separation of variables and exact solution to the dirac equation in curvilinear orthogonal coordinates with cylindrical symmetry in the present article a complete separation of variables in the dirac equation for a free particle is achieved in parabolic cylinder and elliptical coordinates the resulting system of ordinary differential equations is solved and the asymptotic behavior of the spinor solution is discussed

a theory of exact and approximate configurational invariants abstract a theory of integrable hamiltonian systems in two dimensions is formulated and applied the four dimensional phase space problem that is the vanishing of the poisson bracket between another invariant and the hamiltonian is here reduced to the solution of a series of two dimensional configuration space equations the theory is also applicable to more than two dimensions the constraints are found which admit to integrability of the orbits for magnetic or coriolis forces as well as for forces derivable from a potential when a system admits a given invariant the invariant is found for example by quadrature a number of examples including known and apparently previously unknown invariants are given the theory of exact integrals of the motion also can be extended to the derivation of approximate invariants the orbital structure of integrable or approximately integrable systems correlates with the degree maximum power of the velocity of a standardized invariant the theory admits a variational principle among other approximation techniques for the computation of a best approximate invariant the problem of the general cubic potential with one symmetric coordinate v 1 2 ax 2 1 2 by 2 cx 2 y 1 3 dy 3 of which the well studied henon heiles potential is the special case for a b and c d is examined in detail

on separable schrodinger equations we classify 1 3 dimensional schrodinger equations for a particle interacting with the electromagnetic field that are solvable by the method of separation of variables as a result we get 11 classes of the vector potentials of the electromagnetic field a t x a0 t x a t x providing separability of the corresponding schrodinger equations it is established in particular that the necessary condition for the schrodinger equation to be separable is that the magnetic field must be independent of the spatial variables next we prove that any schrodinger equation admitting variable separation into second order ordinary differential equations can be reduced to one of the 11 separable schrodinger equations mentioned above and carry out variable separation in the latter furthermore we apply the results obtained for separating variables in the hamilton jacobi equation

new variable separation approach application to nonlinear diffusion equations the concept of the derivative dependent functional separable solution ddfss as a generalization to the functional separable solution is proposed as an application it is used to discuss the generalized nonlinear diffusion equations based on the generalized conditional symmetry approach as a consequence a complete list of canonical forms for such equations which admit the ddfss is obtained and some exact solutions to the resulting equations are described

integrable and superintegrable systems with spin a system of two particles with spin s 0 and s 12 respectively moving in a plane is considered it is shown that such a system with a nontrivial spin orbit interaction can allow an eight dimensional lie algebra of first order integrals of motion the pauli equation is solved in this superintegrable case and reduced to a system of ordinary differential equations when only one first order integral exists

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

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

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

complex variables for separation of the hamilton jacobi equation on real pseudo riemannian manifolds in this paper the geometric theory of separation of variables for the time independent hamilton jacobi equation is extended to include the case of complex eigenvalues of a killing tensor on pseudo riemannian manifolds this task is performed without complexifying the manifold but just by considering complex valued functions on it the simple formalism introduced in the paper allows us to extend in a very natural way the classical results on separation of variables including levi civita criterion and stackel eisenhart theory to the complex case only orthogonal variables are considered

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

remarks on the connection between the additive separation of the hamilton jacobi equation and the multiplicative separation of the schrodinger equation i the completeness and robertson conditions the fundamental elements of the variable separation theory are revisited including the eisenhart and robertson theorems kalnins miller theory and the intrinsic characterization of the separation of the hamilton jacobi equation in a unitary and geometrical perspective the general notion of complete integrability of first order normal systems of pdes leads in a natural way to completeness conditions for separated solutions of the schrodinger equation and to the robertson condition two general types of multiplicative separation for the schrodinger equation are defined and analyzed they are called free and reduced separation respectively in the free separation the coordinates are necessarily orthogonal while the reduced separation may occur in nonorthogonal coordinates but only in the presence of symmetries killing vectors

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

the classification of complete sets of operators commuting with the dirac operator in minkowski space time under the action of the poincare group p 1 3 the three four and five dimensional vector spaces of formally self adjoint first order matrix differential operators commuting among themselves and with the dirac operator are classified this gives a complete classification of the maximal subspaces of the vector space of first order formally self adjoint matrix differential operators commuting with the dirac operator which form an abelian lie algebra under the commutator

reformulated radial basis neural networks trained by gradient descent this paper presents an axiomatic approach for constructing radial basis function rbf neural networks this approach results in a broad variety of admissible rbf models including those employing gaussian rbfs the form of the rbfs is determined by a generator function new rbf models can be developed according to the proposed approach by selecting generator functions other than exponential ones which lead to gaussian rbfs this paper also proposes a supervised learning algorithm based on gradient descent for training reformulated rbf neural networks constructed using the proposed approach a sensitivity analysis of the proposed algorithm relates the properties of rbfs with the convergence of gradient descent learning experiments involving a variety of reformulated rbf networks generated by linear and exponential generator functions indicate that gradient descent learning is simple easily implementable and produces rbf networks that perform considerably better than conventional rbf models trained by existing algorithms

efficient relighting of rbf based illumination adjustable images an illumination adjustable image iai contains a large number of prerecorded images under various light directions relighting a scene under complicated lighting conditions can be achieved from the iai using the radial basis function rbf approach to represent an iai is proven to be more efficient than using the spherical harmonic approach however to represent high frequency lighting effects we need to use many rbfs hence the relighting speed could be very slow this brief investigates a partial reconstruction scheme for relighting an iai based on the locality of rbfs compared with the conventional rbf and spherical harmonics sh approaches the proposed scheme has a much faster relighting speed under the similar distortion performance

on the new approach to variable separation in the time dependent schrodinger equation with two space dimensions we suggest an effective approach to separation of variables in the schrodinger equation with two space variables using it we classify inequivalent potentials v x1 x2 such that the corresponding schrodinger equations admit separation of variables besides that we carry out separation of variables in the schrodinger equation with the anisotropic harmonic oscillator potential v k1x21 k2x22 and obtain a complete list of coordinate systems providing its separability most of these coordinate systems depend essentially on the form of the potential and do not provide separation of variables in the free schrodinger equation v 0

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

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

generalized symmetries in mechanics and field theories generalized symmetries are introduced in a geometrical and global formalism such a framework applies naturally to field theories and specializes to mechanics generalized symmetries are characterized in a lagrangian context by means of the transformation rules of the poincare cartan form and the generalized nother theorem is applied to obtain conserved quantities first integrals in mechanics in the particular case of mechanics it is shown how to use generalized symmetries to study the separation of variables of hamilton jacobi equations recovering standard results by means of this new method supersymmetries wess zumino model are considered as an intriguing example in field theory

variable separation approach for a differential difference system special toda equation a bilinear variable separation approach is used to construct some special solutions for a differential difference toda equation the semi discrete form of the continuous formula which describes some types of special solutions for many 2 1 dimensional continuous systems is found for a suitable quantity of the differential difference toda equation thus abundant semi discrete localized coherent structures are constructed by appropriately selecting the arbitrary functions

a new class of wavelet networks for nonlinear system identification a new class of wavelet networks wns is proposed for nonlinear system identification in the new networks the model structure for a high dimensional system is chosen to be a superimposition of a number of functions with fewer variables by expanding each function using truncated wavelet decompositions the multivariate nonlinear networks can be converted into linear in the parameter regressions which can be solved using least squares type methods an efficient model term selection approach based upon a forward orthogonal least squares ols algorithm and the error reduction ratio err is applied to solve the linear in the parameters problem in the present study the main advantage of the new wn is that it exploits the attractive features of multiscale wavelet decompositions and the capability of traditional neural networks by adopting the analysis of variance anova expansion wns can now handle nonlinear identification problems in high dimensions

the complexity of computing symmetric functions using threshold circuits abstract this paper considers size depth tradeoffs for threshold circuits computing symmetric functions the size measure used is the number of connections or edges in the threshold circuits as opposed to the number of gates in the circuits the main result is that for all d 2 and n 8 2 d there is a threshold circuit to compute any n input symmetric function which has size o 1 log n 2 d 1 n 1 1 2 d 1 and depth bounded by 6 d 8 as a consequence there is a threshold circuit for any n input symmetric function which has size o n and depth bounded by o log log n a somewhat simpler construction that contains many features of the general solution shows that for all d 1 and n 2 2 d 1 there is a threshold circuit for the n input parity function which has size bounded by 27 2 2 n 1 1 2 d 1 and depth bounded by 2 d

depth size tradeoffs for neural computation the tradeoffs between the depth i e the time for parallel computation and the size i e the number of threshold gates in neural networks are studied the authors focus the study on the neural computations of symmetric boolean functions and some arithmetic functions it is shown that a significant reduction in the size is possible for symmetric functions and some arithmetic functions at the expense of a small constant increase in depth in the process several neural networks which have the minimum size among all the known constructions have been developed results on implementing symmetric functions can be used to improve results about arbitrary boolean functions in particular it is shown that any boolean function can be computed in a depth 3 neural network with o 2 sup n sup 2 threshold gates it is also proven that at least omega 2 sup n sup 3 threshold gates are required

2 1 addition and related arithmetic operations with threshold logic in this paper we investigate the reduction of the size for small depth feed forward linear threshold networks performing binary addition and related functions for n bit operands we propose a depth 3 o n sup 2 log n asymptotic size network for the binary addition with o polynomially bounded weights we propose also a depth 3 addition of optimal o n asymptotic sits network and a depth 2 comparison of o spl radic n asymptotic size network both with o 2 sup spl radic n asymptotic size of weight values for existing architectural formats we show that our schemes with equal or smaller depth networks substantially outperform existing schemes in terms of size and fan in requirements and on occasions in weight requirements

construction of exact dynamical invariants in coupled oscillator problems with a view to obtaining further insight into the theoretical understanding of the problem of coupled harmonic oscillators we carry out the construction of exact dynamical invariants for momentum and time dependent td hamiltonian systems in two dimensions in particular we investigate the systems h1 1 1p 2 2p 2 1x 2 2x 2 2 2 3x1x2 2 3p1p2 h2 1 p 2 1 p 2 2 1 2 x 2 1 x 2 2 f p1x2 p2x1 where the parameters i i i 1 2 3 f may be td while the lie algebraic method is employed for the td forms of h1 and h2 the rationalization method modified here for the momentum dependent case is used for the timeindependent versions of h1 and h2 the role and scope of the invariants so constructed is pointed out

intrinsic characterization of the variable separation in the hamilton jacobi equation the nonorthogonal separation of variables in the hamilton jacobi equation corresponding to a natural hamiltonian h 12gijpipj v with a metric tensor of any signature is intrinsically characterized by geometrical objects on the riemannian configuration manifold killing vectors killing tensors and killing webs comparisons with previous characterizations and some illustrative examples are given

functionally separable solutions to nonlinear wave equations by group foliation method abstract we apply functional separation of variables within the approach of the group foliation method to the nonlinear wave equation with variable speed and external force u t t a x d u u x x b x q u a x 0 a classification of these equations admitting functionally separable solutions is performed and the resulting solutions are obtained in explicit form in many cases

removal of resonances by rotation in linearly degenerate two dimensional oscillator systems a system of two nonlinearly interacting resonant harmonic oscillators is investigated seeking transformation to approximate action angle variables in the vicinity of the equilibrium via the canonical perturbation theory a variety of polynomial perturbations dependent on parameters is considered the freedom of choice of the zero order approximation characteristic of a linearly degenerate resonant system is used to cancel lower order resonant terms in the canonical perturbation series it is found that the cancellation of the resonant terms is only possible for particular values of parameters of the interaction term these special sets of parameters include all the cases with the panleve property

nonlinear evolution equations admitting multilinear variable separable solutions based on the hirota s bilinear transformation method it is shown that a family of nonlinear partial differential equations admit multilinear variable separation solutions this extends the class of nonlinear partial differential equations which admit the multilinear variable separation solutions

on separable pauli equations we classify 1 3 dimensional pauli equations for a spin 12 particle interacting with the electro magnetic field that are solvable by the method of separation of variables as a result we obtain the 11 classes of vector potentials of the electro magnetic field a t x a0 t x a t x providing separability of the corresponding pauli equations it is established in particular that the necessary condition for the pauli equation to be separable into second order matrix ordinary differential equations is its equivalence to the system of two uncoupled schrodinger equations in addition the magnetic field has to be independent of spatial variables we prove that coordinate systems and the vector potentials of the electro magnetic field providing the separability of the corresponding pauli equations coincide with those for the schrodinger equations furthermore an efficient algorithm for constructing all coordinate systems providing the separability of pauli equation with a fixed vector potential of t

integrable and superintegrable quantum systems in a magnetic field integrable quantum mechanical systems with magnetic fields are constructed in two dimensional euclidean space the integral of motion is assumed to be a first or second order hermitian operator contrary to the case of purely scalar potentials quadratic integrability does not imply the separation of variables in the schrodinger equation moreover quantum and classical integrable systems do not necessarily coincide the hamiltonian can depend on the planck constant in a nontrivial manner

on separable solutions of dirac s equation for the electron the properties of coordinate systems that admit separation of the laplacian and hamilton jacobi operators have been thoroughly explored so that the nature of solutions in separable form of laplace s equation the wave equation schrodinger s equation and the hamilton jacobi equation are well understood the corresponding problems for the dirac operator in flat space time have been less completely examined and this paper contains studies intended to produce a more systematic account of possible solutions of dirac s equation because the dirac operator differs from the laplacian in being a first degree differential operator and in having matrix coefficients it is not possible to discuss possible solutions in as general a way and the separable solutions are far less rich than for equations with the laplacian in particular the forms of the potentials for which separable solutions are possible are not for the most part of physical interest although the discussion is confined to coordinates in flat space time some of the procedures are derived from those developed to solve dirac s equation in coordinates with a kerr metric

intertwined isospectral potentials in an arbitrary dimension the method of intertwining with n dimensional nd linear intertwining operator l is used to construct nd isospectral stationary potentials it has been proven that the differential part of l is a series in euclidean algebra generators integrability conditions of the consistency equations are investigated and the general form of a class of potentials respecting all these conditions have been specified for each n 2 3 4 5 the most general forms of 2d and 3d isospectral potentials are considered in detail and construction of their hierarchies is exhibited the followed approach provides coordinate systems which make it possible to perform separation of variables and to apply the known methods of supersymmetric quantum mechanics for 1d systems it has been shown that in choice of coordinates and l there are a number of alternatives increasing with n that enlarge the set of available potentials some salient features of higher dimensional extension as well as some applications of the results are presented

on the construction and training of reformulated radial basis function neural networks presents a systematic approach for constructing reformulated radial basis function rbf neural networks which was developed to facilitate their training by supervised learning algorithms based on gradient descent this approach reduces the construction of radial basis function models to the selection of admissible generator functions the selection of generator functions relies on the concept of the blind spot which is introduced in the paper the paper also introduces a new family of reformulated radial basis function neural networks which are referred to as cosine radial basis functions cosine radial basis functions are constructed by linear generator functions of a special form and their use as similarity measures in radial basis function models is justified by their geometric interpretation a set of experiments on a variety of datasets indicate that cosine radial basis functions outperform considerably conventional radial basis function neural networks with gaussian radial basis functions cosine radial basis functions are also strong competitors to existing reformulated radial basis function models trained by gradient descent and feedforward neural networks with sigmoid hidden units

generalized natural mechanical systems of two degrees of freedom with quadratic integrals this paper is devoted to the problem of constructing integrable mechanical systems with two degrees of freedom whose lagrangians contain terms linear in the velocities and whose second integrals have the form of polynomials of the second degree in the velocity variables in which the coefficients depend only on the coordinates the solution of this problem is known only in the case of reversible systems for which the linear terms do not affect lagrange s equations of motion using the method developed in a previous paper the authors classify all the possible irreversible systems into three types according to a certain normal form of the line element on the configuration space the most general systems of the first two types are constructed several many parameter systems of the third type are also found some of the new cases are found to be generalizations by the introduction of some additional parameters to well known integrable problems in particle and rigid body dynamics mechanical interpretation is also given for some other cases

non homogeneous systems of hydrodynamic type related to quadratic hamiltonians with electromagnetic term abstract we consider a class of non homogeneous systems of hydrodynamic type q t i j i q q x j i q i 1 n which can be related to quadratic hamiltonians with electromagnetic terms whilst it is unlikely that our systems are generally integrable they do possess intriguing properties such as always having a higher conservation law and sometimes a 2 n parameter family of exact solutions in fact our systems coincide with those possessing a conservation law with the densitylbeing a quadratic expression in the first derivatives 1 2 i j 1 n g i j q x i q x j k 1 n a k q x k h we discuss several examples some of which have important applications

pattern recognition with ordered labels abstract we consider a multi class pattern recognition problem with linearly ordered labels and a loss function which is a sum of squared deviations of decisions from true classes it is proved that the optimal in the bayesian sense decision rule is the a posteriori mean which is then rounded to the nearest integer then the plug in type estimator of the optimal decision rule is proposed we also investigate its asymptotic properties finally we discuss briefly the structure of a neural net which approximates the optimal decision rule

superintegrability in three dimensional euclidean space potentials for which the corresponding schrodinger equation is maximally superintegrable in three dimensional euclidean space are studied the quadratic algebra which is associated with each of these potentials is constructed and the bound state wave functions are computed in the separable coordinates

functional separation of variables for laplace equations in two dimensions the authors say that a solution psi of a partial differential equation in two real variables x1 x2 is functionally separable in these variables if psi x1 x2 phi a x1 b x2 for single variable functions phi a b such that phi a b not 0 they classify all possibilities for regular functional separation in local coordinates for equations of the form delta 2 psi f psi x1 x2 where delta 2 is the laplace beltrami operator on a two dimensional riemannian or pseudo riemannian space if the dependence of f on x1 x2 is non trivial then separation can occur for conformal cartesian coordinates on any space if f g psi then for orthogonal coordinates they find that true functional separation i e separation other than additive or multiplicative occurs precisely for cartesian coordinates in the euclidean and pseudo euclidean planes the sine gordon equation provides an example of this separation for many of these cases the separated solutions a b can be expressed in terms of elliptic functions for non orthogonal coordinates and f g psi true functional separation occurs precisely for cartesian coordinates in the pseudo euclidean plane and for a coordinate system on the hyperboloid of one sheet a pseudo riemannian space of constant curvature

direct reduction and differential constraints direct reductions of partial differential equations to systems of ordinary differential equations are in one to one correspondence with compatible differential constraints the differential constraint method is applied to prove that a parabolic evolution equation admits infinitely many characteristic second order reductions but admits a non characteristic second order reduction if and only if it is linearizable

on separable schrodinger maxwell equations we obtain the most general time dependent potential v t x1 x2 enabling separating of variables in the 1 2 dimensional schrodinger equation with the use of this result the four classes of separable schrodinger maxwell equations are constructed

n dimensional integrability from two photon coalgebra symmetry a wide class of hamiltonian systems with n degrees of freedom and endowed with at least n 2 functionally independent integrals of motion in involution is constructed by making use of the two photon lie poisson coalgebra h6 the set of n 2 constants of the motion is shown to be a universal one for all these hamiltonians irrespective of the dependence of the latter on several arbitrary functions and n free parameters within this large class of quasi integrable n dimensional hamiltonians new families of completely integrable systems are identified by finding explicitly a new independent integral through the analysis of the sub coalgebra structure of h6 in particular new completely integrable n dimensional hamiltonians describing natural systems geodesic flows and static electromagnetic hamiltonians are presented

separation of variables and exact solution of the dirac equation in nonstatic minkowski spacetimes using a second order formalism a complete separation of variables in the dirac equation for a free particle in nonstatic orthogonal curvilinear coordinates of the form t f u v x g u v y z is presented it is shown that the dirac equation is separable in eight nonequivalent coordinate systems where the klein gordon equation separates exact solutions of the dirac equation in the systems of coordinates obtained are presented

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

separation of variables in the hydrodynamic stability equations the problem of variable separation in the linear stability equations which govern the disturbance behaviour in viscous incompressible fluid flows is discussed the so called direct approach in which a form of the ansatz for a solution with separated variables as well as a form of reduced odes are postulated from the beginning is applied the results of application of the method are the new coordinate systems and the most general forms of basic flows which permit the postulated form of separation of variables then the basic flows are specified by the requirement that they themselves satisfy the navier stokes equations calculations are made for the 1 3 dimensional disturbance equations written in cartesian and cylindrical coordinates the fluid dynamics interpretation and stability properties of some classes of the exact solutions of the navier stokes equations defined as flows for which the stability analysis can be reduced via separation of variables to the eigenvalue problems of ordinary differential equations are discussed the eigenvalue problems are solved numerically with the help of the spectral collocation method based on chebyshev polynomials for some classes of perturbations the eigenvalue problems can be solved analytically those unique examples of exact explicit solution of the nonparallel unsteady flow stability problems provide a very useful test for numerical methods of solution of eigenvalue problems and for methods used in the hydrodynamic stability theory in general

separation of variables and exact solutions to quasilinear diffusion equations with nonlinear source abstract a solution of a partial differential equation with two real variables t and x is functionally separable in these variables if q u x t for some single variable functions q and in this paper the generalized conditional symmetry approach is used to study the separation of variables of quasilinear diffusion equations with nonlinear source we obtain a complete list of canonical forms for such equations which admit the functionally separable solutions as a result we get broad families of exact solutions to some quasilinear diffusion equations with nonlinear source the behavior and blow up properties of some solutions are described

new integrable hamiltonians with transcendental invariants on presente 3 hamiltoniens integrables a 2 dimensions dont le second invariant i 2 est rationnel ou transcendant en impulsion on trouve un troisieme invariant i 3

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

the general theory of r separation for helmholtz equations we develop the theory of r separation for the helmholtz equation on a pseudo riemannian manifold including the possibility of null coordinates and show that it and not ordinary variable separation is the natural analogy of additive separation for the hamilton jacobi equation we provide a coordinate free characterization of variable separation in terms of commuting symmetry operators

orthogonal least squares learning algorithm for radial basis function networks the radial basis function network offers a viable alternative to the two layer neural network in many applications of signal processing a common learning algorithm for radial basis function networks is based on first choosing randomly some data points as radial basis function centers and then using singular value decomposition to solve for the weights of the network such a procedure has several drawbacks and in particular an arbitrary selection of centers is clearly unsatisfactory the authors propose an alternative learning procedure based on the orthogonal least squares method the procedure chooses radial basis function centers one by one in a rational way until an adequate network has been constructed in the algorithm each selected center maximizes the increment to the explained variance or energy of the desired output and does not suffer numerical ill conditioning problems the orthogonal least squares learning strategy provides a simple and efficient means for fitting radial basis function networks this is illustrated using examples taken from two different signal processing applications

derivative dependent functional separable solutions for the kdv type equations abstract a complete list of canonical forms for a special kdv type of equations which admit derivative dependent functional separable solutions is obtained by using the generalized conditional symmetry approach some exact solutions to the resulting equations are explicitly obtained

on superintegrable symmetry breaking potentials in n dimensional euclidean space we give a graphical prescription for obtaining and characterizing all separable coordinates for which the schr dinger equation admits separable solutions for one of the superintegrable potentials here xn 1 is a distinguished cartesian variable the algebra of second order symmetries of the resulting schr dinger equation is given and for the first potential the closure relations of the corresponding quadratic algebra these potentials are particularly interesting because they occur in all dimensions n 1 the separation of variables problem is highly nontrivial for them and many other potentials are limiting cases

growing radial basis neural networks merging supervised and unsupervised learning with network growth techniques this paper proposes a framework for constructing and training radial basis function rbf neural networks the proposed growing radial basis function grbf network begins with a small number of prototypes which determine the locations of radial basis functions in the process of training the grbf network gross by splitting one of the prototypes at each growing cycle two splitting criteria are proposed to determine which prototype to split in each growing cycle the proposed hybrid learning scheme provides a framework for incorporating existing algorithms in the training of grbf networks these include unsupervised algorithms for clustering and learning vector quantization as well as learning algorithms for training single layer linear neural networks a supervised learning scheme based on the minimization of the localized class conditional variance is also proposed and tested grbf neural networks are evaluated and tested on a variety of data sets with very satisfactory results

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 hamiltonian systems with vector potentials we investigate integrable two dimensional hamiltonian systems with scalar and vector potentials admitting second invariants which are linear or quadratic in the momenta in the case of a linear second invariant we provide some examples of weakly integrable systems in the case of a quadratic second invariant we recover the classical strongly integrable systems in cartesian and polar coordinates and provide some new examples of integrable systems in parabolic and elliptical coordinates

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

on three dimensional quasi stackel hamiltonians a three dimensional integrable generalization of the stackel systems is proposed a classification of such systems is obtained which results in two families the first family is the direct sum of the two dimensional system which is equivalent to the representation of the schottky manakov top in the quasi stackel form and a stackel one dimensional system the second family is probably a new three dimensional system the system of hydrodynamic type which we get from this family in the usual way is a three dimensional generalization of the gibbons tsarev system a generalization of the quasi stackel systems to the case of any dimension is discussed

separation of variables in the kramers equation we consider the problem of separation of variables in the kramers equation admitting a non trivial symmetry group provided the external potential v x is at most quadratic a complete solution of the problem of separation of variables is obtained furthermore we construct solutions of the kramers equation with separated variables in explicit form

dirac equation in external vector fields separation of variables the method of separation of variables in the dirac equation in the external vector fields is developed through the search for exact solutions the essence of the method consists of the separation of the first order matricial differential operators that define the dependence of the dirac bispinor on the related variables but commutation of such operators with the operator of the equations or between them is not assumed this approach which is perfectly justified in the presence of gravitational fields permits one to prove rigorous theorems about necessary and sufficient conditions on the field functions that allow one to separate variables in the dirac equation in analogous investigations by other authors bagrov et al exact solutions of relativistic wave equations nauka novosibirst 1982 for electromagnetic fields an essential demand related to the operators that define the dependence of the bispinor on the separated variables is the demand for the commutation of a complete set of operators betw

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

an axiomatic approach to soft learning vector quantization and clustering this paper presents an axiomatic approach to soft learning vector quantization lvq and clustering based on reformulation the reformulation of the fuzzy c means fcm algorithm provides the basis for reformulating entropy constrained fuzzy clustering ecfc algorithms according to the proposed approach the development of specific algorithms reduces to the selection of a generator function linear generator functions lead to the fcm and fuzzy learning vector quantization algorithms while exponential generator functions lead to ecfc and entropy constrained learning vector quantization algorithms the reformulation of lvq and clustering algorithms also provides the basis for developing uncertainty measures that can identify feature vectors equidistant from all prototypes these measures are employed by a procedure developed to make soft lvq and clustering algorithms capable of identifying outliers in the data set this procedure is evaluated by testing the algorithms generated by linear and exponential generator functions on speech data

new similarity reductions of the boussinesq equation some new similarity reductions of the boussinesq equation which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering are presented these new similarity reductions including some new reductions to the first second and fourth painleve equations cannot be obtained using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method that involves no group theoretical techniques

conditional lie backlund symmetry and reduction of evolution equations we suggest a generalization of the notion of invariance of a given partial differential equation with respect to a lie backlund vector field such a generalization proves to be effective and enables us to construct principally new ansatz reducing evolution type equations to several ordinary differential equations in the framework of the said generalization we obtain principally new reductions of a number of nonlinear heat conductivity equations ut uxx f u ux with poor lie symmetry and obtain their exact solutions it is shown that these solutions cannot be constructed by means of the symmetry reduction procedure

non integrability of homogeneous two dimensional hamiltonians with velocity dependent potential the non integrability of classes of homogeneous two dimensional hamiltonian systems with a polynomial velocity dependent potential is shown on the basis of ziglin s theorem an analytic expression for the trace of the relevant monodromy matrices is presented which fits the numerical data perfectly an application is made to fokker planck hamiltonians with quadratic and with cubic drift terms

comment on integrable potentials with logarithmic integrals of motion the authors analyse and complete the results of an article by ichtiaroglou and voyatzis see ibid vol 21 p 3537 1988 concerning the existence of integrals of motion logarithmic in the momenta for two dimensional velocity dependent hamiltonians they find a new integrable case and establish the generality of some results found by ichtiaroglou and voyatzis which were presented as particular cases

evolving space filling curves to distribute radial basis functions over an input space an evolutionary neural network training algorithm is proposed for radial basis function rbf networks the locations of basis function centers are not directly encoded in a genetic string but are governed by space filling curves whose parameters evolve genetically this encoding causes each group of codetermined basis functions to evolve to fit a region of the input space a network produced from this encoding is evaluated by training its output connections only networks produced by this evolutionary algorithm appear to have better generalization performance on the mackey glass time series than corresponding networks whose centers are determined by k means clustering

quasiseparation of variables in the schrodinger equation with a magnetic field we consider a two dimensional integrable hamiltonian system with a vector and scalar potential in quantum mechanics contrary to the case of a pure scalar potential the existence of a second order integral of motion does not guarantee the separation of variables in the schrodinger equation we introduce the concept of quasiseparation of variables and show that in many cases it allows us to reduce the calculation of the energy spectrum and wave functions to linear algebra

superintegrable systems on 3 dimensional curved spaces eisenhart formalism and separability the eisenhart geometric formalism which transforms an euclidean natural hamiltonian h t v into a geodesic hamiltonian t with one additional degree of freedom is applied to the four families of quadratically superintegrable systems with multiple separabilities in the euclidean plane first the separability and superintegrability of such four geodesic hamiltonians t r r a b c d in a three dimensional curved space are studied and then these four systems are modified with the addition of a potential u r leading to h r t r u r second we study the superintegrability of the four hamiltonians h r h r r where r is a certain position dependent mass that enjoys the same separability as the original system h r all the hamiltonians here studied describe superintegrable systems on non euclidean three dimensional manifolds with a broken spherical symmetry

superintegrable 3d systems in a magnetic field corresponding to cartesian separation of variables we consider three dimensional superintegrable systems in a magnetic field we study the class of such systems which separate in cartesian coordinates in the limit when the magnetic field vanishes i e possess two second order integrals of motion of the cartesian type for such systems we look for additional integrals up to second order in momenta which make these systems minimally or maximally superintegrable and construct their polynomial algebras of integrals and their trajectories we observe that the structure of the leading order terms of the cartesian type integrals should be considered in a more general form than for the case without magnetic field

separation of variables in the nonlinear wave equation we develop a technique making it possible to handle the problem of separation of variables in nonlinear differential equations using it we obtain a number of new two dimensional nonlinear wave equations admitting separation of variables and construct their exact solutions

orthogonal and non orthogonal separation of variables in the wave equation utt uxx v x u 0utt uxx v x u 0 we develop a direct approach to the separation of variables in partial differential equations within the framework of this approach the problem of the separation of variables in the wave equation with time independent potential reduces to solving an over determined system of nonlinear differential equations we have succeeded in constructing its general solution and as a result all potentials v x permitting variable separation have been found for each of them we have constructed all inequivalent coordinate systems providing separability of the equation under study it should be noted that the above approach yields both orthogonal and non orthogonal systems of coordinates

two operand binary adders with threshold logic the central topic of this paper is the implementation of binary adders with threshold logic using a new methodology that introduces two innovations the use of the input and output carries of each bit for obtaining all the sum bits and a modification of the classic carry lookahead adder technique that allows us to obtain the expressions of the generation and propagation carries in a more appropriate way for threshold logic in this way it has been possible to systematize the process of design of a binary adder with threshold logic relating all its important parameters number of bits of the operands depth size maximum fan in and maximum weight the results obtained are an improvement on those published to date and are summarized as follows depth 2 adder s 2n w sub max 2 sup n f sub max 2n 1 depth 3 adder s 4n 2 n spl radic n w sub max 2 n spl radic n f sub max 2 n spl radic n 1 depth d adder asymptotic behavior s o n w sub max o 2 sup d 1 spl radic n f sub max o sup d 1 spl radic n if the weights are bounded by w sub max n sub max o log sup d 1 w sub max d sub min o log n log log w sub max

construction of new integrable hamiltonians in two degrees of freedom a new procedure for deriving integrable hamiltonians and their constants of the motion is introduced we term this procedure the truncation program integrable hamiltonians occurring in the truncation program possess constants of the motion which are polynomials in a perturbation parameter e the relationship between this program and the whittaker program in two degrees of freedom is discussed integrable hamiltonians occurring in the whittaker program a generalization of whittaker s work possess constants of the motion which are polynomials in the momentum coordinates many previously known integrable hamiltonians are derived a new family of integrable double resonance hamiltonians and a new family of integrable hamiltonians of the form p21 p22 2 v q1 q2 are derived

separation of variables in 1 2 dimensional schrodinger equations using our classification of separable schrodinger equations with two space dimensions published in j math phys 36 5506 1995 we give an exhaustive description of the coordinate systems providing their separability furthermore we apply these results to separate variables in the heat hamilton jacobi and fokker planck equations

variable separation for natural hamiltonians with scalar and vector potentials on riemannian manifolds the additive variable separation in the hamilton jacobi equation is studied for a natural hamiltonian with scalar and vector potentials on a riemannian manifold with positive definite metric the separation of this hamiltonian is related to the separation of a suitable geodesic hamiltonian over an extended riemannian manifold thus the geometrical theory of the geodesic separation is applied and the geometrical characterization of the separation is given in terms of killing webs killing tensors and killing vectors the results are applicable to the case of a nondegenarate separation on a manifold with indefinite metric where no null essential separable coordinates occur

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

generalized micz kepler system duality polynomial and deformed oscillator algebras we present the quadratic algebra of the generalized micz kepler system in three dimensional euclidean space e3 and its dual the four dimensional singular oscillator in four dimensional euclidean space e4 we present their realization in terms of a deformed oscillator algebra using the daskaloyannis construction the structure constants are in these cases functions not only of the hamiltonian but also of other integrals commuting with all generators of the quadratic algebra we also present a new algebraic derivation of the energy spectrum of the micz kepler system on the three sphere s3 using a quadratic algebra these results point out also that results and explicit formula for structure functions obtained for quadratic cubic and higher order polynomial algebras in the context of two dimensional superintegrable systems may be applied to superintegrable systems in higher dimensions with and without monopoles

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

dynamics of learning near singularities in layered networks we explicitly analyze the trajectories of learning near singularities in hierarchical networks such as multilayer perceptrons and radial basis function networks which include permutation symmetry of hidden nodes and show their general properties such symmetry induces singularities in their parameter space where the fisher information matrix degenerates and odd learning behaviors especially the existence of plateaus in gradient descent learning arise due to the geometric structure of singularity we plot dynamic vector fields to demonstrate the universal trajectories of learning near singularities the singularity induces two types of plateaus the on singularity plateau and the near singularity plateau depending on the stability of the singularity and the initial parameters of learning the results presented in this letter are universally applicable to a wide class of hierarchical models detailed stability analysis of the dynamics of learning in radial basis function networks and multilayer perceptrons will be presented in separate work