on the isomorphism problem for even artin groups abstract an even artin group is a group which has a presentation with relations of the form s t n t s n with n 1 with a group g we associate a lie z algebra tg r g this is the usual lie algebra defined from the lower central series truncated at the third rank for each even artin group g we determine a presentation for tg r g by means of this presentation we obtain information about the diagram of g we then prove an isomorphism criterion for coxeter matrices that ensures that the diagram of g is uniquely determined by this information let d 2 we show that if two even artin groups g and h having presentations with relations of the form s t d k t s d k with k 0 are such that tg r g tg r h then g and h have the same presentation up to permutation of the generators on the other hand we show an example of two non isomorphic even artin groups g and h such that tg r g tg r h

error bounds on the power method for determining the largest eigenvalue of a symmetric positive definite matrix abstract let a be a positive definite symmetric matrix we wish to determine the largest eigenvalue 1 we consider the power method i e that of choosing a vector v0 and setting vk akv0 then the rayleigh quotients rk avk vk vk vk usually converge to 1 as k here u v denotes their inner product in this paper we give two methods for determining how close rk is to 1 they are both based on a bound on 1 rk involving the difference of two consecutive rayleigh quotients and a quantity k while we do not know how to directly calculate k we can given an algorithm for giving a good upper bound on it at least with high probability this leads to an upper bound for 1 rk which is proportional to 2 1 2k which holds with a prescribed probability the prescribed probability being an arbitrary 0 with the upper bound depending on

a boundary blow up problem with a nonlocal reaction abstract in this paper we deal with several kinds of boundary blow up problems whose main feature is exhibiting a nonlocal reaction term which depends on the integral of some function of the concentration u as a representative of such problems we consider u f u 1 1 g u in u on where is a bounded smooth domain of r n and the boundary condition is understood as u x when dist x 0 we find necessary and sufficient conditions for the existence of positive solutions in some cases we also obtain uniqueness or multiplicity of solutions other classes of nonlocal dependence of the reaction term are also discussed

on the least eigenvalue of cacti with pendant vertices abstract a connected graph is a cactus if any two of its cycles share at most one common vertex we determine the unique graph with minimum least eigenvalue among the cacti with fixed numbers of vertices and pendant vertices

k π 1 and word problems for infinite type artin tits groups and applications to virtual braid groups let be a coxeter graph let w s be its associated coxeter system and let a be its associated artin tits system we regard w as a reflection group acting on a real vector space v let i be the tits cone and let e be the complement in i iv of the reflecting hyperplanes recall that salvetti charney and davis have constructed a simplicial complex having the same homotopy type as e we observe that if t subset s then t naturally embeds into we prove that this embedding admits a retraction pi t omega gamma to omega gamma t and we deduce several topological and combinatorial results on parabolic subgroups of a from a family mathcal s of subsets of s having certain properties we construct a cube complex we show that has the same homotopy type as the universal cover of e and we prove that is cat 0 if and only if mathcal s is a flag complex we say that x subset s is free of infinity if x has no edge labeled by we show that if e gamma x is aspherical and a x has a solution to the word problem for all x subset s free of infinity then e is aspherical and a has a solution to the word problem we apply these results to the virtual braid group vb n in particular we give a solution to the word problem in vb n and we prove that the virtual cohomological dimension of vb n is n 1

the least eigenvalue of a graph with cut vertices in this paper we characterize the unique graph whose least eigenvalue attains the minimum among all connected graphs of fixed order and given number of cut vertices and then obtain a lower bound for the least eigenvalue of a connected graph in terms of the number of cut vertices during the discussion we also get some results for the spectral radius of a connected bipartite graph and its upper bound

a jacobi davidson method for solving complex symmetric eigenvalue problems we discuss variants of the jacobi davidson method for solving the generalized complex symmetric eigenvalue problem the jacobi davidson algorithm can be considered as an accelerated inexact rayleigh quotient iteration we show that it is appropriate to replace the euclidean inner product in mathbb c n with an indefinite inner product the rayleigh quotient based on this indefinite inner product leads to an asymptotically cubically convergent rayleigh quotient iteration advantages of the method are illustrated by numerical examples we deal with problems from electromagnetics that require the computation of interior eigenvalues the main drawback that we experience in these particular examples is the lack of efficient preconditioners

the least eigenvalue of the complements of trees let tnc be the set of the complements of trees of order n in this paper we characterize the unique graph whose least eigenvalue attains the minimum among all graphs in tnc

nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints hamiltoniens sousquadratiques conditions aux limites infinies et donnees d explosion hamiltoniens superquadratiques formulation viscosite des conditions aux limites le probleme ergodique commande stochastique optimale avec contraintes d etat

a note on the strong maximum principle and the compact support principle abstract in this note we are concerned with the strong maximum principle smp and the compact support principle csp for non negative solutions to quasilinear elliptic inequalities of the form div a u u g u f u 0 in and div a u u g u f u 0 in r n b r 0 respectively we give new conditions on the data a g f to obtain smp and csp when these conditions are particularized to the m laplacian and pure power nonlinearities we completely classify the data according to the validity of the csp or the smp in doing so we clarify the general situation and we consider a case not covered in the literature

on the inequality f x d 2u ge f u g u du q we consider fully nonlinear degenerate elliptic equations with zero and first order terms we provide a priori upper bounds and characterize the existence of entire subsolutions under growth conditions on the lower order coefficients which extend the classical keller osserman condition for semilinear equations

boundary blow up elliptic problems with nonlinear gradient terms by karamata regular variation theory and perturbation method we show the exact asymptotical behaviour of solutions near the boundary to nonlinear elliptic problems u u q b x g u u 0 in u where is a bounded domain with smooth boundary in rn q 0 g c1 0 g 0 0 g is regularly varying at infinity with index with 0 and b is nonnegative nontrivial in which may be vanishing on the boundary

sharp lower bounds of the least eigenvalue of planar graphs let g be a simple graph with n 3 vertices and orientable genus g and non orientable genus h we define the euler characteristic g of a graph g by g max 2 2g 2 h let g be the least eigenvalue of the adjacency matrix a of g in this paper we obtain the following lower bounds of g g 2 n g in particular if g is the planar graph then g 2n 4 the equality holds if and only if g k2 n 2 further we have same result of series parallel graph

boundary behavior of large solutions to the monge ampere equations with weights abstract this paper is concerned with the monge ampere equation det d 2 u x b x f u x x where is a strictly convex bounded smooth domain in r n with n 2 and b c which is positive in but may be vanishing on the boundary we find a new structure condition on f which plays a crucial role in the boundary behavior of strictly convex large solutions our results are obtained in a more general setting than those in cirstea and trombetti 2008 12 where f is regularly varying at infinity with index p n

optimal uniqueness theorems and exact blow up rates of large solutions in this paper we prove some optimal uniqueness results for large solutions of a canonical class of semilinear equations under minimal regularity conditions on the weight function in front of the non linearity and combine these results with the localization method introduced in lopez gomez the boundary blow up rate of large solutions j differential equations 195 2003 25 45 to prove that any large solution l of u a x up p 1 a 0 must satisfy limx x0l x fx0 dist x i0 pp 1p 1p 1p 1p 1foreachx0 where fx0 t t 0sfx01p 1 p 1p 1ds i0 limt 0fx0 t fx0 t fx0 t 2 and fx0 is any smooth extension of the boundary normal section of a at x0 i e fx0 t a x0 tnx0 t 0 t 0 subsequently nx0 stands for the outward unit normal at x0 therefore the theory can be extended to cover the general case when fx0 l1p 1

discrete approximations for complex kac moody groups abstract we construct a map from the classifying space of a discrete kac moody group over the algebraic closure of the field with p elements to the classifying space of a complex topological kac moody group and prove that it is a homology equivalence at primes q different from p this generalizes a classical result of quillen friedlander and mislin for lie groups as an application we construct unstable adams operations for general kac moody groups compatible with the frobenius homomorphism our results rely on new integral homology decompositions for certain infinite dimensional unipotent subgroups of discrete kac moody groups

artin groups and infinite coxeter groups if the associated coxeter group 7 is finite for the braid group b the associated coxeter group is the symmetric group s brieskorn and saito 4 and independently deligne 6 have investigated artin groups of finite type in 1969 garside 7 solved the conjugacy problem for braid groups brieskorn this work was supported by nsf grants mcs 7902512 and mcs 7703452 the authors would like to thank the referee for an extremely careful reading of the paper and many helpful sugges tions

proof of the k pi 1 conjecture for affine artin groups we prove the k pi 1 conjecture for affine artin groups the complexified complement of an affine reflection arrangement is a classifying space this is a long standing problem due to arnol d pham and thom our proof is based on recent advancements in the theory of dual coxeter and artin groups as well as on several new results and constructions in particular we show that all affine noncrossing partition posets are el shellable we use these posets to construct finite classifying spaces for dual affine artin groups we introduce new cw models for the orbit configuration spaces associated with arbitrary coxeter groups we construct finite classifying spaces for the braided crystallographic groups introduced by mccammond and sulway

general uniqueness results and variation speed for blow up solutions of elliptic equations let omega be a smooth bounded domain in r n we prove general uniqueness results for equations of the form delta u au b x f u in omega subject to u infty on partial omega our uniqueness theorem is established in a setting involving karamata s theory on regularly varying functions which is used to relate the blow up behavior of u x with f u and b x where b equiv 0 on partial omega and a certain ratio involving b is bounded near partial omega a key step in our proof of uniqueness uses a modification of an iteration technique due to safonov

eigenvalue interlacing and weight parameters of graphs eigenvalue interlacing is a versatile technique for deriving results in algebraic combinatorics in particular it has been successfully used for proving a number of results about the relation between the adjacency matrix or laplacian spectrum of a graph and some of its properties for instance some characterizations of regular partitions and bounds for some parameters such as the independence and chromatic numbers the diameter the bandwidth etc have been obtained for each parameter of a graph involving the cardinality of some vertex sets we can define its corresponding weight parameter by giving some weights that is the entries of the positive eigenvector to the vertices and replacing cardinalities by square norms the key point is that such weights regularize the graph and hence allow us to define a kind of regular partition called pseudo regular intended for general graphs here we s aow how to use interlacing for proving results about some weight parameters and pseudo regular partitions of a graph for instance generalizing a well known result of lovfisz it is shown that the weight shannon capacity 6 of a connected graph f with n vertices and adjacency matrix eigenvalues 2j 2 2 satisfies

some large scale matrix computation problems abstract there are numerous applications in physics statistics and electrical circuit simulation where it is required to bound entries and the trace of the inverse and the determinant of a large sparse matrix all these computational tasks are related to the central mathematical problem studied in this paper namely bounding the bilinear form u t f a v for a given matrix a and vectors u and v where f is a given smooth function and is defined on the spectrum of a we will study a practical numerical algorithm for bounding the bilinear form where the matrix a is only referenced through matrix vector multiplications a monte carlo method is also presented to efficiently estimate the trace of the inverse and the determinant of a large sparse matrix

the deligne complex for the four strand braid group this paper concerns the homotopy type of hyperplane arrangements associated to infinite coxeter groups acting as reflection groups on c n a long standing conjecture states that the complement of such an arrangement should be aspherical some partial results on this conjecture were previously obtained by the author and m davis in this paper we extend those results to another class of coxeter groups the key technical result is that the spherical deligne complex for the 4 strand braid group is cat 1

on estimating the largest eigenvalue with the lanczos algorithm the lanczos algorithm applied to a positive definite matrix produces good approximations to the eigenvalues at the extreme ends of the spectrum after a few iterations in this note we utilize this behavior and develop a simple algorithm which computes the largest eigenvalue the algorithm is especially economical if the order of the matrix is large and the accuracy requirements are low the phenomenon of misconvergence is discussed some simple extensions of the algorithm are also indicated finally some numerical examples and a comparison with the power method are given

generalized harnack inequality for nonhomogeneous elliptic equations this paper is concerned with nonlinear elliptic equations in nondivergence form f d 2 u du x 0 where f has a drift term which is not lipschitz continuous under this condition the equations are nonhomogeneous and nonnegative solutions do not satisfy the classical harnack inequality this paper presents a new generalization of the harnack inequality for such equations as a corollary we obtain the optimal harnack type of inequality for p x harmonic functions which quantifies the strong minimum principle

a matrix exponential spatial specification we introduce the matrix exponential as a way of modelling spatially dependent data the matrix exponential spatial specification mess simplifies the log likelihood allowing a closed form solution to the problem of maximum likelihood estimation and greatly simplifies the bayesian estimation of the model the mess can produce estimates and inferences similar to those from conventional spatial autoregressive models but has analytical computational and interpretive advantages we present maximum likelihood and bayesian approaches to the estimation of this spatial model specification along with methods of model comparisons over different explanatory variables and spatial specifications

keller osserman type conditions for some elliptic problems with gradient terms in this paper we consider the elliptic boundary blow up problems u g u f u in u on where is a smooth bounded domain and the functions f and g are increasing and continuous our main concern will be to prove both existence and nonexistence of nonnegative solutions depending on new integral conditions of keller osserman type involving f and g we show in particular that the problem with a minus sign may have solutions inclusive for some functions g with slightly superquadratic growth at infinity that is somehow not expected we also obtain uniqueness of nonnegative solutions in some cases

differentiation of matrix functionals using triangular factorization in various applications it is necessary to differentiate a matrix functional w a x where a x is a matrix depending on a parameter vector x usually the functional itself can be readily computed from a triangular factorization of a x this paper develops several methods that also use the triangular factorization to efficiently evaluate the first and second derivatives of the functional both the full and sparse matrix situations are considered there are similarities between these methods and algorithmic differentiation however the methodology developed here is explicit leading to new algorithms it is shown how the methods apply to several applications where the functional is a log determinant including spline smoothing covariance selection and restricted maximum likelihood

on the distribution of eigenvalues of a simple undirected graph abstract for a simple undirected graph g n let i g n be the ith largest eigenvalue of g n this paper presents mainly the following 1 for n 4 if g n is incomplete then n 2 n g n 1 1 4 n 3 n 1 2 2 seven sufficient and necessary conditions such that 2 g n 1 3 3 g n 1 implies that j g n 1 j 3 4 n 1

groups generated by reflections and aspherical manifolds not covered by euclidean space a coxeter system r v is a group r a coxeter group together with a set of generators v such that each element of v has order two and such that all relations in r are consequences of relations of the form vw m v w 1 where v w e v and m v w denotes the order of vw the m v w s which are positive integers or oo obviously determine the coxeter system up to isomorphism the list of finite coxeter groups is short and well known however general coxeter groups are much more flexible in fact the set v and the m v w s can be specified arbitrarily at least if v is finite subject only to the conditions m v v 1 and m v w m w v 2 if v w suppose that r v is a coxeter system that x is a hausdorff space and that xv v v is a locally finite family of closed subspaces indexed by v the xv are called the panels of x 2 there is a classical method for constructing a transformation group from these data for each x e x let v x denote the set of v in v such that x e xv for each subset s of v let rs be the subgroup generated by s and let xs be the face of x defined by

the least eigenvalue of unicyclic graphs with n vertices and k pendant vertices abstract let u n k be the set of unicyclic graphs with n vertices and k pendant vertices in this paper we determine the unique graph with the minimal least eigenvalue among all graphs in u n k the work is related with that of guo s g guo the spectral radius of unicyclic and bicyclic graphs with n vertices and k pendant vertices linear algebra appl 408 2005 78 85 which determined the unicyclic graph with the maximal spectral radius in u n k we can observe that the extremal graph on the least eigenvalue is different from that on the spectral radius

extreme eigenfunctions of adjacency matrices for planar graphs employed in spatial analyses mathematical properties of extreme eigenfunctions of popular geographic weights matrices used in spatial statistics are explored and applications of these properties are presented three theorems are proposed and proved these theorems pertain to the popular binary geographic weights matrix an adjacency matrix based upon a planar graph they uncover relationships between the determinant of this matrix and its extreme eigenvalues regression and the minimum eigenvalue of this matrix and the eigenvectors of a row standardized asymmetric version of this matrix and its symmetric similarity matrix counterpart in addition a conjecture is posited pertaining to estimation of the largest eigenvalue of the binary geographic weights matrix when the estimate obtained with the oldest and well known method of matrix powering begins to oscillate between two trajectories in its convergence an algorithm is outlined for calculating the extreme eigenvalues of geographic weights matrices based upon planar graphs and applications results for selected very large adjacency matrices are reported

computing f a b via least squares polynomial approximations given a certain function f various methods have been proposed in the past for addressing the important problem of computing the matrix vector product f a b without explicitly computing the matrix f a such methods were typically developed for a specific function f a common case being that of the exponential this paper discusses a procedure based on least squares polynomials that can in principle be applied to any continuous function f the idea is to start by approximating the function by a spline of a desired accuracy then a particular definition of the function inner product is invoked that facilitates the computation of the least squares polynomial to this spline function since the function is approximated by a polynomial the matrix a is referenced only through a matrix vector multiplication in addition the choice of the inner product makes it possible to avoid numerical integration as an important application we consider the case when f t sqrt t and a is a sparse symmetric positive definite matrix which arises in sampling from a gaussian process distribution the covariance matrix of the distribution is defined by using a covariance function that has a compact support at a very large number of sites that are on a regular or irregular grid we derive error bounds and show extensive numerical results to illustrate the effectiveness of the proposed technique

hyperplane complements of large type let f2 be an open convex subset of a finite dimensional real vectorspace v and m a collection of linear hyperplanes we shall suppose that m is locally finite on f2 this means that each point of f2 has a neighborhood meeting only finitely many hyperplanes in m as in 2 chap 5 f2 is partitioned by m in facets recalling that a plinth is a codimension 2 facet we shall say that m is of large type on f2 if for each plinth p its multiplicity m m p c m 3 the main result of the paper is the following

a fast eigenvalue algorithm for hankel matrices abstract we present an algorithm that can find all the eigenvalues of an n n complex hankel matrix in o n 2 log n operations our scheme consists of an o n 2 log n lanczos type tridiagonalization procedure and an o n qr type diagonalization method

classification of radial solutions for semilinear elliptic systems with nonlinear gradient terms abstract we are concerned with the classification of positive radial solutions for the system u v p v f u where p 0 and f c 1 0 is a nondecreasing function such that f t 0 for all t 0 we show that in the case where the system is posed in the whole space r n such solutions exist if and only if 1 0 s f t d t p 2 p 1 d s this is the counterpart of the keller osserman condition for the case of single semilinear equation similar optimal conditions are derived in case where the system is posed in a ball of r n if f t t q q 1 using dynamical system techniques we are able to describe the behaviour of solutions at infinity in case where the system is posed in the whole r n or around the boundary in case of a ball

a randomized algorithm for approximating the log determinant of a symmetric positive definite matrix abstract we introduce a novel algorithm for approximating the logarithm of the determinant of a symmetric positive definite spd matrix the algorithm is randomized and approximates the traces of a small number of matrix powers of a specially constructed matrix using the method of avron and toledo 1 from a theoretical perspective we present additive and relative error bounds for our algorithm our additive error bound works for any spd matrix whereas our relative error bound works for spd matrices whose eigenvalues lie in the interval 1 1 with 0 1 1 the latter setting was proposed in 16 from an empirical perspective we demonstrate that a c implementation of our algorithm can approximate the logarithm of the determinant of large matrices very accurately in a matter of seconds

existence and non existence of solutions to elliptic equations with a general convection term in this paper we consider the nonlinear elliptic problem u u g u h x in u 0 on where is a smooth bounded domain of n 0 g is an arbitrary c 1 increasing function and h c 1 is non negative we completely analyse the existence and non existence of positive classical solutions in terms of the parameter we show that there exist solutions for every when 0 and the integral 1 g s d s or when 0 and the integral s g s d s conversely when the respective integrals converge and h is non trivial on existence depends on the size of moreover non existence holds for large our proofs mainly rely on comparison arguments and on the construction of suitable supersolutions in annuli our results include some cases where the function g is superquadratic and existence still holds without assuming any smallness condition on

log det approximation based on uniformly distributed seeds and its application to gaussian process regression maximum likelihood estimation mle of hyperparameters in gaussian process regression as well as other computational models usually and frequently requires the evaluation of the logarithm of the determinant of a positive definite matrix denoted by c hereafter in general the exact computation of logdetc is of o n 3 operations where n is the matrix dimension the approximation of logdetc could be developed with o n 2 operations based on power series expansion and randomized trace estimator in this paper the accuracy and effectiveness of using uniformly distributed seeds for logdetc approximation are investigated the research shows that uniform seed based approximation is an equally good alternative to gaussian seed based approximation having slightly better approximation accuracy and smaller variance gaussian process regression examples also substantiate the effectiveness of such a uniform seed based log det approximation scheme

parabolic subgroups of artin groups abstract let a be an artin system for x we denote by a x the subgroup of a generated by x such a group is called a parabolic subgroup of a we reprove van der lek s theorem a parabolic subgroup of an artin group is an artin group we give an algorithm which decides whether two parabolic subgroups of an artin group are conjugate let a be a finite type artin group and let a x be a parabolic subgroup with connected associated coxeter graph the quasi centralizer of a x in a is the set of in a such that x 1 x we prove that the commensurator of a x in a is equal to the normalizer of a x in a and that this group is generated by a x and the quasi centralizer of a x in a moreover if a x is not of type d l l 4 and l even then this group is generated by a x and the centralizer of a x in a

solvability of nonlinear elliptic equations with gradient terms abstract we study the solvability in the whole euclidean space of coercive quasi linear and fully nonlinear elliptic equations modeled on u g u f u u 0 where f and g are increasing continuous functions we give conditions on f and g which guarantee the availability or the absence of positive solutions of such equations in r n our results considerably improve the existing ones and are sharp or close to sharp in the model cases in particular we completely characterize the solvability of such equations when f and g have power growth at infinity we also derive a solvability statement for coercive equations in general form

on the structure of stochastic matrices with a subdominant eigenvalue near 1 abstract an n n irreducible stochastic matrix p can possess a subdominant eigenvalue say 2 p near 1 in this article we clarify the relationship between the nearness of these eigenvalues and the near uncoupling some authors say nearly completely decomposable of p we prove that for fixed n if 2 p is sufficiently close to 1 then p is nearly uncoupled we then provide examples which show that 2 p must in general be remarkably close to 1 before such uncoupling occurs

fast recursive least squares adaptive filtering by fast fourier transform based conjugate gradient iterations recursive least squares rls estimations are used extensively in many signal processing and control applications in this paper we consider rls with sliding data windows involving multiple rank k updating and downdating computations the least squares estimator can be found by solving a near toeplitz matrix system at each step our approach is to employ the preconditioned conjugate gradient method with circulant preconditioners to solve such systems here we iterate in the time domain using toeplitz matrix vector multiplications and precondition in the fourier domain so that the fast fourier transform fft is used throughout the computations the circulant preconditioners are derived from the spectral properties of the given input stochastic process when the input stochastic process is stationary we prove that with probability 1 the spectrum of the preconditioned system is clustered around 1 and the method converges superlinearly provided that a sufficient number of data samples are taken i e

completions branched covers artin groups and singularity theory we study the curvature of metric spaces and branched covers of riemannian manifolds with applications in topology and algebraic geometry here curvature bounds are expressed in terms of the cat inequality we prove a general cat extension theorem giving sufficient conditions on and near the boundary of a locally cat metric space for the completion to be cat we use this to prove that a branched cover of a complete riemannian manifold is locally cat if and only if all tangent spaces are cat 0 and the base has sectional curvature bounded above by we also show that the branched cover is a geodesic space using our curvature bound and a local asphericity assumption we give a sufficient condition for the branched cover to be globally cat and the complement of the branch locus to be contractible we conjecture that the universal branched cover of cn over the mirrors of a finite coxeter group is cat 0 conditionally on this conjecture we use our machinery to prove the arnoldpham thom conjecture on k 1 spaces for artin groups also conditionally we prove the asphericity of moduli spaces of amply lattice polarized k3 surfaces and of the discriminant complements of all the unimodal hypersurface singularities in arnold s hierar

randomized matrix free trace and log determinant estimators we present randomized algorithms for estimating the trace and determinant of hermitian positive semi definite matrices the algorithms are based on subspace iteration and access the matrix only through matrix vector products we analyse the error due to randomization for starting guesses whose elements are gaussian or rademacher random variables the analysis is cleanly separated into a structural deterministic part followed by a probabilistic part our absolute bounds for the expectation and concentration of the estimators are non asymptotic and informative even for matrices of low dimension for the trace estimators we also present asymptotic bounds on the number of samples columns of the starting guess required to achieve a user specified relative error numerical experiments illustrate the performance of the estimators and the tightness of the bounds on low dimensional matrices and on a challenging application in uncertainty quantification arising from bayesian optimal experimental design

expanding the asymptotic explosive boundary behavior of large solutions to a semilinear elliptic equation the main goal of this paper is to study the asymptotic expansion near the boundary of the large solutions of the equation delta u lambda u m f in omega where lambda 0 m 1 f is an element of c omega f 0 and omega is an open bounded set of r n n 1 with boundary smooth enough roughly speaking we show that the number of explosive terms in the asymptotic boundary expansion of the solution is finite but it goes to infinity as in goes to 1 we prove that the expansion consists in two eventual geometrical and non geometrical parts separated by a term independent on the geometry of partial derivative omega but dependent on the diffusion for low explosive sources the non geometrical part does not exist all coefficients depend on the diffusion and the geometry of the domain by means of well known properties of the distance function dist x partial derivative omega for high explosive sources the preliminary coefficients relative to the non geometrical part are independent on omega and the diffusion finally the geometrical part does not exist for very high explosive sources consists in two eventual geometrical and non geometrical parts separated by a term independent on the geometry of partial omega but dependent on the diffusion for low explosive sources the non geometrical part does not exist all coefficients depend on the diffusion and the geometry of the domain by means of well known properties of the distance function rm dist x partial omega dist x for high explosive sources the preliminary coefficients relative to the non geometrical part are independent on omega and the diffusion finally the geometrical part does not exist for very high explosive sources

estimating extremal eigenvalues and condition numbers of matrices the paper describes probabilistic algorithms which may be used to make rough estimates of the largest and smallest eigenvalues of a positive definite matrix and the condition number of a nonsingular matrix in the 2 norm given varepsilon 0 and a prescribed relative error the algorithms compute estimates which with probability at least 1 varepsilon have relative errors less than that prescribed in particular the method gives a reliable way to estimate the condition number of a matrix of large degree

graphs for which the least eigenvalue is minimal ii let g be a connected graph whose least eigenvalue g is minimal among the connected graphs of prescribed order and size we show first that either g is complete or g is a simple eigenvalue in the latter case the sign pattern of a corresponding eigenvector determines a partition of the vertex set and we study the structure of g in terms of this partition we find that g is either bipartite or the join of two graphs of a simple form

boundary behavior for large solutions to elliptic equations with singular weights abstract in this paper we analyze the boundary behavior of large positive solutions to some semilinear elliptic equations which include a singular weight the most important point is that the growth of the solutions can be determined in terms of the solution to a one dimensional first order equation we also consider the questions of existence and uniqueness of positive solutions

groups quasi isometric to right angled artin groups we characterize groups quasi isometric to a right angled artin group raag g with finite outer automorphism group in particular all such groups admit a geometric action on a cat 0 cube complex that has an equivariant fibering over the davis building of g this characterization will be used in forthcoming work of the first author to give a commensurability classification of the groups quasi isometric to certain raags

cohomology of affine artin groups and applications the result of this paper is the determination of the cohomology of artin groups of type an b n and a n with non trivial local coefficients the main result is an explicit computation of the cohomology of the artin group of type b n with coefficients over the module q q 1 t 1 here the first n 1 standard generators of the group act by q multiplication while the last one acts by t multiplication the proof uses some technical results from previous papers plus computations over a suitable spectral sequence the remaining cases follow from an application of shapiro s lemma by considering some well known inclusions we obtain the rational cohomology of the artin group of affine type an as well as the cohomology of the classical braid group br n with coefficients in the n dimensional representation presented in tong yang and ma 1996 the topological counterpart is the explicit construction of finite cw complexes endowed with a free action of the artin groups which are known to be k 1 spaces in some cases including finite type groups particularly simple formulas for the euler characteristic of these orbit spaces are derived

the vertex edge independence number vertex edge cover number and the least eigenvalue of a graph in this paper we characterize the unique graph whose least eigenvalue attains the minimum among all graphs of a fixed order and a given vertex edge independence number or vertex edge cover number and get some bounds for the vertex edge independence number vertex edge cover number of a graph in terms of the least eigenvalue of the graph

symmetrical subgroups of artin groups we show that the subgroup fixed by a group of symmetries of an artin system a s is itself an artin group under the hypothesis that the deligne complex associated to a admits a suitable cat 0 metric such a metric is known to exist for all artin groups of type fc which include all the finite type artin groups as well as many infinite types we also recover the previously known analogous result for an arbitrary coxeter system w s

on the asymptotic equivalence of circulant and toeplitz matrices any sequence of uniformly bounded n times n hermitian toeplitz matrices h n is asymptotically equivalent to a certain sequence of n times n circulant matrices c n derived from the toeplitz matrices in the sense that left h n c n right f o sqrt n as n rightarrow infty this implies that certain collective behaviors of the eigenvalues of each toeplitz matrix are reflected in those of the corresponding circulant matrix and supports the utilization of the computationally efficient fast fourier transform instead of the karhunen loeve transform in applications like coding and filtering in this paper we study the asymptotic performance of the individual eigenvalue estimates we show that the asymptotic equivalence of the circulant and toeplitz matrices implies the individual asymptotic convergence of the eigenvalues for certain types of toeplitz matrices we also show that these estimates asymptotically approximate the largest and smallest eigenvalues for more general classes of toeplitz matrices

boundary blow up elliptic problems of bieberbach and rademacher type with nonlinear gradient terms abstract by a perturbation method and constructing comparison functions we show the exact asymptotic behaviour of solutions near the boundary to nonlinear elliptic problems u u q b x e u x u where is a bounded domain with smooth boundary in r n q 0 b is non negative and non trivial in which may be vanishing on the boundary

monte carlo estimates of the log determinant of large sparse matrices maximum likelihood estimates of parameters of some spatial models require the computation of the log determinant of positive definite matrices of the formi d whered is a large sparse matrix with eigenvalues in 1 1 and where 0 1 with extremely large matrices the usual direct methods of obtaining the log determinant require too much time and memory we propose a monte carlo estimate of the log determinant this estimate is simple to program very sparing in its use of memory easily computed in parallel and can estimate log det i d for many values of simultaneously using this estimator we estimate the log determinant for a 1 000 000 1 000 000 matrixd for 100 values of in 23 1 min on 133 mhz pentium with 64 mb of memory using matlab

metric systolicity and two dimensional artin groups we introduce the notion of metrically systolic simplicial complexes we study geometric and large scale properties of such complexes and of groups acting on them geometrically we show that all two dimensional artin groups act geometrically on metrically systolic complexes as direct corollaries we obtain new results on two dimensional artin groups and all their finitely presented subgroups we prove that the conjugacy problem is solvable and that the dehn function is quadratic we also show several large scale features of finitely presented subgroups of two dimensional artin groups lying background for further studies concerning their quasi isometric rigidity

existence and blow up rate of large solutions of p x laplacian equations with gradient terms abstract in this paper we investigate boundary blow up solutions of the problem p x u f x u k x u m x in u x as d x 0 where p x u div u p x 2 u is called the p x laplacian our results extend the previous work 25 of y liang q h zhang and c s zhao from the radial case to the non radial setting and 46 due to q h zhang and d motreanu from the assumption that k x u x m x is a small perturbation to the case in which k x u m x is a large perturbation we provide an exact estimate of the pointwise different behavior of the solutions near the boundary in terms of d x and in terms of the growth of the exponents furthermore the comparison principle is no longer applicable in our context since f x is not assumed to be monotone in this paper

a general scheme for log determinant computation of matrices via stochastic polynomial approximation abstract we study the approximation of determinant for large scale matrices with low computational complexity this paper develops a generalized stochastic polynomial approximation frame as well as a stochastic legendre approximation algorithm to calculate log determinants of large scale positive definite matrices based on the prior eigenvalue distributions the generalized frame is implemented by weighted l 2 orthogonal polynomial expansions with an efficient recursion formula and matrix vector multiplications so the proposed scheme is efficient both in computational complexity and data storage respective error bounds are given in theory which guarantee the convergence of the proposed algorithms we illustrate the effectiveness of our method by numerical experiments on both synthetic matrices and counting spanning trees

the distribution of eigenvalues of graphs abstract we show that every limit point of the kth largest eigenvalues of graphs is a limit point of the k 1 th largest eigenvalues and we find out the smallest limit point of the kth largest eigenvalues and an upper bound of the limit points of the kth smallest eigenvalues for k 4 we prove that there exists a gap beyond the smallest limit point in which no point is the limit point of the kth largest eigenvalues for the third largest eigenvalues of a graph g with at least three vertices we obtain that 1 3 g 1 3 g 1 5 2 consequently if gc is not a complete bipartite graph plus isolated vertices 3 g 3 d n where d n is the complement of the double star s 1 n 3

the boundary blow up rate of large solutions abstract in this paper we ascertain the blow up rate of the large solutions of a class of sublinear elliptic boundary value problems with a weight function in front of the nonlinearity that vanishes on the boundary of the underlying domain at different rates according to the point of the boundary x all previous results in the literature assumed the decay rate of the underlying weight function to be the same at any point of this hypothesis substantially simplified the mathematical analysis of the problem as it allowed constructing global sub and supersolutions in an open neighborhood of obtaining general results requires localizing at each particular point of the boundary making particularly involved the mathematical analysis of the problem

eigenfunction properties and approximations of selected incidence matrices employed in spatial analyses abstract mathematical properties of eigenfunctions of selected incidence matrices appearing in spatial statistics formulae are summarized seven theorems are proposed and proved and three conjectures are posited results summarized here allow the determinant of massively large n n geographic weights matrices to be accurately approximated in addition the behavior of eigenfunctions for graphs affiliated with a linear configuration of connected nodes are better understood

uniqueness of positive solutions for a boundary blow up problem abstract in this paper we prove the uniqueness of the positive solution for the boundary blow up problem u f u in u on where is a c 2 bounded domain in r n under the hypotheses that f t is nondecreasing in t 0 and f t t p is increasing for large t and some p 1 we also consider the uniqueness of a related problem when the equation includes a nonnegative weight a x

boundary blow up elliptic problems with nonlinear gradient terms and singular weights by karamata regular variation theory a perturbation method and construction of comparison functions we show the exact asymptotic behaviour of solutions near the boundary to nonlinear elliptic problems u u q b x g u u 0 in u where is a bounded domain with smooth boundary in n q 0 g c 1 0 is increasing on 0 g 0 0 g is regularly varying at infinity with positive index and b is non negative in and is singular on the boundary

a lanczos bidiagonalization algorithm for hankel matrices abstract this paper presents a fast algorithm for bidiagonalizing a hankel matrix an m n hankel matrix is reduced to a real bidiagonal matrix in o m n n log m n floating point operations flops using the lanczos method with modified partial orthogonalization and reset schemes to improve its stability performance improvement is achieved by exploiting the hankel structure as fast hankel matrix vector multiplication is used the accuracy and efficiency of the algorithm are demonstrated by our numerical experiments

fast direct methods for gaussian processes a number of problems in probability and statistics can be addressed using the multivariate normal gaussian distribution in the one dimensional case computing the probability for a given mean and variance simply requires the evaluation of the corresponding gaussian density in the n dimensional setting however it requires the inversion of an n times n covariance matrix c as well as the evaluation of its determinant det c in many cases such as regression using gaussian processes the covariance matrix is of the form c sigma 2 i k where k is computed using a specified covariance kernel which depends on the data and additional parameters hyperparameters the matrix c is typically dense causing standard direct methods for inversion and determinant evaluation to require mathcal o n 3 work this cost is prohibitive for large scale modeling here we show that for the most commonly used covariance functions the matrix c can be hierarchically factored into a product of block low rank updates of the identity matrix yielding an mathcal o n log 2 n algorithm for inversion more importantly we show that this factorization enables the evaluation of the determinant det c permitting the direct calculation of probabilities in high dimensions under fairly broad assumptions on the kernel defining k our fast algorithm brings many problems in marginalization and the adaptation of hyperparameters within practical reach using a single cpu core the combination of nearly optimal scaling in terms of problem size with high performance computing resources will permit the modeling of previously intractable problems we illustrate the performance of the scheme on standard covariance kernels