the symmetric Darlington synthesis of a p x p rational symmetric
Schur function S with the constraint that the extension is of size 2p x 2p.
Under the assumption that S is strictly contractive in at least one point of
the imaginary axis, we determine the minimal McMillan degree of the extension.
In particular, we show that it is generically given by the number of zeros of
odd multiplicity of I-SS*. A constructive characterization of all such
extensions is provided in terms of a symmetric realization of S and of the
outer spectral factor of I-SS*. The authors's motivation for the problem stems
from Surface Acoustic Wave filters where physical constraints on the
electro-acoustic scattering matrix naturally raise this mathematical issue
Edgeworth Expansion of the Largest Eigenvalue Distribution Function of GUE Revisited
We derive expansions of the resolvent
Rn(x;y;t)=(Qn(x;t)Pn(y;t)-Qn(y;t)Pn(x;t))/(x-y) of the Hermite kernel Kn at the
edge of the spectrum of the finite n Gaussian Unitary Ensemble (GUEn) and the
finite n expansion of Qn(x;t) and Pn(x;t). Using these large n expansions, we
give another proof of the derivation of an Edgeworth type theorem for the
largest eigenvalue distribution function of GUEn. We conclude with a brief
discussion on the derivation of the probability distribution function of the
corresponding largest eigenvalue in the Gaussian Orthogonal Ensemble (GOEn) and
Gaussian Symplectic Ensembles (GSEn)
Semiseparable integral operators and explicit solution of an inverse problem for the skew-self-adjoint Dirac-type system
Inverse problem to recover the skew-self-adjoint Dirac-type system from the
generalized Weyl matrix function is treated in the paper. Sufficient conditions
under which the unique solution of the inverse problem exists, are formulated
in terms of the Weyl function and a procedure to solve the inverse problem is
given. The case of the generalized Weyl functions of the form
, where is a strictly proper rational
matrix function and is a diagonal matrix, is treated in greater
detail. Explicit formulas for the inversion of the corresponding semiseparable
integral operators and recovery of the Dirac-type system are obtained for this
case
A unified approach to Darboux transformations
We analyze a certain class of integral equations related to Marchenko
equations and Gel'fand-Levitan equations associated with various systems of
ordinary differential operators. When the integral operator is perturbed by a
finite-rank perturbation, we explicitly evaluate the change in the solution. We
show how this result provides a unified approach to Darboux transformations
associated with various systems of ordinary differential operators. We
illustrate our theory by deriving the Darboux transformation for the
Zakharov-Shabat system and show how the potential and wave function change when
a discrete eigenvalue is added to the spectrum.Comment: final version that will appear in Inverse Problem
Variation of discrete spectra for non-selfadjoint perturbations of selfadjoint operators
Let B=A+K where A is a bounded selfadjoint operator and K is an element of
the von Neumann-Schatten ideal S_p with p>1. Let {\lambda_n} denote an
enumeration of the discrete spectrum of B. We show that \sum_n
\dist(\lambda_n, \sigma(A))^p is bounded from above by a constant multiple of
|K|_p^p. We also derive a unitary analog of this estimate and apply it to
obtain new estimates on zero-sets of Cauchy transforms.Comment: Differences to previous version: Extended Introduction, new Section
5, additional references. To appear in Int. Eq. Op. Theor
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