discontinuous galerkin finite element methods for time dependent hamilton jacobi bellman equations with cordes coefficients we propose and analyse a fully discrete discontinuous galerkin time stepping method for parabolic hamilton jacobi bellman equations with cordes coefficients the method is consistent and unconditionally stable on rather general unstructured meshes and time partitions error bounds for both rough and regular solutions in terms of temporal regularity show that the method is arbitrarily high order with optimal convergence rates with respect to the mesh size time interval length and temporal polynomial degree and possibly suboptimal by an order and a half in the spatial polynomial degree numerical experiments on problems with strongly anisotropic diffusion coefficients and early time singularities demonstrate the accuracy and computational efficiency of the method with exponential convergence rates achieved under combined hp and tau q q refinement

grothendieck standard conjectures morphic cohomology and hodge index theorem using morphic cohomology we produce a sequence of conjectures called morphic conjectures which terminates at the grothendieck standard conjecture a a refinement of hodge structures is given and with the assumption of morphic conjectures we prove a hodge index theorem we answer a question of friedlander and lawson by assuming the grothendieck standard conjecture b and prove that the topological filtration from morphic cohomology is equal to the grothendieck arithmetic filtration for some cases

the electromagnetic scattering problem in the time domain for a dissipative slab and a point source using invariant imbedding a vertical magnetic dipole source above an inhomogeneous slab is considered where the permittivity e e z and conductivity z vary with the depth z using the axial symmetry of the problem a hankel transform is introduced besides physical scattering kernels nonphysical scattering kernels are introduced and the direct and inverse transformed problem is solved by means of wave splitting in an invariant imbedding approach the imbedding equations for the nonphysical scattering kernels are simpler than the corresponding equations for the physical kernels and they are solved using existing techniques for the imbedding equations for normally incident plane waves the solution for the physical kernels is obtained by establishing relations between the two sets of kernels these relations are given by volterra equations of the second kind and thus are very easy to solve numerically some applications and extensions of the results derived in the paper are discussed

wave splitting and the reflection operator for the wave equation in r3 the problem of wave splitting in a nonhomogeneous medium in r3 is considered previous results for wave splitting in a planar stratified medium can be generalized to a general nonhomogeneous medium with sufficiently smooth velocity the wave equation is factorized into an up and down going wave system using certain integral and integral differential operators the equation for the reflection operator which relates the up going wave to a down going wave is then obtained and certain properties of the reflection operator are deduced

self correspondences of the generic fibre of lefschetz pencils and the leray filtration abstract in this paper we give a detailed analysis of the interaction between homological self correspondences of the general fibre y k t of the lefschetz fibration x p 1 of a lefschetz pencil on a smooth projective variety x k and the leray filtration of we derive the result that if the standard conjecture b y holds then the operator x p x n 1 is algebraic where p x n 1 is defined as the inverse of l on l p n 1 x and 0 on l k p j x for 1 n 1 k j in the course of our proof we see that under the above assumption the kunneth projectors x i for i n 1 n n 1 are algebraic

recent developments in numerical methods for fully nonlinear second order partial differential equations this article surveys the recent developments in computational methods for second order fully nonlinear partial differential equations pdes a relatively new subarea within numerical pdes due to their ever increasing importance in mathematics itself e g differential geometry and pdes and in many scientific and engineering fields e g astrophysics geostrophic fluid dynamics grid generation image processing optimal transport meteorology mathematical finance and optimal control numerical solutions to fully nonlinear second order pdes have garnered a great deal of interest from the numerical pde and scientific communities significant progress has been made for this class of problems in the past few years but many problems still remain open this article intends to introduce these current advancements and new results to the siam community and generate more interest in numerical methods for fully nonlinear pdes

l algebre de hopf et le groupe de galois motiviques d un corps de caracteristique nulle ii this is the second article of a series of two aiming at constructing and studying motivic galois groups in the context of triangulated motives these motivic galois groups were constructed in the first article and their algebras of regular functions were described concretely in terms of differential forms or algebraic cycles in the present article we gather important complements to the first one in the first part of this article we describe the link between the motivic galois group and the usual galois group of a subfield of in the second part we develop the basis of a theory of ramification for the motivic galois groups by constructing motivic versions of the decomposition and inertia groups associated to a geometric valuation we also introduce the relative motivic galois group of an extension k k which measures the difference between the motivic galois groups of k and k the latter is more accessible than its absolute counterpart in effect we show that it is a quotient of the pro algebraic completion of the topological fundamental pro group of a the pro variety homk k at least when the extension k k is of finite type

on the green functions technique and phase velocity approximation of axially symmetric fields in stratified media in this paper a numerical method is developed to solve a one dimensional inverse scattering problem associated with a pair of two coupled partial differential equations these equations are for the green functions that can be used for the optimization of electromagnetic energy within selected regions of a medium in which the propagation is taking place the method is based upon constructing the nth degree interpolation polynomial to approximate the green functions and the phase velocity function using legendre gauss lobatto collocation points an example is given to demonstrate the accuracy of the developed method

factorization of a dissipative wave equation and the green functions technique for axially symmetric fields in a stratified slab the time domain scattering problem for axially symmetric fields and a dissipative stratified slab is considered explicit representations are derived for the wave splitting operators that factorize the dissipative hankel transformed wave equation the green functions approach associated with this splitting is developed partial differential equations pdes for the green functions are derived and used to derive the imbedding equations for the reflection and transmission kernels numerical results for the internal fields are given in the direct problem and the phase velocity is reconstructed in the inverse problem

motives and filtrations on chow groups in this paper sequential to sa we discuss a philosophy of bloch and beilinson concerning filtrations on chow groups of algebraic vareties which provides us with a good understanding on the structure of at least certain portions of chow groups let k be a fixed ground field and let c be the category of smooth projective schemes over k with a fixed suitable weil cohomology theory cf notations x in mathcal c rightsquigarrow h x mathop oplus limits i in z h i x

motivation for hodge cycles given two smooth projective varieties x and y x is defined to motivate y if the motive of y is contained in the tensor category generated by x some techniques are given for checking this condition it is shown that in a number of cases moduli spaces of sheaves over curves or surfaces are motivated by the underlying curve or surface this is used to check the hodge and related conjectures for some of these examples

noncommutative motives numerical equivalence and semi simplicity making use of hochschild homology we introduce the correct category nnum k f of noncommutative numerical motives over a base ring k and with coefficients in a field f we prove that nnum k f is abelian semi simple and that grothendieck s category num k q of numerical motives embeds into nnum k q after being factored out by the action of the tate object as an application we obtain an alternative proof of jannsen s celebrate semi simplicity result which uses the noncommutative world instead of a classical weil cohomology

the electromagnetic inverse problem in the time domain for a dissipative slab and a point source using invariant imbedding reconstruction of the permittivity and conductivity abstract he s and s strom the electromagnetic inverse problem in the time domain for a dissipative slab and a point source using invariant imbedding reconstruction of the permittivity and conductivity journal of computational and applied mathematics 42 1992 137 155 we consider the electromagnetic inverse problem for a point source above an inhomogeneous dissipative slab of permittivity z and conductivity z where z is the depth two inversion algorithms based on the invariant imbedding equations derived in previous work are used to reconstruct both the permittivity and the conductivity both algorithms use two sided reflection data and one of them also uses transmission data results are presented for clean and noisy data

noncommutative motives in positive characteristic and their applications abstract let k be a base field of positive characteristic making use of topological periodic cyclic homology we start by proving that the category of noncommutative numerical motives over k is abelian semi simple as conjectured by kontsevich then we establish a far reaching noncommutative generalization of the weil conjectures originally proven by dwork and grothendieck in the same vein we establish a far reaching noncommutative generalization of the cohomological interpretations of the hasse weil zeta function originally proven by hesselholt as a third main result we prove that the numerical grothendieck group of every smooth proper dg category is a finitely generated free abelian group as claimed without proof by kuznetsov then we introduce the noncommutative motivic galois super groups and following an insight of kontsevich relate them to their classical commutative counterparts finally we explain how the motivic measure induced by berthelot s rigid cohomology can be recovered from the theory of noncommutative motives

motives and siegel modular forms for a motive m over q we define the fundamental periods of m using invariant theory our definition generalizes deligne s periods we show that if a motive m is constructed from motives m 1 m 2 m n by a standard algebraic operation then the fundamental periods of m can be expressed as monomials of the fundamental periods of m 1 m 2 m n applying this theory we discuss two hypothetical motives attached to a siegel modular form we show that a siegel modular form of degree m has at most m 1 period invariants

analysis of the green s function approach to one dimensional inverse problems i one parameter reconstruction an analysis of a green s function approach based on wave splitting to the one dimensional electromagnetic inverse problem is given the green s functions refer to split components of the fundamental solution the linear system of equations for the green s functions is shown to be well posed for the inverse problem and stability estimates theorems on existence uniqueness local correctness and convergence are given

consistency of generalized finite difference schemes for the stochastic hjb equation we analyze a class of numerical schemes for solving the hjb equation for stochastic control problems which enters the framework of markov chain approximations and generalizes the usual finite difference method the latter is known to be monotonic and hence valid only if the scaled covariance matrix is dominant diagonal we generalize this result by given the set of neighboring points allowed to enter the scheme showing how to compute effectively the class of covariance matrices that is consistent with this set of points we perform this computation for several cases in dimensions 2 3 and 4

discontinuous galerkin finite element approximation of nondivergence form elliptic equations with cordes coefficients nondivergence form elliptic equations with discontinuous coefficients do not generally possess a weak formulation thus presenting an obstacle to their numerical solution by classical finite element methods we propose a new hp version discontinuous galerkin finite element method for a class of these problems which satisfy the cordes condition it is shown that the method exhibits a convergence rate that is optimal with respect to the mesh size h and suboptimal with respect to the polynomial degree p by only half an order numerical experiments demonstrate the accuracy of the method and illustrate the potential of exponential convergence under hp refinement for problems with discontinuous coefficients and nonsmooth solutions

motives with exceptional galois groups and the inverse galois problem we construct motivic l adic representations of textup gal overline mathbb q mathbb q into exceptional groups of type e 7 e 8 and g 2 whose image is zariski dense this answers a question of serre the construction is uniform for these groups and is inspired by the langlands correspondence for function fields as an application we solve new cases of the inverse galois problem the finite simple groups e 8 mathbb f ell are galois groups over mathbb q for large enough primes l

emergent spacetime from modular motives the program of constructing spacetime geometry from string theoretic modular forms is extended to calabi yau varieties of dimensions three and four as well as higher rank motives modular forms on the worldsheet can be constructed from the geometry of spacetime by computing the l functions associated to omega motives of calabi yau varieties generated by their holomorphic n forms via galois representations the modular forms that emerge in this way are related to characters of the underlying rational conformal field theory the converse problem of constructing space from string theory proceeds in the class of diagonal theories by determining the motives associated to modular forms in the category of pure motives with complex multiplication the emerging picture suggests that the l function can be viewed as defining a map between the geometric category of motives and the category of conformal field theories on the worldsheet

polarizations and grothendieck s standard conjectures we prove that grothendieck s hodge standard conjecture holds for abelian varieties in arbitrary characteristic if the hodge conjecture holds for complex abelian varieties of cm type for abelian varieties with no exotic algebraic classes we prove the hodge standard conjecture unconditionally

integral motives and special values of zeta functions for each field k we define a category of rationally decomposed mixed motives with z coefficients when k is finite we show that the category is tannakian and we prove formulas relating the behaviour of zeta functions near integers to certain ext groups

on the regularity theory of fully nonlinear parabolic equations ii recently m crandall and p l lions 3 developed a very successful method for proving the existence of solutions of nonlinear second order partial differential equations their method called the theory of viscosity solutions also applies to fully nonlinear equations in which even the second order derivatives can enter in nonlinear fashion solutions produced by the viscosity method are guaranteed to be continuous but not necessarily smooth here we announce smoothness results for viscosity solutions our methods extend those of 1 we obtain krylov safonov i e c estimates 8 c 1 schauder c and w estimates for viscosity solutions of uniformly parabolic equations in general form the results can be viewed as a priori estimates on the classical c solutions our method produces in particular regularity results for a broad new array of nonlinear heat equations including the bellman equation 6

a finite element method for nonlinear elliptic problems we present a galerkin method with piecewise polynomial continuous elements for fully nonlinear elliptic equations a key tool is the discretization proposed in lakkis and pryer 2011 allowing us to work directly on the strong form of a linear pde an added benefit to making use of this discretization method is that a recovered finite element hessian is a byproduct of the solution process we build on the linear method and ultimately construct two different methodologies for the solution of second order fully nonlinear pdes benchmark numerical results illustrate the convergence properties of the scheme for some test problems as well as the monge ampere equation and the pucci equation

analysis of the green s function approach to one dimensional inverse problems ii simultaneous reconstruction of two parameters the wave splitting green s function approach to one dimensional electromagnetic inverse problems of simultaneous reconstructions with different types of scattering data is analyzed theorems of existence uniqueness and stability are given for the following types of scattering data reflection and transmission data for one round trip two sided reflection data for one round trip reflection data for two round trips

motives and admissible representations of automorphism groups of fields some of basic properties of the groups of automorphisms of algebraically closed fields and of their smooth representations are studied in characteristic zero grothendieck motives modulo numerical equivalence are identified with a full subcategory in the category of graded smooth representations of certain automorphism groups of algebraically closed fields

noncommutative geometry and motives the thermodynamics of endomotives we combine aspects of the theory of motives in algebraic geometry with noncommutative geometry and the classification of factors to obtain a cohomological interpretation of the spectral realization of zeros of l functions the analogue in characteristic zero of the action of the frobenius on l adic cohomology is the action of the scaling group on the cyclic homology of the cokernel in a suitable category of motives of a restriction map of noncommutative spaces the latter is obtained through the thermodynamics of the quantum statistical system associated to an endomotive a noncommutative generalization of artin motives semigroups of endomorphisms of algebraic varieties give rise canonically to such endomotives with an action of the absolute galois group the semigroup of endomorphisms of the multiplicative group yields the bost connes system from which one obtains through the above procedure the desired cohomological interpretation of the zeros of the riemann zeta function in the last section we also give a lefschetz formula for the archimedean local l factors of arithmetic varieties

on motivic decompositions arising from the method of bialynicki birula recently v chernousov s gille and a merkurjev have obtained a decomposition of the motive of an isotropic smooth projective homogeneous variety analogous to the bruhat decomposition using the method of a bialynicki birula and a corollary which is essentially due to s del bano i generalize this decomposition to the case of a possibly anisotropic smooth projective variety homogeneous under the action of an isotropic reductive group this answers a question of n karpenko