47 research outputs found
A Brauer's theorem and related results
Given a square matrix A, a Brauer's theorem [Brauer A., Limits for the characteristic roots of a matrix. IV. Applications to stochastic matrices, Duke Math. J., 1952, 19(1), 75-91] shows how to modify one single eigenvalue of A via a rank-one perturbation without changing any of the remaining eigenvalues. Older and newer results can be considered in the framework of the above theorem. In this paper, we present its application to stabilization of control systems, including the case when the system is noncontrollable. Other applications presented are related to the Jordan form of A and Wielandt's and Hotelling's deflations. An extension of the aforementioned Brauer's result, Rado's theorem, shows how to modify r eigenvalues of A at the same time via a rank-r perturbation without changing any of the remaining eigenvalues. The same results considered by blocks can be put into the block version framework of the above theorem. © 2012 Versita Warsaw and Springer-Verlag Wien.This work is supported by Fondecyt 1085125, Chile, the Spanish grant DGI MTM2010-18228 and the Programa de Apoyo a la Investigacion y Desarrollo (PAID-06-10) of the UPV.Bru GarcĂa, R.; CantĂł Colomina, R.; Soto, RL.; Urbano Salvador, AM. (2012). A Brauer's theorem and related results. Central European Journal of Mathematics. 10(1):312-321. https://doi.org/10.2478/s11533-011-0113-0S312321101Brauer A., Limits for the characteristic roots of a matrix. IV. Applications to stochastic matrices, Duke Math. J., 1952, 19(1), 75–91Crouch P.E., Introduction to Mathematical Systems Theory, Mathematik-Arbeitspapiere, Bremen, 1988Delchamps D.F., State-Space and Input-Output Linear Systems, Springer, New York, 1988Hautus M.L.J., Controllability and observability condition of linear autonomous systems, Nederl. Akad. Wetensch. Indag. Math., 1969, 72, 443–448Kailath T., Linear Systems, Prentice Hall Inform. System Sci. Ser., Prentice Hall, Englewood Cliffs, 1980Langville A.N., Meyer C.D., Deeper inside PageRank, Internet Math., 2004, 1(3), 335–380Perfect H., Methods of constructing certain stochastic matrices. II, Duke Math. J., 1955, 22(2), 305–311Saad Y., Numerical Methods for Large Eigenvalue Problems, Classics Appl. Math., 66, SIAM, Philadelphia, 2011Soto R.L., Rojo O., Applications of a Brauer theorem in the nonnegative inverse eigenvalue problem, Linear Algebra Appl., 2006, 416(2–3), 844–856Wilkinson J.H., The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 196
Line Graphs of Weighted Networks for Overlapping Communities
In this paper, we develop the idea to partition the edges of a weighted graph
in order to uncover overlapping communities of its nodes. Our approach is based
on the construction of different types of weighted line graphs, i.e. graphs
whose nodes are the links of the original graph, that encapsulate differently
the relations between the edges. Weighted line graphs are argued to provide an
alternative, valuable representation of the system's topology, and are shown to
have important applications in community detection, as the usual node partition
of a line graph naturally leads to an edge partition of the original graph.
This identification allows us to use traditional partitioning methods in order
to address the long-standing problem of the detection of overlapping
communities. We apply it to the analysis of different social and geographical
networks.Comment: 8 Pages. New title and text revisions to emphasise differences from
earlier paper
Asymptotic Expansions for Stationary Distributions of Perturbed Semi-Markov Processes
New algorithms for computing of asymptotic expansions for stationary
distributions of nonlinearly perturbed semi-Markov processes are presented. The
algorithms are based on special techniques of sequential phase space reduction,
which can be applied to processes with asymptotically coupled and uncoupled
finite phase spaces.Comment: 83 page
The Projection Method for Reaching Consensus and the Regularized Power Limit of a Stochastic Matrix
In the coordination/consensus problem for multi-agent systems, a well-known
condition of achieving consensus is the presence of a spanning arborescence in
the communication digraph. The paper deals with the discrete consensus problem
in the case where this condition is not satisfied. A characterization of the
subspace of initial opinions (where is the influence matrix) that
\emph{ensure} consensus in the DeGroot model is given. We propose a method of
coordination that consists of: (1) the transformation of the vector of initial
opinions into a vector belonging to by orthogonal projection and (2)
subsequent iterations of the transformation The properties of this method
are studied. It is shown that for any non-periodic stochastic matrix the
resulting matrix of the orthogonal projection method can be treated as a
regularized power limit of Comment: 19 pages, 2 figure
The life and work of A.A. Markov
The Russian mathematician A.A. Markov (1856-1922) is known for his work in number theory, analysis, and probability theory. He extended the weak law of large numbers and the central limit theorem to certain sequences of dependent random variables forming special classes of what are now known as Markov chains. For illustrative purposes Markov applied his chains to the distribution of vowels and consonants in A.S. Pushkin's poem "Eugeny Onegin". At present, much more important applications of Markov chains have been discovered. Here we present an overview of Markov's life and his work on the chains. © 2004 Elsevier Inc. All rights reserved
The life and work of A.A. Markov
The Russian mathematician A.A. Markov (1856-1922) is known for his work in number theory, analysis, and probability theory. He extended the weak law of large numbers and the central limit theorem to certain sequences of dependent random variables forming special classes of what are now known as Markov chains. For illustrative purposes Markov applied his chains to the distribution of vowels and consonants in A.S. Pushkin's poem "Eugeny Onegin". At present, much more important applications of Markov chains have been discovered. Here we present an overview of Markov's life and his work on the chains. © 2004 Elsevier Inc. All rights reserved