on the use of quarter point boundary elements for stress intensity factor computations this communication studies a procedure for stress intensity factor computations using traction singular quarter point boundary elements opening mode stress intensity factors are computed from the tractions nodal values at the crack tip a comparison is made between the factors calculated using this procedure and those obtained by previously recommended methods which made use of the nodal values of the displacements the proposed procedure was seen to be less discretization sensitive than any other of the considered methods accurate results were obtained even in the case of coarse meshes

asymptotic behavior of solutions to time dependent hartree equations on considere le comportement asymptotique pour t grand des solutions du probleme de cauchy pour l equation iu t u f u 2 u t x r r u o x x x r 3 ou f u 2 r 1 u 2 r 3 u t y 2 x y dy

smoothing effect for some schrodinger equations abstract we study the cauchy problem for the following schrodinger equation in r n n n i t u 1 2 u v 1 u v 2 u 2 u t x r r n u 0 x r n where v 1 v 1 x 1 x 1 v 2 v 2 x k 2 3 k x k k r 1 k 3 0 1 min 2 n 2 0 2 3 min 2 n we prove the existence uniqueness and smoothing effect of global solutions of with not necessarily in the sobolev space h1 rn

two and three dimensional soil consolidation by bem via particular integral abstract particular integral formulations are presented for two and three dimensional soil consolidation poroelasticity analysis the steady state poroelasticity equation is used as the complementary solution and a set of global shape functions is considered to approximate the transient terms of the poroelasticity equation to obtain the required particular integrals some numerical results for four example problems are given and compared with their analytical solutions or the results from the convolution method developed earlier by dargush and banerjee int j numer meth engng 28 1989 2423 j appl mech 58 1991 28

alternate bem formulations for 2 and 3 d anisotropic thermoelasticity abstract direct boundary element formulations are presented for the elastostatic analysis of two and three dimensional general anisotropic media subject to known temperature distributions the thermal effects are incorporated in the analysis through a volume integral or alternatively using particular integrals a multi region boundary element approach is adopted in which particular integrals are derived assuming a quadratic temperature distribution in each sub region a multiple regression scheme is used for the best fit quadratic temperature distribution in each sub region for temperature data given at discrete points the multi region analysis capability of the numerical implementation enhances its capacity to model discontinuous temperature fields as well as piecewise homogeneous media such as laminated composites the formulations presented are incorporated in a general purpose system and are illustrated with examples

connection between gaussian periods and cyclic units this paper finds that all known parametric families of units in real quadratic cubic quartic and sextic fields with prime conductor are linear combinations of gaussian periods and exhibits these combinations this approach is used to find new units in the real quintic field for prime conductors p n4 5n3 15n2 25n 25

hypersingular quarter point boundary elements for crack problems the present paper deals with the study and effective implementation for stress intensity factor computation of a mixed boundary element approach based on the standard displacement integral equation and the hypersingular traction integral equation expressions for the evaluation of the hypersingular integrals along general curved quadratic line elements are presented the integration is carried out by transformation of the hypersingular integrals into regular integrals which are evaluated by standard quadratures and simple singular integrals which are integrated analytically the generality of the method allows for the modelling of curved cracks and the use of straight line quarter point elements the stress intensity factors can be computed very accurately from the crack opening displacement at collocation points extremely close to the crack tip several examples with different crack geometries are analyzed the computed results show that the proposed approach for stress intensity factors evaluation is simple produces very accurate solutions and has little dependence on the size of the elements near the crack tip

remarks on the repulsive wigner poisson system we present an estimate for the space time integral of classical solutions to the repulsive wigner poisson system we use markowich s formalism between wigner possion and schrodinger poisson systems through this formalism and morawetz interaction potentials we derive the same a priori estimate given by chae and ha for the repulsive vlasov poisson system

a variable stiffness type elastoplastic boundary element formulation for planar anisotropic media abstract an initial stress boundary element algorithm has been presented for two dimensional anisotropic elastoplasticity a quadratic stress yield criterion with strain hardening recently derived and applied to boron aluminum composites is adopted the details relating to numerical evaluation of singular surface and volume integrals are discussed a new generation iterative newton raphson algorithm has been adopted to solve the nonlinear boundary element equations the validity of the present formulation has been established with the help of examples

first hit analysis of algorithms for computing quadratic irregularity the author has previously extended the theory of regular and irregular primes to the setting of arbitrary totally real number fields it has been conjectured that the bernoulli numbers or alternatively the values of the riemann zeta function at odd negative integers are uniformly distributed modulo p for every p this is the basis of a well known heuristic given by siegel estimating the frequency of irregular primes so far analyses have shown that if q d is a real quadratic field then the values of the zeta function d 1 2m q d 1 2m at negative odd integers are also distributed as expected modulo p for any p we use this heuristic to predict the computational time required to find quadratic analogues of irregular primes with a given order of magnitude we also discuss alternative ways of collecting large amounts of data to test the heuristic

analytical integrations and sifs computation in 2d fracture mechanics analytical integrations in the framework of linear elastic problems modelled by means of boundary integral equations have been considered in a previous publication int j numer methods eng 2002 53 7 1695 1719 the present note aims at extending the subject to linear elastic fracture mechanics in such a context special shape functions have been recently proposed siam j appl math 1998 58 428 455 in order to increase accuracy in stress intensity factors approximation the closed form solution for integrals that arise from the boundary element method is a goal of the present work exploiting the analytical integrations asymptotical analysis around the crack tip are made possible with the purpose of formulating a coherent and accurate correlation between approximated stress intensity factors and crack opening displacements over the crack tip straight special elements copyright 2006 john wiley sons ltd

treatment of body force volume integrals in bem by exact transformation for 2 d anisotropic elasticity in the boundary element method bem based on the direct formulation body force effects manifest themselves as an additional volume integral term in the boundary integral equation bie the numerical solution of the integral equation with this term destroys the notion of the bem as atruly boundary solution method this paper discusses the treatment of this volume integral for two dimensional anisotropic elasticity with body forces present the analytical basis for transforming this integral exactly into boundary ones is presented for geometrically convex regions this restores the application of the bem to such problems as a truly boundary solution technique numerical examples are presented to demonstrate the veracity of the transformation and implementation 1997 by john wiley sons ltd

three dimensional green s functions in anisotropic piezoelectric solids abstract explicit expressions for three dimensional extended green s displacements in general anisotropic piezoelectric solids are derived a very efficient procedure for the numerical evaluation of the derivatives of the extended green s displacements is also proposed numerical comparisons are carried out for a transversely isotropic piezoelectric solid for which exact closed form solutions are available it is found that the extended green s displacements and their derivatives obtained with the present explicit formulation are in perfect agreement with the exact closed form solutions these green s functions can be used in the boundary integral equations for piezoelectric solids of general anisotropy and for subsequent numerical solutions of these equations by means of the boundary element method

nonlinear scattering with nonlocal interaction we consider the scattering problem for the hartree type equation in n withn 2 i frac partial u partial t frac 1 2 delta u v u 2 u where v x sum limits j 1 2 lambda j x gamma j lambda 1 lambda 2 ne 0 0 lambda j in mathbb r gamma j 0 and denotes the convolution in n we prove the existence of wave operators inh0 k l2 n x k l2 n for any positive integerk under the assumption 1 1 2 2 this is an optimal result in the sense that the existence of wave operators breaks down if min 1 2 1 the case where 1 1 2 2 is also treated according to the sign of 2

on the khintchine constant we present rapidly converging series for the khintchine constant and for general khintchine means of continued fractions we show that each of these constants can be cast in terms of an efficient free parameter series each series involving values of the riemann zeta function rationals and logarithms of rationals we provide an alternative polylogarithm series for the khintchine constant and indicate means to accelerate such series we discuss properties of some explicit continued fractions constructing specific fractions that have limiting geometric mean equal to the khintchine constant we report numerical evaluations of such special numbers and of various khintchine means in particular we used an optimized series and a collection of fast algorithms to evaluate the khintchine constant to more than 7000 decimal places k 1

multiple reciprocity boundary element analysis of two dimensional anisotropic thermoelasticity involving an internal arbitrary non uniform volume heat source abstract in the direct boundary element method bem formulation of anisotropic thermoelasticity thermal loads manifest themselves as additional volume integral terms in the boundary integral equation bie conventionally this requires internal cell discretisation throughout the whole domain in this paper the multiple reciprocity method in bem analysis is employed to treat the general 2d thermoelasticity problem when the thermal loading is due to an internal non uniform volume heat source by successively performing the volume to surface integral transformation the general formulation of the associated bie for the problem is derived the successful implementation of such a scheme is illustrated by three numerical examples

determination of interior point stresses in two dimensional bem thermoelastic analysis of anisotropic bodies abstract in conventional boundary element method bem analysis the displacements and stresses at an interior point of an elastic body are obtained using the respective somigliana s identities after the boundary solutions for the displacements and tractions have been solved for this paper presents the derivation of somigliana s identity for the strains at any interior point of an anisotropic domain subjected to thermal effects from these strains the corresponding stresses at the interior point may be calculated directly using the duhamel neumann relation this identity is obtained in terms of integrals over the boundary of the solution domain only difficulties arising from the exact transformation of the volume integral term associated with thermal loading in bem analysis into surface integrals in the anisotropic case are discussed and the procedures to overcome them are described three numerical examples are presented to demonstrate the veracity of the formulations developed

crandall s computation of the incomplete gamma function and the hurwitz zeta function with applications to dirichlet l series this paper extends tools developed by crandall 2012 16 to provide robust high precision methods for computation of the incomplete gamma function and the lerch transcendent we then apply these to the corresponding computation of the hurwitz zeta function and so of dirichlet l series and character polylogarithms

how can we construct abelian galois extensions of basic number elds recall the almost tautological but very useful way we have of dealing with an irreducible polynomial over a field for example in the case of x5 x 1 irreducible over the rational field q we happily adjoin a root to our base field by merely forming l the quotient of the polynomial ring q x modulo the ideal generated by x5 x 1 then l is a field of algebraic numbers and the image of x is indeed a root of x5 x 1 in l

on a wave equation with a cubic convolution abstract we study the well posedness of the cauchy problem and the asymptotic behavior of solutions of the nonlinear wave equation u tt u m 2 u v u 2 0 in euclidean space

computational strategies for the riemann zeta function we provide a compendium of evaluation methods for the riemann zeta function presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new we concentrate primarily on practical computational issues such issues depending on the domain of the argument the desired speed of computation and the incidence of what we call value recycling c 2000 elsevier science b v all rights reserved

three dimensional crack analysis using singular boundary elements a multi domain method of solving three dimensional elastic crack problems in an infinite elastic body using the boundary element method is proposed the displacement and traction behaviours near a crack front are incorporated in special crack elements the elimination of singularities arising from the term combined with kelvin s kernel for displacement in the integrals is discussed in detail stress intensity factors of modes i ii and iii are obtained directly from crack front nodal values without any extrapolation as in some other methods no differentiation of conventional boundary integral equations with kelvin s tensor kernels is necessary in the current approach this method is applicable to cracks of arbitrary shape infinite bodies are modelled precisely as such not approximated as large finite bodies numerical solutions of stress intensity factors are given for several problems involving a penny shaped crack

advanced boundary element analysis of two and three dimensional problems of elasto plasticity an advanced formulation of the boundary element method has been developed for inelastic analysis based on an initial stress approach the iterative solution algorithm makes use of an accelerated initial stress approach in which the past history of initial stresses are used to obtain an initial estimate for the current increment in the present analysis the geometry and functions are represented by higher order quadratic shape functions to model complex geometries and rapid functional variations accurately the methods of numerical integration of the kernels particularly the singular type are substantially improved by devising suitable automatic sub segmentation routines that incorporate the recent developments in mapping procedures the formulations have been implemented for two dimensional plane stress plane strain and three dimensional elasto plasticity problems

2d and 3d transient heat conduction analysis by bem via particular integrals abstract particular integral formulations are presented for 2d and 3d transient potential flow heat conduction analysis the results of the analysis are compared with an alternative formulation developed using the volume integral conversion approach although the mathematical foundation of the two methods are different both formulations are shown to produce almost identical results for the particular integral formulation the steady state heat conduction equation is used as the complementary solution and two global shape functions gsfs are considered to approximate the transient term of the heat conduction equation the numerical results for three example problems are given and compared with their analytical solutions

body loads in scaled boundary finite element method the solution of the scaled boundary finite element equation in displacement with body loads is derived the non homogeneous term caused by the body loads is processed using the technique of variation of parameters integrals in the radial direction arise which can however be evaluated explicitly for concentrated loads and loads varying as power functions in the radial coordinate in these cases no additional approximations are introduced the scaled boundary finite element method thus remains a semi analytical fundamental solution less boundary element method based on finite elements

a new boundary element formulation for two and three dimensional thermoelasticity using particular integrals new two and three dimensional boundary element formulations are developed for steady state and transient uncoupled thermoelasticity these new procedures differ from previous work in that additional surface or volume integration is not required to incorporate thermal loads in the analysis instead thermal body forces are introduced in the boundary element system via particular integrals the present formulation is implemented in a general purpose multi region system and examples are presented to demonstrate the accuracy and versatility of the method

two and three dimensional transient thermoelastic analysis by bem via particular integrals abstract particular integral formulations are presented for two and three dimensional transient uncoupled thermoelastic analysis these formulations differ from previous particular integral formulations in that the equation of uncoupled thermoelasticity including the heat conduction equation is fully satisfied to obtain particular integrals the equation of the steady state thermoelasticity is now used as the complementary solution and two global shape functions are considered to approximate the transient term of the heat conduction equation so that two sets of particular integrals could be derived the numerical results for both sets of particular integrals are given for three example problems and compared with their analytical solutions

time decay of solutions to the cauchy problem for time dependent schrodinger hartree equations we consider the time dependent schrodinger hartree equation iu t delta u left frac 1 r u 2 right u lambda frac u r t x in mathbb r times mathbb r 3 1 u 0 x phi x in sigma 2 2 x in mathbb r 3 2 where 0 and sigma 2 2 g in l 2 parallel g parallel sigma 2 2 2 sum limits a leqq 2 parallel d a g parallel 2 2 sum limits beta leqq 2 parallel x beta g parallel 2 2 infty

the twenty fourth fermat number is composite we have shown by machine proof that f24 2224 1 is composite the rigorous pepin primality test was performed using independently developed programs running simultaneously on two different physically separated processors each program employed a floating point fft based discrete weighted transform dwt to effect multiplication modulo f24 the final respective pepin residues obtained by these two machines were in complete agreement using intermediate residues stored periodically during one of the floating point runs a separate algorithm for pure integer negacyclic convolution verified the result in a wavefront paradigm by running simultaneously on numerous additional machines to effect piecewise verification of a saturating set of deterministic links for the pepin chain we deposited a final pepin residue for possible use by future investigators in the event that a proper factor of f24 should be discovered herein we report the more compact traditional selfridge hurwitz residues for the sake of completeness we also generated a pepin residue for f23 and via the suyama test determined that the known cofactor of this number is composite

two end variable singularity boundary elements and their applications in crack crack interaction problems summary two new boundary elements have been proposed for simulation of variable order singularities at the two ends of an element in two dimensions the rst can model the variable order strain singularity at both the ends of the element the second element can do both the strain and traction singularities simultaneously the elements are useful for studying the interaction of singularities as in the case of multiple neighbouring cracks in a domain they are employed here for the computation of stress intensity factors sifs in the crack crack interaction problems to improve the accuracy of such computations further a modi ed crack closure integral mcci based method for mechanical and or thermal loading is presented examples of mode i crack and mixed mode problems under mechanical loading are studied to illustrate the performance of the proposed elements and the mcci based calculations the e ects of the order of gauss quadrature associated with such elements on the accuracy of the sifs are also reported copyright 2000 john wiley sons ltd

time decay for some schrodinger equations on considere le probleme de cauchy pour l equation de schrodinger dans r n n n i t u u 2 v 1 u v 2 u 2 u t x r r n u o x x x r n ou v 1 v 1 x 1 x u 1 1 0 0 1 min 2 n 2 v 2 v 2 x k 2 3 k x u k k 0 0 k min 2 n on etudie les proprietes de decroissance temporelles des solutions

quintic polynomials and real cyclotomic fields with large class numbers we study a family of quintic polynomials discoverd by emma lehmer we show that the roots are fundamental units for the corresponding quintic fields these fields have large class numbers and several examples are calculated as a consequence we show that for the prime p 641491 the class number of the maximal real subfield of the pth cyclotomic field is divisible by the prime 1566401 in an appendix we give a characterization of the simplest quadratic cubic and quartic fields

on modular representations of bar q q arising from modular forms where g is the galois group gai i q and f is a finite field of characteristic i 3 suppose that p is modular of level n i e that it arises from a weight 2 newform of level dividing n and trivial nebentypus character then p is an odd representation the matrix p c where c is a complex conjugation in g has eigenvalues 1 1 since 1 and 1 are distinct in f p is absolutely irreducible and has a model over every subfield of f containing the set trace p we assume that f has been chosen so that it is generated by this set assume that p is a prime which exactly divides n we write p i n and restrict p to a decomposition group of g for the prime p view the restriction as a representation p p of gal t p qp we say that p is finite at p 34 or 35 w if there is a finite flat f vector space scheme h over zp for which the action of gal t p qp on the f vector space h 1 p gives pp i f p 9 is finite at p if and only if pp is unramified it is clear that p is finite at p whenever p is already modular of level n p in w of 34 cf 35 serre conjectured the converse i e that if p is finite then p is modular of level n p soon thereafter mazur 1 20 proved this conjecture in the case

cyclotomic invariants for primes to one million our recent computation of cyclotomic invariants for primes between 125000 and 150000 was extended to one million no new phenomena appear this note is a sequel to our recent report 2 on the computation of certain cyclotomic invariants for primes p between 125000 and 150000 that work was based on the table of irregular primes supplied by tanner and wagstaff see 4 meanwhile the extension of this table to p 0 let hn and an denote the class number and p class group respectively of the cyclotomic field kn generated by the pn lth roots of 1 and let h and anstand for their minus parts denote by rp the index of irregularity of p that is the number of p divisible bernoulli numbers bt with t e i 2 4 p 3 1 our computations now allow us to state that 1 an z pn lz rp n 0 1 ordp hj ordp b2b4 bp 3 for all p 106 where ordp a denotes the exponent of p in the canonical decomposition of a note that the computations of 1 together with earlier computations by wagstaff et al imply that anan for these p from this and 1 it follows among other things that the a invariant of the 7p extension u 0 kn over ko equals rp in this range it is worth pointing out that the results in 1 also imply the truth of fermat s last theorem for all prime exponents up to one million we now describe briefly the contents of our computations let p t be an irregular pair i e a pair with t e i and p i bt as in 2 put si e at1 qa s2 e atqa2 o a p 2 o a p 2 received by the editor april 15 1991 1991 mathematics subject classification primary 1 r18 1 1b68 1 1r23 1 1r29 1 y40

irregular primes to one million using fast algorithms for power series inversion based on the fast fourier transform and multisectioning of power series we have calculated all irregular primes up to one million including their indices of irregularity and associated irregular pairs using this data we verified that fermat s last theorem and vandiver s conjecture are true for these primes two primes with index of irregularity five were already known we find that there are nine other primes less than one million with index five and that the prime 527377 is the unique prime less than one million with index six a pair of integers p k is said to be an irregular pair if p is a prime k is an even integer satisfying 2 p are known see 8 our calculations show that vandiver s conjecture is true for all p 106 the table of irregular pairs could also be used to calculate iwasawa invariants for the corresponding primes we did not do this as part of our calculations see 9 or 4 for a discussion of this problem previous computations of irregular pairs have used algorithms that take 0 p2 arithmetic operations for each prime p the bernoulli numbers are defined in received by the editor april 30 1991 and in revised form september 10 1991 1991 mathematics subject classification primary 11b68 secondary 1 1d41 65t20 1 1y99 the first author was partially supported by national science foundation grant dms 9012989 d 1992 american mathematical society 0025 5718 92 1 00 25 per page

analytical transformation of the volume integral in the boundary integral equation for 3d anisotropic elastostatics involving body force abstract in the boundary element method bem it is well known that the presence of body force shall give rise to an additional volume integral that conventionally requires domain discretization for numerical computations to restore the bem s distinctive notion of boundary discretization the present work analytically transforms the volume integral to surface ones for the body force effect in the 3d anisotropic elasticity on applying green s theorem new fundamental solutions with explicit forms of fourier series are introduced to facilitate the volume to surface transformation the coefficients of the fourier series representations are determined by solving a banded matrix formulated from integrations of the constrained equation of no doubt such an approach has fully restored the boundary element method as a truly boundary solution technique for analyzing 3d anisotropic elasticity involving body force at the end numerical verifications of the volume to surface integral transformation are presented also such an approach has been implemented in an existing bem code for demonstrating the implementation numerical examples are presented with comparisons with ansys analysis to the author s knowledge this is the first work in the open literature that reports the successful transformation for 3d anisotropic elasticity

new congruences for the bernoulli numbers on etablit une nouvelle congruence pour calculer les nombres de bernoulli modulo un nombre premier on developpe un programme d ordinateur fonctionnant sur le systeme cyber205 on verifie le grand theoreme de fermat pour tous les exposants jusqu a 150000

boundary element analysis in thermoelasticity with and without internal cells in this paper a set of internal stress integral equations is derived for solving thermoelastic problems a jump term and a strongly singular domain integral associated with the temperature of the material are produced in these equations the strongly singular domain integral is then regularized using a semi analytical technique to avoid the requirement of discretizing the domain into internal cells domain integrals included in both displacement and internal stress integral equations are transformed into equivalent boundary integrals using the radial integration method rim two numerical examples for 2d and 3d respectively are presented to verify the derived formulations copyright 2003 john wiley sons ltd

p johnson homomorphisms and pro p groups abstract we propose an approach to study non abelian iwasawa theory using the idea of johnson homomorphisms in low dimensional topology we introduce arithmetic analogues of johnson homomorphisms maps called the p johnson homomorphisms maps associated to the zassenhaus filtration of a pro p galois group over a z p extension of a number field we give their cohomological interpretation in terms of massey products in galois cohomology