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Vyacheslav L. Girko - One of the best experts on this subject based on the ideXlab platform.

  • Canonical Equation K 59 and Universality Law for Random Matrices ( A + UB )( A + UB )*. Arcsine Law
    Theory of Stochastic Canonical Equations, 2020
    Co-Authors: Vyacheslav L. Girko
    Abstract:

    In this chapter we apply the REFORM method for the deduction of the system of Canonical Equations for normalized spectral functions of the matrices (A n + U n B n )(A n + U n B n )*, where A n and B n are nonrandom matrices and U n is a random unitary matrix from the class of matrices C11. The limit distribution function of normalized spectral functions of some matrices A n + B n U n can be found on the basis of these Equations and the regularized V(Victory)- transform $$ \frac{1}{n}\;1n det [{I_n}\alpha \; + ({A_n}\;{\rm{ + }}{U_n}{B_n} - {I_n}\tau )\;({A_n}\;{\rm{ + }}{U_n}{B_n} - {I_n}\tau )*] , \alpha > 0, \;\tau = t + is, $$ where α > 0 and τ is a complex number.

  • Twenty Five Years of Stochastic Canonical Equation K 40 for Normalized Spectral Functions of Ace-Gram Matrices
    Theory of Stochastic Canonical Equations, 2020
    Co-Authors: Vyacheslav L. Girko
    Abstract:

    Twenty-five years ago in [Gir12, p.269] the general formula for the Stieltjes transform of limit normalized spectral function µ(u) of eigenvalues of the large order ACE-Gram matrices $$ \int_0^\infty {(1} + tu{)^{ - 1}}\;{\rm{d}}\mu {\rm{(}}u{\rm{) = }}\int_0^1 x {d_x}{G_t}(x),\;t > 0, $$ was found, where G t (x) satisfies the so-called Stochastic Canonical Equation for ACE-Gram matrices (Gram matrices with asymptotically constant entries): $$ {G_t}\;(x) = \;P\;\left\{ {\left. {\frac{1}{{1\; + \;t{\theta _1}\;{\rm{\{ }}\omega {\rm{, (1 + }}t{\theta _2}{\rm{\{ (*), }}{G_t}{\rm{(*)\} }}{{\rm{)}}^{ - 1}}{\rm{\} }}}}\;{\rm{ < }}x} \right\}} \right., $$ the random functional θ 1{ω, ✱} is given on the set of bounded random real continuous functions, the random functional θ 2{ω, G t (✱)} is given on the set of distribution functions G t (x) and these functionals are mutually independent.

  • Canonical Equation K 2 . Necessary and Sufficient Modified Lindeberg’s Condition. The Wigner and Cubic Laws
    Theory of Stochastic Canonical Equations, 2020
    Co-Authors: Vyacheslav L. Girko
    Abstract:

    In Chapter 2, we continue to study the normalized spectral functions of random symmetric matrices for the case where the entries of all matrices have equal variances. In this case, the system of Canonical Equations K 1 takes a simple form and sometimes it is possible to find its solution.

  • Stochastic Canonical Equation K 43 for Normalized Spectral Functions of Random Gram Block Matrices
    Theory of Stochastic Canonical Equations, 2020
    Co-Authors: Vyacheslav L. Girko
    Abstract:

    In this chapter, we consider nonsymmetric block matrices of the form $$ {\tilde \Xi _{p1\;{\rm{x}}\;p2}} = {\Xi _{p1q1\;{\rm{x }}p2q2}} = \;{\rm{(}}\Xi _{ij}^{(p1,\;{\rm{p2}})}{\rm{)}}_{i = 1,\; \ldots \;{\rm{, }}p{\rm{1}}}^{j = 1,\; \ldots \;{\rm{, }}p{\rm{2}}} $$ with complex matrices \( \Xi _{ij}^{({p_1},\;{p_2})} \) of size q 1 x q 2. We find stochastic Canonical Equations for resolvents of corresponding Gram matrices and consider the case where the expectation of random blocks \( \Xi _{ij}^{({p_1},\;{p_2})} \) do not exist.

  • Stochastic Canonical Equation K 45 for Normalized Spectral Functions of Random Matrices Pencil
    Theory of Stochastic Canonical Equations, 2020
    Co-Authors: Vyacheslav L. Girko
    Abstract:

    In the same manner as in the previous chapters, we consider n.s.f. of the roots of the characteristic Equation det {ΞΞ* - zHH*} = 0, where Ξ and H are some random matrices of the same dimension. Under appropriate conditions, the system of stochastic Canonical Equations K 45 for the limit of the n.s.f. of the matrix ΞΞ*(HH)−1 is found.

Ricard V Sole - One of the best experts on this subject based on the ideXlab platform.

  • adaptive dynamics of unstable cancer populations the Canonical Equation
    Evolutionary Applications, 2018
    Co-Authors: Ricard V Sole, Guim Aguadegorgorio
    Abstract:

    : In most instances of tumour development, genetic instability plays a role in allowing cancer cell populations to respond to selection barriers, such as physical constraints or immune responses, and rapidly adapt to an always changing environment. Modelling instability is a nontrivial task, since by definition evolving instability leads to changes in the underlying landscape. In this article, we explore mathematically a simple version of unstable tumour progression using the formalism of adaptive dynamics (AD) where selection and mutation are explicitly coupled. Using a set of basic fitness landscapes, the so-called Canonical Equation for the evolution of genetic instability on a minimal scenario associated with a population of unstable cells is derived. We obtain explicit expressions for the evolution of mutation probabilities, and the implications of the model on further experimental studies and potential mutagenic therapies are discussed.

  • adaptive dynamics of unstable cancer populations the Canonical Equation
    bioRxiv, 2017
    Co-Authors: Guim Aguade, Ricard V Sole
    Abstract:

    In most instances of tumour development, genetic instability plays a role in allowing cancer cell populations to respond to selection barriers, such as physical constraints or immune responses, and rapidly adapt to an always changing environment. Modelling instability is a nontrivial task, since by definition evolving changing instability leads to changes in the underlying landscape. In this paper we explore mathematically a simple version of unstable tumor progression using the formalism of Adaptive Dynamics (AD) where selection and mutation are explicitly coupled. Using a set of basic fitness landscapes, the so called Canonical Equation for the evolution of genetic instability on a minimal scenario associated to a population of unstable cells is derived. The implications and potential extensions of this model are discussed.

Guim Aguadegorgorio - One of the best experts on this subject based on the ideXlab platform.

  • adaptive dynamics of unstable cancer populations the Canonical Equation
    Evolutionary Applications, 2018
    Co-Authors: Ricard V Sole, Guim Aguadegorgorio
    Abstract:

    : In most instances of tumour development, genetic instability plays a role in allowing cancer cell populations to respond to selection barriers, such as physical constraints or immune responses, and rapidly adapt to an always changing environment. Modelling instability is a nontrivial task, since by definition evolving instability leads to changes in the underlying landscape. In this article, we explore mathematically a simple version of unstable tumour progression using the formalism of adaptive dynamics (AD) where selection and mutation are explicitly coupled. Using a set of basic fitness landscapes, the so-called Canonical Equation for the evolution of genetic instability on a minimal scenario associated with a population of unstable cells is derived. We obtain explicit expressions for the evolution of mutation probabilities, and the implications of the model on further experimental studies and potential mutagenic therapies are discussed.

Michael I Tribelsky - One of the best experts on this subject based on the ideXlab platform.

  • patterns in dissipative systems with weakly broken continuous symmetry
    Physical Review E, 2008
    Co-Authors: Michael I Tribelsky
    Abstract:

    : Patterns in dissipative systems with weakly broken symmetry are studied based upon the simplest Canonical Equation (generalized Nikolaevskiy model). A generic cubic dispersion Equation governing stability of steady spatially periodic patterns is derived and analyzed. A domain of stable states in the space of the problem parameters (stability balloon) is obtained. It is shown that the domain is characterized by unusual scaling properties, so that its different parts obey different scalings. The results obtained may be applied to describe instabilities of advancing fronts and interfaces, pattern formation in reaction-diffusion systems, nonlinear evolution of seismic waves, and other phenomena.

José M. F. Moura - One of the best experts on this subject based on the ideXlab platform.

  • Spectral statistics of lattice graph structured, non-uniform percolations
    2017 IEEE International Conference on Acoustics Speech and Signal Processing (ICASSP), 2017
    Co-Authors: Stephen Kruzick, José M. F. Moura
    Abstract:

    Design of filters for graph signal processing benefits from knowledge of the spectral decomposition of matrices that encode graphs, such as the adjacency matrix and the Laplacian matrix, used to define the shift operator. For shift matrices with real eigenvalues, which arise for symmetric graphs, the empirical spectral distribution captures the eigenvalue locations. Under realistic circumstances, stochastic influences often affect the network structure and, consequently, the shift matrix empirical spectral distribution. Nevertheless, deterministic functions may often be found to approximate the asymptotic behavior of empirical spectral distributions of random matrices. This paper uses stochastic Canonical Equation methods developed by Girko to derive such deterministic equivalent distributions for the empirical spectral distributions of random graphs formed by structured, non-uniform percolation of a D-dimensional lattice supergraph. Included simulations demonstrate the results for sample parameters.