nondegenerate 2d complex euclidean superintegrable systems and algebraic varieties a classical or quantum superintegrable system is an integrable n dimensional hamiltonian system with potential that admits 2n 1 functionally independent constants of the motion polynomial in the momenta the maximum possible if the constants are all quadratic the system is second order superintegrable the kepler coulomb system is the best known example such systems have remarkable properties multi integrability and multi separability an algebra of higher order symmetries whose representation theory yields spectral information about the schrodinger operator deep connections with special functions and with qes systems for complex riemannian spaces with n 2 the structure and classification of second order superintegrable systems is complete here however we present a new and conceptually simpler approach to the classification for complex euclidean 2 space in which the possible superintegrable systems with nondegenerate potentials correspond to points on an algebraic variety specifically we determine a variety in six variables subject to two cubic and one quartic polynomial constraints each point on the variety corresponds to a superintegrable system the euclidean group e 2 acts on the variety such that two points determine the same superintegrable system if and only if they lie on the same leaf of the foliation

quadratic algebra as a hidden symmetry of the hartmann potential it is shown that operators commuting with the hamiltonian of the hartmann potential form the quadratic hahn algebra qh 3 the structure of this algebra and its finite dimensional representations are described an analysis of these representations is applied to obtain all the relevant physical results energy spectrum degree of degeneration and overlap functions

global boundedness of classical solutions to a two species cancer invasion haptotaxis model with tissue remodeling abstract in this paper we deal with a haptotaxis cancer invasion model describing the migration and proliferation of two families of cancer cells the epithelial mesenchymal transition between two families cancer cells the dynamics of matrix degrading enzymes and the evolution and re modeling of the extracellular matrix under appropriate regularity assumptions on initial data by making some a priori estimates and applying iterative techniques we establish the global existence and uniform boundedness of the unique classical solution in two dimensional spatial domain for arbitrary cancer cells proliferation rates and in three dimensional spatial setting for large cancer cells proliferation rates these results improve and extend previously known ones

polynomial poisson algebras for classical superintegrable systems with a third order integral of motion we present polynomial poisson algebras for the eight classical potentials in two dimensional euclidian space that separate in cartesian coordinates and allow a third order integral of motion some of the classical superintegrable potentials do not coincide with quantum ones but are their singular limits we show that all bounded trajectories in these potentials are periodic

high density limit theorems for infinite systems of unscaled branching brownian motions the fluctuations of an infinite system of unscaled branching brownian motions in rd are shown to converge weakly under a spatial central limit normalization when the initial density of particles tends to infinity the limit is a generalized gaussian process m which can be written as m mi m where mi is the fluctuation limit of a poisson system of brownian motions obtained by martin lof and mi arises from the spatial central limit normalization of the demographic variation process of the system in the critical case mi and m are independent and m coincides with the generalized ornstein uhlenbeck process found by dawson and by holley and stroock as the renormalization limit of an infinite system of critical branching brownian motions when d 2 3 generalized langevin equations for m mi and m are given

superintegrability in a two dimensional space of nonconstant curvature a hamiltonian with two degrees of freedom is said to be superintegrable if it admits three functionally independent integrals of the motion this property has been extensively studied in the case of two dimensional spaces of constant possibly zero curvature when all the independent integrals are either quadratic or linear in the canonical momenta in this article the first steps are taken to solve the problem of superintegrability of this type on an arbitrary curved manifold in two dimensions this is done by examining in detail one of the spaces of revolution found by g koenigs we determine that there are essentially three distinct potentials which when added to the free hamiltonian of this space have this type of superintegrability separation of variables for the associated hamilton jacobi and schrodinger equations is discussed the classical and quantum quadratic algebras associated with each of these potentials are determined

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

completeness of multiseparable superintegrability on the complex 2 sphere 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 here we demonstrate for nondegenerate potentials how to establish a complete list of such potentials that are superintegrable on the complex 2 sphere using essentially algebraic means we classify all such potentials that admit a pair of second order constants of motion here nondegenerate means that the potentials depend on four independent parameters the method of proof generalizes to other spaces and dimensions we show for the 2 sphere that all these superintegrable systems possess the remarkable property that they correspond to quadratic algebras and we work out the detailed structure relations and their quantum analogues

coherent states on horospheric three dimensional lobachevsky space in the paper it is shown that due to separation of variables in the laplace beltrami operator hamiltonian of a free quantum particle in horospheric and quasi cartesian coordinates of three dimensional lobachevsky space it is possible to introduce standard conventional according to perelomov generalized coherent states and their applications springer verlag 1986 p 320 coherent states some problems oscillator on horosphere charged particle in analogy of constant uniform magnetic field where coherent states are suitable for treating were considered

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

a law of large numbers for moderately interacting diffusion processes we consider two special models of interacting diffusion processes and derive in the limit as the number of different processes tends to infinity and the interaction is rescaled in a suitable moderate way a law of large numbers for the empirical processes as limit dynamics we obtain certain nonlinear diffusion equations

a particle system with explosions law of large numbers for the density of particles and the blow up time consider a system of independent random walks in the discrete torus with creation annihilation of particles and possible explosion of the total number of particles in finite time rescaling space and rates for diffusion creation annihilation of particles we obtain a strong law of large numbers for the density of particles in the supremum norm the limiting object is a classical solution to the semilinear heat equation t u xx u f u if f u u p 1 p 3 we also obtain a law of large numbers for the explosion time

exact and quasiexact solvability of second order superintegrable quantum systems i euclidean space preliminaries we show that second order superintegrable systems in two dimensional and three dimensional euclidean space generate both exactly solvable es and quasiexactly solvable qes problems in quantum mechanics via separation of variables and demonstrate the increased insight into the structure of such problems provided by superintegrability a principal advantage of our analysis using nondegenerate superintegrable systems is that they are multiseparable most past separation of variables treatments of qes problems via partial differential equations have only incorporated separability not multiseparability also we propose another definition of es and qes the quantum mechanical problem is called es if the solution of schrodinger equation can be expressed in terms of hypergeometric functions fnm and is qes if the schrodinger equation admits polynomial solutions with coefficients necessarily satisfying a three term or higher order of recurrence relations in three dimensions we give an example of a system tha

functional central limit theorem for brownian particles in domains with robin boundary condition abstract we rigorously derive non equilibrium space time fluctuation for the particle density of a system of reflected diffusions in bounded lipschitz domains in r d the particles are independent and are killed by a time dependent potential which is asymptotically proportional to the boundary local time we generalize the functional analytic framework introduced by kotelenez 20 21 to deal with time dependent perturbations our proof relies on dirichlet form method rather than the machineries derived from kotelenez s sub martingale inequality our result holds for any symmetric reflected diffusion for any bounded lipschitz domain and for any dimension d 1

group theory of the smorodinsky winternitz system the three degrees of freedom smorodinsky winternitz system is a degenerate or super integrable hamiltonian that possesses five functionally independent globally defined and single valued integrals of the motion in both classical and quantum mechanics this is explained in terms of a forced degeneracy imposed as a consequence of the invariance of the hamiltonian under a group of symmetry transformations isomorphic to the three dimensional unitary unimodular group su 3 in turn this degeneracy group is embedded in a larger group of transformations that maps all the bound energy levels among each other the so called dynamical group all the bound state eigenfunctions act as basis functions for a single irreducible representation of the dynamical group so in common with the hydrogen atom and the harmonic oscillator the quantum mechanics of the smorodinsky winternitz system may be completely solved within the framework of group theory alone

a two parameter family of travelling waves with a singular barrier arising from the modelling of extracellular matrix mediated cellular invasion abstract invasive cells variously show changes in adhesion protease production and motility in this paper the authors develop and analyse a model for malignant invasion brought about by a combination of proteolysis and haptotaxis a common feature of these two mechanisms is that they can be produced by contact with the extracellular matrix through the mediation of a class of surface receptors called integrins an unusual feature of the model is the absence of cell diffusion by seeking travelling wave solutions the model is reduced to a system of ordinary differential equations which can be studied using phase plane analysis the authors demonstrate the presence of a singular barrier in the phase plane and a hole in this singular barrier which admits a phase trajectory the model admits a family of travelling waves which depend on two parameters i e the tissue concentration of connective tissue and the rate of decay of the initial spatial profile of the invading cells the slowest member of this family corresponds to the phase trajectory which goes through the hole in the singular barrier using a power series method the authors derive an expression relating the minimum wavespeed to the tissue concentration of the extracellular matrix which is arbitrary the model is applicable in a wide variety of biological settings which combine haptotaxis with proteolysis by considering various functional forms the authors show that the key mathematical features of the particular model studied in the early parts of the paper are exhibited by a wider class of models which characterise the behaviour of invading cells

second order superintegrable systems in conformally flat spaces ii the classical two dimensional stackel transform this paper is one of a series that lays the groundwork for a structure and classification theory of second order superintegrable systems both classical and quantum in conformally flat spaces here we study the stackel transform or coupling constant metamorphosis as an invertible mapping between classical superintegrable systems on different spaces through the use of this tool we derive and classify for the first time all two dimensional 2d superintegrable systems the underlying spaces are exactly those derived by koenigs in his remarkable paper giving all 2d manifolds with zero potential that admit at least three second order symmetries our derivation is very simple and quite distinct we also show that every superintegrable system is the stackel transform of a superintegrable system on a constant curvature space

quadratic poisson algebras of two dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems the integrals of motion of the classical two dimensional superintegrable systems with quadratic integrals of motion close in a restrained quadratic poisson algebra whose the general form is investigated each classical superintegrable problem has a quantum counterpart a quantum superintegrable system the quadratic poisson algebra is deformed into a quantum associative algebra the finite dimensional representations of this algebra are calculated by using a deformed parafermion oscillator technique it is shown that the finite dimensional representations of the quadratic algebra are determined by the energy eigenvalues of the superintegrable system the calculation of energy eigenvalues is reduced to the solution of algebraic equations which are universal that is for all two dimensional superintegrable systems with quadratic integrals of motion

completeness of superintegrability in two dimensional constant curvature spaces we classify the hamiltonians h px2 py2 v x y of all classical superintegrable systems in two dimensional complex euclidean space with two additional second order constants of the motion we similarly classify the superintegrable hamiltonians h j12 j22 j32 v x y z on the complex two sphere where x2 y2 z2 1 this is achieved in all generality using properties of the complex euclidean group and the complex orthogonal group

comparison methods for a class of function valued stochastic partial differential equations a comparison theorem is derived for a class of function valued stochastic partial differential equations spde s with lipschitz coefficients driven by cylindrical and regular hilbert space valued brownian motions moreover we obtain necessary and sufficient conditions for the positivity of the mild solutions of the spde s where the sufficiency follows from the comparison theorem thereby it is e g possible to identify a class of spde s which can serve as stochastic space time models for the density of particles as a consequence we can construct unique mild solutions of spde s on the cone of positive functions with non lipschitz drift parts including the case of arbitrary polynomialsr x withr o o and leading negative coefficient

global solutions and asymptotic behavior for a parabolic degenerate coupled system arising from biology abstract in this paper we will focus on a parabolic degenerate system with respect to unknown functions u and w on a bounded domain of the two dimensional euclidean space this system appears as a mathematical model for some biological processes global existence and uniqueness of a nonnegative classical holder continuous solution are proved the last part of the paper is devoted to the study of the asymptotic behavior of the solutions

limit theorems for age structured populations we study populations consisting of individuals whose birth and death rates depend on both their age and the age structure of the whole population in particular as the population size tends to infinity we derive a law of large numbers and a central limit theorem for the empirical processes

density dependent limits for a nonlinear reaction diffusion model a reaction diffusion model constructed on a grid by linking nonlinear density dependent birth and death processes through particle diffusion is studied by proving two laws of large numbers and a fluctuation theorem under varying assumptions on the density parameter

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

comparison of stochastic and deterministic models of a linear chemical reaction with diffusion particles placed in n cells on the unit interval give birth or die according to linear rates adjacent cells are coupled by diffusion with a rate proportional to n2 cell numbers are divided by a density parameter to represent concentrations and the resulting space time markov process is compared to a corresponding deterministic model the solution to a partial differential equation the models are viewed as hilbert space valued processes and compared by means of a law of large numbers and central limit theorem new and nearly optimal results are obtained by exploiting the ornstein uhlenbeck type structure of the stochastic model

unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two dimensional manifold in this paper we prove that the two dimensional superintegrable systems with quadratic integrals of motion on a manifold can be classified by using the poisson algebra of the integrals of motion there are six general fundamental classes of superintegrable systems analytic formulas for the involved integrals are calculated in all the cases all the known superintegrable systems are classified as special cases of these six general classes

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

separation of variables on n dimensional riemannian manifolds i the n sphere sn and euclidean n space rn the following problem is solved what are all the different separable coordinate systems for the laplace beltrami eigenvalue equation on the n sphere sn and euclidean n space rn and how are they constructed this is achieved through a combination of differential geometric and group theoretic methods a graphical procedure for construction of these systems is developed that generalizes vilenkin s construction of polyspherical coordinates the significance of these results for exactly soluble dynamical systems on these manifolds is pointed out the results are also of importance for the analysis of the special functions appearing in the separable solutions of the laplace beltrami eigenvalue equation on these manifolds

invariant classification of second order conformally flat superintegrable systems in this paper we continue the work of kalnins et al in classifying all second order conformally superintegrable laplace type systems over conformally flat spaces using tools from algebraic geometry and classical invariant theory the results obtained show through stackel equivalence that the list of known nondegenerate superintegrable systems over three dimensional conformally flat spaces is complete in particular a seven dimensional manifold is determined such that each point corresponds to a conformal class of superintegrable systems this manifold is foliated by the nonlinear action of the conformal group in three dimensions two systems lie in the same conformal class if and only if they lie in the same leaf of the foliation this foliation is explicitly described using algebraic varieties formed from representations of the conformal group the proof of these results rely heavily on grobner basis calculations using the computer algebra software packages maple and singular

stability of solutions of chemotaxis equations in reinforced random walks in this paper we consider a nonlinear system of differential equations consisting of one parabolic equation and one ordinary differential equation the system arises in chemotaxis a process whereby living organisms respond to chemical substance by moving toward higher or lower concentrations of the chemical substance or by aggregating or dispersing we prove that stationary solutions of the system are asymptotically stable

maximal regularity for evolution equations by interpolation and extrapolation on considere le probleme aux valeurs initiales pour l equation d evolution abstraite u t b t u t f t 0 t t ou b t est pour tout t le generateur infinitesimal d un semi groupe analytique dans un b espace e on considere le cas ou le domaine d b t de b t ne depend pas de t

the complete whittaker theorem for two dimensional integrable systems and its application the authors prove for a particle moving in a plane under the influence of a conservative force that when the motion is constrained by a second invariant quadratic in the velocities then the potential allows separability of the hamilton jacobi equation in rectangular polar elliptical cylinder or parabolic cylinder coordinates this link shows the intimate connection between quadratic invariants and the two dimensional hamilton jacobi equation they give examples of the utility of parabolic cylinder coordinates in cases of recent study

a density dependent chemotaxis haptotaxis system modeling cancer invasion abstract this paper deals with a chemotaxis haptotaxis model of cancer invasion of tissue initially proposed by chaplain and lolas the model consists of three reaction diffusion taxis partial differential equations describing interactions between cancer cells matrix degrading enzymes and the host tissue the equation for cell density includes two bounded nonlinear density dependent chemotactic and haptotactic sensitivity functions in the presence of logistic damping we prove the global existence of a unique classical solution to this model by some delicate a priori estimate techniques

superintegrable systems in darboux spaces almost all research on superintegrable potentials concerns spaces of constant curvature in this paper we find by exhaustive calculation all superintegrable potentials in the four darboux spaces of revolution that have at least two integrals of motion quadratic in the momenta in addition to the hamiltonian these are two dimensional spaces of nonconstant curvature it turns out that all of these potentials are equivalent to superintegrable potentials in complex euclidean 2 space or on the complex 2 sphere via coupling constant metamorphosis or equivalently via stackel multiplier transformations we present a table of the results

nondegenerate three dimensional complex euclidean superintegrable systems and algebraic varieties a classical or quantum second order superintegrable system is an integrable n dimensional hamiltonian system with potential that admits 2n 1 functionally independent second order constants of the motion polynomial in the momenta the maximum possible such systems have remarkable properties multi integrability and multiseparability an algebra of higher order symmetries whose representation theory yields spectral information about the schrodinger operator deep connections with special functions and with quasiexactly solvable systems here we announce a complete classification of nondegenerate i e four parameter potentials for complex euclidean 3 space we characterize the possible superintegrable systems as points on an algebraic variety in ten variables subject to six quadratic polynomial constraints the euclidean group acts on the variety such that two points determine the same superintegrable system if and only if they lie on the same leaf of the foliation there are exactly ten nondegenerat

global dynamics of a hyperbolic parabolic model arising from chemotaxis we prove global existence and qualitative behavior of classical solutions for a hyperbolic parabolic system describing chemotaxis on bounded domains it is shown that classical solutions to the initial boundary value problem of the one dimensional model exist globally in time for large initial data and the solutions converge to constant equilibrium states exponentially in time which rigorously demonstrates the collapsing of cell populations in chemotaxis moreover similar results are established for the multidimensional model when the initial data are small

linear parabolic differential equations as limits of space time jump markov processes abstract a class of linear parabolic differential equations on a bounded domain in rn is obtained as the class of deterministic limits of space time jump markov processes xn these xn describe particle systems which are spatially inhomogeneous due to diffusion and random change in the number of particles the deviation of xn from its deterministic limit is measured by a central limit theorem this fluctation limit is a distribution valued generalized ornstein uhlenbeck process and can be represented as the mild solution of a stochastic partial differential equation whose driving term is the sum of two independent gaussian martingales arising from diffusion and change in the number of particles respectively

wilson polynomials and the generic superintegrable system on the 2 sphere for complex two dimensional riemannian spaces every classical or quantum second order superintegrable system can be obtained from a single generic 3 parameter potential on the complex 2 sphere by delicate limit operations and through stackel transforms between manifolds here we derive families of finite and infinite dimensional irreducible representations of the corresponding quadratic quantum algebra for the 2 sphere and point out their role in explaining the degeneracy of the energy eigenspaces corresponding to bound state and continuous spectra of quantum and wave equation analogs of this system the algebra is exactly the one that describes the wilson and racah polynomials in their full generality

blow up criterion for a hyperbolic parabolic system arising from chemotaxis abstract in this paper we investigate a hyperbolic parabolic system arising from chemotaxis a blow up criterion of local strong solutions to the model in r d is established based on the scaling invariance of the system as a by product we prove a global in time well posedness result when d 1 for large initial data

langevin equations for l valued gaussian processes and fluctuation limits of infinite particle systems l gaussian processes of a certain class are shown to satisfy generalized langevin equations examples are fluctuation limits of several infinite particle systems in particular infinite particle branching brownian motions with immigration under various scalings and the voter model with hydrodynamic scaling

tightness of probabilities on c lbrack 0 1 rbrack mathscr y and d lbrack 0 1 rbrack mathscr y let c mathscr i c lbrack 0 1 rbrack mathscr i be the space of all continuous mappings of lbrack 0 1 rbrack to mathscr i where mathscr l is the topological dual of the schwartz space mathscr i of all rapidly decreasing functions let c be the banach space of all continuous functions on lbrack 0 1 rbrack for each varphi in mathscr i pi varphi is defined by pi varphi x in c mathscr i rightarrow x varphi in c given a sequence of probability measures p n on c mathscr i such that for each varphi in mathscr i p n pi 1 varphi is tight in c we prove that p n itself is tight in c mathscr i a similar result is proved for the space of all right continuous mappings of lbrack 0 1 rbrack to mathscr i

on the derivation of reaction diffusion equations as limit dynamics of systems of moderately interacting stochastic processes we consider systems of moderately interacting particles which are divided into a finite number of different subpopulations and show that in the limit as the population size tends to infinity the empirical processes of the subpopulations converge to the solution of a system of reaction diffusion equations

superintegrability and associated polynomial solutions euclidean space and the sphere in two dimensions in this work we examine the basis functions for those classical and quantum mechanical systems in two dimensions which admit separation of variables in at least two coordinate systems we do this for the corresponding systems defined in euclidean space and on the two dimensional sphere we present all of these cases from a unified point of view in particular all of the special functions that arise via variable separation have their essential features expressed in terms of their zeros the principal new results are the details of the polynomial bases for each of the nonsubgroup bases not just the subgroup cartesian and polar coordinate cases and the details of the structure of the quadratic algebras we also study the polynomial eigenfunctions in elliptic coordinates of the n dimensional isotropic quantum oscillator

convergence of s valued processes and space time random fields abstract the mapping x x of d 0 t s r d into s r d 1 defined by x 0 t x t t dt s r d 1 is shown to be continuous this is used to prove that for a tight sequence of processes x n inn in d 0 t s r d weak convergence of x n n in s r d 1 implies weak convergence of x n when 0 t is replaced by 0 the space s r d 1 is not appropriate in general and we investigate the question of what is the smallest nuclear space of distributions containing s r d 1 that can be used our motivation comes from the asymptotic analysis of particle systems

superintegrable n 2 systems quadratic constants of motion and potentials of drach the properties of superintegrable systems in two degrees of freedom possessing three independent globally defined constants of motion are studied using as an approach the existence of hidden symmetries and the generalized noether s theorem the potentials are obtained as solution of a system of two partial differential equations first the case of standard lagrangians is studied and then the method is applied to the case of lagrangians with a pseudo euclidean kinetic term finally the results are related with other approaches and with a family of potentials admitting a second integral of motion cubic in the velocities obtained by drach

quantum superintegrable systems with quadratic integrals on a two dimensional manifold there are two classes of quantum integrable systems on a manifold with quadratic integrals the liouville and the lie integrable systems as it happens in the classical case the quantum liouville quadratic integrable systems are defined on a liouville manifold and the schrodinger equation can be solved by separation of variables in one coordinate system the lie integrable systems are defined on a lie manifold and are not generally separable ones but can be solved therefore there are superintegrable systems with two quadratic integrals of motion not necessarily separable in two coordinate systems the quantum analogs of the two dimensional superintegrable systems with quadratic integrals of motion on a manifold are classified by using the quadratic associative algebra of the integrals of motion there are six general fundamental classes of quantum superintegrable systems corresponding to the classical ones analytic formulas for the involved integrals are calculated in all the cases all the known quan

a fluctuation theorem for moderately interacting diffusion processes we study the asymptotic behaviour of the fluctuations of the empirical processes of the second example of moderately interacting diffusion processes of 3 and show that in the limit as the number of different processes tends to infinity these fluctuations behave gaussian

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

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

energy type estimates and global solvability in a two dimensional chemotaxis haptotaxis model with remodeling of non diffusible attractant this paper deals with the coupled chemotaxis haptotaxis model ut u u v u w u 1 u w x t 0 0 v u v x t 0 wt vw w 1 w u x t 0 which was initially proposed by chaplain and lolas 2006 10 to describe the interactions between cancer cells the matrix degrading enzyme and the host tissue in a process of cancer cell invasion of tissue extracellular matrix here r2 is a bounded domain with smooth boundary and and are positive parameters as compared to previous mathematical studies the novelty here consists of allowing for positive values of reflecting processes of self remodeling of the extracellular matrix under zero flux boundary conditions it is shown that for any sufficiently smooth initial data u0 w0 satisfying first order compatibility conditions the model admits a unique global smooth solution a crucial ingredient in the proof is an energy like inequality which given t 0 yields boundedness of u t in llogl this serves as a starting point for a bootstrap argument used to derive higher regularity estimates sufficient for global extensibility of solutions

hidden symmetry separation of variables and interbasis expansions in the two dimensional hydrogen atom expansions for each fundamental basis of the hydrogen atom over two others are found and an additional integral of motion corresponding to an elliptic basis is determined rcpresentations of the elliptic basis as a superposition of polar and parabolic states are obtained certain interesting limiting cases are investigated

systems of interacting diffusions with partial annihilation through membranes we introduce an interacting particle system in which two families of reflected diffusions interact in a singular manner near a deterministic interface ii this system can be used to model the transport of positive and negative charges in a solar cell or the population dynamics of two segregated species under competition a related interacting random walk model with discrete state spaces has recently been introduced and studied in chen and fan 2014 in this paper we establish the functional law of large numbers for this new system thereby extending the hydrodynamic limit in chen and fan 2014 to reflected diffusions in domains with mixed type boundary conditions which include absorption harvest of electric charges we employ a new and direct approach that avoids going through the delicate bbgky hierarchy

high density limit theorems for nonlinear chemical reactions with diffusion the solutionx of a nonlinear reaction diffusion equation on then dimensional unit cubes is approximated by a space time jump markov processx v n law of large numbers lln x v n is constructed on a grids n ons ofn cells wherev is proportional to the initial number of particles in each cell the deviation ofx v n fromx is computed by a central limit theorem clt the assumptions on the parametersv n are for the lln asn and for the clt frac n upsilon to 0 asn the limity y x in the clt which is a generalized ornstein uhlenbeck process is represented as the mild solution of a linear stochastic partial differential equation spde and its best possible state spaces are described the problem of stationary solutions ofy x in dependence ofx is also investigated

fine structure for 3d second order superintegrable systems three parameter potentials a classical or quantum superintegrable system of second order is an integrable n dimensional hamiltonian system with potential that admits 2n 1 functionally independent constants of the motion quadratic in the momenta the maximum possible for n 3 on conformally flat spaces with nondegenerate i e four parameter potentials the extreme case we have worked out the structure and classified most of the possible spaces and potentials here we extend the analysis to a more degenerate class of functionally linearly independent superintegrable systems the three parameter potential case we show that for true three parameter potentials the algebra of constants of the motion no longer closes at order 6 but still all such systems are stackel transforms of systems on complex euclidean space or the complex 3 sphere this is a significant step towards the complete structure analysis of all types of second order superintegrable systems