the q laplace operator and q harmonic polynomials on the quantum vector space the aim of this paper is to study q harmonic polynomials on the quantum vector space generated by q commuting elements x1 x2 xn they are defined as solutions of the equation qp 0 where p is a polynomial in x1 x2 xn and the q laplace operator q is determined in terms of q derivatives the projector hm am hm is constructed where am and hm are the spaces of homogeneous of degree m polynomials and q harmonic polynomials respectively by using these projectors a q analog of classical associated spherical harmonics is constructed they constitute an orthonormal basis of hm a q analog of separation of variables is given representations of the nonstandard q deformed algebra uq son which plays the role of the rotation group so n in the case of classical harmonic polynomials on the spaces hm are explicitly constructed

parameter estimation of delay dynamical system from a scalar time series under external noise we introduce an adaptive learning rules for estimating all unknown parameters of delay dynamical system using a scalar time series sufficient condition for synchronization is derived using krasovskii lyapunov theory this scheme is highly applicable in secure communication since multiple messages can be transmitted through multiple parameter modulations one of the advantage of this method is that parameter estimation is also possible even when only one time series of the transmitter is available we present numerical examples for mackey glass system with periodic delay time which are used to illustrate the validity of this scheme the corresponding numerical results and the effect of external noise are also studied

self similar transport in incomplete chaos particle chaotic dynamics along a stochastic web is studied for three dimensional hamiltonian flow with hexagonal symmetry in a plane two different classes of dynamical motion obtained by different values of a control parameter and corresponding to normal and anomalous diffusion have been considered and compared it is shown that the anomalous transport can be characterized by powerlike wings of the distribution function of displacement flights which are similar to l evy flights approximate trappings of orbits near the boundary layer of islands and anomalous behavior of the moments of a distribution function considered as a function of the number of the moment the main result is related to the self similar properties of different topological and dynamical characteristics of the particle motion this self similarity appears in the weierstrass like random walk process that is responsible for the anomalous transport exponent in the mean moment dependence on t this exponent can be expressed as a ratio of fractal dimensions of space and time sets in the weierstrass like process an explicit form for the expression of the anomalous transport exponent through the local topological properties of orbits has been given

complex earthquakes and teichmuller theory it is known that any two points in teichmuller space are joined by an earthquake path in this paper we show any earthquake path r t s extends to a proper holomorphic mapping of a simplyconnected domain d into teichmuller space where r d c these complex earthquakes relate weil petersson geometry projective structures pleated surfaces and quasifuchsian groups using complex earthquakes we prove grafting is a homeomorphism for all 1 dimensional teichmuller spaces and we construct bending coordinates on bers slices and their generalizations in the appendix we use projective surfaces to show the closure of quasifuchsian space is not a topological manifold

a second order accurate numerical approximation for the fractional diffusion equation fractional order diffusion equations are generalizations of classical diffusion equations treating super diffusive flow processes in this paper we examine a practical numerical method which is second order accurate in time and in space to solve a class of initial boundary value fractional diffusive equations with variable coefficients on a finite domain an approach based on the classical crank nicholson method combined with spatial extrapolation is used to obtain temporally and spatially second order accurate numerical estimates stability consistency and therefore convergence of the method are examined it is shown that the fractional crank nicholson method based on the shifted grunwald formula is unconditionally stable a numerical example is presented and compared with the exact analytical solution for its order of convergence

combinatorial quantization of the hamiltonian chern simons theory ii motivated by a recent paper of fock and rosly 6 we describe a mathematically precise quantization of the hamiltonian chern simons theory we introduce the chern simons theory on the lattice which is expected to reproduce the results of the continuous theory exactly the lattice model enjoys the symmetry with respect to a quantum gauge group using this fact we construct the algebra of observables of the hamiltonian chern simons theory equipped with a operation and a positive inner product

estimating parameters of a nonlinear dynamical system a method based on a modified newton raphson scheme is presented to estimate parameters of a nonlinear dynamical system from the time series data of the variables the method removes some of the problems associated with the standard synchronization based methods an important achievement of this method is that it is possible to determine the exact form of dynamical equations for systems with quadratic nonlinearity

a generalized spectral collocation method with tunable accuracy for variable order fractional differential equations we generalize existing jacobi gauss lobatto collocation methods for variable order fractional differential equations using a singular approximation basis in terms of weighted jacobi polynomials of the form 1 pm x mu p j a b x where mu 1 in order to derive the differentiation matrices of the variable order fractional derivatives we develop a three term recurrence relation for both integrals and derivatives of these weighted jacobi polynomials hence extending the three term recurrence relationship of jacobi polynomials the new spectral collocation method is applied to solve fractional ordinary and partial differential equations with endpoint singularities we demonstrate that the singular basis enhances greatly the accuracy of the numerical solution by properly tuning the parameter mu even for cases where we do not know explicitly the form of singularity in the solution at the boundaries

dual generators of the fundamental group and the moduli space of flat connections we define the dual of a set of generators of the fundamental group of an oriented 2 surface sg n of genus g with n punctures and the associated surface sg n d with a disc d removed this dual is another set of generators related to the original generators via an involution and has the properties of a dual graph in particular it provides an algebraic prescription for determining the intersection points of a curve representing a general element of the fundamental group 1 sg n d with the representatives of the generators and the order in which these intersection points occur on the generators we apply this dual to the moduli space of flat connections on sg n and show that when expressed in terms of both the holonomies along a set of generators and their duals the poisson structure on the moduli space takes a particularly simple form using this description of the poisson structure we derive explicit expressions for the poisson brackets of general wilson loop observables associated with closed embedded curves on the surface and determine the associated flows on phase space we demonstrate that the observables constructed from the pairing in the chern simons action generate infinitesimal dehn twists and show that the mapping class group acts by poisson isomorphisms

symplectic structure of the moduli space of flat connection on a riemann surface we consider the canonical symplectic structure on the moduli space of flat g connections on a riemann surface of genus g with a marked points for a being a semisimple lie algebra we obtain an explicit efficient formula for this symplectic form and prove

adaptive approximation method for joint parameter estimation and identical synchronization of chaotic systems we introduce a numerical approximation method for estimating an unknown parameter of a primary chaotic system which is partially observed through a scalar time series specifically we show that the recursive minimization of a suitably designed cost function that involves the dynamic state of a fully observed secondary system and the observed time series can lead to the identical synchronization of the two systems and the accurate estimation of the unknown parameter the salient feature of the proposed technique is that the only external input to the secondary system is the unknown parameter which needs to be adjusted we present numerical examples for the lorenz system which show how our algorithm can be considerably faster than some previously proposed methods

preconditioned iterative methods for fractional diffusion equation in this paper we are concerned with numerical methods for the solution of initial boundary value problems of anomalous diffusion equations of order 1 2 the classical crank nicholson method is used to discretize the fractional diffusion equation and then the spatial extrapolation is used to obtain temporally and spatially second order accurate numerical estimates two preconditioned iterative methods namely the preconditioned generalized minimal residual preconditioned gmres method and the preconditioned conjugate gradient for normal residual preconditioned cgnr method are proposed to solve relevant linear systems numerical experiments are given to illustrate the efficiency of the methods

center of quantum algebra uq so 3 we analyze the structure of the center of the quantum algebra uq so 3 this structure as expected depends substantially on the deformation parameter q when q is not a root of unity there exists only one casimir element which is the deformation of ordinary casimir element known from so3 the center has a structure of a ring of polynomials of one variable when qn 1 there are three more casimir elements of the form of polynomials in algebra generators uq so 3 can be seen as a finite dimensional module over the commutative ring of polynomials in these three casimir elements knowing three parametrical family of n dimensional irreducible representations we prove that the dimension of this module is n3 all four casimir elements are no longer algebraically independent we present the explicit description of the central variety that is polynomial dependence between these four casimir elements this dependence differs for n 2m 1 n 2 2m 1 and n 4m

representations of the q deformed algebra uq so4 we study the nonstandard q deformation uq so4 of the universal enveloping algebra u so4 obtained by deforming the defining relations for skew symmetric generators of u so4 this algebra is used in quantum gravity and algebraic topology we construct a homomorphism of uq so4 to the certain nontrivial extension of the drinfeld jimbo quantum algebra uq sl2 2 and show that this homomorphism is an isomorphism by using this homomorphism we construct irreducible finite dimensional representations of the classical type and of the nonclassical type for the algebra uq so4 it is proved that for q not a root of unity each irreducible finite dimensional representation of uq so4 is equivalent to one of these representations we prove that every finite dimensional representation of uq so4 for q not a root of unity is completely reducible it is shown how to construct by using the homomorphism tensor products of irreducible representations of uq so4 note that no hopf algebra structure is known fo

two linear transformations each tridiagonal with respect to an eigenbasis of the other let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v satisfying both conditions below 1 i there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal 2 ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal we call such a pair a leonard pair on v refining this notion a bit we introduce the concept of a leonard system we give a complete classification of leonard systems integral to our proof is the following result we show that for any leonard pair a a on v there exists a sequence of scalars taken from k such that both 0 a a2a aa a a a2 aa a a a 0 a a 2a a aa aa 2 a a aa a where r s means rs sr the sequence is uniquely determined by the leonard pair if the dimension of v is at least 4 we conclude by showing how leonard systems correspond to q racah and related polynomials from the askey scheme

the fractional order governing equation of levy motion a governing equation of stable random walks is developed in one dimension this fokker planck equation is similar to and contains as a subset the second order advection dispersion equation ade except that the order a of the highest derivative is fractional e g the 1 65th derivative fundamental solutions are levy s a stable densities that resemble the gaussian except that they spread proportional to time 1 a have heavier tails and incorporate any degree of skewness the measured variance of a plume undergoing levy motion would grow faster than fickian plume at a rate of time 2 a where 0 a 2 the equation is parsimonious since the parameters are not functions of time or distance the scaling behavior of plumes that undergo levy motion is accounted for by the fractional derivatives which are appropriate measures of fractal functions in real space the fractional derivatives are integrodifferential operators so the fractional ade describes a spatially nonlocal process that is ergodic and has analytic solutions for all time and space

a class of second order difference approximations for solving space fractional diffusion equations a class of second order approximations called the weighted and shifted grunwald difference wsgd operators are proposed for riemann liouville fractional derivatives with their effective applications to numerically solving space fractional diffusion equations in one and two dimensions the stability and convergence of our difference schemes for space fractional diffusion equations with constant coefficients in one and two dimensions are theoretically established several numerical examples are implemented to test the efficiency of the numerical schemes and confirm the convergence order and the numerical results for variable coefficients problem are also presented

the projective unitary irreducible representations of the poincare group in 1 2 dimensions a complete analysis is given of the projective unitary irreducible representations of the poincare group in 1 2 dimensions applying the mackey theorem and using an explicit formula for the universal covering group of the lorentz group in 1 2 dimensions explicit formulas are given for all representations

estimating model parameters by chaos synchronization using chaos synchronization and a proposed iterative method of parameter adaptation we precisely estimate the model parameters of chaotic systems and synchronize two chaotic systems with originally mismatching model parameters this parameter adaptation method can be applied to a spatiotemporal chaotic system with a one way coupled map lattice as a biomedical application this method is capable of estimating the asymmetric tension parameter of a vocal fold model

on the classification of irreducible finite dimensional representations of uq so3 algebra in an earlier work m havlicek et al j math phys 40 2135 1999 we defined for any finite dimension five nonequivalent irreducible representations of the nonstandard deformation uq so3 of the lie algebra so3 where q is not a root of unity for each dimension only one of them called classical admits limit q 1 in the first part of this paper we show that any finite dimensional irreducible representation is equivalent to some of these representations in the case qn 1 we derive new casimir elements of uq so3 and show that a dimension of any irreducible representation is not higher than n these elements are casimir elements of uq som for all m and even of uq isom 1 due to inonu wigner contraction according to the spectrum of one of the generators the representations are found to belong to two main disjoint sets we give full classification and explicit formulas for all representations from the first set we call them nonsingular representations if n is odd we have full classificatio

quantum deformations of su 2 explicit representations of two deformations of su 2 one in the cartan basis and one in the cartesian basis are constructed the cartan deformation subsumes previously known cases while the cartesian deformation appears to be new in the latter case explicit reversible mappings in terms of representations of su 2 and a casimir for the algebra are constructed

multigrid method for fractional diffusion equations the fractional diffusion equation is discretized by the implicit finite difference scheme with the shifted grunwald formula the scheme is unconditionally stable and the coefficient matrix possesses the toeplitz like structure a multigrid method is proposed to solve the resulting system meanwhile the fast toeplitz matrix vector multiplication is utilized to lower the computational cost with only o nlogn complexity where n is the number of the grid points numerical experiments are given to demonstrate the efficiency of the method

representations of the cyclically symmetric q deformed algebra soq 3 an algebra homomorphism from the nonstandard q deformed cyclically symmetric algebra uq so3 to the extension uq sl2 of the hopf algebra uq sl2 is constructed not all irreducible representations of uq sl2 can be extended to representations of uq sl2 composing the homomorphism with irreducible representations of uq sl2 we obtain representations of uq so3 not all of these representations of uq so3 are irreducible reducible representations of uq so3 are decomposed into irreducible components in this way we obtain all irreducible representations of uq so3 when q is not a root of unity a part of these representations turns into irreducible representations of the lie algebra so3 when q 1 representations of the other part have no classical analog using the homomorphism it is shown how to construct tensor products of finite dimensional representations of uq so3 irreducible representations of uq so3 when q is a root of unity are constructed some of them are obtained from irreducible re

failure of parameter identification based on adaptive synchronization techniques in this paper several examples as well as their numerical simulations are provided to show some possible failures of parameter identification based on the so called adaptive synchronization techniques these failures might arise not only when the synchronized orbit produced by the driving system is designed to be either some kind of equilibrium or to be some kind of periodic orbit but also when this orbit is deliberately designed to be chaotic the reason for emergence of these failures is theoretically analyzed in the paper and the boundedness of all trajectories generated by the coupled systems is rigorously proved moreover synchronization techniques are proposed to realize complete synchronization and unknown parameter identification in a class of systems where nonlinear terms are not globally lipschitz in addition unknown parameter identification is studied in coupled systems with time delays

finite difference approximations for fractional advection dispersion flow equations fractional advection dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium in this paper we develop practical numerical methods to solve one dimensional fractional advection dispersion equations with variable coefficients on a finite domain the practical application of these results is illustrated by modeling a radial flow problem use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site

invariants of 2 1 gravity in 1 2 we established and dicussed the algebra of observables for 2 1 gravity at both the classical and quantum level here our treatment broadens and extends previous results to any genusg with a systematic discussion of the centre of the algebra the reduction of the number of independent observables to 6g 6 g 1 is treated in detail with a precise classification forg 1 andg 2

parameter evaluation from time sequences using chaos synchronization unknown parameters in nonlinear equations are estimated from chaotic time sequences using chaos synchronization the method is based on a random optimization method the parameters are randomly searched for in a sequential manner as the degree of the chaos synchronization is increased the method is applied for the parameter evaluation in the lorenz equation and the lang kobayashi model for the chaotic semiconductor laser

grafting and poisson structure in 2 1 gravity with vanishing cosmological constant we relate the geometrical construction of 2 1 spacetimes via grafting to phase space and poisson structure in the chern simons formulation of 2 1 dimensional gravity with vanishing cosmological constant on manifolds of topology mathbb r times s g where sg is an orientable two surface of genus g 1 we show how grafting along simple closed geodesics is implemented in the chern simons formalism and derive explicit expressions for its action on the holonomies of general closed curves on sg we prove that this action is generated via the poisson bracket by a gauge invariant observable associated to the holonomy of we deduce a symmetry relation between the poisson brackets of observables associated to the lorentz and translational components of the holonomies of general closed curves on sg and discuss its physical interpretation finally we relate the action of grafting on the phase space to the action of dehn twists and show that grafting can be viewed as a dehn twist with a formal parameter satisfying 2 0

twisted clebsch gordan coefficients for suq 2 a new approach to the clebsch gordan problem for suq 2 is proposed the twisted clebsch gordan coefficients tcgc are defined to be the overlaps between twisted connected and unconnected bases being the q analogues of rotated bases in standard su 2 the explicit expression for tcgc is found in terms of the basic hypergeometric function 4 phi 3 racah q polynomials

normalized leonard pairs and askey wilson relations abstract let v denote a vector space with finite positive dimension and let a a denote a leonard pair on v as is known the linear transformations a a satisfy the askey wilson relations a 2 a aa a a a 2 aa a a a a 2 a i a 2 a a aa aa 2 a a aa a a 2 a i for some scalars the scalar sequence is unique if the dimension of v is at least 4 if c c t t are scalars and t t are not zero then ta c t a c is a leonard pair on v as well these affine transformations can be used to bring the leonard pair or its askey wilson relations into a convenient form this paper presents convenient normalizations of leonard pairs by the affine transformations and exhibits explicit askey wilson relations satisfied by them

use of synchronization and adaptive control in parameter estimation from a time series a technique is introduced for estimating unknown parameters when a time series of only one variable from a multivariate nonlinear dynamical system is given the technique employs a combination of two different control methods linear feedback for synchronizing system variables and adaptive control and is based on dynamic minimization of synchronization error the technique is shown to work even when the unknown parameters appear in the evolution equations of the variables other than the one for which the time series is given the technique not only establishes that explicit detailed information about all system variables and parameters is contained in a scalar time series but also gives a way to extract it out under suitable conditions illustrations are presented for lorenz and rossler systems and a nonlinear dynamical system in plasma physics also it is found that the technique is reasonably stable against noise in the given time series and the estimated value of a parameter fluctuates around the correct value with the error of estimation growing linearly with the noise strength for small noise

on a recursive method for the estimation of unknown parameters of partially observed chaotic systems we investigate a recently proposed method for on line parameter estimation and synchronization in chaotic systems this novel technique has been shown effective to estimate a single unknown parameter of a primary chaotic system with known functional form that is only partially observed through a scalar time series it works by periodically updating the parameter of interest in a secondary system with the same functional form as the primary one but no explicit coupling between their dynamic variables in order to minimize a suitably defined cost function in this paper we review the basics of the method and investigate its robustness and new extensions in particular we study the performance of the novel technique in the presence of noise either observational i e an additive contamination of the observed time series or dynamical i e a random perturbation of the system dynamics and when there is a mismatch between the primary and secondary systems numerical results including comparisons with other techniques are presented finally we investigate the extension of the original method to perform the estimation of two unknown parameters and illustrate its effectiveness by means of computer simulations

synchronization based estimation of all parameters of chaotic systems from time series by a simple combination of adaptive scheme and linear feedback with the updated feedback strength for a large class of chaotic systems it is proved rigorously by using the invariance principle of differential equations that all unknown model parameters can be estimated dynamically this approach supplies a systematic and analytical procedure for estimating parameters from time series and it is simple to implement in practice in addition this method is quite robust against the effect of noise and able to respond rapidly to changes in operating parameters of the experimental system lorenz and r ossler hyperchaos systems are used to illustrate the validity of this technique

statistical methods of parameter estimation for deterministically chaotic time series we discuss the possibility of applying some standard statistical methods the least square method the maximum likelihood method and the method of statistical moments for estimation of parameters to deterministically chaotic low dimensional dynamic system the logistic map containing an observational noise a segmentation fitting maximum likelihood ml method is suggested to estimate the structural parameter of the logistic map along with the initial value x 1 considered as an additional unknown parameter the segmentation fitting method called piece wise ml is similar in spirit but simpler and has smaller bias than the multiple shooting previously proposed comparisons with different previously proposed techniques on simulated numerical examples give favorable results at least for the investigated combinations of sample size n and noise level besides unlike some suggested techniques our method does not require the a priori knowledge of the noise variance we also clarify the nature of the inherent difficulties in the statistical analysis of deterministically chaotic time series and the status of previously proposed bayesian approaches we note the trade off between the need of using a large number of data points in the ml analysis to decrease the bias to guarantee consistency of the estimation and the unstable nature of dynamical trajectories with exponentially fast loss of memory of the initial condition the method of statistical moments for the estimation of the parameter of the logistic map is discussed this method seems to be the unique method whose consistency for deterministically chaotic time series is proved so far theoretically not only numerically

fractional order anisotropic diffusion for image denoising this paper introduces a new class of fractional order anisotropic diffusion equations for noise removal these equations are euler lagrange equations of a cost functional which is an increasing function of the absolute value of the fractional derivative of the image intensity function so the proposed equations can be seen as generalizations of second order and fourth order anisotropic diffusion equations we use the discrete fourier transform to implement the numerical algorithm and give an iterative scheme in the frequency domain it is one important aspect of the algorithm that it considers the input image as a periodic image to overcome this problem we use a folded algorithm by extending the image symmetrically about its borders finally we list various numerical results on denoising real images experiments show that the proposed fractional order anisotropic diffusion equations yield good visual effects and better signal to noise ratio

finite dimensional irreducible modules of the universal askey wilson algebra since the introduction of askey wilson algebras by zhedanov in 1991 the classification of the finite dimensional irreducible modules of askey wilson algebras remains open a universal analog triangle q of the askey wilson algebras was recently studied in this paper we consider a family of infinite dimensional triangle q modules by the universal property of these triangle q modules we classify the finite dimensional irreducible triangle q modules when q is not a root of unity

modular leonard triples abstract let k denote a field and let v denote a vector space over k of finite positive dimension an ordered triple a a a of linear operators on v is said to be a leonard triple whenever for each b a a a there exists a basis of v with respect to which the matrix representing b is diagonal and the matrices representing the other two operators are irreducible tridiagonal a leonard triple a a a is said to be modular whenever for each b a a a there exists an antiautomorphism of end v which fixes b and swaps the other two operators we classify the modular leonard triples up to isomorphism

structure identification based on steady state control experimental results and applications we report experimental results on structure identification of nonlinear systems by a steady state control method the idea underlying the method is to drive the nonlinear system to steady state by applying a suitable feedback control input it turns out experimentally that this control based structure identification method can be used for some applications such as estimation of initial conditions and state variables of nonlinear systems and structure identification of some special elements two attractors of the chua oscillator are presented to illustrate the reliability of the suggested techniques under the hypotheses of measurable state variables and physical access to the system for implementing the proportional feedback

the unitary irreducible representations of the quantum heisenberg algebra the goal of this work is to describe the irreducible representations of the quantum heisenberg algebra and the unitary irreducible representation of one of its real forms the solution of this problem is obtained through the investigation of theleft spectrum of the quantum heisenberg algebra using the result about spectra of generic algebras of skew differential operators cf r

linear covariance algebra for slq 2 the askey wilson algebra aw 3 is shown to serve as a hidden covariance algebra for quantum algebra slq 2 the generators of aw 3 are chosen to be linear combinations of slq 2 generators with operator valued coefficients

realization of parameters identification in only locally lipschitzian dynamical systems with multiple types of time delays most of the existing results for identifying parameters in dynamical systems through adaptive coupling techniques always require a uniform lipschitz condition on system vector fields however it is not this requirement but the only locally lipschitz condition that is more suitable for many nonlinear systems in real applications also time delays sometimes are essential for modeling real systems so that in addition to the discrete form time delays in the distributed form could be considered this paper thus considers the problem of parameters identification in only locally lipschitzian dynamical systems with multiple types of time delays through adaptive coupling techniques in particular the paper proposes some linear independence conditions for realization of parameters identification in both nonautonomous and autonomous systems and draws a comparison between these conditions and the persistent of excitation condition which is used frequently in the literature moreover the paper proposes an index t

a direct o nlog2n finite difference method for fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that can not be modeled accurately by the second order diffusion equations because of the nonlocal property of fractional differential operators the numerical methods have full coefficient matrices which require storage of o n 2 and computational cost of o n 3 where n is the number of grid points in this paper we develop a fast finite difference method for fractional diffusion equations which only requires storage of o n and computational cost of o nlog 2n while retaining the same accuracy and approximation property as the regular finite difference method numerical experiments are presented to show the utility of the method

application of a fractional advection dispersion equation abstract a transport equation that uses fractional order dispersion derivatives hasfundamental solutions that are le vy s a stable densities these densities represent plumesthat spread proportional to time 1 a have heavy tails and incorporate any degree ofskewness the equation is parsimonious since the dispersion parameter is not a functionof time or distance the scaling behavior of plumes that undergo le vy motion isaccounted for by the fractional derivative a laboratory tracer test is described by adispersion term of order 1 55 while the cape cod bromide plume is modeled by anequation of order 1 65 to 1 8 1 introduction anomalous or non fickian dispersion has been an activearea of research in the physics community since the introduc tion of continuous time random walks ctrw by montrolland weiss 1965 these random walks extended the predictivecapability of models built on the stochastic process of brown ian motion which is the basis for the classical advection dispersion equation ade the ctrw assign a joint space time distribution called the transition density to individualparticle motions when the tails are heavy enough i e powerlaw non fickian dispersion results for all time scales andspace scales

wellposedness of variable coefficient conservative fractional elliptic differential equations fractional diffusion equations describe phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations fractional differential equations raise mathematical difficulties that have not been encountered in the analysis of second order differential equations there are two properties of fractional differential operators that make the analysis of fractional differential equations more complicated than that for second order differential equations these are i fractional differential operators are nonlocal operators and ii the adjoint of a fractional differential operator is not the negative of itself the wellposedness of a galerkin weak formulation to fractional elliptic differential equations with a constant diffusivity coefficient and the error analysis for corresponding finite element methods were proved previously many subsequent works were carried out to extend the analysis to other numerical methods a constant diffusivity coefficient has been assume

parameter estimation in spatially extended systems the karhunen loeve and galerkin multiple shooting approach parameter estimation for spatiotemporal dynamics for coupled map lattices and continuous time domain systems is shown using a combination of multiple shooting karhunen loeve decomposition and galerkin s projection methodologies the resulting advantages in estimating parameters have been studied and discussed for chaotic and turbulent dynamics using small amounts of data from subsystems availability of only scalar and noisy time series data effects of space time parameter variations and in the presence of multiple time scales

the quantum algebra uq sl2 and its equitable presentation abstract we show that the quantum algebra u q sl 2 has a presentation with generators x 1 y z and relations x x 1 x 1 x 1 q x y q 1 y x q q 1 1 q y z q 1 z y q q 1 1 q z x q 1 x z q q 1 1 we call this the equitable presentation we show that y respectively z is not invertible in u q sl 2 by displaying an infinite dimensional u q sl 2 module that contains a nonzero null vector for y respectively z we consider finite dimensional u q sl 2 modules under the assumption that q is not a root of 1 and char k 2 where k is the underlying field we show that y and z are invertible on each finite dimensional u q sl 2 module we display a linear operator that acts on finite dimensional u q sl 2 modules and satisfies 1 x y 1 y z 1 z x on these modules we define using the q exponential function

synchronization based approach for estimating all model parameters of chaotic systems the problem of dynamic estimation of all parameters of a model representing chaotic and hyperchaotic systems using information from a scalar measured output is solved the variational calculus based method is robust in the presence of noise enables online estimation of the parameters and is also able to rapidly track changes in operating parameters of the experimental system the method is demonstrated using the lorenz rossler chaos and hyperchaos models its possible application in decoding communications using chaos is discussed

the accuracy and stability of cn wsgd schemes for space fractional diffusion equation abstract in this paper we discuss the accuracy and the stability of the cn wsgd when it is applied to the initial boundary value problem for a space fractional diffusion equation in the cn wsgd scheme the weighted and shifted grunwald difference wsgd scheme is used to discretize the space fractional derivative and the crank nicolson cn scheme is used to discretize the temporal derivative there is a free parameter in a cn wsgd scheme that affects the stability and the accuracy of the scheme in general a cn wsgd scheme has temporally and spatially second order accuracy and there exists a scheme that has spatially third order accuracy however the third order accuracy scheme is not always stable in this paper we present and prove some theoretical results old and new on the stability of several cn wsgd schemes and consider choosing a value for the parameter such that the corresponding cn wsgd scheme is unconditioned stable and has optimal upper bound for the accuracy numerical results are presented to verify the theoretical results

2 1 gravity for genus 1 in 1 we analysed the algebra of observables for the simple case of a genus 1 initial data surface 2 for 2 1 de sitter gravity here we extend the analysis to higher genus we construct for genus 2 the group of automorphismsh of the homotopy group 1 induced by the mapping class group the grouph induces a groupd of canonical transformations on the algebra of observables which is related to the braid group for 6 threads

harmonic analysis on the quantum lorentz group using a continuation of 6j symbols of su q 2 with complex spins we give a new description of the unitary representations of sl q 2 and find explicit expressions for the characters of unitary representations of sl q 2 we prove a plancherel theorem for the quantum lorentz group

dunkl shift operators and bannai ito polynomials abstract we consider the most general dunkl shift operator l with the following properties i l is of first order in the shift operator and involves reflections ii l preserves the space of polynomials of a given degree iii l is potentially self adjoint we show that under these conditions the operator l has eigenfunctions which coincide with the bannai ito polynomials we construct a polynomial basis which is lower triangular and two diagonal with respect to the action of the operator l this allows to express the bi polynomials explicitly we also present an anti commutator aw 3 algebra corresponding to this operator from the representations of this algebra we derive the structure and recurrence relations of the bi polynomials we introduce new orthogonal polynomials referred to as the complementary bi polynomials as an alternative q 1 limit of the askey wilson polynomials these complementary bi polynomials lead to a new explicit expression for the bi polynomials in terms of the ordinary wilson polynomials