Linear maps preserving regional eigenvalue location

Let M(n, C) be the vector space of n × n complex matrices and let G(r,s,t) be the set of all matrices in M(n, C) having r eigenvalues with positive real parts eigenvalues with negative real part and t eigenvalues with zero real part. In particularG(0,n,0) is the set of stable matrices. We investigate the set of linear operators on M(n, C) that map G(r,s,t) into itself. Such maps include, but are not always limited to similarities, transposition, and multiplication by a positive constant. The proof of our results depends on a characterization of nilpotent matrices in terms of matrices in a particular G(r,s,t), and an extension of a result about the existence of a matrix with prescribed eigenstructure and diagonal entries. Each of these results is of independent interest. Moreover, our char-acterization of nilpotent matrices is sufficiently general to allow us to determine the preservers of many other "inertia classes."     

Nonlinear maps in spaces of distributions

This research was started while the author was visiting University of Pisa, University of Ferrara and I.C.T.P., Trieste in the autumn of 1988     

Second eigenvalue of transition matrix associated to iterated maps

We study the properties of the second eigenvalue of the transition matrix in unimodal and bimodal map’s families. We use in particular the relation with the mixing rate and apply this to classify families of isentropic bimodal maps. We present also a study which show the convergence of this quantity in those families.     

On limit distributions for intermediate order statistics under power normalization

In this paper the work of Chibisov (1964) for the intermediate order statistics, under linear normalization, is extended to the power normalization. The possible limits and the corresponding domains of attraction are derived.     

The asymptotic power of rank tests under scale-al ternatives including contaminated distributions

Summary The alternative hypothesis of translated scale for the classical non-parametric hypothesis of equality of two distribution functions in the two-sample problem is extended to a scale-alternative including contamination. The asymptotic power of rank tests and the two-sampleF-test under contiguous sequences of the alternatives is derived and asymptotic relative efficiency of these rank tests with respect to theF-test is investigated. It is found that some of the rank tests have reasonably high asymptotic powers satisfied enough.     

Third-order power comparisons for a class of tests for multivariate linear hypothesis under general distributions

The purpose of this paper is, in multivariate linear regression model (Part I) and GMANOVA model (Part II), to investigate the effect of nonnormality upon the nonnull distributions of some multivariate test statistics under normality. It is shown that whatever the underlying distributions, the difference of local powers up to order N⁻¹ after either Bartlett’s type adjustment or Cornish-Fisher’s type size adjustment under nonnormality coincides with that in Anderson [An Introduction to Multivariate Statistical Analysis, 2nd ed. and 3rd ed., Wiley, New York, 1984, 2003] under normality. The derivation of asymptotic expansions is based on the differential operator associated with the multivariate linear regression model under general distributions. The performance of higher-order results in finite samples, including monotone Bartlett’s type adjustment and monotone Cornish-Fisher’s type size adjustment, is examined using simulation studies.     

The smallest positive eigenvalue of graphs under perturbation

The effect on the smallest positive eigenvalue of a bipartite graph is studied when the graph is perturbed by attaching a pendant vertex at one of its vertices. Let $${\widehat{T}}(v)$$ be the graph obtained by attaching a pendant at vertex v of T. We characterize the vertices v such that the smallest positive eigenvalue of $${\widehat{T}}(v)$$ is equal or greater than that of T. As an application, we obtain the pairs of nonisomorphic noncospectral trees having the same smallest positive eigenvalue.     

Order isomorphism under higher power shift and generic form of renormalization group equations in multimodal maps

A framework of the sequence renormalization in symbolic space, which degenerates for order patterns of block shifts, is presented in this paper. The order isomorphism of both the higher power shift on a block sequence and the shift on a sequence is the topological and metric foundation of renormalization group (RG) operator. Based on the order relation with many compositions, the generic form of RG equations in multimodal maps is derived.     

Surjectivity of power maps of real algebraic groups

In this paper we study the surjectivity of the power maps g?gn for real points of algebraic groups defined over reals. The results are also applied to study the exponentiality of such groups.     

Power iteration and inverse power iteration for eigenvalue complementarity problem

In this paper, an inverse complementarity power iteration method (ICPIM) for solving eigenvalue complementarity problems (EiCPs) is proposed. Previously, the complementarity power iteration method (CPIM) for solving EiCPs was designed based on the projection onto the convex cone K. In the new algorithm, a strongly monotone linear complementarity problem over the convex cone K is needed to be solved at each iteration. It is shown that, for the symmetric EiCPs, the CPIM can be interpreted as the well‐known conditional gradient method, which requires only linear optimization steps over a well‐suited domain. Moreover, the ICPIM is closely related to the successive quadratic programming (SQP) via renormalization of iterates. The global convergence of these two algorithms is established by defining two nonnegative merit functions with zero global minimum on the solution set of the symmetric EiCP. Finally, some numerical simulations are included to evaluate the efficiency of the proposed algorithms.     

Stein estimation under elliptical distributions

In a subclass of elliptical distributions, Stein estimators are robust in estimating the mean vector and the regression parameters in a linear regression model. Unbiased estimates of bias and risk are also given for the regression model.     

The least eigenvalue of signless Laplacian of graphs under perturbation

We investigate how the least eigenvalue of the signless Laplacian of a graph changes by relocating a bipartite branch from one vertex to another vertex, and minimize the least eigenvalue of the signless Laplacian among the class of connected graphs with fixed order which contains a given non-bipartite graph as an induced subgraph.     

Equilibria of nonconvex valued maps under constraints

In the paper the notion of n-tangency for set-valued maps defined on a subset of a Banach space is considered. The existence of equilibria of upper semicontinuous map being n-tangent to a sleek retract with the nontrivial Euler characteristic is established.     

Limit distributions of multivariate kurtosis and moments under Watson rotational symmetric distributions

The limit distribution of Mardia's measure of sample multivariate kurtosis is derived for a wide class of multivariate distributions which includes both the family of elliptical and Watson rotational symmetric distributions. Explicit expressions are given for the higher-order moments of Watson rotational symmetric distributions. The problem of constructing approximate confidence intervals for the kurtosis parameter is also discussed.     

Quantization of probability distributions under norm-based distortion measures II: Self-similar distributions

For a probability measure P on Rd and n∈N consider where the infimum is taken over all subsets α of Rd with card(α)?n and V is a nondecreasing function. Under certain conditions on V, we derive the precise n-asymptotics of en for self-similar distributions P and we find the asymptotic performance of optimal quantizers using weighted empirical measures.     

Automorphism invariant Cartan subgroups and power maps of disconnected groups

We study the geometry and the periodic geodesics of a compact Lorentzian manifold that has a Killing vector field which is timelike somewhere. Using a compactness argument for subgroups of the isometry group, we prove the existence of one timelike non self-intersecting periodic geodesic. If the Killing vector field is nowhere vanishing, then there are at least two distinct periodic geodesics; as a special case, compact stationary manifolds have at least two periodic timelike geodesics. We also discuss some properties of the topology of such manifolds. In particular, we show that a compact manifold M admits a Lorentzian metric with a nowhere vanishing Killing vector field which is timelike somewhere if and only if M admits a smooth circle action without fixed points.