continuous time random walks on networks with vertex and time dependent forcing we have investigated the transport of particles moving as random walks on the vertices of a network subject to vertex and time dependent forcing we have derived the generalized master equations for this transport using continuous time random walks characterized by jump and waiting time densities as the underlying stochastic process the forcing is incorporated through a vertex and time dependent bias in the jump densities governing the random walking particles as a particular case we consider particle forcing proportional to the concentration of particles on adjacent vertices analogous to self chemotactic attraction in a spatial continuum our algebraic and numerical studies of this system reveal an interesting pair aggregation pattern formation in which the steady state is composed of a high concentration of particles on a small number of isolated pairs of adjacent vertices the steady states do not exhibit this pair aggregation if the transport is random on the vertices i e without forcing the manifestation of pair aggregation on a transport network may thus be a signature of self chemotactic like forcing

aggregation blowup and collapse the abc s of taxis in reinforced random walks in many biological systems movement of an organism occurs in response to a diffusible or otherwise transported signal and in its simplest form this can be modeled by diffusion equations with advection terms of the form first derived by patlak bull of math biophys 15 1953 pp 311 338 however other systems are more accurately modeled by random walkers that deposit a nondiffusible signal that modifies the local environment for succeeding passages in these systems one example of which is the myxobacteria the question arises as to whether aggregation is possible under suitable hypotheses on the transition rules and the production of a control species that modulates the transition rates davis probab theory related fields 84 1990 pp 203 229 has studied this question for a certain class of random walks and here we extend this analysis to the continuum limit of such walks we first derive several general classes of partial differential equations that depend on how the movement rules are

eigenvalue bounds for transformations of solvable potentials we study smooth transformations of potentials for which exact bound state solutions of schrodinger s equation are known eigenvalue approximation formulae are obtained which provide lower or upper energy bounds according to whether the transformation function g is convex or concave detailed results are presented for perturbed coulomb potentials of the form and

subdiffusive master equation with space dependent anomalous exponent and structural instability we derive the fractional master equation with space dependent anomalous exponent we analyze the asymptotic behavior of the corresponding lattice model both analytically and by monte carlo simulation we show that the subdiffusive fractional equations with constant anomalous exponent in a bounded domain 0 l are not structurally stable with respect to the nonhomogeneous variations of parameter in particular the gibbs boltzmann distribution is no longer the stationary solution of the fractional fokker planck equation whatever the space variation of the exponent might be we analyze the random distribution of in space and find that in the long time limit the probability distribution is highly intermediate in space and the behavior is completely dominated by very unlikely events we show that subdiffusive fractional equations with the nonuniform random distribution of anomalous exponent is an illustration of a black swan the low probability event of the small value of the anomalous exponent that completely dominates the long time behavior of subdiffusive systems

a counter example to the theorem of hiemer and snurnikov a planar polygonal billiard mathcal p is said to have the finite blocking property if for every pair o a of points in mathcal p there exists a finite number of blocking points b 1 b n such that every billiard trajectory from o to a meets one of the b i s as a counter example to a theorem of hiemer and snurnikov we construct a family of rational billiards that lack the finite blocking property

pt symmetric pseudo perturbation recipe an imaginary cubic oscillator with spikes the pseudo perturbative shifted l expansion technique pslet is shown to be applicable in the non hermitian symmetric context the construction of bound states for several symmetric potentials is presented with special attention paid to v r ir3 ir oscillators

the exact solution of two new types of schrodinger equation a compact solution of a special biconfluent heun equation is obtained by means of a laplace transform the eigenvalues and eigenfunctions of two new types of schrodinger equation are deduced by appropriate variable transformations the associated potential function in each case can be arranged to have a double well

the shifted 1 n expansion and the energy eigenvalues of the hulthen potential for l not 0 the shifted 1 n expansion has been used to obtain the energy values of the hulthen potential for non zero values of the orbital angular momentum the analytic expression for the energies en l yield fairly accurate results for a wide range of values of n l and the screening parameter lambda the energy values obtained by this method have been compared and found to be in excellent agreement with pade approximation calculations

on topological entropy of billiard tables with small inner scatterers abstract we present in this paper an approach to studying the topological entropy of a class of billiard systems in this class any billiard table consists of strictly convex domain in the plane and strictly convex inner scatterers combining the concept of anti integrable limit with the theory of lyusternik shnirel man we show that a billiard system in this class generically admits a set of non degenerate anti integrable orbits which corresponds bijectively to a topological markov chain of arbitrarily large topological entropy the anti integrable limit is the singular limit when scatterers shrink to points in order to get around the singular limit and so as to apply the implicit function theorem on auxiliary circles encircling these scatterers we define a length functional whose critical points are well defined at the anti integrable limit and give rise to billiard orbits when the scatterers are not points consequently we prove the topological entropy of the first return map to the scatterers can be made arbitrarily large provided the inner scatterers are sufficiently small

turing instability in sub diffusive reaction diffusion systems linear stability characteristics of an anomalous sub diffusive activator inhibitor system are investigated through a mesoscopic model where anomaly ensues by a memory integro differential operator with an equal anomaly exponent for all species it is shown that monotonic instability known from normal diffusion persists in the anomalous system thereby allowing turing pattern occurrence presence of anomaly stabilizes the system by diminution of the range of diffusion coefficients ratio involving instability in addition the maximal growth rate is diminished proving the existence of an absolutely stable anomalous system

eigenvalues of the schrodinger equation with coulomb potentials plus linear and harmonic radial terms analytic approximations for the eigenvalues have been obtained using the two point quasifractional approximants method here we extend and generalize previous work where the method was applied to determine eigenenergies of two dimensional coulomb potentials in the presence of a magnetic field of arbitrary strength we have also shown the advantage of our results with respect to other calculations using the shifted 1 n method or pade approximants

two interacting electrons in a uniform magnetic field and a parabolic potential the general closed form solution we present an analytical analysis of the two dimensional schrodinger equation for two interacting electrons subjected to a homogeneous magnetic field and confined by a two dimensional external parabolic potential we have found the general closed form expression for the eigenstates of the problem and its corresponding eigenenergies for particular values of magnetic field and spatial confinement length the mathematical framework is just based on a rigorous solution of the three term recursion relation among the coefficients that arises from the series solution of biconfluent heun bhe equation connected with the radial part of the schrodinger equation for the internal motion it is also shown that by vanishing of coulomb repulsion strength the obtained explicit analytical solutions of bhe equation reduces to the well known polynomials satisfying the associated laguerre differential equation furthermore in the presence of this interaction the results are compared with those previously obtained in th

the asymptotic iteration method for the eigenenergies of the schrodinger equation with the potential v r z r gr λr2 the eigenenergies of the schrodinger equation with coulomb potential plus linear and harmonic radial terms are investigated using the perturbation technique within the framework of the asymptotic iteration method the obtained eigenenergies are compared with those obtained from the shifted 1 n expansion method the moment method and also with the exact values obtained from the super symmetric method it is found that asymptotically this method gives accurate results for arbitrary g z and

a basis for variational calculations in d dimensions in this paper we derive expressions for matrix elements i hj for the hamiltonian h qa q rq in d 2 dimensions the basis functions in each angular momentum subspace are of the form the matrix elements are given in terms of the gamma function for all d the significance of the parameters t and p and scale s are discussed applications to a variety of potentials are presented including potentials with singular repulsive terms of the form r 0 perturbed coulomb potentials d r br ar2 and potentials with weak repulsive terms such as r2 r4 0

quantum states of a sextic potential hidden symmetry and quantum monodromy a known terminating polynomial ansatz for a finite sub set of states of the sextic central potential v a b c r ar2 br4 cr6 c 0 subject to integer related parameter constraints is confirmed to apply in two as well as three dimensions a hidden symmetry relating the eigenstates of v a b c r to those of v a b c r is used to establish the boundary between the ansatz sub set and the infinite set of higher states connections between the radial wavefunctions rn a b c r and rn a b c r also provide semiclassical insight into the origin of the parameter constraint finally the ansatz eigenvalue dispositions are related to the phenomenon of quantum monodromy

pattern formation on networks with reactions a continuous time random walk approach we derive the generalized master equation for reaction diffusion on networks from an underlying stochastic process the continuous time random walk ctrw the nontrivial incorporation of the reaction process into the ctrw is achieved by splitting the derivation into two stages the reactions are treated as birth death processes and the first stage of the derivation is at the single particle level taking into account the death process while the second stage considers an ensemble of these particles including the birth process using this model we have investigated different types of pattern formation across the vertices on a range of networks importantly the ctrw defines the laplacian operator on the network in a non ad hoc manner and the pattern formation depends on the structure of this laplacian here we focus attention on ctrws with exponential waiting times for two cases one in which the rate parameter is constant for all vertices and the other where the rate parameter is proportional to the vertex degree this results in nonsymmetric and symmetric ctrw laplacians respectively in the case of symmetric laplacians pattern formation follows from the turing instability however in nonsymmetric laplacians pattern formation may be possible with or without a turing instability

dynamics of reaction diffusion systems in a subdiffusive regime in this paper we examine the dynamics of reaction diffusion systems with fractional time derivatives it is shown that in these conditions diffusion is anomalous in the sense that the mean square displacement r 2 ensuremath sim t ensuremath gamma where ensuremath gamma l1 a situation known as subdiffusion we study the conditions for the appearance of a diffusion driven instability and show that the restrictive conditions for a turing instability are relaxed this implies that systems whose kinetics are not of the activator inhibitor kind can have a turing instability and a modulated final state we demonstrate our results with numerical calculations in two dimensions using a generic turing model

polygonal billiards with small obstacles the technique of unfolding a polygonal billiard table is used to answer certain questions concerning the illumination problem the main problem addressed is how many point obstacles would suffice to block any billiard path between two points of the polygon the answer can then be generalized from point obstacles to small neighborhoods of points

some exact solutions for the potential r 1 2λr 2λ2r2 the authors show how to obtain an infinite number of exact solutions for the excited states of an s wave hydrogen atom with the polynomial perturbation 2 lambda r 2 lambda 2r2 for certain specific values of lambda the energy eigenvalues in such cases satisfy en 1 2 2n 3 mod lambda mod with the corresponding wavefunctions being given by products of exp mod lambda mod r lambda mod lambda mod r2 and polynomials of nth degree in r

physical pictures of transport in heterogeneous media advection dispersion random walk and fractional derivative formulations 1 the basic conceptual picture and theoretical basis for development of transport equations in porous media are examined the general form of the governing equations is derived for conservative chemical transport in heterogeneous geological formations for single realizations and for ensemble averages of the domain the application of these transport equations is focused on accounting for the appearance of non fickian anomalous transport behavior the general ensemble averaged transport equation is shown to be equivalent to a continuous time random walk ctrw and reduces to the conventional forms of the advection dispersion equation ade under highly restrictive conditions fractional derivative formulations of the transport equations both temporal and spatial emerge as special cases of the ctrw in particular the use in this context of levy flights is critically examined in order to determine chemical transport in field scale situations the ctrw approach is generalized to nonstationary systems we outline a practical numerical scheme similar to those used with extended geological models to account for the often important effects of unresolved heterogeneities

an efficient chebyshev lanczos method for obtaining eigensolutions of the schrodinger equation on a grid a grid method for obtaining eigensolutions of bound systems is presented in this the block lanczos method is applied to a cheby shev approximation of exp h where is the range of eigenvalues we are interested in with this choice a preferential convergence of the eigenvectors corresponding to low lying eigenvalues ofhis achieved the method is used to solve a variety of one two and three dimensional problems to apply the kinetic energy operator we use the fast sine transform instead of the fast fourier transform thus fullfilling a priori the box boundary conditions we further extend the chebyshev approximation to treat general functions of matrices thus allowing its application to cases for which no analytical expressions of the expansion coefficients are available

some first excited energy levels for the generalized killingbeck potential with the differential quadratic method the ground energy and some low lying excited energies and the corresponding wavefunctions for the generalized killingbeck potential are investigated within the context of the recently introduced differential quadratic method highly accurate energy eigenvalues and an excellent agreement were found compared with the power series expansion and the supersymmetric perturbation theory

stochastic turing patterns on a network the process of stochastic turing instability on a scale free network is discussed for a specific case study the stochastic brusselator model the system is shown to spontaneously differentiate into activator rich and activator poor nodes outside the region of parameters classically deputed to the deterministic turing instability this phenomenon as revealed by direct stochastic simulations is explained analytically and eventually traced back to the finite size corrections stemming from the inherent graininess of the scrutinized medium

numerically complemented analytic method for solving the time independent one dimensional schrodinger equation a general method of solving the one dimensional schrodinger equation is developed the first step is to construct an exactly solvable reference potential of several smoothly joined morse type components which should be a good approximation to a given potential the exact solutions for that reference hamiltonian are then combined with a nonperturbative approach r g gordon j chem phys 51 14 1969 which enables us to numerically solve the energy eigenvalue problem for the original potential to any desired accuracy a full description of the analytical procedures is given and examples of both exact and numerical solutions are presented

the turing bifurcation in network systems collective patterns and single differentiated nodes abstract we study the emergence of patterns in a diffusively coupled network that undergoes a turing instability our main focus is the emergence of stable solutions with a single differentiated node in systems with large and possibly irregular network topology based on a mean field approach we study the bifurcations of such solutions for varying system parameters and varying degree of the differentiated node such solutions appear typically before the onset of turing instability and provide the basis for the complex scenario of multistability and hysteresis that can be observed in such systems moreover we discuss the appearance of stable collective patterns and present a codimension two bifurcation that organizes the interplay between collective patterns and patterns with single differentiated nodes

subdiffusion chemotaxis and anomalous aggregation we propose a nonlinear random walk model which is suitable for the analysis of both chemotaxis and anomalous subdiffusive transport we derive the master equations for the population density for the case when the transition rate for a random walk depends on residence time chemotactic substance and population density we introduce the anomalous chemotactic sensitivity and find an anomalous aggregation phenomenon so we suggest a different explanation of the well known effect of chemotactic collapse

anomalous metapopulation dynamics on scale free networks we model transport of individuals across a heterogeneous scale free network where a few weakly connected nodes exhibit heavy tailed residence times using the empirical law of the axiom of cumulative inertia and fractional analysis we show that anomalous cumulative inertia overpowers highly connected nodes in attracting network individuals this fundamentally challenges the classical result that individuals tend to accumulate in high order nodes the derived residence time distribution has a nontrivial u shape which we encounter empirically across human residence and employment times

from continuous time random walks to the fractional fokker planck equation we generalize the continuous time random walk ctrw to include the effect of space dependent jump probabilities when the mean waiting time diverges we derive a fractional fokker planck equation ffpe this equation describes anomalous diffusion in an external force field and close to thermal equilibrium we discuss the domain of validity of the fractional kinetic equation for the force free case we compare between the ctrw solution and that of the ffpe

shifted 1 n expansion and exact solutions for the potential v r z r gr λr2 the authors have found an infinite number of exact solutions for the hydrogenic atom in the external potential v r gr r2 not only for an s wave state but for higher waves as well from supersymmetric considerations the general schrodinger equation has been treated by the shifted 1 n expansion method the eigenvalues obtained from the shifted 1 n expansion are compared with those obtained by bessis et al 1987 and also with the supersymmetric exact values

fractional master equations and fractal time random walks fractional master equations containing fractional time derivatives of order 0 ensuremath le 1 are introduced on the basis of a recent classification of time generators in ergodic theory it is shown that fractional master equations are contained as a special case within the traditional theory of continuous time random walks the corresponding waiting time density ensuremath psi t is obtained exactly as ensuremath psi t mathit t mathrm ensuremath omega mathrm ensuremath 1 c mathit e mathrm ensuremath omega mathrm ensuremath omega mathit t mathrm ensuremath omega c where mathit e mathrm ensuremath omega mathrm ensuremath omega x is the generalized mittag leffler function this waiting time distribution is singular both in the long time as well as in the short time limit

fractional chemotaxis diffusion equations we introduce mesoscopic and macroscopic model equations of chemotaxis with anomalous subdiffusion for modeling chemically directed transport of biological organisms in changing chemical environments with diffusion hindered by traps or macromolecular crowding the mesoscopic models are formulated using continuous time random walk equations and the macroscopic models are formulated with fractional order differential equations different models are proposed depending on the timing of the chemotactic forcing generalizations of the models to include linear reaction dynamics are also derived finally a monte carlo method for simulating anomalous subdiffusion with chemotaxis is introduced and simulation results are compared with numerical solutions of the model equations the model equations developed here could be used to replace keller segel type equations in biological systems with transport hindered by traps macromolecular crowding or other obstacles

the eigenenergies of the wave function through the non variational galerkin b spline approach the non variational galerkin b spline method has been applied to the solution of the schrodinger equation for the vibration and rotation bound states of diatomic molecules this method allows the construction of the full hamiltonian matrix of relatively modest size the eigenspectra obtained are compared with the analytic and modified shifted 1 n expansion cases

hydrogenic atoms in the external potential v r gr λr2 exact solutions and ground state eigenvalue bounds using moment methods an infinite number of exact solutions for the ground and excited states of an s wave hydrogenic atom with the perturbation v r gr lambda r2 are constructed subject to special relations between g lambda and the nuclear charge z a powerful method of stieltjes moments is employed to generate rapidly converging upper and lower bounds to the ground state energy e0 z g lambda for arbitrary g z and positive lambda the accuracies of these bounds are relatively insensitive to the strength of the perturbation the moment method also generates the exact s wave ground state solutions mentioned above

new class of conditionally exactly solvable potentials in quantum mechanics motivated by an idea of dutra 1993 we obtain a new class of one dimensional conditionally exactly solvable potentials for which the entire spectra can be obtained in an algebraic manner provided one of the potential parameters is assigned a fixed negative value it is shown that using shape invariant potentials as input one may generate different classes of such potentials even in more than one dimension we also illustrate that wkb and supersymmetry inspired wkb methods provide very good approximations for these potentials with the latter doing comparatively better

generalized master equations for random walks with time dependent jump sizes in this work we develop a unified generalized master equation gme framework that extends the theory of continuous time random walks to include the cases when the jump sizes may have a delayed dependence on time and are not restricted to any particular class of distributions we compare and contrast analytical and numerical behavior of the corresponding master equations including the instantaneous vs delayed jump dependence on time and exponential vs mittag leffler interarrival times with the latter leading to fractional evolution equation we provide existence and uniqueness proofs for the resulting gmes