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

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

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

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

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

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

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

harmonic maps with defects two problems concerning maps with point singularities from a domain c 3 tos2 are solved the first is to determine the minimum energy of when the location and topological degree of the singularities are prescribed in the second problem is the unit ball and g is given on we show that the only cases in whichg x x minimizes the energy isg const org x rx withr a rotation extensions of these problems are also solved e g points are replaced by holes 3 s2 is replaced by n sn 1 or by n pn 1 the latter being appropriate for the theory of liquid crystals

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

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 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

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

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

density of smooth functions between two manifolds in sobolev spaces abstract we give some results concerning density or non density of smooth functions between two compact manifolds m n and n k in sobolev spaces w 1 p m n n k in particular we study the case n s k

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

partial regularity of a minimizer of the relaxed energy for biharmonic maps in this paper we study the relaxed energy for biharmonic maps from an m dimensional domain into spheres for an integer m 5 by an approximation method we prove the existence of a minimizer of the relaxed energy of the hessian energy and that the minimizer is biharmonic and smooth outside a singular set sigma of finite m 4 dimensional hausdorff measure when m 5 we prove that the singular set sigma is i rectifiable moreover we also prove a rectifiability result for the concentration set of a sequence of stationary harmonic maps into manifolds crown copyright c 2011 published by elsevier inc all rights reserved

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

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

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

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

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

punctual hilbert schemes and resolutions of multiple point singularities if f x yp is a morphism of smooth complex analytic varieties with n p then the multiple points of order k of f i n the target are those y e yp with k preimages each preimage counted with proper multiplicity it is clear that to understand the geometry o f f it is necessary to understand these multiple point sets and their closures much work has also been done recently to calculate characteristic class formulas for these sets and their relatives in the source off cf 1 10 17 one approach to the problem of calculating the characteristic class formula of a singularity involves finding resolutions of the singularities 4 15 18 traditionally in singularities of maps one uses multi jet bundles to study and control multiple point singularities jet bundles often enter also in the constructions made to resolve singularities unfortunately as will be discussed in detail later multi jet bundles are difficult to work with if not useless when the problem involves the closure of the set of multiple points in this paper we give a new construction using punctual hilbert schemes which we offer as an alternative to multi jets in the study of multiple point singularities as an illustration of its usefulness we use it to find a resolution of the closure of the triple point set of any good map f theorem 2 3 and of the multiple point set of a g o o d f o f any order provided the m a p f has kernel rank at most 2 theorem 1 6 recall that the kernel rank of f a t x is the dimension of the kernel of df x this construction also provides a useful starting point for finding resolutions of multiple point sets for general f by reducing the problem of resolving the singularity of f t o the problem of resolving the singularities of the corresponding hilbert schemes in a later paper we will show how a modification of our construction allows one to resolve triple points in the source as well

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

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

partial regularity for stationary harmonic maps into spheres in an interesting recent paper 12 f hi lein has shown that any weakly harmonic mapping from a two dimensional surface into a sphere is smooth i present here a kind of generalization to higher dimensions asserting in effect that a stationary harmonic mapping from an open subset of nn n 3 into a sphere is smooth except possibly for a closed singular set of n 2 dimensional hausdorff measure zero my proof crucially depends upon several of h l in s observations as streamlined by p l lions to state the result precisely let us suppose that m n 2 u is a smooth open subset of nn and s m 1 denotes the unit sphere in nm a function u in the sobolev space hi u rm u u 1 urn belongs to h i u s ml provided t u 1 a e in u

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 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

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

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

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

enumerating stationary multiple points given a map f x y an r fold point off is a point xi of x such that there exist points x2 x of x which all have the same image underf we say that an r fold point is strict if x x are all distinct points of x and stationary if some of the xi have coalesced to lie infinitely near the infinitely near points determine tangent directions along the iibref f x in the sense that they constitute ramification points of f this paper enlarges on the work of 9 to develop under appropriate hypotheses enumerative formulas for loci of arbitrary stationary r fold points in terms of invariants of the map multiple point theory has a long and rich history we may trace work back to clebsch 1864 who gave a formula for the number of nodes of a plane curve in terms of the degree and genus of the curve perhaps the most famous example of a stationary multiple point result is the riemann hurwitz formula let f x y be a finite separable surjective map between smooth projective curves the riemann hurwitz theorem gives a formula for deg r where r is the ramification divisor of j it may be easily seen that deg r is simply a weighted number of stationary doublepoints off modern multiple point theory is concerned with developing formulas for general classes of maps the subject has been of considerable interest in differential geometry and topology as well as in algebraic geometry surveys of the work in these fields can be found in lo 9 introduction or 8 chap v pp 365 3911 in algebraic geometry multiple point theory has presently culiminated in two basic approaches the hilbert scheme see lo for a survey and the method of iteration we shall only be concerned with the latter technique in the body of this work both approaches share a 149 oool 8708 87 7 50

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

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

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

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

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

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

axially symmetric harmonic maps minimizing a relaxed energy here we obtain various results on the class of axially symmetric harmonic maps from b3 to s2 we find some new classes of non minimizing harmonic maps exhibiting unusual singular behavior optimal partial regularity estimates are obtained for mappings which minimize among axially symmetric maps various relaxed energies which have been studied in 4 and 11

the resolution of the yang mills plateau problem in super critical dimensions abstract we study the minimization problem for the yang mills energy under fixed boundary connection in supercritical dimension n 5 we define the natural function space a g in which to formulate this problem in analogy to the space of integral currents used for the classical plateau problem the space a g can be also interpreted as a space of weak connections on a real measure theoretic version of reflexive sheaves from complex geometry we prove the existence of weak solutions to the yang mills plateau problem in the space a g we then prove the optimal regularity result for solutions of this plateau problem on the way to prove this result we establish a coulomb gauge extraction theorem for weak curvatures with small yang mills density this generalizes to the general framework of weak l 2 curvatures previous works of meyer riviere and tao tian in which respectively a strong approximability property and an admissibility property were assumed in addition

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

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

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

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 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

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

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

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

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

regularity of minimizers of relaxed problems for harmonic maps we prove that every minimizer on h1 s2 of the relaxed energy u 2 8 l u where 0 1 and l u is the length of a minimal connection connecting the singularities of u is smooth except at a finite number of points

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

asymptotic behavior of the regularized minimizer of an energy functional in higher dimensions abstract this paper is concerned with the asymptotic behavior of the regularized minimizer u e u e 1 u e 2 u e n 1 of an energy functional e e u g 1 n g u n d x 1 2 e n g u n 1 2 d x when e 0 where g r n is a bounded domain the author proves w 1 n convergence of minimizers to the map u n u n 0 where u n is an n harmonic map in addition the author also gives the relation between the zeros of u e 1 2 u e 2 2 u e n 2 and the singularities of u n qualitatively

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

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

boundary regularity and the dirichlet problem for harmonic maps in a previous paper 10 we developed an interior regularity theory for energy minimizing harmonic maps into riemannian manifolds in the first two sections of this paper we prove boundary regularity for energy minimizing maps with prescribed dirichlet boundary condition we show that such maps are regular in a full neighborhood of the boundary assuming appropriate regularity on the manifolds the boundary and the data the reader may refer to theorem 2 7 for a statement of the precise result it is not surprising that the boundary regularity is actually stronger than the partial regularity we obtained for the interior this is due to the fact that there are no nontrivial smooth harmonic maps from hemispheres s which map the boundary s 9s to a point i j n 2 and is analogous to the fact that in certain cases we were able to obtain complete regularity in the interior many authors have worked on boundary regularity for this general type of problem we mention hildebrandt and widman 5 and hamilton 4 as having obtained important results specifically for harmonic maps morrey had obtained the boundary regularity for domain dimension n 2 in conjunction with his investigation of the plateau problem in riemannian manifolds 8 in 3 of this paper we observe that the direct method gives solvability of the dirichlet problem under reasonable hypotheses on the manifolds we give as an application an amusing proof of a theorem of sacks and uhlenbeck 9 on the existence of minimal 2 spheres representing the second homotopy group of a manifold the same method gives smooth harmonic representations for k n for a certain class of manifolds n these are characterized by the nonexistence of lower dimensional harmonic spheres whose homogeneous extensions are minimal see proposition 3 4 in the last section of the paper we discuss approximation of l maps by smooth maps we give a simple example of an l map from the three dimensional ball to the two sphere which is not an l limit of continuous maps we