strong traces for solutions to scalar conservation laws with general flux in this paper we consider bounded weak solutions u of scalar conservation laws not necessarily of class bv defined in a subset omega subset mathbb r times mathbb r we define a strong notion of trace at the boundary of omega reached by l1 convergence for a large class of functionals of u g u the functionals g depend on the flux function of the conservation law and on the boundary of omega the result holds for a general flux function and a general subset

entropy formulation for parabolic degenerate equations with general dirichlet boundary conditions and application to the convergence of fv methods this paper is devoted to the analysis and the approximation of parabolic hyperbolic degenerate problems defined on bounded domains with nonhomogeneous boundary conditions it consists of two parts the first part is devoted to the definition of an original notion of entropy solutions to the continuous problem which can be adapted to define a notion of measure valued solutions or entropy process solutions the uniqueness of such solutions is established in the second part the convergence of the finite volume method is proved this result relies on weak estimates and on the theorem of uniqueness of the first part it also entails the existence of a solution to the continuous problem

equivariant stable homotopy and segal s burnside ring conjecture analogous results were proved later for ko in the generality of compact lie groups by atiyah and segal 8 and for kfq the algebraic k theory spectrum associated to the finite field fq by rector 29 using quillen s 26 computation of it kfq in each case the answer involves an appropriate completed representation ring of g and the cohomology theory in question is constructed from the permutative category of finite dimensional vector spaces over a field see 31 if one considers cohomology theories constructed from other permutative categories one expects to find analogous computations in terms of a completed representation ring of g in the given category appropriately defined in particular stable cohomotopy ts is constructed from the category of finite sets 31 and for this category the analogue of the representation ring is a well known object the burnside ring a g 13 a g is a commutative ring with augmentation so one may speak of a g the completed burnside ring moreover there is

divergence measure fields and hyperbolic conservation laws we analyze a class of l infty vector fields called divergence measure fields we establish the gauss green formula the normal traces over subsets of lipschitz boundaries and the product rule for this class of l infty fields then we apply this theory to analyze l infty entropy solutions of initial boundary value problems for hyperbolic conservation laws and to study the ways in which the solutions assume their initial and boundary data the examples of conservation laws include multidimensional scalar equations the system of nonlinear elasticity and a class of m x m systems with affine characteristic hypersurfaces the analysis in l infty also extends to l p

heisenberg algebras and rational double affine hecke algebras in this paper we categorify the heisenberg action on the fock space via the category o of cyclotomic rational double affine hecke algebras this permits us to relate the filtration by the support on the grothendieck group of o to a representation theoretic grading defined using the heisenberg action this implies a recent conjecture of etingof

well posedness of general boundary value problems for scalar conservation laws in this paper we investigate well posedness for the problem u t div ph u f on 0 t times om om subset r n with initial condition u 0 cdot u 0 on om and with general dissipative boundary conditions varphi u cdot nu in beta t x u on 0 t times ptl om here for a e t x in 0 t times ptl om beta t x cdot is a maximal monotone graph on r this includes as particular cases dirichlet neumann robin obstacle boundary conditions and their piecewise combinations as for the well studied case of the dirichlet condition one has to interprete the it formal boundary condition given by beta by replacing it with the adequate it effective boundary condition such effective condition can be obtained through a study of the boundary layer appearing in approximation processes such as the vanishing viscosity approximation we claim that the formal boundary condition given by beta should be interpreted as the effective boundary condition given by another monotone graph tilde beta which is defined from beta by the projection procedure we describe we give several equivalent definitions of entropy solutions associated with tilde beta and thus also with beta for the notion of solution defined in this way we prove existence uniqueness and l 1 contraction monotone and continuous dependence on the graph beta convergence of approximation procedures and stability of the notion of entropy solution are illustrated by several results

products and duality in waldhausen categories the natural transformatione from l theory to the tate cohomol ogy of z 2 acting on k theory commutes with external products corollary the tate cohomology of z 2 acting on the k theory of any ring with inrvolu tion is a generalized eilenberg mac lane spectrum and it is 4 periodic categories with cofibrations and weak equivalences waldhausen categories for short were introduced in wald generalizing earlier constructions due to quillen and others waldhausen defined the k theory spectrum k c of a waldhausen category c it was realized early on vo that suitable notions of spanier whitehead duality in c give rise to involutions on k c we formalized this idea in wwd by introducing the notion of a spanier whitehead product sw product for short in c to some extent this will be recalled below when c is equipped with an sw product d and c 0 satisfies certain conditions 2 of wwd then the following are defined

extended divergence measure fields and the euler equations for gas dynamics a class of extended vector fields called extended divergence measure fields is analyzed these fields include vector fields in l p and vector valued radon measures whose divergences are radon measures such extended vector fields naturally arise in the study of the behavior of entropy solutions of the euler equations for gas dynamics and other nonlinear systems of conservation laws a new notion of normal traces over lipschitz deformable surfaces is developed under which a generalized gauss green theorem is established even for these extended fields an explicit formula is obtained to calculate the normal traces over any lipschitz deformable surface suitable for applications by using the neighborhood information of the fields near the surface and the level set function of the lipschitz deformation surfaces as an application we prove the uniqueness and stability of riemann solutions that may contain vacuum in the class of entropy solutions of the euler equations for gas dynamics

the homotopy fixed point theorem and the quillen lichtenbaum conjecture in hermitian k theory abstract we settle two conjectures for computing higher grothendieck witt groups also known as hermitian k groups of noetherian schemes x under some mild conditions it is shown that the comparison map from the hermitian k theory of x to the homotopy fixed points of k theory under the natural z 2 action is a 2 adic equivalence we also prove that the mod 2 comparison map between the hermitian k theory of x and its etale version is an isomorphism on homotopy groups in the same range as for the quillen lichtenbaum conjecture in k theory applications compute higher grothendieck witt groups of complex algebraic varieties and rings of 2 integers in number fields and hence values of dedekind zeta functions

strong traces for solutions of multidimensional scalar conservation laws in this paper we consider multidimensional scalar conservation laws without bv estimates defined in a subset d we show that with a non degeneracy hypothesis on the flux we can define a strong notion of trace at the boundary of reached by l 1 convergence

coassembly and the k theory of finite groups abstract we study the k theory and swan theory of the group ring r g when g is a finite group and r is any ring or ring spectrum in this setting the well known assembly map for k r g has a companion called the coassembly map we prove that their composite is the equivariant norm of k r this gives a splitting of both assembly and coassembly after k n localization a new map between whitehead torsion and tate cohomology and a partial computation of k theory of representations in the category of spectra

ergodic theorems for individual random sequences in the framework of the kolmogorov s approach to the substantiation of the probability theory and information theory on the base of the theory of algorithms we try to formulate probabilistic laws i e statements of the form p a 1 where a is some formula in pointwise form if is random then a holds nevertheless not all proofs of such laws can be directly translated into the algorithmic form in 11 two examples have been distinguished birkhoff s 2 ergodic theorem and shannon mcmillan breiman theorem of information theory 1 in this paper an analysis of algorithmic effectiveness of these theorems is given we prove that birkhoff s ergodic theorem is indeed in some strong sense nonconstructive at the same time the claim to formulate probabilistic laws for algorithmically random sequences is not so restrictive we present the versions of these laws for individual random sequences

on the well posedness of entropy solutions to conservation laws with a zero flux boundary condition we study a zero flux type initial boundary value problem for scalar conservation laws with a genuinely nonlinear flux we suggest a notion of entropy solution for this problem and prove its well posedness the asymptotic behavior of entropy solutions is also discussed

nonhomogeneous dirichlet problems for degenerate parabolic hyperbolic equations the aim of the paper is to give a formulation for the initial boundary value problem of parabolic hyperbolic type in the case of nonhomogeneous boundary data a 0 here u u x t with x t q 0 t where is a bounded domain in n with smooth boundary and t 0 the function b is assumed to be nondecreasing allowing degeneration zones where b is constant is locally lipschitz continuous and g l 0 b after introducing the definition of an entropy solution to the above problem in the spirit of otto 14 we prove uniqueness of the solution in the proposed setting moreover we prove that the entropy solution previously defined can be obtained as the limit of solutions of regularized equations of nondegenerate parabolic type specifically the diffusion function b is approximated by functions b that are strictly increasing

well posedness for entropy solutions to multidimensional scalar conservation laws with a strong boundary condition in this paper we give a simple proof of well posedness of multidimensional scalar conservations laws with a strong boundary condition the proof is based on a result of strong trace for solutions of scalar conservation laws and kinetic formulation

a categorification of twisted heisenberg algebras abstract we categorify a quantized twisted heisenberg algebra associated to a finite subgroup of sl 2 c

twisted quantum affine algebras we give a highest weight classification of the finite dimensional irreducible representations of twisted quantum affine algebras as in the untwisted case such representations are in one to one correspondence with n tuples of monic polynomials in one variable but whereas in the untwisted case n is the rank of the underlying finite dimensional complex simple lie algebra in the twisted case n is the rank of the subalgebra of fixed by the diagram automorphism the way in which such an n tuple determines a representation is also more complicated than in the untwisted case

on strongly degenerate convection diffusion problems modeling sedimentation consolidation processes abstract we investigate initial boundary value problems for a quasilinear strongly degenerate convection diffusion equation with a discontinuous diffusion coefficient these problems come from the mathematical modeling of certain sedimentation consolidation processes the existence of entropy solutions belonging to bv is shown by the vanishing viscosity method the existence proof for one of the models includes a new regularity result for the integrated diffusion coefficient new uniqueness proofs for entropy solutions are also presented these proofs rely on a recent extension to second order equations of kru kov s method of doubling the variables the application to a sedimentation consolidation model is illustrated by two numerical examples

the stability of foliations of orientable 3 manifolds covered by a product we examine the relationship between codimension one foliations that are covered by a trivial product of hyperplanes and the branched surfaces that can be constructed from them we present a sufficient condition on a branched surface constructed from a foliation to guarantee that all cl perturbations of the foliation are covered by a trivial product of hyperplanes we also show that a branched surface admits a strictly positive weight system if and only if it can be constructed from a fibration over s1

computational limits to nonparametric estimation for ergodic processes a new negative result for nonparametric distribution estimation of binary ergodic processes is shown the problem of estimation of distribution with any degree of accuracy is studied then it is shown that for any countable class of estimators there is a zero entropy binary ergodic process that is inconsistent with the class of estimators our result is different from other negative results for universal forecasting scheme of ergodic processes

limits to consistent on line forecasting for ergodic time series this article concerns problems of time series forecasting under the weakest of assumptions related results are surveyed and are points of departure for the developments here some of which are new and others are new derivations of previous findings the contributions in this study are all negative showing that various plausible prediction problems are unsolvable or in other cases are not solvable by predictors which are known to be consistent when mixing conditions hold

a boundary value problem for a class of anisotropic degenerate parabolic hyperbolic equations we consider a mixed type boundary value problem for a class of degenerate parabolic hyperbolic equations namely we consider a cartesian product domain and split its boundary into two parts in one of them we impose a dirichlet boundary condition in the other we impose a neumann condition we apply a normal trace formula for l 2 divergence measure fields to prove a new strong trace property in the part of the boundary where the neumann condition is imposed we prove the existence and uniqueness of the entropy solution

branched surfaces and attractors i dynamic branched surfaces we show how using ideas of r f williams about branched sur faces hyperbolic attractors of flows on three manifolds may be classified up to topological equivalence on an isolating neighborhood by a finite combinatorial object a swaddled graph

entropy solutions for nonlinear degenerate problems we consider a class of elliptic hyperbolic degenerate equations g u delta b u divg phi u f with dirichlet homogeneous boundary conditions and a class of elliptic parabolic hyperbolic degenerate equations g u t delta b u divg phi u f with homogeneous dirichlet conditions and initial conditions existence of entropy solutions for both problems is proved for nondecreasing continuous functions g and b vanishing at zero and for a continuous vectorial function satisfying rather general conditions comparison and uniqueness of entropy solutions are proved for g and b continuous and nondecreasing and for continuous

twisted vertex representations and spin characters we establish a new group theoretic realization of the basic representations of the twisted affine and twisted toroidal algebras in the same spirit of our new approach to the mckay correspondence our vertex operator construction provides a unified description to the character tables for the spin cover of the wreath product of the twisted hyperoctahedral groups and an arbitrary finite group

equivariant k theory generalized symmetric products and twisted heisenberg algebra for a space x acted on by a finite group the product space xn affords a natural action of the wreath product n n sn the direct sum of equivariant k groups were shown earlier by the author to carry several interesting algebraic structures in this paper we study the k groups of n equivariant clifford supermodules on xn we show that is a hopf algebra and it is isomorphic to the fock space of a twisted heisenberg algebra twisted vertex operators make a natural appearance the algebraic structures on x when is trivial and x is a point specialize to those on a ring of symmetric functions with the schur q functions as a linear basis as a by product we present a novel construction of k theory operations using the spin representations of the hyperoctahedral groups

twisted vertex representations of quantum affine algebras recent interests in quantum groups are stimulated by their marvelous relations with quantum yang baxter equations conformal field theory invariants of links and knots and q hypergeometric series besides understanding the reason of the appearance of quantum groups in both mathematics and theoretical physics there is a natural problem of finding q deformations or quantum analogues of known structures quantum groups were first defined by drinfeld 2 and jimbo 9 also see 4 as a q deformation of the universal enveloping algebras of the kacmoody algebras in the work of trigonometric solutions of yang baxter equations in the same spirit it was shown in 13 14 that there exists a 1 1 correspondence between the integrable highest weight representations of symmetrizable kac moody algebras and those of the corresponding quantum groups where both spaces have the same dimension in the case of generic q i e q is not a root of unity moreover one can be very explicit in the case of quantum gl n to write down the irreducible highest weight representations quantum affine algebras are the quantum groups associated to affine lie algebras following drinfeld s realization 3 of q analog of loop algebras the vertex representation of untwisted simply laced quantum affine algebras was constructed in frenkel jing 6 which is a q deformation of frenkel kac 7 and segal 15 construction in the theory of affine lie algebras subsequently the same was done for the quantum affine algebra of type b in 1 in the present work we construct vertex representations of quantum affine algebras twisted by an automorphism of the dynkin diagram which generalizes certain important cases in the ordinary twisted vertex operator calculus 5