transformation groups for soliton equations iv a new hierarchy of soliton equations of kp type abstract a new approach to soliton equations based on functions or hirota s dependent variables vertex operators and the clifford algebra of free fermions is applied to study a new hierarchy of kadomtsev petviashvili type equations the bkp hierarchy the infinite dimensional orthogonal group acts on the space of bkp functions the sawada kotera equation is obtained as a reduction of bkp its infinitesimal transformations constitute the euclidean lie algebra a2 2

strong solutions to the three dimensional compressible viscoelastic fluids abstract the existence and uniqueness of the local strong solution to the three dimensional compressible viscoelastic fluids near the equilibrium are established in addition to the uniform estimates on the velocity some essential uniform estimates on the density and the deformation gradient are also obtained

travelling fronts in nonlocal models for phase separation in an external field we consider the one dimensional nonlocal evolution equation derived by de masi et al 1995 for ising systems with glauber dynamics kac potentials and magnetic field we prove the existence of travelling fronts their uniqueness modulo translations among the monotone profiles and their linear stability for all the admissible values of the magnetic field for which the underlying spin system exhibits a stable and metastable phase

framed torsion free sheaves on cp2 hilbert schemes and representations of infinite dimensional lie algebras abstract we construct representations of both heiseberg and clifford algebras on the equivariant cohomology of the moduli space of framed torsion free sheaves on p 2 as well as on the equivariant cohomology of the hilbert scheme of points on the resolution of a type a simple singularity this generalizes the earlier work of nakajima grojnowski and others

almost global existence for 2 d incompressible isotropic elastodynamics author s lei z sideris tc zhou y abstract 2015 american mathematical society we consider the cauchy problem for 2 d incompressible isotropic elastodynamics standard energy methods yield local solutions on a time interval 0 t for initial data of the form u0 where t depends only on some sobolev norm of u0 we show that for such data there exists a unique solution on a time interval 0 exp t provided that is sufficiently small this is achieved by careful consideration of the structure of the nonlinearity the incompressible elasticity equation is inherently linearly degenerate in the isotropic case in other words the equation satisfies a null condition this is essential for time decay estimates the pressure which arises as a lagrange multiplier to enforce the incompressibility constraint is estimated in a novel way as a nonlocal nonlinear term with null structure the proof employs the generalized energy method of klainerman enhanced by weighted l2 estimates and the ghost weight introduced by alinhac

bosonic and fermionic realizations of the affine algebra g hat l_n we give an explicit description of all inequivalent heisenberg subalgebras of the affine lie algebra g hat l n mathbb c and the associated vertex operator constructions of the level one integrable highest weight representations of this algebra the construction uses multicomponent fermionic fields and yields a correspondence between bosons elements of the heisenberg subalgebra and fermions

vertex operator representation of some quantum tori lie algebras we are defining the trigonometric lie subalgebras in bar x infty bar a infty bar b infty bar c infty bar d infty which are the natural generalization of the well known sin lie algebra the embedding formulas into bar x infty are introduced these algebras can be considered as some lie algebras of quantum tori an irreducible representation ofa b series of trigonometric lie algebras is constructed special cases of the trigonometric lie factor algebras which can be considered as a quantum preserving lie algebra structure deformation of the kac moody algebras are considered

pseudoholomorphic strips in symplectizations ii fredholm theory and transversality this paper is part of a larger program the investigation of the chord problem in three dimensional contact geometry the main tool will be pseudoholomorphic strips in the symplectisation of a three dimensional contact manifold with two totally real submanifolds l0 l1 as boundary conditions the submanifolds l0 and l1 do not intersect transversally in this paper we will develop a nonlinear fredholm theory that guarantees the existence of a family of embedded pseudoholomorphic strips near a given one with suitable properties c 2003 wiley periodicals inc

principal vertex operator representations for toroidal lie algebras we introduce the principal vertex operator representations for the toroidal lie algebras generalizing the construction for the affine kac moody algebras we also represent the derivations of the toroidal algebras and introduce analogs of the sugawara operators

variational description of gibbs non gibbs dynamical transitions for spin flip systems with a kac type interaction we continue our study of gibbs non gibbs dynamical transitions in the present paper we consider a system of ising spins on a large discrete torus with a kac type interaction subject to an independent spin flip dynamics infinite temperature glauber dynamics we show that in accordance with the program outlined in van enter et al moscow math j 10 687 711 2010 in the thermodynamic limit gibbs non gibbs dynamical transitions are equivalent to bifurcations in the set of global minima of the large deviation rate function for the trajectories of the empirical density conditional on their endpoint more precisely the time evolved measure is non gibbs if and only if this set is not a singleton for some value of the endpoint a partial description of the possible scenarios of bifurcation is given leading to a characterization of passages from gibbs to non gibbs and vice versa with sharp transition times our analysis provides a conceptual step up from our earlier work on gibbs non gibbs dynamical transitions for the curie weiss model where the mean field interaction allowed us to focus on trajectories of the empirical magnetization rather than the empirical density

well posedness in critical spaces for incompressible viscoelastic fluid system abstract in this paper we study the well posedness in critical spaces of incompressible viscoelastic fluid system of oldroyd model precisely for this system with initial data in besov space of critical regularity we prove the existence and uniqueness of the local solution which is also shown to exist globally in time provided the initial data is small under certain norms

on secondary bifurcations for some nonlinear convolution equations etude de quelques equations de convolution non lineaires sur le tore a d dimensions conditions d existence de bifurcations secondaires

symmetric function products and plethysms and the boson fermion correspondence we use the boson fermion correspondence for s and q functions to establish some interesting properties concerning outer products and plethysms of s functions or q functions by power sum symmetric functions the techniques which are developed are also applied to computing the inverse kostka foulkes matrix which is the transition matrix between hall littlewood symmetric functions and s functions in some simple cases

weight vectors of the basic a1 1 module and the littlewood richardson rule the basic representation of a1 1 is studied the weight vectors are represented in terms of schur functions a suitable base of any weight space is given the littlewood richardson rule appears in the linear relations among weight vectors

on the stationary solutions of the full compressible navier stokes equations and its stability with respect to initial disturbance this paper is concerned with the existence uniqueness and nonlinear stability of stationary solutions to the cauchy problem of the full compressible navier stokes equations effected by external force of general form in r 3 c 2007 elsevier inc all rights reserved

the strong macdonald conjecture and hodge theory on the loop grassmannian we prove the strong macdonald conjecture of hanlon and feigin for reductive groups g in a geometric reformulation we show that the dolbeault cohomology hq x op of the loop grassmannian x is freely generated by de rham s forms on the disk coupled to the indecomposables of h bg equating the two euler characteristics gives an identity independently known to macdonald m which generalises ramanujan s 1 1 sum for simply laced root systems at level 1 we also find a strong form of bailey s 4 4 sum failure of hodge decomposition implies the singularity of x and of the algebraic loop groups some of our results were announced in t2

vector generation functions q spectral functions of hyperbolic geometry and vertex operators for quantum affine algebras we investigate the concept of q replicated argument in symmetric functions with its connection to spectral functions of hyperbolic geometry this construction suffices for vector generation functions in the form of q series and string theory we hope that the mathematical side of the construction can be enriched by ideas coming from physics

global existence for the compressible viscoelastic system with zero shear viscosity in three dimensions abstract we study the three dimensional compressible viscoelastic system with zero shear viscosity the weighted l 2 energy derived from the local energy decay of the linearized system and sobolev type inequalities are applied to control the nonlinear terms and hence the global existence with small perturbations is established

γ conformal algebras conformal algebra is an axiomatic description of the operator product expansion of chiral fields with simple poles at finitely many points we classify these algebras and their representations in terms of lie algebras and their representations with an action of the group to every conformal algebra and a character of we associate a lie algebra generated by fields with the ope with simple poles examples include twisted affine kac moody algebras the sin algebra which is a conformal analogue of the general linear algebra and its analogues the algebra of pseudodifferential operators on the circle etc

the n component kp hierarchy and representation theory it is the aim of the present article to give all formulations of the n component kp hierarchy and clarify connections between them the generalization to the n component kp hierarchy is important because it contains many of the most popular systems of soliton equations like the davey stewartson system for n 2 the two dimensional toda lattice for n 2 the n wave system for n 3 and the darboux egoroff system it also allows us to construct natural generalizations to the davey stewartson and toda lattice systems of course the inclusion of all these systems in the n component kp hierarchy allows us to construct their solutions by making use of vertex operators

bosonization in background of metric and gauge the paper extends the two dimensional bosonization formulas on a riemann surface to the case with abelian gauge field the result is used to bosonize the massless thirring model and as an alternative route to the chiral bosonization

a blow up criterion for 3d compressible viscoelastic flow with large initial data abstract in this paper we consider a cauchy problem for the three dimensional compressible viscoelastic flow with large initial data we establish a blow up criterion for the strong solutions in terms of the gradient of velocity only which is similar to the beale kato majda criterion for ideal incompressible flow cf beale et al 1984 20 and the blow up criterion for the compressible navier stokes equations cf huang et al 2011 21

vertex operator realization of symplectic and orthogonal s functions it is shown that symplectic and orthogonal schur functions can be realized by the action of the modes of certain vertex operators on the function the properties of these vertex operators allow one to reproduce the basic propreties of these types of symmetric functions the vertex operator modes are shown to obey free fermionic relations which provide applications such as calculating products and plethysms as well as generating hirota type partial differential equations pdes which have symplectic s functions as functions

diffusion wave phenomena and lp decay estimates of solutions of compressible viscoelastic system abstract we consider the system of equations describing motion of compressible viscoelastic fluids in a whole space we investigate the large time behavior of solutions around a motionless state and obtain the l p decay estimates of solutions for 1 p provided that the initial data is sufficiently close to the motionless state in addition we clarify the diffusion wave phenomena caused by the sound wave and the elastic shear wave

twisted γ lie algebras and their vertex operator representations let be a generic subgroup of the multiplicative group c of nonzero complex numbers we define a class of lie algebras associated to called twisted lie algebras which are natural generalizations of the twisted affine lie algebras starting from an arbitrary even sublattice q of zn and an arbitrary finite order isometry of zn preserving q we construct a family of twisted vertex operators acting on generalized fock spaces which afford irreducible representations for certain twisted lie algebras as an application this recovers a number of known vertex operator realizations for infinite dimensional lie algebras such as twisted affine lie algebras extended affine lie algebras of type a trigonometric lie algebras of series a and b unitary lie algebras and bc graded lie algebras

asymptotic dynamics non critical and critical fluctuations for a geometric long range interacting model we study the dynamics of geometric spin system on the torus with long range interaction as the number of particles goes to infinity the process converges to a deterministic dynamical magnetization field that satisfies an euler equation law of large numbers its stable steady states are related to the limits of the equilibrium measures gibbs states of the finite particle system a related equation holds for the magnetization densities for which the property of propagation of chaos also is established we prove a dynamical central limit theorem with an infinite dimensional ornstein uhlenbeck process as a limiting fluctuation process at the critical temperature of a ferromagnetic phase transition both a tighter quantity scaling and a time scaling is required to obtain convergence to a one dimensional critical fluctuation process with constant magnetization fields which has a non gaussian invariant distribution similarly at the phase transition to an antiferromagnetic state with frequencyp0 the fluctuation process with critical scaling converges to a two dimensional critical fluctuation process which consists of fields with frequencyp0 and has a non gaussian invariant distribution on these fields finally we compute the critical fluctuation process in the infinite particle limit at a triple point where a ferromagnetic and an antiferromagnetic phase transition coincide

efficient numerical path following beyond critical points the paper presents a numerical pathfollowing technique based on an implicit change of parametrization using qr decomposition an automatic steplength control for tangent continuation is derived on a theoretical basis in the suggested approach turning points do not play any exceptional role for the actual computation of simple bifurcation points a special algorithm based on the augmented system of moore is developed this treatment includes imperfect bifurcations and permits an easy way of branch switching an efficient linear equation solver is worked out in connection with the qr decomposition finally numerical comparisons are included that clearly document the efficiency of the newly developed algorithms

on the well posedness of the ideal incompressible viscoelastic flow in the critical besov spaces abstract in this paper we study the viscoelastic fluid system of the oldroyd model by means of the fourier frequency localization and bony paraproduct decomposition we establish the local well posedness for the incompressible oldroyd system in the critical besov spaces under the ideal case furthermore as a byproduct we obtain a beale kato majida type criterion beale et al 1984

a brylinski filtration for affine kac moody algebras abstract braverman and finkelberg have recently proposed a conjectural analogue of the geometric satake isomorphism for untwisted affine kac moody groups as part of their model they conjecture that at dominant weights lusztig s q analog of weight multiplicity is equal to the poincare series of the principal nilpotent filtration of the weight space as occurs in the finite dimensional case we show that the conjectured equality holds for all affine kac moody algebras if the principal nilpotent filtration is replaced by the principal heisenberg filtration the main body of the proof is a lie algebra cohomology vanishing result we also give an example to show that the poincare series of the principal nilpotent filtration is not always equal to the q analog of weight multiplicity finally we give some partial results for indefinite kac moody algebras

time decay rates for the compressible viscoelastic flows abstract in this paper we investigate the time decay rates of the global smooth solutions to the compressible viscoelastic flows in r 3 the global existence and uniqueness of smooth solutions are obtained under the assumption that the h 3 norm of the initial data is small but the higher order derivatives can be arbitrarily large moreover if the initial data is bounded in h s 0 s 3 2 or b 2 s 0 s 3 2 we establish the optimal decay rates of the smooth solutions by a regularity interpolation trick and delicate energy methods

realization of the level one standard _ 2 1 modules in this paper we study the level one standard or irreducible integrable highest weight modules for the affine symplectic lie algebras in particular we give concrete realizations of all level one standard modules for the affine symplectic lie algebras of even rank introduction in recent years explicit realizations of nontrivial representations of affine lie algebras have attracted the attention of many researchers because of their surprising connections with different areas of mathematics and physics among these realizations the principal realizations and the untwisted homogeneous realizations have proved to be especially useful in 1978 lepowsky and wilson 15 constructed the basic modules level one standard modules i e irreducible integrable highest weight modules for a in the principal realization in 1 o this construction was generalized to the level one standard modules for all simply laced affine lie algebras frenkel and kac 5 and segal 26 have given explicit construction of these modules in the untwisted realization the constructions of these modules in the general realization are given in 9 12 since then some higher level standard modules for certain simply laced affine lie algebras have also been constructed e g 14 17 18 19 21 22 25 however less is known regarding explicit constructions of standard modules for nonsimply laced affine lie algebras that is affine lie algebras of type b c f and g in the last section of 10 kac kazhdan lepowsky and wilson constructed reducible level one modules for the non simply laced affine lie algebras in the principal realization by taking fixed points of dynkin diagraminduced automorphisms recently in the untwisted realization the analogous representations again reducible have been constructed in 3 and 6 the level one standard modules for the affine orthogonal lie algebras b 1 have been n explicitly realized in 4 8 13 and 20 from different view points in 20 mandia has constructed the level one standard modules for the affine lie algebras f 1 and g 1 in 23 the author gave explicit constructions of the level one standard modules for the affine symplectic lie algebras of rank 3 and 4 received by the editors july 13 1989 presented to the 847th meeting of the ams pheonix arizona january 14 1989 1980 mathematics subject classification 1985 revision primary 17b65 1990 american mathematical society 0002 9947 90 1 00 25 per page

agt correspondence ding iohara algebra at roots of unity and lepowsky wilson construction it was recently conjectured that the agt correspondence between the instanton counting on and the two dimensional field theories with the conformal symmetry algebra can be considered as a root of unity limit of its k theoretic analogue from this point of view the algebra and a special basis in its representation are limits of the ding iohara algebra and the macdonald polynomials respectively in this paper we confirm this conjecture for the special case r 1 we uncover the implicit symmetry in this limit we also found that the vertex operators in the special basis have factorized aflt form

global solutions for incompressible viscoelastic fluids we prove the existence of both local and global smooth solutions to the cauchy problem in the whole space and the periodic problem in the n dimensional torus for the incompressible viscoelastic system of oldroyd b type in the case of near equilibrium initial data the results hold in both two and three dimensional spaces the results and methods presented in this paper are also valid for a wide range of elastic complex fluids such as magnetohydrodynamics liquid crystals and mixture problems

the wronskian formalism for linear differential equations and pade approximations on demontre des theoremes generaux sur la determination des meilleures approximations rationnelles des solutions d equations differentielles lineaires a coefficients fonction rationnelle

vertex operators arising from the homogeneous realization for we use the underlying fock space for the homogeneous vertex operator representation of the affine lie algebra to construct a family of vertex operators as an application an irreducible module for an extended affine lie algebra of type a n 1 coordinatized by a quantum torus q of 2 variables or 3 variables is obtained moreover this module turns out to be a highest weight module which is an analog of the basic module for affine lie algebras

affine demazure modules and t fixed point subschemes in the affine grassmannian let g be a simple algebraic group defined over c and t be a maximal torus of g for a dominant coweight of g the t fixed point subscheme formula of the schubert variety formula in the affine grassmannian grg is a finite scheme we prove that for all such if g is of type a or d and for many of them if g is of type e there is a natural isomorphism between the dual of the level one affine demazure module corresponding to and the ring of functions twisted by certain line bundle on grg of formula we use this fact to give a geometrical proof of the frenkel kac segal isomorphism between basic representations of affine algebras of a d e type and lattice vertex algebras

vertex operator representation of the soliton tau functions in the an 1 toda models by dressing transformations we study the relation between the group algebraic approach and the dressing symmetry one to the soliton solutions of the an 1 toda field theory in 1 1 dimensions originally solitons in the affine toda models were found by olive turok and underwood single solitons are created by exponentials of elements which ad diagonalize the principal heisenberg subalgebra alternatively babelon and bernard exploited the dressing symmetry to reproduce the known expressions for the fundamental tau functions in the sine gordon model in this paper we show the equivalence between these two methods to construct solitons in the an 1 toda models

interfaces and typical gibbs configurations for one dimensional kac potentials we consider a one dimensional ising spin system with a ferromagnetic kac potential j r j having compact support we study the system in the limit 0 below the lebowitz penrose critical temperature where there are two distinct thermodynamic phases with different magnetizations we prove that the empirical spin average in blocks of size 1 for any positive converges as 0 to one of the two thermodynamic magnetizations uniformly in the intervals of size p for any given positivep 1 we then show that the intervals where the magnetization is approximately constant have lengths of the order of exp c 1 c 0 and that when normalized they converge to independent variables with exponential distribution we show this by proving large deviation estimates and applying the ventsel and friedlin methods to gibbs random fields finally if the temperature is low enough we characterize the interface namely the typical magnetization pattern in the region connecting the two phases

initial boundary value problems for the compressible viscoelastic fluid abstract this paper is concerned with the initial boundary value problems of the system modeling the compressible viscoelastic fluid of oldroyd type the global in time solution is proved to exist uniquely near the equilibrium state in h 2

initial boundary value problems for the equations of motion of compressible viscous and heat conductive fluids the equations of motion of compressible viscous and heat conductive fluids are investigated for initial boundary value problems on the half space and on the exterior domain of any bounded region the global solution in time is proved to exist uniquely and approach the stationary state ast provided the prescribed initial data and the external force are sufficiently small

on hydrodynamics of viscoelastic fluids in this paper we study a hydrodynamic system describing fluids with viscoelastic properties after a brief examination of the relations between several models we shall concentrate on a few analytical issues concerning them in particular we establish local existence and global existence with small initial data of classical solutions for an oldroyd system without an artificially postulated damping mechanism

on the initial boundary value problem of the incompressible viscoelastic fluid system in this paper we shall establish the local well posedness of the initial boundary value problem of the viscoelastic fluid system of the oldroyd model we shall also prove that the local solutions can be extended globally and that the global solutions decay exponentially fast as time goes to infinity provided that the initial data are sufficiently close to the equilibrium state c 2007 wiley periodicals inc

correction factors for kac moody groups and t deformed root multiplicities we study a correction factor for kac moody root systems which arises in the theory of p adic kac moody groups in affine type this factor is known and its explicit computation is the content of the macdonald constant term conjecture the data of the correction factor can be encoded as a collection of polynomials m lambda in mathbb z t indexed by positive imaginary roots lambda at t 0 these polynomials evaluate to the root multiplicities so we consider m lambda to be a t deformation of mathrm mult lambda we generalize the peterson algorithm and the berman moody formula for root multiplicities to compute m lambda as a consequence we deduce fundamental properties of m lambda

well posedness for the dumbbell model of polymeric fluids the dumbbell model is a coupled hydrodynamic kinetic model for polymeric fluids in which the configurations of the dumbbells are described by stochastic differential equations we prove well posedness of this model by deriving directly a priori estimates on the stochastic model our results can be used to analyze stochastic simulation methods such as the ones that are based on brownian configuration fields

the t analog of the basic string function for twisted affine kac moody algebras abstract we study lusztig s t analog of weight multiplicities associated to level one representations of twisted affine kac moody algebras an explicit closed form expression is obtained for the corresponding t string function using constant term identities of macdonald and cherednik the closed form involves the generalized exponents of the graded pieces of the twisted affine algebra considered as modules for the underlying finite dimensional simple lie algebra this extends previous work on level 1 t string functions for the untwisted simply laced affine kac moody algebras

well posedness of the free boundary problem in compressible elastodynamics abstract we study the free boundary problem for the flow of a compressible isentropic inviscid elastic fluid at the free boundary moving with the velocity of the fluid particles the columns of the deformation gradient are tangent to the boundary and the pressure vanishes outside the flow domain we prove the local in time existence of a unique smooth solution of the free boundary problem provided that among three columns of the deformation gradient there are two which are non collinear vectors at each point of the initial free boundary if this non collinearity condition fails the local in time existence is proved under the classical rayleigh taylor sign condition satisfied at the first moment by constructing an hadamard type ill posedness example for the frozen coefficients linearized problem we show that the simultaneous failure of the non collinearity condition and the rayleigh taylor sign condition leads to rayleigh taylor instability

structural stability of a 1d compressible viscoelastic fluid model abstract this paper is concerned with a compressible viscoelastic fluid model proposed by ottinger although the model has a convex entropy the hessian matrix of the entropy does not symmetrize the system of first order partial differential equations due to the non conservative terms in the constitutive equation we show that the corresponding 1d model is symmetrizable hyperbolic and dissipative and satisfies the kawashima condition based on these we prove the global existence of smooth solutions near equilibrium and justify the compatibility of the model with the navier stokes equations

large deviation estimates for a conditional probability distribution applications to random interaction gibbs measures let x i y i d be independent identically distributed random variables with arbitrary distribution we show that for almost every y i i the conditional law of the empirical field given y i i satisfies to large deviation inequalities this applies to the study of gibbs measures with random interaction in the case of some mean field models as well as of short range summable interaction we show that the pressure is nonrandom and is given by a variational formula these random gibbs measures have the same large deviation rate which does not depend on the particular realization of the interaction their local behaviour is described in terms of conditional probabilities given the interaction of solutions to the variational formula