spline reproducing kernels on r and error bounds for piecewise smooth lbv problems abstract reproducing kernel method for approximating solutions of linear boundary value problems is valid in hilbert spaces composed of continuous functions but its convergence is not satisfactory without additional smoothness assumptions we prove 2nd order uniform convergence for regular problems with coefficient piecewise of sobolev class h2 if the coefficients are globally of class h2 more refined phantom boundary nsc rkhs method is derived and the order of convergence rises to 3 or 4 according to whether the problem is piecewise of class h3 or h4 the algorithms can be successfully applied to various non local linear boundary conditions e g of simple integral form the paper contains also a new explicit formula for general spline reproducing kernels in hm a b if the inner product f g m i m f i g i f m g m depends on any fixed reference point a b the piecewise nsc rkhs methods are then applied to two example regular lbv problems in h3 and h5 exactness of the resulting numerical solutions the degree of convergence and their dependency of the reference point a b are presented in attached figures

collective motions in globally coupled tent maps with stochastic updating we study a generalization of globally coupled maps where the elements are updated with probability p when p is below a threshold p c the collective motion vanishes and the system is the stationary state in the large size limit we present the linear stability analysis

first class selectors for upper semi continuous multifunctions by a multtfunction x y we mean a function defined on the space x and whose values are subsets of the space y a point x0 in x is a point of upper semi continuity for provided x0 belongs to the interior of x q x c g whenever g is an open set in y containing x0 q is said to be upper semi continuous if it is upper semi continuous at ach point of x in this case x g x c g is open in x whenever g is open in y and x x nf is closed in x w h enever f is closed in y interchanging the roles of open and closed sets in this definition we get the concept of a lower semi continuous multifunction a selector for is a single valued function cp x y with q x e cd x for each x in x in this paper we are primarily concerned with the problem of determin ing the best possible selector cp for an upper semi continuous multifunction defined on a metric space x and taking only non empty values in a second metric space y very elementary examples show that such multi functions will in general fail to have continuous selectors e g take 4p x 0 foro x o 1 and x 1 for x 1 hence an optimal selector in our case would be one that is the pointwise limit of a sequence of continuous functions or more generally a first class bore1 function i e cp g is an 9 set in x for each open set g c y here we will prove

a new reproducing kernel hilbert space method for solving nonlinear fourth order boundary value problems this paper presents a new reproducing kernel hilbert space method for solving nonlinear fourth order boundary value problems it is a relatively new analytical technique the solution obtained by using the method takes the form of a convergent series with easily computable components this paper will present a numerical comparison between our method and other methods for solving an open fourth order boundary value problem presented by scott and watts the method is also applied to a nonlinear fourth order boundary value problem the numerical results demonstrate that the new method is quite accurate and efficient for fourth order boundary value problems

a new algorithm for a class of singular boundary value problems abstract in this paper we present a new algorithm to solve a class of singular boundary value problems in the reproducing kernel space w 2 3 0 1 the algorithm is efficiently applied to solving some model problems with the comparison between the numerical solutions and the exact solutions it is demonstrated by the numerical examples that this algorithm is of high precision

baire class 1 selectors for upper semicontinuous set valued maps let t be a metric space and x a banach space let f t x be a set valued map assuming arbitrary values and satisfying the upper semicontinuity condition t t f t c o is closed for each weakly closed set c in x then there is a sequence of norm continuous functions converging pointwise in the norm to a selection for f we prove a statement of similar precision and generality when x is a metric space

fast variants of the nsc rkhs algorithm for solving linear boundary value problems the fast algorithms work in the sobolev hilbert spaces on a finite interval 0 t and arise from the normal spline collocation method introduced in 4 by gorbunov and developed within the last 7years by chinese mathematicians as a reproducing kernel hilbert space method for a natural h 4 class of mth order linear problems ly f two extended variants of the method applied to an operator h m 2 h 2 yield 4th order uniform approximations of any normal solution v together with its derivatives up to order m 1 i e v b v n b c 0

a note on singular nonlinear boundary value problems in this paper we study the existence and multiplicity of solutions of the following operator equation in banach space e u au 0 u p where is a parameter and p a cone of banach space e under certain conditions on the operator a we find such that the operator equation has at least two solutions for 0 as an application we investigate the existence and multiplicity of positive solutions of a singular second order boundary value problem in addition we briefly outline an application of our results which simplifies a previous theorem that appeared in the literature

effects of microscopic disorder on the collective dynamics of globally coupled maps this paper studies the effect of independent additive noise on the synchronous dynamics of large populations of globally coupled maps our analysis is complementary to the approach taken by teramae and kuramoto j teramae y kuramoto strong desynchronizing effects of weak noise in globally coupled systems phys rev e 63 2001 036210 who pointed out the anomalous scaling properties preceding the loss of coherence we focus on the macroscopic dynamics that remains deterministic at any noise level and differs from the microscopic one using properly defined order parameters an analytical approach is proposed for describing the collective dynamics in terms of an approximate low dimensional system the systematic derivation of the macroscopic equations provides a link between the microscopic features of the population single element dynamics and noise distribution and the properties of the emergent behaviour the macroscopic bifurcations induced by noise are compared to those originating from parameter mismatches within the population

representation of the exact solution for infinite system of linear equations in this paper representation of the exact solution for the infinite system of linear equations is given and we give a sufficient and necessary condition for the infinite system of linear equations which has solutions

on a theorem of choquet and dolecki abstract the results of this paper clarify and extend slightly the previous work of dolecki and lechicki c r acad sci paris 293 1981 219 221 j math anal appl 88 1982 547 584 and hansell jayne rogers and the author math z 189 1985 297 318 let x y be hausdorff spaces and f x y an upper semicontinuous set valued map a subset k of f x is said to be a peak of f at x if for every open set v containing k there exists a neighbourhood u of x such that f u f x t v criteria choquet dolecki theorems are given in order that f has the smallest possible peak it turns out that in unexpectedly general situations an upper semicontinuous map f has for every x in x a peak which is the smallest possible at x and moreover compact

solving singular two point boundary value problem in reproducing kernel space in this paper we present a new method for solving singular two point boundary value problem for certain ordinary differential equation having singular coefficients its exact solution is represented in the form of series in reproducing kernel space in the mean time the n term approximation u n x to the exact solution u x is obtained and is proved to converge to the exact solution some numerical examples are studied to demonstrate the accuracy of the present method results obtained by the method are compared with the exact solution of each example and are found to be in good agreement with each other

a simple nsc rkhs algorithm for solving linear boundary value problems error bounds and implementations abstract the simple algorithm is in essence a normal spline collocation method introduced by v k gorbunov in 1989 the paper presents a reproducing kernel hilbert space approach to the method that gives rise to basic error inequalities for a natural h 2 class of mth order linear problems the approximation of any normal solution v is c m 1 uniform i e v a t v n a t c a n 2 v h m 2 for a m where n is diameter of the partition of the domain by nodes simplicity of the algorithm in sobolev spaces is justified by explicit formulas for the kernel functions discovered by gorbunov and v v petrishchev in 2003 and underestimated in the opinion of the author as an application numerical solutions in h 5 and in h 6 are compared with those computed by other authors the simple method is then applied to a fifth order nonlocal boundary value problem in h 6 and in h 7

borel selectors for upper semi continuous multi valued functions on presente un theoreme de selection plus fin qui s applique dans plusieurs espaces de banach non separables

analytical solution of metapopulation dynamics in a stochastic environment we study a discrete stochastic linear metapopulation model to understand the effect of risk spreading by dispersion we calculate analytically the stable distribution of populations in different habitats the simultaneous distribution of populations in habitats has a complicated self similar structure but the population in each habitat follows a log normal distribution a class of discrete stochastic matrix models was mostly dealt with numerically our analytical predictions are robust in the wide range of parameters qualitative predictions of the current results should hold in the case of multiple habitats we thus conclude that environmental stochasticity always promotes dispersal