boundary methods for solving elliptic problems with singularities and interfaces boundary approximation techniques are described for solving homogeneous self adjoint elliptic equations piecewise expansions into particular solutions are used which approximate both the boundary and interface conditions in a least squares sense convergence of such approximations is proved and error estimates are derived in a natural norm numerical experiments are reported for the singular motz problem which yield extremely accurate solutions with only a modest computational effort

the high accurate block grid method for solving laplace s boundary value problem with singularities a high accurate difference analytical method is introduced for the solution of the mixed boundary value problem for laplace s equation on graduated polygons the polygon can have broken sections and be multiply connected the uniform estimate of the error of the approximate solution is of order o h6 whereas it is of order o h 6 r j p lambda j for the errors of p order derivatives p 1 2 in a finite neighborhood of reentry vertices here h is the mesh step rj is the distance from the current point to the vertex in question lambda j 1 a alpha j and a 1 or 2 depending on the types of boundary conditions further alpha j pi is the value of the interior angle at the considered vertex numerical experiments are illustrated in section ref sec8 to support the analysis made

eigenvalues of the laplacian in two dimensions the eigenvalue problem for the laplace operator in two dimensions is classical in mathematics and physics nevertheless computational methods for estimating the eigenvalues are still of much current interest particularly in applications to acoustic and electromagnetic waveguides although our primary interest is with the computational methods there are a number of theoretical results on the behavior of the eigenvalues and eigenfunctions that are useful for understanding the methods and in addition are of interest in themselves these results are discussed first and then the various computational methods that have been used to estimate the eigenvalues are reviewed with particular emphasis on methods that give error bounds some of the more powerful techniques available are illustrated by applying them to a model problem

on solution uniqueness of elliptic boundary value problems in this paper we consider the problem of solution uniqueness for the second order elliptic boundary value problem by looking at its finite element or finite difference approximations we derive several equivalent conditions which are simpler and easier than the boundedness of the entries of the inverse matrix given in yamamoto et al t yamamoto s oishi q fang discretization principles for linear two point boundary value problems ii numer funct anal optim 29 2008 213 224 the numerical experiments are provided to support the analysis made strictly speaking the uniqueness of solution is equivalent to the existence of nonzero eigenvalues in the corresponding eigenvalue problem and this condition should be checked by solving the corresponding eigenvalue problems an application of the equivalent conditions is that we may discover the uniqueness simultaneously while seeking the approximate solutions of elliptic boundary equations

some results on kwong functions and related inequalities abstract we investigate some relations between kwong functions and operator monotone functions as an application we present an arithmetic geometric mean type inequality by showing that for two continuous functions f g on 0 such that h t f t g t is a kwong function and f t g t t any positive matrices a b and any matrix x it holds that f a x g b g a x f b a x x b for each unitarily invariant norm

error and stability analysis of boundary methods for elliptic problems with interfaces in boundary methods piecewise particular solutions are employed to solve a given elliptic equation within subdomains of some region of interest a boundary approximation is then obtained by satisfying the interior and exterior boundary conditions in a least squares sense we examine convergence derive error norm bounds for approximate solutions and conduct a stability analysis of the associated algebraic problem the aim of this analysis is to help choosing good partitions of subdomains

superconvergence and asymptotic expansion for semidiscrete bilinear finite volume element approximation of the parabolic problem abstract we first derive the asymptotic expansion of the bilinear finite volume element for the linear parabolic problem by employing the energy embedded method on uniform grids and then obtain a high accuracy combination pointwise formula of the derivatives for the finite volume element approximation based on the above asymptotic expansion furthermore we prove that the approximate derivatives have the convergence rate of order two numerical experiments confirm the theoretical results

decision tree approximations of boolean functions decision trees are popular representations of boolean functions we show that given an alternative representation of a boolean function f say as a read once branching program one can find a decision tree t which approximates f to any desired amount of accuracy moreover the size of the decision tree is at most that of the smallest decision tree which can represent f and this construction can be obtained in quasi polynomial time we also extend this result to the case where one has access only to a source of random evaluations of the boolean function f instead of a complete representation in this case we show that a similar approximation can be obtained with any specified amount of confidence as opposed to the absolute certainty of the former case this latter result implies proper pac learnability of decision trees under the uniform distribution without using membership queries

the simplified hybrid combined methods for laplace s equation with singularities abstract in li and liang 1983 the simplified hybrid combined method is presented for combining the ritz galerkin method and the finite element method in this paper we will apply this method to solve singularity problems of laplace s equation error bounds and stability analyses will be provided while taking into account the integration approximation along the coupling boundary a significant coupling relation between the ritz galerkin and the finite element method has been found for the laplace equation with singularities an optimal rate of convergence has also been achieved numerical experiments have been carried out for solving the benchmark problem motz s problem to verify the theoretical results

boundary approximation methods for solving elliptic problems on unbounded domains abstract boundary approximation methods with partial solutions are presented for solving a complicated problem on an unbounded domain with both a crack singularity and a corner singularity also an analysis of partial solutions near the singular points is provided these methods are easy to apply have good stability properties and lead to highly accurate solutions hence boundary approximation methods with partial solutions are recommended for the treatment of elliptic problems on unbounded domains provided that piecewise solution expansions in particular asymptotic solutions near the singularities and infinity can be found

some results on matrix monotone functions abstract let a and b be hermitian matrices we say that a b if a b is nonnegative definite a function 0 r is said to be matrix monotone m m if a b 0 implies that a b matrix convex and concave functions can be similarly defined in this paper several known results some in sharper forms are given alternative proofs these include the following a a positive function t is m m iff t t is m m b if t is m m then both t t and t t are matrix convex and are almost always strictly convex on the other hand t is always concave and almost always strictly concave the concavity of m m functions and an inequality of hansen are consequences of a more general inequality on m m functions some complementary inequalities are also established we then prove an extension of lyapunov s theorem let a and p be positive definite matrices and either or 1 be a m m function then the unique matrix solution x of the matrix equation ax xa a p p a is positive definite a similar result holds for a related equation most of the results are obtained by exploiting the integral representation of m m functions

a p extension of the symmetric boundary element method abstract this paper describes an implementation of the symmetric galerkin boundary element method for plane potential and elasticity problems on domains with polygonal boundaries the method makes use of the hyper singular boundary integral equations for the normal derivatives of the unknown function we discuss an integration technique which is sufficiently robust to handle p extensions on geometrically graded meshes computer times required compare favorably with corresponding finite element methods we present numerical results obtained by a uniform p extension for various mixed boundary value problems we demonstrate that the same convergence rates can be obtained by the symmetric variational boundary element method as with p extensions of the finite element method on the basis of the p extension process a reliable error estimate based on extrapolation from a sequence of p levels can be obtained this is a unique feature of the variational methods as compared to the classical collocation approach the error estimate remains valid even for boundary element analysis employing substructures and mixed discretizations with boundary elements and finite elements

on the definiteness of the solutions of certain matrix equations abstract it is shown that for each dimension n 2 there is a real number t n such that for all t 2 t n the unique n n solution x of the matrix equation a 2 x xa 2 taxa b must be positive semidefinite for any positive definite a and positive semidefinite b this generalizes the classical lyapunov result it is established that t n 8 4 for n 2 3 4 respectively and that t n 2 for all n upper bounds are given for n up to 6 related equations are also discussed

modal series expansion of eigensolutions for closed and open periodic waveguides an efficient approach is presented to compute the eigensolutions of closed and open radiating periodically loaded waveguides by modal series expansion the eigensolutions of the homogeneous background waveguides serve as basis functions for the fields in the periodic boundaries the corresponding scattering matrix of an isolated periodic unit cell is determined by commercial software solvers the scattering matrix is transformed into the transfer matrix which can be used to formulate an eigenproblem open field problems are converted into a closed equivalence while the characteristics of the original open configurations are preserved even for radiating leaky waveguiding structures consequently the original continuous spectrum of modes is represented by a discrete and proper set of modes an eigenproblem must still be solved but it is considerably reduced in size and complexity for configurations where relatively few expansion modes are sufficient

a nonconforming combination for solving elliptic problems with interfaces abstract nonconforming combinations are provided for solving interface problems of elliptic equations in these approaches the ritz galerkin method with particular solutions is used for the part of a solution domain where there are interface singular points and the conventional finite element method is used for the rest of the solution domain in addition admissible functions chosen are constrained to be continuous only at the element nodes on the common boundary of the subdomains error bounds are derived in the sobolev norms and numerical experiments are given for solving a model interface problem of the equation u u 0 moreover a significant coupling relation l 1 o ln h is found for interface problems by using the nonconforming combinations where l 1 is the total number of particular solutions used in the ritz galerkin method and h is the maximal boundary length of triangular elements in the finite element method

exact learning boolean functions via the monotone theory we study the learnability of boolean functions from membership and equivalence queries we develop the monotone theory that proves 1 any boolean function is learnable in polynomial time in its minimal disjunctive normal form size its minimal conjunctive normal form size and the number of variables n in particular 2 decision trees are learnable our algorithms are in the model of exact learning with membership queries and unrestricted equivalence queries the hypotheses to the equivalence queries and the output hypotheses are depth 3 formulas

numerical implementation of the edem for modified helmholtz bvps on annular domains in a recent paper by the current authors a new methodology called the extended domain eigenfunction method edem was proposed for solving elliptic boundary value problems on annular like domains in this paper we present and investigate one possible numerical algorithm to implement the edem this algorithm is used to solve modified helmholtz bvps on annular like domains two examples of annular like domains are studied the results and performance are compared with those of the well known boundary element method bem the high accuracy of the edem solutions and the superior efficiency of the edem over the bem make edem an excellent alternate candidate to use in the animation industry where speed is a predominant requirement and by the scientific community where accuracy is the paramount objective

model order reduction for meshfree solution of poisson singularity problems summary model order reduction mor techniques for enriched reproducing kernel meshfree methods are proposed for analysis of poisson problems with mild and strong singularities the employment of an integrated singular basis function method isbfm in conjunction with the selection of harmonic near tip asymptotic basis functions leads to a galerkin formulation in which the non smooth near tip basis functions appear only on the boundaries away from the singularity point this approach avoids the need of integrating the derivatives of non smooth functions near the singularity point and yields a discrete system that allows effective mor procedures under this framework a decomposed reduction method equipped with two distinct projections for smooth and non smooth parts of the finite dimensional space is proposed compared with the uniform reduction approach using a single projection operator the decomposed projection on isbfm discrete system preserves the singularity behavior of the fine scale solution in its lower dimensional approximation analytical error estimation and stability analysis show that isbfm with the decomposed projection can achieve better accuracy with only a slight increase of condition number compared with the uniform reduction approach in the reduced order solution of singularity problems the numerical tests validate the effectiveness of the proposed methods copyright 2014 john wiley sons ltd

a finite element method for stokes equation using discrete singularity expansion the numerical method presented in this paper for stokes equation with corner singularity includes mainly two steps firstly we solve a simple eigenvalue problem which is one dimension less than the original problem to obtain the discrete expansion of the singularity near the corner secondly we combine the approximation of the singularity and standard finite element basis functions to construct special finite element spaces and solve the original problem in the special spaces on a conventional mesh the numerical examples show the effectiveness of this method

zhan s inequality on a g mean inequalities abstract in this paper we show some generalizations of zhan s inequality by virtue of hadamard product as follows let a b x be n n matrices such that a b are positive definite and f g be two continuous functions on 0 k max f g a b where a is the spectrum of a if f x g x is kwong i e the matrix f g i f g j i j is positive semidefinite for any positive numbers 1 n then for any 0 i a 1 2 f a x g b g a x f b b 1 2 k 2 1 2 2 r 0 a x b 4 1 r 0 t 2 a 2 x t a x b x b 2 holds for 2 t 2 2 where r 0 min 1 2 1 r 1 1 r i i a 1 2 f a x g b g a x f b b 1 2 2 k t 2 a 2 x t a x b x b 2 holds for any 2 t 2 where is any unitarily invariant norm

error analysis of the trefftz method for solving laplace s eigenvalue problems for solving laplace s eigenvalue problems we propose new algorithms using the trefftz method tm i e the boundary approximation method bam by means of degeneracy of numerical helmholtz equations since piecewise particular solutions can be fully adopted the new algorithms benefit high accuracy of eigenvalues and eigenfunctions low cost in cpu time and computer storage also the algorithms can be applied to solve the problems with multiple interfaces and singularities in this paper error estimates are derived for the approximate eigenvalues and eigenfunctions obtained numerical experiments for smooth and singular solutions are reported in this paper to show significance of the algorithms proposed and to verify the theoretical results made

error analysis of the method of fundamental solutions for linear elastostatics for linear elastostatics in 2d the trefftz methods i e the boundary methods using the particular solutions and the fundamental solutions satisfying the cauchy navier equation lead to the method of particular solutions mps and the method of fundamental solutions mfs respectively in this paper the mixed types of the displacement and the traction boundary conditions are dealt with and both the direct collocation techniques and the lagrange multiplier are used to couple the boundary conditions the former is just the mfs and the mps and the latter is also called the hybrid trefftz method htm in jirousek 1978 1992 1996 1 3 in bogomolny 1985 4 and li 2009 5 the error analysis of the mfs is given for laplace s equation and in li 2012 6 the error bounds of both mps and htm using particular solutions ps are provided for linear elastostatics in this paper our efforts are devoted to explore the error analysis of the mfs and the htm using fundamental solutions fs the key analysis is to derive the errors between fs and ps of the linear elastostatics where the expansions of the fs in li et al 2011 7 are a basic tool in analysis then the optimal convergence rates can be achieved for the mfs and the htm using fs recently the mfs has been developed with numerous reports in computation the analysis is behind the analysis of the mfs for linear elastostatics in this paper may narrow the existing gap between computation and theory of the mfs

block method for problems on l shaped domains an extremely accurate exponentially convergent solution is presented for both symmetric and non symmetric laplacian problems on l shaped domains by using one block version of the block method bm a simple and highly accurate formula for computing the stress intensity factor is given comparisons with various results in the literature are included

learning decision trees from random examples needed for learning abstract we define the rank of a decision tree and show that for any fixed r the class of all decision trees of rank at most r on n boolean variables is learnable from random examples in time polynomial in n and linear in 1 and log 1 where is the accuracy parameter and is the confidence parameter using a suitable encoding of variables rivest s polynomial learnability result for decision lists can be interpreted as a special case of this result for rank 1 as another corollary we show that decision trees on n boolean variables of size polynomial in n are learnable from random examples in time linear in n o log n 1 and log 1 as a third corollary we show that boolean functions that have polynomial size dnf expressions for both their positive and their negative instances are learnable from random examples in time linear in n o log n 2 1 and log 1

a singular function boundary integral method for laplacian problems with boundary singularities a singular function boundary integral method for laplacian problems with boundary singularities is analyzed in this method the solution is approximated by the truncated asymptotic expansion for the solution near the singular point and the dirichlet boundary conditions are weakly enforced by means of lagrange multiplier functions the resulting discrete problem is posed and solved on the boundary of the domain away from the point of singularity the main result of this paper is the proof of convergence of the method in particular we show that the method approximates the generalized stress intensity factors i e the coefficients in the asymptotic expansion at an exponential rate a numerical example illustrating the convergence of the method is also presented