on numerical cubatures of singular surface integrals in boundary element methods we present and analyze methods for the accurate and efficient evaluation of weakly cauchy and hypersingular integrals over piecewise analytic curved surfaces in 3 the class of admissible integrands includes all kernels arising in the numerical solution of elliptic boundary value problems in three dimensional domains by the boundary integral equation method the possibly not absolutely integrable kernels of boundary integral operators in local coordinates are pseudohomogeneous with analytic characteristics depending on the local geometry of the surface at the source point this rules out weighted quadrature approaches with a fixed singular weight for weakly singular integrals it is shown that duffy s triangular coordinates leadalways to a removal of the kernel singularity also asymptotic estimates of the integration error are provided as the size of the boundary element patch tends to zero these are based on the rabinowitz richter estimates in connection with an asymptotic estimate of domains of analyticity in 2 it is further shown that the modified extrapolation approach due to lyness is in the weakly singular case always applicable corresponding error and asymptotic work estimates are presented for the weakly singular as well as for cauchy and hypersingular integrals which e g arise in the study of crack problems we analyze a family of product integration rules in local polar coordinates these rules are hierarchically constructed from finite part integration formulas in radial and gaussian formulas in angular direction again we show how the rabinowitz richter estimates can be applied providing asymptotic error estimates in terms of orders of the boundary element size

classical dirichlet forms on topological vector spaces closability and a cameron martin formula abstract we prove a sufficient condition for the closability of classical dirichlet forms on l 2 e which is also necessary if all components of the dirichlet form are closable here e is a locally convex topological vector space and a not necessarily quasi invariant probability measure on e the same condition is shown to imply the existence of a closed extension whose domain can be described explicitly in the special case where is quasi invariant with respect to certain vectors in e our result generalises previous theorems on closability in addition we prove a cameron martin type formula for a large class of measures if e is finite dimensional our characterisation of closability is the analogue of the corresponding one dimensional result applications to quantum fields and the connection with the well studied case of abstract wiener spaces are discussed

a simple and accurate scheme based on complex space c to calculate boundary integrals of 2d boundary elements method abstract in this work a semi analytical algorithm is presented to calculate the boundary integrals of higher order which appear in boundary elements method bem in fact treating singularity and near singularity of the boundary integrals using complex space c is the main aim of this paper the integrals are computed for linear quadratic cubic and other higher order elements when the geometry of the boundary elements is curved the main advantages of the new algorithm are its applicability simplicity and high accuracy which enable the conventional higher order bem to solve poisson s problems accurately the potentials at the interior points very close to boundary can be evaluated by the scheme developed in this report some test problems are given and numerical simulations are presented numerical results demonstrate that the new algorithm proposed in the current paper can effectively handle singular and near singular boundary integrals of bem specially for solving partial differential equations which arise in thin body problems

a sinh transformation for evaluating two dimensional nearly singular boundary element integrals a new transformation technique is introduced for evaluating the two dimensional nearly singular integrals which arise in the solution of laplace s equation in three dimensions using the boundary element method when the source point is very close to the element of integration the integrals are evaluated using in a product fashion a transformation which has recently been used to evaluate one dimensional near singular integrals this sinh transformation method automatically takes into account the position of the projection of the source point onto the element and also the distance b between the source point and the element the method is straightforward to implement and when it is compared with a number of existing techniques for evaluating two dimensional near singular integrals it is found that the sinh method is superior to the existing methods considered both for potential integrals across the full range of b values considered 0 0 01 for smaller values of b the use of the lmethod is recommended for flux integrals copyright 2006 john wiley sons ltd

dynamics of bimeromorphic maps of surfaces we classify bimeromorphic self maps f x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of compact kahler surfaces x in terms of their actions f h 1 1 x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i on cohomology we observe that the growth rate of f n is invariant under bimeromorphic conjugacy and that by conjugating one can always arrange that f n f n we show that the sequence f n can be bounded grow linearly grow quadratically or grow exponentially in the first three cases we show that after conjugating f is an automorphism virtually isotopic to the identity f preserves a rational fibration or f preserves an elliptic fibration respectively in the last case we show that there is a unique up to scaling expanding eigenvector for f that is nef and that f is bimeromorphically conjugate to an automorphism if and only if 2 0 we go on in this case to construct a dynamically natural positive current representing and we study the growth rate of periodic orbits of f we conclude by illustrating our results with a particular family of examples

the method of steepest descent for estimating quadrature errors this work presents an application of the method of steepest descent to estimate quadrature errors the method is used to provide a unified approach to estimating the truncation errors which occur when gauss legendre quadrature is used to evaluate the nearly singular integrals that arise as part of the two dimensional boundary element method the integrals considered here are of the form 1 1 h x d x x a 2 b 2 where h x is a well behaved function 0 and 1 a 1 since 0 b 1 the integral is nearly singular with a sharply peaked integrand the method of steepest descent is used to estimate the generally large truncation errors that occur when gauss legendre quadrature is used to evaluate these integrals as well as to estimate the much lower errors that occur when gauss legendre quadrature is performed on such integrals after a sinh transformation has been applied the new error estimates are highly accurate in the case of the transformed integral and are shown to be comparable to those found in previous work by elliott and johnston 2007 one advantage of the new estimates is that they are given by just one formula each for the un transformed and the transformed integrals rather than the much larger set of formulae in the previous work another advantage is that the new method applies over a much larger range of values than the previous method

dynamics of polynomial mappings of c 2 we study the dynamics of polynomial self mappings f of c 2 we construct for a large class of mappings an invariant measure which is mixing and of maximal entropy h f max log d t f log 1 f where d t f is the topological degree of f and 1 f its first dynamical degree to achieve this we look at the meromorphic extensions of f to smooth minimal compactifications of c 2 when a good compactification is found we construct an f invariant green current t which contains many dynamical informations when d t f 1 f 1 the measure is obtained as p dd c vt where v is a partial green function defined on the support of t when 1 t t where t is a globally defined f invariant current

application of sigmoidal transformations to weakly singular and near singular boundary element integrals accurate numerical determination of line integrals is fundamental to reliable implementation of the boundary element method for a source point distant from a particular element standard gaussian quadrature is adequate as well as being the technique of choice however when the integrals are weakly singular or nearly singular source point near the element this technique is no longer adequate here a co ordinate transformation technique based on sigmoidal transformations is introduced to evaluate weakly singular and near singular integrals a sigmoidal transformation has the effect of clustering the integration points towards the endpoints of the interval of integration the degree of clustering is governed by the order of the transformation comparison of this new method with existing co ordinate transformation techniques shows that more accurate evaluation of these integrals can be obtained based on observations of several integrals considered guidelines are suggested for the order of the sigmoidal transformations copyright 1999 john wiley sons ltd

an algorithm for the approximate solution of integral equations of mellin type the cruciform crack problem of elasticity gives rise to an integral equation of the second kind on 0 1 whose kernel has a fixed singularity at 0 0 we introduce a transformation of 0 1 onto itself such that an arbitrary number of derivatives vanish at the end points 0 and 1 if the transformed kernel is dominated near the origin by a mellin kernel then we have given conditions under which the use of a modified euler maclaurin quadrature rule and the nystrom method gives an approximate solution which converges to the exact solution of the original equation the method is illustrated with a numerical example

dynamical study while searching equilibrium solutions in n body problem the theory of complex dynamics is usually applied to compare the global convergence properties of different iterative methods by obtaining the attraction basins for simple polynomial equations in the complex domain however in this work we use it in quite another context the study of a nontrivial nonlinear system that describes the motion of interacting bodies in celestial mechanics namely newtonian planar circular restricted four body problem and its relative equilibrium solutions these have been investigated from a dynamical point of view new properties of the solutions of this system have been obtained practical guidelines for efficient search of relative equilibrium solutions of n body problem have been given

ito s lemma without non anticipatory conditions a stochastic integral with respect to brownian motion which extends ito s integral to anticipatory integrands is constructed and investigated this stochastic integral is different from the skorokhod integral the ito lemma is proved for this integral

numerical p version refinement studies for the regularized stress bem in the development of the boundary element method bem and the finite element method fem researchers have typically selected similar basis functions that is both methods typically employ low order interpolations such as piece wise linear or piece wise quadratic and rely on h version refinement to increase accuracy as required in the case of the fem the decision to use low order elements is made for computational efficiency as an attractive compromise between local modeling accuracy and sparseness of the resulting linear system however in many bem formulations low order elements may be the only practical choice given the complexity of using analytic integration formulae in conjunction with special integral interpretations unlike their efficient use in the fem fine meshes of low order elements in the bem are highly inefficient from a computational standpoint given the dense nature of bem systems moreover owing to singularities in the kernel functions the bem should be expected to benefit more so than the fem from very high levels of local accuracy through the use of regularized algorithms which only require numerical integration p version refinement in the bem is easily extended to include any set of basis functions with no significant increase in programming complexity numerical results show that by using interpolations as high as 12th and 16th order one can expect reductions in error by as many as five orders of magnitude over comparable algorithms based on similar system size for two dimensional problems it is also shown that for a given level of error one can expect reductions in system size by an order of magnitude thus leading to a reduction in computational expense for conventional algorithms by three orders of magnitude copyright 2003 john wiley sons ltd

a self adaptive co ordinate transformation for efficient numerical evaluation of general boundary element integrals almost all general purpose boundary element computer packages include a curved geometry modelling capability thus numerical quadrature schemes play an important role in the efficiency of programming the technique the present work discusses this problem in detail and introduces efficient means of computing singular or nearly singular integrals currently found in two dimensional axisymmetric and three dimensional applications emphasis is given to a new third degree polynomial transformation which was found greatly to improve the accuracy of gaussian quadrature scheme s within the near singularity range the procedure can easily be implemented into existing be codes and presents the important feature of being self adaptive i e it produces a variable lumping of the gauss stations toward the singularity depending on the minimum distance from the source point to the element the self adaptiveness of the scheme also makes it inactive when not useful large source distances which makes it very safe for general usage

real arnold complexity versus real topological entropy for birational transformations we consider a family of birational measure preserving transformations of two variables depending on one parameter for which simple rational expressions with integer coefficients for the exact expression of the dynamical zeta function have been conjectured together with an equality between the multiplicative rate of growth of the arnold complexity and the exponential of the topological entropy this identification takes place for the birational mapping seen as a mapping bearing on two complex variables acting in a complex projective space we revisit this identification between these two quite universal complexities by considering now the mapping as a mapping bearing on two real variables the definitions of the two previous topological complexities arnold complexity and topological entropy are modified according to this real variables point of view most of the universality is lost however the results presented here are again in agreement with an identification between the multiplicative rate of growth of some real arnold complexity and the exponential of some real topological entropy a detailed analysis of this real arnold complexity as a function of the parameter of this family of birational transformations of two variables is given one can also slightly modify the definition of the dynamical zeta function introducing a real dynamical zeta function associated with the counting of the real cycles only similarly one can also introduce some real arnold complexity generating functions we show that several of these two real generating functions seem to have the same singularities furthermore we actually conjecture several simple rational expressions for them yielding again algebraic values for the exponential of the real topological entropy in particular when the parameter of our family of birational transformations becomes large we obtain two interesting compatible nontrivial rational expressions these rational results for real mappings cannot be understood in terms of any obvious markov s partition or symbolic dynamics hyperbolic systems interpretation the birational transformation is far from being hyperbolic it is measure preserving it would be useful to know whether this relation between the arnold complexity and the topological entropy as well as the rationality of the degree generating functions and dynamical zeta functions are a consequence of the measure preserving property of the mapping

analytic version of test functionals fourier transform and a characterization of measures in white noise calculus it is shown that the space j of test white noise functionals has an analytic version a which is an algebra as well as a topological linear space topologized by the projective limit of a sequence ap p n of banach spaces with respective norm given by ap supz cj p z exp 2 1 z p2 where sj p denotes the complexification of j p furthermore it is shown that the a topology and the j topology are equivalent in the course of the proof it is also shown that the space jp is isometrically isomorphic to the bargmann segal analytic functions on bj p under s transform employing this new version we are able to define the fourier transform of a generalized white noise functional as the adjoint of fourier wiener transform t1 i with parameter 1 i and moreover we show that every measure in j always satisfies a certain growth condition

a complete error analysis for the evaluation of a two dimensional nearly singular boundary element integral an important aspect of numerical integration is to have some knowledge of the truncation error for a given number of integration points in this paper we determine estimates for these errors in the application of gauss legendre quadrature to evaluate numerically two dimensional integrals which arise in the boundary element method expressions for the truncation errors developed here require the approximate evaluation of two integrals in the complex plane the second integral which has been termed the remainder of the remainder was assumed small by the authors in a previous attempt in developing error estimates however here this integral is included and it is evaluated using a novel approach for the choice of contour we consider examples where ignoring the remainder of the remainder was a reasonable assumption and also consider cases where this remainder dominates the error finally it is shown for each of the integrals considered that these new error estimates agree very closely with the actual quadrature error

capacity and quantum mechanical tunneling we connect the notion of capacity of sets in the theory of symmetric markov process and dirichlet forms with the notion of tunneling through the boundary of sets in quantum mechanics in particular we show that for diffusion processes the notion appropriate to a boundary without tunneling is more refined than simply capacity zero we also discuss several examples in d

a new method for numerical evaluation of nearly singular integrals over high order geometry elements in 3d bem this work presents a new method for numerical computation of the two dimensional nearly singular integrals by using the eight node second order quadrilateral surface elements in 3d bem a new indirect regularized boundary element formulation excluding the cpv cauchy principal value and hfp hadamard finite part integrals is proposed based on this a new approximation formula of the distance from the fixed calculation point to a generic point of the aforementioned surface geometry elements is developed firstly and then the exponential transformation which has been widely employed in 2d bem is extended to 3d bem to remove the near singularities of integrands for considered integrals several numerical examples are given to verify the high efficiency and the stability of the proposed scheme

post critical set and non existence of preserved meromorphic 2 forms we present a family of birational transformations in cp2 depending on two or three parameters which do not generically preserve meromorphic 2 forms with the introduction of the orbit of the critical set vanishing condition of the jacobian also called the post critical set we get some new structures some non analytic 2 form which reduce to meromorphic 2 forms for particular subvarieties in the parameter space on these subvarieties the iterates of the critical set have a polynomial growth in the degrees of the parameters while one has an exponential growth out of these subspaces the analysis of our birational transformation in cp2 is first carried out using the diller favre criterion in order to find the complexity reduction of the mapping the integrable cases are found the identification between the complexity growth and the topological entropy is once again verified we perform plots of the post critical set as well as calculations of lyapunov exponents for many orbits confirming that generically no meromorphic 2 form can be preserved for this mapping these birational transformations in cp2 which generically do not preserve any meromorphic 2 form are extremely similar to other birational transformations we previously studied which do preserve meromorphic 2 forms we note that these two sets of birational transformations exhibit totally similar results as far as topological complexity is concerned but drastically different results as far as a more probabilistic approach of dynamical systems is concerned lyapunov exponents with these examples we see that the existence of a preserved meromorphic 2 form explains most of the numerical discrepancies between the topological and probabilistic approaches of dynamical systems

analysis of three dimensional anisotropic heat conduction problems on thin domains using an advanced boundary element method abstract in this paper an advanced boundary element method bem is developed for solving three dimensional 3d anisotropic heat conduction problems in thin walled structures the troublesome nearly singular integrals which are crucial in the applications of the bem to thin structures are calculated efficiently by using a nonlinear coordinate transformation method for the test problems studied promising bem results with only a small number of boundary elements have been obtained when the thickness of the structure is in the orders of micro scales 10 6 which is sufficient for modeling most thin walled structures as used in for example smart materials and thin layered coating systems the advantages disadvantages as well as potential applications of the proposed method as compared with the finite element method fem are also discussed

three dimensional stress analysis of thin structures using a boundary element method with sinh transformation for nearly singular integrals abstract in this work a three dimensional 3d boundary element method was established with an efficient nonlinear coordinate transformation scheme namely sinh transformation to evaluate nearly singular integrals in boundary integral formulations second order quadrilateral surface elements were developed based on this method to accurately describe the geometry of thin structures the elastic behaviors of selected thin structures were then computed by using the 3d boundary element model to demonstrate the accuracy and efficiency of this approach a number of testing examples i e the 3d kirsch problem the thin spherical shell problem the ellipsoidal vessel problem with non uniform thickness and the hollow circular cylinder problem were numerically studied to test the established method remarkable accuracy and efficiency were found in the developed approach through the comparison to the numerical results and analytical solutions reported in the literature

on positive generalized functionals abstract a class of spaces of test and generalized functionals over gaussian spaces is constructed and two theorems of bochner s type are proved

real and complex dynamics of a family of birational maps of the plane the golden mean subshift we describe the real dynamics of a family of birational mappings of the plane by combining complex intersection theory and techniques from smooth dynamical systems we are able to give an essentially complete account of the behavior of both wandering and nonwandering orbits in particular the golden mean subshift provides a topological model for the dynamics on the nonwandering set while the mappings are not hyperbolic they are shown to possess many of the structures associated with hyperbolicity

a sinh transformation for evaluating nearly singular boundary element integrals an implementation of the boundary element method requires the accurate evaluation of many integrals when the source point is far from the boundary element under consideration a straightforward application of gaussian quadrature suffices to evaluate such integrals when the source point is on the element the integrand becomes singular and accurate evaluation can be obtained using the same gaussian points transformed under a polynomial transformation which has zero jacobian at the singular point a class of integrals which lies between these two extremes is that of nearly singular integrals here the source point is close to but not on the element and the integrand remains finite at all points however instead of remaining flat the integrand develops a sharp peak as the source point moves closer to the element thus rendering accurate evaluation of the integral difficult this paper presents a transformation based on the sinh function which automatically takes into account the position of the projection of the source point onto the element which we call the nearly singular point and the distance from the source point to the element the transformation again clusters the points towards the nearly singular point but does not have a zero jacobian implementation of the transformation is straightforward and could easily be included in existing boundary element method software it is shown that for the two dimensional boundary element method several orders of magnitude improvement in relative error can be obtained using this transformation compared to a conventional implementation of gaussian quadrature asymptotic estimates for the truncation errors are also quoted copyright 2004 john wiley sons ltd

an improved exponential transformation for nearly singular boundary element integrals in elasticity problems abstract this paper presents an improved exponential transformation for nearly singular boundary element integrals in elasticity problems the new transformation is less sensitive to the position of the projection point compared with the original transformation in our work the conventional distance function is modified into a new form in the polar coordinate system based on the refined distance function an improved exponential transformation is proposed in the polar coordinate system moreover to perform integrations on irregular elements an adaptive integration scheme considering both the element shape and the projection point associated with the improved transformation is proposed furthermore when the projection point is located outside the integration element another nearest point is introduced to subdivide the integration elements into triangular or quadrilateral patches of fine shapes numerical examples are presented to verify the proposed method results demonstrate the accuracy and efficiency of our method

rational dynamical zeta functions for birational transformations we propose a conjecture for the exact expression of the unweighted dynamical zeta function for a family of birational transformations of two variables depending on two parameters this conjectured function is a simple rational expression with integer coecients this yields an algebraic value for the topological entropy furthermore the generating function for the arnold complexity is also conjectured to be a rational expression with integer coecients with the same singularities as for the dynamical zeta function this leads at least in this example to an equality between the asymptotic of the arnold complexity and the exponential of the topological entropy we also give a semi numerical method to eectively compute the arnold complexity c 1999 published by elsevier science b v all rights reserved

mittag leffler analysis i construction and characterization abstract we construct an infinite dimensional analysis with respect to non gaussian measures of mittag leffler type which we call mittag leffler measures it turns out that the well known wick ordered polynomials in gaussian analysis cannot be generalized to this non gaussian case instead of using wick ordered polynomials we prove that a system of biorthogonal polynomials called appell system is applicable to the mittag leffler measures therefore we are able to introduce a test function and a distribution space as an application we construct donsker s delta in a non gaussian setting as a weak integral in the distribution space

error analysis for a sinh transformation used in evaluating nearly singular boundary element integrals in the two dimensional boundary element method one often needs to evaluate numerically integrals of the form 1 1g x j x f x a 2 b 2 dx where j 2 is a quadratic g is a polynomial and f is a rational logarithmic or algebraic function with a singularity at zero the constants a and b are such that 1

factorization properties of birational mappings abstract we analyse birational mappings generated by transformations on q q matrices which correspond respectively to two kinds of transformations the matrix inversion and a permutation of the entries of the q q matrix remarkable factorization properties emerge for quite general involutive permutations it is shown that factorization properties do exist even for birational transformations associated with noninvolutive permutations of entries of q q matrices and even for more general transformation which are rational transformations but no longer birational the existence of factorization relations independent of q the size of the matrices is underlined the relations between the polynomial growth of the complexity of the iterations the existence of recursions in a single variable and the integrability of the mappings are sketched for the permutations yielding these properties all these results show that permutations of the entries of the matrix yielding factorization properties are not so rare in contrast the occurrence of recursions in a single variable or of the polynomial growth of the complexity are of course less frequent but not completely exceptional

a general algorithm for evaluating nearly singular integrals in anisotropic three dimensional boundary element analysis abstract this paper presents a new general method for the evaluation of nearly singular boundary element integrals arising in anisotropic three dimensional 3d boundary element analysis it is shown that the original nearly singular integrals can be transformed into using a sinh function an element by element sum of regular integrals each one expressed in terms of intrinsic local coordinates as a consequence the actual computation can be performed by using standard n n gaussian quadrature and the procedure can be easily included in any existing computer code this new method has full generality and therefore can be applied to a wide class of integrals the numerical results demonstrate the accuracy and efficiency of the method along with its insensitivity to the location of the nearly singular points it is shown that several orders of magnitude improvement in relative errors can be obtained using this transformation when compared to a straightforward implementation of gaussian quadrature

on the contraction property of energy forms on infinite dimensional space abstract we prove that certain energy forms on infinite dimensional space with polynomial domain have the contraction or dirichlet property some consequences are discussed

growth complexity spectrum of some discrete dynamical systems we first study the iteration of birational mappings generated by the composition of the matrix inversion and of a permutation of the entries of 3 3 matrices and consider the degree d n of the numerators or denominators of the corresponding successive rational expressions for the nth iterate the growth of this degree is generically exponential with n d n n is called the growth complexity we introduce a semi numerical analysis which enables to compute these growth complexities for all the 9 possible birational transformations these growth complexities correspond to a spectrum of eighteen algebraic values we then drastically generalize these results replacing permutations of the entries by homogeneous polynomial transformations of the entries possibly depending on many parameters again it is shown that the associated birational or even rational transformations yield algebraic values for their growth complexities 1999 elsevier science b v all rights reserved

the iterated sinh transformation a sinh transformation has recently been proposed to improve the numerical accuracy of evaluating nearly singular integrals using gauss legendre quadrature it was shown that the transformation could improve the accuracy of evaluating such integrals which arise in the boundary element method by several orders of magnitude here this transformation is extended in an iterative fashion to allow the accurate evaluation of similar types of integrals that have more spiked integrands results show that one iteration of this sinh transformation is preferred for nearly weakly singular integrals whereas two iterations lead to several orders of magnitude improvement in the evaluation of nearly strongly singular integrals the same observation applies when considering integrals of derivatives of the two dimensional boundary element kernel however for these integrals more iterations are required as the distance from the source point to the boundary element decreases

a general algorithm for the numerical evaluation of nearly singular integrals on 3d boundary element a general numerical method is proposed to compute nearly singular integrals arising in the boundary integral equations bies the method provides a new implementation of the conventional distance transformation technique to make the result stable and accurate no matter where the projection point is located the distance functions are redefined in two local coordinate systems a new system denoted as a b is introduced here firstly its implementation is simpler than that of the polar system and it also performs efficiently then a new distance transformation is developed to remove or weaken the near singularities to perform integration on irregular elements an adaptive integration scheme is applied numerical examples are presented for both planar and curved surface elements the results demonstrate that our method can provide accurate results even when the source point is very close to the integration element and can keep reasonable accuracy on very irregular elements furthermore the accuracy of our method is much less sensitive to the position of the projection point than the conventional method

dirichlet forms and white noise analysis we use the white noise calculus as a framework for the introduction of dirichlet forms in infinite dimensions in particular energy forms associated with positive generalized white noise functionals are considered and we prove criteria for their closability if the forms are closable we show that their closures are markovian in the sense of fukushima

energy and invariant measures for birational surface maps given a birational self map of a compact complex surface it is useful to find an invariant measure that relates the dynamics of the map to its action on cohomology under a very weak hypothesis on the map we show how to construct such a measure the main point in the construction is to make sense of the wedge product of two positive closed 1 1 currents we are able to do this in our case because local potentials for each current have finite energy with respect to the other our methods also suffice to show that the resulting measure is mixing does not charge curves and has nonzero lyapunov exponents

estimates of the error in gauss legendre quadrature for double integrals error estimates are a very important aspect of numerical integration it is desirable to know what level of truncation error might be expected for a given number of integration points here we determine estimates for the truncation error when gauss legendre quadrature is applied to the numerical evaluation of two dimensional integrals which arise in the boundary element method two examples are considered one where the integrand contains poles when its definition is extended into the complex plane and another which contains branch points in both cases we obtain error estimates which agree with the actual error to at least one significant digit

invariant curves for birational surface maps we classify invariant curves for birational surface maps that are expanding on cohomology when the expansion is exponential the arithmetic genus of an invariant curve is at most one this implies severe constraints on both the type and number of irreducible components of the curve in the case of an invariant curve with genus equal to one we show that there is an associated invariant meromorphic two form

real polynomial diffeomorphisms with maximal entropy tangencies like the henon diffeomorphisms of r2 the quadratic maps of r have also played a central role in the field of dynamical systems these two families of dynamical systems fit together naturally which raises the question of the extent to which the dynamics is similar one difference is that our knowledge of these quadratic maps is much more thorough than our knowledge of these quadratic diffeomorphisms substantial progress in the theory of quadratic maps has led to a rather complete theoretical picture of their dynamics and an understanding of how the dynamics varies with the parameter despite significant recent progress in the theory of henon diffeomorphisms due to benedicks and carleson and many others there are still many phenomena that are not nearly so well understood in this two dimensional setting as they are for quadratic maps one phenomenon which illustrates the difference in the extent of our knowledge in dimensions one and two is the dependence of the complexity of the system on parameters in one dimension the nature of this dependence is understood and the answer is summarized by the principle of monotonicity loosely stated this is the assertion that the complexity of fa does not

real topological entropy versus metric entropy for birational measure preserving transformations abstract we consider a family of birational measure preserving transformations of two complex variables depending on one parameter for which simple rational expressions for the dynamical zeta function have been conjectured together with an equality between the topological entropy and the logarithm of the arnold complexity divided by the number of iterations similar results have been obtained for the adaptation of these two concepts to dynamical systems of real variables yielding to introduce a real topological entropy and a real arnold complexity we try to compare here the kolmogorov sinai metric entropy and this real arnold complexity or real topological entropy on this particular example of a one parameter dependent birational transformation of two variables more precisely we analyze using an infinite precision calculation the lyapunov characteristic exponents for various values of the parameter of the birational transformation in order to compare these results with the ones for the real arnold complexity we find a quite surprising result for this very birational example and in fact for a large set of birational measure preserving mappings generated by involutions the lyapunov characteristic exponents seem to be equal to zero or at least extremely small for all the orbits we have considered and for all values of the parameter birational measure preserving transformations generated by involutions could thus allow to better understand the difference between the topological description and the probabilistic description of discrete dynamical systems many birational measure preserving transformations generated by involutions seem to provide examples of discrete dynamical systems which can be topologically chaotic while they are metrically almost quasi periodic heuristically this can be understood as a consequence of the fact that their orbits seem to form some kind of transcendental foliation of the two dimensional space of variables

the discrete painleve i equations transcendental integrability and asymptotic solutions the integrability of the discrete painleve i equation dp i is reviewed and its integrability studied we establish the existence of a conserved quantity which is algebraic in the case of the autonomous dp i equation and it is argued that the non autonomous dp i map has a non algebraic invariant our analysis leads to among other results the construction of asymptotic solutions with interesting structures

a birational mapping with a strange attractor post critical set and covariant curves we consider some two dimensional birational transformations one of them is a birational deformation of the henon map for some of these birational mappings the post critical set i e the iterates of the critical set is infinite and we show that this gives straightforwardly the algebraic covariant curves of the transformation when they exist these covariant curves are used to build the preserved meromorphic 2 form one may also have an infinite post critical set yielding a covariant curve which is not algebraic transcendental for two of the birational mappings considered the post critical set is finite and we claim that there is no algebraic covariant curve and no preserved meromorphic 2 form for these two mappings with finite post critical sets attracting sets occur and we show that they pass the usual tests lyapunov exponents and the fractal dimension for being strange attractors the strange attractor of one of these two mappings is unbounded

a bi cubic transformation for the numerical evaluation of the cauchy principal value integrals in boundary methods the numerical strategies employed in the evaluation of singular integrals existing in the cauchy principal value cpv sense are undoubtedly one of the key aspects which remarkably affect the performance and accuracy of the boundary element method bem thus a new procedure based upon a bi cubic co ordinate transformation and oriented towards the numerical evaluation of both the cpv integrals and some others which contain different types of singularity is developed both the ideas and some details involved in the proposed formulae are presented obtaining rather simple and attractive expressions for the numerical quadrature which are also easily embodied into existing bem codes some illustrative examples which assess the stability and accuracy of the new formulae are included