discretization on non orthogonal quadrilateral grids for inhomogeneous anisotropic media two classes of discretization methods are proposed for control volume formulations on quadrilateral grids in two space dimensions curvilinear grids are considered as a special case the methods are applicable for any system of conservation laws where the flux is defined by a gradient law like darcy s law for porous media flow a strong feature of the methods is the ability to handle media inhomogeneities in combination with full tensor anisotropy and or non orthogonality of the grid further properties of the methods will be discussed and examples of their use will be presented

black box multigrid for nonsymmetric problems one major problem with the multigrid method has been that each new grid configuration has required a major programming effort to develop a code that specifically handles that grid configuration such a penalty is not required for methods like sor iccg etc in these methods one need only specify the matrix problem no matter what the grid configuration in previous work this defect was overcome and a code written in which one needed only to specify the logically rectangular positive definite matrix problem the code did everything else necessary to set up the auxiliary coarser problems to achieve a multigrid solution in this paper the previous work is extended to a single nonsymmetric equation

on the master symmetries related to certain classes of integrable hamiltonian systems we present the complete classification of master symmetries related to a non degenerate hamiltonian system that is integrable in the arnol d liouville s sense it is shown that a integrable hamiltonian system cannot have generators of degree greater than 2 specific properties of the master symmetries classified in terms of the action angle variables are investigated

the numerical solution of diffusion problems in strongly heterogeneous non isotropic materials a new second order finite difference algorithm for the numerical solution of diffusion problems in strongly heterogeneous and non isotropic media is constructed on problems with rough coefficients or highly nonuniform grids the new algorithm is superior to all other algorithms we have compared it with for problems with smooth coefficients on smooth grids the method is comparable with other second order methods the new algorithm is formulated for logically rectangular grids and is derived using thesupport operators method a key idea in deriving the method was to replace the usual inner product of vector functions by an inner product weighted by the inverse of the material properties tensor and to use the flux operator defined as the material properties tensor times the gradient rather than the gradient as one of the basic first order operators in the support operators method the discrete analog of the flux operator must also be the negative adjoint of the discrete divergence in an inner product that is a discrete analog of the continuum inner product the resulting method is conservative and the discrete analog of the variable coefficient laplacian is symmetric and negative definite on nonuniform grids in addition on any grid the discrete divergence is zero on constant vectors the null space for the gradient is the constant functions and when the material properties are piecewise constant the discrete flux operator is exact for piecewise linear functions we compare the methods on some of the most difficult examples to be found in the literature

classical and quantum superintegrability with applications a superintegrable system is roughly speaking a system that allows more integrals of motion than degrees of freedom this review is devoted to finite dimensional classical and quantum superintegrable systems with scalar potentials and integrals of motion that are polynomials in the momenta we present a classification of second order superintegrable systems in two dimensional riemannian and pseudo riemannian spaces it is based on the study of the quadratic algebras of the integrals of motion and on the equivalence of different systems under coupling constant metamorphosis the determining equations for the existence of integrals of motion of arbitrary order in real euclidean space e2 are presented and partially solved for the case of third order integrals a systematic exposition is given of systems in two and higher dimensional space that allow integrals of arbitrary order the algebras of integrals of motions are not necessarily quadratic but close polynomially or rationally the relation between superintegrability and the classification of orthogonal polynomials is analyzed

universal spectral form factor for chaotic dynamics we consider the semiclassical limit of the spectral form factor k of fully chaotic dynamics starting from the gutzwiller type double sum over classical periodic orbits we set out to recover the universal behaviour predicted by random matrix theory both for dynamics with and without time reversal invariance for times smaller than half the heisenberg time th f 1 we extend the previously known expansion to include the cubic term beyond confirming the random matrix behaviour of individual spectra the virtue of that extension is that the diagrammatic rules come in sight which determine the families of orbit pairs responsible for all orders of the expansion

incomplete beta function expansions of the solutions to the confluent heun equation several expansions of the solutions to the confluent heun equation in terms of incomplete beta functions are constructed a new type of expansion involving certain combinations of the incomplete beta functions as expansion functions is introduced the necessary and sufficient conditions when the derived expansions are terminated thus generating closed form solutions are discussed it is shown that termination of a beta function series solution always leads to a solution that is necessarily an elementary function

calogero sutherland eigenfunctions with mixed boundary conditions and conformal field theory correlators we construct certain eigenfunctions of the calogero sutherland hamiltonian for particles on a circle with mixed boundary conditions that is the behavior of the eigenfunction as neighbouring particles collide depend on the pair of colliding particles this behavior is generically a linear combination of two types of power laws depending on the statistics of the particles involved for fixed ratio of each type at each pair of neighboring particles there is an eigenfunction the ground state with lowest energy and there is a discrete set of eigenstates and eigenvalues the excited states and the energies above this ground state we find the ground state and special excited states along with their energies in a certain class of mixed boundary conditions interpreted as having pairs of neighboring bosons and other particles being fermions these particular eigenfunctions are characterised by the fact that they are in direct correspondence with correlation functions in boundary conformal field theory we expect that they have applications to measures on certain configurations of curves in the statistical o n loop model the derivation although completely independent from results of conformal field theory uses ideas from the coulomb gas formulation

extended integrability and bi hamiltonian systems the current notion of integrability of hamiltonian systems was fixed by liouville in a famous 1855 paper it describes systems in a 2k dimensional phase space whose trajectories are dense on tori q or wind on toroidal cylinders m q m within liouville s construction the dimension q cannot exceed k and is the main invariant of the system in this paper we generalize liouville integrability so that trajectories can be dense on tori k of arbitrary dimensions q 1 2k 1 2k and an additional invariant v 2 q k v 2 q 2 can be recovered the main theorem classifies all k k 1 2 canonical forms of hamiltonian systems that are integrable in a newly defined broad sense an integrable physical problem having engineering origin is presented the notion of extended compatibility of two poisson structures is introduced the corresponding bi hamiltonian systems are shown to be integrable in the broad sense

theory of tensor invariants of integrable hamiltonian systems i incompatible poisson structures this paper develops a new theory of tensor invariants of a completely integrable non degenerate hamiltonian system on a smooth manifoldmn the central objects in this theory are supplementary invariant poisson structurespc which are incompatable with the original poisson structurep1 for this hamiltonian system a complete classification of invariant poisson structures is derived in a neighbourhood of an invariant toroidal domain this classification resolves the well known inverse problem that was brought into prominence by magri s 1978 paper deveoted to the theory of compatible poisson structures applications connected with the kam theory with the kepler problem with the basic integrable problem of celestial mechanics and with the harmonic oscillator are pointed out a cohomology is defined for dynamical systems on smooth manifolds the physically motivated concepts of dynamical compatibility and strong dynamical compatibility of pairs of poisson structures are introduced to study the diversity of pairs of poisson structures incompatible in magri s sense it is proved that if a dynamical systemv preserves two strongly dynamically compatible poisson structuresp1 andp2 in a general position then this system is completely integrable such a systemv generates a hierarchy of integrable dynamical systems which in general are not hamiltonian neither with respect top1 nor with respect top2 necessary conditions for dynamical compatibility and for strong dynamical compatibility are derived which connect these global properties with new local invariants of an arbitrary pair of incompatible poisson structures

m matrix flux splitting for general full tensor discretization operators on structured and unstructured grids diagonal tensor flux approximations are commonly used in fluid dynamics this approximation introduces an o 1 error in flux whenever the coordinate system is nonaligned with the principal axes of the tensor which is particularly common when employing curvilinear gridding in general a consistent full tensor flux approximation leads to a significant increase in support and consequent size of the jacobian matrix after decomposition of a general full tensor flux into a diagonal tensor flux together with cross terms time split semi implicit stable full tensor flux approximations are introduced with in a general finite volume formalism enabling the standard diagonal tensor jacobian matrix structure to be retained for single phase flow impes and standard block fully implicit formulations while ensuring spatial consistency of the discretization for both structured and unstructured grids stability of the scheme is proven for constant elliptic coefficients the results presented demonstrate the benefits of the method for multiphase flow within a fully implicit framework on structured and unstructured grids

analytical results for a modified trigonometric razavy potential we have presented analytical results for the modified trigonometric razavy potential with an additional relative phase between their components it is shown that the analytical solutions can be given in terms of the heun double confluent functions based on the properties of the heun double confluent functions the transfer matrix has been constructed to compute energy bands

sub riemannian geometry of the coefficients of univalent functions abstract we consider coefficient bodies m n for univalent functions based on the lowner kufarev parametric representation we get a partially integrable hamiltonian system in which the first integrals are kirillov s operators for a representation of the virasoro algebra then m n are defined as sub riemannian manifolds given a lie poisson bracket they form a grading of subspaces with the first subspace as a bracket generating distribution of complex dimension two with this sub riemannian structure we construct a new hamiltonian system to calculate regular geodesics which turn to be horizontal lagrangian formulation is also given in the particular case m 3

analysis of quantum semigroups with gks lindblad generators i simple generators semigroups describing the time evolution of open quantum systems in finite dimensional spaces have generators of a special form known as lindblad generators the simple generators characterized by only one operator are analyzed the complete set of all the stationary states is presented in detail including a formula to calculate a stationary state from the generating operator also the opposite task can be fulfilled to construct an evolution leading to a prescribed stationary state

matrix dependent prolongations and restrictions in a blackbox multigrid solver multigrid methods are studied for the solution of linear systems resulting from the 9 point discretization of a general linear second order elliptic partial differential equation in two dimensions the rate of convergence of standard multigrid methods often deteriorates when the coefficients in the differential equation are discontinuous or when dominating first order terms are present these difficulties may be overcome by choosing the prolongation and restriction operators in a special way a novel way to do this is proposed as a result a blackbox solver written in standard fortran 77 has been developed numerical experiments for several hard test problems are described and comparison is made with other algorithms the standard mg method and a method introduced by kettler a significant improvement of robustness and efficiency is found

new quasi exactly solvable double well potentials a new three parameter family of quasi exactly solvable double well potentials is introduced we show that the solutions of this family of double well potentials are expressed in terms of the heun confluent functions under a certain parameter condition some of the bound state wavefunctions and associated energies can be found exactly in explicit form in particular we develop an analytical method to derive the conditions for the energy eigenvalues of the bound states it is also shown that our analytical results can be applied to construct exact solutions of the nonlinear schrodinger equation with double well potentials and spatially localized nonlinearities

sandpile model on an optimized scale free network on euclidean space deterministic sandpile models are studied on a cost optimized barabasi albert ba scale free network whose nodes are the sites of a square lattice for the optimized ba network the sandpile model has the same critical behaviour as the btw sandpile whereas for the un optimized ba network the critical behaviour is mean field like

on finite element methods for heterogeneous elliptic problems dealing with variational formulations of second order elliptic problems with discontinuous coefficients we recall a single field minimization problem of an extended functional presented by bevilacqua et al bevilacqua l feijoo r a rojas l f 1974 a variational principle for the laplace operator with application in the torsion of composite rods international journal of solids structuring 10 1091 1102 which we associate with the basic idea supporting discontinuous galerkin finite element methods we review residual based stabilized mixed methods applied to darcy flow in homogeneous porous media and extend them to heterogeneous media with an interface of discontinuity for smooth interfaces the proposed formulations preserve the continuity of the flux and exactly imposes the constraint between the tangent components of darcy velocity on the interface convergence studies for a heterogeneous and anisotropic porous medium confirm the same rates of convergence predicted for homogeneous problem with smooth solutions

a quasi positive family of continuous darcy flux finite volume schemes with full pressure support a new family of flux continuous locally conservative finite volume schemes is presented for solving the general tensor pressure equation of subsurface flow in porous media the new schemes have full pressure continuity imposed across control volume faces previous families of flux continuous schemes are point wise continuous in pressure and flux when applying the earlier point wise flux continuous schemes to strongly anisotropic full tensor fields their failure to satisfy a maximum principle as with other fem and finite volume methods can result in loss of local stability for high anisotropy ratios which can cause strong spurious oscillations in the numerical pressure solution an m matrix analysis reveals the upper limits for guaranteeing a maximum principle for general 9 point schemes and aids in the design of schemes that minimize the occurrence of spurious oscillations in the discrete pressure field the full pressure continuity schemes are shown to possess a larger range of flux continuous schemes than the previous point wise counter parts for strongly anisotropic full tensor cases it is shown that the full quadrature range possessed by the new schemes permits these schemes to exploit quadrature points previously out of range that are shown to minimize spurious oscillations in discrete pressure solutions the new formulation leads to a more robust quasi positive family of flux continuous schemes applicable to general discontinuous full tensor fields

vertex centered and cell centered multigrid for interface problems abstract cell centered and vertex centered multigrid methods for solving interface problems are studied these methods differ in the location of the nodes in the grids and in the transfer operators it is shown how by means of stencil notation a compact and precise description can be given of the transfer and coarse grid operators a structured fortran description of the fundamental multigrid algorithm with only one goto statement is presented numerical results of several test problems with strong discontinuities in equation coefficients are presented storage and work requirements are discussed

thomas rotation and the mixed state geometric phase it is shown that uhlmann s parallel transport of purifications along a path of mixed states represented by 2 2 density matrices is just the path ordered product of thomas rotations these rotations are invariant under hyperbolic translations inside the bloch sphere that can be regarded as the poincare ball model of hyperbolic geometry a general expression for the mixed state geometric phase for an arbitrary geodesic triangle in terms of the bures fidelities is derived the formula gives back the solid angle result well known from studies of the pure state geometric phase it is also shown that this mixed state anholonomy can be reinterpreted as the pure state non abelian anholonomy of entangled states living in a suitable restriction of the quaternionic hopf bundle in this picture uhlmann s parallel transport is just the pancharatnam transport of quaternionic spinors

univalent functions and integrable systems we study one parameter expanding evolution families of simply connected domains in the complex plane described by infinite systems of evolution parameters these evolution parameters in some cases admit hamiltonian formulation and lead to integrable systems one example of such parameters is complex moments for the laplacian growth that form a whitham toda integrable hierarchy another example we deal with is related to expanding coefficient bodies for conformal maps given by lowner subordination chains the coefficients bodies are proved to form a liouville partially integrable hamiltonian system for each fixed index and the first integrals are obtained we also discuss the contact structure of this system

cell centered multigrid for interface problems abstract a multigrid method is presented for cell centered discretizations of elliptic partial differential equations the method works both for smooth and strongly discontinuous coefficients even though in contrast with earlier works the prolongation and restriction operators do not depend on the equation

a finite volume method for approximating 3d diffusion operators on general meshes a finite volume method is presented for discretizing 3d diffusion operators with variable full tensor coefficients this method handles anisotropic non symmetric or discontinuous variable tensor coefficients while distorted non matching or non convex n faced polyhedron meshes can be used for meshes of polyhedra whose faces have not more than four edges the associated matrix is positive definite and symmetric if the diffusion tensor is symmetric a second order resp first order accuracy is numerically observed for the solution resp gradient of the solution

local flux mimetic finite difference methods we develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients on polyhedral meshes to approximate the velocity vector variable the method uses two degrees of freedom per element edge in two dimensions and n degrees of freedom per n gonal mesh face in three dimensions to approximate the pressure scalar variable the method uses one degree of freedom per element a specially chosen quadrature rule for the l 2 product of vector functions allows for a local flux elimination and reduction of the method to a cell centered finite difference scheme for the pressure unknowns under certain assumptions first order convergence is proved for both variables and second order convergence is proved for the pressure the assumptions are verified on simplicial meshes for a particular quadrature rule that leads to a symmetric method for general polyhedral meshes non symmetric methods are constructed based on quadrature rules that are shown to satisfy some of the assumptions numerical results confirm the theory

a systematic study on the exact solution of the position dependent mass schrodinger equation an algebraic method of constructing potentials for which the schrodinger equation with position dependent mass can be solved exactly is presented a general form of the generators of su 1 1 algebra has been employed with a unified approach to the problem our systematic approach reproduces a number of earlier results and also leads to some novelties we show that the solutions of the schrodinger equation with position dependent mass are free from the choice of parameters for position dependent mass two classes of potentials are constructed that include almost all exactly solvable potentials

about the existence of recursion operators for completely integrable hamiltonian systems near a liouville torus it is shown that the existence of a recursion operator near a liouville torus of a completely integrable hamiltonian system is not always satisfied and that such an existence implies very strong conditions on the hamiltonian and on the actions

hierarchical dobinski type relations via substitution and the moment problem we consider the transformation properties of integer sequences arising from the normal ordering of exponentiated boson a a 1 monomials of the form exp a ra r 1 2 under the composition of their exponential generating functions they turn out to be of sheffer type we demonstrate that two key properties of these sequences remain preserved under substitutional composition a the property of being the solution of the stieltjes moment problem and b the representation of these sequences through infinite series dobinski type relations we present a number of examples of such composition satisfying properties a and b we obtain new dobinski type formulae and solve the associated moment problem for several hierarchically defined combinatorial families of sequences

the effect of pitchfork bifurcations on the spectral statistics of hamiltonian systems we present a quantitative semiclassical treatment of the effects of bifurcations on the spectral rigidity and the spectral form factor of a hamiltonian quantum system defined by two coupled quartic oscillators which on the classical level exhibits mixed phase space dynamics we show that the signature of a pitchfork bifurcation is two fold beside the known effect of an enhanced periodic orbit contribution due to its peculiar hbar dependence at the bifurcation we demonstrate that the orbit pair born at the bifurcation gives rise to distinct deviations from universality slightly above the bifurcation this requires a semiclassical treatment beyond the so called diagonal approximation our semiclassical predictions for both the coarse grained density of states and the spectral rigidity are in excellent agreement with corresponding quantum mechanical results

discretization on unstructured grids for inhomogeneous anisotropic media part i derivation of the methods discretization methods are proposed for control volume formulations on polygonal and triangular grid cells in two space dimensions the methods are applicable for any system of conservation laws where the flow density is defined by a gradient law like darcy s law for porous media flow a strong feature of the methods is the ability to handle media inhomogeneities in combination with full tensor anisotropy this paper gives a derivation of the methods and the relation to previously published methods is also discussed a further discussion of the methods including numerical examples is given in the companion paper part ii siam j sci comput pp 1717 1736

geometrical interpretation of the back propagation algorithm for the perceptron abstract it is well known that the gradient descent algorithm works well for the perceptron when the solution to the perceptron problem exists because the cost function has a simple shape with just one minimum in the conjugate weight space working in the conjugate space we show that if a perceptron solution does not exist the cost function of a perceptron with d inputs and n patterns has an average o n d relative minima for large n in this case finding the best solution the solution with the minimum number of errors becomes a challenging problem for any local search algorithm

the three dimensional noncommutative gross neveu model this work is dedicated to the study of the noncommutative gross neveu model as is known in the canonical weyl moyal approach the model is inconsistent basically due to the separation of the amplitudes into planar and nonplanar parts we prove that if instead a coherent basis representation is used the model becomes renormalizable and free of the aforementioned difficulty we also show that although the coherent state procedure breaks lorentz symmetry in odd dimensions in the gross neveu model this breaking can be kept under control by assuming the noncommutativity parameters to be small enough we also make some remarks on ordering prescriptions used in the literature

hamiltonian time dependent mechanics the usual formulation of time dependent mechanics implies a given splitting y r m of an event space y this splitting however is broken by any time dependent transformation including transformations between inertial frames the goal is the frame covariant formulation of time dependent mechanics on a bundle y r whose fibration y m is not fixed its phase space is the vertical cotangent bundle v y provided with the canonical 3 form and the corresponding canonical poisson structure an event space of relativistic mechanics is a manifold z whose fibration z r is not fixed

boundary critical behaviour at m axial lifshitz points the special transition for the case of a surface plane parallel to the modulation axes the critical behaviour of d dimensional semi infinite systems with n component order parameter is studied at an m axial bulk lifshitz point whose wave vector instability is isotropic in an m dimensional subspace of r d field theoretic renormalization group methods are utilized to examine the special surface transition in the case where the m potential modulation axes with 0 m d 1 are parallel to the surface the resulting scaling laws for the surface critical indices are given the surface critical exponent sp t he

the pointwise product in weyl quantization we study the product of bracken 1 which is the weyl quantized version of the pointwise product of functions in phase space we prove that it is not compatible with the algebras of finite rank and hilbert schmidt operators by solving the linearization problem for the special hermite functions we are able to express the product in terms of the component operators mediated by the linearization coefficients this is applied to finite rank operators and their matrices and operators whose symbols are radial and angular distributions

four electrons in a two leg hubbard ladder exact ground states in the case of a two leg hubbard ladder we present a procedure which allows the exact deduction of the ground state for the four particle problem in an arbitrary large lattice system in a tractable manner which involves only a reduced hilbert space region containing the ground state in the presented case the method leads to nine analytic linear and coupled equations providing the ground state the procedure which is also applicable to few particle problems and other systems is based on an r space representation of the wavefunctions and construction of symmetry adapted orthogonal basis wave vectors describing the hilbert space region containing the ground state once the ground state is deduced a complete quantum mechanical characterization of the studied state can be given since the analytic structure of the ground state becomes visible during the use of the method it is important not only to the understanding of theoretical aspects connected to exact descriptions or potential numerical approximation scheme developments but is also relevant for a large number of potential technological application possibilities placed between nano devices and quantum calculations where the few particle behaviour and deep understanding are important key aspects

dobinski type relations some properties and physical applications we introduce a generalization of the dobinski relation through which we define a family of bell type numbers and polynomials for all these sequences we find the weight function of the moment problem and give their generating functions we provide a physical motivation of this extension in the context of the boson normal ordering problem and its relation to an extension of the kerr hamiltonian

l2 series solutions of the dirac equation for power law potentials at rest mass energy we obtain solutions of the three dimensional dirac equation for radial power law potentials at rest mass energy as an infinite series of square integrable functions these are written in terms of the confluent hypergeometric function and chosen such that the matrix representation of the dirac operator is tridiagonal the wave equation results in a three term recursion relation for the expansion coefficients of the spinor wavefunction which is solved in terms of orthogonal polynomials these are modified versions of the meixner pollaczek polynomials and of the continuous dual hahn polynomials the choice depends on the values of the angular momentum and the power of the potential

a petrov galerkin finite element method for interface problems arising in sensitivity computations continuous sensitivity equation methods have been applied to a variety of applications ranging from optimal design to fast algorithms in computational fluid dynamics to the quantification of uncertainty in order to make use of these methods for interface problems one needs fast and accurate numerical methods for computing sensitivities for problems defined by partial differential equations with solutions that have spatial discontinuities such as shocks and interfaces in this paper we develop a discontinuous petrov galerkin finite element scheme for solving the sensitivity equation resulting from a 1d interface problem the 1d example is sufficient to motivate the theoretical and computational issues that arise when one derives the corresponding boundary value problem for the sensitivities in particular the sensitivity boundary value problem must be formulated in a very weak sense and the resulting variational problem provides a natural framework for developing and analyzing numerical schemes numerical examples are presented to illustrate the benefits of this approach

submersions fibrations and bundles when does a submersion have the homotopy lifting property when is it a locally trivial fibre bundle we establish characterizations in terms of consistency in the topology of the neighbouring fibres

maximal superintegrability of benenti systems for a class of hamiltonian systems naturally arising in the modern theory of separation of variables we establish their maximal superintegrability by explicitly constructing the additional integrals of motion

an analysis of conservative finite difference schemes for differential equations with discontinuous coefficients abstract a class of monotone conservative schemes is derived for the boundary value problem related to the sturm liouville operator au k x u x q x u x with discontinuous coefficient k k x the discrete analogous of the law of conservation are compared for the finite element and finite difference approaches in the class of discontinuous coefficients the necessary condition for conservativeness of the finite difference scheme is derived the obtained one parametric family of conservative schemes permits one to construct new conservative schemes the examples presented for different discontinuous coefficients and results show how the conservativeness conditions need to be taken into account in numerical solving boundary value problems

the liouville arnold nekhoroshev theorem for non compact invariant manifolds under certain conditions generalized action angle coordinates can be introduced near non compact invariant manifolds of completely and partially integrable hamiltonian systems

new quasi exactly solvable periodic potentials we present two new classes of quasi exactly solvable periodic potentials it is shown that their solutions are given in terms of a known special function when the parameters in the two periodic potentials satisfy a certain condition some of the bloch waves and associated energies can be found exactly and in closed form at either the center or the edge of the first brillouin zone two fundamental sets of solutions have been constructed to calculate all the bloch waves and energy bands

bi hamiltonian partially integrable systems given a first order dynamical system possessing a commutative algebra of dynamical symmetries we show that under certain conditions there exists a poisson structure on an open neighborhood of its regular not necessarily compact invariant manifold which makes this dynamical system into a partially integrable hamiltonian system this poisson structure is by no means unique bi hamiltonian partially integrable systems are described in some detail as an outcome we state the conditions of quasiperiodic stability the kam theorem for partially integrable hamiltonian systems

on the variation in the cohomology of the symplectic form of the reduced phase space is called the momentum mapping of the hamiltonian t action given 1 1 the condition 1 2 just means that t acts along the fibers of j for the basic definitions and properties of non commutative hamiltonian group actions see am the results of this paper can easily be extended to hamiltonian actions of arbitrary compact connected lie groups by applying our results to the action of its maximal toms and using the equivariance of the momentum mapping for some more details see the remarks at the end of sect 2 we will assume throughout this paper that the momentum map is proper that is j i u is compact for each compact subset u of t now let r be a regular value of j that is t j tmm t is surjective for all meye j 1 4 then ye is a smooth submanifold of m compact because

convex hull of n planar brownian paths an exact formula for the average number of edges we establish an exact formula for the average number of edges appearing on the boundary of the global convex hull of n independent brownian paths in the plane this requires the introduction of a counting criterion which amounts to cutting off edges that are in a specific sense small the main argument consists in a mapping between planar brownian convex hulls and configurations of constrained independent linear brownian motions this new formula is confirmed by retrieving an existing exact result on the average perimeter of the boundary of brownian convex hulls in the plane

multi grid solution of three dimensional problems with discontinuous coefficients a three dimensional multi grid algorithm is developed and compared to an efficient three dimensional incomplete cholesky conjugate gradient iccg algorithm on a suite of test problems with discontinouous and or anisotropic coefficients the multi grid algorithm used plane relaxation and a definition of coarse grid operators which is an extension of that used for two dimensional problems with discontinuous coefficients numerical results show that the three dimensional results are not nearly as good as the two dimensional results on similar problems

integrable potentials on spaces with curvature from quantum groups a family of classical integrable systems defined on a deformation of the two dimensional sphere hyperbolic and anti de sitter spaces is constructed through hamiltonians defined on the non standard quantum deformation of a sl 2 poisson coalgebra all these spaces have a non constant curvature that depends on the deformation parameter z as particular cases the analogues of the harmonic oscillator and kepler coulomb potentials on such spaces are proposed another deformed hamiltonian is also shown to provide superintegrable systems on the usual sphere hyperbolic and anti de sitter spaces with a constant curvature that exactly coincides with z according to each specific space the resulting potential is interpreted as the superposition of a central harmonic oscillator with either two more oscillators or centrifugal barriers the non deformed limit z 0 of all these hamiltonians can then be regarded as the zero curvature limit contraction which leads to the corresponding super integrable systems on the flat euclidean and minkowskian spaces

some notes on perceptron learning the author extends the geometrical approach to the perceptron and shows that given n examples learning is of maximal difficulty when the number of inputs d is such that n 5d he then presents a new perceptron algorithm that takes advantage of the peculiarities of the cost function in his tests it is more than two times faster than the standard algorithm more importantly it does not have fixed parameters like the usual learning constant eta but it adapts them to the cost function he shows that there exist an optimal choice for beta the steepness of the transfer function he presents also a brief systematic study of the parameters eta and beta of the standard perceptron algorithm

semiclassical approach to parametric spectral correlation with spin 1 2 the spectral correlation of a chaotic system with spin 1 2 is universally described by the gse gaussian symplectic ensemble of random matrices in the semiclassical limit in semiclassical theory the spectral form factor is expressed in terms of the periodic orbits and the spin state is simulated by the uniform distribution on a sphere in this paper instead of the uniform distribution we introduce brownian motion on a sphere to yield the parametric motion of the energy levels as a result the small time expansion of the form factor is obtained and found to be in agreement with the prediction of parametric random matrices in the transition within the gse universality class moreover by starting the brownian motion from a point distribution on the sphere we gradually increase the effect of the spin and calculate the form factor describing the transition from the gaussian orthogonal ensemble class to the gse class

critical phenomena at perfect and non perfect surfaces the effect of imperfections on surface critical properties is studied for ising models with nearest neighbour ferromagnetic couplings on simple cubic lattices in particular results of monte carlo simulations for flat perfect surfaces are compared to those for flat surfaces with random weak or strong interactions between neighbouring spins in the surface layer and for surfaces with steps of monoatomic height surface critical exponents at the ordinary transition in particular are found to be robust against these perturbations

intermediate statistics for a system with symplectic symmetry the dirac rose graph we study the spectral statistics of the dirac operator on a rose shaped graph a graph with a single vertex and all bonds connected at both ends to the vertex we formulate a secular equation that generically determines the eigenvalues of the dirac rose graph which is seen to generalize the secular equation for a star graph with neumann boundary conditions we derive approximations to the spectral pair correlation function at large and small values of spectral spacings in the limit as the number of bonds approaches infinity and compare these predictions with results of numerical calculations our results represent the first example of intermediate statistics from the symplectic symmetry class

boundary critical behaviour at m axial lifshitz points of semi infinite systems with a surface plane perpendicular to a modulation axis semi infinite d dimensional systems with an m axial bulk lifshitz point are considered whose d 1 dimensional surface hyper plane is oriented perpendicular to one of the m modulation axes an n component 4 field theory describing the bulk and boundary critical behaviour when a the hamiltonian can be taken to have o n symmetry and b spatial anisotropies breaking its euclidean symmetry in the m dimensional coordinate subspace of potential modulation directions may be ignored is investigated the long distance behaviour at the ordinary surface transition is mapped onto a field theory with the boundary conditions that both the order parameter and its normal derivative n vanish at the surface plane the boundary operator expansion is utilized to study the short distance behaviour of near the surface its leading contribution is found to be controlled by the boundary operator 2 n the field theory is renormalized for dimensions d below the upper critical dimension d m 4 m 2 with a corresponding surface source term 2 n added the anomalous dimension of this boundary operator is computed to first order in d d the result is used in conjunction with scaling laws to estimate the value of the single independent surface critical exponent ord l1 for d 3 our estimate for the case m n 1 of a uniaxial lifshitz point in ising systems is in reasonable agreement with the published monte carlo results

discretisation and multigrid solution of elliptic equations with mixed derivative terms and strongly discontinuous coefficients this paper introduces a new conservative cell centred finite volume scheme for the accurate solution of elliptic diffusion equations with strongly varying coefficients the discretisation is designed to model a wide range of heterogeneities and anisotropies and in particular the case where the diffusivity is represented by a nondiagonal matrix which may occur if the medium is anisotropic in a general direction such problems arise for example in oil reservoir simulation when renormalisation techniques are used to model the reservoir geology it is in this context that the method is described although it is applicable to a far wider class of problems in heat conduction electrostatics and potential theory the paper also describes the application of a mulitigrid scheme for efficient numerical solution of the algebraic equations derived using the new discretisation

on linear separability of random subsets of hypercube vertices the classical cover results on linear separability of points in rd are a milestone in neural network theory nevertheless they are not valid for digital input networks because in this case the points are not in general position vertices ofa d dimensional hypercube i show here that for large d all cover findings can be extended to this case it is also shown that for n o d i the number of linear separations of n random hypercube vertices tends to that of n points in general position

noncommutative integrability on noncompact invariant manifolds the mishchenko fomenko theorem on noncommutative completely integrable hamiltonian systems on a symplectic manifold is extended to the case of noncompact invariant submanifolds

geometric quantization of time dependent completely integrable hamiltonian systems a time dependent completely integrable hamiltonian system is quantized with respect to time dependent action angle variables near an instantly compact regular invariant manifold its hamiltonian depends only on action variables and has a time independent countable energy spectrum

unstructured grid optimization for improved monotonicity of discrete solutions of elliptic equations with highly anisotropic coefficients multipoint flux approximation mpfa techniques are commonly applied for discretizing the porous media flow equations within the context of finite volume numerical procedures although these methods can be applied to heterogeneous anisotropic systems on generally unstructured grids the inverse of the resulting linear operator can suffer from a loss of monotonicity at high permeability anisotropy ratios resulting in spurious oscillations of the pressure solution the purpose of this paper is to develop a method for optimizing unstructured grids in two and three dimensions such that the monotonicity behavior of the mpfa technique is significantly improved the method employs anisotropic triangulation and can be readily combined with permeability upscaling procedures results are presented for a variety of examples and the technique is shown to perform well on problems involving complex grid point distributions and heterogeneous permeability fields with strong anisotropy ratios of o 100 oscillation free pressure solutions are achieved in all cases considered even for examples in which the original grid shows large oscillations

isochronic potentials and new family of superintegrable systems a general scheme for constructing superintegrable systems for separated hamiltonians in an arbitrary number of degrees of freedom is presented the resulting family contains previously known superintgrable systems with separated hamiltonians in cartesian coordinates at least however in general the models belonging to the family admit additional integrals which are nonpolynomial functions of momenta an application of the method for the construction of superintegrable models of liouville type is described

simulation of miscible displacement in porous media by a modified uzawa s algorithm combined with a characteristic method abstract miscible displacement in porous media is modeled by a nonlinear coupled system of two partial differential equations the pressure equation which is elliptic and the concentration equation which is parabolic but usually convection dominated in this paper we focus on some computational difficulties in the simulation of this problem in order to obtain accurate approximation for velocity we implement a modified uzawa s algorithm for the pressure equation with a full permeability tensor including strong heterogeneity and anisotropy for handling the convection dominance over diffusion we employ a modified method of characteristics for the concentration equation numerical simulation results show that the combination of the modified method of characteristics and the modified uzawa s algorithm is a good choice for the problem in particular the modified method of characteristics dramatically reduces numerical diffusion and non physical oscillation in the approximate concentration and the modified uzawa s algorithm leads to positive definite linear systems with velocity unknowns only and exhibits the accuracy of mixed finite element methods

an exact trace formula and zeta functions for an infinite quantum graph with a non standard weyl asymptotics we study a quantum hamiltonian that is given by the negative laplacian and an infinite chain of like potentials with strength 0 on the half line and which is equivalent to a one parameter family of laplacians on an infinite metric graph this graph consists of an infinite chain of edges with the metric structure defined by assigning an interval in 0 ln to each edge with length we show that the one parameter family of quantum graphs possesses a purely discrete and strictly positive spectrum for each 0 and prove that the dirichlet laplacian is the limit of the one parameter family in the strong resolvent sense the spectrum of the resulting dirichlet quantum graph is also purely discrete the eigenvalues are given by n n2 with multiplicities d n where d n denotes the divisor function we can thus relate the spectral problem of this infinite quantum graph to dirichlet s famous divisor problem and infer the non standard weyl asymptotics for the eigenvalue counting function based on an exact trace formula the voronoi summation formula we derive explicit formulae for the trace of the wave group the heat kernel the resolvent and for various spectral zeta functions these results enable us to establish a well defined renormalized secular equation and a selberg like zeta function defined in terms of the classical periodic orbits of the graph for which we derive an exact functional equation and prove that the analogue of the riemann hypothesis is true

universal integrals for superintegrable systems on n dimensional spaces of constant curvature an infinite family of classical superintegrable hamiltonians defined on the n dimensional spherical euclidean and hyperbolic spaces are shown to have a common set of 2n 3 functionally independent constants of the motion among them two different subsets of n integrals in involution including the hamiltonian can always be explicitly identified as particular cases we recover in a straightforward way most of the superintegrability properties of the smorodinsky winternitz and generalized kepler coulomb systems on spaces of constant curvature and we introduce as well new classes of quasi maximally superintegrable potentials on these spaces results presented here are a consequence of the poisson coalgebra symmetry of all the hamiltonians together with an appropriate use of the phase spaces associated with poincare and beltrami coordinates

combined compact difference scheme for linear second order partial differential equations with mixed derivative a combined compact difference scheme is proposed for linear second order partial differential equations with mixed derivative the scheme is based on a nine point stencil at the interior with sixth order accurate local truncation error fourier analysis is used to analyze the spectral resolution of the proposed scheme numerical tests demonstrate at least sixth order convergence rate with dirichlet boundary condition and fifth order with robin boundary condition a bonus is that high reynolds numbers do not interfere with the order of accuracy

hierarchical high order finite element spaces in h div ω h1 ω for a stabilized mixed formulation of darcy problem abstract the classical dual mixed finite element method for flow simulations is based on h div conforming approximation spaces for the flux which guarantees continuous normal components on element interfaces and discontinuous approximations in l 2 for the pressure however stability and convergence can only be obtained for compatible approximation spaces stabilized finite element methods may provide an alternative stable procedure to avoid this kind of delicate balance the main purpose of this paper is to present a high order finite element methodology to solve the darcy problem based on the combination of an unconditionally stable mixed finite element method with a hierarchical methodology for the construction of finite dimensional subspaces of h div and h 1 the chosen stabilized method is free of mesh dependent stabilization parameters and allows for the use of different high order finite element approximations for the flux and the pressure variables without requiring any compatibility constraint as required in mixed methods for these problems convergence studies are presented comparing the numerical solutions obtained for different approximation orders on quadrilateral elements with the ones given by classical mixed formulation with raviart thomas elements

a rotated monotone difference scheme for the two dimensional anisotropic drift diffusion equation a rotated upwind discretization scheme is presented for the discretization of the steady state two dimensional anisotropic drift diffusion equation taking into account the local characteristic nature of the solution the notable features of the algorithm lie in the projection of the governing partial differential equation onto two orthogonal axes to yield a vanishing mixed derivative term and in the utilization of the upwind flow of information as a result the limiting behaviour of the elliptic equation is preserved in the discretization the scheme produces an m matrix which guarantees an oscillation free solution and which enables the discrete system to be solved using standard iterative solvers the method is illustrated by numerical examples for which the analytical solutions are known

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

an anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems heterogeneous anisotropic diffusion problems arise in the various areas of science and engineering including plasma physics petroleum engineering and image processing standard numerical methods can produce spurious oscillations when they are used to solve those problems a common approach to avoid this difficulty is to design a proper numerical scheme and or a proper mesh so that the numerical solution validates the discrete counterpart dmp of the maximum principle satisfied by the continuous solution a well known mesh condition for the dmp satisfaction by the linear finite element solution of isotropic diffusion problems is the non obtuse angle condition that requires the dihedral angles of mesh elements to be non obtuse in this paper a generalization of the condition the so called anisotropic non obtuse angle condition is developed for the finite element solution of heterogeneous anisotropic diffusion problems the new condition is essentially the same as the existing one except that the dihedral angles are now measured in a metric depending on the diffusion matrix of the underlying problem several variants of the new condition are obtained based on one of them two metric tensors for use in anisotropic mesh generation are developed to account for dmp satisfaction and the combination of dmp satisfaction and mesh adaptivity numerical examples are given to demonstrate the features of the linear finite element method for anisotropic meshes generated with the metric tensors

numerical inversion of a laplace transform solution of a diffusion equation with a mixed derivative term we consider a diffusion equation with a mixed derivative term modelling the transient flow of a second grade fluid the same diffusion equation with a mixed derivative term also comes up in the modelling of heat conduction in which the history of the temperature gradient is included the equations differ in the sign of the coefficient of the mixed derivative term we show that the method of lines solution is stable for the transient flow of a second grade fluid and unstable for heat conduction with the temperature gradient history included a laplace transform solution for the heat conduction case is then investigated the laplace transform solution is inverted numerically using a fourier based method a stable solution is obtained the solutions are compared and discussed

global action angle coordinates for completely integrable systems with noncompact invariant submanifolds the obstruction to the existence of global action angle coordinates of abelian and noncommutative non abelian completely integrable systems with compact invariant submanifolds has been studied we extend this analysis to the case of noncompact invariant submanifolds

hamiltonian systems admitting a runge lenz vector and an optimal extension of bertrand s theorem to curved manifolds bertrand s theorem asserts that any spherically symmetric natural hamiltonian system in euclidean 3 space which possesses stable circular orbits and whose bounded trajectories are all periodic is either a harmonic oscillator or a kepler system in this paper we extend this classical result to curved spaces by proving that any hamiltonian on a spherically symmetric riemannian 3 manifold which satisfies the same conditions as in bertrand s theorem is superintegrable and given by an intrinsic oscillator or kepler system as a byproduct we obtain a wide panoply of new superintegrable hamiltonian systems the demonstration relies on perlick s classification of bertrand spacetimes and on the construction of a suitable globally defined generalization of the runge lenz vector

le probleme general des variables actions angles feuilletages symplectiquement complets monodromie des fibrations isotropes symplectiquement completes structure locale et theoreme des variables action angles classe de chern realisation d une variete de poisson au voisinage d une feuille compacte