comment on the paper a class of methods based on non polynomial spline functions for the solution of a special fourth order boundary value problems with engineering applications abstract comment on the paper a class of methods based on non polynomial spline functions for the solution of a special fourth order boundary value problems with engineering applications in applied mathematics and computation 174 2006 1169 1180 the paper considered a class of two point boundary value problems of the form 1 d 4 y d x 4 f x y g x a x b 2 y a a 1 y b a 2 y a b 1 y b b 2 where f x and g x are continuous on a b and a i and b i i 1 2 are finite real constants here we correct some mistake in derivation of non polynomial spline boundary formulas truncation errors convergence analysis and computational experiments

computations in non commutative iwasawa theory we study special values of l functions of elliptic curves over mathbb q twisted by artin representations that factor through a false tate curve extension mathbb q mu p infty sqrt p infty m mathbb q in this setting we explain how to compute l functions and the corresponding iwasawa theoretic invariants of non abelian twists of elliptic curves our results provide both theoretical and computational evidence for the main conjecture of non commutative iwasawa theory

organizing the arithmetic of elliptic curves abstract suppose that e is an elliptic curve defined over a number field k p is a rational prime and k is the maximal z p power extension of k in previous work b mazur k rubin elliptic curves and class field theory in ta tsien li ed proceedings of the international congress of mathematicians icm 2002 vol ii higher education press beijing 2002 pp 185 195 b mazur k rubin pairings in the arithmetic of elliptic curves in j cremona et al eds modular curves and abelian varieties progress in mathematics vol 224 2004 pp 151 163 we discussed the possibility that much of the arithmetic of e over k i e the mordell weil groups and their p adic height pairings the shafarevich tate groups and their cassels pairings over all finite extensions of k in k can be described efficiently in terms of a single skew hermitian matrix with entries drawn from the iwasawa algebra of k k in this paper using work of nekovar j nekovar selmer complexes preprint available at http www math jussieu fr nekovar pu we show that under not too stringent conditions such an organizing matrix does in fact exist we also work out an assortment of numerical instances in which we can describe the organizing matrix explicitly

the optimal p version approximation of singularities on polyhedra in the boundary element method we consider elliptic boundary value problems on three dimensional polyhedral domains we describe the three types of singular components that arise in the solution edge vertex and edge vertex singularities we analyze the approximation of the traces of each of each of these different types of singularities by polynomial subspaces on the boundary as the polynomial degree p tends to infinity the p version we establish asymptotic rates of convergence that are optimal in the h t norm over the boundary for 0 leq t leq 1 we apply these results to a galerkin boundary element method bem formulation for a model neumann problem obtaining the optimal convergence rate for the p version bem over polyhedral domains

euler systems for rankin selberg convolutions of modular forms we construct a euler system in the cohomology of the tensor product of the galois representations attached to two modular forms using elements in the higher chow groups of products of modular curves we use these elements to prove a finiteness theorem for the strict selmer group of the galois representation when the associated p adic rankin selberg l function is nonvanishing at s 1

p adaptive boundary elements this paper presents the implementation of an adaptive philosophy to plane potential problems using the direct boundary element method after some considerations about the state of the art and a discussion of the standard approach features the possibility of separately treating the modelling of variables and their interpolation through hierarchical shape functions is analysed then the proposed indicators and estimators are given followed by a description of a small computer program written for an ibm pc finally some examples show the kind of results to be expected

the p version of the boundary element method for weakly singular operators on piecewise plane open surfaces we study piecewise polynomial approximations in negative order sobolev norms of singularities which are inherent to neumann data of elliptic problems of second order in polyhedral domains the worst case of exterior crack problems in three dimensions is included as an application we prove an optimal a priori error estimate for the p version of the bem with weakly singular operators on polyhedral lipschitz surfaces and piecewise plane open screens

heegner points on mumford tate curves 1 shimura curves 417 2 heegner points 429 3 the regulator term 441 4 the conjecture 445 5 applications and re nements 447

construction of equilibrated singular basis functions without a priori knowledge of analytical singularity order in this paper a novel method is presented to construct singular basis functions for solving harmonic and bi harmonic problems with weak singularities such bases are found without any knowledge of the singularity order the singular bases are constructed by choosing a series as a tensor product of chebyshev polynomials and trigonometrical functions in radial and angular directions respectively and applying weak form of the governing equation with such features the singular bases are categorized as the equilibrated basis functions the constructed singular functions can be utilized as a complementary part to the smooth part of the approximation in the solution of problems with singularities to demonstrate the efficiency of employing such singular bases they are used in a boundary node method through the solution of some examples selected from the well known literature the capability of the method is shown it will be demonstrated that the main function and its derivatives are excellently approximated at very close neighborhood of the singular point the method may especially be found useful for those who research on the extended finite element method or similar ideas

iwasawa theory of elliptic curves at supersingular primes over ℤ p extensions of number fields in this paper we make a study of the iwasawa theory of an elliptic curve at a supersingular prime p along an arbitrary zp extension of a number field k in the case when p splits completely in k generalizing work of kobayashi 9 and perrin riou 17 we define restricted selmer groups and invariants we then derive asymptotic formulas describing the growth of the selmer group in terms of these invariants to be able to work with non cyclotomic zp extensions a new local result is proven that gives a complete description of the formal group of an elliptic curve at a supersingular prime along any ramified zp extension of qp

derived p adic heights and p adic l functions if e is an elliptic curve defined over a number field and p is a prime of good ordinary reduction for e a theorem of rubin relates the p adic height pairing on the p power selmer group of e to the first derivative of a cohomologically defined p adic l function attached to e bertolini and darmon have defined a sequence of derived p adic heights in this paper we give an alternative definition of the p adic height pairing and prove a generalization of rubin s result relating the derived heights to higher derivatives of p adic l functions we also relate degeneracies in the derived heights to the failure of the selmer group of e over a z p extension to be semi simple as an iwasawa module generalizing results of perrin riou

the optimal rate of convergence of the p version of the boundary element method in two dimensions we introduce the jacobi weighted besov and sobolev spaces in the one dimensional setting in the framework of these spaces we analyze lower and upper bounds for approximation errors in the p version of the boundary element method for hypersingular and weakly singular integral operators on polygons we prove the optimal rate of convergence for the p version in the energy norms of and respectively

the optimal convergence rate of the p version of the finite element method optimal error estimates for the p version of the finite element method in two dimensions are proven for the case when u in h k omega or u has singularities induced by the corners of the domain the case of nonhomogenous essential boundary conditions is also analyzed

on hp error estimation in the bem for a three dimensional helmholtz exterior problem the purpose of the paper is to give an a priori convergence estimate for the hp adaptive boundary element method the method is used to solve an exterior helmholtz boundary value problem with the robin s boundary condition in the three dimensional space the mathematical model describes an acoustic wave scattering problem on a bounded elastic solid where time harmonic process is assumed the estimate consists of three parts which correspond to influence of three errors caused by interpolation of functions geometry and an incident wave

solution of the system of fourth order boundary value problems using non polynomial spline technique non polynomial spline is used for the numerical solutions of a system of fourth order boundary value problems associated with obstacle unilateral and contact problems in addition to the boundary conditions corresponding to the second derivatives the boundary conditions corresponding to the first derivatives are also considered two examples one for each case of boundary conditions are considered for the numerical illustration of the method developed it has been observed that the method developed in this paper is better than the other collocation finite difference and spline methods

the h p and h p versions of the finite element methods in 1 dimension part iii the adaptive h p version abstract the paper is the third and final part in the series of three devoted to the detailed analysis of the three basic versions of the finite element method in one dimension the first part analyzed the p version and the second part concentrated on the h and h p version this paper analyzes a theoretical frame of the adaptive h p version and based on it the authors provide concrete algorithm for the one dimensional problem it is proven that in the case that the solution has x sub alpha type singularity the adaptive algorithm give an exponential rate of convergence very close to the optimal one analyzed in the second part of the paper additional keyword error analysis

derived heights and generalized mazur tate regulators 2 algebraic preliminaries 4 2 1 notations and conventions 4 2 2 duality 5 2 2 1 local duality 5 2 2 2 global duality 6 2 3 assumptions 8 2 4 preliminary calculations 10

the h p version of the boundary element method for transmission problems with piecewise analytic data this paper analyzes the rate of convergence of the h p version of the boundary element galerkin method for transmission problems with piecewise analytic data based on the regularity of solutions of integral equations in terms of countably nonmed spaces we design the geometric mesh on the interface gamma and the boundary element subspace containing piecewise polynomials with varying degrees we prove that the boundary element galerkin solution converges exponentially fast to the solution of the integral equation in the h 1 2 gamma times h 1 2 gamma norm

isogemetric analysis and symmetric galerkin bem isogeometric approach applied to boundary element methods is an emerging research area see e g simpson et al 2012 33 in this context the aim of the present contribution is that of investigating from a numerical point of view the symmetric galerkin boundary element method sgbem devoted to the solution of 2d boundary value problems for the laplace equation where the boundary and the unknowns on it are both represented by b splines de boor 2001 9 we mainly compare this approach which we call iga sgbem with a curvilinear sgbem aimi et al 1999 2 which operates on any boundary given by explicit parametric representation and where the approximate solution is obtained using lagrangian basis both techniques are further compared with a standard conventional sgbem approach aimi et al 1997 1 where the boundary of the assigned problem is approximated by linear elements and the numerical solution is expressed in terms of lagrangian basis several examples will be presented and discussed underlying benefits and drawbacks of all the above mentioned approaches

multilevel additive schwarz method for the h p version of the galerkin boundary element method we study a multilevel additive schwarz method for the h p version of the galerkin boundary element method with geometrically graded meshes both hypersingular and weakly singular integral equations of the first kind are considered as it is well known the h p version with geometric meshes converges exponentially fast in the energy norm however the condition number of the galerkin matrix in this case blows up exponentially in the number of unknowns m we prove that the condition number p of the multilevel additive schwarz operator behaves like o mlog 2 m as a direct consequence of this we also give the results for the 2 level preconditioner and also for the h p version with quasi uniform meshes numerical results supporting our theory are presented

on the birch and swinnerton dyer conjecture let f be a finite extension of the rational field q if e is an elliptic curve defined over f then the mordell weil group e f of points on e with coordinates in f is a finitely generated abelian group let l e f s be the hasseweil zeta function of e over f which at least in the case where e has complex multiplication is known to be an analytic function on the entire complex plane birch and swinnerton dyer have conjectured that l e f s has a zero at s l of order precisely equal to the rank of e f over z the strongest result known in support of this conjecture is the following theorem of coates and wiles 4

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

the treatment of nonhomogeneous dirichlet boundary conditions by thep version of the finite element method the paper addresses the problem of the implementation of nonhomogeneous essential dirichlet type boundary conditions in thep version of the finite element method

additive schwarz algorithms for the p version of the galerkin boundary element method we study some additive schwarz algorithms for the p version galerkin boundary element method applied to some weakly singular and hypersingular integral equations of the first kind both non overlapping and overlapping methods are considered we prove that the condition numbers of the additive schwarz operators grow at most as 1 log p 2 independently of h where p is the degree of the polynomials used in the galerkin boundary element schemes and h is the mesh size thus we show that additive schwarz methods which were originally designed for finite element discretisation of differential equations are also efficient preconditioners for some boundary integral operators which are non local operators

on rubin s variant of the p adic birch and swinnerton dyer conjecture ii let e q be an elliptic curve with complex multiplication by the ring of integers of an imaginary quadratic field k in 1991 by studying a certain special value of the katz two variable p adic l function lying outside the range of p adic interpolation k rubin formulated a p adic variant of the birch and swinnerton dyer conjecture when e k is infinite and he proved that his conjecture is true for e k of rank one when e k is finite however the statement of rubin s original conjecture no longer applies and the relevant special value of the appropriate p adic l function is equal to zero in this paper we extend our earlier work and give an unconditional proof of an analogue of rubin s conjecture when e k is finite

the approximation theory for the p version of the finite element method the purpose of this article is to present the approximation theory underlying the p version of the finite element method by exploiting the relationship between polynomial approximation and certain weighted sobolev spaces which are identified as the domains of positive real powers of the legendre operator this one dimensional result is generalized via a tensor product construction to yield a nonconforming piecewise polynomial approximation result in the usual unweighted sobolev spaces on triangulated domains of r n it is then shown that essentially the same result holds for approximation by conforming piecewise polynomials provided that the function being approximated possesses the same degree of conformality across the common boundaries of adjacent simplices and the same homogeneous boundary conditions inverse results are given for the special case of approximation in l 2

a hypersingular boundary integral method for two dimensional screen and crack problems we analyze hypersingular integral equations on a curved open smooth arc in r 2 that model either curved cracks in an elastic medium or the scattering of acoustic and elastic waves at a hard screen by using the mellin transformation we obtain sharp regularity results for the solution of these equations in sobolev spaces in the form of singular expansions in particular we show that the expansions do not contain logarithmic singularities

arithmetic of formal groups and applications i universal norm subgroups the theory of the norm map for the formal multiplicative group is well known as local class field theory in 1972 mazur 14 in his generalization of iwasawa theory to abelian varieties over number fields made quite clear that it is absolutely crucial for such a global theory first to understand the norm map for the formal groups attached to abelian varieties he achieved this understanding in case of formal groups of multiplicative type by using the theory of proalgebraic groups later on lubin and rosen 12 gave a completely elementary approach to mazur s results in the meantime hazewinkel 8 had settled the case of one dimensional commutative formal lie groups his method consists in an explicit and very complicated study of the properties of the power series coefficients of the logarithm of a formal group law by means of higher ramification theory in the same spirit vvedenskij 18 and konovalov 10 reproved and slightly extended hazewinkel s results let k q be a finite extension with ring of integers r and residue class field and koo k be a ramified zv extension with ring of integers r and f gal koo k let i r denote a smooth connected commutative formal r group of finite dimension d i e a commutative formal lie group if k k is the intermediate layer of degree p in koo k r its ring of integers and g gal k k its galois group then we know from the description of ff as a formal lie group that

numerical methods for unilateral problems abstract the theory of variational inequalities besides being elegant and synthetic also provides a unified general framework for applying the numerical methods for solving many unrelated free boundary value problems in this paper we describe numerical experience on the use of variational inequalities and pade approximants to obtain approximate solutions to a class of unilateral boundary value problems of elasticity like those describing the equilibrium configuration of an elastic beam stretched over an elastic obstacle in the case of a known obstacle these problems can be alternatively formulated as nonlinear boundary value problems without constraints for which the technique of pade approximants can be successfully employed the variational inequality formulation is used to discuss the problem of uniqueness and existence of the solution

fine selmer groups of elliptic curves over p adic lie extensions the aim of this paper is to discuss variants for elliptic curves of some deep conjectures of classical cyclotomic iwasawa theory let f be a finite extension of q p an odd prime number and f cyc the cyclotomic zp extension of q let k f cyc denote the maximal unramified abelian p extension of f cyc in which every prime of f cyc above p splits completely and define y f cyc to be the galois group of k f cyc over f cyc put g f cyc f and write for the iwasawa algebra of iwasawa proved that y f cyc is always a finitely generated torsion module and he conjectured 12 that in fact y f cyc is always a finitely generated zp module at present this conjecture has only been proven when f is abelian over q see 5 26 perhaps surprisingly it does not seem to have been pointed out in the literature that there is a precise analogue of iwasawa s conjecture for elliptic curves over f cyc let e be an elliptic curve over f and let

algorithms for the arithmetic of elliptic curves using iwasawa theory we explain how to use results from iwasawa theory to obtain information about p parts of tate shafarevich groups of specific elliptic curves over q our method provides a practical way to compute x e q p in many cases when traditional p descent methods are completely impractical and also in situations where results of kolyvagin do not apply e g when the rank of the mordell weil group is greater than 1 we apply our results along with a computer calculation to show that x e q p 0 for the 1 534 422 pairs e p consisting of a non cm elliptic curve e over q with conductor 30 000 rank 2 and good ordinary primes p with 5 p 1000 and surjective mod p representation

the p adic gross zagier formula for elliptic curves at supersingular primes let p be a prime number and let e be an elliptic curve defined over of conductor n let k be an imaginary quadratic field with discriminant prime to pn such that all prime factors of n split in k b perrin riou established the p adic gross zagier formula that relates the first derivative of the p adic l function of e over k to the p adic height of the heegner point for k when e has good ordinary reduction at p in this article we prove the p adic gross zagier formula of e for the cyclotomic p extension at good supersingular prime p our result has an application for the full birch and swinnerton dyer conjecture suppose that the analytic rank of e over is 1 and assume that the iwasawa main conjecture is true for all good primes and the p adic height pairing is not identically equal to zero for all good ordinary primes then our result implies the full birch and swinnerton dyer conjecture up to bad primes in particular if e has complex multiplication and of analytic rank 1 the full birch and swinnerton dyer conjecture is true up to a power of bad primes and 2

exceptional zero formulae and a conjecture of perrin riou let a mathbf q be an elliptic curve with split multiplicative reduction at a prime p we prove an analogue of a conjecture of perrin riou relating p adic beilinson kato elements to heegner points in a mathbf q and a large part of the rank one case of the mazur tate teitelbaum exceptional zero conjecture for the cyclotomic p adic l function of a more generally let f be the weight two newform associated with a let f infty be the hida family of f and let l p f infty k s be the mazur kitagawa two variable p adic l function attached to f infty we prove a p adic gross zagier formula expressing the quadratic term of the taylor expansion of l p f infty k s at k s 2 1 as a non zero rational multiple of the extended height weight of a heegner point in a mathbf q

the p version of the finite element method for constraint boundary conditions abstract this paper addresses the implementation of general constraint boundary conditions for a system of equations by the p version of the finite element method by constraint boundary conditions the authors mean conditions where some relation between the components is prescribed at the boundary optimal error bounds are proven keywords convergence sobolev spaces

integral iwasawa theory of galois representations for non ordinary primes in this paper we study the iwasawa theory of a motive whose hodge tate weights are 0 or 1 thence in practice of a motive associated to an abelian variety at a non ordinary prime over the cyclotomic tower of a number field that is either totally real or cm in particular under certain technical assumptions we construct sprung type coleman maps on the local iwasawa cohomology groups and use them to define integral p adic l functions and one unconditionally and other conjecturally cotorsion selmer groups this allows us to reformulate perrin riou s main conjecture in terms of these objects in the same fashion as kobayashi s iwasawa theory for supersingular elliptic curves by the aid of the theory of coleman adapted kolyvagin systems we develop here we deduce parts of perrin riou s main conjecture from an explicit reciprocity conjecture

the h p and h p versions of the finite element method in 1 dimension this paper is the first one in the series of three which are addressing in detail the properties of the three basic versions of the finite element method in the one dimensional setting the main emphasis is placed on the analysis when the exact solution has singularity of x type the first part analyzes thep version the second theh version and generalh p version and the final third part addresses the problems of the adaptiveh p version

on the structure of signed selmer groups let f be a number field unramified at an odd prime p and f infty be the mathbf z p cyclotomic extension of f generalizing kobayashi plus minus selmer groups for elliptic curves buyukboduk and lei have defined modified selmer groups called signed selmer groups for certain non ordinary mathrm gal overline f f representations in particular their construction applies to abelian varieties defined over f with good supersingular reduction at primes of f dividing p assuming that these selmer groups are cotorsion mathbf z p mathrm gal f infty f modules we show that they have no proper sub mathbf z p mathrm gal f infty f module of finite index we deduce from this a number of arithmetic applications on studying the euler poincare characteristic of these selmer groups we obtain an explicit formula on the size of the bloch kato selmer group attached to these representations furthermore for two such representations that are isomorphic modulo p we compare the iwasawa invariants of their signed selmer groups

analytic regularity of stokes flow on polygonal domains in countably weighted sobolev spaces we investigate the analytic regularity of the stokes problem in a polygonal domain r2 with straight sides and piecewise analytic data we establish a shift theorem in weighted sobolev spaces of arbitrary order with explicit control of the order dependence of the constants the shift theorem in the framework of countably weighted sobolev spaces implies in particular interior analyticity and gevrey type analytic regularity near the corners

solution of elastic scattering problems in linear acoustics using h p boundary element methods abstract a simplified model problem reduced from the general class of elastic scattering problems is analyzed both classical helmholtz and hypersingular formulations are discussed and a general h p boundary element method based on the burton miller combination of both approaches is formulated a simple l 2 residual a posteriori error estimate is discussed as a basis of adaptive h or p refinements

solution of the system of fourth order boundary value problems using cubic spline a system of fourth order boundary value problems associated with obstacle unilateral and contact problems is solved using cubic spline since the solution of system of fourth order bvp with boundary conditions in terms of values of function and first derivatives is not found in the literature so in this paper the boundary conditions are considered in terms of values of function and not only in first derivatives but in terms of second derivatives as well three examples are considered for the numerical illustration of the method developed it has been observed that the method developed in this paper is better than the other collocation finite difference and spline methods

a class of methods based on non polynomial spline functions for the solution of a special fourth order boundary value problems with engineering applications we use a quintic non polynomial spline functions to develop a numerical method for computing approximations to the solution of a system of fourth order boundary value problems associated with plate deflection theory we show that the present family of methods gives better approximations and generalize all the existing finite difference and spline functions based methods up to order six convergence of the methods is shown through standard convergence analysis numerical exampls are given to illustrate the applicability and efficiency of the new method

on the convergence of the p version of the boundary element galerkin method abstract the authors consider various physical problems which may be formulated in terms of integral equations of the first kind including the two dimensional screen neumann and dirichlet problems in acoustics and crack problems in elasticity sharp regularity results for the solutions are available for these problems proven is the convergence of the p version for some galerkin boundary element schemes based on the integral equation formulations it is shown that the rate of convergence obtained by our method is twice that for the usual h version

solution of elastic scattering problems in linear acoustics using h p boundary element method abstract the work is a continuation of 1 a simplified 3d model problem reduced from the general class of elastic scattering problems is analyzed focusing attention on such practical details of the 3d implementation as a triangles based h p approximation and corresponding data structure integration of singular integrals and its effect on theoretically optimal rates of convergence a posteriori error estimation and sample adaptive strategies

convergence of cubic spline approach to the solution of a system of boundary value problems abstract we use cubic spline to derive some consistency relations which are then used to develop a numerical method for the solution of a system of fourth order boundary value problems associated with obstacle unilateral and contact problems it is known that a class of variational inequalities related to contact problems in elastostatics can be characterized by a sequence of variational inequations which are solved using some numerical method boundary formula of order o h 8 are formulated the most common approach for convergence analysis are using monotonicity of the coefficient matrix but here we study a new approach and give the convergence of prescribed method so that the matrix associated with the system of linear equations that arises is not required to be monotone numerical examples are given to show the applicability and efficiency of our method

on the p adic height of heegner cycles 0 1 in gr za gross and zagier proved a remarkable formula which relates the first derivative of the l function of a modular form f of weight 2 on 0 n over a suitable imaginary quadratic field and the neron tate height of a heegner point over the same quadratic field on the f part of the jacobian of the modular curve x0 n later perrin riou pr 2 proved a p adic version of this formula kolyvagin s method of euler systems ko combined with the formula of gross zagier proves the conjecture of birch and swinnerton dyer up to a controlled rational factor for all modular elliptic curves over q with analytic rank 1

the p version of the finite element method in the p version of the finite element method the triangulation is fixed and the degree p of the piecewise polynomial approximation is progressively increased until some desired level of precision is reached in this paper we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation these properties lead to an a priori estimate of the asymptotic rate of convergence of the p version the estimate shows that the p version gives results which are not worse than those obtained by the conventional finite element method called the h version in which h represents the maximum diameter of the elements when quasi uniform triangulations are employed and the basis for comparison is the number of degrees of freedom furthermore in the case of a singularity problem we show under conditions which are usually satisfied in practice that the rate of convergence of the p version is twice that of the h version with quasi uniform mesh inve

computing canonical heights using arithmetic intersection theory the canonical height on an abelian variety a defined over a global field k is an object of fundamental importance in the study of the arithmetic of a for many applications it is required to compute p for a given point p a k for instance given generators of a subgroup of the mordell weil group a k of finite index this is necessary for most known approaches to the computation of generators of the mordell weil group a k furthermore the regulator of a k which appears in the statement of the conjecture of birch and swinnerton dyer is defined in terms of the canonical height and thus we need the ability to compute canonical heights in order to gather numerical evidence for the conjecture in the case of positive rank

a sixth order multiderivative method for two beam problems a sixth order method is developed and analysed for the numerical solution of the linear fourth order boundary value problem for which the boundary conditions are given in terms of functional values and i first order derivatives the clamped clamped beam problem or ii second order derivatives the simple simple beam problem the method is tested on a number of problems from the literature

on the h p version of the boundary element method for symm s integral equation on polygons abstract in this paper we present the numerical implementation of and numerical experiments for the galerkin approximation of symm s integral equation using the h p and h p methods numerical results obtained using an adaptive algorithm are also given the theoretical results for these methods are summarized and are compared with the experimental results

mathematical foundation of adaptive boundary element methods abstract in this paper we review some mathematical aspects of adaptive boundary element methods in brief and give some a posteriori local error estimates for the h version and the p version

the h p boundary element method for solving 2 and 3 dimensional problems abstract in this survey paper we report on recent developments of the hp version of the boundary element method we consider weakly singular and hypersingular integral equations on polygons and polyhedral surfaces we show that the galerkin solutions computed with the hp version on geometric meshes converge exponentially fast towards the exact solutions of the integral equations the implementation of the hp version of the boundary element method is discussed and adaptive algorithms are given model problems for the coupling of finite elements and boundary elements are also considered numerical results are presented which underline the theoretical results