a system of matrix equations and a linear matrix equation over arbitrary regular rings with identity abstract in this paper we consider the classical system of matrix equations a 1 xb 1 c 1 a 2 xb 2 c 2 over r an arbitrary regular ring with identity necessary and sufficient conditions for the existence and the expression of the general solution to the system are derived as an application the linear matrix equation axb cyd e over r is considered

quaternion singular value decomposition based on bidiagonalization to a real or complex matrix using quaternion householder transformations we present a practical and efficient means to compute the singular value decomposition svd of a real or complex quaternion matrix a based on bidiagonalization of a to a real or complex bidiagonal matrix b using quaternionic householder transformations computation of the svd of b using an existing subroutine library such as lapack provides the singular values of a the singular vectors of a are obtained trivially from the product of the householder transformations and the real or complex singular vectors of b we show in the paper that left and right quaternionic householder transformations are different because of the non commutative multiplication of quaternions and we present formulae for computing the householder vector and matrix in each case

estimating the embedding dimension abstract well reconstructed dynamics should be described as a mapping we propose to search for the lowest embedding dimension for which the property of a continuous mapping is satisfied the images of close points are close we demonstrate the effectiveness of the approach by finding the lowest embedding dimension of several chaotic systems embedded by the method of delays

ideals with bounded approximate identities in fourier algebras abstract we make use of the operator space structure of the fourier algebra a g of an amenable locally compact group to prove that if h is any closed subgroup of g then the ideal i h consisting of all functions in a g vanishing on h has a bounded approximate identity this result allows us to completely characterize the ideals of a g with bounded approximate identities we also show that for several classes of locally compact groups including all nilpotent groups i h has an approximate identity with norm bounded by 2 the best possible norm bound

bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations in this paper we consider bisymmetric and centrosymmetric solutions to certain matrix equations over the real quaternion algebra necessary and sufficient conditions are obtained for the matrix equation ax c and the following systemsa 1x c 1 a 1x c 1 xb 3 c 3 a 2x c 2 to have bisymmetric solutions and the systema 1x c 1 a 3xb 3 c 3 to have centrosymmetric solutions the expressions of such solutions of the matrix and the systems mentioned above are also given moreover a criterion for a quaternion matrix to be bisymmetric is established and some auxiliary results on other sets over are also mentioned

remarks on the schur complement abstract first a generalization of the schur determinantal formula is given using properties of quasidirect sums of matrices a new characterization of the schur complement is proved

a pair of simultaneous linear matrix equations a1xb1 c1 a2xb2 c2 and a matrix programming problem necessary and sufficient conditions are obtained for a pair of matrix equations a1xb1 c1 a2xb2 c2 on a general field f to have a common solution along with the expression for a general common solution when certain conditions hold common solutions of minimum rank are described a matrix programming problem is solved en route

reliability of the 0 1 test for chaos in time series analysis it has been considered of key importance to determine whether a complex time series measured from the system is regular deterministically chaotic or random recently gottwald and melbourne have proposed an interesting test for chaos in deterministic systems their analyses suggest that the test may be universally applicable to any deterministic dynamical system in order to fruitfully apply their test to complex experimental data it is important to understand the mechanism for the test to work and how it behaves when it is employed to analyze various types of data including those not from clean deterministic systems we find that the essence of their test can be described as to first constructing a random walklike process from the data then examining how the variance of the random walk scales with time by applying the test to three sets of data corresponding to i 1 f ensuremath alpha noise with long range correlations ii edge of chaos and iii weak chaos we show that the test mis classifies i both deterministic and weakly stochastic edge of chaos and weak chaos as regular motions and ii strongly stochastic edge of chaos and weak chaos as well as 1 f ensuremath alpha noise as deterministic chaos our results suggest that while the test may be effective to discriminate regular motion from fully developed deterministic chaos it is not useful for exploratory purposes especially for the analysis of experimental data with little a priori knowledge a few speculative comments on the future of multiscale nonlinear time series analysis are made

operator space structure and amenability for figa talamanca herz algebras column and row operator spaces which we denote by col and row respectively over arbitrary banach spaces were introduced by the first named author for hilbert spaces these definitions coincide with the usual ones given a locally compact group g and p p 1 with 1p 1p 1 we use the operator space structure on cb col lp g to equip the figa talamanca herz algebra ap g with an operator space structure turning it into a quantized banach algebra moreover we show that for p q 2 or 2 q p and amenable g the canonical inclusion aq g ap g is completely bounded with cb norm at most kg2 where kg is grothendieck s constant as an application we show that g is amenable if and only if ap g is operator amenable for all and equivalently for one p 1 this extends a theorem by ruan

property t and rigidity for actions on banach spaces we study property t and the fixed point property for actions on lp and other banach spaces we show that property t holds when l2 is replaced by lp and even a subspace quotient of lp and that in fact it is independent of 1 p we show that the fixed point property for lp follows from property t when 1 p 2 e for simple lie groups and their lattices we prove that the fixed point property for lp holds for any 1 p if and only if the rank is at least two finally we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive banach spaces

a system of matrix equations with five variables in this paper we give some necessary and sufficient conditions for the consistence of the system of quaternion matrix equations a 1 x c 1 y b 1 d 1 a 2 w c 2 z b 2 d 2 a 3 v c 3 v b 3 c 4 a 4 v b 4 c 5 a 5 x y b 5 c 6 w z d 6 e 6 v f 6 g 6 and constitute an expression of the general solution to the system when it is solvable the outcomes of this paper encompass some recognized results in the collected works in addition we establish an algorithm and a numerical example to illustrate the theory constructed in the paper

the general solution to a system of real quaternion matrix equations in this paper we consider the system of matrix equations a 1x c 1 a 2x c 2 a 3xb 3 c 3 and a 4xb 4 c 4 over the real quaternion algebra a necessary and sufficient condition for the existence and the expression of the general solution to the system are given as particular cases the corresponding results on other systems over are also obtained

what are schur complements anyway abstract given a matrix m a b c d the schur complements of a in m are the matrices of the form s d cab where a is a generalized inverse of a we survey several recent characterizations of schur complements and discuss where they arose and how they are related

more on maximal and minimal ranks of schur complements with applications the maximal and minimal ranks of the schur complement d cgb are determined with respect to generalized inverse g of a matrix a where g is taken respectively as least squares g inverse a 1 3 minimum norm g inverse a 1 4 as well as a 1 2 3 a 1 2 4 and a 1 3 4 various consequences are also presented

filter products of c0 semigroups and ultraproduct representations for lie groups we provide a new approach to filter products of c0 semigroups and prove a spectral theorem for the generator and its filter product in a similar fashion we construct ultraproducts of strongly continuous unitary representations of locally compact groups and study spectral theoretic connections between the representations and their ultraproducts in the case of lie groups our investigations are extended to the infinitesimal representation

on the density of the fourier stieltjes algebra abstract let g be a locally compact group and 1 p let a p g denote the figa talamanca herz algebra and ma p g its pointwise multiplier algebra in this paper we shall introduce two new spaces of multipliers of a p g defined as coefficient functions of certain representations of g on some interpolation couples then we shall show that in the amenable case the fourier stieltjes algebra b g is dense in ma p g and that ma p g as an ordered space is a complete invariant for g

a system of real quaternion matrix equations with applications abstract let h be the real quaternion algebra and h n m denote the set of all n m matrices over h let p h n n and q h m m be involutions i e p 2 i q 2 i a matrix a h n m is said to be p q symmetric if a paq this paper studies the system of linear real quaternion matrix equations a 1 x 1 c 1 x 1 b 1 c 2 a 2 x 2 c 3 x 2 b 2 c 4 a 3 x 1 b 3 a 4 x 2 b 4 c c we present some necessary and sufficient conditions for the existence of a solution to this system and give an expression of the general solution to the system when the solvability conditions are satisfied as applications we discuss the necessary and sufficient conditions for the system a a x c a xb b c b a c xb c c c to have a p q symmetric solution we also show an expression of the p q symmetric solution to the system when the solvability conditions are met moreover we provide an algorithm and a numerical example to illustrate our results the findings of this paper extend some known results in the literature

the dimension of chaotic attractors dimension is perhaps the most basic property of an attractor in this paper we discuss a variety of different definitions of dimension compute their values for a typical example and review previous work on the dimension of chaotic attractors the relevant definitions of dimension are of two general types those that depend only on metric properties and those that depend on the frequency with which a typical trajectory visits different regions of the attractor both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value which we call the dimension of the natural measure and all of the metric dimensions take on a common value which we call the fractal dimension furthermore the dimension of the natural measure is typically equal to the lyapunov dimension which is defined in terms of lyapunov numbers and thus is usually far easier to calculate than any other definition because it is computable and more physically relevant we feel that the dimension of the natural measure is more important than the fractal dimension

extreme ranks of the solution to a consistent system of linear quaternion matrix equations with an application abstract in this paper we establish the maximal and minimal ranks of the solution to the consistent system of quaternion matrix equations a 1 x c 1 a 2 x c 2 a 3 xb 3 c 3 and a 4 xb 4 c 4 which was investigated recently by wang q w wang the general solution to a system of real quaternion matrix equations comput math appl 49 2005 665 675 moreover corresponding results on some special cases are presented as an application a necessary and sufficient condition for the invariance of the rank of the general solution to the system mentioned above is presented some previous known results can be regarded as the special cases of this paper

state space reconstruction in the presence of noise takens theorem demonstrates that in the absence of noise a multidimensional state space can be reconstructed from a scalar time series this theorem gives little guidance however about practical considerations for reconstructing a good state space we extend takens treatment applying statistical methods to incorporate the effects of observational noise and estimation error we define the distortion matrix which is proportional to the conditional covariance of a state given a series of noisy measurements and the noise amplification which is proportional to root mean square time series prediction errors with an ideal model we derive explicit formulae for these quantities and we prove that in the low noise limit minimizing the distortion is equivalent to minimizing the noise amplification we identify several different scaling regimes for distortion and noise amplification and derive asymptotic scaling laws when the dimension and lyapunov exponents are sufficiently large these scaling laws show that no matter how the state space is reconstructed there is an explosion in the noise amplification from a practical point of view determinism is lost and the time series is effectively a random process in the low noise large data limit we show that the technique of local singular value decomposition is an optimal coordinate transformation in the sense that it achieves the minimum distortion in a state space of the lowest possible dimension however in numerical experiments we find that estimation error complicates this issue for local approximation methods we analyze the effect of reconstruction on estimation error derive a scaling law and suggest an algorithm for reducing estimation errors

extreme ranks of a linear quaternion matrix expression subject to triple quaternion matrix equations with applications abstract in this paper we establish the formulas of the maximal and minimal ranks of the quaternion matrix expression c 4 a 4 xb 4 where x is a variant quaternion matrix subject to quaternion matrix equations a 1 x c 1 xb 2 c 2 a 3 xb 3 c 3 as applications we give a new necessary and sufficient condition for the existence of solutions to the system of matrix equations a 1 x c 1 xb 2 c 2 a 3 xb 3 c 3 a 4 xb 4 c 4 which was investigated by wang q w wang a system of four matrix equations over von neumann regular rings and its applications acta math sin 21 2 2005 323 334 by rank equalities in addition extremal ranks of the generalized schur complement d ca b with respect to an inner inverse a of a which is a common solution to quaternion matrix equations a 1 x c 1 xb 2 c 2 are also considered some previous known results can be viewed as special cases of the results of this paper

extremal ranks of a quaternion matrix expression subject to consistent systems of quaternion matrix equations with applications abstract assume that the linear quaternion matrix expression f x 1 x 2 a a 3 x 1 b 3 a 4 x 2 b 4 where x 1 x 2 are variant quaternion matrices in this paper we derive the maximal and minimal ranks of f x 1 x 2 subject to consistent systems of quaternion matrix equations a 1 x 1 c 1 x 1 b 1 c 2 and a 2 x 2 c 3 x 2 b 2 c 4 moreover corresponding results on some special cases are presented as applications we give necessary and sufficient conditions for the existence of solutions to some systems of quaternion matrix equations some previous known results can be regarded as the special cases of this paper

ranks and the least norm of the general solution to a system of quaternion matrix equations we in this paper first establish a new expression of the general solution to the consistent system of linear quaternion matrix equations a1x1 c1 a2x2 c2 a3x1b1 a4x2b2 c3 which was investigated recently q w wang h x chang c y lin p skew symmetric common solutions to a pair of quaternion matrix equations appl math comput 195 2008 721 732 then derive the maximal and minimal ranks and the least norm of the general solution to the system mentioned above some previous known results can be viewed as special cases of the results of this paper

p skew symmetric common solutions to a pair of quaternion matrix equations abstract an n n quaternion matrix a is termed p symmetric or p skewsymmetric if a pap or a pap where p is an n n nontrivial quaternion involution in this paper we first establish necessary and sufficient conditions for the existence and the expression of the general solution to the system of quaternion matrix equations a 1 x 1 c 1 a 2 x 2 c 2 a 3 x 1 b 1 a 4 x 2 b 2 c b then use the results on the system mentioned above to give necessary and sufficient conditions for the existence and the representations of p symmetric and p skewsymmetric solutions to the system of quaternion matrix equations a a x c a and a b xb b c b furthermore we establish representations of p symmetric and p skewsymmetric quaternion matrices a numerical example is presented to illustrate the results of this paper

direct dynamical test for deterministic chaos and optimal embedding of a chaotic time series we propose here a local exponential divergence plot which is capable of providing an alternative means of characterizing a complex time series the suggested plot defines a time dependent exponent and a plus exponent based on their changes with the embedding dimension and delay time a criterion for estimating simultaneously the minimal acceptable embedding dimension the proper delay time and the largest lyapunov exponent has been obtained when redefining the time dependent exponent lambda k curves on a series of shells we have found that whether a linear envelope to the lambda k curves exists can serve as a direct dynamical method of distinguishing chaos from noise

right eigenvalues for quaternionic matrices a topological approach abstract we apply the lefschetz fixed point theorem to show that every square matrix over the quaternions has right eigenvalues we classify them and discuss some of their properties such as an analogue of jordan canonical form and diagonalization of elements of the compact symplectic group sp n

right eigenvalue equation in quaternionic quantum mechanics we study the right eigenvalue equation for quaternionic and complex linear matrix operators defined in n dimensional quaternionic vector spaces for quaternionic linear operators the eigenvalue spectrum consists of n complex values for these operators we give a necessary and sufficient condition for the diagonalization of their quaternionic matrix representations our discussion is also extended to complex linear operators whose spectrum is characterized by 2n complex eigenvalues we show that a consistent analysis of the eigenvalue problem for complex linear operators requires the choice of a complex geometry in defining inner products finally we introduce some examples of the left eigenvalue equations and highlight the main difficulties in their solution

hypercomplex correlation techniques for vector images correlation techniques have been applied to almost every area of signal processing over the past century yet their use has in general been limited to scalar signals while there have been implementations in multichannel applications these can be characterized as a combination of single channel processes true vector correlation techniques with global and interchannel measures have only recently been demonstrated and are still in their infancy by comparison this paper describes our work on vector correlation based on the use of hypercomplex fourier transforms and presents for the first time a unified theory behind the information contained in the peak of a vector correlation response by using example applications for color images we also demonstrate some of the practical implications together with our latest results

rank equalities related to the generalized inverse at s 2 with applications abstract a variety of rank formulas of some matrix expressions and certain partitioned matrices with respect to the generalized inverse a t s 2 are established some necessary and sufficient conditions are given by using the rank formulas presented in this paper for two three and four ordered matrices to be independent in the generalized inverse a t s 2 as special cases necessary and sufficient conditions are derived for two three and four ordered matrices to be independent in the weighted moore penrose inverse and the drazin inverse some known results can be regarded as the special cases of the results in this paper

explicit representation formulas for the minimum norm least squares solutions of some quaternion matrix equations abstract within the framework of the theory of the column and row determinants we obtain explicit representation formulas analogs of cramer s rule for the minimum norm least squares solutions of quaternion matrix equations ax b xa b and axb d

quaternions and matrices of quaternions abstract we give a brief survey on quaternions and matrices of quaternions present new proofs for certain known results and discuss the quaternionic analogues of complex matrices the methods of converting a quaternion matrix to a pair of complex matrices and homotopy theory are emphasized

multifractal analysis of evolving noise associated with unstable plastic flow plastic flow instability attracts increasing interest as a self organization phenomenon showing various dynamical regimes including deterministic chaos and self organized criticality the analysis of the associated nonrandom noise drastic jumps of the mechanical stress however confronts the variation of the noise average parameters due to the evolution of the dislocation microstructure the present paper examines some limitations of the multifractal approach to the study of the evolving noise the applicability of the multifractal analysis to practical situations is proven using the example of discontinuous deformation curves observed under conditions of the portevin le ch telier effect in an a1mg alloy as well as model signals generated by stretching multifractal cantor sets it is found that the smooth trends in the stress serration parameters may narrow the range of the scale invariant behavior associated with the multifractal structure but do not essentially mask it

the common solution to six quaternion matrix equations with applications abstract we in this paper establish necessary and sufficient conditions for the existence of and an explicit expression for a common solution to six classical linear quaternion matrix equations a 1 x c 1 xb 1 c 2 a 2 x c 3 xb 2 c 4 a 3 xb 3 c 5 and a 4 xb 4 c 6 as applications symmetric persymmetric and centrosymmetric solutions to certain systems of quaternion matrix equations are presented furthermore we give an example to illustrate the result of this paper some known results can be regarded as the special cases of this paper

solutions to optimization problems on ranks and inertias of a matrix function with applications abstract consider the hermitian matrix function f x a 3 b 3 x b 3 x subject to a consistent system of matrix equations 0 1 a 1 x c 1 a 2 xb 2 c 2 where means conjugate transpose in this paper we first establish explicit expansion formulas to calculate the global maximal and minimal ranks and inertias of the hermitian matrix function f x then we use the derived formulas to give necessary and sufficient conditions for system 0 1 to have re nonnegative definite re nonpositive definite re positive definite and re negative definite solutions moreover as another application of the derived formulas we establish necessary and sufficient conditions for the solvability to the system of matrix equations 0 2 a 1 x c 1 a 2 xb 2 c 2 b 3 x b 3 x a 3 and provide an expression of the general solution to 0 2 when it is solvable the findings of this paper widely extend the known results in the literature

on solutions of the quaternion matrix equation ax b and their applications in color image restoration by using the complex representation of quaternion matrices and the moore penrose generalized inverse we derive the expressions of the least squares solution with the least norm the least squares pure imaginary solution with the least norm and the least squares real solution with the least norm for the quaternion matrix equation ax b respectively finally we discuss their applications in color image restoration

the matrix equations ax c xb d for the pair of matrix equations ax c xb d this paper gives common solutions of minimum possible rank and also other feasible specified ranks

upper and lower bounds for ranks of matrix expressions using generalized inverses abstract the maximal and minimal ranks of the matrix expression a 1 b 1 x 1 c 1 b 2 x 2 c 2 with respect to x 1 and x 2 are presented as applications the maximal and minimal ranks of a 1 b 1 xc 1 subject to a consistent matrix equation b 2 xc 2 a 2 are also determined in addition the maximal and minimal ranks of the schur complement d ca b with respect to the generalized inverse a of a and their various consequences are also considered

littlewood s algorithm and quaternion matrices abstract a strengthened form of schur s triangularization theorem is given for quaternion matrices with real spectrum for complex matrices it was given by littlewood it is used to classify projectors a 2 a and self annihilating operators a 2 0 on a quaternion unitary space and examples of unitarily wild systems of operators on such a space are presented littlewood s algorithm for reducing a complex matrix to a canonical form under unitary similarity is extended to quaternion matrices whose eigenvalues have geometric multiplicity 1

cohen host type idempotent theorems for representations on banach spaces and applications to figa talamanca herz algebras abstract let g be a locally compact group and let r g denote the ring of subsets of g generated by the left cosets of open subsets of g the cohen host idempotent theorem asserts that a set lies in r g if and only if its indicator function is a coefficient function of a unitary representation of g on some hilbert space we prove related results for representations of g on certain banach spaces we apply our cohen host type theorems to the study of the figa talamanca herz algebras a p g with p 1 for arbitrary g we characterize those closed ideals of a p g that have an approximate identity bounded by 1 in terms of their hulls furthermore we characterize those g such that a p g is 1 amenable for some and equivalently for all p 1 these are precisely the abelian groups

high order amplitude equation for steps on the creep curve we consider a model proposed by one of the authors for a type of plastic instability found in creep experiments which reproduces a number of experimentally observed features the model consists of three coupled nonlinear differential equations describing the evolution of three types of dislocations the transition to the instability has been shown to be via hopf bifurcation leading to limit cycle solutions with respect to physically relevant drive parameters here we use a reductive perturbative method to extract an amplitude equation of up to seventh order to obtain an approximate analytic expression for the order parameter the analysis also enables us to obtain the bifurcation phase diagram of the instability we find that while supercritical bifurcation dominates the major part of the instability region subcritical bifurcation gradually takes over at one end of the region these results are compared with the known experimental results approximate analytic expressions for the limit cycles for different types of bifurcations are shown to agree with their corresponding numerical solutions of the equations describing the model the analysis also shows that high order nonlinearities are important in the problem this approach further allows us to map the theoretical parameters to the experimentally observed macroscopic quantities

on fourier algebra homomorphisms let g be a locally compact group and let b g be the dual space of c g the group c algebra of g the fourier algebra a g is the closed ideal of b g generated by elements with compact support the fourier algebras have a natural operator space structure as preduals of von neumann algebras given a completely bounded algebra homomorphism a g b h we show that it can be described in terms of a piecewise affine map y g with y in the coset ring of h as follows f f ony 0offy when g is discrete and amenable this extends a similar result by host we also show that in the same hypothesis the range of a completely bounded algebra homomorphism a g a h is as large as it can possibly be and it is equal to a well determined set the same description of the range is obtained for bounded algebra homomorphisms this time when g and h are locally compact groups with g abelian