finite size corrections to poisson approximations in general renewal success processes abstract consider a renewal process and let k 0 denote the random duration of a typical renewal cycle assume that on any renewal cycle a rare event called success can occur such successes lend themselves naturally to approximation by poisson point processes if each success occurs after a random delay however poisson convergence can be relatively slow because each success corresponds to a time interval not a point if k is an arithmetic variable a finite size correction fsc is known to speed poisson convergence by providing a second subdominant term in the appropriate asymptotic expansion this paper generalizes the fsc from arithmetic k to general k genomics applications require this generalization because they have already heuristically applied the fsc to p values involving absolutely continuous distributions the fsc also sharpens certain results in queuing theory insurance risk traffic flow and reliability theory

construction of classical superintegrable systems with higher order integrals of motion from ladder operators we construct integrals of motion for multidimensional classical systems from ladder operators of one dimensional systems this method can be used to obtain new systems with higher order integrals we show how these integrals generate a polynomial poisson algebra we consider a one dimensional system with third order ladder operators and found a family of superintegrable systems with higher order integrals of motion we obtain also the polynomial algebra generated by these integrals we calculate numerically the trajectories and show that all bounded trajectories are closed

superintegrability and higher order integrals for quantum systems we refine a method for finding a canonical form of symmetry operators of arbitrary order for the schrodinger eigenvalue equation h 2 v e on any 2d riemannian manifold real or complex that admits a separation of variables in some orthogonal coordinate system the flat space equations with potentials v x iy k 1 x iy k 1 in cartesian coordinates and v r2 r2cos 2k r2sin 2k the tremblay turbiner and winternitz system in polar coordinates have each been shown to be classically superintegrable for all rational numbers k we apply the canonical operator method to give a constructive proof that each of these systems is also quantum superintegrable for all rational k we develop the classical analog of the quantum canonical form for a symmetry it is clear that our methods will generalize to other hamiltonian systems

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

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

a new superintegrable hamiltonian we identify a new superintegrable hamiltonian in three degrees of freedom obtained as a reduction of pure keplerian motion in six dimensions the new hamiltonian is a generalization of the keplerian one and has the familiar 1 r potential with three barrier terms preventing the particle crossing the principal planes in three degrees of freedom there are five functionally independent integrals of motion and all bound classical trajectories are closed and strictly periodic the generalization of the laplace runge lenz vector is identified and shown to provide functionally independent isolating integrals they are quartic in the momenta and do not arise from separability of the hamilton jacobi equation a formulation of the system in action angle variables is presented

master symmetries non hamiltonian symmetries and superintegrability of the generalized smorodinsky winternitz system the first part of the paper is devoted to the theory of master symmetries using the geometric formalism as an approach it is shown that certain superintegrable systems are endowed with this property as a consequence of the existence of a family of master symmetries in the second part the properties of dynamical but non hamiltonian symmetries are studied it is proved that the higher order superintegrability of the generalized smorodinsky winternitz system is a consequence of the existence of symplectic symmetries not preserving the hamiltonian function

a new approach to the higher order superintegrability of the tremblay turbiner winternitz system the higher order superintegrability of systems separable in polar coordinates is studied using an approach that was previously applied for the study of the superintegrability of a generalized smorodinsky winternitz system the idea is that the additional constant of motion can be factorized as the product of powers of two particular rather simple complex functions here denoted by m and n this technique leads to a proof of the superintegrability of the tremblay turbiner winternitz system and to the explicit expression of the constants of motion a second family related with the first one of superintegrable systems is also studied

reduction of superintegrable systems the anisotropic harmonic oscillator we introduce a 2n parametric family of maximally superintegrable systems in n dimensions obtained as a reduction of an anisotropic harmonic oscillator in a 2n dimensional configuration space these systems possess closed bounded orbits and integrals of motion which are polynomial in the momenta they generalize known examples of superintegrable models in the euclidean plane

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

polynomial constants of motion for calogero type systems in three dimensions we give an explicit and concise formula for higher degree polynomial first integrals of a family of calogero type hamiltonian systems in dimension three these first integrals together with the already known ones prove the maximal superintegrability of the systems

n 2 supersymmetric extension of the tremblay turbiner winternitz hamiltonians on a plane the family of tremblay turbiner winternitz hamiltonians hk on a plane corresponding to any positive real value of k is shown to admit an supersymmetric extension of the same kind as that introduced by freedman and mende for the calogero problem and based on an superalgebra the irreducible representations of the latter are characterized by the quantum number specifying the eigenvalues of the first integral of motion xk of hk bases for them are explicitly constructed the ground state of each supersymmetrized hamiltonian is shown to belong to an atypical lowest weight state irreducible representation

a class of superintegrable systems of calogero type we show that the three body calogero model with inverse square potentials can be interpreted as a maximally superintegrable and multiseparable system in euclidean three space as such it is a special case of a family of systems involving one arbitrary function of one variable

on the superintegrability of calogero moser sutherland model the superintegrability of calogero moser sutherland hyperbolic model and nonsuperintegrability of trigonometric one are shown

superintegrable deformations of superintegrable systems quadratic superintegrability and higher order superintegrability the superintegrability of four hamiltonians hr hr r a b c d where hr are known hamiltonians and is a certain function defined on the configuration space and depended on a parameter is studied the new hamiltonians and the associated constants of motion jri i 1 2 3 are continous functions of the parameter the first part is concerned with separability and quadratic superintegrability the integrals of motion are quadratic in the momenta and the second part is devoted to the existence of higher order superintegrability the results obtained in the second part are related with the tremblay turbiner winternitz and the post winternitz systems

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

third order superintegrable systems separable in parabolic coordinates in this paper we investigate superintegrable systems which separate in parabolic coordinates and admit a third order integral of motion we give the corresponding determining equations and show that all such systems are multi separable and so admit two second order integrals the third order integral is their lie or poisson commutator we discuss how this situation is different from the cartesian and polar cases where new potentials were discovered which are not multi separable and which are expressed in terms of painleve transcendents or elliptic functions

lagrangian formalism for nonlinear second order riccati systems one dimensional integrability and two dimensional superintegrability the existence of a lagrangian description for the second order riccati equation is analyzed and the results are applied to the study of two different nonlinear systems both related with the generalized riccati equation the lagrangians are non natural and the forces are not derivable from a potential the constant value e of a preserved energy function can be used as an appropriate parameter for characterizing the behavior of the solutions of these two systems in the second part the existence of two dimensional versions endowed with superintegrability is proved the explicit expressions of the additional integrals are obtained in both cases finally it is proved that the orbits of the second system that represents a nonlinear oscillator can be considered as nonlinear lissajous figures

two dimensional quantum gravity and quasiclassical integrable hierarchies the main results for the two dimensional quantum gravity conjectured from the matrix model or integrable approach are presented in the form to be compared with the worldsheet or liouville approach in a spherical limit the integrable side for minimal string theories is completely formulated using simple manipulations with two polynomials based on residue formulae from quasiclassical hierarchies explicit computations for particular models are performed and certain delicate issues of nontrivial relations among them are discussed they concern the connections between different theories obtained as expansions of basically the same stringy solution to dispersionless kadomtsev petviashvili hierarchy in different backgrounds characterized by nonvanishing background values of different times being the simplest known example of the change of the quantum numbers of physical observables when moving to a different point in the moduli space of the theory

an infinite family of solvable and integrable quantum systems on a plane an infinite family of exactly solvable and integrable potentials on a plane is introduced it is shown that all already known rational potentials with the above properties allowing separation of variables in polar coordinates are particular cases of this family the underlying algebraic structure of the new potentials is revealed as well as its hidden algebra we conjecture that all members of the family are also superintegrable and demonstrate this for the first few cases a quasi exactly solvable and integrable generalization of the family is found

third order superintegrable systems separating in polar coordinates a complete classification of quantum and classical superintegrable systems in e2 is presented that allow the separation of variables in polar coordinates and admit an additional integral of motion of order 3 in the momentum new quantum superintegrable systems are discovered for which the potential is expressed in terms of the sixth painleve transcendent or in terms of the weierstrass elliptic function

objective method for estimating asymptotic parameters with an application to sequence alignment sequence alignment is an indispensable computational tool in modern molecular biology the model underlying biological sequence alignment is of interest to physicists because it approximates the statistical mechanics of dna and protein annealing while bearing an intimate relationship to models of directed polymers in random media recent methods for determining the statistics of random sequence alignments have reduced the computation time to less than 1 s opening up some interesting possibilities for online computation with biological search engines before implementation however the methods required an objective technique for computing regression coefficients pertinent to an asymptotic regime typically physicists estimate parameters pertinent to an asymptotic regime subjectively they eyeball their data estimate the asymptotic regime where the regression model holds with reasonable accuracy and then regress data only within the estimated asymptotic regime our publicly available computer program arrp replaces the subjective assessment of the asymptotic regime with an objective change point detection method increasing confidence in the scientific objectivity of the parameter estimates asymptotic regression has potential applications across most of physics

periodic orbits for an infinite family of classical superintegrable systems we show that all bounded trajectories in the two dimensional classical system with the potential are closed for all integer and rational values of k the period is and does not depend on k this agrees with our earlier conjecture suggesting that the quantum version of this system is superintegrable