the existence of countably many positive solutions for singular multipoint boundary value problems abstract in this paper we study the existence of countably many positive solutions for a singular multipoint boundary value problem by using fixed point index theory and the leggett williams fixed point theorem sufficient conditions for the existence of countably many positive solutions are established

fast and accurate evaluation of double surface integrals in time domain integral equation formulations a novel integration approach is developed for efficiently evaluating double surface integrals in time domain integral equation formulations the approach mainly consists of an angular integration scheme for inner 2 d integrals and a modified outer integration scheme the approach proposed can handle the singularity and near hypersingularity of the 4 d integrals and is sufficiently general for arbitrary types of basis functions over planar domains as several numerical results will demonstrate the proposed approach has high accuracy and rapid convergence rate for a variety of integral kernels

polar integration for exact space time quadrature in time domain integral equations a space time polar quadrature technique for numerical integration of green s function interactions in time domain integral equations is presented the method transforms 2 d surface space time integrals associated with vector and scalar potentials to a 1 d integral that is performed using gauss legendre integration the advantage of the presented technique compared to standard 2 d gaussian quadrature is that time delays between each section of the source basis function and the observation point are accounted for exactly in an analytic manner this ensures highly accurate temporal behavior of the green s function interactions thereby contributing to the stability of the overall time domain integral equations

global properties of invariant measures abstract we study global regularity properties of invariant measures associated with second order differential operators in r n under suitable conditions we prove global boundedness of the density sobolev regularity a harnack inequality and pointwise upper and lower bounds

a singular boundary value problem for the one dimensionalp laplacian abstract the singular boundary value problem formula is studied in this paper whereg s s p 2s p 1 the singularity may appear atu 0 and att 0 ort 1 and the functionfmay be discontinuous the authors prove that for anyp 1 and for any positive nonincreasing functionfand nonnegative measurable functionkwith some integrability conditions the above mentioned problem has a unique solution also the properties of the solution are discussed in the paper

a generalized multi point boundary value problem for second order ordinary differential equations let f 0 1 x r 2 r be a function satisfying caratheodory s conditions and e t l 1 0 1 let x i t j 0 1 a i b j r i 1 2 m 2 j 1 2 n 2 0 x 1 x 2 x m 2 1 0 t 1 t 2 t n 2 1 be given this paper is concerned with the problem of existence of a solution for the generalized multi point boundary value problems x t f t x t x t e t 0

the solution of multipoint boundary value problems by the optimal homotopy asymptotic method in this work we solve multipoint boundary value problems using the optimal homotopy asymptotic method oham the proposed method is tested upon several problems from the literature and the results are compared with the available exact solution this method provides easy tools to control the convergence region of approximating solution series where ever necessary

nodal solutions for second order m point boundary value problems with nonlinearities across several eigenvalues abstract we study the existence of nodal solutions of the m point boundary value problem u f u 0 0 t 1 u 0 0 u 1 i 1 m 2 i u i where i q i 1 2 m 2 with 0 1 2 m 2 1 and i r i 1 2 m 2 with i 0 and 0 i 1 m 2 i 1 we give conditions on the ratio f s s at infinity and zero that guarantee the existence of nodal solutions the proofs of our main results are based upon bifurcation techniques

multiple positive solutions for some four point boundary value problems with p laplacian this paper is devoted to the existence of multiple positive solutions for some nonlinear singular four point boundary value problems with p laplacian under appropriate assumptions on the nonlinear terms by applying several known fixed point theorems sufficient conditions for the existence of at least three positive solutions are established

the existence of countably many positive solutions for one dimensional p laplacian with infinitely many singularities on the half line we study the existence of countably many positive solutions of boundary value problems on the half line for differential equations of second order the fixed point index theory and a new fixed point theorem in cones are used

stable electric field tdie solvers via quasi exact evaluation of mot matrix elements prior theoretical studies and experience confirm that the stability of marching on in time mot solvers pertinent to the analysis of scattering from free standing three dimensional perfect electrically conducting surfaces hinges on the accurate evaluation of mot matrix elements resulting from a galerkin discretization of the underlying time domain integral equation tdie unfortunately the accurate evaluation of the four dimensional spatial integrals involved in the expressions for these matrix elements is prohibitively expensive when performed by computational means here a method that permits the quasi exact evaluation of mot matrix elements is presented specifically the proposed method permits the analytical evaluation of three out of the four spatial integrations leaving only one integral to be evaluated numerically since the latter has finite range and a piecewise smooth integrand it can be evaluated to very high accuracy using standard quadrature rules as a result the proposed method permits the fast evaluation of mot matrix elements with arbitrary user specified accuracy extensive numerical experiments show that an mot solver for the electric field tdie that uses the proposed quasi exact method is stable for a very wide range of time step sizes and yields solutions that decay exponentially after the excitation vanishes

numerical integration of functions with boundary singularities abstract in this paper we deal with the problem of constructing efficient rules for the numerical evaluation of integrals of functions which are very smooth everywhere in the domain of integration except at the boundaries where they possess mild singularities in particular we consider integrals defined on bounded intervals or on triangles integrals of this type appear for example in the numerical solution of singular and weakly singular integral equations by boundary element methods

solution of time domain electric field integral equation using the laguerre polynomials in this paper we propose a numerical method to obtain a solution for the time domain electric field integral equation td efie for arbitrary shaped conducting structures this method does not utilize the customary marching on in time mot solution method often used to solve a hyperbolic partial differential equation instead we solve the wave equation by expressing the transient behaviors in terms of laguerre polynomials by using these causal orthonormal basis functions for the temporal variation the time derivatives can be handled analytically in order to solve the wave equation we introduce two separate testing procedures a spatial and temporal testing by introducing first the galerkin temporal testing procedure the mot procedure is replaced by a recursive relation between the different orders of the weighted laguerre polynomials the other novelty of this approach is that through the use of the entire domain laguerre polynomials for the expansion of the temporal variation of the current the spatial and the temporal variables can be separated and the temporal variables can be integrated out for convenience we use the hertz vector as the unknown variable instead of the electric current density to verify our method we compare the results of a td efie and inverse fourier transform of a frequency domain efie

adaptive drift estimation for nonparametric diffusion model we consider a nonparametric diffusion process whose drift and diffusion coefficients are nonparametric functions of the state variable the goal is to estimate the unknown drift coefficient we apply a locally linear smoother with a data driven bandwidth choice the procedure is fully adaptive and nearly optimal up to a log log factor the results about the quality of estimation are nonasymptotic and do not require any ergodic or mixing properties of the observed process

spectral properties of p laplacian problems with neumann and mixed type multi point boundary conditions abstract we consider the boundary value problem consisting of the p laplacian equation 1 p u p u on 1 1 where p 1 p s s p 1 sgn s for s r r together with the multi point boundary conditions 2 p u 1 i 1 m i p u i or 3 u 1 i 1 m i u i or a mixed pair of these conditions with one condition holding at each of x 1 and x 1 in 2 3 m 1 are integers i 1 1 1 i m and the coefficients i satisfy i 1 m i 1 we term the conditions 2 and 3 respectively neumann type and dirichlet type boundary conditions since they reduce to the standard neumann and dirichlet boundary conditions when 0 given a suitable pair of boundary conditions a number is an eigenvalue of the corresponding boundary value problem if there exists a non trivial solution u an eigenfunction the spectrum of the problem is the set of eigenvalues in this paper we obtain various spectral properties of these eigenvalue problems we then use these properties to prove rabinowitz type global bifurcation theorems for related bifurcation problems and to obtain nonresonance conditions in terms of the eigenvalues for the solvability of related inhomogeneous problems

linear and nonlinear second order problems with sturm liouville type multi point boundary conditions abstract we consider the nonlinear boundary value problem consisting of the equation u f u on 1 1 where f r r is continuous together with general sturm liouville type multi point boundary conditions at 1 we obtain rabinowitz type global bifurcation results and then use these to obtain nodal solutions of the problem we conclude with a nonresonance result for an inhomogeneous form of the problem these results rely on the spectral properties of the eigenvalue problem consisting of the equation u u on 1 1 together with the multi point boundary conditions in a previous paper it was shown that under certain optimal conditions the basic spectral properties of this eigenvalue problem are similar to those of the standard sturm liouville problem with single point boundary conditions in particular for each integer k 0 there exists a unique simple eigenvalue k whose eigenfunctions have oscillation count equal to k where the oscillation count was defined in terms of a complicated prufer angle construction unfortunately it seems to be difficult to apply the prufer angle construction to the nonlinear problem accordingly in this paper we use alternative non optimal oscillation counting methods to obtain the required spectral properties of the linear problem and these are then applied to the nonlinear problem to yield the results mentioned above

the existence of countably many positive solutions for nonlinear singular m point boundary value problems on the half line in this paper by introducing a new operator improving and generating a p laplace operator for some p 1 we study the existence of countably many positive solutions for nonlinear boundary value problems on the half line f u a t f u t 0 0r is the increasing homeomorphism and positive homomorphism and f 0 0 we show the sufficient conditions for the existence of countably many positive solutions by using the fixed point index theory and a new fixed point theorem in cones

p laplacian problems with jumping nonlinearities we consider the p laplacian boundary value problem 1 p u x f x u x u x a e x 0 1 2 c00u 0 c01u 0 c10u 1 c11u 1 where p 1 is a fixed number p s s p 2s s r and for each j 0 1 cj0 cj1 0 the function f 0 1 r2 r is a caratheodory function satisfying for x s t 0 1 r2 x p s e x s t f x s t x p s e x s t s 0 where l1 0 1 and e has the form e x s t x e s t with l1 0 1 0 e 0 and limr e r r1 p 0 this condition allows the nonlinearity in 1 to behave differently as u such a nonlinearity is often termed jumping related to 1 2 is the problem 3 p u a p u b p u p u in 0 1 together with 2 where a b l1 0 1 r and u x max u x 0 for x 0 1 this problem is positively homogeneous and jumping values of for which 2 3 has a nontrivial solution u will be called half eigenvalues while the corresponding solutions u will be called half eigenfunctions we show that a sequence of half eigenvalues exists the corresponding half eigenfunctions having certain nodal properties and we obtain certain spectral and degree theoretic properties of the set of half eigenvalues these properties lead to existence and nonexistence results for the problem 1 2 we also consider a related bifurcation problem and obtain a global bifurcation result similar to the well known rabinowitz global bifurcation theorem this then leads to a multiplicity result for 1 2 when the functions a and b are constant the set of half eigenvalues is closely related to the fucik spectrum of the problem and equivalent solvability results are obtained using the two approaches however when a and b are not constant the half eigenvalue approach yields stronger results

high degree efficient symmetrical gaussian quadrature rules for the triangle gaussian quadrature is required for the computation of matrices based on the isoparametric formulztion of the finite element method a brief review of existing quadrature rules for the triangle is given and the method for the determination of high degree efficient symmetrical rules for the triangle is discussed new quadrature rules of degree 12 20 are presented and a short fortran program is included

quadrature over a pyramid or cube of integrands with a singularity at a vertex a simple transformation is introduced which facilitates the evaluation of integrals over certain square based pyramids or cubes in cases where the integrand has a singularity at a vertex

triple positive solutions to third order three point bvp with increasing homeomorphism and positive homomorphism in this paper we investigate the existence of triple positive solutions for nonlinear differential equations boundary value problems with increasing homeomorphism and positive homomorphism operator by using fixed point theorems in cones we establish results on the existence of three positive solutions with suitable growth conditions imposed on the nonlinear term as applications two examples are given to demonstrate our result the conclusions in this paper essentially extend and improve the known results

solving time domain electric field integral equation without the time variable an improved testing procedure using the marching on in order method to solve the time domain electric field integral equation td efie for conducting objects using the laguerre polynomials is presented exact temporal testing is performed before the spatial testing therefore the retarded terms composed of the spatial and the temporal variables can be analytically separated the uniqueness of this testing procedure is that the time variable can be analytically integrated out and the accuracy can be improved this paper is then an improvement over the earlier marching on in order method in addition this methodology is quite different from the conventional marching on in time algorithm as the present method leads to a set of final equations which need to be numerically solved containing only the spatial variables therefore there is no requirement to have a courant stability condition in this procedure how the singular integrals are treated is also discussed several examples are simulated both for radiation and scattering problem the results are compared with the inverse discrete fourier transform of the frequency domain data and they agree well

optimal interval lengths for nonlocal boundary value problems associated with third order lipschitz equations abstract for the third order differential equation y f x y y y where f x y 1 y 2 y 3 is lipschitz continuous in terms of y i i 1 2 3 we obtain optimal bounds on the length of intervals on which there exist unique solutions of certain nonlocal three and four point boundary value problems these bounds are obtained through an application of the pontryagin maximum principle from the theory of optimal control

spectral properties and nodal solutions for second order m point boundary value problems abstract we consider the m point boundary value problem consisting of the equation u f u on 0 1 where f r r is c 1 with f 0 0 together with the boundary conditions u 0 0 u 1 i 1 m 2 i u i where m 3 i 0 1 and i 0 for i 1 m 2 with i 1 m 2 i 1 we first show that the spectral properties of the linearisation of this problem are similar to the well known properties of the standard sturm liouville problem with separated boundary conditions with a minor modification to deal with the multi point boundary condition these spectral properties are then used to prove a rabinowitz type global bifurcation theorem for a bifurcation problem related to the above problem finally we use the global bifurcation theorem to obtain nodal solutions that is sign changing solutions with a specified number of zeros of the above problem under various conditions on the asymptotic behaviour of f

efficient evaluation of double surface integrals in time domain integral equation formulations a new integration approach is presented for accurately calculating time domain efie mfie and cfie matrix elements over triangular domains it mainly consists of a radial integration scheme for handling weakly singular and near hypersingular inner integrals and some new smoothing techniques for treating outer two dimensional 2 d integrals the proposed approach has sufficient generality and efficiency for solving time domain integral equations tdie with arbitrary types of temporal basis functions and temporal discretization schemes such as marching on in time mot marching on in degree mod and finite difference delay modeling convolution quadrature fddm cq etc the numerical results for calculating some typical integrals are given to demonstrate its capability with high accuracy and rapid convergence rate achieved

some multipoint boundary value problems of neumann dirichlet type involving a multipoint p laplace like operator abstract let and be two increasing homeomorphisms from r onto r with 0 0 0 0 let f 0 1 r r r be a function satisfying caratheodory s conditions and for each i i 1 2 m 2 let a i r r be a continuous function with i 1 m 2 a i 0 1 i 0 1 0 1 2 m 2 1 in this paper we first prove a suitable continuation lemma of leray schauder type which we use to obtain several existence results for the m point boundary value problem u f t u u t 0 1 u 0 0 u 1 i 1 m 2 u i a i u i we note that this problem is at resonance in the sense that the associated m point boundary value problem u t 0 t 0 1 u 0 0 u 1 i 1 m 2 u i a i u i has the non trivial solution u t where r is an arbitrary constant vector in view of the assumption i 1 m 2 a i 0 1

impulsive boundary value problems involving the one dimensional p laplacian abstract by using the classical fixed point index theorem for compact maps some sufficient conditions for the existence of multiple positive solutions for a class of second order impulsive differential equations with p laplacian are established which are complement of previously known results

exact constants for pointwise adaptive estimation under the riesz transform we consider nonparametric estimation of a multivariate function and its partial derivatives at a fixed point when the riesz transform of the function is observed in gaussian white noise we assume that the unknown function belongs to some sobolev class and construct an estimation procedure which achieves the best asymptotic minimax risk when the smoothness of the function is unknown

exact adaptive pointwise drift estimation for multidimensional ergodic diffusions the problem of pointwise adaptive estimation of the drift coefficient of a multivariate diffusion process is investigated we propose an estimator which is sharp adaptive on scales of sobolev smoothness classes the analysis of the exact risk asymptotics allows to identify the impact of the dimension and other influencing values such as the geometry of the diffusion coefficient of the prototypical drift estimation problem for a large class of multidimensional diffusion processes we further sketch generalizations of our results to arbitrary diffusions satisfying suitable bernstein type inequalities

a high order locally corrected nystrom scheme for charge current integral equations a high order nystrom scheme is developed for electromagnetic em scattering problems of smooth objects the implementation is based on different well known charge current formulations this study aims to develop a novel quadrature scheme to compute different singular integrals that appear on integral equation methods in ems the proposed quadrature scheme is used on a locally corrected nystrom method and applied to charge current integral equations to obtain a fast accurate and a low frequency stable method to solve em scattering problems the proposed method allows obtaining accuracy in the range 4 10 digits in the near electric and magnetic scattered field for certain simple geometries and a wide range of frequencies we also do a preliminary research on nonsmooth geometries obtaining a certain degree of accuracy depending on the particular charge current integral equation used

time domain integral equation analysis of scattering from composite bodies via exact evaluation of radiation fields a novel marching on in time and poggio miller chang harrington wu tsai pmchwt based time domain integral equation solver for analyzing transient scattering from composite multiregion structures comprising homogeneous penetrable volumes and perfectly conducting surfaces is presented to render the marching on in time procedure stable observer and source integrals are approximated via multipoint collocation and evaluated in closed form the proposed scheme avoids the costly evaluation of five dimensional integrals required in full space time galerkin schemes and proves to be stable for all targets investigated including coated perfect electrically conducting surfaces and multiregion penetrable volumes

multiple positive solutions for multipoint boundary value problems with sign changing nonlinearity abstract in this paper we consider the multipoint boundary value problem for one dimensional p laplacian p u q t f t u 0 t 0 1 subject to the boundary value conditions u 0 i 1 n i u i u 1 i 1 n i u i where p s s p 2s p 1 i 0 1 with 0 1 2 n 1 and i i 0 satisfy 0 i 1 n i i 1 n i 1 the nonlinear term f may change sign using a fixed point theorem for operators in a cone we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem

evaluation of weakly singular integrals via generalized cartesian product rules based on the double exponential formula various weakly singular integrals over triangular and quadrangular domains arising in the mixed potential integral equation formulations are computed with the help of novel generalized cartesian product rules the proposed integration schemes utilize the so called double exponential quadrature rule originally developed for the integration of functions with singularities at the endpoints of the associated integration interval the final formulas can easily be incorporated in the context of singularity subtraction singularity cancellation and fully numerical methods often used for the evaluation of multidimensional singular integrals the performed numerical experiments clearly reveal the superior overall performance of the proposed method over the existing numerical integration methods

numerical integration scheme for the near singular green function gradient on general triangles a new near singularity cancellation transformation quadrature scheme for triangle domains is presented it is tailored to the green function gradient kernel static and dynamic multiplied with higher order weighting functions on curvilinear triangles the transformation is analytically invertible attention is paid to ensuring accuracy in both tangential and normal components such integrals must be routinely evaluated in modern method of moments mom implementations the scheme follows the standard three subtriangle splitting approach of which the benefits are single parameter accuracy control a geometry independent total number of quadrature points and implementation simplicity extensive numerical results show superior efficiency and reliability to existing three subdomain schemes consistent dependable error convergence is demonstrated with very low variation in accuracy for fixed quadrature order and singularity height the scheme is suitable for practical use by mom developers

boundary data smoothness for solutions of nonlocal boundary value problems for second order differential equations abstract under certain conditions solutions of the boundary value problem y f x y y y x 1 y 1 and y x 2 i 1 m r i y i y 2 are differentiated with respect to boundary conditions where a x 1 1 m x 2 b r 1 r m r and y 1 y 2 r

potential integrals for uniform and linear source distributions on polygonal and polyhedral domains formulas for the potentials due to uniform and linearly varying source distributions defined on simply shaped domains are systematically developed and presented domains considered are infinite planar strips infinite cylinders of polygonal cross sections planar surfaces with polygonal boundaries and volumetric regions with polyhedral boundaries the expressions obtained are compact in form and their application in the numerical solution of electromagnetics problems by the method of moments is illustrated

non resonance conditions for semilinear sturm liouville problems with jumping non linearities abstract we consider the sturm liouville boundary value problem p x u x q x u x f x u x h x x 0 c00u 0 c01u 0 0 c10u c11u 0 where p c1 0 q c0 0 with p x 0 x 0 c2i0 c2i1 0 i 0 1 h l2 0 and f 0 r r is a caratheodory function we assume that the rate of growth of f x is at most linear as but the asymptotic behaviour may be different as so the non linearity is termed jumping conditions for existence of solutions of this problem are usually expressed in terms of non resonance with respect to the standard fucik spectrum in this paper we give conditions for both existence and non existence of solutions in terms of a slightly different idea of the spectrum these conditions extend the usual fucik spectrum conditions

on the near interaction elements in integral equation solvers for electromagnetic scattering by three dimensional thin objects the accuracy of near interaction ni evaluation is investigated for hypersingular and strongly singular kernels in solving electromagnetic em integral equations we first define the ni between a field point and a source triangular patch using a nearness factor nf and show the accuracy of numerical integrations for the ni integrals with hypersingular or strongly singular kernels it is found that there exists a transition area or turning point before and after which the convergence behaviors of those integrations are greatly different we define the nf value at the 1 relative error as the turning point nf 0 and define nf les nf 0 as the ni range we find that nf 0 1 4 could be a bound of ni range for the triangles with a quality factor q 0 35 and analytical formulas should be used in evaluating ni elements within the bound numerical examples for em scattering by three dimensional 3d thin objects are used to demonstrate the significance of ni treatments for the accurate solutions

the existence of symmetric positive solutions for sturm liouville like four point boundary value problem with a p laplacian operator abstract in this paper the sturm liouville like four point boundary value problem with a p laplacian operator p u t a t f t u t 0 t 0 1 u 0 u 0 u 1 u 0 is studied where p s s p 2 s p 1 by the use of fixed point index theory leray schauder degree and upper and lower solution method some existence nonexistence and multiplicity results of symmetric positive solutions are acquired

on the numerical evaluation of the singular integrals of scattering theory in a previous work the authors introduced a scheme for the numerical evaluation of the singular integrals which arise in the discretization of certain weakly singular integral operators of acoustic and electromagnetic scattering that scheme is designed to achieve high order algebraic convergence and high accuracy when applied to operators given on smoothly parameterized surfaces this paper generalizes the approach to a wider class of integral operators including many defined via the cauchy principal value operators of this type frequently occur in the course of solving scattering problems involving boundary conditions on tangential derivatives the resulting scheme achieves high order algebraic convergence and approximately 12 digits of accuracy

sharp adaptive estimation of linear functionals we consider estimation of a linear functional t f where f is an unknown function observed in gaussian white noise we find asymptotically sharp adaptive estimators on various scales of smoothness classes in multidimensional situations the results allow evaluating explicitly the effect of dimension and treating general scales of classes furthermore we establish a connection between sharp adaptation and optimal recovery namely we propose a scheme that reduces the construction of sharp adaptive estimators on a scale of functional classes to a solution of the corresponding optimization problem

existence of multiple positive solutions for multipoint boundary value problems with a one dimensional p laplacian abstract in this paper we consider the multipoint boundary value problem for a one dimensional p laplacian p u a t f t u 0 t 0 1 u 0 0 u 1 i 1 m 2 a i u i where p s s p 2 s p 1 0 1 2 m 2 1 a i 0 for i 1 2 m 3 and a m 2 0 using a fixed point theorem for operators on a cone we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem

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

asymptotic statistical equivalence for ergodic diffusions the multidimensional case asymptotic local equivalence in the sense of le cam is established for inference on the drift in multidimensional ergodic diffusions and an accompanying sequence of gaussian shift experiments the nonparametric local neighbourhoods can be attained for any dimension provided the regularity of the drift is sufficiently large in addition a heteroskedastic gaussian regression experiment is given which is also locally asymptotically equivalent and which does not depend on the centre of localisation for one direction of the equivalence an explicit markov kernel is constructed

half eigenvalues and the fucik spectrum of multi point boundary value problems abstract we consider the nonlinear boundary value problem consisting of the equation 1 u f u h a e on 1 1 where h l 1 1 1 together with the multi point dirichlet type boundary conditions 2 u 1 i 1 m i u i where m 1 are integers 1 m 0 1 m 1 1 m and we suppose that i 1 m i 1 we also suppose that f r r is continuous and 0 f lim s f s s we assume that these limits exist we allow f f such a nonlinearity f is said to be jumping related to 1 is the equation 3 u a u b u on 1 1 where a b 0 and u x max u x 0 for x 1 1 the problem 2 3 is positively homogeneous and jumping regarding a b as fixed values of a b for which 2 3 has a non trivial solution u will be called half eigenvalues while the corresponding solutions u will be called half eigenfunctions we show that a sequence of half eigenvalues exists the corresponding half eigenfunctions having specified nodal properties and we obtain certain spectral and degree theoretic properties of the set of half eigenvalues these properties lead to solvability and non solvability results for the problem 1 2 the set of half eigenvalues is closely related to the fucik spectrum of the problem which we briefly describe equivalent solvability and non solvability results for 1 2 are obtained from either the half eigenvalue or the fucik spectrum approach

existence of multiple positive solutions for nonlinear m point boundary value problems abstract in this paper we afford some sufficient conditions to guarantee the existence of multiple positive solutions for the nonlinear m point boundary value problem for the one dimensional p laplacian p u a t f t u 0 t 0 1 u 0 0 u 1 i 1 m 2 a i u i

on the numerical integration of the linear shape functions times the 3 d green s function or its gradient on a plane triangle formulas for the static and dynamic potentials due to linearly varying source distributions defined on a planar triangle are presented the general linear variation of the source is represented in terms of three linear basis functions each one associated with a different vertex of the triangular domain the singular kernels considered are given by the 3 d green s function and its gradient both for the static and dynamic case in the static case the evaluation of potentials is performed analytically and compact form results are given in the dynamic case the results are given in a form suited for numerical integration

positive solutions of nonlinear third order m point bvp for an increasing homeomorphism and homomorphism with sign changing nonlinearity in this paper several existence theorems of positive solutions are established for a nonlinear m point boundary value problem for the following third order differential equations f u a t f t u t 0 email protected 0 1 f u 0 email protected i 1m 2a email protected u x i u 0 0 u 1 email protected i 1m 2b iu x i where f email protected r is an increasing homeomorphism and homomorphism and f 0 0 the nonlinear term f may change sign as an application an example to demonstrate our results is given

spectral properties of second order multi point p laplacian boundary value problems abstract we consider the multi point boundary value problem p u p u on 1 1 u 1 i 1 m i u i where p 1 p s s p 1 sgn s for s r r m 1 are integers i 1 1 1 i m and the coefficients i satisfy i 1 m i 1 a number r is said to be an eigenvalue of the above problem if there exists a non trivial solution u the spectrum is the set of eigenvalues in this paper we obtain some basic spectral and degree theoretic properties of this eigenvalue problem these results have numerous applications to more general problems as an example a rabinowitz type global bifurcation theorem is briefly described

sharp adaptive estimation of the drift function for ergodic diffusions the global estimation problem of the drift function is considered for a large class of ergodic diffusion processes the unknown drift s is supposed to belong to a nonparametric class of smooth functions of order k 1 but the value of k is not known to the statistician a fully data driven procedure of estimating the drift function is proposed using the estimated risk minimization method the sharp adaptivity of this procedure is proven up to an optimal constant when the quality of the estimation is measured by the integrated squared error weighted by the square of the invariant density

second order three point boundary value problems with jumping non linearities abstract we consider the non linear three point boundary value problem consisting of the equation 1 u f u h a e on 0 1 where h l 1 0 1 together with the boundary conditions 2 u 0 0 u 1 u where 0 1 the function f r r is continuous and we assume that the following limits exist and are finite f lim s f s s we allow f f such a non linearity f is said to be jumping related to 1 is the equation 3 u a u b u u on 0 1 where a b r and u x max u x 0 for x 0 1 the problem 2 and 3 is positively homogeneous and jumping values of for which 2 and 3 has a non trivial solution u will be called half eigenvalues while the corresponding solutions u will be called half eigenfunctions we show that a sequence of half eigenvalues exists the corresponding half eigenfunctions having specified nodal properties and we obtain certain spectral and degree theoretic properties of the set of half eigenvalues these properties lead to solvability and non solvability results for the problem 1 and 2 the set of half eigenvalues is closely related to the fucik spectrum of the problem and equivalent solvability and non solvability results for 1 and 2 are obtained from either the half eigenvalue or the fucik spectrum approach

transportation information inequalities for markov processes in this paper one investigates the transportation information tci inequalities tc i for all probability measures on a metric space mathcal x d where is a given probability measure tc is the transportation cost from to with respect to the cost function c x y on mathcal x 2 i is the fisher donsker varadhan information of with respect to and 0 0 is a left continuous increasing function using large deviation techniques it is shown that tci is equivalent to some concentration inequality for the occupation measure of a reversible ergodic markov process related to i the tensorization property of tci and comparisons of tci with poincare and log sobolev inequalities are investigated several easy to check sufficient conditions are provided for special important cases of tci and several examples are worked out

second order multi point problems with variable coefficients abstract in this paper we consider the eigenvalue problem consisting of the equation u r u on 1 1 where r c 1 1 1 r 0 and r together with the multi point boundary conditions u 1 i 1 m i u i where m 1 are integers and for i 1 m i r i 1 1 with i 1 i 1 we show that if the coefficients i r are sufficiently small depending on r then the spectral properties of this problem are similar to those of the usual separated problem but if the coefficients i are not sufficiently small then these standard spectral properties need not hold the spectral properties of such multi point problems have been obtained before for the constant coefficient case r 1 but the variable coefficient case has not been considered previously apart from the existence of principal eigenvalues some nonlinear multi point problems are also considered we obtain a partial rabinowitz type result on global bifurcation from the eigenvalues and various nonresonance conditions for the existence of general solutions and also of nodal solutions these results rely on the spectral properties of the linear problem

the existence of countably many positive solutions for nonlinear singular m point boundary value problems in this paper we study the existence of countably many positive solutions for nonlinear singular boundary value problem f u a t f u t 0 0r is an increasing homeomorphism and positive homomorphism and f 0 0 x i 0 1 with 0 0 and has countably many singularities in 0 12 we show that there exist countably many positive solutions by using the fixed point index theory and a new fixed point theorem in cones

pointwise and sup norm sharp adaptive estimation of functions on the sobolev classes the problem of nonparametric function estimation in the gaussian white noise model is considered it is assumed that the unknown function belongs to one of the sobolev classes with an unknown regularity parameter asymptotically exact adaptive estimators of functions are proposed on the scale of sobolev classes with respect to pointwise and sup norm risks it is shown that unlike the case of l 2 risk a loss of efficiency under adaptation is inevitable here bounds on the value of the loss of efficiency are obtained

double positive solutions of three point boundary value problems for p laplacian dynamic equations on time scales a new double fixed point theorem is applied to investigate the existence of at least double positive solutions of three point boundary value problems for p laplacian dynamic equations on a time scale