higher order first integrals in classical mechanics we present a practical algorithm for computing first integrals of motion which are polynomial in the momenta for natural hamiltonian systems defined in a flat pseudo riemannian space of arbitrary dimension and signature we then apply our algorithm to explore the integrability of two physical systems first we study the holt potential in two dimensions and derive analogous potentials which admit an additional first integral quartic in the momenta second we analyze a class of cylindrically symmetric potentials in three dimensional euclidean space and recover known families of second order maximally superintegrable systems

group invariant classification of separable hamiltonian systems in the euclidean plane and the o 4 symmetric yang mills theories of yatsun we present a new and effective method of determining separable coordinate systems for natural hamiltonians with two degrees of freedom in flat riemannian space the method is based on intrinsic properties of the associated killing tensors and their invariants under the group of rigid motions e 2 applications to the hamiltonian systems derived by the late v a yatsun from o 4 symmetric yang mills theories are presented in addition an equivalence between separability of two dimensional hamiltonian systems and the existence of pfaffian quasi bi hamiltonian representations is specified

superintegrable three body systems on the line we consider classical three body interactions on a euclidean line depending on the reciprocal distance of the particles and admitting four functionally independent quadratic in the momentum first integrals these systems are multiseparable superintegrable and equivalent up to rescalings to a one particle system in the three dimensional euclidean space common features of the dynamics are discussed we show how to determine quantum symmetry operators associated with the first integrals considered here but do not analyze the corresponding quantum dynamics the conformal multiseparability is discussed and examples of conformal first integrals are given the systems considered here in generality include the calogero wolfes and other three body interactions widely studied in mathematical physics

explicit h2 estimates and pointwise bounds for solutions of second order elliptic boundary value problems abstract it is well known that the h 2 norm and the c 0 norm of a function u h 2 where r n is a bounded domain n 3 can be estimated in terms of a given uniformly elliptic second order differential operator l and some boundary operator b applied to u if certain regularity assumptions are satisfied if these bounds shall be used for numerical purposes the constants occurring in the estimates must be known explicitly the main goal of the present article is the computation of such explicit constants for simplicity of presentation we restrict ourselves to the case where l u u c x u as an application we prove an existence and inclusion result for nonlinear boundary value problems

folner sequences and finite operators abstract this article analyzes folner sequences of projections for bounded linear operators and their relationship to the class of finite operators introduced by williams in the 70s we prove that each essentially hyponormal operator has a proper folner sequence i e an increasing folner sequence of projections strongly converging to 1 in particular any quasinormal any subnormal any hyponormal and any essentially normal operator has a proper folner sequence moreover we show that an operator is finite if and only if it has a proper folner sequence or if it has a non trivial finite dimensional reducing subspace we also analyze the structure of operators which have no folner sequence and give examples of them for this analysis we introduce the notion of strongly non folner operators which are far from finite block reducible operators in some uniform sense and show that this class coincides with the class of non finite operators

a numerical verification method for solutions of boundary value problems with local uniqueness by banach s fixed point theorem in this paper we propose a method to prove the existence and the local uniqueness of solutions to infinite dimensional fixed point equations using computers choosing a set which possibly includes a solution we transform it by an approximate linearization of the operator appearing in the equation then we calculate the radii of the transformed set in order to check sufficient conditions for banach s fixed point theorem this method is applied to elliptic problems and numerical examples are given

crosswind smear and pointwise errors in streamline diffusion finite element methods for a model convection dominated singularly perturbed convection diffusion prob lem it is shown that crosswind smear in the numerical streamline diffusion finite element method is minimized by introducing a judicious amount of artificial crosswind diffusion the ensuing method with piecewise linear elements converges with a pointwise accuracy of almost h 5 4 under local smoothness assumptions 1 introduction the streamline diffusion method is a finite element method for convection dominated convection diffusion problems which combines formal high accuracy with decent stability properties the method was introduced in the case of stationary problems by hughes and brooks 7 cf raithby and torrance 14 and wahlbin 15 for earlier thoughts in this direction the mathematical analysis of the method was started in johnson and navert 8 and continued with extensions to e g time dependent problems in navert 12 johnson navert and pitkaranta 9 and johnson and saranen 10 in these papers local error estimates in l2 of order o h k 1 2 in regions of smoothness with piecewise polynomial finite elements of degree k were derived together with estimates stating as a typical example that in the zero diffusion limit a sharp discontinuity in the exact solution across a streamline will be captured in a numerical crosswind layer of width 0 h1 2 essentially the purpose of the present paper is first to improve the result just mentioned on numerical crosswind smear to 0 h3 4 the improvement from 0 h1 2 to 0 h3 4 is obtained by adding a small amount 0 h3 2 of artificial crosswind diffusion to the method in the piecewise linear case k 1 this does not destroy the known o h3 2 accuracy in l2 in smooth regions using our first result we then obtain our second main result localized pointwise error estimates of order 0 h5 4 in regions of smoothness the previously known best pointwise error estimate in the piecewise linear situation is 0 h1 2 another consequence is a global l estimate of order 0 h 2 in the presence of typical crosswind and downwind singularities we shall consider the model problem of finding u u x y such that

a computer assisted proof for the kolmogorov flows of incompressible viscous fluid a computer assisted proof of non trivial steady state solutions for the kolmogorov flows is described the method is based on the infinite dimensional fixed point theorem using newton like operator this paper also proposes a numerical verification algorithm which generates automatically on a computer a set including the exact non trivial solution with local uniqueness all discussed numerical results take into account the effects of rounding errors in the floating point computations

the super separability of the three body inverse square calogero system the geometrical theory of the variable separation for the hamilton jacobi equation is applied to the classical three body inverse square calogero system it is proved that this system is separable in infinitely many inequivalent ways related to five different kinds of separable webs in the euclidean three space and the corresponding systems of independent first integrals in involution are computed

on computational proofs of the existence of solutions to nonlinear parabolic problems abstract this paper is an extension of the preceding study nakao this journal 1991 in which we described a numerical verification method of the solution for one space dimensional parabolic problems to the several space dimensional case here numerical verification means the automatic proof of the existence of solutions to the problems by some numerical techniques on a computer we reformulate the verification condition for nonlinear parabolic initial boundary value problems using the fixed point problem of a compact operator on certain function spaces as in the preceding study based upon a simple c 0 finite element approximation and its constructive a priori error estimates a numerical verification procedure is presented with some numerical examples

dynamical invariance algebra of the hartmann potential the accidental degeneracy occurring in the quantum mechanical treatment of the ring shaped potential v eta sigma 2r 1 1 2q eta 2 sigma 2 r sin theta 2 is explained by constructing an su 2 dynamical invariance algebra the schrodinger equation is solved in parabolic coordinates written in the framework of the kustaanheimo stiefel transformation and the hamilton jacobi equations are solved in ordinary parabolic coordinates all finite trajectories are found to be periodic

automatic differentiation for fourier series and the radii polynomial approach abstract in this work we develop a computer assisted technique for proving existence of periodic solutions of nonlinear differential equations with non polynomial nonlinearities we exploit ideas from the theory of automatic differentiation in order to formulate an augmented polynomial system we compute a numerical fourier expansion of the periodic orbit for the augmented system and prove the existence of a true solution nearby using an a posteriori validation scheme the radii polynomial approach the problems considered here are given in terms of locally analytic vector fields i e the field is analytic in a neighborhood of the periodic orbit hence the computer assisted proofs are formulated in a banach space of sequences satisfying a geometric decay condition in order to illustrate the use and utility of these ideas we implement a number of computer assisted existence proofs for periodic orbits of the planar circular restricted three body problem pcrtbp

an efficient numerical verification method for the kolmogorov problem of incompressible viscous fluid some computer assisted proofs of nontrivial steady state solutions for the kolmogorov flows are presented the method is based on the infinite dimensional fixed point theorem using a newton like operator with a numerical verification algorithm that automatically generates a set that includes the exact nontrivial solution when discussing the numerical results we consider the effects of rounding errors in the floating point computations this is a continuation of our study that was presented in watanabe 2009

second order superintegrable systems in conformally flat spaces i two dimensional classical structure theory this paper is part of a series that lays the groundwork for a structure and classification theory of second order superintegrable systems both classical and quantum in real or complex conformally flat spaces here we consider classical superintegrable systems with nondegenerate potentials in three dimensions we show that there exists a standard structure for such systems based on the algebra of 3 3 symmetric matrices and that the quadratic algebra always closes at order 6 we show that the spaces of truly second third fourth and sixth order constants of the motion are of dimension 6 4 21 and 56 respectively and we construct explicit bases for the fourth and sixth order constants in terms of products of the second order constants

superintegrability of the calogero moser system constants of motion master symmetries and time dependent symmetries the classical n dimensional calogero moser system is a maximally superintegrable system endowed with a rich variety of symmetries and constants of motion in the first part of the article some properties related with the existence of several families of constants of motion are analyzed in the second part the master symmetries and the time dependent symmetries of this system are studied

invariant classification of orthogonally separable hamiltonian systems in euclidean space the problem of the invariant classification of the orthogonal coordinate webs defined in euclidean space is solved within the framework of felix klein s erlangen program the results are applied to the problem of integrability of the calogero moser model

an extension of the classical theory of algebraic invariants to pseudo riemannian geometry and hamiltonian mechanics we develop a new approach to the study of killing tensors defined in pseudo riemannian spaces of constant curvature that is ideologically close to the classical theory of invariants the main idea which provides the foundation of the new approach is to treat a killing tensor as an algebraic object determined by a set of parameters of the corresponding vector space of killing tensors under the action of the isometry group the spaces of group invariants and conformal group invariants of valence two killing tensors defined in the minkowski plane are described the group invariants which are the generators of the space of invariants are applied to the problem of classification of orthogonally separable hamiltonian systems defined in the minkowski plane transformation formulas to separable coordinates expressed in terms of the parameters of the corresponding space of killing tensors are presented the results are applied to the problem of orthogonal separability of the drach superintegrable potentials

constructive a priori error estimates for a full discrete approximation of the heat equation in this paper we consider the constructive a priori error estimates for a full discrete numerical solution of the heat equation our method is based on the finite element galerkin method with an interpolation in time that uses the fundamental solution for semidiscretization in space the present estimates play an essential role in the numerical verification method of exact solutions for the nonlinear parabolic equations this implies that by utilizing the present results we could get the guaranteed a posteriori error estimates for various kinds of nonlinear evolutional problems our results can also be considered as an explicit optimal estimate with the limited regularity of solutions which should be unknown up to the present

global smooth solution curves using rigorous branch following in this paper we present a new method for rigorously computing smooth branches of zeros of nonlinear operators f l 1 b 1 l 2 b 2 where b 1 and b 2 are banach spaces the method is first introduced for parameter continuation and then generalized to pseudo arclength continuation examples in the context of ordinary partial and delay differential equations are given

hamiltonians separable in cartesian coordinates and third order integrals of motion we present in this article all hamiltonian systems in e 2 that are separable in cartesian coordinates and that admit a third order integral both in quantum and in classical mechanics many of these superintegrable systems are new and it is seen that there exists a relation between quantum superintegrable potentials invariant solutions of the korteweg de vries equation and the painleve transcendents

numerical method for verifying the existence and local uniqueness of a double turning point for a radially symmetric solution of the perturbed gelfand equation a numerical verification method to confirm the existence and local uniqueness of a double turning point for a radially symmetric solution of the perturbed gelfand equation is presented using certain systems of equations corresponding to a double turning point we derive a sufficient condition for its existence whose satisfaction can be verified computationally we describe verification procedures and give a numerical example as a demonstration

numerical existence and uniqueness proof for solutions of nonlinear hyperbolic equations abstract we consider a numerical method to verify the existence and uniqueness of the solutions of nonlinear hyperbolic problems with guaranteed error bounds using a c 1 finite element solution and an inequality constituting a bound on the norm of the inverse operator of the linearized operator we numerically construct a set of functions which satisfy the hypothesis of banach s fixed point theorem for a continuous map on l p space in a computer we present detailed verification procedures and give some numerical examples

c algebras generated by subnormal operators abstract this paper gives necessary and sufficient conditions on a hilbert space operator a in order that c a the c algebra generated by a be generated by an irreducible essentially normal subnormal operator several specialized applications of this result are given and methods of constructing irreducible subnormal operators are developed

operators with c algebra generated by a unilateral shift si t est un operateur sur un espace de hilbert on donne des conditions necessaires et suffisantes sur t pour que c t la c etoile algebre generee par t soit generee par un decalage unilateral d une certaine multiplicite

a numerical verification method for the existence of weak solutions for nonlinear boundary value problems abstract in this paper we consider a numerical technique which enables us to verify the existence of solutions for nonlinear two point boundary value problems bvp we formulate the problem as a fixed point of a newton like operator and present a verification algorithm by computer based on sadovskii s fixed point theorem in order to compute the newton type iterative sequences in infinite dimensional space two concepts the rounding and the rounding error are introduced also some numerical examples are illustrated we emphasize that since the present method uses the weak formulation of bvp and the finite element approximations instead of the existing validation techniques it is aslo applicable in principle to the bvp for partial differential equations as it stands

quantum superintegrability and exact solvability in n dimensions a family of maximally superintegrable systems containing the coulomb atom as a special case is constructed in n dimensional euclidean space two different sets of n commuting second order operators are found overlapping in the hamiltonian alone the system is separable in several coordinate systems and is shown to be exactly solvable it is solved in terms of classical orthogonal polynomials the hamiltonian and n further operators are shown to lie in the enveloping algebra of a hidden affine lie algebra

subnormal and hyponormal generators of c algebras for a hilbert space operator a that is not a normal operator we give some necessary conditions on the thickness of the spectrum of a for c a to have a subnormal generator when a is irreducible and essentially normal whether or not c a has a subnormal generator depends only on the spectral picture of a we show that under certain conditions on the essential spectrum and fredholm index function of a that c a has a subnormal or a hyponormal generator

existence and enclosure results for continua of solutions of parameter dependent nonlinear boundary value problems we propose a numerical method for computing enclosures for continua of solutions of nonlinear elliptic boundary value problems depending on a real parameter at the same time proving the existence of a smooth solution continuum the method is extended to turning point problems by change of parameters

numerical verification of solutions for variational inequalities of the second kind abstract the purpose of this paper is to present an approach to the numerical proof of existence of solutions for the problem of the flow of a viscous plastic fluid in a pipe using the finite element approximations and the explicit a priori error estimates for the problem of the flow of a viscous plastic fluid in a pipe we present an effective verification procedure that through numerical computation generates a set which includes the exact solution further a numerical example is presented

exact solvability of superintegrable systems it is shown that all four superintegrable quantum systems on the euclidean plane possess the same underlying hidden algebra sl 3 the gauge rotated hamiltonians as well as their integrals of motion once rewritten in appropriate coordinates preserve a flag of polynomials this flag corresponds to highest weight finite dimensional representations of the sl 3 algebra realized by first order differential operators