group classification of schrodinger equations with position dependent mass maximal kinematical invariance groups of the schrodinger equations with position dependent mass pdm and arbitrary potential are classified it is demonstrated that there exist seven classes of such equations possessing non equivalent continuous symmetry group three of these classes include arbitrary functions while the remaining ones are defined up to arbitrary parameters in particular for the case of a constant mass the class missing in the boyer classification boyer 1974 helv phys acta 47 450 is indicated a constructive test of non equivalence of a pdm system to a constant mass system is proposed

deformed shape invariance and exactly solvable hamiltonians with position dependent effective mass known shape invariant potentials for the constant mass schrodinger equation are taken as effective potentials in a position dependent effective mass pdem one the corresponding shape invariance condition turns out to be deformed its solvability imposes the form of both the deformed superpotential and the pdem a lot of new exactly solvable potentials associated with a pdem background are generated in this way a novel and important condition restricting the existence of bound states whenever the pdem vanishes at an end point of the interval is identified in some cases the bound state spectrum results from a smooth deformation of that of the conventional shape invariant potential used in the construction in others one observes a generation or suppression of bound states depending on the mass parameter values the corresponding wavefunctions are given in terms of some deformed classical orthogonal polynomials

spiral waves in reaction diffusion equations we consider the reaction diffusion system begin gathered r t nabla 2 r r left 1 r 2 vec nabla theta cdot vec nabla theta right hfill r theta t r nabla 2 theta 2 vec nabla r cdot vec nabla theta qr 3 hfill end gathered this system governs the solutions of reaction diffusion systems near a hopf bifurcation point in two spatial dimensions we use formal asymptotic expansions to contruct one armed and multi armed archimedean spiral waves for small values of the parameter q we then show that the one armed spiral waves are probably stable and the multi armed ones are unstable for q small next by numerical continuation methods we construct spiral waves for all q these calculations show that the one armed spiral waves are unstable when q 1 397 cdots we also find the explicit one dimensional analogues of spiral waves for all q and show that they are unstable for q small

a new class of quasi exactly solvable potentials with a position dependent mass a new class of quasi exactly solvable potentials with a variable mass in the schrodinger equation is presented we have derived a general expression for the potentials including natanzon confluent potentials the general solution of the schrodinger equation is determined and the eigenstates are expressed in terms of the orthogonal polynomials

interactions of solitons in nonintegrable systems direct perturbation method and applications abstract problems of approximate description of soliton interaction in nonintegrable systems are discussed it is shown that in an arbitrary lagrangian system in the first nontrivial order of the small parameter asymptotic scheme for a system of solitons with close velocities a lagrangian may be introduced analogous to that for classical particles with a pair interaction potential a classification of possible types of soliton interaction is given and criteria of the existence of bound states are formulated a possibility of generalization of approximated description of soliton interaction in the multidimensional case is noted

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

an exactly solvable schrodinger equation with finite positive position dependent effective mass the solution of the one dimensional schrodinger equation is discussed in the case of position dependent mass the general formalism is specified for potentials that are solvable in terms of generalized laguerre polynomials and mass functions that are positive and bounded on the whole real x axis the resulting four parameter potential is shown to belong to the class of implicit potentials closed expressions are obtained for the bound state energies and the corresponding wave functions including their normalization constants the constant mass case is obtained by a specific choice of the parameters it is shown that this potential contains both the harmonic oscillator and the morse potentials as two distinct limiting cases and that the original potential carries several characteristics of these two potentials possible generalizations of the method are outlined

superintegrable systems with position dependent mass first order integrals of motion for schrodinger equations with position dependent masses are classified eighteen classes of such equations with non equivalent symmetries are specified they include integrable superintegrable and maximally superintegrable systems among them is a system invariant with respect to the lie algebra of lorentz group and a system whose integrals of motion form algebra so 4 three of the obtained systems are solved exactly

stable particle like solutions of multidimensional nonlinear fields abstract stable localized structures of two and three dimensional fields are investigated a theory of the elementary particle interaction the formation of their bound states and dynamics under the action of external perturbations is considered for scalar and complex fields

group analysis of self organizing maps based on functional mri using restricted frechet means studies of functional mri data are increasingly concerned with the estimation of differences in spatio temporal networks across groups of subjects or experimental conditions unsupervised clustering and independent component analysis ica have been used to identify such spatio temporal networks while these approaches have been useful for estimating these networks at the subject level comparisons over groups or experimental conditions require further methodological development in this paper we tackle this problem by showing how self organizing maps soms can be compared within a frechean inferential framework here we summarize the mean som in each group as a frechet mean with respect to a metric on the space of soms the advantage of this approach is twofold firstly it allows the visualization of the mean som in each experimental condition secondly this frechean approach permits one to draw inference on group differences using permutation of the group labels we consider the use of different distance functions and introduce one extension of the classical sum of minimum distance smd between two soms which take into account the spatial pattern of the fmri data the validity of these methods is illustrated on synthetic data through these simulations we show that the two distance functions of interest behave as expected in the sense that the ones capturing temporal and spatial aspects of the soms are more likely to reach significance under simulated scenarios characterized by temporal spatial and spatio temporal differences respectively in addition a re analysis of a classical experiment on visually triggered emotions demonstrates the usefulness of this methodology in this study the multivariate functional patterns typical of the subjects exposed to pleasant and unpleasant stimuli are found to be more similar than the ones of the subjects exposed to emotionally neutral stimuli in this re analysis the group level som output units with the smallest sample jaccard indices were compared with standard glm group specific z score maps and provided considerable levels of agreement taken together these results indicate that our proposed methods can cast new light on existing data by adopting a global analytical perspective on functional mri paradigms

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

oscillon type structures and their interaction in a swift hohenberg model abstract motivated by the observation of localized circular excitations oscillons in vertically vibrated granular layers p umbanhowar f melo h swinney nature 382 1996 793 we numerically investigate an extension of a swift hohenberg model that exhibits a subcritical transition to square patterns for sufficiently strong damping of the basic state stable oscillon structures are found the localization mechanism is quite general and is due to non adiabatic effects bound structures of oscillons of equal and opposite polarity are found with bound states of like polarity being less robust much of the phenomena are consistent with the experimental observations and suggest that oscillons are not specific to patterns in granular media or to parametrically driven systems experimental tests are suggested that would determine whether this minimal framework is sufficient to describe the phenomena

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

a class of exactly solvable potentials related to the jacobi polynomials the author investigates a family of solvable potentials related to the jacobi polynomials this one dimensional potential family depends on three parameters and is restricted to the domain xo so it can be interpreted as the radial part of a central potential in three dimensions with l 0 closed expressions are obtained for the bound state energy spectrum and the wavefunctions the supersymmetric partner of this potential is also determined and it is found not to belong to the same potential family it is shown that this potential family is a special subclass of the general six parameter natanzon potential class and similarities with another subclass the ginocchio potentials are pointed out some aspects of supersymmetric quantum mechanics and shape invariance are also discussed in connection with the potential family under study

perturbation methods in stability and norm analysis of spatially periodic systems we consider systems governed by partial differential equations with spatially periodic coefficients over unbounded domains these spatially periodic systems are considered as perturbations of spatially invariant ones and we develop perturbation methods to study their stability and cal h 2 system norm the operator lyapunov equations characterizing the cal h 2 norm are studied by using a special frequency representation and formulas are given for the perturbation expansion of their solution the structure of these equations allows for a recursive method of solving for the expansion terms our analysis provides conditions that capture possible resonances between the periodic coefficients and the spatially invariant part of the system these conditions can be regarded as useful guidelines when spatially periodic coefficients are to be designed to increase or decrease the cal h 2 norm of a spatially distributed system the developed perturbation framework also gives simple conditions for checking whether a spatially periodic operator generates a holomorphic c 0 semigroup and thus satisfies the spectrum determined growth condition

analysis of fmri data using improved self organizing mapping and spatio temporal metric hierarchical clustering the self organizing mapping som and hierarchical clustering hc methods are integrated to detect brain functional activation functional magnetic resonance imaging fmri data are first processed by som to obtain a primary merged neural nodes image and then by hc to obtain further brain activation patterns the conventional euclidean distance metric was replaced by the correlation distance metric in som to improve clustering and merging of neural nodes to improve the use of spatial and temporal information in fmri data a new spatial distance node coordinates in the 2 d lattice and temporal correlation correlation degree of each time course in the exemplar matrix are introduced in hc to merge the primary som results two simulation studies and two in vivo fmri data that both contained block design and event related experiments revealed that brain functional activation can be effectively detected and that different response patterns can be distinguished using these methods our results demonstrate that the improved som and hc methods are clearly superior to the statistical parametric mapping spm independent component analysis ica and conventional som methods in the block design especially in the event related experiment as revealed by their performance measured by receiver operating characteristic roc analysis our results also suggest that the proposed new integrated approach could be useful in detecting block design and event related fmri data

unified treatment of the coulomb and harmonic oscillator potentials in d dimensions quantum mechanical models and practical calculations often rely on some exactly solvable models like the coulomb and the harmonic oscillator potentials the d dimensional generalized coulomb potential contains these potentials as limiting cases thus it establishes a continuous link between the coulomb and harmonic oscillator potentials in various dimensions we present results which are necessary for the utilization of this potential as a model and practical reference problem for quantum mechanical calculations we define a hilbert space basis the generalized coulomb sturmian basis and calculate the green s operator on this basis and also present an su 1 1 algebra associated with it we formulate the problem for the one dimensional case too and point out that the complications arising due to the singularity of the one dimensional coulomb problem can be avoided with the use of the generalized coulomb potential

interaction between vegetation patches and gaps a self organized response to water scarcity abstract the dynamics of ecological systems are often described by integrodifferential equations that incorporate nonlocal interactions associated with facilitative competitive interactions between plants and seed dispersion in the weak gradient limit these models can be reduced to a simple partial differential equation in the form of a nonvariational swift hohenberg equation in this contribution we perform this reduction for any type of kernels provided that their taylor series converge some parameters such as linear and nonlinear diffusion coefficients are affected by the spatial form of the kernel in particular gaussian and exponential kernels are used to evaluate all coefficients of the reduced model this weak gradient approximation is greatly useful for the investigation of periodic and localized vegetation patches and gaps based on this simple model we investigate the interaction between two well separated patches and gaps in the case of patches the interaction is always repulsive as a consequence bounded states of patches are excluded however when two gaps are close to one another they start to interact through their oscillatory tails the interaction alternates between attractive and repulsive depending on the distance separating them this allows for the stabilization of bounded gaps and clusters of them the analytical formula of the interaction potential is derived for both patches and gaps interactions and checked by numerical investigation of the model equation this volume is dedicated to professor ehud meron on the occasion of his sixtieth birthday we take this opportunity to express our warmest and most sincere wishes to him

a systematic study on the exact solution of the position dependent mass schrodinger equation an algebraic method of constructing potentials for which the schrodinger equation with position dependent mass can be solved exactly is presented a general form of the generators of su 1 1 algebra has been employed with a unified approach to the problem our systematic approach reproduces a number of earlier results and also leads to some novelties we show that the solutions of the schrodinger equation with position dependent mass are free from the choice of parameters for position dependent mass two classes of potentials are constructed that include almost all exactly solvable potentials

fourth order superintegrable systems separating in cartesian coordinates i exotic quantum potentials a study is presented of two dimensional superintegrable systems separating in cartesian coordinates and allowing an integral of motion that is a fourth order polynomial in the momenta all quantum mechanical potentials that do not satisfy any linear differential equation are found they do however satisfy nonlinear odes we show that these equations always have the painleve property and integrate them in terms of known painleve transcendents or elliptic functions

statistical parametric network analysis of functional connectivity dynamics during a working memory task abstract network analysis has become a tool of choice for the study of functional and structural magnetic resonance imaging mri data little research however has investigated connectivity dynamics in relation to varying cognitive load in fmri correlations among slow

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

a lie algebraic approach to effective mass schrodinger equations we use lie algebraic techniques to obtain exact solutions of the effective mass schrodinger equation in particular we use the su 1 1 algebra both as a spectrum generating algebra and as a potential algebra to obtain exact solutions of effective mass schrodinger equations corresponding to a number of potentials we also discuss the construction of isospectral hamiltonians for which both the mass and the potential are different

frequency analysis and norms of distributed spatially periodic systems we investigate several fundamental aspects of the theory of linear distributed systems with spatially periodic coefficients we develop a spatial frequency domain representation analogous to the lifted or frequency response operator representation for linear time periodic systems using this representation we introduce the notion of the h2 norm for this class of systems and provide algorithms for its computation a stochastic interpretation of the h2 norm is given in terms of spatially cyclostationary random fields and spectral correlation density operators when the periodic coefficients are viewed as feedback modifications of spatially invariant systems we show how they can stabilize or destabilize the original systems in a manner analogous to vibrational control or parametric resonance in time periodic systems two examples from physics are provided to illustrate the main results

fronts pulses sources and sinks in generalized complex ginzberg landau equations an important clement in the long time dynamics of pattern forming systems is a class of solutions we will call coherent structures these are states that are either themselves localized or that consist of domains of regular patterns connected by localized defects or interfaces this paper summarizes and extends recent work on such coherent structures in the one dimensional complex ginzburg landau equation and its generalizations for which rather complete information can be obtained on the existence and competition of fronts pulses sources and sinks for the special subclass of uniformly translating structures the solutions are derived from a set of ordinary differential equations that can be interpreted as a flow in a three dimensional phase space fixed points of the flow correspond to the two basic building blocks of coherent structures uniform amplitude states and evanescent waves whose amplitude decreases smoothly to zero a study of the stability of the fixed points under the flow leads to results on the existence and multiplicity of the different coherent structures the dynamical analysis of the original partial differential equation focusses on the competition between pulses and fronts and is expressed in terms of a set of conjectures for front propagation that generalize the marginal stability and pinch point approaches of earlier authors these rules together with an exact front solution whose dynamics plays an important role in the selection of patterns yield an analytic expression for the upper limit of the range of existence of pulse solutions as well as a determination of the regions of parameter space where uniformly translating fron t solutions can exist extensive numerical simulations show consistency with these rules and conjectures for the existence of fronts and pulses in the parameter ranges where no uniformly translating fronts can exist examples are shown of irregularly spreading fronts that generate strongly chaotic regions as well as nonuniformly translating fronts that lead to uniform amplitude states recent perturbative treatments based on expansions about the nonlinear schr6dinger equation are generalized to perturbations of the cubic quintic and derivative schr idinger equations for which both pulses and fronts exist in the unperturbed system comparison of the results with the exact solutions shows that the perturbation theory only yields a subset of the relevant solutions nevertheless those that are obtained are found to be consistent with the general conjectures and in particular they provide an analytic demonstration of front pulse competition while the discussion of the competition between fronts and pulses focusses on the complex ginzburg landau equation with quintic terms and a subcritical bifurcation a number of results are also presented for the cubic equation in particular the existence of a family of moving source solutions derived by bekki and nozaki for this equation contradicts the naive counting arguments we attribute this contradiction to a hidden symmetry of the solution but have not been able to show explicitly how this symmetry affects the phase space orbits

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