the dirichlet problem for nonlinear second order elliptic equations iii functions of the eigenvalues of the hessian on etudie le probleme de dirichlet dans un domaine borne de r n a frontiere lisse f d 2 u dans u sur

a priori estimate for convex solutions to special lagrangian equations and its application we derive a priori interior hessian estimates for special lagrangian equations when the potential is convex when the phase is very large we show that continuous viscosity solutions are smooth in the interior of the domain c 2008 wiley periodicals inc

interior estimates for solutions of the quadratic hessian equations in dimension three abstract we derive a priori c 2 interior estimates in dimension three for viscosity solutions of the quadratic hessian equation 2 d 2 u f x provided that the positive f is bounded and its holder semi norm is small actually we relax the holder continuity condition of f to be of integral sense

is gauss quadrature better than clenshaw curtis we compare the convergence behavior of gauss quadrature with that of its younger brother clenshaw curtis seven line matlab codes are presented that implement both methods and experiments show that the supposed factor of 2 advantage of gauss quadrature is rarely realized theorems are given to explain this effect first following o hara and smith in the 1960s the phenomenon is explained as a consequence of aliasing of coefficients in chebyshev expansions then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log z 1 z 1 in the complex plane gauss quadrature corresponds to pade approximation at z infty clenshaw curtis quadrature corresponds to an approximation whose order of accuracy at z infty is only half as high but which is nevertheless equally accurate near 1 1

explicit gradient estimates for minimal lagrangian surfaces of dimension two we derive explicit uniform a priori interior hessian and gradient estimates for special lagrangian equations of all phases in dimension two

hessian estimates for special lagrangian equations with critical and supercritical phases in general dimensions we derive a priori interior hessian estimates for special lagrangian equation with critical and supercritical phases in general higher dimensions our unified approach leads to sharper estimates even for the previously known three dimensional and convex solution cases

hessian and gradient estimates for three dimensional special lagrangian equations with large phase we obtain a priori interior hessian and gradient estimates for special lagrangian equations with phase larger than a critical value in dimension three gradient estimates are also derived for critical and super critical phases in general dimensions

uniform approximation to cauchy principal value integrals with logarithmic singularity abstract an approximation of clenshaw curtis type is given for cauchy principal value integrals of logarithmically singular functions i f c 1 1 f x log x c x c d x c 1 1 with a given function f using a polynomial p n of degree n interpolating f at the chebyshev nodes we obtain an approximation i p n c i f c we expand p n in terms of chebyshev polynomials with o n log n computations by using the fast fourier transform our method is efficient for smooth functions f for which p n converges to f fast as n grows and so simple to implement this is achieved by exploiting three term inhomogeneous recurrence relations in three stages to evaluate i p n c for f z analytic on the interval 1 1 in the complex plane z the error of the approximation i p n c is shown to be bounded uniformly using numerical examples we demonstrate the performance of the present method

basic hopf algebras and quantum groups this paper investigates the structure of basic finite dimensional hopf algebras h over an algebraically closed field k the algebra h is basic provided h modulo its jacobson radical is a product of the field k in this case h is isomorphic to a path algebra given by a finite quiver with relations necessary conditions on the quiver and on the coalgebra structure are found in particular it is shown that only the quivers gamma g w given in terms of a finite group g and sequence w w 1 w 2 ldots w r of elements of g in the following way can occur the quiver gamma g w has vertices v g g in g and arrows a i g colon v g 1 to v w ig 1 mid g in g w i in w where the set w 1 w 2 ldots w r is closed under conjugation with elements in g and for each g in g the sequences w and gw 1g 1 gw 2g 1 ldots gw rg 1 are the same up to a permutation we show how k gamma g w is a kg bimodule and study properties of the left and right actions of g on the path algebra furthermore it is shown that the conditions we find can be used to give the path algebras k gamma g w themselves a hopf algebra structure for an arbitrary field k the results are also translated into the language of coverings finally a new class of finite dimensional basic hopf algebras are constructed over a not necessarily algebraically closed field most of which are quantum groups the construction is not characteristic free all the quivers gamma g w where the elements of w generates an abelian subgroup of g are shown to occur for finite dimensional hopf algebras the existence of such algebras is shown by explicit construction for closely related results of cibils and rosso see ci r

procedures for computing one and two dimensional integrals of functions with rapid irregular oscillations a collocation procedure for efficient integration of rapidly oscillatory functions is presented the integration problem is transformed into a certain o d e problem and this is solved by a collocation technique the method is also extended to two dimensional integration and some numerical results are appended showing the efficiency of the method in handling difficult cases of rapid irregular oscillations 1 the procedure for one dimensional integrals we consider integrals of the form a 1 1 1 f x e iq x dx where f is smooth and nonoscillatory and iq x i b a 1 two practical methods for evaluating rapidly oscillatory integrals are described in 1 the use of approximation as in filon s method 3 4 and the speedup method of longman 5 formally both methods are applicable to any integral of the form 1 1 but their best performance is for the case of a constant frequency q w in this note we present an efficient method which is applicable for cases of varying frequency q using only a small number of values of f and q in a b and the values q a and q b the proposed method follows the spirit of filon s method it is based upon the fact that if f were of the form 1 2 f x iq x p x p x l l1p x a x b then the integral could be evaluated directly as 1 3 j b iq x p x pt x eiq x dx f d p x eiq x dx b e qbp a eiqa equation 1 2 can be considered as a differential equation for p x and any solution of this equation can be used in 1 3 for evaluating i the general solution of this equation is 1 4 p x e iq x f t eiq t dt cj received march 24 1981 revised july 29 1981 1980 mathematics subject classification primary 65d30 secondary 65d32 1982 american mathematical society 0025 5718 8 1 ooo 0 1 23 03 25 531 this content downloaded from 157 55 39 104 on sun 19 jun 2016 06 36 01 utc all use subject to http about jstor org terms

the exterior dirichlet problem for hessian quotient equations abstract in this paper using a reverse maclaurin inequality we established the existence theorem of the exterior dirichlet problem for hessian quotient equations with a stronger asymptotic behavior at infinity when k n 1 2 we improve the result in l m dai the dirichlet problem for hessian quotient equations in exterior domains j math anal appl 380 2011 87 93 for s k l with k l 3 to the result with any 0 l k

filon clenshaw curtis rules for a class of highly oscillatory integrals with logarithmic singularities in this work we propose and analyse a numerical method for computing a family of highly oscillatory integrals with logarithmic singularities for these quadrature rules we derive error estimates in terms of n the number of nodes k the rate of oscillations and a sobolev like regularity of the function we prove that the method is not only robust but the error even decreases for fixed n as k increases practical issues about the implementation of the rule are also covered in this paper by a writing down ready to implement algorithms b analysing the numerical stability of the computations and c estimating the overall computational cost we finish by showing some numerical experiments which illustrate the theoretical results presented in this paper

an o 1 integration scheme for three dimensional surface scattering problems we present an accurate method of o 1 complexity with respect to frequency i e a method that to achieve a prescribed error tolerance requires a bounded computational cost for arbitrarily high frequencies for the computation of singular oscillatory integrals arising in the boundary integral formulation of problems of acoustic scattering by surfaces in three dimensional space like the two dimensional counterpart of this algorithm which we introduced recently and which is applicable to scattering by curves in the plane the present method is based on a combination of two main elements 1 a high frequency ansatz for the unknown density in a boundary integral formulation of the problem and 2 an extension of the ideas of the method of stationary phase to allow for o 1 high order accurate integration of oscillatory functions the techniques we introduce to implement an efficient o 1 integrator in the present three dimensional context differ significantly from those used in the earlier two dimensional algorithm in particular in the present text we introduce an efficient canonical hybrid analytic numerical algorithm which in addition to allowing for integration of oscillatory functions around both singular points and points of stationary phase can handle the significant difficulty that arises as singular points and one or more stationary points approach each other within the two dimensional scattering surface we include numerical results illustrating the behavior of the integration algorithm on sound soft spheres with diameters of up to 5000 wavelengths in such cases for a single integral the algorithm yields accuracies of the order of three digits in computational times of less than two seconds in a preliminary full scattering simulation we present a solution with two digits of accuracy in the surface density was obtained in about three hours running time in a single 1 5ghz amd athlon processor for a sphere of 500 wavelengths in diameter

error bounds for approximation in chebyshev points this paper improves error bounds for gauss clenshaw curtis and fejer s first quadrature by using new error estimates for polynomial interpolation in chebyshev points we also derive convergence rates of chebyshev interpolation polynomials of the first and second kind for numerical evaluation of highly oscillatory integrals preliminary numerical results show that the improved error bounds are reasonably sharp

numerical integration of oscillatory airy integrals with singularities on an infinite interval abstract this work is devoted to the quadrature rules and asymptotic expansions for two classes of highly oscillatory airy integrals on an infinite interval we first derive two important asymptotic expansions in inverse powers of the frequency then based on structure characteristics of the two asymptotic expansions in inverse powers of the frequency both the so called filon type method and the more efficient clenshaw curtis filon type method are introduced and analyzed the required moments in the former can be explicitly expressed by the meijer g functions the latter can be implemented in o n log n operations based on fast fourier transform fft and fast computation of the modified moments here we can construct two useful recurrence relations for computing the required modified moments accurately with the help of the airy s equation and some properties of the chebyshev polynomials particularly we also provide their error analyses in inverse powers of the frequency furthermore the presented error analysis shows the advantageous property that the accuracy improves greatly as increases numerical examples are provided to illustrate the efficiency and accuracy of the proposed methods

a hybrid collocation method for volterra integral equations with weakly singular kernels the commonly used graded piecewise polynomial collocation method for weakly singular volterra integral equations may cause serious round off error problems due to its use of extremely nonuniform partitions and the sensitivity of such time dependent equations to round off errors the singularity preserving nonpolynomial collocation method is known to have only local convergence to overcome the shortcoming of these well known methods we introduce a hybrid collocation method for solving volterra integral equations of the second kind with weakly singular kernels in this hybrid method we combine a singularity preserving nonpolynomial collocation method used near the singular point of the derivative of the solution and a graded piecewise polynomial collocation method used for the rest of the domain we prove the optimal order of global convergence for this method the convergence analysis of this method is based on a singularity expansion of the exact solution of the equations we prove that the solutions of such equations can be decomposed into two parts with one part being a linear combination of some known singular functions which reflect the singularity of the solutions and the other part being a smooth function a numerical example is presented to demonstrate the effectiveness of the proposed method and to compare it to the graded collocation method

computation of integrals with oscillatory singular factors of algebraic and logarithmic type in this paper we present the clenshaw curtis filon methods and the higher order methods for computing many classes of oscillatory integrals with algebraic or logarithmic singularities at the two endpoints of the interval of integration the methods first require an interpolant of the nonoscillatory and nonsingular parts of the integrands in n 1 clenshaw curtis points then the required modified moments can be accurately and efficiently computed by constructing some recurrence relations moreover for these quadrature rules their absolute errors in inverse powers of the frequency are given the presented methods share the advantageous property that the accuracy improves greatly for fixed n as increases numerical examples show the accuracy and efficiency of the proposed methods

a compactness approach to hessian estimates for special lagrangian equations with supercritical phase abstract we derive interior hessian estimates for special lagrangian equations with supercritical phase in general dimension through a compactness approach

the neumann problem of special lagrangian equations with supercritical phase abstract in this paper we establish the global c 2 estimates of the neumann problem of special lagrangian equations with supercritical phase and the existence theorem by the method of continuity

a bernstein problem for special lagrangian equations where is are the eigenvalues of the hessian d2u namely any global convex solution to 1 1 in rn must be a quadratic polynomial recall the classical result any global convex solution in rn to the laplace equation u 1 n c or the monge ampere equation log det d2u log 1 log n c must be quadratic equation 1 1 originates from special lagrangian geometry hl the lagrangian graph x u x rn rn is called special when the argument of the complex number 1 1 1 1 1 n is constant c or u satisfies 1 1 and it is special if and only if x u x is a minimal surface in rn rn hl theorem 2 3 proposition 2 17 in terms of minimal surface our result is the following

a special lagrangian type equation for holomorphic line bundles let l be a holomorphic line bundle over a compact kahler manifold x motivated by mirror symmetry we study the deformed hermitian yang mills equation on l which is the line bundle analogue of the special lagrangian equation in the case that x is calabi yau we show that this equation is the euler lagrange equation for a positive functional and that solutions are unique global minimizers we provide a necessary and sufficient criterion for existence in the case that x is a kahler surface for the higher dimensional cases we introduce a line bundle version of the lagrangian mean curvature flow and prove convergence when l is ample and x has non negative orthogonal bisectional curvature

quadrature of integrands with a logarithmic singularity a quadrature rule is presented that is exact for integrands of the form 7t z m c loxic on the interval 0 1 where n and tu are polynomi als the computed weights and abscissae are given for one through seven point rules in particular the four point rule is exact for integral operators with log arithmically singular kernel on a cubic b spline basis and it is expected these results shall prove useful for numerical applications of weighted residual finite element methods in the formulation of solutions to initial value and boundary value problems for a partial differential equation by means of integral operators one is led in a natural way to integrals with singular kernel functions this is due to the singularity of the associated fundamental solution and the fact that this fundamental solution or some derivative of it often is used for the kernel of the integral representation the problems arising in mathematical physics in two spatial dimensions typically lead to kernels with a logarithmic singularity for example the wave equation c2ay is reduced by the form i if e to the helmholtz equation aip k2 p 0 where k2 co2 c2 and r2 and which in two dimensions has a fundamental solution involving

liouville property and regularity of a hessian quotient equation we are concerned with properties of convex solutions to the hessian quotient equation sn c d2 p 1 k n as our first main result we prove some regularity of strong solutions while the second states that for p 1 an entire convex solution with a quadratic growth bound must be a quadratic polynomial

on quadrature of highly oscillatory integrals with logarithmic singularities in this paper a quadrature rule is discussed for highly oscillatory integrals with logarithmic singularities at the same time its error depends on the frequency and the computation of its moments are given the new rule is implemented by interpolating f at chebyshev nodes and singular point where the interpolation polynomial satisfies some conditions numerical experiments conform the efficiency for obtaining the approximations

the exterior dirichlet problem for fully nonlinear elliptic equations related to the eigenvalues of the hessian in this paper we establish the existence theorem for the exterior dirichlet problems for a class of fully nonlinear elliptic equations which are related to the eigenvalues of the hessian matrix with prescribed asymptotic behavior at infinity this extends the previous results on monge ampere equation and k hessian equation to more general cases in particular including the special lagrangian equation

a bernstein problem for special lagrangian equations in exterior domains abstract we establish quadratic asymptotics for solutions to special lagrangian equations with supercritical phases or with semiconvexity on solutions in exterior domains the method for convex case is based on an exterior evans krylov or exterior liouville type result for general fully nonlinear elliptic equations toward constant asymptotics of bounded hessian and also certain rotation arguments toward hessian bound our unified approach also leads to quadratic asymptotics for convex solutions to monge ampere equations previously known quadratic hessian equations and inverse harmonic hessian equations over exterior domains the semiconvex case is based on allard almgren s uniqueness of tangent cones

quiver approaches to quasi hopf algebras we provide a quiver setting for quasi hopf algebras generalizing the hopf quiver theory as applications we obtain some general structure theorems in particular the quasi hopf analog of the cartier theorem and the cartier gabriel decomposition theorem

on quadrature for cauchy principal value integrals of oscillatory functions the authors develop an algorithm for the numerical evaluation of cauchy principal value integrals of oscillatory functions the method is based on an interpolatory procedure at the zeros of the orthogonal polynomials with respect to a jacobi weight a numerically stable procedure is obtained and the corresponding algorithm can be implemented in a fast way yielding satisfactory numerical results bounds of the error and of the amplification factor are also provided

regularity theory and superalgebraic solvers for wire antenna problems we consider the problem of evaluating the current distribution j z that is induced on a straight wire antenna by a time harmonic incident electromagnetic field the scope of this paper is twofold one of its main contributions is a regularity proof for a straight wire occupying the interval 1 1 in particular for a smooth time harmonic incident field this theorem implies that j z i z sqrt 1 z 2 where i z is an infinitely differentiable function the previous state of the art in this regard placed i in the sobolev space w 1 p p 1 the second focus of this work is on numerics we present three superalgebraically convergent algorithms for the solution of wire problems two based on hallen s integral equation and one based on the pocklington integrodifferential equation both our proof and our algorithms are based on two main elements 1 a new decomposition of the kernel of the form g z f 1 z ln z f 2 z where f 1 z and f 2 z are analytic functions on the real line and 2 removal of the end point square root singularities by means of a coordinate transformation the hallen and pocklington based algorithms we propose converge superalgebraically faster than mathcal o n m and mathcal o m m for any positive integer m where n and m are the numbers of unknowns and the number of integration points required for construction of the discretized operator respectively in previous studies at most the leading order contribution to the logarithmic singular term was extracted from the kernel and treated analytically the higher order singular derivatives were left untreated and the resulting integration methods for the kernel exhibit mathcal o m 3 convergence at best a rather comprehensive set of tests we consider shows that in many cases to achieve a given accuracy the numbers n of unknowns required by our codes are up to a factor of five times smaller than those required by the best solvers previously available the required number m of integration points in turn can be several orders of magnitude smaller than those required in previous methods in particular four digit solutions were found in computational times of the order of four seconds and in most cases of the order of a fraction of a second on a contemporary personal computer much higher accuracies result in very small additional computing times

a variational theory of the hessian equation by studying a negative gradient flow of certain hessian functionals we establish the existence of critical points of the functionals and consequently the existence of ground states to a class of nonhomogenous hessian equations to achieve this we derive uniform first and second order a priori estimates for the elliptic and parabolic hessian equations our results generalize well known results for semilinear elliptic equations and the monge ampere equation c 2001 john wiley sons inc

hessian estimates for the sigma 2 equationin dimension 3 we derive a priori interior hessian estimates for the special lagrangian equation 2 d1 in dimension3 c 2008 wiley periodicals inc

a practical algorithm for computing cauchy principal value integrals of oscillatory functions abstract a new automatic quadrature scheme is proposed for evaluating cauchy principal value integrals of oscillatory functions 1 1 f x exp i x x 1 dx 1 1 r the desired approximation is obtained by expanding the function f in the series of chebyshev polynomials of the first kind and then by constructing the indefinite integral for a properly modified integrand to overcome the singularity the method is proved to converge uniformly with respect to both and for any function f satisfying max 1 x 1 f x

singular solutions to special lagrangian equations with subcritical phases and minimal surface systems we construct singular solutions to special lagrangian equations with subcritical phases and minimal surface systems a priori estimate breaking families of smooth solutions are also produced correspondingly a priori estimates for special lagrangian equations with certain convexity are largely known by now

bosonization for dual quasi bialgebras and preantipode abstract to every dual quasi bialgebra h and every bialgebra r in the category of yetter drinfeld modules over h one can associate a dual quasi bialgebra called bosonization in this paper using the fundamental theorem we characterize as bosonizations the dual quasi bialgebras with a projection onto a dual quasi bialgebra with a preantipode as an application we investigate the structure of the graded coalgebra gr a associated to a dual quasi bialgebra a with the dual chevalley property e g a is pointed

on the convergence rate of clenshaw curtis quadrature for integrals with algebraic endpoint singularities abstract in this paper we are concerned with clenshaw curtis quadrature for integrals with algebraic endpoint singularities an asymptotic error expansion and convergence rate are derived by combining a delicate analysis of the chebyshev coefficients of functions with algebraic endpoint singularities and the aliasing formula of chebyshev polynomials numerical examples are provided to confirm our analysis

quivers quasi quantum groups and finite tensor categories we study finite quasi quantum groups in their quiver setting developed recently by the first author we obtain a classification of finite dimensional pointed majid algebras of finite corepresentation type or equivalently a classification of elementary quasi hopf algebras of finite representation type over the field of complex numbers by the tannaka krein duality principle this provides a classification of the finite tensor categories in which every simple object has frobenius perron dimension 1 and there are finitely many indecomposable objects up to isomorphism some interesting information of these finite tensor categories is given by making use of the quiver representation theory

fast non polynomial interpolation and integration for functions with logarithmic singularities abstract a fast non polynomial interpolation is proposed in this paper for functions with logarithmic singularities it can be executed fast with the discrete cosine transform based on this interpolation a new quadrature is proposed for a kind of logarithmically singular integrals the interpolation and integration errors are also analyzed numerical examples of the interpolation and integration are shown to validate the efficiency of the proposed new interpolation and the new quadrature

quiver hopf algebras abstract in this paper we study subhopfalgebras of graded hopf algebra structures on a path coalgebra k q c we prove that a hopf structure on some subhopfquivers can be lifted to a hopf structure on the whole hopf quiver if q is a schurian hopf quiver then we classified all simple pointed subhopfalgebras of a graded hopf structure on k q c we also prove a dual gabriel theorem for pointed hopf algebras