a graph theoretic model of symmetric givens operations and its implications abstract symmetric givens operations a gag t g a givens rotation matrix are a basic tool in many matrix computations especially for eigenvalue eigenvector computations a graph theoretic model of these operations is given for symmetric matrices analogous to the graph theoretic model of cholesky factorization using this model it is shown that unless there is accidental cancellation it is impossible to reduce a range of different matrix classes to tridiagonal form in o n 2 givens operations these classes include arrowhead matrices pentadiagonal matrices and cyclic tridiagonal matrices

on the spectra of some weighted rooted trees and applications abstract let t be a weighted rooted tree of k levels such that 1 the vertices in level j have a degree equal to d k j 1 for j 1 2 k and 2 the edges joining the vertices in level j with the vertices in level j 1 have a weight equal to w k j for j 1 2 k 1 we give a complete characterization of the eigenvalues of the laplacian matrix and adjacency matrix of t they are the eigenvalues of leading principal submatrices of two nonnegative symmetric tridiagonal matrices of order k k moreover we give some results concerning their multiplicities by application of the above mentioned results we derive upper bounds on the largest eigenvalue of any weighted tree and the spectra of some weighted bethe trees

on the spectral radius of unicyclic graphs with prescribed degree sequence we consider the set of unicyclic graphs with prescribed degree sequence in this set we determine the unique graph with the largest spectral radius or index with respect to the adjacency matrix in addition we give a conjecture about the unique graph with the largest index in the set of connected graphs with prescribed degree sequence

factoring matrices with a tree structured sparsity pattern abstract let a be a matrix whose sparsity pattern is a tree with maximal degree d max we show that if the columns of a are ordered using minimum degree on a a then factoring a using a sparse lu with partial pivoting algorithm generates only o d max n fill requires only o d max n operations and is much more stable than lu with partial pivoting on a general matrix we also propose an even more efficient and just as stable algorithm called sibling dominant pivoting this algorithm is a strict partial pivoting algorithm that modifies the column preordering locally to minimize fill and work it leads to only o n work and fill more conventional column pre ordering methods that are based usually implicitly on the sparsity pattern of a a are not as efficient as the approaches that we propose in this paper

on the characteristic and laplacian polynomials of trees abstract we find the characteristic polynomials of adjacency and laplacian matrices of arbitrary unweighted rooted trees in term of vertex degrees using the concept of the rooted product of graphs our result generalizes a result of rojo and soto o rojo r soto the spectra of the adjacency matrix and laplacian matrix for some balanced trees linear algebra appl 403 2005 97 117 on a special class of rooted unweighted trees namely the trees such that their vertices in the same level have equal degrees

a note on the multiplicities of graph eigenvalues abstract let g be a graph with vertex set 1 n and let h be the graph obtained by attaching one pendant path of length k i at vertex i i 1 r 1 r n for a real symmetric matrix a whose graph is h let m a denote the multiplicity of an eigenvalue of a from a result in da fonseca 2005 7 we know that m a n in this note we characterize the case m a n we also give two upper bounds on eigenvalue multiplicity of trees and unicyclic graphs which are generalizations of some results in rowlinson 2010 10

treediagonal matrices and their inverses abstract a generalization of tridiagonal matrices is considered namely treediagonal matrices which have nonzero off diagonal elements only in positions where the adjacency matrix of a tree has nonzero elements some properties of treediagonal matrices are given and their inverses are characterized and shown to have an interesting structure

homotopy theory of associative rings a kind of unstable homotopy theory on the category of associative rings without unit is developed there are the notions of fibrations homotopy in the sense of karoubi path spaces puppe sequences etc one introduces the notion of a quasi isomorphism or weak equivalence for rings and shows that similar to spaces the derived category obtained by inverting the quasi isomorphisms is naturally left triangulated also homology theories on rings are studied these must be homotopy invariant in the algebraic sense meet the mayer vietoris property and plus some minor natural axioms to any functor from rings to pointed simplicial sets a homology theory is associated in a natural way if and fibrations are the gl fibrations one recovers karoubi villamayor s functors kvi i 0 if is quillen s k theory functor and fibrations are the surjective homomorphisms one recovers the non negative homotopy k theory in the sense of weibel technical tools we use are the homotopy information for the category of simplicial functors on rings and the bousfield localization theory for model categories the machinery developed in the paper also allows to give another definition for the triangulated category kk constructed by cortinas and thom g cortinas a thom bivariant algebraic k theory preprint math kt 0603531 the latter category is an algebraic analog for triangulated structures on operator algebras used in kasparov s kk theory

on the spectral radius of weighted trees with fixed diameter and weight set abstract the spectrum of weighted graphs are often used to solve the problems in the design of networks and electronic circuits we first give some perturbational results on the spectral radius of weighted graphs when some weights of edges are modified then we derive the weighted tree with the largest spectral radius in the set of all weighted trees with fixed diameter and weight set furthermore an open problem of spectral radius on weighted paths is solved h z yang g z hu y hong bounds of spectral radii of weighted tree tsinghua science and technology 8 2003 517 520

bounds on the kth eigenvalues of trees and forests abstract we prove the upper bound k t n k 1 for the kth largest eigenvalue of a tree t 1 k n 2 and show that this bound is best possible when n 0 mod k we further obtain the strict inequality t 1 k 1 t 1 k t t 1 where t n k for the case n 0 mod k 2 k n 2 our upper bound is also proved to be the best possible upper bound for the kth largest eigenvalues of the forests in all cases

on multiple eigenvalues of trees abstract let t be a tree of order n 6 with as a positive eigenvalue of multiplicity k star complements are used to show that i if k n 3 then 1 ii if 1 then without restriction on k t has k 1 pendant edges that form an induced matching the results are used to identify the trees with a non zero eigenvalue of maximum possible multiplicity

gaussianization of the spectra of graphs and networks theory and applications abstract matrix functions of the adjacency matrix are very useful for understanding important structural properties of graphs and networks such as communicability node centrality bipartivity and many more they are also intimately related to the solution of differential equations describing dynamical processes on graphs and networks here we propose a new matrix function based on the gaussianization of the adjacency matrix of a graph this function gives more weight to a selected reference eigenvalue ref which may be located in any region of the graph spectra we show here that this matrix function can be derived from physical models that consider the interactions between nearest and next nearest neighbors in the graph we first obtain a few mathematical results for the trace of this matrix function when ref 1 h 1 for simple graphs as well as for random graphs we also provide a combinatorial interpretation of this index in terms of subgraphs in the graph and in terms of the competition pressure among agents in a complex system finally we apply this index to the study of magnetic properties of molecules emerging due to spin interactions as well as to studying the temporal evolution of the international trade network in the period 1992 2002 in both cases we give a clear phenomenological interpretation of the processes described

the spectra of a graph obtained from copies of a generalized bethe tree abstract we generalize the concept of a bethe tree as follows we say that an unweighted rooted tree is a generalized bethe tree if in each level the vertices have equal degree if b k is a generalized bethe tree of k levels then we characterize completely the eigenvalues of the adjacency matrix and laplacian matrix of a graph b k r obtained from the union of r copies of b k and the cycle c r connecting the r vertex roots moreover we give results on the multiplicity of the eigenvalues on the spectral radii and on the algebraic conectivity

stability of acyclic matrices abstract necessary and sufficient conditions for stability of acyclic matrices whose principal minors are nonnegative are given also a characterization is given of additive d stable acyclic matrices

on the multiplicity of eigenvalues of trees abstract let t be a tree of order n 6 with 1 as a positive eigenvalue of multiplicity k rowlinson 10 obtained a nice bound for k in terms of n that is k n 3 it seems slightly imperfect that the bound n 3 for k is not tight thus we are motivated to give a sharp upper bound for k in terms of n applying the theory of star set and star complement we prove that k n 4 3 if 2 is an integer at least 2 and the extremal trees t satisfying k n 4 3 are characterized completely as a corollary we prove that if n 16 then k n 4 3 for an arbitrary positive eigenvalue 1

the multiplicity of laplacian eigenvalue two in unicyclic graphs abstract let g be a graph and l g be the laplacian matrix of g in this paper we explicitly determine the multiplicity of laplacian eigenvalue 2 for any unicyclic graph containing a perfect matching

computing singular values of diagonally dominant matrices to high relative accuracy for a row diagonally dominant matrix if all of its off diagonal entries and its diagonally dominant parts which are defined for each row as the absolute value of the diagonal entry subtracted by the sum of the absolute values of off diagonal entries in that row are accurately known we develop an algorithm that computes all the singular values including zero ones if any with relative errors in the order of the machine precision when the matrix is also symmetric with positive diagonals i e a symmetric positive semi definite diagonally dominant matrix our algorithm computes all eigenvalues to high relative accuracy rounding error analysis will be given and numerical examples will be presented to demonstrate the high relative accuracy of the algorithm

on the kth laplacian eigenvalues of trees with perfect matchings abstract let t n be the set of all trees of order n with perfect matchings in this paper we prove that for any tree t t n its k th largest laplacian eigenvalue k t satisfies k t 2 when n 2 k and k t n 2 k 2 n 2 k 2 4 2 when n 2 k moreover this upper bound is sharp when n 0 mod 2 k

highly accurate symmetric eigenvalue decomposition and hyperbolic svd let g be a m n real matrix with full column rank and let j be a n n diagonal matrix of signs jii 1 1 the hyperbolic singular value decomposition hsvd of the pair g j is defined as g u v 1 where u is orthogonal is positive definite diagonal and v is j orthogonal matrix vtjv j we analyze when it is possible to compute the hsvd with high relative accuracy this essentially means that each computed hyperbolic singular value is guaranteed to have some correct digits even if they have widely varying magnitudes we show that one sided j orthogonal jacobi method method computes the hsvd with high relative accuracy more precisely let b gd 1 where d is diagonal such that the columns of b have unit norms essentially we show that the computed hyperbolic singular values of the pair g j will have log10 e min b correct decimal digits where e is machine precision we give the necessary relative perturbation bounds and error analysis of the algorithm our numerical tests confirmed all theoretical results for the symmetric non singular eigenvalue problem hx x we analyze the two step algorithm which consists of factorization h gjgt followed by the computation of the hsvd of the pair g j here g is square and non singular let b dg where d is diagonal such that the rows of b have unit norms and let b be defined as above essentially we show that the computed eigenvalues of h will have log10 e 2min b e min b correct decimal digits this accuracy can be much higher then the one obtained by the classical qr and jacobi methods applied to h where the accuracy depends on the spectral condition number of h particularly if the matrices b and b are well conditioned and we are interested in the accurate computation of tiny eigenvalues again we give the perturbation and error bounds and our theoretical predictions are confirmed by a series of numerical experiments we also give the corresponding results for eigenvectors and hyperbolic singular vectors

graphs with least eigenvalue 2 ten years on abstract the authors monograph spectral generalizations of line graphs was published in 2004 following the successful use of star complements to complete the classification of graphs with least eigenvalue 2 guided by citations of the book we survey progress in this area over the past decade some new observations are included

asymptotic behavior of the integrated density of states of acoustic operators with random long range perturbations in this paper we study the behavior of the integrated density of states of random acoustic operators of the form a 1 when is considered as an anderson type long range perturbations of some periodic function the behavior of the integrated density of states of a in the vicinity of the internal spectral edges is given

on the orthogonality of eigenvectors computed by divide and conquer techniques a detailed analysis on the accuracy issues in calculating the eigensystems of rank 1 perturbed diagonal systems is presented such calculations are the core of the divide and conquer technique prop

on the spectra of some graphs like weighted rooted trees abstract let g be a weighted rooted graph of k levels such that for j 2 k 1 each vertex at level j is adjacent to one vertex at level j 1 and all edges joining a vertex at level j with a vertex at level j 1 have the same weight where the weight is a positive real number 2 if two vertices at level j are adjacent then they are adjacent to the same vertex at level j 1 and all edges joining two vertices at level j have the same weight 3 two vertices at level j have the same degree 4 there is not a vertex at level j adjacent to others two vertices at the same level we give a complete characterization of the eigenvalues of the laplacian matrix and adjacency matrix of g they are the eigenvalues of leading principal submatrices of two nonnegative symmetric tridiagonal matrices of order k k and the roots of some polynomials related with the characteristic polynomial of the referred submatrices by application of the above mentioned results we derive an upper bound on the largest eigenvalue of a graph defined by a weighted tree and a weighted triangle attached by one of its vertices to a pendant vertex of the tree

bivariant algebraic k theory we show how methods from k theory of operator algebras can be applied in a completely algebraic setting to define a bivariant m1 stable homotopy invariant excisive k theory of algebras over a fixed unital ground ring h a b 7 kk a b which is universal in the sense that it maps uniquely to any other such theory it turns out kk is related to c weibel s homotopy algebraic k theory kh we prove that if h is commutative and a is central as an h bimodule then kk h a kh a we show further that some calculations from operator algebra kk theory such as the exact sequence of pimsner voiculescu carry over to algebraic kk

semiregular trees with minimal laplacian spectral radius a semiregular tree is a tree where all non pendant vertices have the same degree among all semiregular trees with fixed order and degree a graph with minimal adjacency laplacian spectral radius is a caterpillar counter examples show that the result cannot be generalized to the class of trees with a given non constant degree sequence

why finite element discretizations can be factored by triangular hierarchical matrices although the asymptotic complexity of direct methods for the solution of large sparse finite element systems arising from second order elliptic partial differential operators is far from being optimal these methods are often preferred over modern iterative methods this is mainly due to their robustness in this article it is shown that an approximate lu decomposition exists which can be computed in the algebra of hierarchical matrices with almost linear complexity and with the same robustness as the classical lu decomposition low precision approximations may be used for preconditioning iterative solvers as a byproduct we prove that schur complements of stiffness matrices can be approximated with almost linear complexity

designing optimal multiplex networks for certain laplacian spectral properties we discuss the design of interlayer edges in a multiplex network under a limited budget with the goal of improving its overall performance we analyze the following three problems separately first we maximize the smallest nonzero eigenvalue also known as the algebraic connectivity second we minimize the largest eigenvalue also known as the spectral radius and finally we minimize the spectral width maximizing the algebraic connectivity requires identical weights on the interlayer edges for budgets less than a threshold value however for larger budgets the optimal weights are generally nonuniform the dual formulation transforms the problem into a graph realization embedding problem that allows us to give a fuller picture namely before the threshold budget the optimal realization is one dimensional with nodes in the same layer embedded to a single point while beyond the threshold the optimal embeddings generally unfold into spaces with dimension bounded by the multiplicity of the algebraic connectivity finally for extremely large budgets the embeddings again revert to lower dimensions minimizing the largest eigenvalue is driven by the spectral radius of the individual networks and its corresponding eigenvector before a threshold the total budget is distributed among interlayer edges corresponding to the nodal lines of this eigenvector and the optimal largest eigenvalue of the laplacian remains constant for larger budgets the weight distribution tends to be almost uniform in the dual picture the optimal graph embedding is one dimensional and nonhomogeneous at first with the nodes corresponding to the layer with the largest spectral radius distributed on a line according to its eigenvector while the other layer is embedded at the origin beyond this threshold the optimal embedding expands to be multidimensional and for larger values of the budget the two layers fill the embedding space finally we show how these two problems are connected to minimizing the spectral width

computing the null space of finite element problems we present a method for computing the null space of finite element models including models with equality constraints the method is purely algebraic it requires access to the element matrices but not to the geometry or material properties of the model theoretical considerations show that under certain conditions both the amount of computation and the amount of memory required by our method scale linearly with model size memory scales linearly but computation scales quadratically with the dimension of the null space our experiments confirm this the method scales extremely well on 3 dimensional model problems in general large industrial models do not satisfy all the conditions that the theoretical results assume however experimentally the method performs well and outperforms an established method on industrial models including models with many equality constraints the accuracy of the computed null vectors is acceptable but the method is usually less accurate than a more naive and computationally much more expensive method

note on the kth laplacian eigenvalues of trees with perfect matchings abstract denote by k t the k th laplacian eigenvalue of tree t let t n be the set of all trees of order n with perfect matchings in this paper we characterize the tree t in t 4 t with 2 t t 2 t 2 4 2 and the tree in t 6 t with 3 t t 2 t 2 4 2 these results affirm problem of j x li w c shiu a chang 2010 5 for the cases k 2 and k 3 respectively

sharp upper bounds on the spectral radius of graphs abstract let g be a simple connected graph with n vertices m edges and degree sequence d 1 d 2 d n the spectral radius g of graph g is the largest eigenvalue of its adjacency matrix in this paper we present some sharp upper bounds of the spectral radius in terms of the degree sequence of graphs

on computing accurate singular values and eigenvalues of matrices with acyclic graphs abstract it is known that small relative perturbations in the entries of a bidiagonal matrix only cause small relative perturbations in its singular values independentof the values of the matrix entries in this paper we show that a matrix has this property if and only if its associated bipartite graph is acyclic we also show how to compute the singular values of such a matrix to high relative accuracy the same algorithm can compute eigenvalues of symmetric matrices with acyclic graphs with tiny componentwise relative backward error this class includes tridiagonal matrices arrow matrices and exponentially many others

eigenvalue multiplicity in triangle free graphs abstract let g be a connected triangle free graph of order n 5 with 1 0 as an eigenvalue of multiplicity k 1 we show that if d is the maximum degree in g then k n d 1 moreover if k n d 1 then either a g is non bipartite and k 2 3 1 2 2 1 with equality only if g is strongly regular or b g is bipartite and k d 1 with equality only if g is a bipolar cone in each case we discuss the extremal graphs that arise

on the characteristic polynomial of a special class of graphs and spectra of balanced trees abstract let h be a simple graph with n vertices and g be a sequence of n rooted graphs g 1 g 2 g n godsil and mckay c d godsil b d mckay a new graph product and its spectrum bull austral math soc 18 1978 21 28 defined the rooted product h g of h by g by identifying the root vertex of g i with the ith vertex of h and determined the characteristic polynomial of h g in this paper we prove a general result on the determinants of some special matrices and as a corollary determine the characteristic polynomials of adjacency and laplacian matrices of h g rojo and soto o rojo r soto the spectra of the adjacency matrix and laplacian matrix for some balanced trees linear algebra appl 403 2005 97 117 computed the characteristic polynomials and the spectrum of adjacency and laplacian matrices of a class of balanced trees as an application of our results we obtain their conclusions by a simple method

determinantal formulae for matrices with sparse inverses abstract the determinant of a matrix is expressed in terms of certain of its principal minors by a formula which can be read off from the graph of the inverse of the matrix the only information used is the zero pattern of the inverse and each zero pattern yields one or more corresponding formulae for the determinant

graph invertibility and median eigenvalues abstract let g w be a weighted graph with a weight function w e g r 0 a weighted graph g w is invertible to a new weighted graph if its adjacency matrix is invertible graph inverses have combinatorial interests and can be applied to bound median eigenvalues of graphs such as have physical meanings in this paper we characterize the inverse of a weighted graph based on its sachs subgraphs that are spanning subgraphs with only k 2 or cycles or loops as components the characterization can be used to find the inverse of a weighted graph based on its structures instead of its adjacency matrix if a graph has its spectra split about the origin i e half of eigenvalues are positive and half of them are negative then its median eigenvalues can be bounded by estimating the largest and smallest eigenvalues of its inverse we characterize graphs with a unique sachs subgraph and prove that these graphs has their spectra split about the origin if they have a perfect matching as applications we show that the median eigenvalues of stellated graphs of trees and corona graphs belong to different halves of the interval 1 1

the spectra of the adjacency matrix and laplacian matrix for some balanced trees abstract let t be an unweighted rooted tree of k levels such that in each level the vertices have equal degree let dk j 1 denotes the degree of the vertices in the level j we find the eigenvalues of the adjacency matrix and of the laplacian matrix of t they are the eigenvalues of principal submatrices of two nonnegative symmetric tridiagonal matrices of order k k the codiagonal entries for both matrices are d j 1 2 j k 1 and d k while the diagonal entries are zeros in the case of the adjacency matrix and dj 1 j k in the case of the laplacian matrix moreover we give some results concerning to the multiplicity of the above mentioned eigenvalues

the minimal spectral radius of graphs with a given diameter the spectral radius of a graph i e the largest eigenvalue of its corresponding adjacency matrix plays an important role in modeling virus propagation in networks in fact the smaller the spectral radius the larger the robustness of a network against the spread of viruses among all connected graphs on n nodes the path pn has minimal spectral radius however its diameter d i e the maximum number of hops between any pair of nodes in the graph is the largest possible namely d n 1 in general communication networks are designed such that the diameter is small because the larger the number of nodes traversed on a connection the lower the quality of the service running over the network this leads us to state the following problem which connected graph on n nodes and a given diameter d has minimal spectral radius in this paper we solve this problem explicitly for graphs with diameter d fenced 1 2 fenced frac n 2 n 3 n 2 n 1 moreover we solve the problem for almost all graphs on at most 20 nodes by a computer search 2007 elsevier inc all rights reserved

lower eigenvalue bounds for singular pencils of matrices abstract beauwens procedure for obtaining lower eigenvalue bounds for regular pencils of matrices a b is simplified and extended to the singular case the theory is then compared through a particular perturbed modified incomplete factorization with notay s generalization of another approach initiated by gustafsson and developed by axelsson and barker and by wilmet

the minimal laplacian spectral radius of trees with a given diameter for a graph g its laplacian matrix is the difference of the diagonal matrix of its vertex degrees and its adjacency matrix let t n d be the set of trees on n vertices with diameter d in this paper for email protected 1 2 3 4 n 3 n 2 n 1 trees with minimal laplacian spectral radii in the set t n d are characterized

a low concentration asymptotic expansion for the density of states of a random schrodinger operator with poisson disorder abstract in this paper we study the density of states of a random schrodinger operator of the formh v wherev is a poisson potential i e a poisson random field of concentration we show thatn d the density of states ofh admits an asymptotic expansion in when 0 then we use this expansion to deduce the behaviour of the integrated density of states ofh in the energy interval 0 when goes to 0 dans ce travail nous etudions la densite d etats d un operateur de schrodinger aleatoire de la formeh v ouv est un potentiel de poisson i e un champ aleatoire de poisson de concentration nous demontrons l existence d un developpement asymptotique en pourn d la densite d etat deh ceci quand 0 puis nous utilisons ce developpement pour estimer la taille de la densite d etats integree deh dans l intervalle d energie 0 quand tend vers 0

the spectrum of the laplacian matrix of a balanced binary tree abstract let l b k be the laplacian matrix of an unweighted balanced binary tree b k of k levels we prove that spectrum of l b k is l b k j 1 k 1 t j s k where for 1 j k 1 t j is the j j principal submatrix of the tridiagonal k k matrix s k s k 1 2 0 0 2 3 2 0 2 0 3 2 0 0 2 2 we derive that the multiplicity of each eigenvalue of t j 1 j k 1 as an eigenvalue of l b k is at least 2 k j 1 finally for each t j using some results in electron j linear algebra 6 2000 62 we obtain lower and upper bounds for its smallest eigenvalue and an upper bound for its largest eigenvalue in particular we give upper bounds for the second largest eigenvalue and for the largest eigenvalue of l b k

relative perturbation theory iv sin 2θ theorems abstract the double angle theorems of davis and kahan bound the change in an invariant subspace when a hermitian matrix a is subject to an additive perturbation a a a a this paper supplies analogous results when a is subject to a congruential or multiplicative perturbation a a d ad the relative gaps that appear in the bounds involve the spectrum of only one matrix either a or a in contrast to the gaps that appear in the single angle bounds the double angle theorems do not directly bound the difference between the old invariant subspace s and the new one s but instead bound the difference between s and its reflection j s where the mirror is s and j reverses s the orthogonal complement of s the double angle bounds are proportional to the departure from the identity and from orthogonality of the matrix d def d 1 jdj note that d is invariant under the transformation d d for 0 whereas the single angle theorems give bounds proportional to d s departure from the identity and from orthogonality the corresponding results for the singular value problem when a nonsquare matrix b is perturbed to b d 1 bd 2 are also presented

recent directions in matrix stability abstract matrix stability has been intensively investigated in the past two centuries we review work that has been done in this topic focusing on the great progress that has been achieved in the last decade or two we start with classical stability criteria of lyapunov routh and hurwitz and lienard and chipart we then study recently proven sufficient conditions for stability with particular emphasis on p matrices we investigate conditions for the existence of a stable scaling for a given matrix we review results on other types of matrix stability such as d stability additive d stability and lyapunov diagonal stability we discuss the weak principal submatrix rank property shared by lyapunov diagonally semistable matrices we also discuss the uniqueness of lyapunov scaling factors maximal lyapunov scaling factors cones of real positive semidefinite matrices and their applications to matrix stability and inertia preserving matrices

characterization of acyclic d stable matrices abstract necessary and sufficient conditions for d stability of acyclic matrices are given special cases of this characterization are recent results on tridiagonal matrices

automated adaptive multilevel solver abstract this paper presents an automated adaptive multilevel solver for linear or linearized system of equations the multilevel aspect of the solver is aimed at securing an optimal rate of convergence while keeping the size of the coarsest problem sufficiently small to ensure that the direct portion of the solution does not dominate the total computational cost adaptivity in terms of a priori selection of the number of levels one or more and construction of the optimal multilevel preconditioner is the key to the robustness of the method the number of levels is selected on the basis of estimated conditioning sparsity of the factor and available memory the auxiliary coarse models if required are automatically constructed on the basis of spectral characteristics of individual aggregates groups of neighboring elements an obstacle test consisting of twenty industry and model problems was designed to i determine the optimal values of computational parameters and to ii compare the adaptive multilevel scheme with existing state of the art equation solvers including the multifrontal solver 17 with the mmd reordering scheme and the pcg solver with the nearly optimal modified incomplete cholesky factorization preconditioner

sequential and parallel triangulating algorithms for elimination game and new insights on minimum degree elimination game is a well known algorithm that simulates gaussian elimination of matrices on graphs and it computes a triangulation of the input graph the number of fill edges in the computed triangulation is highly dependent on the order in which elimination game processes the vertices and in general the produced triangulations are neither minimum nor minimal in order to obtain a triangulation which is close to minimum the minimum degree heuristic is widely used in practice but until now little was known on the theoretical mechanisms involved in this paper we show some interesting properties of elimination game in particular that it is able to compute a partial minimal triangulation of the input graph regardless of the order in which the vertices are processed this results in a new algorithm to compute minimal triangulations that are sandwiched between the input graph and the triangulation resulting from elimination game one of the strengths of the new approach is that it is easily parallelizable and thus we are able to present the first parallel algorithm to compute such sandwiched minimal triangulations in addition the insight that we gain through elimination game is used to partly explain the good behavior of the minimum degree algorithm we also give a new algorithm for producing minimal triangulations that is able to use the minimum degree idea to a wider extent

controllability inertia and stability for tridiagonal matrices abstract criteria are given for the controllability of certain pairs of tridiagonal matrices these criteria may be used with the chen wimmer theorem to obtain inertia results also a characterization is given of those nonsingular tridiagonal matrices with certain principal minors nonnegative which are positive stable this extends a previous characterization of the real d stable tridiagonal matrices

relative perturbation techniques for singular value problems a technique is presented for deriving bounds on the relative change in the singular values of a real matrix or the eigenvalues of a real symmetric matrix due to a perturbation as well as bounds on the angles between the unperturbed and perturbed singular vectors or eigenvectors the class of perturbations considered consists of all delta b for which b delta b d l bd r for some nonsingular matrices d l and d r this class includes componentwise relative perturbations of a bidiagonal or biacyclic matrix and perturbations that annihilate the off diagonal block in a block triangular matrix many existing relative perturbation and deflation bounds are derived from results for this general class of perturbations also some new relative perturbation and deflation results for the singular values and vectors of biacyclic triangular and shifted triangular matrices are presented

the integrated density of states in strong magnetic fields we consider three dimensional schrodinger operators with constant magnetic fields and ergodic electric potentials we study the strong magnetic field asymptotic behaviour of the integrated density of states distinguishing between the asymptotics far from the landau levels and the asymptotics near a given landau level

on the weighted trees with given degree sequence and positive weight set abstract we determine the unique weighted tree with the largest spectral radius with respect to the adjacency and laplacian matrix in the set of all weighted trees with a given degree sequence and positive weight set moreover we also derive the weighted trees with the largest spectral radius with respect to the matrices mentioned above in the sets of all weighted trees with a given maximum degree or pendant vertex number and so on

on the spectra of certain rooted trees abstract let t be an unweighted tree with vertex root v which is the union of two trees t 1 v 1 e 1 t 2 v 2 e 2 such that v 1 v 2 v and t 1 and t 2 have the property that the vertices in each of their levels have equal degree we characterize completely the eigenvalues of the adjacency matrix and of the laplacian matrix of t they are the eigenvalues of symmetric tridiagonal matrices whose entries are given in terms of the vertex degrees moreover we give some results about the multiplicity of the eigenvalues applications to some particular trees are developed

implicit standard jacobi gives high relative accuracy we prove that the jacobi algorithm applied implicitly on a decomposition a xdx t of the symmetric matrix a where d is diagonal and x is well conditioned computes all eigenvalues of a to high relative accuracy the relative error in every eigenvalue is bounded by o epsilon kappa x where epsilon is the machine precision and kappa x equiv x 2 cdot x 1 2 is the spectral condition number of x the eigenvectors are also computed accurately in the appropriate sense we believe that this is the first algorithm to compute accurate eigenvalues of symmetric indefinite matrices that respects and preserves the symmetry of the problem and uses only orthogonal transformations

perturbation theory for the ldu factorization and accurate computations for diagonally dominant matrices we present a structured perturbation theory for the ldu factorization of row diagonally dominant matrices and we use this theory to prove that a recent algorithm of ye math comp 77 264 2195 2230 2008 computes the l d and u factors of these matrices with relative errors less than 14n 3 u where u is the unit roundoff and n n is the size of the matrix the relative errors for d are componentwise and for l and u are normwise with respect the max norm a m max ij a ij these error bounds guarantee that for any diagonally dominant matrix a we can compute accurately its singular value decomposition and the solution of the linear system ax b for most vectors b independently of the magnitude of the traditional condition number of a and in o n 3 flops

proof of a problem on laplacian eigenvalues of trees abstract denote by k l t the k th laplacian eigenvalue of a tree t let t k 2 t be the set of all trees of order 2 t k with perfect matchings in this note the trees t in t k 2 t with k l t t 2 t 2 4 2 are characterized which solves problem of j x li w c shiu and a chang in 3 completely

localization for some continuous random hamiltonians in d dimensions abstract we prove the existence with probability one of an interval of pure point spectrum for some families of continuous random schrodinger operators in d dimensions for anderson like models with positive short range single site potentials we also prove that the corresponding eigenfunctions decay exponentially and that the integrated density of states is lipschitz continuous for the other families of random potentials that we study we show that the corresponding eigenfunctions decay faster than an inverse power of x which depends upon the decay rate of the single site potential to obtain these results we develop an extension of the classical aronszajn donoghue theory for a class of relatively compact perturbations and a spectral averaging method which extends kotani s trick to these more general families of operators

on the two conjectures of graffiti we study the laplacian eigenvalues of trees on n vertices with independence number and describe all extremal graphs that attain the maximal laplacian spectral radius and algebraic connectivity then the results are used to confirm two conjectures of graffiti wow conjectures 584 and 636 on the relationship between the laplacian eigenvalues and the independence number of a graph

a relation between the matching number and laplacian spectrum of a graph abstract let g be a graph its laplacian matrix is the difference of the diagonal matrix of its vertex degrees and its adjacency matrix in this paper we generalize a result in r merris port math 48 3 1991 and obtain the following result let g be a graph and m g be a maximum matching in g then the number of edges in m g is a lower bound for the number of laplacian eigenvalues of g exceeding 2

fluctuations of principal eigenvalues and random scales we investigate the principal dirichlet eigenvalue of the laplacian with soft poissonian obstacles in large boxes of d 2 with the help of our recent version of the method of enlargement of obstacles 18 we derive quantitative confidence intervals for these eigenvalues we also provide less quantitative estimates which however point out the correct size of fluctuations and indicate a stiffness in their behavior in the two dimensional case we derive geometric controls which relate these eigenvalues to certain empty circular droplets our results also have natural applications to the study of the location of minima of certain intermittent random variational problems motivated by 13 17

potentials of random walks on trees abstract in this article we characterize inverse m matrices and potentials whose inverses are supported on trees in the symmetric case we show they are a hadamard product of tree ultrametric matrices generalizing a result by gantmacher and krein 12 done for inverse tridiagonal matrices we also provide an algorithm that recognizes when a positive matrix w has an inverse m matrix supported on a tree this algorithm has quadratic complexity we also provide a formula to compute w 1 which can be implemented with a linear complexity finally we also study some stability properties for hadamard products and powers

stochastic preconditioning for diagonally dominant matrices this paper presents a new stochastic preconditioning approach for large sparse matrices for the class of matrices that are rowwise and columnwise irreducibly diagonally dominant we prove that an incomplete rm ldl t factorization in a symmetric case or an incomplete ldu factorization in an asymmetric case can be obtained from random walks and used as a preconditioner it is argued that our factor matrices have better quality i e better accuracy size trade offs than preconditioners produced by existing incomplete factorization methods therefore a resulting preconditioned krylov subspace iterative solver requires less computation than traditional methods to solve a set of linear equations with the same error tolerance the advantage increases for larger and denser matrices these claims are verified by numerical tests and we provide techniques that can potentially extend the theory to non diagonally dominant matrices

resonances in one dimension and fredholm determinants we discuss resonances for schrodinger operators in whole and half line problems one of our goals is to connect the fredholm determinant approach of froese to the fourier transform approach of zworski another is to prove a result on the number of antibound states namely in a half line problem there are an odd number of antibound states between any two bound states

some results on the index of unicyclic graphs we identify in some classes of unicyclic graphs of fixed order and girth those graphs whose index i e the largest eigenvalue is maximal besides some lower and upper bounds on the indices of the graphs being considered are provided

lifshitz tails for random schrodinger operators with negative singular poisson potential we develop a method of asymptotic study of the integrated density of states ids n e of a random schrodinger operator with a non positive attractive poisson potential the method is based on the periodic approximations of the potential instead of the dirichlet neumann bracketing used before this allows us to derive more precise bounds for the rate of approximations of the ids by the ids of respective periodic operators and to obtain rigorously for the first time the leading term of log n e as e for the poisson random potential with a singular single site impurity potential in particular for the screened coulomb impurities dislocations etc

kv theory of categories quillen has constructed a k theory k c for nice categories one of which is the category of projective r modules we construct a theory kv c for the nice categories parametrized by rings when applied to projective modules we recover the karoubi villamayor k theory kv r as an application we show that the cartan map from k r to g r factors through the groups kv r we also compute kv for the categories of faithful projectives and azumaya algebras generalizing results of bass the purpose of this paper is to provide enough tools to make the karoubivillamayor groups kv r as manipulable as quillen s groups k r since the kv are much closer to a cohomology theory than the k it should be easier to compute the values of the groups kv r the gersten anderson spectral sequence relating these groups should then allow us to detect nontrivial elements in k r and to look for new structural results quillen has defined a k theory for two kinds of nice categories exact categories and symmetric monoidal categories with faithful translations now a fundamental property of karoubi villamayor functors should be that i they depend on a ring r and ii they do not change if r is replaced by r t in order to accomodate this property we need to consider categories parametrized by rings fortunately there is an abundance of such categories in q we encounter the exact categories m p hs and mp in b1 we encounter the symmetric monoidal categories iso p fp pic az quad etc after some preliminaries the definitions are given in 2 the groups kv are the homotopy groups of an infinite loop space bisimplicially defined we show that they possess all the expected properties cofinality homotopy invariance long exact sequences and an analogue of the gersten anderson spectral sequence with care the tools of characteristic filtration resolution devissage localization etc all go through in this setting one unexpected benefit of thi approach is that the cartan map k r g r factors through the groups kv r for all noetherian rings r this is proven in 3 we conclude by computing the groups kv s for many of the symmetric monoidal categories mentioned above the idea is that in these cases we can construct an analogue of gl providing a simplicial group delooping of the relevant received by the editors december 6 1979 and in revised form june 24 1980 1980 mathematics subject classification primary 18f25 secondary 13d15 16a54

nonsingularity criteria for general combinatorially symmetric matrices we present two criteria for nonsingularity of matrices over general fields the first applies to irreducible acyclic matrices with zero diagonal the second to arbitrary combinatorially symmetric matrices both involve graphs of matrices and a closure operation on sets of vertices of the graphs we focus attention not on the entries of the matrices under consideration but rather on patterns of zero entries of null vectors of the matrices our criteria are of theoretical rather than of computational interest

on the spectral radius of trees abstract in this paper we investigated the spectral radius of trees and obtained the following result let t be a tree on n vertices let m t denote one maximum matching of t and m t i let t i be a tree as shown in fig 1 then t 1 2 n i 1 n i 1 2 4 n 2i 1 and equality holds if and only if t t i

a parallel algorithm for computing the eigenvalues of a symmetric tridiagonal matrix a parallel algorithm called polysection is presented for computing the eigenvalues of a symmetric tridiagonal matrix the method is based on a quadratic recurrence in which the characteristic polynomial is constructed on a binary tree from polynomials whose degree doubles at each level intervals that contain exactly one zero are determined by the zeros of polynomials at the previous level which ensures that different processors compute different zeros the signs of the polynomials at the interval endpoints are determined a priori and used to guarantee that all zeros are found the use of finite precision arithmetic may result in multiple zeros however in this case the intervals coalesce and their number determines exactly the multiplicity of the zero for an n x n matrix the eigenvalues can be determined in o log squared n time with n squared processors and o n time with n processors the method is compared with a parallel variant of bisection that requires o n squared time on a single processor o n time with n processors and o log n time with n squared processors

internal lifshitz tails for schrodinger operators with random potentials in this short note we prove that for periodic schrodinger operators perturbed by an anderson type random potential with long range single site potential the density of states always show a lifshitz tail at a band edge thus we correct a mistake made in an earlier paper f klopp duke math j 98 335 396 1999

improved bounds for the laplacian energy of bethe trees abstract for k d 2 a bethe tree is a rooted tree with k levels which the root vertex has degree d the vertices from level 2 to k 1 have degree d 1 and the vertices at the level k are pendent vertices so et al using a theorem by ky fan have obtained both upper and lower bounds for the laplacian energy of bipartite graphs we shall employ the above mentioned theorem to obtain new and improved bounds for the laplacian energy in the case of bethe trees

effective preconditioning through minimum degree ordering interleaved with incomplete factorization in this paper we study a kind of effective preconditioning technique which interleaves the incomplete cholesky ic factorization with an approximate minimum degree ordering an ic factorization algorithm derived from ikj version gaussian elimination is proposed and some details on implementation are presented then we discuss the ways to compute the degrees of the unnumbered nodes exactly and approximately using the concept of element absorbing when used in conjunction with conjugate gradient algorithm the new preconditioners usually lead to fast convergence the numerical experiments show that the interleaving of symbolic ordering and numerical ic factorization will generate better preconditioners than those generated by the ic factorization without ordering or with purely symbolic ordering ahead of the factorization

the evolution of the minimum degree ordering algorithm over the past fifteen years the implementation of the minimum degree algorithm has received much study and many important enhancements have been made to it this paper describes these various enhancements their historical development and some experiments showing how very effective they are in improving the execution time of the algorithm a shortcoming is also presented that exists in all of the widely used implementations of the algorithm namely that the quality of the ordering provided by the implementations is surprisingly sensitive to the initial ordering for example changing the input ordering can lead to an increase or decrease of as much as a factor of three in the cost of the subsequent numerical factorization this sensitivity is caused by the lack of an effective tie breaking strategy and the authors experiments illustrate the importance of developing such a strategy