normal forms and gauge symmetry of local dynamics a systematic procedure is proposed for deriving all the gauge symmetries of the general not necessarily variational equations of motion for the variational equations this procedure reduces to the dirac bergmann algorithm for the constrained hamiltonian systems with certain extension it remains applicable beyond the scope of dirac s conjecture even though no pairing exists between the constraints and the gauge symmetry generators in general nonvariational dynamics certain counterparts still can be identified of the first and second class constraints without appealing to any poisson structure it is shown that the general local gauge dynamics can be equivalently reformulated in an involutive normal form the last form of dynamics always admits the brst embedding which does not require the classical equations to follow from any variational principle

approximations of singular integral equations on lyapunov contours in banach spaces a general approximation method is investigated for the numerical evaluation of the singular integral equations on lyapunov contours defined in banach spaces the method consists in the application of the faber polynomials and the faber laurent expansion first some theorems are proved for the approximation of functions in a complex domain while these are defined in the banach space h c c 0 c 1 where c denotes a closed lyapunov contour these results are further used in order to prove the existence and uniqueness of the solutions for the systems on which the singular integral equations are reduced

numerical analysis of newton projection methods for nonlinear singular integral equations abstract numerical methods of newton iteration type for the solution of two classes of nonlinear cauchy singular integral equations are studied the existence of appropriate initial approximations the solvability of the linearized discrete equations and the weighted l2 convergence of the solutions of the discrete equations to a solution of the original problem are shown

regularity and symmetries of nonholonomic systems lagrangian systems with nonholonomic constraints may be considered as singular differential equations defined by some constraints and some multipliers the geometry solutions symmetries and constants of motion of such equations are studied within the framework of linearly singular differential equations some examples are given in particular the well known singular lagrangian of the relativistic particle which with the nonholonomic constraint v2 c2 yields a regular system

differential algebraic systems as differential equations on manifolds based on the theory of differential equations on manifolds existence and uniqueness results are proved for a class of mixed systems of differential and algebraic equations as they occur in various applications both the autonomous and nonautonomous case are considered moreover a class of algebraically incomplete systems is introduced for which existence and uniqueness results only hold on certain lower dimensional manifolds this class includes systems for which the application of ode solvers is known to lead to difficulties finally some solution approach based on continuation techniques is outlined

the pseudohyperbolic metric and bergman spaces in the ball the pseudohyperbolic metric is developed for the unit ball of c and is applied to a study of uniformly discrete sequences and bergman spaces of holomorphic functions on the ball

uniformly discrete sequences in the ball abstract we verify a conjecture of p duren and r weir concerning uniformly discrete sequences in the unit ball in c n

a geometric treatment of implicit differential algebraic equations abstract a differential geometric approach for proving the existence and uniqueness of implicit differential algebraic equations is presented it provides for a significant improvement of an earlier theory developed by the authors as well as for a completely intrinsic definition of the index of such problems the differential algebraic equation is transformed into an explicit ordinary differential equation by a reduction process that can be abstractly defined for specific submanifolds of tangent bundles here called reducible submanifolds local existence and uniqueness results for differential algebraic equations then follow directly from the final stage of this reduction by means of an application of the standard theory of ordinary differential equations

on the stability of the classical collocation method for cauchy type singular integral equations abstract the classical collocation method for cauchy type singular integral equations of the second kind on an interval using gaussian quadrature rules with respect to jacobi nodes is investigated in general the nodes and the weights of the quadrature rules can only be evaluated approximately in the present paper the question is answered how precise our knowledge about them must be in order to ensure that the corresponding algebraic system of equations is solvable yet and its solution is close to the solution of the exact system

on the numerical resolution of the generalized airfoil equation with possio kernel in this paper we consider the so called generalized airfoil singular integral equation with a smooth input function and present ad hoc quadrature rules to compute efficiently the elements of collocation and galerkin matrices

sinc nystro m method for numerical solution of a dominant system of cauchy singular integral equations given on a piecewise smooth contour a numerical method is proposed for obtaining an approximate solution to a dominant system of cauchy singular integral equations dscsies taken over a piecewise smooth contour l consisting of a finite number of arcs l j first the dscsies are reduced to a system of fredholm integral equations of the second kind which are then solved approximately by nystrom s method based on a sinc quadrature rule it is shown that the error of an n point approximation converges to zero at the rate o e cn 1 2 as n to infty provided that the coefficients of the dscsies are analytic in neighborhoods d j of each arc l j and satisfy a lipschitz condition in each d j

the discrete sloan iterate for the generalized airfoil equation abstract we have defined and established the superconvergence of the sloan iterate obtained from a polynomial galerkin approximation for solving the generalized airfoil equation when all the integrals involved are evaluated by using gaussian quadratures with the same number of nodes as basis elements the resulting discrete sloan iterate becomes the nystrom approximation obtained by using the same rules from this it follows that the discrete sloan iterate v n and the discrete galerkin approximation v n agree at the quadrature nodes thus v n converges no faster locally than v n for practical purposes computational superconvergence does not occur

the perturbed galerkin method for cauchy singular integral equations with constant coefficients we have considered the numerical solution of cauchy singular integral equations with constant coefficients via a polynomial based galerkin method l 2 and uniform convergence results are proved under holder continuity conditions on the kernel and right hand side and taking into account the quadrature errors produced when inner products are replaced by numerical quadratures these results generalize those of miel who considered only the case of equations of the first kind with index 0

heisenberg langevin and current equations via the recurrence relations approach some years ago the heisenberg equation of motion was formally solved by the recurrence relations approach it is shown here that the langevin equation represents a structural property of the recurrence relations the langevin equation is useful for studying the time evolution of the current the resulting current current correlation function is compared with luttinger s phenomenological theory geometric interpretations are made for the conductivity and the dielectric function

co isotropic and legendre lagrangian submanifolds and conformal jacobi morphisms the notion of a co isotropic and legendre lagrangian submanifold of a jacobi manifold is given a characterization of conformal jacobi morphisms and conformal jacobi infinitesimal transformations is obtained as co isotropic and legendre lagrangian submanifolds of jacobi manifolds

toeplitz operators and carleson measures in strongly pseudoconvex domains abstract we study mapping properties of toeplitz operators associated to a finite positive borel measure on a bounded strongly pseudoconvex domain d c n in particular we give sharp conditions on the measure ensuring that the associated toeplitz operator maps the bergman space a p d into a r d with r p generalizing and making more precise results by cuckovic and mcneal to do so we give a geometric characterization of carleson measures and of vanishing carleson measures of weighted bergman spaces in terms of the intrinsic kobayashi geometry of the domain generalizing to this setting results obtained by kaptanoglu for the unit ball

generalized hamiltonian dynamics i formulation on t q tq the dirac bergmann generalized hamiltonian dynamics for a degenerate lagrangian system is formulated on the whitney sum t q tq of the phase space t q and the velocity space tq over the configuration space q the formulation is related to those on t q and tq some ambiguities concerning generalized dynamics that have appeared in the literature are clarified

a general framework for solving riemann hilbert problems numerically a new numerical framework for the approximation of solutions to matrix valued riemann hilbert problems is developed based on a recent method for the homogeneous painleve ii riemann hilbert problem we demonstrate its effectiveness by computing solutions to other painleve transcendents an implementation in mathematica is made available online

complex hyperbolic geometry and hilbert spaces with complete pick kernels abstract suppose h is a finite dimensional reproducing kernel hilbert space of functions on x if h has the complete pick property then there is an isometric map from x with the metric induced by h into complex hyperbolic space ch n with its pseudohyperbolic metric we investigate the relationships between the geometry of x and the function theory of h and its multiplier algebra

a fast and well conditioned spectral method for singular integral equations abstract we develop a spectral method for solving univariate singular integral equations over unions of intervals by utilizing chebyshev and ultraspherical polynomials to reformulate the equations as almost banded infinite dimensional systems this is accomplished by utilizing low rank approximations for sparse representations of the bivariate kernels the resulting system can be solved in o m 2 n operations using an adaptive qr factorization where m is the bandwidth and n is the optimal number of unknowns needed to resolve the true solution the complexity is reduced to o m n operations by pre caching the qr factorization when the same operator is used for multiple right hand sides stability is proved by showing that the resulting linear operator can be diagonally preconditioned to be a compact perturbation of the identity applications considered include the faraday cage and acoustic scattering for the helmholtz and gravity helmholtz equations including spectrally accurate numerical evaluation of the far and near field solution the julia software package singularintegralequations jl implements our method with a convenient user friendly interface

on the collocation methods for singular integral equations with hilbert kernel in the present paper we introduce some singular integral operators singular quadrature operators and discretization matrices of singular integral equations with hilbert kernel these results both improve the classical theory of singular integral equations and develop the theory of singular quadrature with hilbert kernel then by using them a unified framework for various collocation methods of numerical solutions of singular integral equations with hilbert kernel is given under the framework it is very simple and obvious to obtain the coincidence theorem of collocation methods then the existence and convergence for constructing approximate solutions are also given based on the coincidence theorem

a hamilton jacobi theory for implicit differential systems in this paper we propose a geometric hamilton jacobi theory for systems of implicit differential equations in particular we are interested in implicit hamiltonian systems described in terms of lagrangian submanifolds of tt q generated by morse families the implicit character implies the nonexistence of a hamiltonian function describing the dynamics this fact is here amended by a generating family of morse functions which plays the role of a hamiltonian a hamilton jacobi equation is obtained with the aid of this generating family of functions to conclude we apply our results to singular lagrangians by employing the construction of special symplectic structures

on approximate solution of linear singular integral equations we investigate some properties of solutions of a class of linear singular integral equations the basic approach consists in replacing the singular kernel by a family of regular fredholm kernels projection techniques are used to discuss the convergence properties of solutions of these fredholm equations

first order equations of motion for classical mechanics a global canonical first order equation of motion is derived for any mechanical system obeying newton s second law the existence of a lagrangian is not assumed but the properties of the canonical equation are similar to those of the hamiltonian formulation the choice of map f from velocity space to phase space is not determined by the condition that the first order equation of motion be equivalent to a second order equation on configuration space and therefore is left open to be selected on the basis of other considerations the canonical equation is a covector or 1 form equation on the whitney sum t q tq and contains the second order equation condition restriction to the graph of the map f and newton s equation of motion in first order form the last is related to newton s second order equations by the consistency condition that the motions not lead off graph f in t q tq the first order equation of motion can be projected onto phase space if the map f can be inverted

integrability of implicit differential equations the problem of integrability of differential equations is discussed examples and integrability criteria are given an algorithm for extracting the integrable part of an implicit differential equation is formulated a procedure for generating a class of submanifolds of the cotangent bundle is defined this procedure is then used for generating implicit differential equations in the phase space of a mechanical system integrability criteria of such equations are established and an extraction algorithm is formulated

the classical collocation method for singular integral equations first of all a method of collocation which we call classical collocation is described for the approximate solution of complete singular integral equations with cauchy kernel taken over the arc 1 1 secondly we demonstrate that under reasonable conditions the approximate solutions converge to the solution of the original equation

regularity properties of fiber derivatives associated with higher order mechanical systems the aim of this work is to study fiber derivatives associated to lagrangian and hamiltonian functions describing the dynamics of a higher order autonomous dynamical system more precisely given a function in t t k 1 q we find necessary and sufficient conditions for such a function to describe the dynamics of a kth order autonomous dynamical system thus being a kth order hamiltonian function then we give a suitable definition of hyper regularity for these higher order hamiltonian functions in terms of their fiber derivative in addition we also study an alternative characterization of the dynamics in lagrangian submanifolds in terms of the solutions of the higher order euler lagrange equations

a note on solving cauchy integral equations of the first kind by iterations abstract the successive approximation method was applied for the first time by n i ioakimidis to solve practical cases of a cauchy singular integral equation the airfoil one in this paper we study a more general case we prove the convergence of the method in this general case the proposed method has been tested for two kernels which are particularly important in practice finally some numerical examples illustrate the accuracy of the method

discrete projection methods for cauchy singular integral equations with constant coefficients we consider proving the mean square and uniform convergence of various discrete galerkin and collocation methods for cauchy singular integral equations with constant coefficients for galerkin s method the convergence rates improve those given in an earlier paper while for collocation methods the results appear to be new in particular we consider product quadrature methods for equations with discontinuous coefficients and develop and prove the convergence of a new algorithm for equations of the first kind with index zero and logarithmically singular kernels

morse families in optimal control problems we geometrically describe optimal control problems in terms of morse families in the hamiltonian framework these geometric structures allow us to recover the classical first order necessary conditions for optimality and the starting point to run an integrability algorithm moreover the integrability algorithm is adapted to optimal control problems in such a way that the trajectories originated by discontinuous controls are also obtained from the hamiltonian viewpoint we obtain the equations of motion for optimal control problems in the lagrangian formalism by means of a proper lagrangian submanifold singular optimal control problems and overdetermined ones are also studied in the paper

rates of convergence for the method of classical collocation for solving singular integral equations the method of classical collocation for the approximate solution of the complete singular integral equation with cauchy kernel taken over the arc 1 1 has been described previously by the author in siam j numer anal 19 1982 pp 816 832 in this paper we consider in greater detail the urates of convergence for both discrete and global errors in the approximate solution of this equation this analysis gives a better convergence theorem than that given in the earlier paper the paper concludes with a discussion on the necessity of a particular condition for discrete convergence

a rational function approximation of the singular eigenfunction of the monoenergetic neutron transport equation demonstrates how a proper rational fraction approximation to the singular eigenfunction of the neutron transport theory can be constructed based on the properties of generalised functions and singular integral equations the parameters of the approximant are determined by a proper use of the orthogonality integrals satisfied by the case eigenfunctions this ensures the convergence of the approximant to its exact singular distributional form use of lebesgue integrable spaces made in the analysis leads to a new possibility of approximating functions in lp 1 p infinity and also of finding approximate solutions of singular integral equations

a generalization of chetaev s principle for a class of higher order nonholonomic constraints the constraint distribution in nonholonomic mechanics has a double role on the one hand it is a kinematic constraint that is it is a restriction on the motion itself on the other hand it is also a restriction on the allowed variations when using d alembert s principle to derive the equations of motion we will show that many systems of physical interest where d alembert s principle does not apply can be conveniently modeled within the general idea of the principle of virtual work by the introduction of both kinematic constraints and variational constraints as being independent entities this includes for example elastic rolling bodies and pneumatic tires also d alembert s principle and chetaev s principle fall into this scheme we emphasize the geometric point of view avoiding the use of local coordinates which is the appropriate setting for dealing with questions of global nature like reduction

sinc nystrom method for numerical solution of one dimensional cauchy singular integral equation given on a smooth arc in the complex plane on developpe une methode numerique basee sur les fonctions sinc pour la solution approchee d une equation integrale singuliere de cauchy a une dimension sur un arc regulier du plan complexe

a generalized lagrange equation in implicit form for non conservative mechanics different geometric formulations are obtained for a generalized lagrange equation non reducible to normal form and encompassing non conservative dynamics

rates of convergence for collocation with jacobi polynomials for the airfoil equation abstract in this paper we consider for the classical airfoil equation a collocation method based on jacobi polynomials the error is estimated in suitable nonstandard sobolev norms which are defined in such a way to respect the singularity structure of the exact solution furthermore our numerical experiments underline our optimal theoretical error estimates

a sinc hunter quadrature rule for cauchy principal value integrals a sine function approach is used to derive a new hunter type quadrature rule for the evaluation of cauchy principal value integrals of cer tain analytic functions integration over a general arc in the complex plane is considered special treatment is given to integrals over the interval 1 1 it is shown that the quadrature error is of order 0 e where n is the number of nodes used and where c is a positive constant which is indepen dent of n an application of the rule to the approximate solution of cauchy singular integral equations is also discussed numerical examples are included to illustrate the performance of the rule

symmetries and infinitesimal symmetries of singular differential equations the aim of this paper is to study symmetries of linearly singular differential equations namely equations that cannot be written in normal form because the derivatives are multiplied by a singular linear operator the concept of geometric symmetry of a linearly singular differential equation is introduced as a transformation that preserves the geometric data that define the problem it is proved that such symmetries are essentially equivalent to dynamic symmetries that is transformations mapping solutions into solutions similar results are given for infinitesimal symmetries to study the invariance of several objects under the flows of vector fields a careful study of infinitesimal variations is performed with a special emphasis on infinitesimal vector bundle automorphisms

application of quadrature rules for cauchy type integrals to the generalized poincare bertrand formula the classical poincare bertrand transposition formula for the inversion of the order of integration in repeated cauchy type integrals is generalized in accordance with a new interpretation of cauchy type integrals next the gauss jacobi quadrature rule is applied in a particular case of the generalized poincare bertrand formula to both members of this formula and it is proved that this formula still remains valid after the approximation of the integrals by quadrature sums two simple applications of this result one concerning the convergence of a quadrature rule for repeated cauchy type integrals and the other the numerical solution of singular integral equations are made further generalizations and applications of the present results follow easily

higher order conditions for singular lagrangian dynamics for a kth order singular lagrangian the presymplectic equation plus the different mth order differential equation conditions that can be considered in the lagrangian space yield different dynamics which are studied this leads to a new classification of the constraints this study is also performed in the intermediate formalisms which can be defined between the lagrangian and the hamiltonian ones the corresponding classification schemes are then related

a fast algorithm for solving the generalized airfoil equation abstract the present paper is devoted to the quadrature method for cauchy type singular integral equations on 1 1 with an additional logarithmic singularity of the kernel an algorithm being based on this method is given whose realization only requires a computational work of o n ln n operations in order to determine n unknown coefficients of the approximate solution furthermore in a scale of weighted sobolev like spaces on 1 1 having analogous properties like the classical sobolev spaces on the unit circle the order of convergence is optimal

higher order lagrangian systems geometric structures dynamics and constraints in order to study the connections between lagrangian and hamiltonian formalisms constructed from a perhaps singular higher order lagrangian some geometric structures are constructed intermediate spaces between those of lagrangian and hamiltonian formalisms partial ostrogradski s transformations and unambiguous evolution operators connecting these spaces are intrinsically defined and some of their properties studied equations of motion constraints and arbitrary functions of lagrangian and hamiltonian formalisms are thoroughly studied in particular all the lagrangian constraints are obtained from the hamiltonian ones once the gauge transformations are taken into account the true number of degrees of freedom is obtained both in the lagrangian and hamiltonian formalisms and also in all the intermediate formalisms herein defined

geometric formulation of mechanical systems subjected to time dependent one sided constraints a geometrical setting for time dependent impulsive constraints is given this description permits us to treat in a systematic way a wide family of mechanical systems subjected to time dependent constraints like moving walls etc if the impulsive constraints remain then an affine projector can be defined giving the new initial data in terms of the old ones several examples are discussed

the singular value decomposition of the gauss chebyshev and lobatto chebyshev methods for cauchy singular integral equations we obtain a closed form expression for the singular values and singular vectors of two matrices arising in the approximation of cauchy singular integral equations csie by the gauss chebyshev and lobatto chebyshev quadratures we also derive their singular value decomposition we use this decomposition to investigate the convergence of an iterative method for csie s proposed by ioakimidis an iterative algorithm for the numerical solution of singular integral equations j computational physics 43 164 176 1981 we show that this iterative method converges only under very restrictive conditions

a remark on the application of interpolatory quadrature rules to the numerical solution of singular integral equations abstract a cauchy type singular integral equation of the first or the second kind can be numerically solved either directly or after its reduction by the usual regularization procedure to an equivalent fredholm integral equation of the second kind the equivalence of these two methods that is the equivalence both of the systems of linear algebraic equations to which the singular integral equation is reduced and of the natural interpolation formulae is proved in this paper for a class of cauchy type singular integral equations of the first kind and of the second kind but with constant coefficients for general interpolatory quadrature rules under sufficiently mild assumptions the present results constitute an extension of a series of previous results concerning only gaussian quadrature rules based on the corresponding orthogonal polynomials and their properties

constrained hamiltonian systems as implicit differential equations generalized hamiltonian systems derived from degenerate lagrangians are presented as implicit differential equations rather than families of hamiltonian vector fields restricted to constraint subsets of the phase space systems considered are more general than the ones described by dirac they are generated by morse families of functions and not by constrained hamiltonians integrability of such systems is analysed examples are given