rank revealing qr factorizations abstract an algorithm is presented for computing a column permutation pi and a qr factorization a pi qr of an m by n m or n matrix a such that a possible rank deficiency of a will be revealed in the triangular factor r having a small lower right block for low rank deficient matrices the algorithm is guaranteed to reveal the rank of a and the cost is only slightly more than the cost of one regular qr factorization a posteriori upper and lower bounds on the singular values of a are derived and can be used to infer the numerical rank of a keywords qr factorization rank deficient matrices least squares computation subset selection rank singular values

multiple solutions of the quasirelativistic choquard equation we prove existence of multiple solutions to the quasirelativistic choquard equation with a scalar potential

inertia controlling methods for general quadratic programming active set quadratic programming qp methods use a working set to define the search direction and multiplier estimates in the method proposed by fletcher in 1971 and in several subsequent mathematically equivalent methods the working set is chosen to control the inertia of the reduced hessian which is never permitted to have more than one nonpositive eigenvalue such methods will be called inertia controlling this paper presents an overview of a generic inertia controlling qp method including the equations satisfied by the search direction when the reduced hessian is positive definite singular and indefinite recurrence relations are derived that define the search direction and lagrange multiplier vector through equations related to the karush kuhn tucker system discussion is included of connections with inertia controlling methods that maintain an explicit factorization of the reduced hessian matrix

a note on the undirected rural postman problem polytope this note refers to the article by g ghiani and g laporte a branch and cut algorithm for the undirected rural postman problem math program 87 2000 we show that some conditions for the facet defining property of the basic non trivial inequalities are not sufficient and that the rural postman problem polytope is more complex even when focusing on canonical inequalities only

lu decompositions of generalized diagonally dominant matrices using the simple vehicle ofm matrices the existence and stability oflu decompositions of matricesa which can be scaled to diagonally dominant possibly singular matrices are investigated bounds on the growth factor for gaussian elimination ona are derived motivation for this study is provided in part by applications to solving homogeneous systems of linear equationsax 0 arising in markov queuing networks input output models in economics and compartmental systems wherea or a is an irreducible singularm matrix this paper extends earlier work by funderlic and plemmons and by varga and cai

on the augmented system approach to sparse least squares problems we study the augmented system approach for the solution of sparse linear least squares problems it is well known that this method has better numerical properties than the method based on the normal equations we use recent work by arioli et al 1988 to introduce error bounds and estimates for the components of the solution of the augmented system in particular we find that using iterative refinement we obtain a very robust algorithm and our estimates of the error are accurate and cheap to compute the final error and all our error estimates are much better than the classical or skeel s error analysis 1979 indicates moreover we prove that our error estimates are independent of the row scaling of the augmented system and we analyze the influence of the bjorck scaling 1967 on these estimates we illustrate this with runs both on large scale practical problems and contrived examples comparing the numerical behaviour of the augmented systems approach with a code using the normal equations these experiments show that while the augmented system approach with iterative refinement can sometimes be less efficient than the normal equations approach it is comparable or better when the least squares matrix has a full row and is in any case much more stable and robust

positive solutions for some non autonomous schrodinger poisson systems abstract in this paper we study the schrodinger poisson system sp u u k x x u a x u p 1 u x r 3 k x u 2 x r 3 with p 3 5 assuming that a r 3 r and k r 3 r are nonnegative functions such that lim x a x a 0 lim x k x 0 and satisfying suitable assumptions but not requiring any symmetry property on them we prove the existence of positive solutions

nodal solutions for the choquard equation abstract we consider the general choquard equations u u i u p u p 2 u where i is a riesz potential we construct minimal action odd solutions for p n n n n 2 and minimal action nodal solutions for p 2 n n 2 we introduce a new minimax principle for least action nodal solutions and we develop new concentration compactness lemmas for sign changing palais smale sequences the nonlinear schrodinger equation which is the nonlocal counterpart of the choquard equation does not have such solutions

ground state and bounded state solution for the nonlinear fractional choquard schrodinger poisson system in this paper we study a class of fractional choquard schrodinger poisson system under some suitable assumptions by using a variational approach we establish the existence of non negative ground state solution and also discuss the nonexistence of ground states moreover we prove the existence of bounded state by using a linking theorem

algorithms for almost block diagonal linear systems codes for solving systems of ordinary differential equations for use in the method of lines for partial differential equations pdes usually provide only a banded system solver in this context a frequently occurring structure is almost block diagonal abd solving abd systems by imposing banded structure introduces fill in and is inefficient though robust efficient abd software has been developed and used in packages for solving boundary value problems with separated boundary conditions for ordinary differential equations bvodes it has not been generally exploited in pde software the situation with bordered almost block diagonal system software for bvodes with nonseparated boundary conditions is less satisfactory this survey draws on material from a variety of sources particularly p amodio et al numer linear algebra appl 7 2000 pp 275 317 and b garrett and i gladwell j comput methods sci engrg 1 2001 pp 75 98 and the references therein

fractional laplacian in conformal geometry abstract in this note we study the connection between the fractional laplacian operator that appeared in the recent work of caffarelli and silvestre and a class of conformally covariant operators in conformal geometry

classification of positive solitary solutions of the nonlinear choquard equation in this paper we settle the longstanding open problem concerning the classification of all positive solutions to the nonlinear stationary choquard equation delta u u 2u left frac 1 x u 2 right 0 quad u in h 1 mathbb r 3 which can be considered as a certain approximation of the hartree fock theory for a one component plasma as explained in lieb and lieb simon s papers starting from 1970s we first prove that all the positive solutions of this equation must be radially symmetric and monotone decreasing about some fixed point interestingly to use the new method of moving planes introduced by chen li ou we deduce the problem into an elliptic system as a key step we transform this differential system into a system of integral equations with the help of riesz and bessel potentials and then use the method of a moving plane in an integral form next using radial symmetry we deduce the uniqueness result from lieb s work our argument can be adapted well to study the radial symmetry of positive solutions of the equation in the generalized form u k 1 f left u k 2 u right

rounding error and perturbation bounds for the cholesky and ldl t factorizations abstract componentwise rounding error and perturbation bounds for the cholesky and ldl t factorizations are derived the precision of the computed cholesky and ldl t factorizations is given

floating point perturbations of hermitian matrices abstract we consider the perturbation properties of the eigensolution of hermitian matrices for the matrix entries and the eigenvalues we use the realistic floating point error measure a a recently demmel and veselic considered the same problem for a positive definite matrix h showing that the floating point perturbation theory holds with constants depending on the condition number of the matrix a dhd where aii 1 and d is a diagonal scaling we study the general hermitian case along the same lines thus obtaining new classes of well behaved matrices and matrix pairs our theory is applicable to the already known class of scaled diagonally dominant matrices as well as to matrices given by factors like those in symmetric indefinite decompositions we also obtain norm estimates for the perturbations of the eigenprojections and show that some of our techniques extend to non hermitian matrices however unlike in the positive definite case we are still unable to describe simply the set of all well behaved hermitian matrices

computing the singular value decomposition with high relative accuracy we analyze when it is possible to compute the singular values and singular vectors of a matrix with high relative accuracy this means that each computed singular value is guaranteed to have some correct digits even if the singular values have widely varying magnitudes this is in contrast to the absolute accuracy provided by conventional backward stable algorithms which in general only guarantee correct digits in the singular values with large enough magniturds it is of interest to compute the tiniest singular values with several correct digits because in some cases such as finite element problems and quantum mechanics it is the smallest singular values that have physical meaning and should be determined accurately by the data many recent papers have identified special classes of matrices where high relative accuracy is possible since it is not possible in general the perturbation theory and algorithms for these matrix classes have been quite different motivating us to seek a common perturbation theory and common algorithm we provide these in this paper and show that high relative accuracy is possible in many new cases as well the briefest way to describe our results is that we can compute the svd to high relative accuracy provided we can compute a high accuracy pivoted ldu decomposition we provide many examples of matrix classes permitting such an ldu decomposition

existence of ground state solutions for the nonlinear fractional schrodinger poisson system with critical sobolev exponent abstract in this paper we study the existence of ground state solutions for the nonlinear fractional schrodinger poisson system with critical sobolev exponent s u v x u u u q 1 u u 2 s 2 u in r 3 t u 2 in r 3 where r is a parameter 1 q 2 s 1 3 2 s 3 2 s s t 0 1 and 2 s 2 t 3 under certain assumptions on v x using the method of pohozaev nehari manifold and the arguments of brezis nirenberg the monotonic trick and global compactness lemma we prove the existence of a nontrivial ground state solution

concentrating standing waves for the fractional nonlinear schrodinger equation abstract we consider the semilinear equation e 2 s s u v x u u p 0 u 0 u h 2 s r n where 0 s 1 1 p n 2 s n 2 s v x is a sufficiently smooth potential with inf r v x 0 and e 0 is a small number letting w be the radial ground state of s w w w p 0 in h 2 s r n we build solutions of the form u e x i 1 k w i x i e e where i v i e and the i e approach suitable critical points of v via a lyapunov schmidt variational reduction we recover various existence results already known for the case s 1 in particular such a solution exists around k nondegenerate critical points of v for s 1 this corresponds to the classical results by floer and weinstein 13 and oh 24 25

groundstates of nonlinear choquard equations existence qualitative properties and decay asymptotics we consider a semilinear elliptic problem delta u u i alpha ast abs u p abs u p 2 u quad text in r n where i alpha is a riesz potential and p 1 this family of equations includes the choquard or nonlinear schr odinger newton equation for an optimal range of parameters we prove the existence of a positive groundstate solution of the equation we also establish regularity and positivity of the groundstates and prove that all positive groundstates are radially symmetric and monotone decaying about some point finally we derive the decay asymptotics at infinity of the groundstates

componentwise error analysis for the block lu factorization of totally nonnegative matrices abstract for the first time perturbation bounds including componentwise perturbation bounds for the block lu factorization have been provided by dopico and molera 2005 5 in this paper componentwise error analysis is presented for computing the block lu factorization of nonsingular totally nonnegative matrices we present a componentwise bound on the equivalent perturbation for the computed block lu factorization consequently combining with the componentwise perturbation results we derive componentwise forward error bounds for the computed block factors

regularity of the obstacle problem for a fractional power of the laplace operator given a function and s 0 1 we will study the solutions of the following obstacle problem u in r n s u 0i nr n s u x 0 for those x such that u x x lim x u x 0 we show that when is c 1 s or smoother the solution u is in the space c 1 for every s in the case where the contact set u is convex we prove the optimal regularity result u c 1 s when is only c 1 for a s we prove that our solution u is c 1 for every c 2006 wiley periodicals inc

transformation of facets of the general routing problem polytope we study the class of polytopes associated with the 0 1 variable formulation by g ghiani and g laporte math program 87 2000 pp 467 481 of the rural postman problem and general routing problem we introduce a technique called switching which allows one to obtain new valid or facet defining inequalities from known ones we study the switched forms which can be obtained from two important types of known valid inequalities namely so called path bridge a n letchford eur j oper res 96 1997 pp 317 322 and honeycomb a corberan and j m sanchis eur j oper res 108 1998 pp 538 550 inequalities for the switched path bridge inequalities a generic blueprint for separation routines is developed we give a polynomial time separation algorithm for a special subclass of these inequalities

a branch and cut algorithm for the maximum benefit chinese postman problem the maximum benefit chinese postman problem mbcpp is an np hard problem that considers several benefits associated with each edge one for each time the edge is traversed with a service the objective is to find a closed walk with maximum benefit we propose an ip formulation for the undirected mbcpp and based on the description of its associated polyhedron we propose a branch and cut algorithm and present computational results on instances with up to 1 000 vertices and 3 000 edges

existence of semiclassical ground state solutions for a generalized choquard equation abstract in this paper we study a generalized quasilinear choquard equation e p p u v x u p 2 u e n r n q y f u y x y q x f u in r n where p is the p laplacian operator 1 p n v and q are two continuous real functions on r n 0 n f s is the primate function of f s and e is a positive parameter under suitable assumptions on p and f we establish a new concentration behavior of solutions for the quasilinear choquard equation by variational methods the results are also new for the semilinear case p 2

existence of groundstates for a class of nonlinear choquard equations we prove the existence of a nontrivial solution h n to the nonlinear choquard equation i in n where i is a riesz potential under almost necessary conditions on the nonlinearity in the spirit of berestycki and lions this solution is a groundstate if moreover is even and monotone on 0 then is of constant sign and radially symmetric

semiclassical analysis for pseudo relativistic hartree equations abstract in this paper we study the semiclassical limit for the pseudo relativistic hartree equation e 2 m 2 u v u i u p u p 2 u in r n where m 0 2 p 2 n n 1 v r n r is an external scalar potential i x c n x n is a convolution kernel c n is a positive constant and n 1 p n n for n 3 p 2 our equation becomes the pseudo relativistic hartree equation with coulomb kernel

componentwise analysis of direct factorization of real symmetric and hermitian matrices abstract we derive componentwise error bound for the factorization h gjg t where h is a real symmetric matrix g has full column rank and j is diagonal with 1 s on the diagonal we also derive a componentwise forward error bound that is we bound the difference between the exact and the computed factor g in the cases where such a bound is possible we extend these results to the hermitian case and to the well known bunch parlett factorization finally we prove bounds for the scaled condition of the matrix g and show that the factorization can have the rank revealing property