ray solvable linear systems and ray s2ns matrices abstract ray solvable linear systems and ray s 2 ns matrices are complex generalizations of the sign solvable linear systems and s 2 ns matrices we use the determinantal ray unique matrices instead of ray nonsingular matrices as a generalization of sns matrices to generalize some fundamental results of s 2 ns matrices from the real case to complex case such as the graph theoretical characterization the inverse ray patterns and the upper bound of the number of nonzero entries of s 2 ns matrices the well known characterization of the sign solvable linear systems in terms of the l matrices and s matrices is also generalized to ray solvable linear systems and the relationships between the ray s matrices and real s matrices are investigated some examples are also given to illustrate that some results such as the characterization of the sign inconsistent linear systems do not carry over to the complex case

uniqueness of positive radial solutions for semilinear elliptic equations on annular domains where 5 5a b x rn 0iai x ib is an annular domain in rn n 3 it is well known that the uniqueness problems of solutions of 1 1 and 1 2 are both fundamental and often di8cult during the last decade there has been tremendous progress in studying these problems when 5 is a ball in rn or entire rn n 3 see e g 1 2 5 10 11 however very little was known concerning the annular domain until the work of ni and nussbaum 12 until very recently co man 3 showed that when n 3 f u u up 1ip 3 1 1 and 1 2 have unique positive radial solution for any annular domain see also 13 then it is of interest to extend co man s result to a more general situation for example we may ask the following question

functional clustering by bayesian wavelet methods summary we propose a nonparametric bayes wavelet model for clustering of functional data the wavelet based methodology is aimed at the resolution of generic global and local features during clustering and is suitable for clustering high dimensional data based on the dirichlet process the nonparametric bayes model extends the scope of traditional bayes wavelet methods to functional clustering and allows the elicitation of prior belief about the regularity of the functions and the number of clusters by suitably mixing the dirichlet processes posterior inference is carried out by gibbs sampling with conjugate priors which makes the computation straightforward we use simulated as well as real data sets to illustrate the suitability of the approach over other alternatives

convergence and quotient convergence of iterative methods for solving singular linear equations with index one singular systems with index one arise in many applications such as markov chain modelling in this paper we use the group inverse to characterize the convergence and quotient convergence properties of stationary iterative schemes for solving consistent singular linear systems when the index of the coefficient matrix equals one we give necessary and sufficient conditions for the convergence of stationary iterative methods for such problems next we show that for the stationary iterative method the convergence and the quotient convergence are equivalent

a unified approach to krylov subspace methods for the drazin inverse solution of singular nonsymmetric linear systems abstract consider the linear system ax b where a c n n is a singular matrix in the present work we propose a general framework within which krylov subspace methods for drazin inverse solution of this system can be derived in a convenient way the krylov subspace methods known to us to date treat only the cases in which a is hermitian and its index ind a is unity necessarily in the present work a is not required to be hermitian it can have any type of spectrum and ind a is arbitrary we show that as is the case with nonsingular systems the krylov subspace methods developed here terminate in a finite number of steps that is at most n ind a for one of the methods derived here we also provide an analysis by which we are able to bound the errors the relevant bounds decreasing with increasing dimension of the krylov subspaces involved the results of this paper are applicable to consistent systems as well as to inconsistent ones an interesting feature of the approach to singular systems presented in this work is that it is formulated as a generalization of the standard krylov subspace approach to nonsingular systems indeed our approach here reduces to that relevant for nonsingular systems upon setting ind a 0 everywhere

characterizations iterative method sign pattern and perturbation analysis for the dmp inverse with its applications abstract this paper is a study on main properties and characterizations of the dmp inverse also corresponding representations and computational procedures are derived particularly an upgrade of the bordering method to the case of the dmp inverse is considered as well as applications of the dmp inverse in solving singular linear systems also the dmp inverse of an upper block triangular matrix and its sign pattern are investigated furthermore several computational procedures aimed to approximating the dmp inverse are developed implemented and tested the main of them are the limit representation a revised successive matrix squaring iterations and the gradient neural network gnn dynamical system finally some perturbation bounds and continuity of the dmp inverse are studied

some block matrices with signed drazin inverses abstract the sign pattern of a real matrix m is the 0 1 1 matrix obtained from m by replacing each entry by its sign let q m be the set of real matrices with the same sign pattern as m for any m q m if the drazin inverses of m and m have the same sign pattern then m is said to have signed drazin inverse in this paper we give a complete characterization for a class of anti triangular matrices with signed drazin inverse and present a complete characterization for sign symmetric bipartite matrices with signed drazin inverse

the solution of a problem on matrices having signed generalized inverses abstract a real matrix a is said to have a signed generalized inverse if the sign pattern of its generalized inverse a is uniquely determined by the sign pattern of a in this paper we solve a problem proposed in r a brualdi b l shader matrices of sign solvable linear systems cambridge university press cambridge 1995 b l shader siam j matrix anal appl 16 1995 1056 about the characterizations of the matrices with a special lower triangular blocked form to have a signed generalized inverse using this characterization and the fact that every matrix which has a signed generalized inverse and full column term rank and no zero rows is permutation equivalent to a matrix of this form we give two algorithms for determining whether or not a matrix with full column term rank or a general matrix has a signed generalized inverse

detecting features in spatial point processes with clutter via model based clustering abstract we consider the problem of detecting features such as minefields or seismic faults in spatial point processes when there is substantial clutter we use model based clustering based on a mixture model for the process in which features are assumed to generate points according to highly linear multivariate normal densities and the clutter arises according to a spatial poisson process nonlinear features are represented by several densities giving a piecewise linear representation hierarchical model based clustering provides a first estimate of the features and this is then refined using the em algorithm the number of features is estimated from an approximation to its posterior distribution the method gives good results for the minefield and seismic fault problems software to implement it is available on the world wide web

explicit characterization of the drazin index abstract let b x be the set of bounded linear operators on a banach space x and a b x be drazin invertible an element b b x is said to be a stable perturbation of a if b is drazin invertible and i a b is invertible where i is the identity operator on x a and b are the spectral projectors of a and b respectively under the condition that b is a stable perturbation of a a formula for the drazin inverse b d is derived based on this formula a new approach is provided to the computation of the explicit drazin indices of certain 2 2 operator matrices

index splitting for the drazin inverse and the singular linear system abstract an index splitting of a singular matrix a for the drazin inverse and its relative iterations for the minimal p norm solution of the singular linear system ax b b r a k k ind a are presented some convergence criteria are also given and a method is developed for constructing the index splitting of matrix a

ray patterns of matrices and nonsingularity a complex matrix a is ray nonsingular if det x 0 a f 0 for every matrix x with positive entries a sufficient condition for ray nonsingularity is that the origin is not in the relative interior of the convex hull of the signed transversal products of a the concept of an isolated set of transversals is defined and used to obtain a neces

g outer inverse of banach spaces operators abstract the aim of this paper is to present a new generalized inverse which is called a g outer inverse of operators between two banach spaces as a generalization of g drazin inverses of square matrices and banach space operators several characterizations of this new inverse are given based on the g outer inverse we define and characterize the g outer partial order extending the g drazin partial order to operators between two banach spaces we also relate this new partial order to the minus partial order in the end we study an outer binary relation which is an equivalence relation on the corresponding set

product partition models for normal means abstract i consider probability models for the estimation of normal means that allow for some of the means to be equal these probability models called product partition models specify prior probabilities for a random partition the posterior probability of the partition given the observations has the same form the resulting estimate of the means the product estimate is obtained by conditioning on the partition and summing over all possible partitions the large number of computations involved leads to the use of markov sampling to compute the product estimate i compare the product estimate to other estimates of normal means both in a simulation study and in the prediction of baseball batting averages

graphical description of group inverses of certain bipartite matrices abstract a necessary and sufficient condition for the existence of the group inverse of a certain bipartite matrix is given and an explicit formula is obtained for the group inverse in terms of its block submatrices this form is used to derive a graph theoretic description of the entries of the group inverse of some examples of such a matrix including those corresponding to broom graphs if the group inverse of a nonnegative matrix corresponding to a broom graph exists then it is shown that this group inverse is signed an open question about group inverses of more general bipartite matrices is posed and a summary of cases for which its answer is known is given

matrices with signed generalized inverses abstract a real matrix a is said to have a signed generalized inverse if the sign pattern of its generalized inverse a is uniquely determined by the sign pattern of a in this paper we give complete characterizations of the m n matrices a which have signed generalized inverses for both cases a n and a where a is the term rank of a and without loss of generality we assume n m and thus solve a problem proposed in b l shader siam j matrix anal appl 16 1995 1056 using these characterizations we are also able to show that the property of having a signed generalized inverse for a matrix a is inherited by all the submatrices b of a with b a and is also inherited by all those matrices a 1 with a 1 a which can be obtained from a by replacing some nonzero entries of a by zero we also consider several special cases of a problem proposed in r a brualdi b l shader matrices of sign solvable linear systems cambridge university press cambridge ma 1995 b l shader siam j matrix anal appl 16 1995 1056 about the characterization of the matrices in a special triangular block form to have signed generalized inverses

uniqueness of positive ground state solutions of the logarithmic schrodinger equation we prove the uniqueness of positive ground state solutions of the problem frac d 2 u dr 2 frac n 1 r frac du dr u ln u 0 u r 0 forall r ge 0 and u r u r to 0 0 as r to infty this equation is derived from the logarithmic schrodinger equation rm i psi t delta psi u ln left u 2 right and also from the classical equation frac partial u partial t delta u u left u p 1 right u for each n ge 1 a positive ground state solution is u 0 r exp left frac r 2 4 frac n 2 right 0 le r infty we combine u 0 r with energy estimates and associated ricatti equation estimates to prove that for each n in left 1 9 right u 0 r is the only positive ground state we also investigate the stability of u 0 r several open problems are stated

on digraphs and forbidden configurations of strong sign nonsingular matrices a square real matrix a called a strong sign nonsingular matrix or s2ns matrix if all matrices with the same sign pattern as a nonsingular and the inverses of these matrices all have the same sign pattern a digraph which is the underlying digraph of the signed digraph of an s2ns matrix with a negative main diagonal is called an s2ns digraph in 9 thomassen gave a characterization of strongly connected s2ns digraphs in terms of the forbidden subdigraphs in 2 brualdi and shader constructed minimal forbidden configurations for s2ns digraphs for the general cases where the digraphs considered are not necessarily strongly connected they also proposed the problem about the existence of new minimal forbidden configurations other than those found in 2 9 in this paper we construct infinitely many new basic minimal forbidden configurations and thus obtain the answer this problem we also obtain several necessary conditions for minimal forbidden configurations and give a generalization of thomassen s theorem

elicitation of prior distributions for variable selection problems in regression this paper addresses the problem of quantifying expert opinion about a normal linear regression model when there is uncertainty as to which independent variables should be included in the model opinion is modeled as a mixture of natural conjugate prior distributions with each distribution in the mixture corresponding to a different subset of the independent variables it is shown that for certain values of the independent variables the predictive distribution of the dependent variable simplifies from a mixture of t distributions to a single t distribution using this result a method of eliciting the conjugate distributions of the mixture is developed the method is illustrated in an example

the drazin inverse of an even order tensor and its application to singular tensor equations abstract the notion of the moore penrose inverses of matrices was recently extended from matrix space to even order tensor space with einstein product in the literature in this paper we further study the properties of even order tensors with einstein product we define the index and characterize the invertibility of an even order square tensor we also extend the notion of the drazin inverse of a square matrix to an even order square tensor an expression for the drazin inverse through the core nilpotent decomposition for a tensor of even order is obtained as an application the drazin inverse solution of the singular linear tensor equation a x b will also be included

minimum maximum rank of sign pattern tensors and sign nonsingular tensors abstract in this paper we define the sign pattern tenors minimum maximum rank of sign pattern tenors term rank of tensors and sign nonsingular tensors the necessity and sufficiency for the minimum rank of sign pattern tenors to be 1 is given we show that the maximum rank of a sign pattern tensor is not less than the term rank and the minimum rank of the sign pattern of a sign nonsingular tensor is not less than its dimension we get some characterizations of tensors having sign left or sign right inverses

separation measures and the geometry of bayes factor selection for classification conjugacy assumptions are often used in bayesian selection over a partition because they allow the otherwise unfeasibly large model space to be searched very quickly the implications of such models can be analysed algebraically we use the explicit forms of the associated bayes factors to demonstrate that such methods can be unstable under common settings of the associated hyperparameters we then prove that the regions of instability can be removed by setting the hyperparameters in an unconventional way under this family of assignments we prove that model selection is determined by an implicit separation measure a function of the hyperparameters and the sufficient statistics of clusters in a given partition we show that this family of separation measures has plausible properties the methodology proposed is illustrated through the selection of clusters of longitudinal gene expression profiles

uniqueness of positive radial solutions for δu u up 0 on an annulus abstract we prove uniqueness of positive radial solutions to the semilinear elliptic equation u u u p 0 p 1 subject to the dirichlet boundary condition on an annulus in r n n 3 as a by product our argument also provides a much simpler if not the simplest new proof for the uniqueness of positive solutions to the same problem in a finite ball or in the whole space r n

complex neural network models for time varying drazin inverse two complex zhang neural network znn models for computing the drazin inverse of arbitrary time varying complex square matrix are presented the design of these neural networks is based on corresponding matrix valued error functions arising from the limit representations of the drazin inverse two types of activation functions appropriate for handling complex matrices are exploited to develop each of these networks theoretical results of convergence analysis are presented to show the desirable properties of the proposed complex valued znn models numerical results further demonstrate the effectiveness of the proposed models

uniqueness of positive radial solutions for dirichlet problems on annular domains abstract this paper deals with the uniqueness of positive radial solutions to dirichlet problems on annular domains in r n n 3 as an application we can obtain the results to equation u u p 1 u 0 0 where p 1 1 0 0 0 and 1 0 0

when the sign pattern of a square matrix determines uniquely the sign pattern of its inverse abstract we give a precise characterization in terms of parity digraphs of those square matrices a such that for every matrix b with the same sign pattern as a b 1 exists and has the same sign pattern as a 1

existence of positive nonradial solutions for nonlinear elliptic equations in annular domains we study the existence of positive nonradial solutions of equation a f u 0 in sia u 0 on d la where s x 6 w a 2 and is positive and superlinear at both 0 and oo we use a bifurcation method to show that there is a nonradial bifurcation with mode k at ak 0 1 for any positive integer k if is subcritical and for large k if is supercritical when is subcritical then a nehari type variational method can be used to prove that there exists a e 0 1 such that for any a e a 1 the equation has a nonradial solution on 0 a

nonparametric regression using bayesian variable selection abstract this paper estimates an additive model semiparametrically while automatically selecting the significant independent variables and the appropriate power transformation of the dependent variable the nonlinear variables are modeled as regression splines with significant knots selected from a large number of candidate knots the estimation is made robust by modeling the errors as a mixture of normals a bayesian approach is used to select the significant knots the power transformation and to identify outliers using the gibbs sampler to carry out the computation empirical evidence is given that the sampler works well on both simulated and real examples and that in the univariate case it compares favorably with a kernel weighted local linear smoother the variable selection algorithm in the paper is substantially faster than previous bayesian variable selection algorithms

uniqueness of radially symmetric positive solutions for delta u u u p in an annulus in this article we prove that the semi linear elliptic partial differential equation u u upin u 0in u 0on possesses a unique positive radially symmetric solution here p 1 and is the annulus x rn a x b with n 2 0 a b we also show the positive solution is non degenerate

condition number for the drazin inverse and the drazin inverse solution of singular linear system with their condition numbers in this paper we investigate the condition number of drazin inverse and drazin inverse solution of singular linear system ax b where a is a nxn rank deficient matrix and b a real vector of size n x a real vector let a and b be two positive real numbers when we consider the weighted frobenius norm aa bb p q f on the data we get the formula of condition number of the drazin inverse solution of singular linear system for the normwise condition number the sensitivity of the relative condition number itself is studied the componentwise perturbation is also investigated

relative perturbation bounds for the eigenvalues of diagonalizable and singular matrices application of perturbation theory for simple invariant subspaces perturbation bounds for the relative error in the eigenvalues of diagonalizable and singular matrices are derived by using perturbation theory for simple invariant subspaces of a matrix and the group inverse of a matrix these upper bounds are supplements to the related perturbation bounds for the eigenvalues of diagonalizable and nonsingular matrices

ray pattern of drazin inverses of some matrices abstract let q a be the set of complex matrices with the same ray pattern as a for any a q a if the drazin inverses of a and a have the same ray pattern then a is said to have ray drazin inverse in this paper we give some properties on matrices whose drazin inverses or m p inverses are ray unique and use these properties to obtain necessary and sufficient conditions for some block matrices to have ray drazin inverses

a characterization and representation of the generalized inverse a 2 t s and its applications abstract this paper presents an explicit expression for the generalized inverse a 2 t s based on this we established the characterization the representation theorem and the limiting process for a 2 t s as an application we estimate the error bound of the iterative method for approximating a 2 t s

uniqueness of positive radial solutions for quasilinear elliptic equations in an annulus abstract extending a previous result of tang 1 we prove the uniqueness of positive radial solutions of p u f u 0 subject to dirichlet boundary conditions on an annulus in r n with 2 p n under suitable hypotheses on the nonlinearity f this argument also provides an alternative proof for the uniqueness of positive solutions of the same problem in a finite ball see 9 in the complement of a ball or in the whole space r n see 10 3 11

the moore penrose inverse of matrices with an acyclic bipartite graph abstract the moore penrose inverse of a real matrix having no square submatrix with two or more diagonals is described in terms of bipartite graphs for such a matrix the sign of every entry of the moore penrose inverse is shown to be determined uniquely by the signs of the matrix entries i e the matrix has a signed generalized inverse necessary and sufficient conditions on an acyclic bipartite graph are given so that each nonnegative matrix with this graph has a nonnegative moore penrose inverse nearly reducible matrices are proved to contain no submatrix having two or more diagonals implying that a nearly reducible matrix has a signed generalized inverse furthermore it is proved that the term rank and rank are equal for each submatrix of a nearly reducible matrix

on the solution of singular linear systems of algebraic equations by semiiterative methods for a square matrixt n n where i t is possibly singular we investigate the solution of the linear fixed point problemx tx c by applying semiiterative methods sim s to the basic iterationx0 n xk tck 1 c k 1 such problems arise if one splits the coefficient matrix of a linear systemax b of algebraic equations according toa m n m nonsingular which leads tox m 1nx m 1b tx c even ifx tx c is consistent there are cases where the basic iteration fails to converge namely ift possesses eigenvalues 1 with 1 or if 1 is an eigenvalue oft with nonlinear elementary divisors in these cases and also ifx tx c is incompatible we derive necessary and sufficient conditions implying that a sim tends to a vector hat x which can be described in terms of the drazin inverse of i t we further give conditions under which hat x is a solution or a least squares solution of i t x c

conditional s matrices conditional s matrices and conditional s matrices abbreviated cs matrices and cs matrices were first introduced and studied in math comput modelling 17 1993 141 matrices of sign solvable linear systems cambridge university press 1995 they aregeneralizations of the well known s matrices and s matrices cs matrices and cs matrices play an important role in the study of conditionally sign solvable linear systems we study various properties and recognition criteria of cs matrices we also study several special classes of cs matrices such as square cs matrices barely cs matrices bcs matrices and maximal cs matrices

from boolean to sign pattern matrices abstract a nonnegative sign pattern matrix is a matrix whose entries are from the set 0 a nonnegative sign pattern matrix can also be viewed as a boolean matrix by replacing each entry with 1 in this paper some interesting connections between nonnegative sign pattern matrices and boolean matrices are investigated in particular the relations between the minimum rank and the boolean row or column rank are explored the idempotent boolean matrices that allow idempotence are identified and the nonnegative sign patterns that allow various types of nonnegative or positive generalized inverses are characterized

benchmark priors for bayesian model averaging in contrast to a posterior analysis given a particular sampling model posterior model probabilities in the context of model uncertainty are typically rather sensitive to the speciflcation of the prior in particular difiuse priors on model speciflc parameters can lead to quite unexpected consequences here we focus on the practically relevant situation where we need to entertain a large number of sampling models and we have or wish to use little or no subjective prior information we aim at providing an automatic or benchmark prior structure that can be used in such cases we focus on the normal linear regression model with uncertainty in the choice of re gressors we propose a partly noninformative prior structure related to a natural conjugate g prior speciflcation where the amount of subjective information requested from the user is limited to the choice of a single scalar hyperparameter g0j the consequences of difierent choices for g0j are examined we investigate theoretical properties such as consistency of the implied bayesian procedure links with classical information criteria are provided in addition we examine the flnite sample implications of several choices of g0j in a simulation study the use of the mc 3 algorithm of madigan and york 1995 combined with e cient coding in fortran makes it feasible to conduct large simulations in addition to posterior criteria we shall also compare the predictive performance of difierent priors a classic exam ple concerning the economics of crime will also be provided and contrasted with results in the literature the main flndings of the paper will lead us to propose a benchmark prior speciflcation in a linear regression context with model uncertainty

convergence properties of krylov subspace methods for singular linear systems with arbitrary index abstract krylov subspace methods have been recently considered to solve singular linear systems ax b in this paper we derive the necessary and sufficient conditions guaranteeing that a krylov subspace method converges to a vector adb px0 where ad is the drazin inverse of a and p is the projection p i a d a let k be the index of a we further show that a d b px 0 x 0 r a k 1 n a is a generalized least squares solution of ax b in r a k n a finally we present the convergence bounds for the quasi minimal residual algorithm qmr and transpose free quasi minimal residual algorithm tfqmr the index k of a in this paper can be arbitrary which extends to the main results of freund and hochbruck numer linear algebra appl 1 1994 403 420 that only considers the case k 1

sign pattern matrices and generalized inverses abstract if a is an n n sign pattern matrix then q a denotes the set of all real n n matrices b such that the sign of the entries in b match the corresponding entries in a in this paper we consider various requires allows questions connected with sign pattern matrices and generalized inverses in particular we investigate the class of all square patterns a for which there exist b c q a where bcb b for nonnegative patterns we characterize and show that coincides with the class of all square patterns a for which there exists b q a where b3 b nonnegative square patterns that allow an idempotent and those that allow a 1 3 inverse are each characterized some interesting open questions are also indicated

characterizations of matrices with nonzero signed row compound abstract a real square matrix a is called a sign nonsingular sns matrix if every matrix with the same sign pattern as a is not singular an m n matrix a with term rank m is called to have a nonzero signed row compound provided that each square submatrix of order m of a is an sns matrix or has an identically zero determinant as generalizations of sns matrices s matrices and totally l matrices matrices with nonzero signed row compound have a close relationship with matrices with signed null spaces which are applied to characterize linear systems with signed solutions in this paper matrices with nonzero signed row compound are characterized in terms of their signed bipartite graphs following these results characterizations of matrices with signed null spaces and full row term ranks in terms of their signed bipartite graphs are obtained too the recursive structure of m m 2 row sign balanced rsb maximal 0 1 1 matrices with nonzero signed row compound or with signed null spaces is also characterized

representations for the drazin inverse of 2 2 block operator matrix with singular schur complement abstract in this paper we present formulae for the drazin inverse of 2 2 block operator matrix with generalized schur complement being drazin invertible moreover necessary and sufficient conditions for the existence as well as the expressions for the group inverse are obtained

bayesian cart model search abstract in this article we put forward a bayesian approach for finding classification and regression tree cart models the two basic components of this approach consist of prior specification and stochastic search the basic idea is to have the prior induce a posterior distribution that will guide the stochastic search toward more promising cart models as the search proceeds such models can then be selected with a variety of criteria such as posterior probability marginal likelihood residual sum of squares or misclassification rates examples are used to illustrate the potential superiority of this approach over alternative methods

uniqueness of sign changing radial solutions for δu u u p 1u 0 in some ball and annulus abstract the following dirichlet problem u u u p 1 u 0 in u 0 on is considered where is either an annulus or a ball in r n and p 1 the uniqueness of radial solutions having exactly k 1 nodes is shown for the following cases is a sufficiently thin annulus is a certain small ball n 4 and 1 p n n 2 is the unit ball n 3 and 1 p 3 is any annulus or any ball but p 1 is sufficiently close to 1 and n 3 5 or 7

acute perturbation of the group inverse abstract for any n n complex matrix a let a g be the group inverse of a when a is singular a matrix b a e is said to be an acute perturbation of a if e is small and the spectral radius b g b a g a 1 the acute perturbation coincides with the stable perturbation of the group inverse if the matrix b satisfies condition r b n a 0 n b r a 0 which was introduced by castro gonzalez et al 2008 8 furthermore several examples are provided to illustrate the acute perturbation of the group inverse

orbital stability of standing waves for semilinear wave equations with indefinite energy the orbital stability of standing waves for semilinear wave equations is studied in the case that the energy is indefinite and the underlying space domain is bounded or a compact manifold or whole r n with n 2 the stability is determined by the convexity on of the lowest energy d of standing waves with frequency the arguments rely on the conservation of energy and charge and the construction of suitable invariant manifolds of solution flows

generalized inverses over integral domains in an earlier paper one of the authors showed that a matrix of rank r over an integral domain has a generalized inverse if and only if a linear combination of all the r r minors of the matrix is one in the same paper a procedure for constructing a generalized inverse from such a linear combination was also given in the present paper we show that any reflexive generalized inverse can be obtained by that procedure we also show that if the integral domain is a principal ideal domain every generalized inverse can be obtained by that procedure it is also shown that a matrix a of rank r over an integral domain has moore penrose inverse if and only if the sum of squares of all r r minors of a is an invertible element of the integral domain

the existence and uniqueness of bound state solutions of a semi linear equation we investigate the existence and uniqueness of bounded radially symmetric solutions in r3 of uf u 0 where f u is continuous piecewise linear and has three distinct real zeros we give a detail

uniqueness of positive solutions of a class of o d e with robin boundary conditions abstract we study the uniqueness of positive solutions of the boundary value problem u h t u f t u 0 t a b u a 0 u b 0 where 0 a b h c 0 r and f c 1 0 0 0 satisfy suitable conditions the proof of our main result is based upon the shooting method and the sturm comparison theorem

matrices with doubly signed generalized inverses abstract a real matrix a is said to have a signed generalized inverse if the sign pattern of its generalized inverse a is uniquely determined by the sign pattern of a a is said to have doubly signed generalized inverse if both a and a have a signed generalized inverse matrices with doubly signed generalized inverses are generalizations of those matrices a such that both a and a 1 are s 2 ns matrices which are studied in matrices of sign solvable linear systems cambridge university press cambridge 1995 linear algebra appl 232 1996 97 in this paper we give a complete characterization of matrices with doubly signed generalized inverses we show that if a is a real matrix with no zero rows and no zero columns then a has doubly signed generalized inverses if and only if a is permutation equivalent to a direct sum of matrices of the following types a row or a column with no zero entries fully indecomposable s 2 ns matrices of order 1 or 2 matrices of the blocked form d 1 o x d 2 where d 1 and d 2 are both invertible diagonal matrices

bayesian clustering and product partition models summary we present a decision theoretic formulation of product partition models ppms that allows a formal treatment of different decision problems such as estimation or hypothesis testing and clustering methods simultaneously a key observation in our construction is the fact that ppms can be formulated in the context of model selection the underlying partition structure in these models is closely related to that arising in connection with dirichlet processes this allows a straightforward adaptation of some computational strategies originally devised for nonparametric bayesian problems to our framework the resulting algorithms are more flexible than other competing alternatives that are used for problems involving ppms we propose an algorithm that yields bayes estimates of the quantities of interest and the groups of experimental units we explore the application of our methods to the detection of outliers in normal and student t regression models with clustering structure equivalent to that induced by a dirichlet process prior we also discuss the sensitivity of the results considering different prior distributions for the partitions

representations for the drazin inverses of 2 2 block matrices abstract let m denote a 2 2 block complex matrix a b c d where a and d are square matrices not necessarily with the same orders in this paper explicit representations for the drazin inverse of m are presented under the condition that bd i c 0 for i 0 1 n 1 where n is the order of d

number of nonzero entries of s2ns matrices and matrices with signed generalized inverses abstract a real matrix a has a signed generalized inverse or signed gi if the sign pattern of its generalized inverse a is uniquely determined by the sign pattern of a the notion of matrices having signed gi s is a generalization of the well known notion of strong sns matrices or s 2 ns matrices sharp bounds and characterization of equality for the number of nonzero entries of s 2 ns matrices of order n are given then sharp bounds and characterization of equality for the number of nonzero entries of m n matrices with signed gi s are given

a note on the drazin inverse of an anti triangular matrix abstract in this paper we give formulae for the generalized drazin inverse m d of an anti triangular matrix m under some conditions moreover some particular cases of these results are also considered

product partition models for change point problems product partition models assume that observations in different components of a random partition of the data are independent if the probability distribution of random partitions is in a certain product form prior to making the observations it is also in product form given the observations the product model thus provides a convenient machinery for allowing the data to weight the partitions likely to hold and inference about particular future observations may then be made by first conditioning on the partition and then averaging over all partitions these models apply with special computational simplicity to change point problems where the partitions divide the sequence of observations into components within which different regimes hold we show with appropriate selection of prior product models that the observations can eventually determine approximately the true partition