383 research outputs found
Chern-Simons theory of multi-component quantum Hall systems
The Chern-Simons approach has been widely used to explain fractional quantum
Hall states in the framework of trial wave functions. In the present paper, we
generalise the concept of Chern-Simons transformations to systems with any
number of components (spin or pseudospin degrees of freedom), extending earlier
results for systems with one or two components. We treat the density
fluctuations by adding auxiliary gauge fields and appropriate constraints. The
Hamiltonian is quadratic in these fields and hence can be treated as a harmonic
oscillator Hamiltonian, with a ground state that is connected to the Halperin
wave functions through the plasma analogy. We investigate several conditions on
the coefficients of the Chern-Simons transformation and on the filling factors
under which our model is valid. Furthermore, we discuss several singular cases,
associated with symmetric states.Comment: 11 pages, shortened version, accepted for publication in Phys. Rev.
Quantum process reconstruction based on mutually unbiased basis
We study a quantum process reconstruction based on the use of mutually
unbiased projectors (MUB-projectors) as input states for a D-dimensional
quantum system, with D being a power of a prime number. This approach connects
the results of quantum-state tomography using mutually unbiased bases (MUB)
with the coefficients of a quantum process, expanded in terms of
MUB-projectors. We also study the performance of the reconstruction scheme
against random errors when measuring probabilities at the MUB-projectors.Comment: 6 pages, 1 figur
On a new generalized inverse for matrices of an arbitrary index
[EN] The purpose of this paper is to introduce a new generalized inverse, called DMP inverse, associated with a square complex matrix using its Drazin and Moore-Penrose inverses. DMP inverse extends the notion of core inverse, introduced by Baksalary and Trenkler for matrices of index at most 1 in (Baksalary and Trenkler (2010) [1]) to matrices of an arbitrary index. DMP inverses are analyzed from both algebraic as well as geometrical approaches establishing the equivalence between them. (C) 2013 Elsevier Inc. All rights reserved.This author was partially supported by Ministry of Education of Spain (Grant DGI MTM2010-18228).Malik, SB.; Thome, N. (2014). On a new generalized inverse for matrices of an arbitrary index. Applied Mathematics and Computation. 226:575-580. doi:10.1016/j.amc.2013.10.060S57558022
Determining global mean-first-passage time of random walks on Vicsek fractals using eigenvalues of Laplacian matrices
The family of Vicsek fractals is one of the most important and
frequently-studied regular fractal classes, and it is of considerable interest
to understand the dynamical processes on this treelike fractal family. In this
paper, we investigate discrete random walks on the Vicsek fractals, with the
aim to obtain the exact solutions to the global mean first-passage time
(GMFPT), defined as the average of first-passage time (FPT) between two nodes
over the whole family of fractals. Based on the known connections between FPTs,
effective resistance, and the eigenvalues of graph Laplacian, we determine
implicitly the GMFPT of the Vicsek fractals, which is corroborated by numerical
results. The obtained closed-form solution shows that the GMFPT approximately
grows as a power-law function with system size (number of all nodes), with the
exponent lies between 1 and 2. We then provide both the upper bound and lower
bound for GMFPT of general trees, and show that leading behavior of the upper
bound is the square of system size and the dominating scaling of the lower
bound varies linearly with system size. We also show that the upper bound can
be achieved in linear chains and the lower bound can be reached in star graphs.
This study provides a comprehensive understanding of random walks on the Vicsek
fractals and general treelike networks.Comment: Definitive version accepted for publication in Physical Review
Further properties on the core partial order and other matrix partial orders
This paper carries further the study of core partial order initiated by Baksalary and Trenkler [Core inverse of matrices, Linear Multilinear Algebra. 2010;58:681-697]. We have extensively studied the core partial order, and some new characterizations are obtained in this paper. In addition, simple expressions for the already known characterizations of the minus, the star (and one-sided star), the sharp (and one-sided sharp) and the diamond partial orders are also obtained by using a Hartwig-Spindelbck decomposition.This author was partially supported by Ministry of Education of Spain [grant number DGI MTM2010-18228] and by Universidad Nacional de La Pampa, Argentina, Facultad de Ingenieria [grant number Resol. No 049/11].Malik, SB.; Rueda, LC.; Thome, N. (2014). Further properties on the core partial order and other matrix partial orders. Linear and Multilinear Algebra. 62(12):1629-1648. https://doi.org/10.1080/03081087.2013.839676S162916486212Mitra, S. K., & Bhimasankaram, P. (2010). MATRIX PARTIAL ORDERS, SHORTED OPERATORS AND APPLICATIONS. SERIES IN ALGEBRA. doi:10.1142/9789812838452Baksalary, J. K., & Hauke, J. (1990). A further algebraic version of Cochran’s theorem and matrix partial orderings. Linear Algebra and its Applications, 127, 157-169. doi:10.1016/0024-3795(90)90341-9Baksalary, O. M., & Trenkler, G. (2010). Core inverse of matrices. Linear and Multilinear Algebra, 58(6), 681-697. doi:10.1080/03081080902778222Baksalary, J. K., Baksalary, O. M., & Liu, X. (2003). Further properties of the star, left-star, right-star, and minus partial orderings. Linear Algebra and its Applications, 375, 83-94. doi:10.1016/s0024-3795(03)00609-8Groβ, J., Hauke, J., & Markiewicz, A. (1999). Partial orderings, preorderings, and the polar decomposition of matrices. Linear Algebra and its Applications, 289(1-3), 161-168. doi:10.1016/s0024-3795(98)10108-8Mosić, D., & Djordjević, D. S. (2012). Reverse order law for the group inverse in rings. Applied Mathematics and Computation, 219(5), 2526-2534. doi:10.1016/j.amc.2012.08.088Patrício, P., & Costa, A. (2009). On the Drazin index of regular elements. Open Mathematics, 7(2). doi:10.2478/s11533-009-0015-6Rakić, D. S., & Djordjević, D. S. (2012). Space pre-order and minus partial order for operators on Banach spaces. Aequationes mathematicae, 85(3), 429-448. doi:10.1007/s00010-012-0133-2Tošić, M., & Cvetković-Ilić, D. S. (2012). Invertibility of a linear combination of two matrices and partial orderings. Applied Mathematics and Computation, 218(9), 4651-4657. doi:10.1016/j.amc.2011.10.052Hartwig, R. E., & Spindelböck, K. (1983). Matrices for whichA∗andA†commute. Linear and Multilinear Algebra, 14(3), 241-256. doi:10.1080/03081088308817561Baksalary, O. M., Styan, G. P. H., & Trenkler, G. (2009). On a matrix decomposition of Hartwig and Spindelböck. Linear Algebra and its Applications, 430(10), 2798-2812. doi:10.1016/j.laa.2009.01.015Mielniczuk, J. (2011). Note on the core matrix partial ordering. Discussiones Mathematicae Probability and Statistics, 31(1-2), 71. doi:10.7151/dmps.1134Meyer, C. (2000). Matrix Analysis and Applied Linear Algebra. doi:10.1137/1.978089871951
Necessary conditions for variational regularization schemes
We study variational regularization methods in a general framework, more
precisely those methods that use a discrepancy and a regularization functional.
While several sets of sufficient conditions are known to obtain a
regularization method, we start with an investigation of the converse question:
How could necessary conditions for a variational method to provide a
regularization method look like? To this end, we formalize the notion of a
variational scheme and start with comparison of three different instances of
variational methods. Then we focus on the data space model and investigate the
role and interplay of the topological structure, the convergence notion and the
discrepancy functional. Especially, we deduce necessary conditions for the
discrepancy functional to fulfill usual continuity assumptions. The results are
applied to discrepancy functionals given by Bregman distances and especially to
the Kullback-Leibler divergence.Comment: To appear in Inverse Problem
Spectral Simplicity of Apparent Complexity, Part I: The Nondiagonalizable Metadynamics of Prediction
Virtually all questions that one can ask about the behavioral and structural
complexity of a stochastic process reduce to a linear algebraic framing of a
time evolution governed by an appropriate hidden-Markov process generator. Each
type of question---correlation, predictability, predictive cost, observer
synchronization, and the like---induces a distinct generator class. Answers are
then functions of the class-appropriate transition dynamic. Unfortunately,
these dynamics are generically nonnormal, nondiagonalizable, singular, and so
on. Tractably analyzing these dynamics relies on adapting the recently
introduced meromorphic functional calculus, which specifies the spectral
decomposition of functions of nondiagonalizable linear operators, even when the
function poles and zeros coincide with the operator's spectrum. Along the way,
we establish special properties of the projection operators that demonstrate
how they capture the organization of subprocesses within a complex system.
Circumventing the spurious infinities of alternative calculi, this leads in the
sequel, Part II, to the first closed-form expressions for complexity measures,
couched either in terms of the Drazin inverse (negative-one power of a singular
operator) or the eigenvalues and projection operators of the appropriate
transition dynamic.Comment: 24 pages, 3 figures, 4 tables; current version always at
http://csc.ucdavis.edu/~cmg/compmech/pubs/sdscpt1.ht
Sampling Theorem and Discrete Fourier Transform on the Riemann Sphere
Using coherent-state techniques, we prove a sampling theorem for Majorana's
(holomorphic) functions on the Riemann sphere and we provide an exact
reconstruction formula as a convolution product of samples and a given
reconstruction kernel (a sinc-type function). We also discuss the effect of
over- and under-sampling. Sample points are roots of unity, a fact which allows
explicit inversion formulas for resolution and overlapping kernel operators
through the theory of Circulant Matrices and Rectangular Fourier Matrices. The
case of band-limited functions on the Riemann sphere, with spins up to , is
also considered. The connection with the standard Euler angle picture, in terms
of spherical harmonics, is established through a discrete Bargmann transform.Comment: 26 latex pages. Final version published in J. Fourier Anal. App
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