analysing images of curved surfaces a reflectance map makes the relationship between image intensity and surface orientation explicit trade offs between image intensity and surface orientation emerge which cannot be resolved locally in a single view existing methods for determining surface orientation from a single view embody assumptions about surface curvature the hessian matrix is introduced to represent surface curvature properties of surface curvature are expressed as properties of the hessian matrix for several classes of surface image analysis simplifies this result has already been established for planar surfaces forming trihedral corners similar simplification is demonstrated for developable surfaces and for the subclass of surfaces known as generalized cones these studies help to delineate shape information that can be determined from geometric measurements at object boundaries and shape information that can be determined from intensity measurements over sections of smoothly curved surface a novel technique called photometric stereo is discussed the idea of stereo is to obtain multiple images in order to determine the underlying scene precisely in photometric stereo the viewing direction is constant multiple images are obtained by varying the incident illumination it is shown that this provides sufficient information to determine surface orientation at each image point

integrability of two dimensional homogeneous potentials combining the extended painleve conjecture with yoshida s singularity and stability analyses 1986 it is shown that for two dimensional homogeneous potentials of degree 2m integrability restricts kowalevskaya exponents and integrability coefficients to discrete sets of values this result is made use of in the analysis of integrability of symmetric potentials with m 2 3 and 4 direct construction of additional first integrals is successful only in special cases which can be transformed to known integrable ones

two dimensional integrable potentials with quartic invariants for autonomous two dimensional conservative dynamical systems the author derives four necessary and sufficient conditions which the potential function u x y has to satisfy in order that it is integrable with the second constant of motion quartic in the velocity components the author also develops the method by means of which he finds the quartic invariant for a given potential satisfying these conditions certain degenerate cases leading to pseudo quartic integrals are discussed two examples and a counter example are presented

configurational invariants of hamiltonian systems in this paper we explore the general conditions in order that a two dimensional natural hamiltonian system possess a second invariant which is a polynomial in the momenta and is therefore liouville integrable we examine the possibility that the invariant is preserved by the hamiltonian flow on a given energy hypersurface only weak integrability and derive the additional requirement necessary to have conservation at arbitrary energy strong integrability using null complex coordinates we show that the leading order coefficient of the polynomial is an arbitrary holomorphic function in the case of weak integrability and a polynomial in the coordinates in the strongly integrable one we review the results obtained so far with strong invariants up to degree four and provide some new examples of weakly integrable systems with linear and quadratic invariants

eigenfaces vs fisherfaces recognition using class specific linear projection we develop a face recognition algorithm which is insensitive to large variation in lighting direction and facial expression taking a pattern classification approach we consider each pixel in an image as a coordinate in a high dimensional space we take advantage of the observation that the images of a particular face under varying illumination but fixed pose lie in a 3d linear subspace of the high dimensional image space if the face is a lambertian surface without shadowing however since faces are not truly lambertian surfaces and do indeed produce self shadowing images will deviate from this linear subspace rather than explicitly modeling this deviation we linearly project the image into a subspace in a manner which discounts those regions of the face with large deviation our projection method is based on fisher s linear discriminant and produces well separated classes in a low dimensional subspace even under severe variation in lighting and facial expressions the eigenface technique another method based on linearly projecting the image space to a low dimensional subspace has similar computational requirements yet extensive experimental results demonstrate that the proposed fisherface method has error rates that are lower than those of the eigenface technique for tests on the harvard and yale face databases

first integrals versus configurational invariants and a weak form of complete integrability abstract a confusion over the concept of first integrals which has been created in a recent paper by hall 13 is clarified the clear distinction between first integrals and functions which are first integrals only on a specific fixed hypersurface is discussed hall s terminology of configurational invariants is adopted for the latter case the possible relevance of knowing configurational invariants for a hamiltonian system is illustrated by results concerning a weak form of the theory on complete integrability

polynomial constants of motion in flat space some general results on commuting integrals for a hamiltonian system are given the question of the existence of integrals which are polynomial in the momenta is investigated and the results applied to a variety of mechanical systems

time dependent invariants and quantum mechanics in two dimensions a class of time dependent problems in two space dimensions possessing time dependent invariants bilinear in momenta is considered explicit expression of the potentials and the corresponding invariants are derived quantum mechanics is introduced in these time dependent problems directly through a feynman propagator defined as a path integral involving the classical action the propagators are shown to admit expansions in terms of the eigenfunctions of the corresponding invariant operators equivalence of the present theory to that of lewis and riesenfeld h r lewis jr and w b riesenfeld j math phys 10 1458 1969 is discussed

the weak painleve property as a criterion for the integrability of dynamical systems we investigate the validity of the weak painleve property as an integrability criterion we present an example of a time dependent hamiltonian system which possesses a weak painleve type expansion while presenting a chaotic behavior however this system presents also critical fixed singularities the importance of the latter as far as integrability is concerned is discussed here

atlas of two dimensional irreversible conservative lagrangian mechanical systems with a second quadratic integral this paper aims at the most comprehensive and systematic construction and tabulation of mechanical systems that admit a second invariant quadratic in velocities other than the hamiltonian the configuration space is in general a two dimensional riemannian or pseudo riemannian manifold and the determination of its geometry is a part of the process of solution forces acting on the system include a part derived from a scalar potential and a part derived from a vector potential associated with terms linear in velocities in the lagrangian function of the system the last caused time irreversibility of the system we constructed 41 multiparameter integrable systems of the type described in the title mostly on riemannian manifolds they are mostly new and cover all previously known systems as special cases corresponding to special values of the parameters those include all known cases of motion of a particle in the plane and all known cases in the dynamics of rigid body in the last field we introduce a n

integrable three dimensional systems possessing linear and quadratic integrals of motion in the momenta the total number of three dimensional integrable potentials which possess one linear and one quadratic integral of motion in involution in the momenta is found these potentials which are given by a darboux equation belong to four distinct classes and involve two arbitrary functions in their expression

a class of integrable potentials a class of time independent two dimensional integrable potentials all possessing an invariant of the same general form is constructed one of these potentials is superintegrable its invariants realize the symmetry algebra so 3 for negative energies e 2 for zero energy and so 2 1 for positive energies a transformation of coupling constants reveals that in parabolic coordinates this potential is the harmonic oscillator acted on by constant forces this and another potential in the class may be considered as successive extensions of the kepler potential the analytic properties of these integrable systems in the complex time plane are also discussed

the computation of visible surface representations a computational theory of visible surface representations is developed the visible surface reconstruction process that computes these quantitative representations unifies formal solutions to the key problems of 1 integrating multiscale constraints on surface depth and orientation from multiple visual sources 2 interpolating dense piecewise smooth surfaces from these constraints 3 detecting surface depth and orientation discontinuities to apply boundary conditions on interpolation and 4 structuring large scale distributed surface representations to achieve computational efficiency visible surface reconstruction is an inverse problem a well posed variational formulation results from the use of a controlled continuity surface model discontinuity detection amounts to the identification of this generic model s distributed parameters from the data finite element shape primitives yield a local discretization of the variational principle the result is an efficient algorithm for visible surface reconstruction

integrable systems without the painleve property we examine whether the painleve property is a necessary condition for the integrability of nonlinear ordinary differential equations we show that for a large class of linearizable systems this is not the case in the discrete domain we investigate whether the singularity confinement property is satisfied for the discrete analogues of the non painleve continuous linearizable systems we find that while these discrete systems are themselves linearizable they possess non confined singularities

the master integrable two dimensional system with a quartic second integral in this paper we construct a new integrable natural conservative mechanical system admitting a second invariant of the fourth degree in velocities this system is quite general and involves 21 parameters we also show that all systems with a quartic integral known up to date can be obtained from it as special cases by a relevant choice of the values of parameters the results are applied to problems of particle and rigid body dynamics new integrable cases are obtained as special versions of the new system these cases include motions in a plane lobachevsky plane sphere and surfaces of variable curvature they also include generalizations of the classical cases of kovalevskaya chaplygin and goriatchev with the addition of certain singular terms to the potential

photometric ambient occlusion for intrinsic image decomposition we present a method for computing ambient occlusion ao for a stack of images of a lambertian scene from a fixed viewpoint ambient occlusion a concept common in computer graphics characterizes the local visibility at a point it approximates how much light can reach that point from different directions without getting blocked by other geometry while ao has received surprisingly little attention in vision we show that it can be approximated using simple per pixel statistics over image stacks based on a simplified image formation model we use our derived ao measure to compute reflectance and illumination for objects without relying on additional smoothness priors and demonstrate state of the art performance on the mit intrinsic images benchmark we also demonstrate our method on several synthetic and real scenes including 3d printed objects with known ground truth geometry

on the connection between hyperelliptic separability and painleve integrability we consider systems of odes which are associated with some physically significant examples shallow water equilibrium solutions travelling waves of the harry dym equation a lotka volterra system of competing species and the geodesic flow on the triaxial ellipsoid the first three are shown to share the following properties i they are hyperelliptically separable systems hss and after a suitable nonlinear time transformation become algebraically completely integrable aci and ii they are of the weak painleve type and become full painleve after the application of this transformation the geodesic flow on the other hand although it passes the usual painleve test does not possess a full set of free constants and thus one may not conclude whether it has the painleve property or not this system is also hss and becomes aci after the application of a suitable nonlinear time transformation we also combine our geometric analytical investigation with a numerical analysis of the system in the complex plane and show that there is perfect correspondence between the results of the two approaches this correspondence strengthens the reliability of such numerical studies and helps us better understand their implication in cases where such nonlinear transformations to complete integrability are not available

a new class of integrable systems we present a family of dynamical systems associated with the motion of a particle in two space dimensions these systems possess a second integral of motion quadratic in velocities apart from the hamiltonian and are thus completely integrable they were found through the derivation and subsequent resolution of the integrability condition in the form of a partial differential equation pde for the potential a most important point is that the same pde was derived through considerations on the analytic structure of the singularities of the solutions weak painleve property

integrability and the painleve property for low dimensional systems we examine a system described by two first order nonlinear differential equations from the point of view of integrability the singularity analysis in the complex time plane is used to investigate the painleve property which according to the ablowitz ramani segur conjecture is a prerequisite for integrability for infinite dimensional systems we show that for such low dimensional systems the painleve analysis is still a most useful guide but integrable cases also exist which do not possess the painleve property

painleve property and geometry abstract the painleve property for discrete hamiltonian systems implies the existence of a symplectic manifold which augments the original phase space and on which the flows exist and are analytic for all times the augmented manifold is constructed by expanding the hamilton jacobi equation a complete classification of the types of poles allowed in complex time is given for hamiltonians which separate into the direct product of hyperelliptic curves for such systems bounds on the degrees of the polynomial separating variable change and the other integrals in involution can be found from the pole series and the hamilton jacobi equation it is shown how branching can arise naturally in a painleve system

painleve test and exact similarity solutions of a class of nonlinear diffusion equations for 1 1 dimensional diffusion equations where the diffusion coefficient is an arbitrary power of the dependent variable a painleve test is performed on both the partial differential equation and the similarity reductions the invariance of the painleve property under a large class of similarity reductions is shown for the reductions with the painleve property the explicit similarity solutions are given it is demonstrated how results of a painleve analysis can be used to obtain further exact similarity solutions even in cases without the painleve property

statistical approach to shape from shading reconstruction of three dimensional face surfaces from single two dimensional images the human visual system is proficient in perceiving three dimensional shape from the shading patterns in a two dimensional image how it does this is not well understood and continues to be a question of fundamental and practical interest in this paper we present a new quantitative approach to shape from shading that may provide some answers we suggest that the brain through evolution or prior experience has discovered that objects can be classified into lower dimensional object classes as to their shape extraction of shape from shading is then equivalent to the much simpler problem of parameter estimation in a low dimensional space we carry out this proposal for an important class of three dimensional 3d objects human heads from an ensemble of several hundred laser scanned 3d heads we use principal component analysis to derive a low dimensional parameterization of head shape space an algorithm for solving shape from shading using this representation is presented it works well even on real images where it is able to recover the 3d surface for a given person maintaining facial detail and identity from a single 2d image of his face this algorithm has applications in face recognition and animation

on the hbar2 correction terms in quantum integrability the authors study the question of quantum integrability for two dimensional hamiltonian systems with special attention on the 2 correction terms in the potential a class of hamiltonians of type h 1 2 px2 py2 v4y4 y2 f2 x 6 f0 x with a second invariant of type i px4 a x y px2 b x y pxpy c x y py2 d x y is considered the general solution for f2 involves elliptic integrals for quantum integrability the potential must be modified with 2 dependent terms they construct a point transformation which after coupling constant metamorphosis takes the hamiltonian to a new quantum integrable hamiltonian for which no correction terms are necessary the new system does not in general have a flat space kinetic part

segmentation through variable order surface fitting the solution of the segmentation problem requires a mechanism for partitioning the image array into low level entities based on a model of the underlying image structure a piecewise smooth surface model for image data that possesses surface coherence properties is used to develop an algorithm that simultaneously segments a large class of images into regions of arbitrary shape and approximates image data with bivariate functions so that it is possible to compute a complete noiseless image reconstruction based on the extracted functions and regions surface curvature sign labeling provides an initial coarse image segmentation which is refined by an iterative region growing method based on variable order surface fitting experimental results show the algorithm s performance on six range images and three intensity images

non integrable systems with algebraic singularities in complex time dynamical arguments are presented which suggest that there are non integrable systems without clustering of singularities without infinite singularities or singularities with an infinite number of branches in the complex t plane several examples with only algebraic singularities are studied for which strong numerical evidence is presented for non integrability and infinitely sheeted solutions weak painleve potentials are also analysed from this point of view and all integrable cases are found to possess only finitely sheeted solutions

parrondo effect in quantum coin toss simulations game a game b game c parrondo s games follow this basic structure where a and b are losing games and c is a winning game a phenomenon called parrondo s paradox these games can take on a wider class of definitions and exhibit these paradoxical results in this paper we show three paradoxical cases i the successive tossing of a single fair quantum coin gives a biased result a previously known result ii the random tossing of two quantum coins each with successive biased expectations gives an average random walk position of approximately zero iii the sequential periodic tossing of two quantum coins each with successive negative biased expectations gives an average random walk with positive expectation using these results we then propose a protocol for identifying and classifying quantum operations that span the same hilbert space for a two level quantum system

hamiltonians with high order integrals and the weak painleve concept we examine the singularity structure of the equations of motion associated to integrable two dimensional hamiltonians with second integrals of order higher than 2 we show in these specific examples that the integrability is associated to a singularity expansion of the weak painleve type new cases of integrability are discovered with still higher order integrals which are explicitly computed

semianalytical solution of the random product problem of matrices and discrete time random evolution we consider the product of a large number of matrices chosen randomly with some correlation as a or b we introduce an exact functional equation whose solution gives the limiting distribution and the lyapunov exponent for the product of large number of matrices thus we suggest new semianalytical method to solve the random product problem of correlated matrices the method can be easily generalized to high order matrices as we deal with a larger systems of functional equations we have provided a known numerical algorithm for the solution of the matrix product problem we have also applied our method for a solution of discrete time random evolution our functional equation method provides exact transition point

painleve analysis yoshida s theorems and direct methods in the search for integrable hamiltonians treating as an example two dimensional similarity invariant hamiltonians with polynomial velocity dependent potentials the authors show how painleve analysis yoshida s theorems and direct methods can be combined in the search for integrable hamiltonians new integrable systems are found one of which has a polynomial second invariant of fifth order in the momenta most of the necessary calculations have been performed by applying various reduce packages developed for that purpose

linearization on a submanifold of integrable hamiltonians with polynomials potentials abstract following a method proposed by j weiss we linearize the integrable hamiltonians with quartic potentials as well as the remaining integrable cubic potentials a non generic solution is obtained in each case we explain this phenomenon in terms of a constraint submanifold over which the solution of the system is in fact linearizable we show that the linearization approach can depending on the particular case lead to generic solutions non generic ones or even to no solutions at all

integrability of hamiltonians with third and fourth degree polynomial potentials the weak painleve property as a criterion of integrability is applied to the case of simple hamiltonians describing the motion of a particle in two dimensional polynomial potentials of degree three and four this allows a complete identification of all the integrable cases of cubic potentials in the case of quartic potentials although our results are not exhaustive some new integrable cases are discovered in both cases the integrability is explicited by a direct calculation of the second integral of motion of the system

symmetries of hamiltonian systems with two degrees of freedom we classify the lie point symmetry groups for an autonomous hamiltonian system with two degrees of freedom with the exception of the harmonic oscillator or a free particle where the dimension is 15 we obtain all dimensions between 1 and 7 for each system in the classification we examine integrability

a class of 1 d gas flows soluble by the inverse scattering transform abstract we first show that the euler equations of 1 dimensional adiabatic gas dynamics with arbitrary equation of state present a symmetry s of order three i e such that s3 1 for a particular class of equations of state which may be viewed as representing the geometrical center of the symmetry a higher symmetry exists the sl 3 symmetry in agreement with the results of ref 2 for these equations of state we derive riemann invariants and are thus able to reduce the system of the euler s equations to a mere second order equation versus characteristic coordinates that equation turns out to be soluble by an inverse scattering method with a third order scattering operator as shown in ref 6 we thus provide a first instance of soliton propagation on a nondissipative gaseous background it must be noted however that these are solitons in a mathematical sense only as the space time in which they propagate uniformly does not coincide with ordinary space time

explicit integrability for hamiltonian systems and the painleve conjecture we analyze a class of hamiltonian systems in two dimensions for which we proved that a second constant of motion exists it is shown that using the two first integrals the equations of motion can be written in a form which allows their integration by quadratures an analysis of the equations of motion in this reduced form establishes the behavior of the solutions in the complex time plane it is shown explicitly that the systems belonging to this class possess the weak painleve property i e their solutions in complex time can present singularities of a specific algebraic type

on the integrability of systems of nonlinear ordinary differential equations with superposition principles a new class of solvable nonlinear dynamical systems has been recently identified by the requirement that the ordinary differential equations ode s describing each member of this class possess nonlinear superposition principles these systems of ode s are generally not derived from a hamiltonian and are classified by associated pairs of lie algebras of vector fields in this paper all such systems of n 3 ode s are integrated in a unified way by finding explicit integrals for them and relating them all to a pivotal member of their class the projective riccati equations moreover by perturbing two parametrically driven projective riccati equations thus making them nonsolvable in the above sense evidence is discovered of chaotic behavior on the poincare surface of section in the form of sensitive dependence on initial conditions near a boundary separating bounded from unbounded motion

darboux points and integrability analysis of hamiltonian systems with homogeneous rational potentials abstract we study integrability in the liouville sense for natural hamiltonian systems with a homogeneous rational potential v q strong necessary conditions for the integrability of such systems were obtained by analysis of differential galois groups of variational equations along certain particular solutions these conditions have the form of arithmetic restrictions on eigenvalues of the hessians v d calculated at non zero solutions d of the equation grad v d d such solutions are called proper darboux points it was recently proved that for generic polynomial homogeneous potentials there exist universal relations between eigenvalues of hessians of the potential taken at all proper darboux points the existence of such relations for rational potentials seems to be a difficult question one of the reasons is the presence of points of indeterminacy of the potential and its gradient nevertheless for two degrees of freedom we prove that such a relation exists this result is important because it allows us to show that the set of admissible values for hessian eigenvalues at a proper darboux point for potentials satisfying the necessary conditions for integrability is finite in turn this provides a tool for classification of integrable rational potentials it was also recently shown that for polynomial homogeneous potentials additional necessary conditions for integrability can be deduced from the existence of improper darboux points that is points d that are non zero solutions of grad v d 0 these new conditions also take the form of arithmetic restrictions imposed on eigenvalues of v d in this paper we prove that for rational potentials improper darboux points give the same necessary conditions for integrability

integrable potentials with logarithmic integrals of motion in this paper several examples of integrable potentials with a second invariant which contains a logarithmic part in the velocities are given in most cases the potential depends on the velocities there is however a case where the potential is velocity independent

darboux points and integrability of hamiltonian systems with homogeneous polynomial potential in this paper we study the integrability of natural hamiltonian systems with a homogeneous polynomial potential the strongest necessary conditions for their integrability in the liouville sense have been obtained by a study of the differential galois group of variational equations along straight line solutions these particular solutions can be viewed as points of a projective space of dimension smaller by one than the number of degrees of freedom we call them darboux points we analyze in detail the case of two degrees of freedom we show that except for a radial potential the number of darboux points is finite and it is not greater than the degree of the potential moreover we analyze cases when the number of darboux points is smaller than maximal for two degrees of freedom the above mentioned necessary condition for integrability can be expressed in terms of one nontrivial eigenvalue of the hessian of potential calculated at a darboux point we prove that for a given potential these nontrivial eigenvalues calculated for all darboux points cannot be arbitrary because they satisfy a certain relation which we give in an explicit form we use this fact to strengthen maximally the necessary conditions for integrability and we show that in a generic case for a given degree of the potential there is only a finite number of potentials which satisfy these conditions we also describe the nongeneric cases as an example we give a full list of potentials of degree four satisfying these conditions then investigating the differential galois group of higher order variational equations we prove that except for one discrete family among these potentials only those which are already known to be integrable are integrable we check that a finite number of potentials from the exceptional discrete family are not integrable and we conjecture that all of them are not integrable

invariants at fixed and arbitrary energy a unified geometric approach invariants at arbitrary and fixed energy strongly and weakly conserved quantities for two dimensional hamiltonian systems are treated in a unified way this is achieved by utilizing the jacobi metric geometrization of the dynamics using killing tensors we obtain an integrability condition for quadratic invariants which involves an arbitrary analytic function s z for invariants at arbitrary energy the function s z is a second degree polynomial with real second derivative the integrability condition then reduces to darboux s condition for quadratic invariants at arbitrary energy the four types of classical quadratic invariants for positive definite two dimensional hamiltonians are shown to correspond to certain conformal transformations we derive the explicit relation between invariants in the physical and jacobi time gauges in this way knowledge about the invariant in the physical time gauge enables one to write down directly the components of the corresponding killing tensor for the jacobi metric we also discuss the possibility of searching for linear and quadratic invariants at fixed energy and its connection with the problem of the third integral in galactic dynamics in our approach linear and quadratic invariants at fixed energy can be found by solving a linear ordinary differential equation of the first or second degree respectively

completeness of multiseparable superintegrability in e2 c the possibility that schrodinger s equation with a given potential can separate in more than one coordinate system is intimately connected with the notion of superintegrability examples of this type of system are well known in this paper we demonstrate how to establish a complete list of such potentials using essentially algebraic means our approach is to classify all nondegenerate potentials that admit a pair of second order constants of motion here nondegenerate means that the potentials depend on four independent parameters this is carried out for two dimensional complex euclidean space though the method generalizes to other spaces and dimensions we show that all these superintegrable systems correspond to quadratic algebras and we work out the detailed structure relations and their quantum analogues

two dimensional time dependent hamiltonian systems with an exact invariant we present a direct approach to investigate the existence of an exact invariant for two dimensional hamiltonians in which the potential depends explicitly on time the method is based on an expansion of the invariant in the velocities the problem is solved completely for invariants linear and quadratic in the momenta our results contain as a particular case the results of lewis and leach on one dimensional systems

on the singularity analysis of intersecting separatrices in near integrable dynamical systems abstract it has been recently proved by s l ziglin that transversal intersections of separatrices invariant manifolds in near integrable hamiltonian systems of two degrees of freedom imply the existence of multi valued solutions with infinitely many riemann sheets ziglin s theorem is illustrated here on a simple example and then extended and applied to non hamiltonian analytic flows x f x eg x t with x x y and g x t g x t 2 which for e 0 possess the painleve property on the other hand the theoretically expected logarithmic singularities for e 0 are obtained explicitly in solutions near the intersecting separatrices thus we conjecture that dynamical systems with the painleve property can have no separatrix intersections and hence no strange attractors etc these singularities are then numerically located and found to form certain very interesting chimney patterns in the complex t plane on which they accumulate densely the upper part of the chimneys away from the re t axis is asymptotically quite insensitive to changes in parameter values or initial conditions the singularity pattern itself however becomes denser and each chimney is seen to gradually collapse towards the re t axis as the amplitude of the driving term increases and the motion becomes more chaotic

exact probability distribution functions for parrondo s games we study the discrete time dynamics of brownian ratchet models and parrondo s games using the fourier transform we calculate the exact probability distribution functions for both the capital dependent and history dependent parrondo s games in certain cases we find strong oscillations near the maximum of the probability distribution with two limiting distributions for odd and even number of rounds of the game indications of such oscillations first appeared in the analysis of real financial data but now we have found this phenomenon in model systems and a theoretical understanding of the phenomenon the method of our work can be applied to brownian ratchets molecular motors and portfolio optimization

periodic orbits and nonintegrability of generalized classical yang mills hamiltonian systems the averaging theory of first order is applied to study a generalized yang mills system with two parameters two main results are proved first we provide sufficient conditions on the two parameters of the generalized system to guarantee the existence of continuous families of isolated periodic orbits parameterized by the energy and these families are given up to first order in a small parameter second we prove that for the nonintegrable classical yang mills hamiltonian systems in the sense of liouville arnold which have the isolated periodic orbits found with averaging theory cannot exist in any second first integral of class c1 this is important because most of the results about integrability deals with analytic or meromorphic integrals of motion

an integrable three particle system a study of the existence of some integrable systems with nonlinear constants of motion is presented using the approach of the theory of generalized dynamical or hidden symmetries two lagrangians are considered both obtained by modifying the toda lagrangian first a two particle system is studied and then the results are generalized to a three particle system it is shown that in both cases the lagrangians possess nonlinear constants of motion in involution and thus they are integrable

existence of exponentially unstable periodic solutions and the non integrability of homogeneous hamiltonian systems abstract in hamiltonian systems with polynomial potential of homogeneous degree characteristic multipliers floquet multipliers permit an explicit expression for straight line periodic solutions by use of ziglin s theorem it is shown that the existence of an exponentially unstable straight line periodic solution implies the non integrability of the system

analytic structure of the henon heiles hamiltonian in integrable and nonintegrable regimes the solutions of the henon heiles hamiltonian are investigated in the complex time plane the use of the painleve property i e the property that the only movable singularities exhibited by the solution are poles enables successful prediction of the values of the nonlinear coupling parameter for which the system is integrable special attention is paid to the structure of the natural boundaries that are found in some of the nonintegrable regimes these boundaries have a remarkable self similar structure whose form changes as a function of the nonlinear coupling

hamiltonian maps in the complex plane abstract the standard map is a nonintegrable discrete time analog of the vertical pendulum detailed calculations are presented and illustrated graphically for the standard map at the golden mean frequency the functional dependence of the coordinate q on the canonical angle variable is analtyically continued into the complex plane where natural boundaries are found at constant absolute values of im the boundaries represent the appearance of chaotic motion in the complex plane when the domain of analyticity shrinks to zero the kam invariant curve is destroyed two independent numerical methods with fourier analysis in the angle variable were used one based on a variation annihilation method and the other on a double expansion the results were further checked by direct solution of the complex equations of motion the numerically simpler but intrinsically complex semipendulum and semistandard map are also studied we conjecture that natural boundaries appear in the analogous analytic continuation of the invariant tori or kam surfaces of general nonintegrable systems with analytic hamiltonians

integrals of motion for toda systems with unequal masses we present new integrals of motion for the toda lattice chain of particles in one dimension with exponential interaction for two special cases of boundary conditions the free end lattice with three non equal mass particles and the fixed end lattice for two particles in both cases we use two distinct approaches in order to identify the integrable cases direct search of the integral of motion and group theoretical methods our results are in agreement with the predictions of painleve analysis

exact solutions for a class of integrable henon heiles type systems we study the exact solutions of a class of integrable henon heiles type systems according to the analysis of bountis et al phys rev a 25 1257 1982 these solutions are expressed in terms of two dimensional kleinian functions special periodic solutions are expressed in terms of the well known weierstrass function we extend some of our results to a generalized henon heiles type system with n 1 degrees of freedom

integrable models from mathcal pt symmetric deformations we address the question of whether integrable models allow for symmetric deformations which preserve their integrability for this purpose we carry out the painleve test for symmetric deformations of burgers and the korteweg de vries equations we find that the former equation allows for infinitely many deformations which pass the painleve test for a specific deformation we prove the convergence of the painleve expansion and thus establish the painleve property for these models which are therefore thought to be integrable the korteweg de vries equation does not allow for deformations which pass the painleve test in complete generality but we are able to construct a defective painleve expansion

hamiltonian systems with a certain stability property by requiring a hamiltonian system with several adjustable parameters to have a certain stability property a family is created which includes as its members the integrable cases the results obtained analytically for two hamiltonian systems with two degrees of freedom are presented

singular point analysis and integrals of motion for coupled nonlinear schrodinger equations for a system of two coupled nonlinear schrodinger equations and the corresponding madelung fluid equations we calculate the similarity transformation and reductions to ordinary differential equations the symmetries can be used to classify several types of solutions besides a system of coupled painleve ii equations we obtain a two dimensional quartic hamiltonian system using noether s theorem and performing a painleve test we identify those parameters which lead to regular motion on tori using the results of the p test we obtain heretofore undiscovered integrals of motion for the madelung fluid by direct calculations investigating the dynamics of the system for parameters that are different from those obtained by the painleve test we calculate numerically the surface of section kam tori and study the transition into chaotic behaviour

a graph spectral approach to shape from shading in this paper we explore how graph spectral methods can be used to develop a new shape from shading algorithm we characterize the field of surface normals using a weight matrix whose elements are computed from the sectional curvature between different image locations and penalize large changes in surface normal direction modeling the blocks of the weight matrix as distinct surface patches we use a graph seriation method to find a surface integration path that maximizes the sum of curvature dependent weights and that can be used for the purposes of height reconstruction to smooth the reconstructed surface we fit quadrics to the height data for each patch the smoothed surface normal directions are updated ensuring compliance with lambert s law the processes of height recovery and surface normal adjustment are interleaved and iterated until a stable surface is obtained we provide results on synthetic and real world imagery

direct integration of generalized lie symmetries of nonlinear hamiltonian systems with two degrees of freedom integrability and separability many of the integrable coupled nonlinear oscillator systems are associated with generalized lie symmetries involving velocity dependent terms for a class of systems with two degrees of freedom the authors show that by integrating the characteristic equation associated with the generalized symmetries the required involutive integrals of motion can be obtained explicitly in a straightforward manner almost by inspection and without recourse to noether s theorem further all the separable coordinates can be obtained by integrating a subset of the characteristic equation associated with the coordinate variables alone the explicit examples include the two coupled generalized henon heiles quartic sextic and other polynomial oscillator systems as well as the perturbed kepler system

integrable hamiltonian systems with velocity dependent potentials the integrability of a two dimensional hamiltonian in which the potential depends explicitly on the momenta is investigated hamiltonians of this kind are encountered in the description of the motion of a particle in a magnetic field two integrable classes of potentials are identified and the second integral of motion is constructed for each of them the singularity analysis of the equations of motion is also performed confirming once more the relation between the weak painleve property and integrability

semiclassical spectrum using secular perturbation theory the quartic oscillator we analyse the quartic oscillator using the technique of removal of resonances in a region of parameter space where the system exhibits soft chaos we obtain analytic expressions for all primary resonances and thus obtain a semiclassical spectrum a criterion relating to overlapping resonances gives an estimate about the break down of the removal of resonance method

integrability and orbits in quartic polynomial potentials the effect of enlarging the class of admissible movable singularities in painleve analysis to include all rational algebraic branch points is examined for a class of quartic polynomial potentials eight homogeneous quartic potentials are found in addition to the seven integrable cases given by weak painleve analysis they are examined using various numerical techniques and ziglin s theorem only one of them remains a candidate for integrability which indicates that in general most rational algebraic branch points are incompatible with integrability movable logarithmic singularities also appear to be inconsistent with integrability however the remaining potentials are either regular or nearly regular for the energies examined and are therefore still of interest for numerical purposes the corresponding surfaces of section are found to be particularly simple in structure and belong to a small number of topologically distinct classes their stable and unstable periodic orbit structures are examined to provide information about their regularity and for use in ziglin s theorem there appears to be a correlation between the resonances of the stable periodic orbits and the order of the corresponding movable singularities

a list of all integrable two dimensional homogeneous polynomial potentials with a polynomial integral of order at most four in the momenta we searched for integrable two dimensional homogeneous polynomial potentials with a polynomial first integral by using the so called direct method of searching for first integrals we proved that there exist no polynomial first integrals which are genuinely cubic or quartic in the momenta if the degree of homogeneous polynomial potentials is greater than four

integrability of the toda chain with free boundaries the hamilton function for the toda chain with free boundaries is studied for different interaction and mass parameters for certain systems the connection to simple lie algebras is shown and the integrability is proved by constructing the corresponding lax representations for the four particle system a painleve analysis is presented

classical versus quantum integrability classical integrability and quantum integrability are compared for two degrees of freedom hamiltonian systems we use c number representatives for quantum operators and the moyal bracket for the commutator three different cases are found i the c number representative of the quantum mechanical second invariant is identical to the classical second invariant ii o 2 corrections are needed in the classical second invariant to obtain the quantum invariant and iii also the potential must be deformed by an o 2 term several examples from the henon heiles and holt families of integrable potentials are included

on the complete and partial integrability of non hamiltonian systems abstract the methods of singularity analysis are applied to several third order non hamiltonian systems of physical significance including the lotka volterra equations the three wave interaction and the rikitake dynamo model complete integrability is defined and new completely integrable systems are discovered by means of the painleve property in all these cases we obtain integrals which reduce the equations either to a final quadrature or to an irreducible second order ordinary differential equation ode solved by painleve transcendents relaxing the painleve property we find many partially integrable cases whose movable singularities are poles at leading order with in t t 0 terms entering at higher orders in an n th order generalized rossler model a precise relation is established between the partial fulfillment of the painleve conditions and the existence of n 2 integrals of the motion

incorporating illumination constraints in deformable models for shape from shading and light direction estimation we present a method for the integration of nonlinear holonomic constraints in deformable models and its application to the problems of shape and illuminant direction estimation from shading experimental results demonstrate that our method performs better than previous shape from shading algorithms applied to images of lambertian objects under known illumination it is also more general as it can be applied to non lambertian surfaces and it does not require knowledge of the illuminant direction in this paper 1 we first develop a theory for the numerically robust integration of nonlinear holonomic constraints within a deformable model framework in this formulation we use lagrange multipliers and a baumgarte stabilization approach 1972 2 we also describe a fast new method for the computation of constraint based forces in the case of high numbers of local parameters 3 we demonstrate how any type of illumination constraint from the simple lambertian model to more complex highly nonlinear models can be incorporated in a deformable model framework 4 we extend our method to work when the direction of the light source is not known we couple our shape estimation method with a method for light estimation in an iterative process where improved shape estimation results in improved light estimation and vice versa 5 we perform a series of experiments

new bi hamiltonian systems on the plane we discuss several new bi hamiltonian integrable systems on the plane with integrals of motion of third fourth and sixth orders in momenta the corresponding variables of separation separated relations compatible poisson brackets and recursion operators are also presented in the framework of the jacobi method

lie symmetries of the lorenz model abstract we study the generalized symmetries of the lorenz model to find the parameter values at which one or more time dependent integrals of motion exist in these cases the integrals are found trivially from the symmetries themselves a complete study of the one completely algebraically integrable case shows a the dynamics can be integrated exactly by reducing it first to a lower dimensional system b the symmetry vector field is hamiltonian these facts hold for other dissipative completely integrable dynamical systems as well the analytic study of a natural two form reveals that it is an entire function of time the foliation of phase space induced by the two form for the partially integrable cases has a simple description in terms of the coefficients occurring in the laurent series expansions of the dependent variables

a multiple scales approach to maximal superintegrability in this paper we present a simple algorithmic test to establish if a hamiltonian system is maximally superintegrable or not this test is based on a very simple corollary of a theorem due to nekhoroshev and on a perturbative technique called multiple scales method if the outcome is positive this test can be used to suggest maximal superintegrability whereas when the outcome is negative it can be used to disprove it this method can be regarded as a finite dimensional analog of the multiple scales method as a way to produce soliton equations we use this technique to show that the real counterpart of a mechanical system found by jules drach in 1935 is in general not maximally superintegrable we give some hints on how this approach could be applied to classify maximally superintegrable systems by presenting a direct proof of the well known bertrand s theorem

determining generative models of objects under varying illumination shape and albedo from multiple images using svdand integrability we describe a method of learning generative models of objects from a set of images of the object under different and unknown illumination such a model allows us to approximate the objects appearance under a range of lighting conditions this work is closely related to photometric stereo with unknown light sources and in particular to the use of singular value decomposition svd to estimate shape and albedo from multiple images up to a linear transformation hayakawa 1994 firstly we analyze and extend the svd approach to this problem we demonstrate that it applies to objects for which the dominant imaging effects are lambertian reflectance with a distant light source and a background ambient term to determine that this is a reasonable approximation we calculate the eigenvectors of the svd on a set of real objects under varying lighting conditions and demonstrate that the first few eigenvectors account for most of the data in agreement with our predictions we then analyze the linear ambiguities in the svd approach and demonstrate that previous methods proposed to resolve them hayakawa 1994 are only valid under certain conditions we discuss alternative possibilities and in particular demonstrate that knowledge of the object class is sufficient to resolve this problem secondly we describe the use of surface consistency for putting constraints on the possible solutions we prove that this constraint reduces the ambiguities to a subspace called the generalized bas relief ambiguity gbr which is inherent in the lambertian reflectance function and which can be shown to exist even if attached and cast shadows are present belhumeur et al 1997 we demonstrate the use of surface consistency to solve for the shape and albedo up to a gbr and describe and implement a variety of additional assumptions to resolve the gbr thirdly we demonstrate an iterative algorithm that can detect and remove some attached shadows from the objects thereby increasing the accuracy of the reconstructed shape and albedo

on monoparametric families of orbits sufficient for integrability of planar potentials with linear or quadratic invariants by combining darboux s results 1901 on the direct construction of integrable systems with results from the inverse problem of dynamics the authors prove that if a planar potential admits a monoparametric family of conic sections with constant focal distance or a family of confocal parabolas it is integrable and the second invariant is quadratic in the moments while the existence of a family of straight lines either parallel or intersecting at one point also suffices for integrability the second invariant being linear in the momenta

killing tensors in spaces of constant curvature a killing tensor is one possible way of generalizing the notion of a killing vector on a riemannian or pseudo riemannian manifold it is explained how killing tensors may be identified with functions that are homogeneous polynomials in the fibers on the associated cotangent bundle as such killing tensors may be identified with first integrals of the hamiltonian geodesic flow which are homogeneous polynomials in the momenta again using this identification it is shown that in flat spaces the dimension of the vector space of killing tensors is maximal and that the killing tensors are generated by the killing vectors finally using riemann s model for the metric in spaces of constant curvature a comparison argument is used to show that similar results are valid in that more general context

painleve conjecture revisited the discovery of new integrable two dimensional hamiltonian systems is reported the analytic structure of the solutions makes necessary the generalization of the painlev e conjecture a widely used integrability criterion such a generalization is presented which the authors believe should replace the usual conjecture for two dimensional hamiltonian systems it is indeed compatible with all the systems already found and in addition leads to still new integrable cases

towards a description of complexity of the simplest cosmological systems we study the problem of complex dynamics and nonintegrability integrability of cosmological dynamical systems which are given in the hamiltonian form with indefinite kinetic energy form t g v v where g is a two dimensional pseudo riemannian metric with a lorentzian signature and vtx is a tangent vector at a point x of the configuration space we present examples showing the effectiveness of using a the direct method of construction of linear and quadratic first integrals b the ziglin and yoshida theorems concerning nonintegrability of hamiltonian systems with homogeneous potential functions and c the morales ramis theorem of nonintegrability of hamiltonian systems with complex potential functions

nonexistence and existence of various order integrals for two and three dimensional polynomial potentials the nonexistence of integrals of the motion which are sixth and fourth degree polynomials in the velocities was established for a range of polynomial potentials the first known systematic search for sixth order integrals was performed for cubic and quartic polynomial potentials it revealed that there exist no nondegenerate cases with such integrals for either potential similarly there exists no nondegenerate fifth or sixth degree polynomial potentials possessing quartic invariants a complete list of three dimensional cubic potentials with quadratic integrals is given all these integrable three dimensional potentials can be interpreted as orthogonal superpositions of known integrable two dimensional potentials possessing quadratic integrals they correspond to 3 of 11 possible coordinate systems in which three dimensional potentials separate

an algebraic approach to shape from image problems abstract this paper presents a new method for recovering three dimensional shapes of polyhedral objects from their single view images the problem of recovery is formulated in a constrained optimization problem in which the constraints reflect the assumption that the scene is composed of polyhedral objects and the objective function to be minimized is a weighted sum of quadric errors of surface information such as shading and texture for practical purpose it is decomposed into the two more tractable problems a linear programming problem and an unconstrained optimization problem in the present method the global constraints placed by the polyhedron assumption are represented in terms of linear algebra whereas similar constraints have usually been represented in terms of a gradient space moreover superstrictness of the constraints can be circumvented by a new concept position free incidence structure for this reason the present method has several advantages it can recover the polyhedral shape even if image data are incorrect due to vertex position errors it can deal with perspective projection as well as orthographic projection the number of variables in the optimization problem is very small three or a little greater than three and any kinds of surface information can be incorporated in a unifying manner