boundary states extended symmetry algebra and module structure for certain rational torus models the massless bosonic field compactified on the circle of rational r2 is reexamined in the presense of boundaries a particular class of models corresponding to r2 1 2k is distinguished by demanding the existence of a consistent set of neumann boundary states the boundary states are constructed explicitly for these models and the fusion rules are derived from them these are the ones prescribed by the verlinde formula from the s matrix of the theory in addition the extended symmetry algebra of these theories is constructed which is responsible for the rationality of these theories finally the chiral space of these models is shown to split into a direct sum of irreducible modules of the extended symmetry algebra

c 1 conformal field theories on riemann surfaces we study the theory ofc 1 torus and 2 orbifold models on general riemann surfaces the operator content and occurrence of multi critical points in this class of theories is discussed the partition functions and correlation functions of vertex operators and twist fields are calculated using the theory of double covered riemann surfaces it is shown that orbifold partition functions are sensitive to the torelli group we give an algebraic construction of the operator formulation of these nonchiral theories on higher genus surfaces modular transformations are naturally incorporated as canonical transformations in the hilbert space

limits and degenerations of unitary conformal field theories in the present paper degeneration phenomena in conformal field theories are studied for this purpose a notion of convergent sequences of cfts is introduced properties of the resulting limit structure are used to associate geometric degenerations to degenerating sequences of cfts which as familiar from large volume limits of non linear sigma models can be regarded as commutative degenerations of the corresponding quantum geometries as an application the large level limit of the a series of unitary virasoro minimal models is investigated in detail in particular its geometric interpretation is determined

symmetries of perturbed conformal field theories the symmetries of perturbed conformal field theories are analysed we explain which generators of the chiral algebras of a bulk theory survive a perturbation by an exactly marginal bulk field we also study the behaviour of d branes under current current bulk deformations we find that the branes always continue to preserve as much symmetry as they possibly can i e as much as is preserved in the bulk we illustrate these findings with several examples including permutation branes in wzw models and b type d branes in gepner models

numerical approximation of a time dependent nonlinear space fractional diffusion equation in this article we analyze a fully discrete numerical approximation to a time dependent fractional order diffusion equation which contains a nonlocal quadratic nonlinearity the analysis is performed for a general fractional order diffusion operator the nonlinear term studied is a product of the unknown function and a convolution operator of order 0 convergence of the approximation and a priori error estimates are given numerical computations are included which confirm the theoretical predictions

partition functions of the two dimensional ashkin teller model on the critical line by mapping the ashkin teller model onto a scalar free field theory the author derives the continuum limit of the partition functions in various finite geometries he justifies expressions which have been conjectured recently explaining in particular why the free field lives on an orbifold and why the spin should be described by a twist operator

partiallyu 1 compactified scalar massless field on the compact riemann surface and the bosonic string amplitudes the theory of the partiallyu 1 compactified scalar massless field on the compact riemann surface with nambu goto action is defined the partition function is determined completely by a choice of the finite dimensional approximations the correlation functions are the only correctly defined objects of the theory the averages of the correlation function asymptotic values provide the amplitudes for the compact riemann surfaces of any genus the usual bosonic string amplitudes are the special cases of the above amplitudes

representations of the heisenberg algebra on a riemann surface we formulate reflection positivity for meromorphic functions and for 1 forms on a riemann surface this construction yields representations of the heisenberg algebra on a riemann surface

relating kac moody virasoro and krichever novikov algebras we demonstrate that the kac moody and virasoro like algebras on riemann surfaces of arbitrary genus with two punctures introduced by krichever and novikov are in two ways linearly related to kac moody and virasoro algebras ons1 the two relations differ by a bogoliubov transformation and we discuss the connection with the operator formalism

2 2 superconformal field theories near orbifold points a thorough analysis of the blowing up modes of the 6 based on the lie algebraa2 d4 is presented we discover that the descriptions of these modes in the language of superconformal field theory and calabi yau compactification are not immediately in agreement a solution to this apparent inconsistency is offered which leads to the possibility of differentiably distinct calabi yau manifolds giving isomorphic physics

stability and convergence of a finite volume method for the space fractional advection dispersion equation we consider the space fractional advection dispersion equation which is obtained from the classical advection diffusion equation by replacing the spatial derivatives with a generalised derivative of fractional order we derive a finite volume method that utilises fractionally shifted grunwald formulae for the discretisation of the fractional derivative to numerically solve the equation on a finite domain with homogeneous dirichlet boundary conditions we prove that the method is stable and convergent when coupled with an implicit timestepping strategy results of numerical experiments are presented that support the theoretical analysis

w algebras new rational models and completeness of the c 1 classification two series ofw with two generators are constructed from chiral vertex operators of a free field representation ifc 1 24k there exists aw 2 3k algebra for k 2 and aw 2 8k algebra for k 4 all possible lowest weight representations their characters and fusion rules are calculated proving that these theories are rational it is shown that these non unitary theories complete the classification of all rational theories with effective central chargeceff 1 the results are generalized to the case of extended supersymmetric conformal algebras

string vertices overlap equations τ functions and the hirota equation string vertices v are shown to satisfy a new type of overlap equation of the form v exp ip cdot q i xi i v exp ip cdot q i xi i left frac d xi j d xi i right p 2 2 as well as corresponding equations foran andbn cycles a special case of such an equation when integrated is shown to be the hirota equation for the k p hierarchy

deformations of complex structures and conformal field theories on riemann surfaces the role of complex structure deformations in representing the virasoro algebra associated to any conformal field theory on a genus p 1 compact riemann surface is investigated

theta functions modular invariance and strings we use quillen s theorem and algebraic geometry to investigate the modular transformation properties of some quantities of interest in string theory in particular we show that the spin structure dependence of the chiral dirac determinant on a riemann surface is given by riemann s theta function we use this result to investigate the modular invariance of multiloop heterotic string amplitudes

lie algebra automorphisms the weyl group and tables of shift vectors the canonical lift of the weyl group in the automorphism group of a simple lie algebra is discussed a constructive algorithm is given to compute the shift vector for every weyl group conjugacy class and tables for the simply laced lie algebras of rank 8 are provided applications in physics such as the equivalence of certain orbifold and twisted wzw string models as well as the implication for the moduli space of 2 d conformal field theories are discussed

duality and modular invariance in rational conformal field theories we investigate the polynomial equations which should be satisfied by the duality data for a rational conformal field theory we show that by these duality data we can construct some vector spaces which are isomorphic to the spaces of conformal blocks one can construct explicitly the inner product for the former if one deals with a unitary theory these vector spaces endowed with an inner product are the algebraic reminiscences of the hilbert spaces in a chern simons theory as by products we show that the polynomial equations involving the modular transformations for the one point blocks on the torus are not independent we discuss the solution of structure constants for a physical theory making some assumption we obtain a neat solution and this solution in turn implies that the quantum groups of the left sector and of the right sector must be the same although the chiral algebras need not be the same some examples are given finally we discuss the reconstruction of the quantum group in a rational conformal theory

correlation functions of the 2d sine gordon model a number of two dimensional 2d critical phenomena can be described in terms of the 2d sine gordon model with bosonization several 1d quantum systems can be transformed to the same model however the transition of the 2d sine gordon model the berezinskii kosterlitz thouless bkt transition is essentially different from a second order transition the divergence of the correlation length is more rapid than any power law and there are logarithmic corrections these pathological features make it difficult to determine the bkt transition point and critical indices from finite size calculations in this paper we calculate correlation functions of this model using a real space renormalization technique it is found that several correlation functions or eigenvalues of the corresponding transfer matrix for a finite system become degenerate on the bkt line including the logarithmic corrections by the use of this degeneracy which reflects the hidden su 2 symmetry on the bkt line it is possible to determine the bkt critical line with high precision from a small amount of data and to identify the universality class in addition new universal relations are found these results shed light on the relation between abelian and non abelian bosonization

a log linear model for the birnbaum saunders distribution the birnbaum saunders distribution was derived to model times to failure for metals subject to fatigue in this article we formulate and develop a log linear model for the birnbaum saunders distribution the model may be used for accelerated life testing or to compare the median lives of several populations methods of analyzing data for this log linear model are discussed with maximum likelihood and least squares methods being compared it is found that for commonly occurring conditions the notorious intractability of the birnbaum saunders distribution is not a serious problem because least squares and normal theory procedures provide a reasonable alternative

bosonization on higher genus riemann surfaces we prove the equivalence between certain fermionic and bosonic theories in two spacetime dimensions the theories have fields of arbitrary spin on compact surfaces with any number of handles global considerations require that we add new topological terms to the bosonic action the proof that our prescription is correct relies on methods of complex algebraic geometry

a semi alternating direction method for a 2 d fractional fitzhugh nagumo monodomain model on an approximate irregular domain a fitzhugh nagumo monodomain model has been used to describe the propagation of the electrical potential in heterogeneous cardiac tissue in this paper we consider a two dimensional fractional fitzhugh nagumo monodomain model on an irregular domain the model consists of a coupled riesz space fractional nonlinear reaction diffusion model and an ordinary differential equation describing the ionic fluxes as a function of the membrane potential second we use a decoupling technique and focus on solving the riesz space fractional nonlinear reaction diffusion model a novel spatially second order accurate semi implicit alternating direction method siadm for this model on an approximate irregular domain is proposed third stability and convergence of the siadm are proved finally some numerical examples are given to support our theoretical analysis and these numerical techniques are employed to simulate a two dimensional fractional fitzhugh nagumo model on both an approximate circular and an approximate irregular domain

finite size effects in conformal field theories and non local operators in one dimensional quantum systems the authors generalise the well known connection between critical exponents and finite size effects in conformal field theories to non local operators and find the long wave asymptotics of vacuum expectation values of some non local operators in one dimensional quantum systems

a stable explicitly solvable numerical method for the riesz fractional advection dispersion equations abstract in this paper we present a finite difference scheme for solving the riesz fractional advection dispersion equations rfades the scheme is obtained by using asymmetric discretization technique and modify the shifted grunwald approximation to fractional derivative by calculating the unknowns in differential nodal point sequences at the odd and even time levels the discrete solution of the scheme can be obtained explicitly the computational cost for the scheme at each time step can be o klog k by using the fast matrix vector multiplication with the help of toeplitz structure where k is the number of unknowns we prove that the scheme is solvable and unconditionally stable we derive the error estimates in discrete l2 norm which is optimal in some cases numerical examples are presented to verify our theoretical results

a novel method for nonlinear fractional partial differential equations combination of dtm and generalized taylor s formula in this article a novel numerical method is proposed for nonlinear partial differential equations with space and time fractional derivatives this method is based on the two dimensional differential transform method dtm and generalized taylor s formula the fractional derivatives are considered in the caputo sense several illustrative examples are given to demonstrate the effectiveness of the present method results obtained using the scheme presented here agree well with the analytical solutions and the numerical results presented elsewhere results also show that the numerical scheme is very effective and convenient for solving nonlinear partial differential equations of fractional order

high order unconditionally stable difference schemes for the riesz space fractional telegraph equation in this paper a class of unconditionally stable difference schemes based on the pade approximation is presented for the riesz space fractional telegraph equation firstly we introduce a new variable to transform the original differential equation to an equivalent differential equation system then we apply a second order fractional central difference scheme to discretize the riesz space fractional operator finally we use 1 1 2 2 and 3 3 pade approximations to give a fully discrete difference scheme for the resulting linear system of ordinary differential equations matrix analysis is used to show the unconditional stability of the proposed algorithms two examples with known exact solutions are chosen to assess the proposed difference schemes numerical results demonstrate that these schemes provide accurate and efficient methods for solving a space fractional hyperbolic equation