on the existence and computation of rank revealing lu factorizations abstract by exploring properties of schur complements this paper presents bounds on the existence of rank revealing lu factorizations that are comparable with those of rank revealing qr factorizations the new bounds provide substantial improvement over previously derived bounds this paper also proposes two algorithms using gaussian elimination with a block pivoting strategy to select a subset of columns from a given matrix which has a guaranteed relatively large smallest singular value each of these two algorithms is faster than its orthogonal counterpart for dense matrices if implemented appropriately these algorithms are faster than the corresponding rank revealing qr methods even when the orthogonal matrices are not explicitly updated based on these two algorithms an algorithm using only gaussian elimination for computing rank revealing lu factorizations is introduced

multiple positive solutions for second order m point boundary value problems abstract in this paper the second order m point boundary value problem u t f t u 0 0 t 1 u 0 0 u 1 i 1 m 2 k i u i 0 is studied where k i 0 i 1 2 m 2 0 1 2 m 2 1 we give at first the associated green s function and its properties for the m point boundary value problem and growth conditions are imposed on f which yield the existence of at least three positive solutions by using the five functionals fixed point theorem and the properties

nonzero solutions of hammerstein integral equations with discontinuous kernels using the theory of fixed point index we establish new results for the existence of nonzero solutions of integral equations of the form u t gk t s f s u s ds where g is a compact set in rn and k changes sign so positive solutions may not exist f satisfies caratheodory conditions and k may be discontinuous we apply our results to prove the existence of nontrivial solutions of some nonlocal boundary value problems

positive solutions of nth order m point boundary value problem abstract in this paper we study the existence of positive solutions of n th order m point boundary value problem u n t a t f u t 0 t 0 1 u 0 u 0 u n 2 0 0 u 1 i 1 m 2 i u i where n 2 m 2 0 1 2 m 2 1 i 0 i 1 2 m 2 with i 1 m 2 i i n 1 1 we obtained the existence of positive solutions by using the fixed point theorems of cone expansion and compression of norm type the results obtained in this paper essentially improve and generalize some well known results

a volume associated with m x n matrices abstract let a mxnr with nonzero singular values 1 2 r the volume of a vol a is defined as zero if r 0 and otherwise vola ri 1 i or equivalently vola det2aij summing over all r x r nonsingularsubmatrices aij the matrix volume vol a generalizes the absolute value of the determinant from nonsingular to arbitrary matrices any r dimensional unit cube in r at is mapped under a into a parallelepiped of volume vol a this paper gives properties and applications of vol a in particular the moore penrose inverse of a is a convex combination of ordinary inverses of its maximal nonsingular submatrices a i j ij or is a convex combination of moore penrose inverses of its maximal full rank submatrices a i i j j where the convex weights are proport ional to the squares of volumes of the corresponding submatrices since a a the matrix a is a convex combination of inverses of submatrices of a interestingly the weights of these convex combinations are the above weights in transposed position

twin monotone positive solutions to a singular nonlinear third order differential equation abstract this paper discusses the existence of at least one or two nondecreasing positive solutions for the following singular nonlinear third order differential equation x t t f t x t 0 a t b x a x a x b 0 where 0 is a parameter c a b r f c a b 0 r t may be singular at t a b and f t s may be singular at s 0 green s function and the fixed point theorem of cone expansion and compression type are utilized to illustrate the results four nontrivial examples are included

green s function and positive solutions of nth order m point boundary value problem abstract in this paper we study the expression and properties of green s function for nth order m point boundary value problem u n t a t f u 0 t 0 1 u 0 u 0 u n 2 0 0 u 1 i 1 m 2 i u i where 0 0 with i 1 m 2 i i n 1 1 furthermore we obtained the existence of positive solutions by meas of fixed point index theory

positive solutions of a nonlinear m point boundary value problem let ai 0 for i 1 m 3 and am 2 0 let i satisfy 0 1 2 m 2 1 and m 2i 1 ai i 1 we study the existence of positive solutions to the boundary value problem where a c 0 1 0 and f c 0 0 we show the existence of at least one positive solution if f is either superlinear or sublinear by applying a fixed point theorem in cones

three positive solutions for second order m point boundary value problems for the second order m point boundary value problemu t f t u 0 0 0 i 1 2 m 2 0 x 1 x 2 x m 2 1 growth conditions are imposed on f which yield the existence of at least three positive solutions by using the leggett williams fixed point theorem the associated green s function for the m point boundary value problem is also given

existence of positive solutions for nonlinear third order three point boundary value problems abstract this paper is concerned with the following nonlinear third order three point boundary value problem u t a t f u t 0 t 0 1 u 0 u 0 0 u 1 u where 0 1 and 1 1 first the green s function for the associated linear boundary value problem is constructed and then some useful properties of the green s function are obtained by a new method finally existence results for at least one positive solution for the above problem are established when f is superlinear or sublinear

efficient algorithms for computing a strong rank revealing qr factorization given anm n matrixm withm n it is shown that there exists a permutation fi and an integer k such that the qr factorization myi q ak ckbk reveals the numerical rank of m the k k upper triangular matrix ak is well conditioned ilckll2 is small and bk is linearly dependent on ak with coefficients bounded by a low degree polynomial in n existing rank revealing qr rrqr algorithms are related to such factorizations and two algorithms are presented for computing them the new algorithms are nearly as efficient as qr with column pivoting for most problems and take o ran2 floating point operations in the worst case

positive solutions to a generalized second order three point boundary value problem abstract in this paper by using krasnoselskii fixed point theorem we study the existence of at least or two positive solutions to the nonlinear second order three point boundary value problem y t a t f y t 0 0 t t y 0 y y t y where 0 t 0 t 0 t t are given constants as an application we also give some examples to demonstrate our results

positive solutions of nonlinear singular third order two point boundary value problem abstract in this paper we are concerned with the existence of single and multiple positive solutions to the nonlinear singular third order two point boundary value problem u t a t f u t 0 0 t 1 u 0 u 0 u 1 0 where is a positive parameter under various assumptions on a and f we establish intervals of the parameter which yield the existence of at least one at least two and infinitely many positive solutions of the boundary value problem by using krasnoselskii s fixed point theorem of cone expansion compression type

positive solutions of nonlinear second order m point boundary value problem abstract in this paper by using krasnoselskii s fixed point theorem of cone expansion compression type and under suitable conditions we present the existence of single and multiple positive solutions to the nonlinear second order m point boundary value problem u t a t f u t 0 0 t 1 u 0 0 u 1 i 1 m 2 a i u i where is a positive parameter a i 0 for i 1 2 m 3 and a m 2 0 i satisfy 0 1 2 m 2 1 and i 1 m 2 a i i 1 we derive an explicit interval of such that for any in this interval the existence of at least one positive solution to the boundary value problem is guaranteed and the existence of at least two solutions for in an appropriate interval is also discussed

on the existence of nonnegative solutions to an integral equation with applications to boundary value problems abstract by the use of a guo krasnoselskii theorem in cones existence of positive eigenvalues yielding nonnegative or positive solutions to an integral equation is studied the results are applied to a variety of boundary value problems concerning ordinary differential equations

multiple positive solutions for nth order m point boundary value problems with all derivatives abstract we investigate the existence of positive solutions for the boundary value problem u n t f t u t u t u n 2 t u n 1 t 0 0 t 1 u 0 u 0 u n 2 0 0 u n 1 1 i 1 m 2 k i u n 1 i where f 0 1 r n r is continuous k i 0 i 1 2 m 2 0 i 1 m 2 k i 1 we give at first the associated green s function and some of its properties then imposing growth conditions on f we obtain the existence of at least three positive solutions for the boundary value problem by using the five functional fixed point theorem finally we give an example to demonstrate our result

rank revealing gaussian elimination by the maximum volume concept abstract a gaussian elimination algorithm is presented that reveals the numerical rank of a matrix and identifies a square nonsingular submatrix of maximum dimension the bounds on the singular values of the submatrix and its schur complement are similar to the best known bounds for rank revealing lu factorization but in contrast to existing methods the algorithm does not make use of the normal matrix an implementation for dense matrices is described whose computational cost is roughly twice the cost of an lu factorization with complete pivoting because of its flexibility in choosing pivot elements the algorithm is amenable to implementation with blocked memory access and for sparse matrices

positive solution for m point boundary value problems in this paper we investigate an m point boundary value problem with sign changing nonlinearity the existence of an interval of parameters which ensures the problem has at least one positive solution is determined by constructing available operator and combining the method of lower solution with the method of topology degree moreover the associated green s function for the above problem is also given