the use of pde centres in the local rbf hermitian method for 3d convective diffusion problems in this work an extension is proposed to the local hermitian interpolation lhi method a meshless numerical method based on interpolation with small and heavily overlapping radial basis function rbf systems this extension to the lhi method uses interpolation functions which themselves satisfy the partial differential equation pde to be solved in this way a much improved reconstruction of partial derivatives can be obtained resulting in significantly improved accuracy in many cases the implementation algorithm is described and is validated via three convection diffusion reaction problems for steady and transient situations a crank nicolson implicit time stepping technique is used for the time dependent problems in the proposed approach a form of analytical upwinding is implicitly implemented by the use of the partial differential operator of the governing equation in the interpolation function which includes the desired information about the convective velocity field the implicit upwinding scheme intrinsic to the proposed numerical approach is tested by solving a one dimensional travelling front problem at peclet numbers of 500 1000 2000 5000 and infinity which corresponds to a shock front in the case of infinity in addition the accuracy of the numerical scheme is validated against a one dimensional steady state solution exhibiting strong boundary layer effects and also against a steady and a transient three dimensional convection diffusion problem on irregular datasets all the test cases are validated against the corresponding analytical solutions finally the effect of various interpolation stencil configurations is investigated and some important limitations on local data centre distribution are identified

dynamical noether invariants for time dependent nonlinear systems dynamical invariants are derived for time dependent systems with nonlinear equations of motion including nonharmonic damped systems the concept of a dynamical algebra is discussed and its utility for the construction of dynamical invariants for nonharmonic systems is demonstrated finally we show the existence of dynamical invariants for some nonlinear quantum systems

stable computation of differentiation matrices and scattered node stencils based on gaussian radial basis functions stable computation of differentiation matrices and scattered node stencils based on gaussian radial basis functions

operator splitting methods for pricing american options under stochastic volatility we consider the numerical pricing of american options under heston s stochastic volatility model the price is given by a linear complementarity problem with a two dimensional parabolic partial differential operator we propose operator splitting methods for performing time stepping after a finite difference space discretization the idea is to decouple the treatment of the early exercise constraint and the solution of the system of linear equations into separate fractional time steps with this approach an efficient numerical method can be chosen for solving the system of linear equations in the first fractional step before making a simple update to satisfy the early exercise constraint our analysis suggests that the crank nicolson method and the operator splitting method based on it have the same asymptotic order of accuracy the numerical experiments show that the operator splitting methods have comparable discretization errors they also demonstrate the efficiency of the operator splitting methods when a multigrid method is used for solving the systems of linear equations

wkb equivalent potentials for q deformed harmonic and anharmonic oscillators wkb equivalent potentials wkb ep s giving the same wkb spectrum as the q deformed harmonic oscillator are determined in analytic form for q being complex the wkb ep resembles the poschl teller morse or woods saxon potentials widely used in nuclear hypernuclear and molecular physics for q real the wkb ep still goes to infinity at large x while at small x it resembles a sextic oscillator the wkb ep s of the anharmonic oscillators with uq 2 or suq 1 1 symmetries used for the description of vibrational molecular spectra are also calculated they correspond to q deformed versions of the modified poschl teller potential

ladder operators normal ordering problem for quantum deformed systems and the q p generalization of the stirling and bell numbers we resolve the ladder operators normal ordering problem for strings in the form for q p deformed supersymmetric and shape invariant potential systems where n is a positive integer we provide exact and explicit expressions for their normal form where in all are at the right hand side and find that the solution involves expansion coefficients sequence which corresponds to the generalization for any q p deformed shape invariant potential system of the classical stirling and bell numbers we show that these numbers are given for expressions depending on parameters related to the forms of supersymmetric partner potentials and the system quantum deformation we apply the general formalism to poschl teller morse and scarf potential systems

ground state energy and mass gap of a generalized quantum spin ladder we show that a two leg ladder hamiltonian introduced recently by albeverio and fei can be made to satisfy the hecke algebra as a result we have found an equivalent representation of the eigenspectrum in terms of the spin antiferromagnetic xxz chain at 5 3 the values of thermodynamic quantities such as the ground state energy and mass gap follow from the known xxz results

duality relations for asymmetric exclusion processes we derive duality relations for a class ofu q su 2 symmetric stochastic processes including among others the asymmetric exclusion process in one dimension like the known duality relations for symmetric hopping processes these relations express certainm point correlation functions inn particle systems n m in terms of sums of correlation functions of the same system but with onlym particles for the totally asymmetric case we obtain exact expressions for some boundary density correlation functions the dynamical exponent for these correlators isz 2 which is different from the dynamical exponent for bulk density correlations which is known to bez 3 2

rbf fd formulas and convergence properties the local rbf is becoming increasingly popular as an alternative to the global version that suffers from ill conditioning in this paper we study analytically the convergence behavior of the local rbf method as a function of the number of nodes employed in the scheme the nodal distance and the shape parameter we derive exact formulas for the first and second derivatives in one dimension and for the laplacian in two dimensions using these formulas we compute taylor expansions for the error from this analysis we find that there is an optimal value of the shape parameter for which the error is minimum this optimal parameter is independent of the nodal distance our theoretical results are corroborated by numerical experiments

application of radial basis function with l stable pade time marching scheme for pricing exotic option in this work a high order l stable method for pricing exotic option has been discussed the spatial discretization is done by radial basis function based local grid free method to achieve first order ordinary differential equation followed by fourth order l stable method numerical study with one and two asset problems for digital option butterfly spread and barrier option is carried out with highly accurate results that are in good agreement with those obtained by other numerical methods in the literature

new results for the xx model with boundaries in this paper we study the xx model with particular non diagonal boundaries using a fermionization technique for an extended version of this chain we are able to present a new operator which commutes with the extended and also the original hamiltonian it is a highly non trivial conserved quantity in the spin chain formalism which gives rise to an interesting combinatorial partition function

temperley lieb words as valence bond ground states based on the temperley lieb algebra we define a class of one dimensional hamiltonians with nearest and next nearest neighbour interactions using the regular representation we present ground states of this model as words of the algebra two point correlation functions can be computed employing the temperley lieb relations choosing a spin 1 2 representation of the algebra we obtain a generalization of the q deformed majumdar ghosh model the ground states become valence bond states

the q hermite polynomial and the representations of heisenberg and quantum heisenberg algebras the authors introduce a pair of canonical conjugate q deformed operators d x and discuss the relations between the deformed operators d x and q series the realizations of some lie symmetries heisenberg and quantum heisenberg algebras are given for the operators d and x they show that the q analogous hermite polynomials are representations of heisenberg and quantum heisenberg algebras realized in this way when q is a root of unity the properties of the q analogous hermite polynomials are also discussed

observations on the behavior of radial basis function approximations near boundaries abstract rbf approximations would appear to be very attractive for approximating spatial derivatives in numerical simulations of pdes rbfs allow arbitrarily scattered data generalize easily to several space dimensions and can be spectrally accurate however accuracy degradations near boundaries in many cases severely limit the utility of this approach with that as motivation this study aims at gaining a better understanding of the properties of rbf approximations near the ends of an interval in 1 d and towards edges in 2 d

a q deformed oscillator system with quantum group slq l symmetry a q deformed oscillator system with quantum group slq l symmetry is constructed in terms of the q deformed boson operators the energy level degeneracy associated with the irreducible representation of slq l is analysed especially for the case that q is a root of unity

clifford representations in integrable systems in this paper we analyze integrable systems from a clifford algebra point of view this approach allows us to give a clear representation theoretic exposition of techniques used in spin systems thereby showing their naturality we then extend this approach to the analysis of the xx model with nondiagonal boundaries which is among others related to growing and fluctuating interfaces and stochastic reaction diffusion systems with this rationale it is possible to diagonalize the system and find new hidden conservation laws

exact solution of the deformed biquadratic spin 1 chain we solve for the sped of the deformed biquadratic spin l chain by a generalization of the coordinate bethe ansatz the model is invariant under the quantum group u sl z an6 depending on the deformation parameter q it s hamiltonian may not be hermitian we consider several boundary conditions and find in particular that for periodic ones the complete specmm of the spin 4 xxz model is always contained in the spec of the biquadratic model for free boundary conditions the spectrum is identicd to the xxz model speci only the multiplicities are of course different bethe states are highest weight states of the quantum group except possibly for zerwnergy states

quantum clifford hopf algebras for even dimensions in this paper we study the quantum clifford hopf algebras chq d for even dimensions d and obtain their intertwiner r matrices which are elliptic solutions to the yang baxter equation in the trigonometric limit of these new algebras we find the possibility of connecting with extended supersymmetry we also analyse the corresponding spin chain hamiltonian which leads to suzuki s generalized xy model

local rbf fd solutions for steady convection diffusion problems this paper describes the application of radial basis function rbf based finite difference type scheme rbf fd for solving steady convection diffusion equations numerical studies are made using multiquadric mq rbf by varying the shape parameter in mq the accuracy of the solution is seen to be highly improved for large values of reynolds numbers the developed scheme has been compared with the corresponding finite difference scheme and found that the solutions obtained using the former are non oscillatory copyright 2007 john wiley sons ltd

space time adaptive finite difference method for european multi asset options the multi dimensional black scholes equation is solved numerically for a european call basket option using a priori a posteriori error estimates the equation is discretized by a finite difference method on a cartesian grid the grid is adjusted dynamically in space and time to satisfy a bound on the global error the discretization errors in each time step are estimated and weighted by the solution of the adjoint problem bounds on the local errors and the adjoint solution are obtained by the maximum principle for parabolic equations comparisons are made with monte carlo and quasi monte carlo methods in one dimension and the performance of the method is illustrated by examples in one two and three dimensions

on the critical behaviour of the anisotropic biquadratic spin 1 chain a study is made of the anisotropic biquadratic spin 1 hamiltonian introduced by batchelor et al 1990 this model can be represented as a sum of operators satisfying a temperley lieb algebra its critical behaviour is explored by using the relations between mass gap amplitudes and critical exponents predicted by conformal invariance the model like in the xxz chain is found to exhibit a line of continuously varying exponents by comparing its eigenspectrum with that of the xxz hamiltonian several exact values are predicted

deformed su 2 heisenberg chain the general hamiltonian for the su 2 q invariant arbitrary spin heisenberg chain is presented some of these interactions are shown to satisfy braid group relations and the temperley lieb algebra relations spin 3 2 is explored in more detail and in this case a general solution to the braid conditions is known in the generic sense

the model with boundaries part i diagonalization of the finite chain this is the first of three papers dealing with the xx finite quantum chain with arbitrary not necessarily hermitian boundary terms this extends previous work where the periodic or diagonal boundary terms were considered in order to find the spectrum and wavefunctions an auxiliary quantum chain is examined which is quadratic in fermionic creation and annihilation operators and hence diagonalizable the secular equation is in general complicated but several cases were found when it can be solved analytically for these cases the ground state energies are given the appearance of boundary states is also discussed and in view of the applications considered in the next papers the one and two point functions are expressed in terms of pfaffians

dissipative quantum dynamics solution of the generalized von neumann equation for the damped harmonic oscillator a recently proposed generalized von neumann equation describing dissipative time evolution of quantum systems is applied to a damped and driven harmonic oscillator utilizing lie algebraic methods the nonlinear operator equation is solved and from this solution the behaviour of the mean value x is extracted including both amplitude and phaseshift relative to the driving force a differential equation for x is derived and the damping constant is identified

semi canonical quantisation of dissipative equations in order to avoid the known problems of canonical quantisation of the dissipative equation mx alpha xn delta v delta x 0 a modified approach is proposed it is based on the classically conserved quantity e mx2 2 v x alpha integral xndx rather than on a hamilton function linear quadratic and cubic damping n 1 2 and 3 are discussed explicitly for n 1 and n 3 normalisable one particle states cannot be stable however for quadratic friction which is classically a reversible process the author shows using the example of the harmonic oscillator that stable normalisable eigenstates can exist

a note on graded yang baxter solutions as braid monoid invariants we construct two osp n mod 2m solutions of the graded yang baxter equation by using the algebraic braid monoid approach the factorizable s matrix interpretation of these solutions is also discussed

radial basis function partition of unity methods for pricing vanilla basket options meshfree methods based on radial basis function rbf approximation are becoming widely used for solving pde problems they are flexible with respect to the problem geometry and highly accurate a disadvantage of these methods is that the linear system to be solved becomes dense for globally supported rbfs a remedy is to introduce localisation techniques such as partition of unity rbf partition of unity methods rbf pum allow for a significant sparsification of the linear system and lower the computational effort in this work we apply a global rbf method as well as rbf pum to problems in option pricing we consider one and two dimensional vanilla options in order to price american options we employ a penalty approach a penalty term suitable for american multi asset call options has been designed rbf pum is shown to be competitive compared with a finite difference method and a global rbf method it is as accurate as the global rbf method but significantly faster the results for rbf pum look promising for extension to higher dimensional problems

an imex scheme for pricing options under stochastic volatility models with jumps partial integro differential equation pide formulations are often preferable for pricing options under models with stochastic volatility and jumps especially for american style option contracts we consider the pricing of options under such models namely the bates model and the so called stochastic volatility with contemporaneous jumps svcj model the nonlocality of the jump terms in these models leads to matrices with full matrix blocks standard discretization methods are not viable directly since they would require the inversion of such a matrix instead we adopt a two step implicit explicit imex time discretization scheme the imex cnab scheme where the jump term is treated explicitly using the second order adams bashforth ab method while the rest is treated implicitly using the crank nicolson cn method the resulting linear systems can then be solved directly by employing lu decomposition alternatively the systems can be iterated under a scalable algebraic multigrid amg method f

free fermionic elliptic reflection matrices and quantum group invariance diagonal solutions for the reflection matrices associated to the elliptic r matrix of the eight vertex free fermion model are presented they lead through the second derivative of the open chain transfer matrix to an xy hamiltonian in a magnetic field which is invariant under a quantum deformed clifford hopf algebra

the spin 1 2xxz heisenberg chain the quantum algebra uq sl 2 and duality transformations for minimal models the finite size scaling spectra of the spin 1 2 xxz heisenberg chain with toroidal boundary conditions and an even number of sites provide a projection mechanism yielding the spectra of models with a central charge c 1 including the unitary and nonunitary minimal series taking into account the half integer angular momentum sectors which correspond to chains with an odd number of sites in many cases leads to new spinor operators appearing in the projected systems these new sectors in the xxz chain correspond to new types of frustration lines in the projected minimal models the corresponding new boundary conditions in the hamiltonian limit are investigated for the ising model and the 3 state potts model and are shown to be related to duality transformations which are an additional symmetry at their self dual critical point by different ways of projecting systems we find models with the same central charge sharing the same operator content and modular invariant partition function which however differ in the distribution of operators into sectors and hence in the physical meaning of the operators involved related to the projection mechanism in the continuum there are remarkable symmetry properties of the finite xxz chain the observed degeneracies in the energy and momentummore spectra are shown to be the consequence of intertwining relations involving u sub q sl 2 quantum algebra transformations 35 refs 3 tabs less

q deformations of the o 3 symmetric spin 1 heisenberg chain the authors present the general expression for the spin 1 heisenberg chain invariant under the uq so 3 quantum algebra several physical and mathematical implications are discussed

on the quantization of linearly damped harmonic oscillator using the equation of motion for a linearly damped harmonic oscillator as fundamental equation a condition has been obtained that must be satisfied by its lagrangian and a number of functional forms has been critically investigated the lagrangians that successfully describe a classical damped oscillator have been quantized using feynman s elegant path integral method however it is found that the expectation value of energy of the damped oscillator in all the cases becomes infinitely large for t ruling out the possibility of treating the damped quantum system as a one body problem

finitely correlated states on quantum spin chains we study a construction that yields a class of translation invariant states on quantum spin chains characterized by the property that the correlations across any bond can be modeled on a finite dimensional vector space these states can be considered as generalized valence bond states and they are dense in the set of all translation invariant states we develop a complete theory of the ergodic decomposition of such states including the decomposition into periodic neel ordered states the ergodic components have exponential decay of correlations all states considered can be obtained as local functions of states of a special kind so called purely generated states which are shown to be ground states for suitably chosen finite range vbs interactions we show that all these generalized vbs models have a spectral gap our theory does not require symmetry of the state with respect to a local gauge group in particular we illustrate our results with a one parameter family of examples which are not isotropic except for one special case this isotropic model coincides with the one dimensional antiferromagnet recently studied by affleck kennedy lieb and tasaki

on the role of polynomials in rbf fd approximations radial basis function generated finite difference rbf fd approximations generalize classical grid based finite differences fd from lattice based to scattered node layouts this greatly increases the geometric flexibility of the discretizations and makes it easier to carry out local refinement in critical areas many different types of radial functions have been considered in this rbf fd context in this study we find that i polyharmonic splines phs in conjunction with supplementary polynomials provide a very simple way to defeat stagnation also known as saturation error and ii give particularly good accuracy for the tasks of interpolation and derivative approximations without the hassle of determining a shape parameter in follow up studies we will focus on how to best use these hybrid rbf polynomial bases for fd approximations in the contexts of solving elliptic and hyperbolic type pdes

an rbf fd sparse scheme to simulate high dimensional black scholes partial differential equations abstract the objective of this research is to investigate the numerical solution of high dimensional black scholes partial differential equations pdes to do this the weights of the radial basis function finite difference rbf fd scheme using inverse multi quadric function are used with an emphasis on the hotzone the proposed rbf fd technique is capable to price multi asset options efficiently

the realizations of quantum groups of an 1 and cn types in q deformed oscillator systems at classical and quantum levels in the classical q deformed oscillators system the poisson bracket realizations of the quantum enveloping algebras of an 1 and cn types are given in the symplectic space v without deformation when the oscillators system is canonically quantized the lie bracket realizations of the quantum enveloping algebras of an 1 and cn types are obtained the hopf structures of these quantum enveloping algebras are supplied

four state models and clifford algebras with appropriate boundary conditions the anisotropic xy chain in a magnetic field is known to be invariant under quantum group transformations we generalize this model defining a class of integrable chains with several fermionic degrees of freedom per site in order to maintain the quantum group symmetry a general condition on the parameters of these systems is derived it is shown that the corresponding quantum algebra is a multi parameter deformation of the clifford algebra discussing a special physical example we observe a new type of zero mode

time dependent invariant associated to nonlinear schrodinger langevin equations via the quantum hydrodynamical method a time dependent invariant associated to the quantum dissipative time dependent harmonic oscillator tdho described by two classes of nonlinear schrodinger langevin equations with the following frictional nonlinear terms is constructed i w1 i ln ln which is the schuch chung hartman frictional nonlinear term and ii w2 x x cp 1 c p 1 2 i c which includes the sussmann c 1 the hasse c 1 2 and the albrecht kostin c 0 frictional nonlinear operators the associated invariant found is exact for the schuch chung hartman and hasse models and only approximate for the sussman and albrecht kostin models

quantum spin chains with quantum group symmetry we consider actions of quantum groups on lattice spin systems we show that if an action of a quantum group respects the local structure of a lattice system it has to be an ordinary group even allowing weakly delocalized quasi local tails of the action we find that there are no actions of a properly quantum group commuting with lattice translations the non locality arises from the ordering of factors in the quantum groupc algebra and can be made one sided thus allowing semi local actions on a half chain under such actions localized quantum group invariant elements remain localized hence the notion of interactions invariant under the quantum group and also under translations recently studied by many authors makes sense even though there is no global action of the quantum group we consider a class of such quantum group invariant interactions with the property that there is a unique translation invariant ground state under weak locality assumptions its gns representation carries no unitary representation of the quantum group

radial basis function generated finite differences for option pricing problems abstract in this paper we present a numerical method to price options based on radial basis function generated finite differences rbf fd in space and the backward differentiation formula of order 2 bdf 2 in time we use gaussian rbfs that depend on a shape parameter e the choice of this parameter is crucial for the performance of the method we chose e as const h 1 and we derive suitable values of the constant for different stencil sizes in 1d and 2d this constant is independent of the problem parameters such as the volatilities of the underlying assets and the interest rate in the market in the literature on option pricing with rbf fd a constant value of the shape parameter is used we show that this always leads to ill conditioning for decreasing h whereas our proposed method avoids such ill conditioning we present numerical results for problems in 1d 2d and 3d demonstrating the useful features of our method such as discretization sparsity flexibility in node placement and easy dimensional extendability which provide high computational efficiency and accuracy

fusion procedure for open chains the authors have generalized sklyanin s approach of constructing open integrable quantum spin chains to the case of pt invariant r matrices they formulate a fusion procedure for such chains in particular they show that the fused transfer matrix can be expressed in terms of products of the original transfer matrix and products of certain quantum determinants which can be explicitly evaluated applications of these results include constructing open integrable higher spin chains as well as obtaining functional equations for transfer matrix eigenvalues which may be solved by an analytical bethe ansatz

minimal unitary models and the closed invariant spin chain we consider the hamiltonian of the closed invariant chain we project a particular class of statistical models belonging to the unitary minimal series a particular model corresponds to a particular value of the coupling constant the operator content is derived this class of models has charge dependent boundary conditions in simple cases ising three state potts corresponding hamiltonians are constructed these are non local as the original spin chain

bethe ansatz solution of a closed spin 1 xxz heisenberg chain with quantum algebra symmetry a quantum algebra invariant integrable closed spin 1 chain is introduced and analyzed in detail the bethe ansatz equations as well as the energy eigenvalues of the model are obtained the highest weight property of the bethe vectors with respect to uq sl 2 is proved

spectra of non hermitian quantum spin chains describing boundary induced phase transitions the spectrum of the non hermitian asymmetric xxz chain with additional non diagonal boundary terms is studied the lowest lying eigenvalues are determined numerically for the ferromagnetic chain the temporal correlation length shows the same properties as the spatial correlation length for the antiferromagnetic chain we find a conformal invariant spectrum where the partition function corresponds to the one of a coulomb gas with only magnetic charges this is in contrast to the case of diagonal boundary terms where one finds only electric charges similar results are obtained in a toy model that can be diagonalized analytically in terms of free fermions

a radial basis function hermite finite difference approach to tackle cash or nothing and asset or nothing options abstract a novel local meshfree method is proposed for simulating cash or nothing and asset or nothing options at which the initial condition is discontinuous the novelty of the scheme is in the use of hermite radial basis function rbf interpolation on set of local nodes we also construct the rbf generated hermite finite difference hfd formulas and subsequently obtain their associated differentiation matrices dms the developed accuracy equipped with a shift algorithm for smoothing the discontinuous initial conditions are illustrated with a series of financial tests

conformal invariance and the operator content of the xxz model with arbitrary spin the authors are concerned with the critical properties of the anisotropic heisenberg chain or xxz model with arbitrary integer or half integer spin the eigenspectra of these hamiltonians with periodic boundaries are calculated for finite chains by solving numerically their associated bethe ansatz equations the resulting spectra are found to be in accord with the predictions of conformal invariance and the operator content is identified for lattices with an even and odd number of sites the results for spin 1 and spin 3 2 indicate that the conformal anomaly c for these models in the gapless regime has the value c 3s 1 s independent of the anisotropy and the exponents vary continuously with the anisotropy as in the eight vertex model the operator content of these models indicate that the underlying field theory governing these critical spin s models is described by composite fields formed by the product of gaussian and z n fields with n 2s finally some of the irrelevant operators which produce the leading finite size corrections of the eigenenergies are identified

an algebraic q deformed form for shape invariant systems a quantum deformed theory applicable to all shape invariant bound state systems is introduced by defining q deformed ladder operators we show that these new ladder operators satisfy new q deformed commutation relations in this context we construct an alternative q deformed model that preserves the shape invariance property presented by the primary system q deformed generalizations of morse scarf and coulomb potentials are given as examples

multi dimensional option pricing using radial basis functions and the generalized fourier transform we show that the generalized fourier transform can be used for reducing the computational cost and memory requirements of radial basis function methods for multi dimensional option pricing we derive a general algorithm including a transformation of the black scholes equation into the heat equation that can be used in any number of dimensions numerical experiments in two and three dimensions show that the gain is substantial even for small problem sizes furthermore the gain increases with the number of dimensions