maximization and minimization of the rank and inertia of the hermitian matrix expression a bx bx with applications abstract we give in this paper a group of closed form formulas for the maximal and minimal ranks and inertias of the linear hermitian matrix function a bx bx with respect to a variable matrix x as applications we derive the extremal ranks and inertias of the matrices x x where x is a solution to the matrix equation axb c and then give necessary and sufficient conditions for the matrix equation axb c to have hermitian definite and re definite solutions in addition we give closed form formulas for the extremal ranks and inertias of the difference x 1 x 2 where x 1 and x 2 are hermitian solutions of two matrix equations a 1 x 1 a 1 c 1 and a 2 x 2 a 2 c 2 and then use the formulas to characterize relations between hermitian solutions of the two equations

the common re nnd and re pd solutions to the matrix equations ax c and xb d abstract in this article we consider common re nnd and re pd solutions of the matrix equations ax c and xb d with respect to x where a b c and d are given matrices we give necessary and sufficient conditions for the existence of common re nnd and re pd solutions to the pair of the matrix equations and derive a representation of the common re nnd and re pd solutions to these two equations when they exist the presented examples show the advantage of the proposed approach

explicit solutions to the matrix inverse problem ax b abstract recently wu and cain l wu b cain linear algebra and its applications 236 1996 137 146 gave necessary and sufficient conditions for the matrix equationax b to have a solution with respect toa subject to the condition that the hermitian part ofa is non negative definite in this note we give an easier approach which also provides possible explicit expressions fora further necessary and sufficient conditions are derived when the hermitian part ofa is prescribed to be positive definite thus correcting a result in l wu b cain linear algebra and its applications 236 1996 137 146 theorem 5

random walk in chemical space of cantor dust as a paradigm of superdiffusion we point out that the chemical space of a totally disconnected cantor dust k n ensuremath subset e n is a compact metric space c n with the spectral dimension d s d ensuremath ell ngd where d and d ensuremath ell n are the fractal and chemical dimensions of k n respectively hence we can define a random walk in the chemical space as a markovian gaussian process the mapping of a random walk in c n into k n ensuremath subset e n defines the quenched l evy flight on the cantor dust with a single step duration independent of the step length the equations describing the superdiffusion and diffusion reaction front propagation ruled by the local quenched l evy flight on k n ensuremath subset e n are derived the use of these equations to model superdiffusive phenomena observed in some physical systems in which propagators decay faster than algebraically is discussed

fracture mechanics analysis using the wavelet galerkin method and extended finite element method summary this paper presents fracture mechanics analysis using the wavelet galerkin method and extended finite element method the wavelet galerkin method is a new methodology to solve partial differential equations where scaling wavelet functions are used as basis functions in solid structural analyses the analysis domain is divided into equally spaced structured cells and scaling functions are periodically placed throughout the domain to improve accuracy wavelet functions are superposed on the scaling functions within a region having a high stress concentration such as near a hole or notch thus the method can be considered a refinement technique in fixed grid approaches however because the basis functions are assumed to be continuous in applications of the wavelet galerkin method there are difficulties in treating displacement discontinuities across the crack surface in the present research we introduce enrichment functions in the wavelet galerkin formulation to take into account the discontinuous displacements and high stress concentration around the crack tip by applying the concept of the extended finite element method this paper presents the mathematical formulation and numerical implementation of the proposed technique as numerical examples stress intensity factor evaluations and crack propagation analyses for two dimensional cracks are presented copyright 2012 john wiley sons ltd

re nnd generalized inverses abstract in this paper we give some necessary and sufficient conditions for the existence of re nnd and nonnegative definite 1 3 and 1 4 inverses of a matrix a c n n and completely described these sets moreover we prove that the existence of nonnegative definite 1 3 inverse of a matrix a is equivalent with the existence of its nonnegative definite 1 2 3 inverse and present the necessary and sufficient conditions for the existence of re nnd 1 3 4 inverse of a

crack growth analysis using mesh superposition technique and x fem in this paper a new approach to crack propagation analysis for large scale or complicated geometry structures is presented that is the connection of mesh superposition technique and the extended finite element method the former is a technique that increases the accuracy of analysis locally by superimposing additional mesh of higher resolution onto the mesh that represents the rough deformation of the structure in this technique the boundaries and nodes of both meshes do not have to coincide with each other in the latter technique the discontinuity across the crack segment and singularity around the crack tips are represented in the approximation by enriching the nodal degrees using partition of unity condition this technique does not require meshes to conform to the crack geometry and this enables crack propagation procedure with no re meshing process by connecting both techniques it becomes possible to analyse crack growth models of large scale and complicated shape with highly flexible modelling and no remeshing process by using this approach some numerical examples are analysed and appropriate results are obtained copyright 2007 john wiley sons ltd

combined extended and superimposed finite element method for cracks a combination of the extended finite element method xfem and the mesh superposition method s version fem for modelling of stationary and growing cracks is presented the near tip field is modelled by superimposed quarter point elements on an overlaid mesh and the rest of the discontinuity is implicitly described by a step function on partition of unity the two displacement fields are matched through a transition region the method can robustly deal with stationary crack and crack growth it simplifies the numerical integration of the weak form in the galerkin method as compared to the s version fem numerical experiments are provided to demonstrate the effectiveness and robustness of the proposed method copyright 2004 john wiley sons ltd

clouds fibres and echoes a new approach to studying random walks on fractals up to now the general approach of constructing evolution differential equations to describe random walks on fractals has not succeeded is this because the true probability density function is inherently fractal when plotted in the appropriate similarity variable we find a cloud which is not too smooth further investigation shows that this cloud has a structure that might be overlooked if one is looking for the usual single valued probability density function the cloud is composed of an infinite family of smooth fibres each of which describes the behaviour of the walk on an infinite echo point class the fibres are individually smooth and so are naturally amenable to analysis with differential equations

equations ax c and xb d in rings and rings with involution with applications to hilbert space operators abstract this paper reviews the equations ax c and xb d from a new perspective by studying them in the setting of associative rings with or without involution results for rectangular matrices and operators between different banach and hilbert spaces are obtained by embedding the rectangles into rings of square matrices or rings of operators acting on the same space necessary and sufficient conditions using generalized inverses are given for the existence of the hermitian skew hermitian reflexive antireflexive positive and real positive solutions and the general solutions are described in terms of the original elements or operators new results are obtained and many results existing in the literature are recovered and corrected

fractional diffusion exact representations of spectral functions for all the relevant transformed spaces i e fourier laplace and fourier laplace we present exact solutions of a fractional diffusion equation describing random transport on fractals the potential importance of such spectral representations lies in their applications to interpreting experimental measurements of anomalous diffusion processes in contrast to the well known asymptotic results the exact representations provide a much broader basis for comparison with data

brownian motion on the sierpinski gasket we construct a brownian motion taking values in the sierpinski gasket a fractal subset of 2 and study its properties this is a diffusion process characterized by local isotropy and homogeneity properties we show for example that the process has a continuous symmetric transition density p t x y with respect to an appropriate hausdorff measure and obtain estimates on p t x y

probability distribution of random walks on self avoiding walks the probability distribution of random walks on one dimensional fractal structures generated by random walks rw chains and self avoiding walks saw chains in d dimensional space is studied analytically in the case where is the fractal dimension of the random walk it is shown that there exists an infinite hierarchy of critical dimensions with for rw chains and for saw chains for each term in the expansion of the scaling part of each transition is characterized by its own logarithmic correction

information dimension of stochastic processes on networks relating entropy production to spectral properties we consider discrete stochastic processes modeled by classical master equations on networks the temporal growth of the lack of information about the system is captured by its non equilibrium entropy defined via the transition probabilities between different nodes of the network we derive a relation between the entropy and the spectrum of the master equation s transfer matrix our findings indicate that the temporal growth of the entropy is proportional to the logarithm of time if the spectral density shows scaling in analogy to chaos theory the proportionality factor is called stochastic information dimension and gives a global characterization of the dynamics on the network these general results are corroborated by examples of regular and of fractal networks

waiting times for random walks on regular and fractal lattices we use the exact renormalization procedure introduced by machta phys rev b 24 5260 1981 to evaluate the waiting time density for a walker to jump to one of its nearest neighbors on a decimated lattice in terms of the original waiting time densities on the nondecimated lattice we explicitly identify all the fixed points and their corresponding domain of attraction for the one dimensional case and for the sierpinsky gasket

an explicit application of partition of unity approach to xfem approximation for precise reproduction of a priori knowledge of solution summary the application of the xfem to fracture mechanics is effective because a crack can be modeled independently from the meshes and a complex remeshing procedure can be avoided however the classical xfem has an essential problem in the approximation of partially enriched elements that is blending elements which causes a lack of accuracy for the weighted xfem although the numerical results show the effective improvements it was found that the issue of blending elements still remains upon detailed examination in the present paper the pu xfem is formulated as an explicit application of the partition of unity pu approach to the xfem in order to precisely reproduce a priori knowledge of the solution by enrichment the pu xfem is applied to two dimensional linear fracture mechanics and its effectiveness is verified it is consequently found out that the pu xfem precisely reproduces a priori knowledge of the solution and is therefore effective to completely solve the problem of the blending elements copyright 2013 john wiley sons ltd

adaptive s method for linear elastostatics abstract a methodology has been developed to simultaneously treat phenomena at several scales the key features of the method are i a detailed error diagnosis which identifies critical regions and ii a feedback procedure which overlays portions of the finite element mesh where unacceptable errors have been identified by local finite element mesh es and then subsequently improves the quality of the solution in the global and local response until the desired level of accuracy is achieved the method can be routinely used as a global local method and or as an adaptive technique which provides a measure of reliability of global and local computations numerical experiments for problems in homogeneous and heterogeneous laminated composites media are presented to validate the present formulation

on the propagator of sierpinski gaskets the authors present numerical calculations for the propagator p r t the probability to reach a distance r at time t having started at the origin at t 0 on sierpinski gaskets the results are confronted with approximate analytical expressions it is shown that p r t approximately t ds 2 pi xi where xi is the scaling variable xi r t1 dw in the short xi regime the scaling function follows the form pi xi approximately exp c1 xi dw while for large xi pi xi is given asymptotically by pi xi approximately xi alpha exp c2 xi nu with alpha df dw 2 dw 1 and nu dw dw 1 this result extends the previously derived expressions the numerically observed oscillations which are superimposed on the power law decay of the autocorrelation function p r 0 t are analysed in terms of typical residence times on hierarchical substructures

analytical solutions for diffusion on fractal objects a generalization of the euclidean diffusion equation is proposed for diffusion on fractals on the basis of a scaling argument a renormalization group theory for the green s function and numerical tests conjectures on the applicability to natural fractals such as porous media are presented

description of diffusion on discrete arrays through system of piecewise linear maps abstract it is shown that diffusion s dynamic on any discrete array including fractals can be described by piecewise linear maps the proposed method which is geometric in its nature allows to construct a system of piecewise linear maps based on the nearest neighbor coupling matrix of cosenza and kapral we propose that the dynamics of the map system describes the diffusion dynamics on the array which in our case is the sierpinski gasket due to the existence of an intrinsic homeomorphism between them it is found that there is a one to one correspondence between the partitions of the map and the cells of the fractal this fact allows to generate specific diffusion trajectories on the fractal in order to study the particle dwell time distribution since the method proposed here is completely deterministic the diffusion on the map is anomalous the mean square displacement scales with time to a power very near to 2 it is also shown that as a consequence of the lattice autosimilarity and the finite precision of the calculations this parameter remains fixed for several levels of construction of the sierpinski gasket

an expression of the general common least squares solution to a pair of matrix equations with applications in this paper an explicit representation of the general common least squares solution to the pair of matrix equations a 1xb 1 c 1 and a 2xb 2 c 2 is obtained furthermore we use this result to determine the condition for the existence of a hermitian least squares solution to the matrix equation axb c and the expression of the general hermitian least squares solution is also given special attention is paid to consider the existence of hermitian 1 i inverses of a i 3 4 and the representations of the hermitian generalized inverses are presented finally a numerical example is given to illustrate our result

first passage time survival probability and propagator on deterministic fractals the first passage time density psi r t defined as the probability density for the time spent by a random walker to travel for the first time the distance r that separates the starting site from its nearest neighbours and the survival probability s r t i e the probability that a random walker who starts at a site has not been absorbed by traps located on its nearest neighbours at distance r in the time interval 0 t were calculated for the class of deterministic fractals in which sites are isolated from the rest of the lattice by their nearest neighbours the large xi identical to r square root 2dt 1dw asymptotic expressions for these quantities are psi r t approximately a xi nu 2 dw exp c xi nu and h r t 1 s r t approximately a c dw 1 xi nu 2 exp c xi nu with v dw dw 1 a and c being characteristic constants for each fractal the asymptotic expression for s r t is used to justify that for this class of deterministic fractals the propagator or green function is given asymptotically by p r t t ds 2 xi alpha exp c xi nu for large xi with alpha nu 2 df this value of alpha differs from others proposed recently

continuum diffusion on networks trees with hyperbranched trunks and fractal branches the probability that a random walker returns to its origin for large times scales as t d over 2 where d over is the spectral dimension we calculate d over for a class of tree structures using a renormalization technique on an infinite continued fraction we consider a wide range of homogeneous networks based on replacing the branches of a self similar tree with arbitrary fractals and composite fractals we also consider a new class of inhomogeneous hyperbranched trees

the re positive definite solutions to the matrix inverse problem ax b abstract an n n complex matrix a is termed re positive definite re pd if the real part of x ax is positive for every nonzero complex n vector x this paper is concerned with constructing complex re pd matrices and solving the following matrix inverse problem given complex matrices x and b find the set of all complex re pd matrices a such that ax b