conformal invariance and critical dynamics the author discusses the behaviour of dynamic correlation functions under conformal transformations in particular the exact form of the dynamic scaling function at the critical point for an arbitrary isotropic two dimensional model is obtained with either conserved or nonconserved order parameter

critical phenomena at surfaces in a model of nonequilibrium phase transitions critical phenomena at surfaces are studied in a simple two dimensional model of non equilibrium phase transitions belonging to the class of interacting particle systems the mean field renormalization group approach and numerical simulations are used to determine the phase diagram for this system and some of the critical exponents associated with ordinary special extraordinary and surface phase transitions

representations of the schrodinger group and matrix orthogonal polynomials the representations of the schrodinger group in one space dimension are explicitly constructed in the basis of the harmonic oscillator states these representations are seen to involve matrix orthogonal polynomials in a discrete variable that have charlier and meixner polynomials as building blocks the underlying lie theoretic framework allows for a systematic derivation of the structural formulas recurrence relations difference equations rodrigues formula etc that these matrix orthogonal polynomials satisfy

dynamic correlations in domain growth a 1 n expansion the authors consider the dynamics of the n component ginzburg landau model with nonconserved order parameter model a following a quench from a high temperature equilibrium state to zero temperature the two time correlation function of the order parameter field is found in the 1 n expansion to have the asymptotic scaling form ck t t t d2 t t lambda 2 f k2t k2t for t t with f 0 0 constant the form of the new exponent lambda which is a non trivial function of n and d was given explicitly to o 1 n in a recent letter the purpose of this study is to present a more detailed account of the calculation leading to the o 1 n form for lambda they also examine the role of thermal fluctuations in the ordered phase and the effect of long range initial correlations on the ordering process

on holographic realization of logarithmic galilean conformal algebra we study two dimensional logarithmic galilean conformal algebra lgca by making use of a contraction of topologically massive gravity at critical point we observe that using a naive contraction at the critical point fails to give a well defined theory though contracting the theory while we are approaching the critical point leads to a well behaved expression for two point functions of the energy momentum tensors of lgca

the exotic conformal galilei algebra and nonlinear partial differential equations abstract the conformal galilei algebra cga and the exotic conformal galilei algebra ecga are applied to construct partial differential equations pdes and systems of pdes which admit these algebras we show that there are no single second order pdes invariant under the cga but systems of pdes can admit this algebra moreover a wide class of nonlinear pdes exists which are conditionally invariant under cga it is further shown that there are systems of nonlinear pdes admitting ecga with the realisation obtained very recently in d martelli y tachikawa comments on galilei conformal field theories and their geometric realisation preprint arxiv 0903 5184v2 hep th 2009 moreover wide classes of nonlinear systems invariant under two different 10 dimensional subalgebras of ecga are explicitly constructed and an example with possible physical interpretation is presented

fast analysis of transient acoustic wave scattering from rigid bodies using the multilevel plane wave time domain algorithm the analysis of transient wave scattering from rigid bodies using integral equation based techniques is computationally intensive if carried out using classical schemes the evaluation of the velocity potential on the surface of a three dimensional scatterer represented in terms of ns spatial basis functions for nt time steps requires o ntns2 operations the recently developed plane wave time domain pwtd algorithm permits the rapid evaluation of transient fields that are generated by bandlimited source distributions it has been shown that incorporation of the pwtd algorithm into integral equation based solvers in a two level setting reduces the computational complexity of a transient analysis to o ntns1 5 logns in this paper it is shown that casting the pwtd scheme into a multilevel framework permits the analysis of transient acoustic surface scattering phenomena in o ntnslog2ns operations using o ntns memory numerical examples that demonstrate the efficacy of the multilevel implementation are also presented

a monte carlo study of scaling and universality in 2d directed percolation monte carlo data for the pair correlation function is obtained on five different lattices for the directed percolation problem in two dimensions scaling is tested and found to hold beyond 20 lattice spacings for all models evidence is found to support the universality of the scaling function the values of the exponents gamma v sub and vperpendicular to v sub are extracted from the data the results for all models are summarised by gamma v sub 1 305 or 0 007 and vperpendicular to v sub 0 629 or 0 010 comparison is made with estimates derived by other means

on the identification of quasiprimary scaling operators in local scale invariance the relationship between physical observables defined in lattice models and the associated quasi primary scaling operators of the underlying field theory is revisited in the context of local scale invariance we argue that this relationship is only defined up to a time dependent amplitude and derive the corresponding generalizations of predictions for two time response and correlation functions applications to non equilibrium critical dynamics of several systems with a fully disordered initial state and vanishing initial magnetization including the glauber ising model the frederikson andersen model and the ising spin glass are discussed the critical contact process and the parity conserving non equilibrium kinetic ising model are also considered

the galilean relativistic principle and nonlinear partial differential equations the second order partial differential equations invariant under transformations of galilei rotation scale and projection are described

logarithmic correlators in nonrelativistic conformal field theory we show how logarithmic terms may arise in the correlators of fields which belong to the representation of the schrodinger virasoro algebra or the affine galilean conformal algebra gca we show that in gca only scaling operator can have a jordan form and rapidity cannot we observe that in both algebras logarithmic dependence appears along the time direction alone

meta conformal invariance and the boundedness of two point correlation functions the covariant two point functions derived from ward identities in direct space can be affected by consistency problems and can become unbounded for large time or space separations this difficulty arises for several extensions of dynamical scaling for example schrodinger invariance conformal galilei invariance or meta conformal invariance but not for standard ortho conformal invariance for meta conformal invariance in dimensions which acts as a dynamical symmetry of a simple advection equation these difficulties can be cured by going over to a dual space and an extension of these dynamical symmetries through the construction of a new generator in the cartan sub algebra this provides a canonical interpretation of meta conformally covariant two point functions as correlators galilei conformal correlators can be obtained from meta conformal invariance through a simple contraction in contrast by an analogus construction schrodinger covariant two point functions are causal response functions all these two point functions are bounded at large separations for sufficiently positive values of the scaling exponents

remark on higher derivative mechanics with l conformal galilei symmetry based on the results in a galajinsky and i masterov nucl phys b 866 212 2013 we consider a way to construct a higher derivative mechanical model which possesses the l conformal galilei symmetry the dynamical system describes generalized pais uhlenbeck oscillator in the presence of an effective external field

lie bialgebra structures on the deformative schrodinger virasoro algebras in this paper we investigate lie bialgebra structures on the deformative schrodinger virasoro algebras mainly using the techniques introduced recently which indicate that all cases considered in this paper except one behave different from their centerless ones

a multilevel cartesian non uniform grid time domain algorithm a multilevel cartesian non uniform grid time domain algorithm cngtda is introduced to rapidly compute transient wave fields radiated by time dependent three dimensional source constellations cngtda leverages the observation that transient wave fields generated by temporally bandlimited and spatially confined source constellations can be recovered via interpolation from appropriately delay and amplitude compensated field samples this property is used in conjunction with a multilevel scheme in which the computational domain is hierarchically decomposed into subdomains with sparse non uniform grids used to obtain the fields for both surface and volumetric source distributions the computational cost of cngtda to compute the transient field at n s observation locations from n s collocated sources for n t discrete time instances scales as o n tn slogn s and o n tn slog 2n s in the low and high frequency regimes respectively coupled with marching on in time mot time domain integral equations cngtda can facilitate efficient analysis of large scale time domain electromagnetic and acoustic problems

non local representations of the ageing algebra in higher dimensions the ageing lie algebra and especially its local representations for a dynamical exponent z 2 have played an important role in the description of systems undergoing simple ageing after a quench from a disordered state to the low temperature phase here the construction of representations of for generic values of z is described for any space dimension d 1 generalizing upon earlier results for d 1 the mechanism for the closure of the lie algebra is explained the lie algebra generators contain higher order differential operators or the riesz fractional derivative co variant two time response functions are derived some simple applications to the exactly solvable models of phase separation or interface growth with conserved dynamics are discussed

convergence of the trinomial tree method for pricing european american options in this paper we present a trinomial tree method for pricing options and show its equivalence to certain explicit difference scheme using the numerical analysis and the notion of viscosity solution we prove the uniform convergence of the trinomial tree method for european american options the accuracy and efficiency are shown through numerical simulations

response of non equilibrium systems with long range initial correlations the long time dynamics of the d dimensional spherical model with a non conserved order parameter and quenched from an initial state with long range correlations is studied through the exact calculation of the two time autocorrelation and autoresponse functions in the ageing regime these are given in terms of non trivial universal scaling functions of both time variables at criticality five distinct types of ageing are found depending on the form of the initial correlations while at low temperatures only a single type of ageing exists the autocorrelation and autoresponse exponents are shown to be generically different and to depend on the initial conditions the scaling form of the two time response functions agrees with a recent prediction coming from conformal invariance

conformal carroll groups conformal extensions of levy leblond s carroll group based on geometric properties analogous to those of newton cartan space time are proposed the extensions are labeled by an integer k this framework includes and extends our recent study of the bondi metzner sachs bms and newman unti nu groups the relation to conformal galilei groups is clarified conformal carroll symmetry is illustrated by carrollian photons motion both in the newton cartan and carroll spaces may be related to that of strings in the bargmann space

affine extension of galilean conformal algebra in 2 1 dimensions we show that a class of nonrelativistic algebras including noncentrally extended schrodinger algebra and galilean conformal algebra gca has an affine extension in 2 1 hitherto unknown this extension arises out of the conformal symmetries of the two dimensional complex plane we suggest that this affine form may be the symmetry that explains the relaxation of some classical phenomena toward their critical point this affine algebra admits a central extension and maybe realized in the bulk the bulk realization suggests that this algebra may be derived by looking at the asymptotic symmetry of an anti de sitter ads theory this suggests that ads cft conformal field theory duality may take on a special form in four dimensions

super schrodinger algebra in ads cft we discuss extended super schrodinger algebras obtained as subalgebras of the superconformal algebra psu 2 2 4 the schrodinger algebra with two spatial dimensions can be embedded into so 4 2 in the superconformal case the embedded algebra may be enhanced to the so called super schrodinger algebra in fact we find an extended super schrodinger subalgebra of psu 2 2 4 it contains 24 supercharges i e 3 4 of the original supersymmetries and the generators of so 6 as well as the generators of the original schrodinger algebra in particular the 24 supercharges come from 16 rigid supersymmetries and half of 16 superconformal ones moreover this superalgebra contains a smaller super schrodinger subalgebra which is a supersymmetric extension of the original schrodinger algebra and so 6 by eight supercharges half of 16 rigid supersymmetries it is still a subalgebra even if there are no so 6 generators we also discuss super schrodinger subalgebras of the superconformal algebras osp 8 4 and

a fast and stable solver for acoustic scattering problems based on the nonuniform grid approach a fast and stable boundary element method bem algorithm for solving external problems of acoustic scattering by impenetrable bodies is developed the method employs the burton miller integral equation which provides stable convergence of iterative solvers and a generalized multilevel nonuniform grid mlng algorithm for fast evaluation of field integrals the mlng approach is used here for the removal of computational bottlenecks involved with repeated matrix vector multiplications as well as for the low order basis function regularization of the hyper singular integral kernel the method is used for calculating the fields scattered by large acoustic scatterers including nonconvex bodies with piece wise smooth surfaces as a result the algorithm is capable of accurately incorporating high frequency effects such as creeping waves and multiple edges diffractions in all cases stable convergence of the method is observed high accuracy of the method is demonstrated by comparison with the traditional b

schrodinger invariance and strongly anisotropic critical systems the extension of strongly anisotropic or dynamical scaling to local scale invariance is investigated for the special case of an anisotropy or dynamical exponent z 2 the group of local scale transformation considered is the schrodinger group which can be obtained as the nonrelativistic limit of the conformal group the requirement of schrodinger invariance determines the two point function in the bulk and reduces the three point function to a scaling form of a single variable scaling forms are also derived for the two point function close to a free surface which can be either spacelike or timelike these results are reproduced in several exactly solvable statistical systems namely the kinetic ising model with glauber dynamics lattice diffusion lifshitz points in the spherical model and critical dynamics of the spherical model with a nonconserved order parameter for generic values of evidence from higher order lifshitz points in the spherical model and from directed percolation suggests a simple scaling form of the two point function

nonuniform grid time domain ngtd algorithm for fast evaluation of transient wave fields a novel algorithm to efficiently compute transient wave fields produced by known three dimensional source constellations is proposed the algorithm uses domain decomposition concepts and comprises two steps to be repeated for each subdomain considered in the first step delay and amplitude compensated fields produced by sources residing inside each subdomain are computed at a sparse set of points surrounding the observation domain in the second step total fields in the observer domain are evaluated by interpolation delay and amplitude restoration and aggregation of subdomain fields the proposed scheme is well suited to accelerate the solution of time domain integral equations by marching on in time to carry out time domain physical optics calculations and to realize near to far field transformations of transients moreover the scheme automatically adapts to and takes advantage of special geometrical features of the source observer constellation studied a key benefit when analyzing quasi planar configurations in addition it realizes a seamless transition from the dynamic to the quasi static regime thus facilitating a unified treatment of electrically large and small problems last but not least the scheme is remarkably simple to implement

pricing the american options a closed form simple formula abstract we overcome a major obstacle in the literature in doing we introduce a simple closed form formula for pricing the american options in particular we significantly simplify alghalith s closed form formula for pricing american options in doing so we introduce a formula that does not require the unknown expected consumption this is a vast simplification since the estimation of is challenging that is similar to a european option we only need to know the interest rate and volatility furthermore we derive an exact upper bound for the price

lowest weight representations of super schrodinger algebras in one dimensional space lowest weight modules in particular verma modules over the n 1 2 super schrodinger algebras in 1 1 dimensional spacetime are investigated the reducibility of the verma modules is analyzed via explicitly constructed singular vectors the classification of the irreducible lowest weight modules is given for both massive and massless representations a vector field realization of the n 1 2 super schrodinger algebras is also presented

simple harish chandra modules over super schrodinger algebra in 1 1 dimensional spacetime the n 1 super schrodinger algebra s 1 1 in 1 1 dimensional spacetime contains a subalgebra isomorphic to osp 1 2 module let v be a simple weight module for the n 1 super schrodinger algebra s 1 1 but not a simple osp 1 2 module let supp v if v is neither a highest weight module nor a lowest weight module for s 1 1 we prove that supp v z and all non trivial weight spaces of v have the same dimension we prove that if v is a harish chandra module then it is a highest weight module or lowest weight module or a twisted localization of a highest weight module

automorphisms and verma modules for generalized schrodinger virasoro algebras abstract let f be a field of characteristic 0 g an additive subgroup of f f satisfying g and 2 g we define a class of infinite dimensional lie algebras which are called generalized schrodinger virasoro algebras and denoted by gsv g in this paper the automorphism group and irreducibility of verma modules for gsv g are completely determined

nonrelativistic conformal groups in this work a systematic study of finite dimensional nonrelativistic conformal groups is carried out under two complementary points of view first the conformal killing equation is solved to obtain a whole family of finite dimensional conformal algebras corresponding to each of the galilei and newton hooke kinematical groups some of their algebraic and geometrical properties are studied in a second step among the groups included in these families one can identify for example the contraction of the minkowski conformal group the analog for a nonrelativistic de sitter space or the nonextended schrodinger group

theory of phase ordering kinetics the theory of phase ordering dynamics that is the growth of order through domain coarsening when a system is quenched from the homogeneous phase into a broken symmetry phase is reviewed with the emphasis on recent developments interest will focus on the scaling regime that develops at long times after the quench how can one determine the growth laws that describe the time dependence of characteristic length scales and what can be said about the form of the associated scaling functions particular attention will be paid to systems described by more complicated order parameters than the simple scalars usually considered for example vector and tensor elds the latter are needed for example to describe phase ordering in nematic liquid crystals on which there have been a number of recent experiments the study of topological defects domain walls vortices strings and monopoles provides a unifying framework for discussing coarsening in these dia erent systems

nonlinear galilei invariant pdes with infinite dimensional lie symmetry a class of nonlinear galilei invariant generalizations of the heat equation admitting infinite dimensional lie symmetry is presented lie symmetries and examples of exact solutions are found for some nonlinear equations generalizing the heat equation full sets of the absolute differential invariants for two non equivalent representations of the galilei algebra are constructed

aspects of infinite dimensional l super galilean conformal algebra in this work we construct an infinite dimensional l super galilean conformal algebra which is a generalization of the l 1 algebra found in the literature we give a classification of central extensions the vector field representation the coadjoint representation and the operator product expansion of the infinite dimensional l super galilean conformal algebra keeping possible applications in physics and mathematics in mind

on two dimensional directed percolation extended series expansions for the mean size and the first and second moments of the pair connectedness for both bond and site percolation on the directed square and triangular lattices have been obtained analysis based on differential approximants allows the critical percolation probabilities and exponents to be estimated and as a result the critical exponents are conjectured to be gamma 41 18 nu perpendicular to 79 72 and nu sub 26 15 scaling then gives beta 199 720 alpha 299 360 and delta 1839 199

classification of finite irreducible conformal modules over some lie conformal algebras related to the virasoro conformal algebra in this paper we classify all finite irreducible conformal modules over a class of lie conformal algebras w b with b related to the virasoro conformal algebra explicitly any finite irreducible conformal module over w b is proved to be isomorphic to m with 0 or 0 if b 0 or m with 0 if b 0 as a byproduct all finite irreducible conformal modules over the heisenberg virasoro conformal algebra and the w 2 2 lie conformal algebra are classified finally the same thing is done for the schrodinger virasoro conformal algebra

a six vertex model as a diffusion problem derivation of correlation functions a cellular automaton which describes diffusion of particles with exclusion in one dimension is shown to be equivalent to a six vertex model on a critical line the arrow arrow correlation function of the six vertex model is calculated exactly on this line using a transfer matrix method

classification of harish chandra modules over some lie algebras related to the virasoro algebra abstract in this paper we provide a uniform method to thoroughly classify all harish chandra modules over some lie algebras related to the virasoro algebras we first classify such modules over the lie algebra w s for s 0 1 2 with this result and method we can also do such works for some lie algebras related to the virasoro algebra including the several kinds of schrodinger virasoro lie algebras which are open up to now

classification of finite irreducible conformal modules over a class of lie conformal algebras of block type abstract we classify finite irreducible conformal modules over a class of infinite lie conformal algebras b p of block type where p is a nonzero complex number in particular we obtain that a finite irreducible conformal module over b p may be a nontrivial extension of a finite conformal module over vir if p 1 where vir is a virasoro conformal subalgebra of b p as a byproduct we also obtain the classification of finite irreducible conformal modules over a series of finite lie conformal algebras b n for n 1

on the non relativistic structure of the ads cft superalgebras the property of the conformal algebra to contain the schrodinger algebra in one less space dimension is extended to the supersymmetric case more precisely we determine the non relativistic counterpart of any field theory admissible superconformal algebra even if each type of superalgebra provides a different solution its basis decomposition into two copies of the super schrodinger algebra differing only by their super heisenberg part remains valid in all cases so generalizing a feature already observed in the non supersymmetric conformal case

whittaker modules for the schrodinger witt algebra in this paper whittaker modules for the schrodinger witt algebra sv are defined the whittaker vectors and the irreducibility of the whittaker modules are studied sv has a triangular decomposition according to its cartan subalgebra h sv sv h sv for any lie algebra homomorphism sv c we can define whittaker modules of type when is nonsingular the whittaker vectors the irreducibility and the classification of whittaker modules are completely determined when is singular by constructing some special whittaker vectors we find that the whittaker modules are all reducible moreover we get some more precise results for special

radial fredholm perturbation in the two dimensional ising model and gap exponent relation we consider concentric circular defects in the two dimensional ising model which are distributed according to a generalized fredholm sequence i e at exponentially increasing radii this type of aperiodicity does not change the bulk critical behaviour but introduces a marginal extended perturbation the critical exponent of the local magnetization is obtained through finite size scaling using a corner transfer matrix approach in the extreme anisotropic limit it varies continuously with the amplitude of the modulation and is closely related to the magnetic exponent of the radial hilhorst van leeuwen model through a conformal mapping of the system onto a strip the gap exponent relation is shown to remain valid for such an aperiodic defect

a crossing probability for critical percolation in two dimensions langlands et al considered two crossing probabilities and in their extensive numerical investigations of critical percolation in two dimensions cardy was able to find the exact form of by treating it as a correlation function of boundary operators in the limit of the q state potts model we extend his results to find an analogous formula for which compares very well with the numerical results

lie bialgebra structures on the schrodinger virasoro lie algebra in this paper we shall investigate lie bialgebra structures on the schrodinger virasoro algebra l we found out that not all lie bialgebra structures on the schrodinger virasoro algebra are triangular coboundary which is different from the related known results of some other lie algebras related to the virasoro algebra

on a way to save memory when solving time domain boundary integral equations for acoustic and vibroacoustic applications abstract solving acoustic equations in the time domain possibly coupled with the description of flexible structure dynamics remains attractive as compared to solving the same in the frequency domain this allows for better consideration of local non linearities acoustics structure and the boundary integral formulation also known as bem offers an exact description of the infinite acoustic field based on a simple surface mesh no need for 3d volume discretization some issues remain however the required memory space and computation time continue to grow rapidly when the number of elements of the surface mesh increases in the case of a structure with a regular non slender shape the computational cost measured in terms of required memory space varies by helmholtz number to the power of 4 this paper illustrates how the accelerating method called ngtd helps overcome this difficulty this paper shows the applicability of 2 level ngtd to acoustic and vibroacoustic problems described solely by the hypersingular formulation for surfaces it goes into more detail on some important aspects of the interpolation process and on the memory saving obtained implementation within the mot march on time astryd code shows the benefits of this method the memory requirement shows an estimated trend lower than power 1 35 of the number of surface elements

super galilean conformal algebra in ads cft galilean conformal algebra gca is an inonu wigner iw contraction of a conformal algebra while newton hooke string algebra is an iw contraction of an anti de sitter ads algebra which is the isometry of an ads space it is shown that the gca is a boundary realization of the newton hooke string algebra in the bulk ads the string lies along the direction transverse to the boundary and the worldsheet is ads2 the one dimensional conformal symmetry so 2 1 and rotational symmetry so d contained in the gca are realized as the symmetry on the ads2 string worldsheet and rotational symmetry in the space transverse to the ads2 in adsd 2 respectively it follows from this correspondence that 32 supersymmetric gcas can be derived as iw contractions of superconformal algebras psu 2 2 4 osp 8 4 and osp 8 4 we also derive less supersymmetric gcas from su 2 2 2 osp 4 4 osp 2 4 and osp 8 2

on non linear partial differential equations with an infinite dimensional conditional symmetry abstract the invariance of non linear partial differential equations under a certain infinite dimensional lie algebra a n z in n spatial dimensions is studied the special case a 1 2 was introduced in j stat phys 75 1994 1023 and contains the schrodinger lie algebra sch 1 as a lie subalgebra it is shown that there is no second order equation which is invariant under the massless realizations of a n z however a large class of strongly non linear partial differential equations is found which are conditionally invariant with respect to the massless realization of a n z such that the well known monge ampere equation is the required additional condition new exact solutions are found for some representatives of this class

galilei invariant nonlinear systems of evolution equations all systems of n 1 dimensional quasilinear second order evolution equations invariant under chain of algebras ag 1 n contained in implied by ag1 1 n contained in implied by ag2 1 n are described the results obtained are illustrated by the examples of the nonlinear schrodinger equations hamilton jacobi type systems and reaction diffusion equations

local scale invariances in the bosonic contact and pair contact processes local scale invariance for ageing systems without detailed balance is tested through studying the dynamical symmetries of the critical bosonic contact process and the critical bosonic pair contact process their field theoretical actions can be split into a schrodinger invariant term and a pure noise term it is shown that the two time response and correlation functions are reducible to certain multipoint response functions which depend only on the schrodinger invariant part of the action for the bosonic contact process the representation of the schrodinger group can be derived from the free diffusion equation whereas for the bosonic pair contact process a new representation of the schrodinger group related to a nonlinear schrodinger equation with dimensionful couplings is constructed the resulting predictions of local scale invariance for the two time responses and correlators are completely consistent with the exactly known results in both models

out of equilibrium properties of the semi infinite kinetic spherical model we study the ageing properties of the semi infinite kinetic spherical model at the critical point and in the ordered low temperature phase for both dirichlet and neumann boundary conditions the surface fluctuation dissipation ratio and the scaling functions of two time surface correlation and response functions are determined explicitly in the dynamical scaling regime in the low temperature phase our results show that for the case of dirichlet boundary conditions the value of the non equilibrium surface exponent b1 differs from the usual bulk value of systems undergoing phase ordering

intertwining operators for l conformal galilei algebras and hierarchy of invariant equations the l conformal galilei algebra denoted by is a non semisimple lie algebra specified by a pair of parameters d l the algebra is regarded as a nonrelativistic analogue of the conformal algebra we derive hierarchies of partial differential equations which have invariance of the group generated by with a central extension as kinematical symmetry this is done by developing a representation theory such as verma modules singular vectors of and vector field representations for d 1 2

possible central extensions of non relativistic conformal algebras in 1 1 we investigate possibility of central extension for non relativistic conformal algebras in 1 1 dimension three different forms of charges can be suggested a trivial charge for temporal part of the algebra exists for all elements of l galilei algebra class in attempt to find a central extension as of conformal galilean algebra for other elements of the l galilei class possibility for such extension was excluded for integer and half integer elements of the class we can have an infinite extension of the generalized mass charge for the virasoro like extended algebra for finite algebras a regular charge inspired by schrodinger central extension is possible

evolution equations invariant under two dimensional space time schrodinger group the most general second order evolution equation t f x t x x xx xx 0 invariant under the galilei galilei similitude and schrodinger groups in two dimensions is constructed a preliminary step is a classification of all possible realizations of the corresponding lie algebras of vector fields in r2 c parametrized by x t and applications of this study include the investigation of nonlinear alternatives to quantum mechanics and nonrelativistic classical field theories among the schrodinger invariant equations in particular are found integrable equations linearizable by contact transformations

kinetics of the long range spherical model the kinetic spherical model with long range interactions is studied after a quench to t tc or to t tc for the two time response and correlation functions of the order parameter as well as for composite fields such as the energy density the ageing exponents and the corresponding scaling functions are derived the results are compared to the predictions which follow from local scale invariance

invariant partial differential equations with two dimensional exotic centrally extended conformal galilei symmetry conformal galilei algebras cgas labeled by d l where d is the number of space dimensions and l denotes a spin l representation w r t the x1d530 x1d529 2 subalgebra admit two types of central extensions the ordinary one for any d and half integer l and the exotic central extension which only exists for d 2 and l for both types of central extensions invariant second order partial differential equations pdes with continuous spectrum were constructed by aizawa et al j phys a 46 405204 2013 it was later proved by aizawa et al j math phys 3 031701 2015 that the ordinary central extensions also lead to oscillator like pdes with discrete spectrum we close in this paper the existing gap constructing a new class of second order invariant pdes for the exotic centrally extended cgas they admit a discrete and bounded spectrum when applied to a lowest weight representation these pdes are markedly different with respect to their ordinary counterparts the l 1 case which is the prototype

scaling fields in the two dimensional abelian sandpile model we consider the unoriented two dimensional abelian sandpile model from a perspective based on two dimensional conformal field theory we compute lattice correlation functions for various cluster variables at and off criticality from which we infer the field theoretic description in the scaling limit we find perfect agreement with the predictions of a c 2 conformal field theory and its massive perturbation thereby providing direct evidence for conformal invariance and more generally for a description in terms of a local field theory the question of the height 2 variable is also addressed with however no definite conclusion yet

more on symmetries of the schrodinger equation besides the usual and well known lie symmetries of the schrodinger equation the authors study higher order symmetry operators and point out their impact on classes of solvable potentials when time independent interactions take place for one dimensional space systems they consider the case of nth order symmetry operators and more particularly the third order context

pricing the american options using the black scholes pricing formula abstract we develop a simple exact explicit and analytical solution to the american option partial differential equation pde using the black scholes pricing formula

cellular automata and statistical mechanical models we elaborate on the analogy between the transfer matrix of usual lattice models and the master equation describing the time development of cellular automata transient and stationary properties of probabilistic automata are linked to surface and bulk properties respectively of restricted statistical mechanical systems it is demonstrated that methods of statistical physics can be successfully used to describe the dynamic and the stationary behavior of such automata some exact results are derived including duality transformations exact mappings disorder and linear solutions many examples are worked out in detail to demonstrate how to use statistical physics in order to construct cellular automata with desired properties this approach is considered to be a first step toward the design of fully parallel probabilistic systems whose computational abilities rely on the cooperative behavior of their components

representations of the schrodinger virasoro algebras in this paper it is proved that an irreducible weight module with finite dimensional weight spaces over the schrodinger virasoro algebras is a highest lowest weight module or a uniformly bounded module furthermore indecomposable modules of the intermediate series over these algebras are completely determined