a new algebraization of the lameequation we develop a new way of writing the lamehamiltonian in lie algebraic form this yields in a natural way an explicit formula for both the lamepolynomials and the classical non meromorphic lame functions in terms of chebyshev polynomials and of a certain family of weakly orthogonal polynomials

mechanism of stochastic resonance enhancement in neuronal models driven by 1 f noise noise can assist neurons in the detection of weak signals via a mechanism known as stochastic resonance sr in a previous study phys lett a 243 281 1998 we showed that when colored noise with 1 f ensuremath beta spectrum is added to the fitzhugh nagumo fhn neuronal model the optimal noise variance for sr could be minimized with ensuremath beta ensuremath approx 1 in this study we investigate analytically how the noise color ensuremath beta affects the sr profile in a linearized version of the fhn model we demonstrate that the aforementioned effect of 1 f noise is related to the dynamical characteristics of the model neuron i e the refractory period the low pass filtering effect of the membrane capacitance and the high pass filtering effect of the recovery variable

control of spiking regularity in a noisy complex neural network the effects of spatiotemporally correlated noise on the regularity of spiking oscillations are studied in a network composed of fitz hugh nagumo neurons the spiking regularity of the neural network becomes the best at a moderate noise intensity indicating the occurrence of coherence resonance cr the cr in a watts strogatz small world network is further improved by adding a small fraction of long range connections given a set of temporal correlation constant ensuremath tau and spatial correlation length ensuremath lambda of the noise there exists an optimal network topology randomness at which the spiking oscillations show the best regularity the optimal randomness of the network topology at different ensuremath tau and ensuremath lambda varies in a narrow range changing ensuremath lambda does not affect the optimal ensuremath tau for achieving the most regular spike train whereas varying ensuremath tau the best spiking regularity emerges at different optimal ensuremath lambda

linear operators with invariant polynomial space and graded algebra the irreducible finite dimensional representations of the graded algebras osp j 2 j 1 2 3 are expressed in terms of differential operators some quantum deformations of these algebras are shown to admit similar kinds of representations these are formulated in terms of finite difference operators the results are discussed in the framework of the quasi exactly solvable equations

on the sutherland spin model of b n type and its associated spin chain the b n hyperbolic sutherland spin model is expressed in terms of a suitable set of commuting dunkl operators this fact is exploited to derive a complete family of commuting integrals of motion of the model thus establishing its integrability the dunkl operators are shown to possess a common flag of invariant finite dimensional linear spaces of smooth scalar functions this implies that the hamiltonian of the model preserves a corresponding flag of smooth spin functions the discrete spectrum of the restriction of the hamiltonian to this spin flag is explicitly computed by triangularization the integrability of the hyperbolic sutherland spin chain of b n type associated with the dynamical model is proved using polychronakos s freezing trick

influence of spatiotemporally correlated noise on structure formation in excitable media we discuss the influence of additive spatiotemporally correlated i e colored noise on pattern formation in a two dimensional network of excitable systems the signature of spatiotemporal stochastic resonance stsr is analyzed using cross correlation and information theoretic measures it is found that the stsr behavior is affected by both the spatial and temporal correlations of the noise due to an interplay with the length scales of the deterministic network increasing the spatiotemporal noise correlation shifts the occurrence of stsr to smaller values of the noise variance additionally if the spatial correlation of the noise exceeds that of the network the excitation patterns disappear in favor of cloudy structures directly rendering the underlying spatial noise field

extracting fractal components from time series abstract we present a technique for extracting random fractal components from a given time series in the frequency domain the method was based on characteristics of the fractal time series that the original and the renormalized coarse grained time series had a random phase relationship an ordinary fft based cross spectral method was used to measure the phase relationship the method was tested on simulated data with known fractal and or simple harmonic properties and with a mixture of these the percent fractal components derived by this technique were closely matched to the theoretical values we set the extracted fractal power showed power law dependence expressed as spectral exponent by which we could calculate the hurst exponent h of the time series from it was also shown that the calculated h for the simulated time series gave good estimates for the true h of the random fractal process

stochastic resonance in neuron models periodically stimulated sensory neurons typically exhibit a kind of statistical phase locking to the stimulus they tend to fire at a preferred phase of the stimulus cycle but not at every cycle hence the histogram of interspike intervals isih i e of times between successive firings is multimodal for these neurons with peaks centered at integer multiples of the driving period a particular kind of residence time histogram for a large class of noisy bistable systems has recently been shown to exhibit the major features of the neural data in the present paper we show that an excitable cell model the fitzhugh nagumo equations also reproduces these features when driven by additive periodic and stochastic forces this model exhibits its own brand of stochastic resonance as the peaks of the isih successively go through a maximum when the noise intensity is increased further the presence of a noise induced limit cycle introduces a third time scale in the problem this limit cycle is found to modify qualitatively the phase locking picture e g by suppressing certain peaks in the isih finally the role of noise and possibly of stochastic resonance sr in the neural encoding of sensory information is discussed

the structure of quasi exactly solvable systems the linear differential operators that preserve the n dimensional vector space whose entries are polynomials of fixed but arbitrary degrees in one variable are considered herein they are relevant for the classification of quasi exactly solvable systems of equations generating elements are explicitly constructed and rules for ordering their products are derived some examples of equations that can be reduced to such systems are discussed

scene segmentation from visual motion using global optimization this paper presents results from computer experiments with an algorithm to perform scene disposition and motion segmentation from visual motion or optic flow the maximum a posteriori map criterion is used to formulate what the best segmentation or interpretation of the scene should be where the scene is assumed to be made up of some fixed number of moving planar surface patches the bayesian approach requires first specification of prior expectations for the optic flow field which here is modeled as spatial and temporal markov random fields and secondly a way of measuring how well the segmentation predicts the measured flow field the markov random fields incorporate the physical constraints that objects and their images are probably spatially continuous and that their images are likely to move quite smoothly across the image plane to compute the flow predicted by the segmentation a recent method for reconstructing the motion and orientation of planar surface facets is used the search for the globally optimal segmentation is performed using simulated annealing

the mechanism of stochastic resonance it is shown that a dynamical system subject to both periodic forcing and random perturbation may show a resonance peak in the power spectrum which is absent when either the forcing or the perturbation is absent

the symmetric form of the koekoeks laguerre type differential equation abstract koekoek and koekoek 1991 proved that the laguerre type polynomials ln n x n 0 which are orthogonal on 0 with respect to the measure d n defined on 0 by d n x x e x 1 n x dx satisfy a differential equation l2 4 n y x ny x of order 2 4 when is a nonnegative integer in this paper we announce that the differential expression l2 4 n is symmetrizable on 0 with symmetry factor x e x and explicitly give this formally symmetric expression x e xl2 4 n y

on some polynomial potentials in d dimensions the d dimensional schrodinger s equation is analyzed with regard to the existence of exact solutions for polynomial potentials under certain conditions on the interaction parameters we show that the polynomial potentials v8 r k 18 krk 8 0 and v10 r k 110 krk 10 0 are exactly solvable by examining the polynomial solutions of certain linear differential equations with polynomial coefficients the necessary and sufficient conditions for the existence of these exact solutions are discussed finding accurate solutions for arbitrary values of the potential parameters using the asymptotic iteration method is also presented

on one dimensional schrodinger problems allowing polynomial solutions we discuss the explicit construction of the schrodinger equations admitting a representation through some family of general polynomials almost all solvable quantum potentials are shown to be generated by this approach some generalization has also been performed in higher dimensional problems

criterion for polynomial solutions to a class of linear differential equations of second order we consider the differential equations y 0 x y s0 x y where 0 x s0 x are c functions we prove i if the differential equation has a polynomial solution of degree n 0 then n nsn 1 n 1sn 0 where n n 1 sn 1 0 n 1andsn s n 1 s0 k 1 n 1 2 conversely ii if n n 1 0 and n 0 then the differential equation has a polynomial solution of degree at most n we show that the classical differential equations of laguerre hermite legendre jacobi chebyshev first and second kinds gegenbauer and the hypergeometric type etc obey this criterion further we find the polynomial solutions for the generalized hermite laguerre legendre and chebyshev differential equations

quommutator deformations of spl n 1 we analyze the witten woronowicz s type deformations of the lie superalgebras spl n 1 and obtain deformations parametrized by n n 1 2 independent parameters for n 2 and by three parameters for n 2 for some of these algebras finite dimensional representations can be constructed in terms of finite difference operators providing operators that are relevant for the classification of quasiexactly solvable finite difference equations

umbral calculus discretization and quantum mechanics on a lattice umbral calculus deals with representations of the canonical commutation relations we present a short exposition of it and discuss how this calculus can be used to discretize continuum models and to construct representations of lie algebras on a lattice related ideas appeared in recent publications and we show that the examples treated there are special cases of umbral calculus this observation then suggests various generalizations of these examples a special umbral representation of the canonical commutation relations given in terms of the position and momentum operator on a lattice is investigated in detail

quasi exact solvability resonances and trivial monodromy in ordinary differential equations a correspondence between the sextic anharmonic oscillator and a pair of third order ordinary differential equations is used to investigate the phenomenon of quasi exact solvability for eigenvalue problems involving differential operators with order greater than 2 in particular links with bender dunne polynomials and resonances between independent solutions are observed for certain second order cases and extended to the higher order problems this article is part of a special issue of journal of physics a mathematical and theoretical devoted to quantum physics with non hermitian operators

lie algebras and polynomials in one variable a classification of linear differential and difference equations in one variable having polynomial solutions the generalized bochner problem is presented the idea of the approach is based on consideration of the eigenvalue problem for a polynomial in generators of the algebra sl2 r for differential equations or sl2 r q for finite difference equation given in the finite dimensional projectivized representation

quasiexactly solvable 2 2 matrix equations the conditions under which systems of two differential eigenvalue equations are quasiexactly solvable are investigated these systems reveal a rich set of algebraic structures some of them are explicitly described an example of a quasiexactly solvable system which provides a direct counterpart of the lame equation is studied

neuron dynamics in the presence of 1 f noise interest in understanding the interplay between noise and the response of a nonlinear device cuts across disciplinary boundaries it is as relevant for unmasking the dynamics of neurons in noisy environments as it is for designing reliable nanoscale logic circuit elements and sensors most studies of noise in nonlinear devices are limited to either time correlated noise with a lorentzian spectrum of which the white noise is a limiting case or just white noise we use analytical theory and numerical simulations to study the impact of the more ubiquitous natural noise with a 1 f frequency spectrum specifically we study the impact of the 1 f noise on a leaky integrate and fire model of a neuron the impact of noise is considered on two quantities of interest to neuron function the spike count fano factor and the speed of neuron response to a small steplike stimulus for the perfect nonleaky integrate and fire model we show that the fano factor can be expressed as an integral over noise spectrum weighted by a low pass filter function given by f t f sinc 2 ft this result elucidates the connection between low frequency noise and disorder in neuron dynamics under 1 f noise spike dynamics lacks a characteristic correlation time inducing the leaky and nonleaky models to exhibit nonergodic behavior and the fano factor increasing logarithmically as a function of time we compare our results to experimental data of single neurons in vivo teich heneghan lowen ozaki and kaplan j opt soc am a 14 529 1997 and show how the 1 f noise model provides much better agreement than the usual approximations based on lorentzian noise the low frequency noise however complicates the case for an information coding scheme based on interspike intervals by introducing variability in the neuron response time on a positive note the neuron response time to a step stimulus is remarkably nearly optimal in the presence of 1 f noise an explanation of this effect elucidates how the brain can take advantage of noise to prime a subset of the neurons to respond almost instantly to sudden stimuli

stochastic resonance of a periodically driven neuron under non gaussian noise abstract we investigate the first passage time statistics of the integrate fire neuron model driven by a sub threshold harmonic signal superposed with a non gaussian noise here we considered the noise as the result of a random multiplicative process displaced from the origin by an additive term such a mechanism generates a power law distributed noise whose characteristic decay exponent can be finely tuned we performed numerical simulations to analyze the influence of the noise non gaussian character on the stochastic resonance condition we found that when the noise deviates from gaussian statistics the resonance condition occurs at weaker noise intensities achieving a minimum at a finite value of the distribution function decay exponent we discuss the possible relevance of this feature to the efficiency of the firing dynamics of biological neurons as the present result indicates that neurons would require a lower noise level to detect a sub threshold signal when its statistics departs from gaussian

tridiagonalization and the heun equation it is shown that the tridiagonalization of the hypergeometric operator l yields the generic heun operator m the algebra generated by the operators l m and z l m is quadratic and a one parameter generalization of the racah algebra the racah heun orthogonal polynomials are introduced as overlap coefficients between the eigenfunctions of the operators l and m an interpretation in terms of the racah problem for su 1 1 and separation of variables in a superintegrable system are discussed

quasi exactly solvable problems and sl 2 algebra recently discovered quasi exactly solvable problems of quantum mechanics are shown to be related to the existence of the finite dimensional representations of the groupsl 2 q whereq r c it is proven that the bilinear formh a j j b j stand for the generators allows one to generate a set of quasi exactly solvable problems of different types including those that are already known we get in particular problems in which the spectral riemannian surface containing an infinite number of sheets is split off one or two finite sheet pieces in the general case the transitionh h d2 dx2 v x is realized with the aim of the elliptic functions all known exactly solvable quantum problems with known spectrum and factorized riemannian surface can be obtained in this approach

maximal dimension of invariant subspaces admitted by nonlinear vector differential operators in this paper the dimension of invariant subspaces admitted by m component nonlinear vector differential operators is estimated it is shown that if the m component nonlinear vector differential operators of order k preserves the invariant subspace wn11 wnmm where wnqq is the space generated by solutions of linear ordinary differential equations of order nq q 1 m then max n1 nm 2mk 1 to illustrate the approach examples of nonlinear vector differential operators admitting invariant subspace wn11 wn22 with max n1 n2 9 and three component nonlinear vector differential operators admitting invariant subspace wn11 wn22 wn33 with max n1 n2 n3 13 are presented

aperiodic stochastic resonance stochastic resonance sr is a phenomenon wherein the response of a nonlinear system to a weak periodic input signal is optimized by the presence of a particular level of noise recently we presented a method and theory for characterizing sr type behavior in excitable systems with aperiodic i e broadband input signals phys rev e 52 r3321 1995 we coined the term aperiodic stochastic resonance asr to describe this general type of behavior in that earlier study we demonstrated asr in the fitzhugh nagumo neuronal model here we demonstrate asr in three additional systems a bistable well system an integrate and fire neuronal model and the hodgkin huxley hh neuronal model we present computational and theoretical results for each system in the context of the hh model we develop a general theory for asr in excitable membranes this work clearly shows that sr type behavior is not limited to systems with periodic inputs thus in general noise can serve to enhance the response of a nonlinear system to a weak input signal regardless of whether the signal is periodic or aperiodic textcopyright 1996 the american physical society

noise sensitivity of phase synchronization time in stochastic resonance theory and experiment recent numerical and heuristic arguments have revealed that the average phase synchronization time between the input and the output associated with stochastic resonance is highly sensitive to noise variation in particular there is evidence that this average time exhibits a cusplike behavior as the noise strength varies through the optimal value here we present an explicit formula for the average phase synchronization time in terms of the phase diffusion coefficient and the average frequency difference between the input and the output signals we also provide experimental evidence for the cusplike behavior by using a bistable microelectronic circuit system

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