radial symmetry of positive solutions in nonlinear polyharmonic dirichlet problems we extend the symmetry result of gidas ni nirenberg to semilinear polyharmonic dirichlet problems in the unit ball in the proof we develop a new variant of the method of moving planes relying on fine estimates for the green function of the polyharmonic operator we also consider minimizers for subcritical higher order sobolev embeddings for embeddings into weighted spaces with a radially symmetric weight function we show that the minimizers are at least axially symmetric this result is sharp since we exhibit examples of minimizers which do not have full radial symmetry

quantum symmetric kac moody pairs the present paper develops a general theory of quantum group analogs of symmetric pairs for involutive automorphism of the second kind of symmetrizable kac moody algebras the resulting quantum symmetric pairs are right coideal subalgebras of quantized enveloping algebras they give rise to triangular decompositions including a quantum analog of the iwasawa decomposition and they can be written explicitly in terms of generators and relations moreover their centers and their specializations are determined the constructions follow g letzter s theory of quantum symmetric pairs for semisimple lie algebras the main additional ingredient is the classification of involutive automorphisms of the second kind of symmetrizable kac moody algebras due to kac and wang the resulting theory comprises various classes of examples which have previously appeared in the literature such as q onsager algebras and the twisted q yangians introduced by molev ragoucy and sorba

generalized drinfel d sokolov hierarchies a general approach is adopted to the construction of integrable hierarchies of partial differential equations a series of hierarchies associated to untwisted kac moody algebras and conjugacy classes of the weyl group of the underlying finite lie algebra is obtained the generalized kdv hierarchies of v g drinfel d and v v sokolov are obtained as the special case for the coxeter element various examples of the general formalism are treated in some detail including the fractional kdv hierarchies

a rearrangement based proof for the existence of extremal functions for the sobolev poincare inequality on bn abstract existence of extremals for 1 p n 1 2 is shown for the sobolev poincare inequality on b n using a mean value zero preserving rearrangement

conjugacy classes in affine kac moody groups and principal g bundles over elliptic curves abstract for a simple complex lie group g the connected components of the moduli space of semistable g bundles over an elliptic curve are weighted projective spaces or quotients of weighted projective spaces by a finite group action in this note we will provide a new proof of this result using the invariant theory of affine kac moody groups in particular the action of the twisted coxeter element on the root system of g

a symmetry problem related to wirtinger s and poincare s inequality j u 1 1 u p u q among periodic functions in w 1 1 with mean value zero and for p q 1 the minimizers are scaled in such a way that their maximum is 1 and their minimum is m 0 1 they raise the question if m 1 since this implies symmetries for the minimizers their answer is positive for q 2p and negative for q 1 practically the same question is raised in 2 and 3 if m 1 minimizers must be odd or symmetric in the sense that u x u x it was shown in 3 that for p q 2 the minimizer is symmetric and that for p 2 q it is not symmetric moreover in 2 one can find that the minimizer is symmetric for q 2p and nonsymmetric for for q 4p 1 the nonsymmetry result was derived by egorov in the following way restricted to the following set a

regular conjugacy classes in the weyl group and integrable hierarchies generalized kdv hierarchies associated by drinfeld sokolov reduction with grade 1 regular semisimple elements from non equivalent heisenberg subalgebras of a loop algebra g x c lambda lambda 1 are studied the graded heisenberg subalgebras containing such elements are labelled by the regular conjugacy classes in the weyl group w g of the simple lie algebra g a representative w epsilon w g of a regular conjugacy class can be lifted to an inner automorphism of g given by w exp 2i pi ad i0 m where i0 is the defining vector of an sl2 subalgebra of g the grading is then defined by the operator d m i0 m lambda d d lambda ad i0 and any grade 1 regular element lambda from the heisenberg subalgebra associated with w takes the form lambda c lambda c where i0 c m 1 c and c is included in an sl2 subalgebra containing i0 the largest eigenvalue of adi0 is m 1 except for some cases in f4 e6 7 8 we explain how these lie algebraic results follow from known results and apply them to construct integrable systems if the largest ad i0 eigenvalue is m 1 then using any grade 1 regular element from the heisenberg subalgebra associated with w we can construct a kdv system possessing the standard w algebra defined by i0 as its second poisson bracket algebra for g a classical lie algebra we derive pseudo differential lax operators for those non principal kdv systems that can be obtained as discrete reductions of kdv systems related to gln non abelian toda systems are also considered

a remark on kac wakimoto hierarchies of d type for the kac wakimoto hierarchy constructed from the principal vertex operator realization of the basic representation of the affine lie algebra d 1 n we compute the coefficients of the corresponding hirota bilinear equations and verify the coincidence of these bilinear equations with the ones that are satisfied by givental s total descendant potential of the dn singularity as conjectured by givental and milanov 2005 simple singularities and integrable hierarchies the breadth of symplectic and poisson geometry prog math vol 232 boston birkh user pp 173 201

tau functions and generalized integrable hierarchies the tau function formalism for a class of generalized zero curvature integrable hierarchies of partial differential equations is constructed the class includes the drinfel d sokolov hierarchies a direct relation between the variables of the zero curvature formalism and the tau functions is established the formalism also clarifies the connection between the zero curvature hierarchies and the hirotatype hierarchies of kac and wakimoto

minimax mixed integer symmetric duality for multiobjective variational problems abstract a mond weir type multiobjective variational mixed integer symmetric dual program over arbitrary cones is formulated applying the separability and generalized f convexity on the functions involved weak strong and converse duality theorems are established self duality theorem is proved a close relationship between these variational problems and static symmetric dual minimax mixed integer multiobjective programming problems is also presented

glm equations tau function and scattering data the direct and the inverse scattering problem for affine toda mkdv systems is addressed and is found to develop non standard features within the framework of the inverse scattering method a solution scheme based on the tau function formalism is described the inverse problem is shown to be equivalent to a set of decoupled scalar gelfand levitan marchenko type equations the fredholm grothendieck determinants of the latter are shown to define tau functions in the sense of the kyoto school in particular a simple monodromy formula allows the derivation of trace identities

locating the first nodal linein the neumann problem the location of the nodal line of the first nonconstant neumann eigenfunction of a convex planar domain is specified to within a distance comparable to the inradius this is used to prove that the eigenvalue of the partial differential equation is approximated well by the eigenvalue of an ordinary differential equation whose coefficients are expressed solely in terms of the width of the domain a variant of these estimates is given for domains that are thin strips and satisfy a lipschitz condition

symmetric duality for minimax mixed integer programming problems with pseudo invexity in this paper we consider the pair of symmetric dual multiobjective variational mixed integer programs proposed by chen and yang x chen j yang symmetric duality for minimax multiobjective variational mixed integer programming problems with partial invexity european journal of operational research 181 2007 76 85 and extend some of their results under the assumptions of partial pseudo invexity and separability on the functions involved these results include several results available in literature as special cases

generalized drinfel d sokolov hierarchies ii the hamiltonian structures in this paper we examine the bi hamiltonian structure of the generalized kdv hierarchies we verify that both hamiltonian structures take the form of kirillov brackets on the kac moody algebra and that they define a coordinated system classical extended conformal algebras are obtained from the second poisson bracket in particular we construct the w n l algebras first discussed for the case n 3 and l 2 by polyakov and bershadsky

a loop group method for affine harmonic maps into lie groups we generalize the uhlenbeck segal theory for harmonic maps into compact semi simple lie groups to general lie groups equipped with torsion free bi invariant connection

computation of generalized equivariant cohomologies of kac moody flag varieties abstract in 1998 goresky kottwitz and macpherson showed that for certain projective varieties x equipped with an algebraic action of a complex torus t the equivariant cohomology ring h t x can be described by combinatorial data obtained from its orbit decomposition in this paper we generalize their theorem in three different ways first our group g need not be a torus second our space x is an equivariant stratified space along with some additional hypotheses on the attaching maps third and most important we allow for generalized equivariant cohomology theories e g instead of h t for these spaces we give a combinatorial description of e g x as a subring of e g f i where the f i are certain invariant subspaces of x our main examples are the flag varieties g p of kac moody groups g with the action of the torus of g in this context the f i are the t fixed points and e g is a t equivariant complex oriented cohomology theory such as h t k t or mu t we detail several explicit examples

symmetries in the third painleve equation arising from the modified pohlmeyer lund regge hierarchy we propose a modification of the ablowitz kaup newell segur akns hierarchy that includes the modified pohlmeyer lund regge mplr equation similarity reductions of this hierarchy give the second third and fourth painleve equations especially we present a new lax representation and a complete description of the symmetry of the third painleve equation through the similarity reduction we also show the relation between the tau function of the mplr hierarchy and painleve equations

covering algebras ii isomorphism of loop algebras this paper studies the loop algebras that arise from pairs consisting of a symmetrizable kac moody lie algebra g and a finite order automorphism of g we obtain necessary and sufficient conditions for two such loop algebras to be isomorphic

on the hot spots conjecture of j rauch rodrigo banuelos research partially supported by nsf grant dms 9400854 krzysztof burdzy s research partially supported by nsf grant dms 9322689

second order symmetric duality for nonlinear multiobjective mixed integer programming we formulate two pairs of second order symmetric duality for nonlinear multiobjective mixed integer programs for arbitrary cones by using the concepts of efficiency and second order invexity we establish weak strong converse and self duality theorems for the dual models several known results are obtained as special cases

a note on pseudo invexity and symmetric duality a pair of symmetric dual nonlinear programming problems is presented and duality theorems are established under pseudoinvexity type assumptions on the kernel function this formulation removes certain inconsistencies in a recently introduced primal dual pair and gives the correct proof of various duality theorems

infinite dimensional grassmannian structure of two dimensional quantum gravity we study the infinite dimensional grassmannian structure of 2d quantum gravity coupled to minimal conformal matters and show that there exists a large symmetry thew 1 symmetry using this symmetry structure we prove that the square root of the partition function which is a function of thep reduced kp hierarchy satisfies the vacuum condition of thew 1 algebra we further show that this condition is reduced to the vacuum condition of thew p algebra when the redundant variables for thep reduction are eliminated this mechanism also gives a prescription for extracting thew p algebra from thew 1 algebra

kac moody groups and integrability of soliton equations a new approach to integrability of affine toda field theories and closely related to them kdv hierarchies is proposed the flows of a hierarchy are explicity identified with infinitesimal action of the principal abelian subalgebra of the corresponding affine kac moody algebra on a homogeneous space

minimization problem related to a lyapunov inequality abstract in this study we consider the following minimization problem on a bounded smooth domain in r n s inf u 2 2 u 2 2 u h 1 0 u 2 2 u 0 this minimization problem plays a crucial role in the study of l p lyapunov type inequalities 1 p for linear partial differential equations with neumann boundary conditions in this study we prove the existence of minimizers of s as a consequence we prove the l p lyapunov type inequality in the critical case which was left open in 4

minimax and symmetric duality for nonlinear multiobjective mixed integer programming abstract we formulate two pairs of symmetric duality for nonlinear multiobjective mixed integer programs for arbitrary cones by using the concept of efficiency we establish the weak strong converse and self duality theorems for our symmetric models several known results are obtained as special cases

the solution space of the unitary matrix model string equation and the sato grassmannian the space of all solutions to the string equation of the symmetric unitary one matrix model is determined it is shown that the string equation is equivalent to simple conditions on pointsv1 andv2 in the big cell gr 0 of the sato grassmannian gr this is a consequence of a well defined continuum limit in which the string equation has the simple form open image in new window matrices of differential operators these conditions onv1 andv2 yield a simple system of first order differential equations whose analysis determines the space of all solutions to the string equation this geometric formulation leads directly to the virasoro constraintsln n 0 whereln annihilate the two modified kdv functions whose product gives the partition function of the unitary matrix model

an approach to symmetrization via polarization we prove that the steiner symmetrization of a function can be approximated in l p r n by a sequence of very simple rearrangements which are called polarizations this result is exploited to develop elementary proofs of many inequalities including the isoperimetric inequality in euclidean space in this way we also obtain new symmetry results for solutions of some varia tional problems furthermore we compare the solutions of two boundary value problems one of them having a polarized geometry and we show some point wise inequalities between the solutions this leads to new proofs of well known functional inequalities which compare the solutions of two elliptic or parabolic problems one of them having a steiner symmetrized geometry the method also allows us to investigate the case of equality in the inequalities roughly speaking we prove that the equality sign is valid only if the original problem has the symmetrized geometry

existence unique continuation and symmetry of least energy nodal solutions to sublinear neumann problems we consider the sublinear problem begin aligned left begin array l quad l delta u u q 2 u quad text in omega u nu 0 quad text on partial omega end array right end aligned where omega subset mathbb r n is a bounded domain and 1 le q 2 for q 1 u q 2 u will be identified with mathrm sgn u we establish a variational principle for least energy nodal solutions and we investigate their qualitative properties in particular we show that they satisfy a unique continuation property their zero set is lebesgue negligible moreover if omega is radial then least energy nodal solutions are foliated schwarz symmetric and they are nonradial in case omega is a ball the case q 1 requires special attention since the formally associated energy functional is not differentiable and many arguments have to be adjusted

second order symmetric duality for nonlinear minimax mixed integer programs abstract wolfe type second order minimax mixed integer dual programs are formulated and a symmetric duality theorem is e established under separability and bonvexity boncavity of the kernel function k x y mond weir type symmetric duality is also discussed under weaker bonvexity assumptions moreover self duality theorems for these pairs are obtained assuming k x y to be skew symmetric

least energy nodal solutions of hamiltonian elliptic systems with neumann boundary conditions abstract we study existence regularity and qualitative properties of solutions to the system u v q 1 v in v u p 1 u in u v 0 on with r n bounded in this setting all nontrivial solutions are sign changing our proofs use a variational formulation in dual spaces considering sublinear p q 1 and superlinear p q 1 problems in the subcritical regime in balls and annuli we show that least energy solutions l e s are foliated schwarz symmetric and due to a symmetry breaking phenomenon l e s are not radial functions a key element in the proof is a new l t norm preserving transformation which combines a suitable flipping with a decreasing rearrangement this combination allows us to treat annular domains sign changing functions and neumann problems which are non standard settings to use rearrangements and symmetrizations in particular we show that our transformation diminishes the dual energy and as a consequence radial l e s are strictly monotone we also study unique continuation properties and simplicity of zeros our theorems also apply to the scalar associated model where our approach provides new results as well as alternative proofs of known facts

on a pair of nonlinear mixed integer programming problems abstract we consider maximin and minimax nonlinear mixed integer programming problems which are nonsymmetric in duality sense under weaker pseudo convex pseudo concave assumptions we show that the supremum infimum of the maximin problem is greater than or equal to the infimum supremum of the minimax problem as a particular case this result reduces to the weak duality theorem for minimax and symmetric dual nonlinear mixed integer programming problems further this is used to generalize available results on minimax and symmetric duality in nonlinear mixed integer programming

symmetric duality for minimax nonlinear mixed integer programming abstract a pair of symmetric dual minimax integer programming problems is formulated and an appropriate duality theorem is established under the pseudoconvexity pseudoconcavity assumption on the kernel function k x y this model removes certain inconsistencies in an earlier model and helps to study self duality as well

additional symmetries of generalized integrable hierarchies the non isospectral symmetries of a general class of integrable hierarchies are found by generalizing the galilean and scaling symmetries of the korteweg de vries equation and its hierarchy the symmetries arise in a very natural way from the semi direct product structure of the virasoro algebra and the affine kac moody algebra underlying the construction of the hierarchy in particular the generators of the symmetries are shown to satisfy a subalgebra of the virasoro algebra when a tau function formalism is available the infinitesimal symmetries act directly on the tau functions as moments of virasoro currents some comments are made regarding the role of the non isospectral symmetries and the form of the string equations in matrix model formulations of quantum gravity in two dimensions and related systems

mixed symmetric duality in non differentiable multiobjective mathematical programming two mixed symmetric dual models for a class of non differentiable multiobjective nonlinear programming problems with multiple arguments are introduced in this paper these two mixed symmetric dual models unify the four existing multiobjective symmetric dual models in the literature weak and strong duality theorems are established for these models under some mild assumptions of generalized convexity several special cases are also obtained

invariants of real forms of affine kac moody lie algebras abstract the theory of vogan diagrams which are dynkin diagrams with an overlay of certain additional information allows one to give a rapid classification of finite dimensional real semisimple lie algebras and to make use of this classification in practice this paper develops a corresponding theory of vogan diagrams for almost compact real forms of indecomposable nontwisted affine kac moody lie algebras in this case also the equivalence classes of vogan diagrams correspond to the isomorphism classes of almost compact real forms although the real forms of such algebras had already been classified the theory of vogan diagrams introduces invariants for such algebras and makes it possible to locate a given real form within the classification

generic hecke algebra for renner monoids abstract to each renner monoid r we associate a generic hecke algebra h r over z q which is a deformation of the monoid z algebra of r if m is a finite reductive monoid with borel subgroup b and associated renner monoid r then we obtain the associated iwahori hecke algebra h m b by specialising q in h r and tensoring by c over z as in the classical case of finite reductive groups

τ functions and zero curvature equations of toda akns type the connection between functions and zero curvature equations for the homogeneous construction of the basic module l 0 over the simplest affine kac moody algebra a 1 1 is studied

symmetric duality for minimax multiobjective variational mixed integer programming problems with partial invexity a pair of symmetric dual multiobjective variational mixed integer programs for the polars of arbitrary cones are formulated which some primal and dual variables are constrained to belong to the set of integers under the separability with respect to integer variables and partial invexity assumptions on the functions involved we prove the weak strong converse and self duality theorems to related minimax efficient solution these results include some of available results

infinite flag varieties and conjugacy theorems abstract we study the orbit of a highest weight vector in an integrable highest weight module of the group g associated to a kac moody algebra unk a we obtain applications to the geometric structure of the associated flag varieties and to the algebraic structure of unk a in particular we prove conjugacy theorems for cartan and borel subalgebras of unk a so that the cartan matrix a is an invariant of unk a

the shape of extremal functions for poincare sobolev type inequalities in a ball we study extremal functions for a family of poincare sobolev type inequalities these functions minimize for subcritical or critical p 2 p 2 the quotient u 2 u p u 2 u p among all u h 1 b 0 u h 1 b 0 with b u 0 b u 0 here b is the unit ball in rn r n we show that the minimizers are axially symmetric with respect to a line passing through the origin we also show that they are strictly monotone in the direction of this line in particular they take their maximum and minimum precisely at two antipodal points on the boundary of b we also prove that for p close to 2 minimizers are antisymmetric with respect to the hyperplane through the origin perpendicular to the symmetry axis and that once the symmetry axis is fixed they are unique up to multiplication by a constant in space dimension two we prove that minimizers are not antisymmetric for large p

the witten equation mirror symmetry and quantum singularity theory for any nondegenerate quasi homogeneous hypersurface singularity we describe a family of moduli spaces a virtual cycle and a corresponding cohomological eld theory associated to the singularity this theory is analogous to gromov witten theory and generalizes the theory of r spin curves which corresponds to the simple singularity ar 1 we also resolve two outstanding conjectures of witten the rst conjecture is that ade singularities are self dual and the second conjecture is that the total potential functions of ade singularities satisfy corresponding ade integrable hierarchies other cases of integrable hierarchies are also discussed

tau functions and virasoro symmetries for drinfeld sokolov hierarchies abstract for each drinfeld sokolov integrable hierarchy associated to affine kac moody algebra we obtain a uniform construction of tau function by using tau symmetric hamiltonian densities moreover we represent its virasoro symmetries as linear nonlinear actions on the tau function the relations between the tau function constructed in this paper and those defined for particular cases of drinfeld sokolov hierarchies in the literature are clarified we also show that whenever the affine kac moody algebra is simply laced or twisted the tau function of the drinfeld sokolov hierarchy coincides with the solution of the corresponding kac wakimoto hierarchy constructed from the principal vertex operator realization of the affine algebra

tau functions and dressing transformations for zero curvature affine integrable equations the solutions of a large class of hierarchies of zero curvature equations that includes toda and kdv type hierarchies are investigated all these hierarchies are constructed from affine twisted or untwisted kac moody algebras g their common feature is that they have some special vacuum solutions corresponding to lax operators lying in some abelian up to the central term subalgebra of g in some interesting cases such subalgebras are of the heisenberg type using the dressing transformation method the solutions in the orbit of those vacuum solutions are constructed in a uniform way then the generalized tau functions for those hierarchies are defined as an alternative set of variables corresponding to certain matrix elements evaluated in the integrable highest weight representations of g such definition of tau functions applies for any level of the representation and it is independent of its realization vertex operator or not the particular important cases of generalized mkdv and kdv hierarc