on statistical aspects of deterministic tree like fractals we prove convergence of the partition function of the skewed mass distribution for a class of finite type tree like fractals

characterization and construction of the nearest defective matrix via coalescence of pseudospectral components let a be a matrix with distinct eigenvalues and let w a be the distance from a to the set of defective matrices using either the 2norm or the frobenius norm define the pseudospectrum of a to be the set of points in the complex plane which are eigenvalues of matrices a e with kek and let c a be the supremum of all with the property that has n distinct components demmel and wilkinson independently observed in the 1980s that w a c a and equality was established for the 2 norm by alam and bora in 2005 we give new results on the geometry of the pseudospectrum near points

matrix polynomials with specified eigenvalues abstract this work concerns the distance in the 2 norm from a given matrix polynomial to a nearest polynomial with a specified number of its eigenvalues at specified locations in the complex plane initially we consider perturbations of the constant coefficient matrix only a singular value optimization characterization is derived for the associated distance we also consider the distance in the general setting when all of the coefficient matrices are perturbed in this general setting we obtain a lower bound in terms of another singular value optimization problem the singular value optimization problems derived facilitate the numerical computation of the distances

on a class of skewed self similar and hyperbolic fractals probabilistic recurrent renewal theory is used herein to describe the geometry of a new class of self similar distributions in a skewed multiplicative mass splitting procedure it is shown how this model class can account for a deterministic and conservative treelike structure in connection with recent developments on anomalous multiplicative multifractals coexistence of both micro and macroscopic scales in such objects is a consequence of the underlying renewal phenomena being transient as is shown in a second part of the article various entropylike measures of the intrinsic disorder prevailing in these structures are proposed

fixing multiple eigenvalues by a minimal perturbation abstract suppose we are given an n n matrix m and a set of values i i 1 m m n and we wish to find the smallest perturbation in the 2 norm i e spectral norm m such that m m has the given eigenvalues i some interesting results have been obtained for variants of this problem for fixing two distinct eigenvalues fixing one double eigenvalue and fixing a triple eigenvalue this paper provides a geometric motivation for these results and also motivates their generalization we also present numerical examples both successes and failures of fixing more than two eigenvalues by these generalizations

the structure of decoherence functionals for von neumann quantum histories gell mann and hartle have proposed a significant generalization of quantum theory in which decoherence functionals perform a key role isham linden schreckenberg have given a penetrating analysis of decoherence functionals for l h where h is finite dimensional dimension greater than 2 in this note their conjecture on the significance of boundedness is verified in particular it is shown that when d is a bounded decoherence functional associated with a von neumann algebra a then provided a has no direct summand of type i2 d can be represented as the difference between semi innerproducts on a

on the distance from a matrix polynomial to matrix polynomials with some prescribed eigenvalues abstract consider an n n matrix polynomial p and a set consisting of k n complex numbers recently kokabifar loghmani psarrakos and karbassi studied a weighted spectral norm distance from p to the n n matrix polynomials whose spectra contain the specified set under the assumption that all the entries of are distinct in this paper the case in which some or all of the desired eigenvalues can be multiple is discussed lower and upper bounds for the distance are computed and a perturbation of p associated to the upper bound is constructed a detailed numerical example illustrates the efficiency and validity of the proposed computational method

a formula for computation of the real stability radius this paper presents a readily computable formula for the real stability radius with respect to an arbitrary stability region in the complex plane

a formula for the 2 norm distance from a matrix to the set of matrices with multiple eigenvalues we prove that the 2 norm distance from an n times n matrix a to the matrices that have a multiple eigenvalue lambda is equal to where the singular values sigma k are ordered nonincreasingly therefore the 2 norm distance from a to the set of matrices with multiple eigenvalues is

on the construction of nearest defective matrices to a normal matrix given an n by n normal matrix a having n distinct eigenvalues we describe a simple procedure for constructing a defective matrix nearest to a our construction requires only an appropriate pair of eigenvalues of a and their corresponding eigenvectors

new anomalous multiplicative multifractals left sided ƒ α and the modelling of dla abstract a sharp distinction is drawn between general multiplicative multifractals as originally introduced by the author and the more familiar but more restricted class defined by frisch and parisi and by halsey et al all the general multiplicative multifractals are exactly renormalizable by design however this is important and perhaps suprising it is shown that the scaling relations that serve to define the restricted class fail to extend to certain multiplicative multifractals as a result in addition to the familiar restricted class the general multifractals are shown to allow for a wide variety of anomalous behaviors several of which are described we believe that suitable examples of these anomalies are of the most important concrete applications to fully developed turbulence and to diffusion limited aggregates dla this paper concerns several new classes of multifractals we have recently proposed as tools to model the anomalies of dla they are left sided that is characterized by max with max d 0 related multifractals characterized by max 0 and even by max and min 0 are encountered along the way

nearest matrix with two prescribed eigenvalues abstract given a square complex matrix a and two complex numbers z1 and z2 we find the distance from a to the set of matrices that have z1 z2 as some of their eigenvalues we use the distance between two matrices associated with the spectral norm

locating a nearest matrix with an eigenvalue of prespecified algebraic multiplicity the wilkinson distance of a matrix a is the two norm of the smallest perturbation e so that a e has a multiple eigenvalue malyshev derived a singular value optimization characterization for the wilkinson distance in this work we generalize the definition of the wilkinson distance as the two norm of the smallest perturbation so that the perturbed matrix has an eigenvalue of prespecified algebraic multiplicity we provide a singular value characterization for this generalized wilkinson distance then we outline a numerical technique to solve the derived singular value optimization problems in particular the numerical technique is applicable to malyshev s formula to compute the wilkinson distance as well as to retrieve a nearest matrix with a multiple eigenvalue

the distance from a matrix polynomial to matrix polynomials with a prescribed multiple eigenvalue abstract for a matrix polynomial p and a given complex number we introduce a spectral norm distance from p to the matrix polynomials that have as an eigenvalue of geometric multiplicity at least and a distance from p to the matrix polynomials that have as a multiple eigenvalue then we compute the first distance and obtain bounds for the second one constructing associated perturbations of p

computational aspect to the nearest matrix with two prescribed eigenvalues abstract given a complex square matrix a and two complex numbers 1 and 2 we present a method to calculate the distance from a to the set of matrices x that have 1 and 2 as some of their eigenvalues we also find the nearest matrix x

level set methods for finding critical points of mountain pass type abstract computing mountain passes is a standard way of finding critical points we describe a numerical method for finding critical points that is convergent in the nonsmooth case and locally superlinearly convergent in the smooth finite dimensional case we apply these techniques to describe a strategy for addressing the wilkinson problem of calculating the distance from a matrix to a closest matrix with repeated eigenvalues finally we relate critical points of mountain pass type to nonsmooth and metric critical point theory

nearest southeast submatrix that makes multiple a prescribed eigenvalue part 1 abstract given four complex matrices a b c and d where a c n n and d c m m and given a complex number z 0 what is the spectral norm distance from d to the set of matrices x c m m such that z 0 is a multiple eigenvalue of the matrix a b c x this problem is solved when z 0 is not an eigenvalue of a we also give a conjecture when z 0 is an eigenvalue of a

distance bounds for prescribed multiple eigenvalues of matrix polynomials abstract in this paper motivated by a problem posed by wilkinson we study the coefficient perturbations of a square matrix polynomial to a matrix polynomial that has a prescribed eigenvalue of specified algebraic multiplicity and index of annihilation for an n n matrix polynomial p and a given scalar c we introduce two weighted spectral norm distances e r and e r k from p to the n n matrix polynomials that have as an eigenvalue of algebraic multiplicity at least r and to those that have as an eigenvalue of algebraic multiplicity at least r and maximum jordan chain length exactly k respectively then we obtain a lower bound for e r k and derive an upper bound for e r by constructing an associated perturbation of p

nearest matrix with prescribed eigenvalues and its applications consider an n n matrix a and a set consisting of k n prescribed complex numbers lippert 2010 in a challenging article studied geometrically the spectral norm distance from a to the set of matrices whose spectra included specified set and constructed a perturbation matrix with minimum spectral norm such that a had in its spectrum this paper presents an easy practical computational method for constructing the optimal perturbation by improving and extending the methodology necessary definitions and lemmas of previous related works also some conceivable applications of this issue are provided

fixing two eigenvalues by a minimal perturbation abstract the optimal perturbation m to a matrix m such that m m has a given eigenvalue 0 is given by the eckart young theorem this perturbation is optimal in the sense that m 2 is minimal in this article we present a generalization finding m 2 optimal perturbations of m such that m m has two of given eigenvalues this result also generalizes recent work by malyshev

on neighbouring matrices with quadratic elementary divisors this paper presents new results on matrices with quadratic elementary divisors neighbouring a given matrixa and relates them to those given by kakan it shows that all results of this kind may give severe overestimates of the distance to the nearest defective matrix and that matrices with elementary divisors of degrees higher than two are often at a much greater distance than those with quadratic divisors

structured pseudospectra and structured sensitivity of eigenvalues pseudospectra and structured pseudospectra have been investigated widely in this paper for the matrix spectral norm we give a new equivalent definition of structured pseudospectra of a matrix triplet a b c via the so called restricted singular value decomposition rsvd according to this definition we provide a solution of structured wilkinson s problem which concerns structured sensitivity of eigenvalues of square matrices several examples are given to illustrate our results also we extend the new definition of structured pseudospectra to polynomial eigenvalue problems

numerical computation of an analytic singular value decomposition of a matrix valued function this paper extends the singular value decomposition to a path of matricese t an analytic singular value decomposition of a path of matricese t is an analytic path of factorizationse t x t s t y t t wherex t andy t are orthogonal ands t is diagonal to maintain differentiability the diagonal entries ofs t are allowed to be either positive or negative and to appear in any order this paper investigates existence and uniqueness of analytic svd s and develops an algorithm for computing them we show that a real analytic pathe t always admits a real analytic svd a full rank smooth pathe t with distinct singular values admits a smooth svd we derive a differential equation for the left factor develop euler like and extrapolated euler like numerical methods for approximating an analytic svd and prove that the euler like method converges

on pseudospectra of matrix polynomials and their boundaries in the first part of this paper sections 2 4 the main concern is with the boundary of the pseudospectrum of a matrix polynomial and particularly with smoothness properties of the boundary in the second part sections 5 6 results are obtained concerning the number of connected components of pseudospectra as well as results concerning matrix polynomials with multiple eigenvalues or the proximity to such polynomials

nearest matrix with a prescribed eigenvalue of bounded multiplicities abstract let a c n n and z 0 c be given we compute the distance with respect to the spectral norm from a to the set of matrices that have z 0 as an eigenvalue of algebraic multiplicity at least 3 and geometric multiplicity at least 2 with an additional hypothesis on the maximum of a function furthermore we determine a nearest matrix b to a in this set

on sensitivity of eigenvalues and eigendecompositions of matrices abstract the main purpose of this paper is to provide solutions of two problems on sensitivity of eigenvalues and eigendecompositions of matrices the first problem is due to wilkinson and it concerns finding the distance from an n n matrix with n distinct eigenvalues to the set of matrices having multiple eigenvalues we also describe how to construct a nearest matrix having a multiple eigenvalue the second problem concerns providing a characterization of the stability of eigendecompositions of matrices and is due to demmel