modified hierarchy basis for solving singular boundary value problems abstract in this paper we develop an efficient preconditioning method on the basis of the modified hierarchy basis for solving the singular boundary value problem by the galerkin method after applying the preconditioning method we show that the condition number of the linear system arising from the galerkin method is uniformly bounded in particular the condition number of the preconditioned system will be bounded by 2 for the case q x 0 see eq 1 in the paper numerical results are presented to confirm our theoretical results

a cosine maximization method for the priority vector derivation in ahp abstract the derivation of a priority vector from a pair wise comparison matrix pcm is an important issue in the analytic hierarchy process ahp the existing methods for the priority vector derivation from pcm include eigenvector method ev weighted least squares method wls additive normalization method an logarithmic least squares method lls etc the derived priority vector should be as similar to each column vector of the pcm as possible if a pair wise comparison matrix pcm is not perfectly consistent therefore a cosine maximization method cm based on similarity measure is proposed which maximizes the sum of the cosine of the angle between the priority vector and each column vector of a pcm an optimization model for the cm is proposed to derive the reliable priority vector using three numerical examples the cm is compared with the other prioritization methods based on two performance evaluation criteria euclidean distance and minimum violation the results show that the cm is flexible and efficient

the affine polya szego principle equality cases and stability a brothers ziemer type theorem for the affine polya szego principle and a quantitative affine polya szego principle are established

on the finite element method for elliptic problems with degenerate and singular coefficients we consider dirichlet boundary value problems for second order elliptic equations over polygonal domains the coefficients of the equations under consideration degenerate at an inner point of the domain or behave singularly in the neighborhood of that point this behavior may cause singularities in the solution the solvability of the problems is proved in weighted sobolev spaces and their approximation by finite elements is studied this study includes regularity results graded meshes and inverse estimates applications of the theory to some problems appearing in quantum mechanics are given numerical results are provided which illustrate the theory and confirm the predicted rates of convergence of the finite element approximations for quasi uniform meshes

affine orlicz polya szego principle for log concave functions abstract the affine l p polya szego principle significantly strengthens the classical polya szego principle it is an open problem whether there exists an affine orlicz polya szego principle which includes the affine l p polya szego principle as special case in this paper an affine orlicz polya szego principle for log concave functions is established by using functional steiner symmetrizations

new orlicz affine isoperimetric inequalities abstract the orlicz brunn minkowski theory received considerable attention recently and many results in the l p brunn minkowski theory have been extended to their orlicz counterparts the aim of this paper is to develop orlicz l affine and geominimal surface areas for a single convex body as well as for multiple convex bodies which generalize the l p mixed affine and geominimal surface areas fundamental concepts in the l p brunn minkowski theory our extensions are different from the general affine surface areas by ludwig in 21 moreover our definitions for orlicz l affine and geominimal surface areas reveal that these affine invariants are essentially the infimum supremum of v k l the orlicz mixed volume of k and the polar body of l where l runs over all star bodies and all convex bodies respectively with volume of l equal to the volume of the unit euclidean ball b 2 n properties for the orlicz l affine and geominimal surface areas such as affine invariance and monotonicity are proved related orlicz affine isoperimetric inequalities are also established

the finite element method for a degenerate elliptic equation a class of second order degenerate elliptic boundary value problems is considered two different finite element formulations are studied and compared optimal order error estimates in weighted norms are proved and numerical results are presented

galerkin methods for singular boundary value problems in one space dimension two galerkin type piecewise polynomial approximation procedures based on bilinear forms with different weight functions are analyzed and compared optimal order error estimates are proved and numerical results are presented 1 introduction in this paper we shall discuss galerkin piecewise polynomial approximation methods for the singular two point boundary value problem b 1 1 lu x u x u x q x u x f x x e i 0 1 x

a fourth order finite difference method based on non uniform mesh for a class of singular two point boundary value problems abstract we present a three point finite difference method based on non uniform mesh for a class of singular two point boundary value problem x y f x y y 0 a y 1 b 0 we show that the method based on non uniform mesh provides o h4 convergent approximations this method is illustrated by two numerical examples one is linear and the other is non linear

the monotone iterative method and zeros of bessel functions for nonlinear singular derivative dependent bvp in the presence of upper and lower solutions abstract in this paper we consider a class of nonlinear singular boundary value problems x y x x f x y x x y x 0 0 x 1 y 0 y 1 0 for 1 we assume that the source function f x y x y is lipschitz in x y and one sided lipschitz in y the initial approximations are an upper solution u 0 x and a lower solution v 0 x which can be ordered in one way v 0 x u 0 x or the other u 0 x v 0 x we propose an iterative scheme and establish the existence of solutions bounded by v 0 and u 0 and allow f y to take both positive and negative values the method is constructive in nature and can be used to generate solutions of the nonlinear singular boundary value problems

a uniform mesh finite difference method for a class of singular two point boundary value problems a new finite difference method based on uniform mesh is given for the weakly singular two point boundary value problem x alpha y f x y y 0 a y 1 b 0 alpha 1 under quite general conditions on f and f we show that our present method based on uniform mesh provides o h 2 convergent approximations for all alpha in 0 1 our method is based on one evaluation of f and for alpha 0 it reduces to the classical second order method for y f x y

deriving priorities from pairwise comparison matrices with a novel consistency index abstract it is important to measure the inconsistency level of a pairwise comparison matrix pcm and derive the priority vector in the analytic hierarchy process ahp in the present study a new consistency index is proposed by using the cosine similarity measures of two row column vectors in a pcm it is called as the double cosine similarity consistency index dcsci since the row and column vectors are all considered some interesting properties of dcsci are investigated and the thresholds for inconsistency tolerance level are discussed in detail then following the idea of dcsci we provide a new method for obtaining the priority vector from a pcm through maximizing the sum of the cosine similarity measures between the priority vector and the row column vectors the priority vector is derived by solving the constructed optimization problem by analyzing the proposed double cosine similarity maximization dcsm method it is found that the existing cosine maximization cm method can be retrieved by carrying out numerical examples some comparisons with the existing methods show that the proposed index and method are effective and flexible

a novel numerical approach and its convergence for numerical solution of nonlinear doubly singular boundary value problems this paper deals with two new recursive numerical schemes for the solution of a class of doubly singular two point boundary value problems with different types of boundary conditions we first transform the original problem into an equivalent integral equation to overcome the singular behaviour at the origin and then establish recursive schemes by employing homotopy perturbation method for the solution of the resultant equation we establish the convergence results of the new numerical methods four numerical examples are considered to demonstrate the efficiency and accuracy of the proposed methods the present methods are shown their advantage over some existing methods chawla and katti 1982 pandey and singh 2007 pandey and singh 2004 caglar et al 2009 kumar and aziz 2004 khuri and sayfy 2010 developed for the solution of singular two point boundary value problems moreover the new methods are used to obtain the approximate solutions of the singular boundary value problems arising in various physical problems

the consistency index in reciprocal matrices comparison of deterministic and statistical approaches abstract when checking the inconsistency level of a positive reciprocal matrix saaty uses a deterministic criterion based on two parameters a benchmark the average and a consistency level usually 10 using results from a simulation experiment with 100 000 positive random reciprocal matrices of size varying from 3 to 15 we developed a probabilistic criterion and compare it to saaty s index we found that if a positive reciprocal matrix is consistent according to the deterministic criterion is also consistent according to the probabilistic criterion only if we accept a higher than usual probability of type i error reducing this error implies that the benchmark must be a small percentile of the probability distribution of the consistency index

a note on existence uniqueness results for a class of doubly singular boundary value problems abstract in this paper we establish the existence and uniqueness of solution for nonlinear singular boundary value problem p x y x q x f x y for 0 x b and lim x 0 p x y x 0 1 y b 1 p b y b 1 here p 0 0 and also q is not assumed to be continuous at 0 so the problem may be doubly singular in this note we discuss the cases when y 0 0 at the singular end is not possible but lim x 0 p x y x 0 regions for multiple solutions have also been determined

a fourth order spline method for singular two point boundary value problems this paper describes a new fourth order spline approximation method for the class of singular two point boundary value problems y frac alpha x y f x y 0 0 x 1 y 0 0 y 1 a for all alpha geqslant 1

on the convergence of finite difference methods for weakly regular singular boundary value problems the second order finite difference methods m 1 based on a non uniform mesh and m 2 based on an uniform mesh developed by chawla and katti finite difference methods and their convergence for a class of singular two point boundary value problems numer math 39 1982 341 350 for weakly regular singular boundary value problems p x y f x y 0

a linear degenerate elliptic equation arising from two phase mixtures we consider the linear degenerate elliptic system of two first order equations mathbf u a phi nabla p mathbf g and nabla cdot b phi mathbf u phi p phi 1 2 f where a and b satisfy a 0 b 0 0 and are otherwise positive and the porosity phi ge0 may be zero on a set of positive measure this model equation has a similar degeneracy to that arising in the equations describing the mechanical system modeling the dynamics of partially melted materials e g in the earth s mantle and in polar ice sheets and glaciers in the context of mixture theory phi represents the phase variable separating the solid one phase phi 0 and fluid solid two phase phi 0 regions the equations should remain well posed as phi vanishes so that the free boundary between the one and two phase regions need not be found explicitly two main problems arise first as phi vanishes one equation is lost second it is shown by stability or energy bounds for the solution that the pre

finite element methods for spherically symmetric elliptic equations in this paper we obtain error estimates for the numerical solution by the rayleigh ritz galerkin method with piecewise polynomial subspaces of singular two point boundary value problems which arise from spherically symmetric elliptic equations our results are stated in the natural sobolev norms which are l 2 norms with respect to a weight function the original elliptic problem may be set in a space of arbitrary dimension moreover we can enforce smoothness of the numerical solution at the origin

stability results for the volume of random simplices it is known that for a convex body k in r d of volume one the expected volume of random simplices in k is minimized if k is an ellipsoid and for d 2 maximized if k is a triangle here we provide corresponding stability estimates

a survey on parametric spline function approximation this survey paper contains a large amount of material and indeed can serve as an introduction to some of the ideas and methods for the solution of ordinary and partial differential equations starting from schoenberg s work quart appl math 4 1946 345 369 the parametric spline function which depends on a parameter w 0 is reduces to the ordinary cubic or quintic spline for w 0 a note on parametric spline function approximation which is special case of this work has been published in comp math applics 29 1995 67 73 this article deals with the odd order parametric spline relations

the finite element method for a boundary value problem with strong singularity the existence and uniqueness of the r n generalized solution for the third boundary value problem and the non self adjoint second order elliptic equation with strong singularity are established we construct a finite element method with a basis containing singular functions the rate of convergence of the approximate solution to the r n generalized solution in the norm of the sobolev weighted space is established and finally results of numerical experiments are presented

existence and uniqueness of solutions of a class of two point singular nonlinear boundary value problems abstract el gebeily m a a boumenir and m b m elgindi existence and uniqueness of solutions of a class of two point singular nonlinear boundary value problems journal of computational and applied mathematics 46 1993 345 355 this paper is concerned with the existence and uniqueness of solution of a class of two point singular nonlinear boundary value problems it is shown that the problem has a unique solution only for certain boundary conditions under the assumption that the range of f y has empty intersection with the closure of the spectrum of the singular differential operator where f denotes the nonlinearity

a fourth order finite difference method based on uniform mesh for singular two point boundary value problems we present a new fourth order finite difference method based on uniform mesh for the weakly singular two point boundary value problem x y f x y 0 x 1 y 0 a y 1 b 0 1 our method provides o h4 convergent approximations for all 0 1 for 0 it reduces to the well known fourth order method of numerov for y x y

a uniform mesh finite difference method for a class of singular two point boundary value problems abstract a method is described based on a uniform mesh for the class of singular two point boundary value problems x y f x y 0 x 1 y 0 a y 1 b it is shown to be order h 4 convergent for all 0 1

a new fourth order cubic spline method for second order nonlinear two point boundary value problems abstract bickley 5 had suggested the use of cubic splines for the solution of general linear two point boundary value problems it is well known since then that this method gives only order h 2 uniformly convergent approximations but cubic spline interpolation itself is a fourth order process we present a new fourth order cubic spline method for second order nonlinear two point boundary value problems y f x y y a x b 0 y a 0 y a a 0 y b 1 y b b we generate the solution at the nodal points by a fourth order method and then use conditions of continuity to obtain smoothed approximations for the second derivatives of the solution needed for the construction of the cubic spline solution we show that our method provides order h 4 uniformly convergent approximations over a b the fourth order of the presented method is demonstrated computationally by two examples

the orlicz minkowski problem for torsional rigidity abstract in this paper we prove a variational formula of torsional rigidity under the orlicz sum and then introduce the orlicz mixed torsional rigidity which yields the orlicz brunn minkowski inequality and orlicz minkowski inequality for torsional rigidity as an application we prove the existence of solution to the orlicz minkowski problem for discrete measure and general measure associated with torsional rigidity

affine vs euclidean isoperimetric inequalities abstract it is shown that every even zonal measure on the euclidean unit sphere gives rise to an isoperimetric inequality for sets of finite perimeter which directly implies the classical euclidean isoperimetric inequality the strongest member of this large family of inequalities is shown to be the only affine invariant one among them the petty projection inequality as an application a family of sharp sobolev inequalities for functions of bounded variation is obtained each of which is stronger than the classical sobolev inequality moreover corresponding families of l p isoperimetric and sobolev type inequalities are also established

the dual orlicz brunn minkowski theory this paper introduces the dual orlicz brunn minkowski theory for star sets a radial orlicz addition of two or more star sets is proposed and a corresponding dual orlicz brunn minkowski inequality is established based on a radial orlicz linear combination of two star sets a formula for the dual orlicz mixed volume is derived and a corresponding dual orlicz minkowski inequality proved the inequalities proved yield as special cases the precise duals of the conjectured log brunn minkowski and log minkowski inequalities of boroczky lutwak yang and zhang a new addition of star sets called radial m addition is also introduced and shown to relate to the radial orlicz addition

axiomatic foundation of the analytic hierarchy process this paper contains an axiomatic treatment of the analytic hierarchy process ahp the set of axioms corresponding to hierarchic structures are a special case of axioms for priority setting in systems with feedback which allow for a wide class of dependencies the axioms highlight 1 the reciprocal property that is basic in making paired comparisons 2 homogeneity that is characteristic of people s ability for making comparisons among things that are not too dissimilar with respect to a common property and hence the need for arranging them within an order preserving hierarchy 3 dependence of a lower level on the adjacent higher level 4 the idea that an outcome can only reflect expectations when the latter are well represented in the hierarchy the ahp neither assumes transitivity or the stronger condition of consistency nor does it include strong assumptions of the usual notions of rationality a number of facts are derived from these axioms providing an operational basis for the ahp

lutwak petty projection inequalities for minkowski valuations and their duals abstract lutwak s volume inequalities for polar projection bodies of all orders are generalized to polarizations of minkowski valuations generated by even zonal measures on the euclidean unit sphere this is based on analogues of mixed projection bodies for such minkowski valuations and a generalization of the notion of centroid bodies a new integral representation is used to single out lutwak s inequalities as the strongest among these families of inequalities which in turn are related to a conjecture on affine quermassintegrals in the dual setting a generalization of volume inequalities for intersection bodies of all orders by leng and lu is proved these results are related to grinberg s inequalities for dual affine quermassintegrals

on the existence of solutions of a class of singular nonlinear two point boundary value problems we examine the existence of solutions of the class of singular nonlinear two point boundary value problems y a x y ffi f x y 0 l we show that for every a l a unique solution of the singular two point boundary value problem exists provided u kl where u sup 3f ay and k i is the first positive zero of jta l 2 fk j z is bessel s function of the first kind of order s interestingly k i is a monotonically increasing function of a values of k i for some values of a are tabulated

spline finite difference methods for singular two point boundary value problems in this paper we discuss the construction of a spline function for a class of singular two point boundary value problemx x u f x u u 0 a u 1 b 0 1 or 1 2 the boundary conditions may also be of the formu 0 0 u 1 b three point finite difference methods using the above splines are obtained for the solution of the boundary value problem these methods are of second order and are illustrated by four numerical examples

nonresonant nonlinear singular problems in the limit circle case existence results are presented for the resonant singular boundary value problem tfrac 1 p py prime prime lambda m qy f t y a e on 0 1 with limt 0 py y 1 0 here we donot assume int 0 1 tfrac ds p s infty but only that int 0 1 tfrac 1 p s left int 0 s p x q x dx right tfrac 1 2 ds infty

higher order finite difference method for a class of singular boundary value problems in this paper a fourth order finite difference method for a class of singular boundary value problems is presented the original differential equation is modified at the singular point the fourth order finite difference method is then employed to solve the boundary value problem some model problems are solved and the numerical results are compared with exact solution

wavelet galerkin method for solving nonlinear singular boundary value problems arising in physiology abstract in this study wavelet galerkin method has been developed for solving nonlinear singular boundary value problems associated with physiology science the quasilinearization technique is applied to reduce the given nonlinear problem to a sequence of linear problems we modify the resulting set of differential equations at the singular point then treat this set of boundary value problems by using compactly supported cubic b spline wavelets and galerkin method these wavelets are used as testing and weighting functions the method is computationally attractive and applications are demonstrated through illustrative examples

cone volume measure and stability abstract we prove stability results for two central inequalities involving the cone volume measure of a centered convex body the subspace concentration conditions and the u functional volume inequality

a difference method for singular two point boundary value problems a difference method based on uniform mesh for solving singular two point boundary value problems of the form x ay f x y 0

an analysis of dendy s piecewise polynomial petrov galerkin method for a hyperbolic equation with stagnation points abstract a method of dendy was shown by him in the absence of stagnation points to give optimal order error for a large class of piecewise polynomial trial spaces this result is extended to the situation when stagnation points may be present the analysis contains a result of interest in connection with numerical solution of singular two point boundary value problems

a difference scheme based on non uniform mesh for singular two point boundary value problems a difference scheme based on non uniform mesh for singular two point boundary value problems of the form y 1xy f x y 0 0

on a class of regular singular two point boundary value problems abstract existence of a unique solution for a class of regular singular two point boundary value problems p x y p x f x y 0 x b lim x 0 y x 0 y b b has been established wherep x satisfies i p x 0 on 0 b ii p x c1 0 r and for somer b iii xp x p x is analytic in z z xp x p x b 0 b 1 x b 0 1 and with quite general conditions onf x y these conditions onf x y are sharp which is seen through one example regions for multiple solutions have also been determined

a new finite difference method for a class of singular two point boundary value problems we present a new three point finite difference method based on uniform mesh for a class of singular two point boundary value problem x ay f x y y 0 a y 1 b 0 a 1 we show that the method based on uniform mesh provides o h 4 convergent approximations this method is illustrated by two numerical examples one is linear and another is non linear

the numerical solution of non linear singular boundary value problems arising in physiology a class of non linear singular ordinary differential equations is solved by a new method based on non polynomial cubic spline we use the quesilinearization technique to reduce the given non linear problem to a sequence of linear problems we modify the resulting set of differential equations at the singular point then treat this set of boundary value problems by using non polynomial cubic spline approximation the resulting system of algebraic equations is solved by using a tri diagonal solver computational results are provided to demonstrate the viability of the new method

a second order spline finite difference method for singular two point boundary value problems a second order spline finite difference method based on a non uniform mesh for singular two point boundary value problems of the form x a x au f x u 0

finite difference methods and their convergence for a class of singular two point boundary value problems we discuss the construction of three point finite difference approximations and their convergence for the class of singular two point boundary value problems x y f x y y 0 a y 1 b 0 1 we first establish a certain identity based on general non uniform mesh from which various methods can be derived to obtain a method having order two for all 0 1 we investigate three possibilities by employing an appropriate non uniform mesh over 0 1 we obtain a methodm1 based on just one evaluation off for uniform mesh we obtain two methodsm2 andm3 each based on three evaluations off for 0 m1 andm2 both reduce to the classical second order method based on one evaluation off these three methods are investigated theiro h2 convergence established and illustrated by numerical examples