infinitely divisible nonnegative matrices m matrices and the embedding problem for finite state stationary markov chains abstract this paper explicitly details the relation between m matrices nonnegative roots of nonnegative matrices and the embedding problem for finite state stationary markov chains the set of nonsingular nonnegative matrices with arbitrary nonnegative roots is shown to be the closure of the set of matrices with matrix roots in im the methods presented here employ nothing beyond basic matrix analysis however it answers a question regarding m matrices posed over 30 years ago and as an application a new characterization of the set of all embeddable stochastic matrices is obtained as a corollary

graph theoretic methods for matrix completion problems abstract a pattern is a list of positions in an n n real matrix a matrix completion problem for the class of matrices asks whether every partial matrix whose specified entries are exactly the positions of the pattern can be completed to a matrix we survey the current state of research on matrix completion problems for many subclasses of p 0 matrices including positive definite matrices m matrices inverse m matrices p matrices and matrices defined by various sign symmetry and positivity conditions on p 0 and p matrices graph theoretic techniques used to study completion problems are discussed several new results are also presented including the solution to the m 0 matrix completion problem and the sign symmetric p 0 matrix completion problem

inverse no matrices abstract results similar to those of thomas l markham concerning inverse m matrices are shown to hold for inverse n o matrices a more general result and an alternate method of proof using schur complements are given for one of markham s theorems conditions are given to transform a nonpositive matrix to an n o amtrix by multiplication by householder transformations also an lu factorization of an inverse n o matrix is given

sign patterns of inverse nonnegative matrices abstract we determine here the 0 sign patterns which occur among the inverses of nonsingular entrywise nonnegative matrices these results complete a sequence of work which began with the characterization of all possible sign patterns which occur among inverses of positive matrices and the characterization in this general case is more involved to state than the earlier ones

closure properties of certain positivity classes of matrices under various algebraic operations abstract we present a table indicating whether or not each of five positivity classes of matrices positive definite hermitian matrices m matrices inverse m matrices totally positive matrices and inverse totally positive matrices is closed under each of seven algebraic operations conventional multiplication addition powers extraction of roots hadamard multiplication the hadamard product of one element and the inverse of another and lu factorization three of these 35 facts seem not to have been previously known and key examples and references are given

on the inverse of a special schur complement abstract the inverse of a schur complement is a very useful tool in many algorithms for the computation of the matrix inversion in this paper we study the inverse of a special schur complement cd 1 b we proved that always exist some x and y such that xdy is a inverse of cd 1 b furthermore using minimal rank properties we give some explicit expressions for x y and the inverse of cd 1 b numerical examples also given

monotone finite volume schemes for diffusion equations on unstructured triangular and shape regular polygonal meshes we consider a non linear finite volume fv scheme for stationary diffusion equation we prove that the scheme is monotone i e it preserves positivity of analytical solutions on arbitrary triangular meshes for strongly anisotropic and heterogeneous full tensor coefficients the scheme is extended to regular star shaped polygonal meshes and isotropic heterogeneous coefficients

renormalization in quantum field theory and the riemann hilbert problem ii the β function diffeomorphisms and the renormalization group we showed in part i that the hopf algebra of feynman graphs in a given qft is the algebra of coordinates on a complex infinite dimensional lie group g and that the renormalized theory is obtained from the unrenormalized one by evaluating at 0 the holomorphic part of the riemann hilbert decomposition 1 of the loop g provided by dimensional regularization we show in this paper that the group g acts naturally on the complex space x of dimensionless coupling constants of the theory more precisely the formula g 0 gz 1 z 3 3 2 for the effective coupling constant when viewed as a formal power series does define a hopf algebra homomorphism between the hopf algebra of coordinates on the group of formal diffeomorphisms to the hopf algebra this allows first of all to read off directly without using the group g the bare coupling constant and the renormalized one from the riemann hilbert decomposition of the unrenormalized effective coupling constant viewed as a loop of formal diffeomorphisms this shows that renormalization is intimately related with the theory of non linear complex bundles on the riemann sphere of the dimensional regularization parameter it also allows to lift both the renormalization group and the function as the asymptotic scaling in the group g this exploits the full power of the riemann hilbert decomposition together with the invariance of under a change of unit of mass this not only gives a conceptual proof of the existence of the renormalization group but also delivers a scattering formula in the group g for the full higher pole structure of minimal subtracted counterterms in terms of the residue

finite volume method for coupled subsurface flow problems i darcy problem abstract the article introduces a finite volume method for the darcy problem in heterogeneous anisotropic media the method is based on the mixed formulation for the pressure and its gradient the method is stable despite collocation of both pressure and its gradient at cell centers and demonstrates the first order convergence on numerous benchmarks as well as good monotonicity property the method produces quasi definite matrix which is numerically shown to have good asymptotics of the condition number our flux discretization method is a realization of our more general concept of stable flux discretization for saddle point systems with vector of several unknowns in this paper this vector is composed of pressure and its gradient and the saddle point system is the mixed formulation of the darcy problem

the symmetric m matrix and symmetric inverse m matrix completion problems abstract the symmetric m matrix and symmetric m 0 matrix completion problems are solved and results of johnson and smith linear algebra appl 290 1999 193 are extended to solve the symmetric inverse m matrix completion problem 1 a pattern i e a list of positions in an n n matrix has symmetric m completion i e every partial symmetric m matrix specifying the pattern can be completed to a symmetric m matrix if and only if the principal subpattern r determined by its diagonal is permutation similar to a pattern that is block diagonal with each diagonal block complete or in graph theoretic terms if and only if each component of the graph of r is a clique 2 a pattern has symmetric m 0 completion if and only if the pattern is permutation similar to a pattern that is block diagonal with each diagonal block either complete or omitting all diagonal positions or in graph theoretic terms if and only if every principal subpattern corresponding to a component of the graph of the pattern either omits all diagonal positions or includes all positions 3 a pattern has symmetric inverse m completion if and only if its graph is block clique and no diagonal position is omitted that corresponds to a vertex in a graph block of order 2 the techniques used are also applied to matrix completion problems for other classes of symmetric matrices

on inverse m matrices abstract we generalize a previous result of lewin and neumann and present a characterization of inverse m matrices the approach is mainly graph theoretical

on matrices satisfying a maximum principle with respect to a cone abstract a matrix equation a x y is considered in the space r n that is ordered by a cone k in case k r n it is known that a matrix a is said to satisfy the maximum principle if a is invertible and for each y r n 0 the solution x belongs to r n and is such that xi max1 k nxk and yi 0 for some i this concept is generalized to finitely generated and to circular cones k r n this is achieved by evaluating x and y with the aid of elements of a given base for the polar cone k of k the maximum principle is characterized geometrically by means of the behavior under a 1 of convex boundary parts of a base for k a weighted maximum principle is investigated and an infinite dimensional example is indicated

maximum principle in linear finite element approximations of anisotropic diffusion convection reaction problems a mesh condition is developed for linear finite element approximations of anisotropic diffusion convection reaction problems to satisfy a discrete maximum principle loosely speaking the condition requires that the mesh be simplicial and mathcal o vert varvec b vert infty h vert c vert infty h 2 o b h c h 2 nonobtuse when the dihedral angles are measured in the metric specified by the inverse of the diffusion matrix where h h denotes the mesh size and varvec b b and c c are the coefficients of the convection and reaction terms in two dimensions the condition can be replaced by a weaker mesh condition an mathcal o vert varvec b vert infty h vert c vert infty h 2 o b h c h 2 perturbation of a generalized delaunay condition these results include many existing mesh conditions as special cases numerical results are presented to verify the theoretical findings

a constrained finite element method satisfying the discrete maximum principle for anisotropic diffusion problems nonlinear constrained finite element approximations to anisotropic diffusion problems are considered starting with a standard linear or bilinear galerkin discretization the entries of the stiffness matrix are adjusted so as to enforce sufficient conditions of the discrete maximum principle dmp an algebraic splitting is employed to separate the contributions of negative and positive off diagonal coefficients which are associated with diffusive and antidiffusive numerical fluxes respectively in order to prevent the formation of spurious undershoots and overshoots a symmetric slope limiter is designed for the antidiffusive part the corresponding upper and lower bounds are defined using an estimate of the steepest gradient in terms of the maximum and minimum solution values at surrounding nodes the recovery of nodal gradients is performed by means of a lumped mass l 2 projection the proposed slope limiting strategy preserves the consistency of the underlying discrete problem and the structure of the stiffness matrix symmetry zero row and column sums a positivity preserving defect correction scheme is devised for the nonlinear algebraic system to be solved numerical results and a grid convergence study are presented for a number of anisotropic diffusion problems in two space dimensions

completions of inverse m matrix patterns abstract a list of positions in an n n real matrix that includes all diagonal positions a pattern is said to have inverse m completion if every partial inverse m matrix that specifies exactly these positions can be completed to an inverse m matrix johnson and smith c r johnson r l smith linear algebra and its applications 241 243 1996 655 667 characterize the positionally symmetric patterns that have inverse m completion as those patterns whose graphs are block clique in this paper the restriction of positional symmetry is removed a pattern has inverse m completion if and only if the digraph g of the pattern has the properties a the induced subdigraph of a cycle is a clique and b if g contains both arc i j and a path of length 1 between i and j then the induced subdigraph of the path is a clique furthermore any irreducible pattern with inverse m completion is positionally symmetric and has a block clique digraph any pattern is permutation similar to a block lower triangular pattern with irreducible diagonal blocks and such a pattern has inverse m completion if and only if i the pattern digraph of each diagonal block is block clique ii each subdiagonal block contains at most one position and iii the pattern digraph of the block structure has no alternate path to a single arc

sign preserving of principal eigenfunctions in p1 finite element approximation of eigenvalue problems of second order elliptic operators this paper is concerned with the p1 finite element approximation of the eigenvalue problem of second order elliptic operators subject to the dirichlet boundary condition the focus is on the preservation of the basic properties of the principal eigenvalue and eigenfunctions of the continuous problem it is shown that when the stiffness matrix is an irreducible m matrix the discrete eigenvalue problem maintains almost all of the basic properties such as the smallest eigenvalue being real and simple and the corresponding eigenfunctions being either positive or negative inside the physical domain mesh conditions leading to such a stiffness matrix are also studied a sufficient condition is that the mesh is simplicial interiorly connected and acute when measured in the metric specified by the inverse of the diffusion matrix the acute requirement can be replaced by the delaunay condition in two dimensions numerical results show that when the stiffness matrix is not an m matrix a finite element approximation can be structurally different from the continuous eigenvalue problem the eigenfunctions corresponding to the smallest eigenvalue can change sign over the physical domain and the smallest eigenvalue in modulus can even be complex for the case with nonsymmetric operators

global and local refinement techniques yielding nonobtuse tetrahedral partitions preservation of basic qualitative properties for example the maximum principle ofthe solution of partial differential equations by its finite element approximations is an important goal in mathematical modelling and simulation nonobtuse tetrahedral partitions and linear finite elements guarantee the validity of the discrete analogues of the maximum principle for a wide class of parabolic and elliptic problems in order to get more accurate approximation we often need to refine the used partitions globally or locally in this paper we first propose two variants of global refinement techniques which produce nonobtuse face to face tetrahedral partitions second we present a new local refinement technique which generates nonobtuse face to face tetrahedral partitions in a neighbourhood of a given vertex

on copositive matrices abstract a criterion for copositive matrices is given and for n 3 the set of all copositive matrices is determined in terms of matrix elements copositive matrices are applied to the problem of excluding periodic solutions of certain algebraic differential equations

some properties of schur complements and diagonal schur complements of diagonally dominant matrices in this paper we prove that the diagonal schur complement of a strictly doubly diagonally dominant matrix is strictly doubly diagonally dominant matrix the same holds for the diagonal schur complement of a strictly generalized doubly diagonally dominant matrix and a nonsingular h matrix we point out that under certain assumptions the diagonal schur complement of a strictly doubly doubly product diagonally dominant matrix is also strictly doubly doubly product diagonally dominant further we provide the distribution of the real parts of eigenvalues of a diagonal schur complement of h matrix we also show that the schur complement of a diagonally dominant matrix is not always diagonally dominant by a numerical example and then obtain a sufficient condition to ensure that the schur complement of a diagonally dominant matrix is diagonally dominant

looking for nonnegative solutions of a leontief dynamic model let a be an n n nonnegative real matrix we know that a is monotone if it has a nonnegative inverse moreover if a has an appropriate generalized inverse then we can obtain nonnegative solutions for the linear system ax b where b 0 in this paper using properties about nonnegatives matrices we present sufficient conditions in order that the leontief price model i f p k 1 1 r k fp k gp0 k 1 a0 where f i at 1bt and g i at 1 have nonnegative solutions

on a maximum principle for inverse monotone matrices abstract let 1 n be a given vector with positive coordinates a matrix a is said to satisfy the maximum principle mp if ax y y 0 imply x 0 and max 1 i n i i max i n y i i where n y is the set of indices such that y is positive for an invertible matrix a with positive inverse the mp is characterized geometrically by means of the behavior under a 1 of convex boundary parts of the simplex generated in r n by permissible multiples of the unit coordinate vectors some sufficient conditions and applications to m matrices are given

on schur complement of block diagonally dominant matrices abstract it is well known that the schur complements of strictly diagonally dominant matrices are strictly diagonally dominant d carlson t markham schur complements of diagonally dominant matrices czech math j 29 104 1979 246 251 the same is true of generalized strictly diagonally dominant matrices jianzhou liu yungqing huang some properties on schur complements of h matrix and diagonally dominant matrices linear algebra appl 389 2004 365 380 in this paper this result is extended to the block strictly diagonally dominant matrices and the generalized block strictly diagonally dominant matrices that is it is shown that the schur complement of a block strictly diagonally dominant matrix is a block strictly diagonally dominant matrix and so is the schur complement of a generalized block strictly diagonally dominant matrix

a lie theoretic approach to renormalization motivated by recent work of connes and marcolli based on the connes kreimer approach to renormalization we augment the latter by a combinatorial lie algebraic point of view our results rely both on the properties of the dynkin idempotent one of the fundamental lie idempotents in the theory of free lie algebras and on properties of hopf algebras encapsulated in the notion of associated descent algebras besides leading very directly to proofs of the main combinatorial aspects of the renormalization procedures the new techniques give rise to an algebraic approach to the galois theory of renormalization in particular they do not depend on the geometry underlying the case of dimensional regularization and the riemann hilbert correspondence this is illustrated with a discussion of the bphz renormalization scheme

inverse m matrix completions of patterns omitting some diagonal positions abstract let n 1 n which subsets q of n n have the property that whenever a ij for i j q form a partial inverse m matrix i e a ij 0 and if l l q then the inverse of a ij i j l is an m matrix a ij can be defined for i j q so that the inverse of a a ij is an m matrix in terms of a digraph g with vertex set n and arcs i j q with i j our results say 1 it is necessary that the induced subdigraph of any alternate path to an arc be a clique 2 a cycle need not be contained in a clique if i i q for at least one vertex i of 3 when n 4 it is necessary and sufficient that condition 1 is true and every simple cycle has a vertex i with i i q and 4 the general question can be reduced to the case in which g is strongly connected much as it has been when q contains i i for every i n

nonnegative matrices with power invariant zero patterns abstract if a is a nonsingular m matrix the elements of the sequence a k all have the same zero pattern using the drazin inverse we show that a similar zero pattern invariance property holds for a class of matrices which is larger than the generalized m matrices

on the inverse mean first passage matrix problem and the inverse m matrix problem abstract the inverse mean first passage time problem is given a positive matrix m r n n then when does there exist an n state discrete time homogeneous ergodic markov chain c whose mean first passage matrix is m the inverse m matrix problem is given a nonnegative matrix a then when is a an inverse of an m matrix the main thrust of this paper is to show that the existence of a solution to one of the problems can be characterized by the existence of a solution to the other in so doing we extend earlier results of tetali and fiedler

generalized ultrametric matrices a class of inverse m matrices abstract recently marti nez michon and san marti n introduced the new class of symmetric strictly ultrametric matrices they proved that the inverse of a strictly ultrametric matrix is a strictly row and strictly column diagonally dominant stieltjes matrix here we generalize their result by introducing a class of nonsymmetric matrices calledgeneralized ultrametric matrices we give a necessary and sufficient condition for the regularity of these matrices and prove that the inverse of a nonsingular generalized ultrametric matrix is a row and column diagonally dominant m matrix we establish that a nonnegative matrix is a generalized ultrametric matrix if and only if the matrix is a certain sum of at most rank two matrices moreover we give a characterization of generalized ultrametric matrices based on weighted trees the entries of generalized ultrametric matrices then arise as certain distances between the leaves and the root of the tree

characterizations of inverse m matrices with special zero patterns in this paper we provide some characterizations of inverse m matrices with special zero patterns in particular we give necessary and sufficient conditions for k diagonal matrices and symmetric k diagonal matrices to be inverse m matrices in addition results for triadic matrices tridiagonal matrices and symmetric 5 diagonal matrices are presented as corollaries

exponential renormalisation ii bogoliubov s r operation and momentum subtraction schemes this article aims at advancing the recently introduced exponential method for renormalisation in perturbative quantum field theory it is shown that this new procedure provides a meaningful recursive scheme in the context of the algebraic and group theoretical approach to renormalisation in particular we describe in detail a hopf algebraic formulation of bogoliubov s classical r operation and counterterm recursion in the context of momentum subtraction schemes this approach allows us to propose an algebraic classification of different subtraction schemes our results shed light on the peculiar algebraic role played by the degrees of taylor jet expansions especially the notion of minimal subtraction and oversubtractions

weakened acute type condition for tetrahedral triangulations and the discrete maximum principle we prove that a discrete maximum principle holds for continuous piecewise linear finite element approximations for the poisson equation with the dirichlet boundary condition also under a condition of the existence of some obtuse internal angles between faces of terahedra of triangulations of a given space domain this result represents a weakened form of the acute type condition for the three dimensional case

matrices with a transitive graph and inverse m matrices abstract the relationship between inverse m matrices and matrices whose graph is transitive is studied the results are applied to obtain a new proof of the characterization due to m lewin and m neumann of 0 1 inverse m matrices

monotonic solution of heterogeneous anisotropic diffusion problems anisotropic problems arise in various areas of science and engineering for example groundwater transport and petroleum reservoir simulations the pure diffusive anisotropic time dependent transport problem is solved on a finite number of nodes that are selected inside and on the boundary of the given domain along with possible internal boundaries connecting some of the nodes an unstructured triangular mesh that attains the generalized anisotropic delaunay condition for all the triangle sides is automatically generated by properly connecting all the nodes starting from an arbitrary initial one the control volume of each node is the closed polygon given by the union of the midpoint of each side with the anisotropic circumcentre of each final triangle a structure of the flux across the control volume sides similar to the standard galerkin finite element scheme is derived a special treatment of the flux computation mainly based on edge swaps of the initial mesh triangles is proposed in order to obtain a stiffness m matrix system that guarantees the monotonicity of the solution the proposed scheme is tested using several literature tests and the results are compared with analytical solutions as well as with the results of other algorithms in terms of convergence order computational costs are also investigated

criteria for sign regularity of sets of matrices abstract we consider matrix intervals with respect to a certain checkerboard partial ordering we show that every real matrix contained in a matrix interval is sign regular if two special matrices taken from that matrix interval are sign regular

inverse m matrices abstract this is an attempt at a comprehensive expository study of those nonnegative matrices which happen to be inverses of m matrices and is aimed at an audience conversant with basic ideas of matrix theory a theme is the parallels anddifferences between the class of m matrices and the class of inverse m matrices among the primary tools used are diagonal multiplications the neumann expansion and the form of the inverse of a partitioned matrix some of the results seem not to have appeared before and several unsolved problems are mentioned

hopf algebras renormalization and noncommutative geometry we explore the relation between the hopf algebra associated to the renormalization of qft and the hopf algebra associated to the ncg computations of tranverse index theory for foliations

on maximum principles for monotone matrices abstract for monotone matrices and h matrices maximum principles are defined and necessary and sufficient conditions are derived for these principles to hold

the cutoff method for the numerical computation of nonnegative solutions of parabolic pdes with application to anisotropic diffusion and lubrication type equations the cutoff method which cuts off the values of a function less than a given number is studied for the numerical computation of nonnegative solutions of parabolic partial differential equations a convergence analysis is given for a broad class of finite difference methods combined with cutoff for linear parabolic equations two applications are investigated linear anisotropic diffusion problems satisfying the setting of the convergence analysis and nonlinear lubrication type equations for which it is unclear if the convergence analysis applies the numerical results are shown to be consistent with the theory and in good agreement with existing results in the literature the convergence analysis and applications demonstrate that the cutoff method is an effective tool for use in the computation of nonnegative solutions cutoff can also be used with other discretization methods such as collocation finite volume finite element and spectral methods and for the computation of positive solutions

the schur complements of generalized doubly diagonally dominant matrices as is known the schur complements of diagonally dominant matrices are diagonally dominant the same is true of doubly diagonally dominant matrices the purpose of this paper is to extend the results to the generalized doubly diagonally dominant matrices a proper subset of h matrices that is we show that the schur complement of a generalized doubly diagonally dominant matrix is a generalized doubly diagonally dominant matrix

the quasi principal rank characteristic sequence abstract a minor of a matrix is quasi principal if it is a principal or an almost principal minor the quasi principal rank characteristic sequence qpr sequence of an n n symmetric matrix is introduced which is defined as q 1 q 2 q n where q k is a s or n according as all some but not all or none of its quasi principal minors of order k are nonzero this sequence extends the principal rank characteristic sequences in the literature which only depend on the principal minors of the matrix a necessary condition for the attainability of a qpr sequence is established using probabilistic techniques a complete characterization of the qpr sequences that are attainable by symmetric matrices over fields of characteristic 0 is given

inequalities for m matrices and inverse m matrices in this paper we establish some determinantal inequalities concerning m matrices and inverse m matrices the main results are as follows 1 if a aij is either an n n m matrix or inverse m matrix then for any permutation i1 i2 in of 1 2 n a deta i 1naii s 2n1 ai1i2 ais 1isaisi1 ai1i1ai2i2 aisis b deta i 1naii if and only if a is essentially triangular 2 if a aij is an n n m matrix b bij is an n n inverse m matrix a b denotes the hadamard product of a and b then a b is an m matrix and for any permutation i1 i2 in of 1 2 n det a b det ab s 2naisisdeta i1 i2 is 1 deta i1 is 1 is bisisdetb i1 i2 is 1 detb i1 is 1 is 1

the dominant degree and disc theorem for the schur complement of matrix the theory of schur complement plays an important role in many fields such as control theory and computational mathematics in this paper we obtain some estimates for the diagonally c diagonally and product c diagonally dominant degree on the schur complement of matrices which improve some relative results as application we present some bounds for the eigenvalues of schur complement by the entries of the original matrix instead of those of the schur complement particularly we obtain that the eigenvalues of the schur complements are located in the gerschgorin circles of the original matrices under certain conditions

on monotonicity of difference schemes for computational physics criteria are developed for monotonicity of linear as well as nonlinear difference schemes associated with the numerical analysis of systems of partial differential equations integrodifferential equations etc difference schemes are converted into variational forms that satisfy the boundary maximum principle and also allow the investigation of monotonicity for nonlinear operators using linear patterns sufficient conditions are provided to review the monotonicity of single and coupled difference schemes necessary as well as necessary and sufficient conditions for monotonicity of explicit schemes are also developed the notion of submonotone difference schemes is considered and the associated criteria are developed we discuss the interrelationship between monotonicity submonotonicity and stability some known schemes serve as examples demonstrating the implementation of the developed approaches among these examples we describe the possibility that stable schemes such as total variation diminishing tvd as well as monotonicity preserving can produce spurious oscillations

on perron complements of inverse n0 matrices abstract in this paper we show that the perron complements of irreducible n 0 matrices are n 0 matrices we also demonstrate the perron complements of irreducible inverse n 0 matrices are inverse n 0 matrices with a little restriction in addition we give some related inequalities about inverse n 0 matrices

responses of a hodgkin huxley neuron to various types of spike train inputs numerical investigations have been made of responses of a hodgkin huxley hh neuron to spike train inputs whose interspike interval isi is modulated by deterministic semi deterministic chaotic and stochastic signals as deterministic one we adopt inputs with the time independent isi and with time dependent isi modulated by sinusoidal signal the rossler and lorentz models are adopted for chaotic modulations of isi stochastic isi inputs with the gamma distribution are employed it is shown that distribution of output isi data depends not only on the mean of isis of spike train inputs but also on their fluctuations the distinction of responses to the three kinds of inputs can be made by return maps of input and output isis but not by their histograms the relation between the variations of input and output isis is shown to be different from that of the integrate and fire if model because of the refractory period in the hh neuron

principal pivot transforms properties and applications the principal pivot transform ppt is a transformation of the matrix of a linear system tantamount to exchanging unknowns with the corresponding entries of the right hand side of the system the notion of the ppt is encountered in mathematical programming statistics and numerical analysis among other areas the purpose of this paper is to draw attention to the main properties and uses of ppts make some new observations and motivate further applications of ppts in matrix theory special consideration is given to ppts of matrices whose principal minors are positive 2000 elsevier science inc all rights reserved

dynamical mean field theory of noisy spiking neuron ensembles application to the hodgkin huxley model a dynamical mean field approximation dma previously proposed by the present author h hasegawa phys rev e 67 041903 2003 has been extended to ensembles described by a general noisy spiking neuron model ensembles of n unit neurons each of which is expressed by coupled k dimensional differential equations des are assumed to be subject to spatially correlated white noises the original mathrm kn dimensional stochastic des have been replaced by k k 2 dimensional deterministic des expressed in terms of means and the second order moments of local and global variables the fourth order contributions are taken into account by the gaussian decoupling approximation our dma has been applied to an ensemble of hodgkin huxley hh neurons k 4 for which effects of the noise the coupling strength and the ensemble size on the response to a single spike input have been investigated numerical results calculated by the dma theory are in good agreement with those obtained by direct simulations although the former computation is about a thousand times faster than the latter for a typical hh neuron ensemble with n 100

notes on completely positive matrices abstract let a be a n n symmetric matrix and in the closure of inverse m matrices then a can be factored as a bb t for some nonnegative lower triangular n n matrix b and cp rank a n if a is a positive semidefinite 0 1 matrix then a is completely positive and cp rank a rank a if a is a nonnegative symmetric h matrix then a is completely positive and cp rank a n n 1 2 n n where is the number of connected components of the graph g a

dynamical mean field theory of spiking neuron ensembles response to a single spike with independent noises a semianalytical dynamical mean field theory has been developed for a study of dynamics of an ensemble of n unit fitzhugh nagumo neurons subject to white noises assuming weak noises and the gaussian distribution of state variables we have driven equations of motions for moments of local and global variables dynamical mean field approximation dma has replaced original 2n dimensional stochastic differential equations des by eight dimensional deterministic des whereas the conventional moment method yields n 2n 3 dimensional deterministic des for local variables we have discussed the dependence of the spike firing precision and the synchronization on the noise intensity synaptic coupling and the size of the neuron ensemble the spike timing precision is shown to be improved by increasing the size of the neuron ensemble even when there are no couplings among neurons when the coupling is introduced neurons in ensembles respond to an input spike with a partial synchronization results calculated by our dma theory are in good agreement with those obtained by direct simulations dma theory is extended to a large cluster which can be divided into multiple subclusters according to their functions a model calculation has demonstrated that when the noise intensity is moderate the spike propagation with a fairly precise timing is possible among noisy subclusters with feed forward couplings as in the synfire chain we have compared dma theory with the conventional moment method showing that the former may be alternatively derived from the latter by a reduction in the number of moments with the mean field approximation

on maximum principles for m operators abstract for m matrices a condition to satisfy the maximum principle for inverse column entries is known we generalize this result concerning a more general maximum principle for m operators on r n ordered by some cone as well as to a certain extent for m operators on infinite dimensional ordered normed spaces

geometric aspects of the symmetric inverse m matrix problem abstract we investigate the symmetric inverse m matrix problem from a geometric perspective the central question in this geometric context is which conditions on the k dimensional facets of an n simplex s guarantee that s has no obtuse dihedral angles the simplest of such conditions is that if all triangular facets of s are equilateral then s is regular and thus nonobtuse first we study the properties of an n simplex s whose k facets are all nonobtuse and generalize some classical results by fiedler 14 we prove that if all n 1 facets of an n simplex s are nonobtuse each makes at most one obtuse dihedral angle with another facet this helps to identify a special type of tetrahedron which we will call sub orthocentric with the property that if all tetrahedral facets of s are sub orthocentric then s is nonobtuse rephrased in the language of linear algebra this constitutes a purely geometric proof of the fact that each symmetric ultrametric matrix 28 is the inverse of a weakly diagonally dominant m matrix the geometric proof provides valuable insights that supplement the discrete measure theoretic 24 and the linear algebraic 25 proof the review papers 20 21 support our believe that the linear algebraic perspective on the inverse m matrix problem dominates the literature the geometric perspective however connects sign properties of entries of inverses of a symmetric positive definite matrix to the dihedral angle properties of an underlying simplex and enables an explicit visualization of how these angles and signs can be manipulated this will serve to formulate purely geometric conditions on the k facets of an n simplex s that may render s nonobtuse also for k 3 for this we generalize the class of sub orthocentric tetrahedra that gives rise to the class of ultrametric matrices to sub orthocentric simplices that define symmetric positive definite matrices a with special types of k k principal submatrices for k 3 each sub orthocentric simplices is nonobtuse and we conjecture that any simplex with sub orthocentric facets only is sub orthocentric itself along the way several additional new concepts will be introduced such as vertex gramians simplicial matrix classes the dual hull and the sub orthocentric set of a nonobtuse simplex and nonblocking matrices these concepts may also be of use in a different linear algebraic setting in particular in the context if completely positive matrices for which we will prove some auxiliary results as well

positive path product and inverse m matrices it is known that an inverse m matrix is strict path product but not necessarily vice versa for n 3 it is shown that any square positive matrix may be made strict path product by predictable additions to the diagonal and that any normalized strict path product matrix may be made inverse m by additions to the diagonal that are bounded in terms of the size of the matrix the latter has implications for pairs of inverse m matrices whose hadamard product is inverse m a determinantal inequality relating principal minors and certain associated almost principal minors is derived for normalized inverse m matrices

copositive matrices and definiteness of quadratic forms subject to homogeneous linear inequality constraints abstract a symmetric matrix c is called copositive if the quadratic form x cx is nonnegative for all nonnegative values of the variables x 1 x 2 x n x a known sufficient condition for a quadratic form x qx to be positive unless x 0 subject to the linear inequality constraints ax 0 is that there should exist a copositive matrix c such that q a ca is positive definite the main result of this paper establishes the necessity of this condition for x qx to be merely nonnegative subject to ax 0 the situation is less straightforward the necessity of the existence of a copositive matrix c such that q a ca is positive semidefinite is proved only under various additional hypotheses regarding the size or rank of a and counterexamples are given to show that in general no such matrix may exist even when slater s constraint qualification holds our approach to these existence questions also furnishes certain tests for positivity or mere nonnegativity of x qx subject to ax 0 in which specific symmetric matrices constructed by rational operations from a and q and depending upon a single real parameter v must be tested for positive definiteness or strict copositivity for large values of v this technique is illustrated by several examples

a polynomial time test for m matrices abstract a real square matrix with nonpositive off diagonal elements is a z matrix and a z matrix with nonnegative principal minors is an m matrix using principal pivoting we derive a polynomial time algorithm for testing whether a z matrix is an m matrix our algorithm is much simpler than a recent one proposed by ramamurthy we also study the zero nonzero structure of the inverse of a nonsingular m matrix and compare a 1 and b 1 in the case that a and b are nonsingular m matrices and b a

stochastic resonance of ensemble neurons for transient spike trains wavelet analysis by using the wavelet transformation wt i have analyzed the response of an ensemble of n 1 10 100 and 500 hodgkin huxley neurons to transient m pulse spike trains m i to 3 with independent gaussian noises thecross correlation between the input and output signals is expressed in terms of the wt expansion coefficients the signal to noise ratio snr is evaluated by using the denoising method within the wt by which the noise contribution is extracted from the output signals although the response of a single n 1 neuron to subthreshold transient signals with noises is quite unreliable the transmission fidelity assessed by the cross correlation and snr is shown to be much improved by increasing the value of n a population of neurons plays an indispensable role in the stochastic resonance sr for transient spike inputs it is also shown that in a large scale ensemble the transmission fidelity for suprathreshold transient spikes is not significantly degraded by a weak noise which is responsible to sr for subthreshold inputs

some results on structured m matrices with an application to wireless communications in this paper we study conditions under which a specially structured z matrix is an m matrix we apply the result to a capacity problem in wireless communications

theorems on schur complement of block diagonally dominant matrices and their application in reducing the order for the solution of large scale linear systems abstract we firstly consider the block dominant degree for i ii block strictly diagonally dominant matrix and their schur complements showing that the block dominant degree for the schur complement of an i ii block strictly diagonally dominant matrix is greater than that of the original grand block matrix then as application we present some disc theorems and some bounds for the eigenvalues of the schur complement by the elements of the original matrix further by means of matrix partition and the schur complement of block matrix based on the derived disc theorems we give a kind of iteration called the schur based iteration which can solve large scale linear systems though reducing the order by the schur complement and the numerical example illustrates that the iteration can compute out the results faster

nonstationary stochastic resonance it is by now established that remarkably the addition of noise to a nonlinear system may sometimes facilitate rather than hamper the detection of weak signals this phenomenon usually referred to as stochastic resonance was originally associated with strictly periodic signals but it was eventually shown to occur for stationary aperiodic signals as well however in several situations of practical interest the signal can be markedly nonstationary we demonstrate that the phenomenon of stochastic resonance extends to nonstationary signals as well and thus could be relevant to a wider class of biological and electronic applications building on both nondynamic and aperiodic stochastic resonance our scheme is based on a multilevel trigger mechanism which could be realized as a parallel network of differentiated threshold sensors we find that optimal detection is reached for a number of thresholds of order 10 and that little is gained by going much beyond that number we raise the question of whether this is related to the fact that evolution has favored some fixed numbers of precisely this order of magnitude in certain aspects of sensory perception

notes on f0 matrices abstract the author studies f 0 matrices and shows that n n matrix a f 0 if and only if a ti b with b 0 and n 2 b t n 1 b where k b denotes the maximum of the spectral radii of all k principal submatrices of b also an inverse f 0 matrix problem is considered

wavelet entropy of stochastic processes abstract we compare two different definitions for the wavelet entropy associated to stochastic processes the first one the normalized total wavelet entropy ntws family s blanco a figliola r q quiroga o a rosso e serrano time frequency analysis of electroencephalogram series iii wavelet packets and information cost function phys rev e 57 1998 932 940 o a rosso s blanco j yordanova v kolev a figliola m schurmann e basar wavelet entropy a new tool for analysis of short duration brain electrical signals j neurosci method 105 2001 65 75 and a second introduced by tavares and lucena physica a 357 1 2005 71 78 in order to understand their advantages and disadvantages exact results obtained for fractional gaussian noise 1 1 and fractional brownian motion 1 3 are assessed we find out that the ntws family performs better as a characterization method for these stochastic processes

stochastic resonance in neuron models periodically stimulated sensory neurons typically exhibit a kind of statistical phase locking to the stimulus they tend to fire at a preferred phase of the stimulus cycle but not at every cycle hence the histogram of interspike intervals isih i e of times between successive firings is multimodal for these neurons with peaks centered at integer multiples of the driving period a particular kind of residence time histogram for a large class of noisy bistable systems has recently been shown to exhibit the major features of the neural data in the present paper we show that an excitable cell model the fitzhugh nagumo equations also reproduces these features when driven by additive periodic and stochastic forces this model exhibits its own brand of stochastic resonance as the peaks of the isih successively go through a maximum when the noise intensity is increased further the presence of a noise induced limit cycle introduces a third time scale in the problem this limit cycle is found to modify qualitatively the phase locking picture e g by suppressing certain peaks in the isih finally the role of noise and possibly of stochastic resonance sr in the neural encoding of sensory information is discussed

on classes of inverse z matrices abstract recently a classification of matrices of class z was introduced by fiedler and markham this classification contains the classes of m matrices and the classes of n 0 and f 0 matrices studied by fan g johnson and smith the problem of determining which nonsingular matrices have inverses which are z matrices is called the inverse z matrix problem for special classes of z matrices such as the m and n 0 matrices there exist at least partial results i e special classes of matrices have been introduced for which the inverse of such a matrix is an m matrix or an n 0 matrix here we define a system of classes of matrices for which the inverse of each matrix of each class belongs to one class of the classification of z matrices defined by fiedler and markham moreover certain properties of the matrices of each class are established e g inequalities for the sum of the entries of the inverse and the structure of certain schur complements we also give a necessary and sufficient condition for regularity the class of inverse n 0 matrices given here generalizes the class of inverse n 0 matrices discussed by johnson all results established here can be applied to a class of distance matrices which corresponds to a nonarchimedean metric this metric arises in p adic number theory and in taxonomy

renormalization in quantum field theory and the riemann hilbert problem i the hopf algebra structure of graphs and the main theorem this paper gives a complete selfcontained proof of our result announced in 6 showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the riemann hilbert problem we shall first show that for any quantum field theory the combinatorics of feynman graphs gives rise to a hopf algebra which is commutative as an algebra it is the dual hopf algebra of the enveloping algebra of a lie algebra whose basis is labelled by the one particle irreducible feynman graphs the lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa the corresponding lie group g is the group of characters of we shall then show that using dimensional regularization the bare unrenormalized theory gives rise to a loop where c is a small circle of complex dimensions around the integer dimension d of space time our main result is that the renormalized theory is just the evaluation at z d of the holomorphic part of the birkhoff decomposition of we begin to analyse the group g and show that it is a semi direct product of an easily understood abelian group by a highly non trivial group closely tied up with groups of diffeomorphisms the analysis of this latter group as well as the interpretation of the renormalization group and of anomalous dimensions are the content of our second paper with the same overall title

the linear order complementarity problem the classical complementarity problem in euclidean space can be viewed alternatively as a variational inequality or as a lattice orthogonality problem generalizations of the former have been extensively studied but infinite dimensional analogues of the latter have been largely ignored moreover as we show many well known results about the classical complementarity problem are more appropriately viewed order theoretically this is particularly true of least element solutions which are central to the present study of order complementarity in vector lattices we emphasize that the lattice theoretic descriptions we employ are very useful even in the standard finite dimensional setting

tridiagonal and upper triangular inverse m matrices abstract in this paper we characterize the nonnegative nonsingular tridiagonal matrices belonging to the class of inverse m matrices we give a geometric equivalence for a nonnegative nonsingular upper triangular matrix to be in this class this equivalence is extended to include some reducible matrices

inverse h matrices abstract the notion of inverse m matrices has been quite well studied and many of their properties have been documented in the literature in this article a proposal to consider the larger class of inverse h matrices is made first certain special types of real matrices are identified as inverse h matrices while these matrices are not inverse m matrices in general their definitions are inspired by matrices that are known to be inverse m matrices in the rest of the article motivated by results that hold for inverse m matrices the endeavor will be to attempt at proving corresponding results for inverse h matrices it is shown that for the full class of inverse h matrices many of these extensions are false nevertheless two subclasses of real inverse h matrices are identified where these generalizations are shown to hold

on diagonal schur complements of block diagonally dominant matrices it is known that the diagonal schur complements of strictly diagonally dominant matrices are strictly diagonally dominant matrices j z liu y q huang some properties on schur complements of h matrices and diagonally dominant matrices linear algebra appl 389 2004 365 380 and the same is true for nonsingular h matrices j z liu j c li z t huang x kong some properties of schur complements and diagonal schur complements of diagonally dominant matrices linear algebra appl 428 2008 1009 1030 in this paper we research the properties on diagonal schur complements of block diagonally dominant matrices and prove that the diagonal schur complements of block strictly diagonally dominant matrices are block strictly diagonally dominant matrices and the same holds for generalized block strictly diagonally dominant matrices

monotonicity and positivity of coefficients of power series expansions associated with newton and halley methods for the matrix pth root abstract if a is a matrix with all eigenvalues in the disk z 1 1 the principal pth root of a can be computed by newton s method or halley s method the study of newton s method and halley s method for the matrix pth root can be done through a study of power series expansions of some sequences of scalar functions in this paper we prove monotonicity results for the coefficients in the power series expansions for both newton s method and halley s method and for all integer p 2 the sign patterns of these coefficients can be seen directly from the monotonicity results we then use these monotonicity results and their consequences to obtain some new error estimates in the matrix case to obtain a monotonic convergence result when a is a nonsingular m matrix and to obtain a structure preserving result when a is a nonsingular m matrix or a real nonsingular h matrix with positive diagonal entries

matrices and the linear complementarity problem the paper is a collection of results on the linear complementarity problem q m the results are stated in terms of the matrix m

on the hadamard product of inverse m matrices abstract we investigate the hadamard product of inverse m matrices and present two classes of inverse m matrices that are closed under the hadamard multiplication in the end we give some inequalities on the fan product of m matrices and schur complements

wavelet analysis of discharge dynamics of fusimotor neurons abstract we study the interspike intervals isi time series of the spontaneous fusimotor neuron activity by applying the wavelet transform analysis and confirm the existence of the white noise characteristics of the isi time series this means that the neuron activity may serve as the requisite noisy component for occurrence of the stochastic resonance mechanism in the neural coordination of muscle spindles besides we apply the multifractal formalism adapted for the wavelet transform time series analysis thus we have established the multifractality of the isi data and achieved an additional insight into fusimotor discharge dynamics

z matrices and inverse z matrices abstract we consider z matrices and inverse z matrices i e those nonsingular matrices whose inverses are z matrices recently fiedler and markham introduced a classification of z matrices this classification directly leads to a classification of inverse z matrices among all classes of z matrices and inverse z matrices the classes of m matrices n 0 matrices f 0 matrices and inverse m matrices inverse n 0 matrices and inverse f 0 matrices respectively have been studied in detail here we discuss each single class of z matrices and inverse z matrices as well as considering the whole classes of z matrices and inverse z matrices we establish some common properties of the classes such as eigenvalue bounds and determinant inequalities and we give a new characterization of some classes of z matrices and inverse z matrices moreover we prove that other classes besides those of m matrices n 0 matrices and f 0 matrices consist of matrices whose determinants have the same sign some of the results generalize known results for m matrices n 0 matrices and f 0 matrices and for inverse m matrices inverse n 0 matrices and inverse f 0 matrices however we also show that some properties of the specific classes mentioned above do not hold for all classes of z matrices and inverse z matrices

matrix completion problems for pairs of related classes of matrices abstract for a class x of real matrices a list of positions in an n n matrix a pattern is said to have x completion if every partial x matrix that specifies exactly these positions can be completed to an x matrix if x and x0 are classes that satisfy the conditions any partial x matrix is a partial x0 matrix for any x0 matrix a and e 0 a ei is a x matrix and for any partial x matrix a there exists 0 such that a i is a partial x matrix where is the partial identity matrix specifying the same pattern as a then any pattern that has x0 completion must also have x completion however there are usually patterns that have x completion that fail to have x0 completion this result applies to many pairs of subclasses of p and p0 matrices defined by the same restriction on entries including the classes p p0 matrices weakly sign symmetric p p0 matrices and non negative p p0 matrices it also applies to other related pairs of subclasses of p0 matrices such as the pairs classes of p p0 1 matrices weakly sign symmetric p p0 1 matrices and non negative p p0 1 matrices furthermore any pattern that has weakly sign symmetric sign symmetric non negative p0 completion must also have weakly sign symmetric sign symmetric non negative p0 1 completion although these pairs of classes do not satisfy condition 3 similarly the class of inverse m matrices and its topological closure do not satisfy condition 3 but the conclusion remains true and the matrix completion problem for the topological closure of the class of inverse m matrices is solved for patterns containing the diagonal

information transmission in parallel threshold arrays suprathreshold stochastic resonance the information transmitted through a parallel summing array of noisy threshold elements with a common threshold is considered in particular using theoretical and numerical analysis a recently reported n g stocks phys rev lett 84 2310 2000 form of stochastic resonance termed suprathreshold stochastic resonance ssr is studied in detail ssr is observed to occur in arrays with two or more elements and unlike stochastic resonance sr in a single element gives rise to noise induced information gains that occur independent of the setting of the threshold or the size of the signal the transmitted information is maximized when all thresholds are set to coincide with the signal mean in this situation and for large arrays the noise can enhance performance up to approximately half the theoretical noiseless channel capacity the theory is tested against digital simulation

theoretical analysis of array enhanced stochastic resonance in the diffusively coupled fitzhugh nagumo equation the array enhanced stochastic resonance aesr in the diffusively coupled fitzhugh nagumo equation is investigated the two properties of aesr namely the scaling of the optimal noise intensity and the enhancement of the maximum value of the correlation coefficient as a function of the coupling strength are analyzed theoretically by transforming the dynamics of n elements into that of the mean and the deviation from it it is found that aesr is caused by the correlation between them a low dimensional model that reproduces the above properties is constructed

almost principal minors of inverse m matrices abstract it is well known that if an inverse m matrix has a 0 entry then it must be reducible and thus have many more 0 entries this property is actually a special case of a deeper phenomenon that might be loosely described as relations among vanishing almost principal minors in an inverse m matrix this phenomenon encompasses both minors of nested dimension a certain loose monotonicity and minors of the same size in loosely related positions this phenomenon is limited to almost principal minors and where possible converses and examples are given to show the limit of the extent of this phenomenon it is also shown that if one almost principle minor is contained in another then the magnitude of the former is larger than that of the latter

an anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems heterogeneous anisotropic diffusion problems arise in the various areas of science and engineering including plasma physics petroleum engineering and image processing standard numerical methods can produce spurious oscillations when they are used to solve those problems a common approach to avoid this difficulty is to design a proper numerical scheme and or a proper mesh so that the numerical solution validates the discrete counterpart dmp of the maximum principle satisfied by the continuous solution a well known mesh condition for the dmp satisfaction by the linear finite element solution of isotropic diffusion problems is the non obtuse angle condition that requires the dihedral angles of mesh elements to be non obtuse in this paper a generalization of the condition the so called anisotropic non obtuse angle condition is developed for the finite element solution of heterogeneous anisotropic diffusion problems the new condition is essentially the same as the existing one except that the dihedral angles are now measured in a metric depending on the diffusion matrix of the underlying problem several variants of the new condition are obtained based on one of them two metric tensors for use in anisotropic mesh generation are developed to account for dmp satisfaction and the combination of dmp satisfaction and mesh adaptivity numerical examples are given to demonstrate the features of the linear finite element method for anisotropic meshes generated with the metric tensors

aperiodic stochastic resonance stochastic resonance sr is a phenomenon wherein the response of a nonlinear system to a weak periodic input signal is optimized by the presence of a particular level of noise recently we presented a method and theory for characterizing sr type behavior in excitable systems with aperiodic i e broadband input signals phys rev e 52 r3321 1995 we coined the term aperiodic stochastic resonance asr to describe this general type of behavior in that earlier study we demonstrated asr in the fitzhugh nagumo neuronal model here we demonstrate asr in three additional systems a bistable well system an integrate and fire neuronal model and the hodgkin huxley hh neuronal model we present computational and theoretical results for each system in the context of the hh model we develop a general theory for asr in excitable membranes this work clearly shows that sr type behavior is not limited to systems with periodic inputs thus in general noise can serve to enhance the response of a nonlinear system to a weak input signal regardless of whether the signal is periodic or aperiodic textcopyright 1996 the american physical society

on the spectra of matrices having nonnegative sums of principal minors abstract the localization of the eigenvalues of matrices with nonnegative sums of principal minors is studied the discussion is carried out in particular for p matrices such matrices for which almost all the eigenvalues lie on the left half plane are constructed