on the matsumoto yor property in free probability abstract we study the matsumoto yor property in free probability we prove that the limiting empirical eigenvalue distribution of the gig matrices and the marchenko pastur distribution have the free matsumoto yor property finally we characterize these distributions by regression properties in free probability

on the existence of positive solution for a class of singular systems involving quasilinear operators abstract in this paper we consider the existence of positive solution for the following class of singular systems p u 1 v 1 v 1 q v 1 u 2 u 2 in u v 0 in u v 0 on where r n n 2 is a smooth bounded domain p and q are respectively the p laplacian and q laplacian with p q 1 and 1 2 1 and 2 are positive constants the main tools used are a result due to rabinowitz and a hardy sobolev inequality

adjoint and defect error bounding and correction for functional estimates we present two error estimation approaches for bounding or correcting the error in functional estimates such as lift or drag adjoint methods quantify the error in a particular output functional that results from residual errors in approximating the solution to the partial differential equation defect methods can be used to bound or reduce the error in the entire solution with corresponding improvements to functional estimates both approaches rely on smooth solution reconstructions and may be used separately or in combination to obtain highly accurate solutions with asymptotically sharp error bounds the adjoint theory is presented for both smooth and shocked problems numerical experiments confirm fourth order error estimates for a pressure integral of shocked quasi id euler flow by employing defect and adjoint methods together and accounting for errors in approximating the geometry it is possible to obtain functional estimates that exceed the order of accuracy of the discretization process and the reconstruction approach superconvergent drag estimates are obtained for subsonic euler flow over a lifting airfoil

singular nonlinear boundary value problems for second order ordinary differential equations abstract the boundary value problem y f x y 0 y 0 y 0 0 y 1 y 1 0 is studied under conditions which are modeled on the problem y a x y p 0y 0 y 1 0 or y 0 y 1 0 which occurs in a number of applications the technical arguments involve properties of solutions i e concavity and the use of iterative techniques a new fixed point theorem for cones for decreasing mappings is proved and used it is of interest in its own right since most such theorems are for monotone increasing mappings

asymptotic expansion for the discretization error in linear elliptic boundary value problems on general regions to define a discretization we introduce a set f h of grid points see 2 2 and replace the derivatives in 1 1 by symmetric divided differences with respect to those grid points this device is unproblematic as long as the direct grid neighbours of interior grid points are either interior grid points or boundary points by a proper choice of h that may be obtained e g for rectangular shape regions f2 for all inner grid points for general domains f2 this assumption is no longer satisfied and direct grid neighbours will not always be in and therefore symmetric divided differences are not directly applicable in some points so called irregular mesh points there is a long history to solve this problem 1930 gerschgorin 153 has proposed to use for the unknown function value in a grid neighbour outside of f2 next to an irregular grid point the boundary value of the next boundary point on that grid line obtaining a method of order 1 compared with order 2 available in cases without irregular grid points in 1933 collatz 11 improved this method by using linear interpolation between the irregular grid point and its neighbouring boundary point obtaining a method of order 2 wasow 22 showed that for the methods in 11 15 no asymptotic expansions of the discretization error can exist the 1938 approach of shortley weller 20 may be interpreted as using polynomials of degree 2 interpolating in the

on a class of singular p laplacian boundary value problems abstract we prove the existence and nonexistence of positive solutions for the boundary value problem p u f u in u 0 on where p u div u p 2 u p 1 is a bounded domain in r n with smooth boundary f 0 r is p 1 subhomogeneous at and is allowed to change sign and is a large parameter our approach is based on the method of sub and supersolutions

twin solutions to singular dirichlet problems the existence of two nonnegative solutions to dirichlet second order boundary value problems is established in this paper our nonlinearity may be singular at y 0 t 0 and or t 1

positive radial solutions for a class of quasilinear boundary value problems in a ball abstract we prove the existence and nonexistence of positive radial solutions for the boundary value problems p u h u f u in u 0 on where p u div u p 2 u p 1 is the open unit ball in r n h f 0 r are allowed to be singular at 0 f is asymptotically p linear and is a positive parameter

a theoretical framework for proving accuracy results for deferred corrections general techniques are described for proving accuracy results for deferred correction solutions to differential equations these techniques apply also to computational estimates of the local discretization error the proofs avoid the necessity of demonstrating the existence of asymptotic expansions of the global error in powers of some meshsize parameter

the lukacs theorem and the olkin baker equation abstract the olkin baker functional equation except of being studied inside the theory of functional equations is closely related to the celebrated lukacs characterization of the gamma distribution its deeper understanding in the case of measurable unknown functions is essential to settle a challenging question of multivariate extensions of the lukacs theorem in this paper first we provide a new approach to the additive olkin baker equation which holds almost everywhere on 0 2 with respect to the lebesgue measure on r 2 under measurability assumption second this new approach is adapted to the case when unknown functions are allowed to be non measurable and the complete solution is given in such a general case third the olkin baker equation holding outside of a set from proper linearly invariant ideal of subsets of r 2 is considered

gaussian collocation via defect correction for the numerical integration of boundary value problems for first order ordinary differential systems collocation on gaussian points is known to provide a powerful method in this paper we introduce a defect correction method for the iterative solution of such high order collocation equations the method uses the trapezoidal scheme as the basic discretization and an adapted form of the collocation equations for defect evaluation the error analysis is based on estimates of the contractive power of the defect correction iteration it is shown that the iteration produceso h 2 convergence rates for smooth starting vectors a new result is that the iteration damps all kind of errors so that it can also handle non smooth starting vectors successfully

a three solution theorem for singular nonlinear elliptic boundary value problems abstract we establish a three solution theorem for singular elliptic boundary value problems of the form u f u u in u 0 on where is a bounded domain in r n n 1 with a smooth boundary here f 0 0 is a c 1 function in 0 with f 0 0 and 0 1 in particular if there exist two pairs of sub supersolutions 1 1 and 2 2 where 1 2 1 1 2 1 with 2 2 and 2 2 are strict sub and supersolutions then we establish existence of three solutions u 1 u 2 and u 3 for the above boundary value problem such that u 1 1 2 u 2 2 1 and u 3 1 1 1 2 2 1 our results extend the work in 1 and also in 17 where such multiplicity results were discussed in the non singular case 0 further our results strengthen the multiplicity results in 12 and 13 for such singular problems 0 namely establish an additional third solution we also establish a sub super solution theorem for infinite semipositone problems i e the case when f 0 0

on a singular nonlinear semilinear elliptic problem where k x c2 b v9 a p 0 1 and l is a real parameter such singular elliptic problems arise in the contexts of chemical heterogeneous catalysts nonnewtonian fluids and also the theory of heat conduction in electrically conducting materials see 3 5 8 9 for a detailed discussion obviously 1 1 cannot have a solution u c2 v9 if k x is not vanishing near v however under various appropriate assumptions on k x we will obtain classical solutions of 1 1 for l belonging to a certain range and we will also obtain some uniqueness criteria here a classical solution is a solution u of 1 1 which belongs to c2 v mc v9 with u 0 in v we also study the boundary behaviour of solutions of 1 1 and we will show that the solution u of 1 1 lies in a certain holder class the special case when k x is negative and l 0 has been studied by several authors the existence and uniqueness of the solution were established by crandall rabinowitz and tartar 6 del pino 7 gomes 10 and lazer and mckenna

b convergence properties of defect correction methods i b convergence properties of defect correction methods based on the implicit euler and midpoint schemes are discussed the property ofb convergence means that there exist global error bounds for nonlinear stiff problems independent of their stiffness it turns out that the orders ofb convergence of these methods coincide with the conventional orders of small wherel is a lipschitz constant of the right hand side in part i these assertions are reduced to the validity of the so called hypothesisa which is discussed in greater detail in part ii numerical experiments confirming the theoretical analysis are also given in part ii

existence of positive entire solutions for singular and non singular quasi linear elliptic equation we consider the quasilinear elliptic equation div u p 2 u m x u 0 in domain rn where 0 p 1 under several hypotheses on the m x we obtain the existence of positive entire solutions

sublinear singular elliptic problems with two parameters abstract we establish several existence and nonexistence results for the boundary value problem u k x g u f x u h x in u 0 on where is a smooth bounded domain in r n and are positive parameters h is a positive function while f has a sublinear growth the main feature of this paper is that the nonlinearity g is assumed to be unbounded around the origin our analysis shows the importance of the role played by the decay rate of g combined with the signs of the extremal values of the potential k x on the proofs are based on various techniques related to the maximum principle for elliptic equations

discrete newton methods and iterated defect corrections in this paper we present a general theory for discrete newton methods iterated defect corrections via neighbouring problems and deferred corrections based on asymptotic expansions of the discretization error

asymptotic expansions for the discretization error of least squares solutions of linear boundary value problems on presente des developpements asymptotiques pour l erreur de discretisation des solutions des moindres carres des problemes aux limites lineaires

a matsumoto yor characterization for kummer and wishart random matrices abstract in the paper we resolve positively the conjecture on a characterization of matrix kummer and wishart laws through independence property which was posed in koudou 2012 12 apart from the probabilistic result we determine the general solution of the functional equation associated to the characterization problem under weak assumptions

the existence of three positive solutions of a singular p laplacian problem abstract we establish the existence of three positive solutions of the p laplacian problem which involves a singular nonlinearity three solutions are obtained by using the cutoff argument and the three critical points theorem proved by ricceri 2000 in 15 and bonanno 2003 in 12 provided the nonlinear term is a p 1 sublinear growth at infinity

existence theory for single and multiple solutions to singular positone boundary value problems abstract in this paper we present some old and new existence results for singular boundary value problems our nonlinearity may be singular in its dependent variable

change of measure technique in characterizations of the gamma and kummer distributions abstract if x and y are independent random variables with distributions nd then u x y and v x y are also independent for some transformations and properties of this type are known for many important probability distributions and also related characterization questions have been widely investigated these are questions of the form let x and y be independent and let u and v be also independent are the distributions of x and y necessarily and respectively recently two new properties and characterizations of this kind involving the kummer distribution appeared in the literature for independent x and y with gamma and kummer distributions koudou and vallois in 17 observed that u 1 x y 1 1 x 1 and v x y are also independent and hamza and vallois in 13 observed that u y 1 x and v x 1 y 1 x are independent in 16 and 17 characterizations related to the first property were proved while the characterizations in the second setting have been recently given in 29 these results were not fully satisfactory since in both cases technical assumptions on smoothness properties of densities of x and y were needed in 31 the assumption of independence of u and v in the first setting was weakened to constancy of regressions of u and u 1 given v with no density assumptions however the additional assumption e x 1 was introduced in the present paper we provide a complete answer to the characterization question in both settings without any additional technical assumptions regarding smoothness or existence of moments the approach is first via characterizations exploiting some conditions imposed on regressions of u given v which are weaker than independence but for which moment assumptions are necessary second using a technique of change of measure we show that the moment assumptions can be avoided

positive solutions for infinite semipositone problems with falling zeros abstract we consider the positive solutions to the singular problem p u a u f u c u in u 0 on where 0 1 a 0 and c 0 are constants is a bounded domain with smooth boundary and f 0 r is a continuous function we assume that there exist m 0 a 0 p 1 such that a u m f u a u p for all u 0 a simple example of f satisfying these assumptions is f u u p for any p 1 we use the method of sub supersolutions to prove the existence of a positive solution of p when a 2 1 1 and c is small here 1 is the first eigenvalue of operator with dirichlet boundary conditions we also extend our result to classes of infinite semipositone systems

nonlinear elliptic equations with singular boundary conditions abstract we consider the model problem formula where is a bounded region in r n with smooth boundary q 0 2 and k u u p p 1 or e u we show that there exists a classical solution

thirteen ways to estimate global error various techniques that have been proposed for estimating the accumulated discretization error in the numerical solution of differential equations particularly ordinary differential equations are classified described and compared for most of the schemes either an outline of an error analysis is given which explains why the scheme works or a weakness of the scheme is illustrated

an independence property for the product of gig and gamma laws matsumoto and yor have recently discovered an interesting transformation which preserves a bivariate probability measure which is a product of the generalized inverse gaussian gig and gamma distributions this paper is devoted to a detailed study of this phenomenon let x and y be two independent positive random variables we prove theorem 4 1 that u x y 1 and v x 1 x y 1 are independent if and only if there exists p a b 0 such that y is gamma distributed with shape parameter p and scale parameter 2a 1 and such that x has a gig distribution with parameters p a and b the direct part for a b was obtained in matsumoto and yor the result is partially extended theorem 5 1 to the case where x and y are valued in the cone v of symmetric positive definite r r real matrices as follows under a hypothesis of smoothness of densities we prove that u x y 1 and v x 1 x y 1 are independent if and only if there exists p r 1 2 and a and b in v such that y is wishart distributed with shape parameter p and scale parameter 2a 1 and such that x has a matrix gig distribution with parameters p a and b the direct result is also extended to singular wishart distributions theorem 3 1