the convergence of spline collocation for strongly elliptic equations on curves most boundary element methods for two dimensional boundary value problems are based on point collocation on the boundary and the use of splines as trial functions here we present a unified asymptotic error analysis for even as well as for odd degree splines subordinate to uniform or smoothly graded meshes and prove asymptotic convergence of optimal order the equations are collocated at the breakpoints for odd degree and the internodal midpoints for even degree splines the crucial assumption for the generalized boundary integral and integro differential operators is strong ellipticity our analysis is based on simple fourier expansions in particular we extend results by j saranen and w l wendland from constant to variable coefficient equations our results include the first convergence proof of midpoint collocation with piecewise constant functions i e the panel method for solving systems of cauchy singular integral equations

totally real minimal tori in in this paper we show that all totally real minimal tori in open image in new window correspond to doubly periodic finite gap solutions of the tzitzeica equation open image in new window using the results on the tzitzeica equation in integrable systems theory we describe explicitly these tori by prym theta functions

the collocation method for first kind boundary integral equations on polygonal regions in this paper the collocation method for first kind boundary integral equations by using piecewise constant trial functions with uniform mesh is shown to be equivalent to a projection method for second kind fredholm equations in a certain sense this projection is an interpolation projection by introducing this technique of analysis we particularly consider the case of polygonal boundaries we give asymptotic error estimates in l 2 norm on the boundaries and some superconvergence results for the single layer potential

spline collocation for strongly elliptic equations on the torus we prove convergence and error estimates in sobolev spaces for the collocation method with tensor product splines for strongly elliptic pseudodifferential equations on the torus examples of applications include elliptic partial differential equations with periodic boundary conditions but also the classical boundary integral operators of potential theory on torus shaped domains in three or more dimensions for odd degree splines we prove convergence of nodal collocation for any strongly elliptic operator for even degree splines and midpoint collocation we find an additional condition for the convergence which is satisfied for the classical boundary integral operators our analysis is a generalization to higher dimensions of the corresponding analysis of arnold and wendland 4

a constant mean curvature surface and the dirac operator the shape of a surface with constant mean curvature cmc has been studied in mathematics and physics related to nonlinear integrable theory and harmonic map model theory in the study a fictitious linear dirac type operator appears as a tool of the calculus konopelchenko b g and taimanov i a 1996 j phys a math gen 29 1261 5 in this paper i confine the dirac field defined in to a thin surface embedded in and obtain a proper dirac operator for the thin surface then it completely agrees with the dirac type operator used in the calculus of the cmc surface theory in other words the mathematical dirac type operator is realized by a physical dirac particle

local error estimates for some petrov galerkin methods applied to strongly elliptic equations on curves on etudie les proprietes de convergence locales de quelques methodes de petrov galerkin appliquees aux equations fortement elliptiques sur des courbes regulieres du plan on etablit des estimations d erreurs locales pour une grande echelle de normes de sobolev

isometric immersions of mathbb r 2 into mathbb r 4 and perturbation of hopf tori we produce a new general family of flat tori in mathbb r 4 the first one since bianchi s classical works in the nineteenth century to construct these flat tori obtained via small perturbation of certain hopf tori in mathbb s 3 we first present a global description of all isometric immersions of mathbb r 2 into mathbb r 4 with flat normal bundle

a deformation of tori with constant mean curvature in to those in other space forms it is shown that tori with constant mean curvature in r 3 constructed by wente 7 can be deformed to tori with constant mean curvature in the hyperbolic 3 space or the 3 sphere

universality and complexity in cellular automata abstract cellular automata are discrete dynamical systems with simple construction but complex self organizing behaviour evidence is presented that all one dimensional cellular automata fall into four distinct universality classes characterizations of the structures generated in these classes are discussed three classes exhibit behaviour analogous to limit points limit cycles and chaotic attractors the fourth class is probably capable of universal computation so that properties of its infinite time behaviour are undecidable

revisiting the fradkin vilkovisky theorem the status of the usual statement of the fradkin vilkovisky theorem claiming complete independence of the batalin fradkin vilkovisky path integral on the gauge fixing fermion even within a nonperturbative context is critically reassessed basic but subtle reasons why this statement cannot apply as such in a nonperturbative quantisation of gauge invariant theories are clearly identified a criterion for admissibility within a general class of gauge fixing conditions is provided for a large ensemble of simple gauge invariant systems this criterion confirms the conclusions of previous counter examples to the usual statement of the fradkin vilkovisky theorem comment 21 pages no figures to appear in jnl phys a

asymptotic expansions of the error of spline galerkin boundary element methods in this paper we analyse the existence of asymptotic expansions of the error of galerkin methods with splines of arbitrary degree for the approximate solution of integral equations with logarithmic kernels these expansions are obtained in terms of an interpolation operator and are useful for the application of richardson extrapolation and for obtaining sharper error bounds we also present and analyse a family of fully discrete spline galerkin methods for the solution of the same equations following the analysis of galerkin methods we show the existence of asymptotic expansions of the error

flat surfaces in the hyperbolic 3 space in this paper we give a conformal representation of flat surfaces in the hyperbolic 3 space using the complex structure induced by its second fundamental form we also study some examples and the behaviour at infinity of complete flat ends

lawson s genus two surface and meromorphic connections we investigate the lawson genus 2 surface by methods from integrable system theory to this surface we derive a globally defined meromorphic dpw potential which is determined up to two unknown functions depending only on the spectral parameter

a method of calculation of lifting flows around 2 dimensional corner shaped bodies introduction in this paper we study the flow problem of an incompressible inviscible fluid around a two dimensional corner shaped body generally one uses the potential function of such a flow which is the solution of an exterior neumann problem considering this solution as a single layer potential one then gets a fredholm integral equation of the second kind on the boundary which can be solved by collocation methods we shall use another approach let q represent the interior region of the body q the exterior one and r its boundary then the stream function of the perturbation flow i is the solution of the exterior dirichlet problem

on the convergence of collocation methods for symm s integral equation on open curves on etablit la convergence des methodes de collocation pour l equation integrale de symm sur des courbes ouvertes on confirme experimentalement les resultats theoriques sur les vitesses de convergence asymptotiques

the convergence of even degree spline collocation solution for potential problems in smooth domains of the plane in this article we derive new error estimates for collocation solution of potential type problems by using even degree smooth splines as trial functions it turns out that for smooth potentials the assured convergence is of the same order as by using splines of the odd degreed 1 some numerical examples which conform the theoretical results are presented

a hopf differential for constant mean curvature surfaces in s2 r and h2 r a basic tool in the theory of constant mean curvature cmc surfaces in space forms is the holomorphic quadratic differential discovered by h hopf in this paper we generalize this differential to immersed cmc surfaces in the product spaces s r and h r and prove a corresponding result about the geometry of immersed cmc spheres s in these target spaces

high order methods for linear functionals of solutions of second kind integral equations the authors show that linear functionals e g inner products or point functionals of solutions of second kind fredholm integral equations may be found within o 1 n 2r by very simple computational schemes using the nystrom or quadrature method based on piecewise polynomials of order r on an n 1 point mesh our theory covers a class of nonsmooth wiener hopf type equations and applications include the calculation of crack energy and stress intensity factors in certain linear elastic fracture problems and the use of the boundary integral method to compute various superconvergent solutions to boundary value problems in polygonal domains a proof of the uniform convergence of nystrom s method for certain boundary integral equations on polygonal domains is also included

on the boundary element method for some nonlinear boundary value problems here we analyse the boundary element galerkin method for two dimensional nonlinear boundary value problems governed by the laplacian in an interior or exterior domain and by highly nonlinear boundary conditions the underlying boundary integral operator here can be decomposed into the sum of a monotoneous hammerstein operator and a compact mapping we show stability and convergence by using leray schauder fixed point arguments due to petryshyn and necas

generalized pt symmetry and real spectra the fact that eigenvalues of pt symmetric hamiltonians h can be real for some values of ap arameter and complex for others is explained by showing that the matrix elements ofh a ndhence the secular equation are real not only for pt but also for an ya ntiunitary operator a satisfying a 2k 1w ith k odd the argument is illustrated by a 2 2m atrix hamiltonian and two examples of the generalization are given

spline collocation for convolutional parabolic boundary integral equations summary we consider spline collocation methods for a class of parabolic pseudodifferential operators we show optimal order convergence results in a large scale of anisotropic sobolev spaces the results cover for example the case of the single layer heat operator equation when the spatial domain is a disc

limit theorems for the nonattractive domany kinzel model we study the domany kinzel model which is a class of discrete time markov processes with two parameters p 1 p 2 0 1 2 and whose states are subsets of z the set of integers when p 1 and p 2 2 2 with 0 1 2 the process can be identified with the mixed site bond oriented percolation model on a square lattice with the probabilities of open site a and of open bond for the attractive case 0 p 1 p 2 1 the complete convergence theorem is easily obtained on the other hand the case p 1 p 2 1 0 realizes the rule 90 cellular automaton of wolfram in which starting from the bernoulli measure with density 0 the distribution converges weakly only if e 0 1 2 1 using our new construction of processes based on signed measures we prove limit theorems which are also valid for nonattractive cases with p 1 p 2 1 0 in particular when p 2 0 1 and p 1 is close to 1 the complete convergence theorem is obtained as a corollary of the limit theorems

invariant measures and convergence properties for cellular automaton 184 and related processes our results concern long time limit properties of a deterministic dynamics that is common for a wide class of processes that have been studied so far during at least last two decades the most widely known process from this class is a cellular automaton that acquired number 184 in the classification of s wolfram this ca 184 is being intensively used to model vehicular traffic however our results are mainly derived with help of another process that offers a helpful insight into the studied dynamics it is a so called ballistic annihilation model abbreviated by ba ba is a model for chemical reaction a b inert in ba a and b type particles move in opposite directions with velocities 1 and 1 respectively and annihilate upon collisions certain results concerning ba and ca 184 are also formulated in terms of another process known as a model of surface growth sg for short the surface shape in this process behaves as the integrated profile of particle distribution in ca 184

on the asymptotic convergence of collocation methods we prove quasioptimal and optimal order estimates in various sobolev norms for the approximation of linear strongly elliptic pseudodifferential equations in one independent variable by the method of nodal collocation by odd degree polynomial splines the analysis pertains in particular to many of the boundary element methods used for numerical computation in engineering applications equations to which the analysis is applied include fredholm integral equations of the second kind certain first kind fredholm equations singular integral equations involving cauchy kernels a variety of integro differential equations and two point boundary value problems for ordinary differential equations the error analysis is based on an equivalence which we establish between the collocation methods and certain nonstandard galerkin methods we compare the collocation method with a standard galerkin method using splines of the same degree showing that the galerkin method is quasioptimal in a sobolev space of lower index and furnishes optimal order approximation for a range of sobolev indices containing and extending below that for the collocation method and so the standard galerkin method achieves higher rates of convergence

spline qualocation methods for variable coefficient elliptic equations on curves here the stability and convergence results of oqualocation methods providing additional orders of convergence are extended from the special class of pseudodifferential equations with constant coefficient symbols to general classical pseudodifferential equations of strongly and of oddly elliptic type the analysis exploits localization in the form of frozen coefficients pseudohomogeneous asymptotic symbol representation of classical pseudodifferential operators a decisive commutator property of the qualocation projection and requires qualocation rules which provide sufficiently many additional degrees of precision for the numerical integration of smooth remainders numerical examples show the predicted high orders of convergence

the numerical solution of symm s equation on smooth open arcs by spline galerkin methods abstract we consider the classical fredholm linear integral equation of the first kind with logarithmic kernel on a smooth jordan open arc applying the well known cosine change of variable the arc is reparametrized and the problem is transformed into a new integral equation we investigate the existence of an asymptotic expansion for the error of the galerkin method with splines on a uniform mesh as test trial functions we also analyse a full discretization of the method based on the galerkin collocation method using high order integration formulae to keep the optimal error estimates of the galerkin method in weak norms asymptotic expansions of the error for this method are provided finally we show how these expansions extend to the computation of the potential the expansions of the error in powers of the discretization parameter are useful to obtain a posteriori estimates of the error and to apply richardson extrapolation for acceleration of convergence 1999 elsevier science ltd all rights reserved

stability of a spline collocation method for strongly elliptic multidimensional singular integral equations we consider a spline collocation method for strongly elliptic zero order pseudodifferential equationsp gw au f on a cube 0 1 m utilizing multilinear spline functions which are zero at the boundary we collocate at the meshpoints inside for classical strongly elliptic translation invariant pseudodifferential operators we verify the stability of the considered collocation method inl 2 afterwards form 2 and a right hand sidef h 8 s m 2 we prove an asymptotic convergence estimate

spline qualocation methods for boundary integral equations this paper further develops the qualocation method for the solution of integral equations on smooth closed curves qualocation is a petrov galerkin method in which the outer integrals are performed numerically by special quadrature rules here we allow the use of splines as trial and test functions and prove stability and convergence results for certain qualocation rules when the underlying petrov galerkin method is stable in the common case of a first kind equation with the logarithmic kernel a typical qualocation method employs piecewise linear splines and performs the outer integrals using just two points per interval this method with appropriate choice of the quadrature points and weights has ordero h 5 convergence in a suitable negative norm numerical experiments suggest that these rates of convergence are maintained for integral equations on open intervals when graded meshes are used to cope with unbounded solutions

on the asymptotic convergence of collocation methods with spline functions of even degree we investigate the collocation of linear one dimensional strongly elliptic integro dif ferential or more generally pseudo differential equations on closed curves by even degree polynomial splines the equations are collocated at the respective midpoints subject to uniform nodal grids of the even degree s splines we prove quasioptimal and optimal order asymptotic error estimates in a scale of sobolev spaces the results apply in particular to boundary element methods used for numerical computations in engineering applications the equations considered include fredholm integral equations of the second and the first kind singular integral equations involving cauchy kernels and integro differential equations having convolutional or constant coefficient principal parts respectively the error analysis is based on an equivalence between the collocation and certain varia tional methods with different degree splines as trial and as test functions we further need to restrict our operators essentially to pseudo differential operators having convolutional prin cipal part this allows an explicit fourier analysis of our operators as well as of the spline spaces in terms of trigonometric polynomials providing babuska s stability condition based on strong ellipticity our asymptotic error estimates extend partly those obtained by d n arnold and w l wendland from the case of odd degree splines to the case of even degree splines 1 introduction in this paper we investigate the asymptotic convergence of the collocation method using even degree polynomial splines applied to strongly elliptic systems of pseudo differential equations on closed curves with convolutional prin cipal part the collocation here employs the gauss points of one point integration as collocation points which corresponds to the usual boundary element collocation in applications 5 12 17 19 26 31 34 this is in contrast to the method investigated by g schmidt in 29 where one collocates at the break points of even order splines the asymptotic convergence properties for the standard galerkin method with splines of arbitrary orders are well known 22 for the collocation method however asymptotic convergence up to now has been shown for strongly elliptic systems only in the case of odd order splines by d n arnold and w l wendland 7

condition of boundary integral equations arising from flow computations abstract stating flow boundary conditions for normal and tangential velocities leads to fredholm integral equations of the first and second kind approximating the vortex distribution on pressure and suction side by piecewise linear functions with equidistant knots collocation yields a commonly overdetermined system of linear equations with the vortex densities as unknowns there are several possibilities for collocation the question arises what kind of integral equation and what collocation points are to be preferred our main conclusions based on condition numbers of the resulting linear systems are the equation of the first kind with cauchy singular kernel does not appear to be worse conditioned than the equation of the second kind with regular kernel collocation points for the equation of the first kind are better placed in between the knots for the equation of the second kind at the knots the circulation is mainly determined by the equation of the first kind the mean velocity level by the equation of the second kind employing simultaneously equations of the first and second kind generally improves the solution and allows to cope with aerodynamic profiles of any thickness and loading

applications of ergodic theory and sofic systems to cellular automata abstract we indicate a mathematical framework for analysing the evolution of cellular automata those automata obeying an additive rule are shown to be the same as endomorphisms of a compact abelian group and therefore their statistical and dynamical behavior can be told exactly by using fourier analysis and ergodic theory those obeying certain nonlinear rules are closely tied to a finitely described object called a sofic system but the underlying statistics appear to be more complicated and interesting we conclude by formulating several conjectures about one such system

extrapolation methods for spline collocation solutions of pseudodifferential equations on curves in this article we consider extrapolation methods in approximating smooth linear functionals by means of collocation solutions of boundary integral equations we are able to derive an asymptotic expansion of the error in main applications this expansion covers arbitrary high powers of the discretization parameter if the boundary solution is smooth the expansion gives rise to richardson type extrapolation schemes which require only simple postprocessing of calculated numbers the results are applicable for example in calculating pointwise values in the space domain when solving boundary value problems by means of integral equations in such examples the extrapolation rapidly improves the original rate of the convergence numerical experiments conform our results

lower bounds for critical values of a cancellative model we consider the following one dimensional discrete time cancellative model whose evolution is given by n 1 x n x 1 n x 1 mod 2 with probability p and n 1 0 with probability 1 p concerning critical probabilities pc and pc on a survival probability it is known that 0 706 pc pc 1 under a condition in this paper we give improved lower bounds of 0 771 and 0 781 on pc and pc respectively by finding suitable supermartingales for the model

the cauchy problem for the liouville equation and bryant surfaces abstract we give a construction that connects the cauchy problem for the 2 dimensional elliptic liouville equation with a certain initial value problem for mean curvature one surfaces in hyperbolic 3 space h 3 and solve both of them we construct the unique mean curvature one surface in h 3 that passes through a given curve with a given unit normal along it and provide diverse applications in particular topics such as period problems symmetries finite total curvature planar geodesics rigidity etc are treated for these surfaces

quadrature methods for strongly elliptic equations of negative order on smooth closed curves in this article the authors introduce and analyze a family of simple quadrature methods for strongly elliptic equations of negative order the results cover all the classical pseudodifferential equations of this kind and hence the boundary integral equations appearing in applications the quadrature methods are based on the trapezoidal approximation of the operator with respect to a uniform mesh and on varepsilon collocation they are proved to have the stability and the convergence with the maximal rate o h beta of convergence in general and o h beta 1 for some special values of varepsilon here beta is the order of the operator the method is applicable to systems of boundary integral equations as well in this connection the authors have conducted some numerical experiments which confirm their theoretical results

on some harris fkg type correlation inequalities for a non attractive model we consider the following non attractive one dimensional model whose evolution is given by n 1 x n x 1 n x 1 mod 2 with probability p 0 with probability 1 p in the present letter we prove that some harris fkg type correlation inequalities hold for this model moreover it is shown that a correlation inequality is also correct for the more general non attractive class

embedded constant mean curvature tori in the three sphere we prove that any constant mean curvature embedded torus in the three dimensional sphere is axially symmetric and use this to give a complete classification of such surfaces for any given value of the mean curvature

a spline trigonometric galerkin method and an exponentially convergent boundary integral method we consider a galerkin method for functional equations in one space variable which uses periodic cardinal splines as trial functions and trigonometric polynomials as test functions we analyze the method applied to the integral equation of the first kind arising from a single layer potential formulation of the dirichlet problem in the interior or exterior of an analytic plane curve in constrast to ordinary spline galerkin methods we show that the method is stable and so provides quasioptimal approximation in a large family of hilbert spaces including all the sobolev spaces of negative order as a consequence we prove that the approximate solution to the dirichlet problem and all its derivatives converge pointwise with exponential rate

stable hypersurfaces with constant scalar curvature it is well known that hypersurfaces m n with constant mean curvature in a riemannian manifold overline m n 1 c of constant sectional curvature c are solutions to the variational problem of extremizing the area function for volumepreserving variations

a method for finding bounds on critical values for non attractive interacting particle systems the method is to use a computer to find a function for the process that always has negative expectation by considering all relevant configurations of 0 s and 1 s at the boundary of a finite process in one dimension it is shown that a branching annihilating random walk will die out if the diffusion parameter is greater than 0 176 the method may also be applied to attractive processes and is used for the contact process in one dimension obtaining the same values as ziezold and grillenberger 1988 for up to 10 places in from the boundaries

capillary surfaces in a convex cone we show that a compact embedded hypersurface s subset mathbb r n 1 with constant higher order mean curvature in a convex piecewise smooth cone c which is perpendicular to c is part of a hypersphere also we prove that an embedded disk type constant mean curvature surface s subset mathbb r 3 in a convex polyhedral cone c which makes constant contact angles with c is a spherical cap if c has at most five faces this condition on the number of faces can be dropped if c is a right cone over a regular n gon and the contact angles are the same on s

the geometry of fronts we shall introduce the singular curvature function on cuspidal edges of surfaces which is related to the gauss bonnet formula and which characterizes the shape of cuspidal edges moreover it is closely related to the behavior of the gaussian curvature of a surface near cuspidal edges and swallowtails

on the convergence of collocation methods for boundary integral equations on polygons the integral equations encountered in boundary element methods are frequently solved numerically using collocation with spline trial functions convergence proofs and error estimates for these approximation methods have been only available in the following cases fredholm integral equations of the second kind 4 7 one dimensional pseudodifferential equations and singular integral equations with piecewise smooth coefficients on smooth curves 2 3 17 26 29 and some special results on the classical neumann integral equation of potential theory for polygonal plane domains 5 8 9 here we give convergence proofs for collocation with piecewise linear trial functions for neumann s integral equation and symm s integral equation on plane curves with corners we derive asymptotic error estimates in sobolev norms and analyze the effect of graded meshes 0 introduction in this paper we give convergence proofs and asymptotic error estimates in sobolev norms for collocation with piecewise linear spline trial func tions applied to two basic integral equations of potential theory on plane polygons namely the integral equation of the second kind with the double layer potential neumann s integral equation and the integral equation of the first kind with the simple layer potential symm s integral equation we use an idea of arnold and wendland 2 namely considering dirac delta functions the test functions in the collocation method as second derivatives of piecewise linear functions therefore similar results as presented here should be possible for splines of higher odd order corresponding results for even order splines are not yet available thus for one of the simplest methods of numerically solving dirichlet s problem on a plane domain with corners the midpoint collocation with piecewise constant trial functions for the first kind integral equation with the simple layer potential convergence is still an open problem the method of fourier series that yields the convergence proof in the case of a smooth boundary 27 cannot be applied in the presence of corners we apply the method of local mellin transformation that has previously been used to derive error estimates for galerkin methods for a wide class of operators including those occurring in boundary element methods in acoustics electromag

a simple proof of the stability criterion of gray and griffeath gray and griffeath studied attractive nearest neighbor spin systems on the integers having all 0 s and all 1 s as traps using the contour method they established a necessary and sufficient condition for the stability of the all 1 s equilibrium under small perturbations in this paper we use a renormalized site construction to give a much simpler proof our new approach can be used in many situations as a substitute for the contour method

surfaces on lie groups on lie algebras and their integrability it is shown that the problem of the immersion of a 2 dimensional surface into a 3 dimensional euclidean space as well as then dimensional generalization of this problem is related to the problem of studying surfaces in lie groups and surfaces in lie algebras a particular case of the general formalism presented here implies that any surface can be characterized in terms of 2 2 matrices using an arbitrary parametrization it is also shown that this generality of parametrization is useful for studying integrable surfaces i e surfaces described by integrable equations in particular starting from a suitable lax pair i e a suitable integrable equation it is possible to construct explicitly large classes of integrable surfaces

stability of hypersurfaces of constant mean curvature in riemannian manifolds hypersurfaces m n with constant mean curvature in a riemannian manifold overline m n 1 display many similarities with minimal hypersurfaces of overline m n 1 they are both solutions to the variational problem of minimizing the area function for certain variations in the first case however the admissible variations are only those that leave a certain volume function fixed for precise definitions see sect 2 this isoperimetric character of the variational problem associated to hypersurfaces of constant mean curvature introduces additional complications in the treatment of stability of such hypersurfaces

a quadrature based approach to improving the collocation method the collocation method is a popular method for the approximate solution of boundary integral equations but typically does not achieve the high order of convergence reached by the galerkin method in appropriate negative norms in this paper a quadrature based method for improving upon the collocation method is proposed and developed in detail for a particular case that case involves operators with even symbol such as the logarithmic potential operating on 1 periodic functions a smooth spline trial space of odd degree with constant mesh spacingh 1 n and a quadrature rule with 2n points where ann point quadrature rule would be equivalent to the collocation method in this setting it is shown that a special quadrature rule which depends on the degree of the splines and the order of the operator can yield a maximum order of convergence two powers ofh higher than the collocation method