the use of pde centres in the local rbf hermitian method for 3d convective diffusion problems in this work an extension is proposed to the local hermitian interpolation lhi method a meshless numerical method based on interpolation with small and heavily overlapping radial basis function rbf systems this extension to the lhi method uses interpolation functions which themselves satisfy the partial differential equation pde to be solved in this way a much improved reconstruction of partial derivatives can be obtained resulting in significantly improved accuracy in many cases the implementation algorithm is described and is validated via three convection diffusion reaction problems for steady and transient situations a crank nicolson implicit time stepping technique is used for the time dependent problems in the proposed approach a form of analytical upwinding is implicitly implemented by the use of the partial differential operator of the governing equation in the interpolation function which includes the desired information about the convective velocity field the implicit upwinding scheme intrinsic to the proposed numerical approach is tested by solving a one dimensional travelling front problem at peclet numbers of 500 1000 2000 5000 and infinity which corresponds to a shock front in the case of infinity in addition the accuracy of the numerical scheme is validated against a one dimensional steady state solution exhibiting strong boundary layer effects and also against a steady and a transient three dimensional convection diffusion problem on irregular datasets all the test cases are validated against the corresponding analytical solutions finally the effect of various interpolation stencil configurations is investigated and some important limitations on local data centre distribution are identified

k theory of log schemes ii log syntomic k theory abstract we prove that suitably truncated topological log syntomic etale homotopy k theory of proper semistable schemes in mixed characteristic surjects onto the kummer log etale p adic k theory of the generic fiber as a corollary we get that the log syntomic etale cohomology is a direct factor of the log syntomic etale cohomology of homotopy k theory sheaves the proofs use p adic hodge theory computations of p adic nearby cycles

error estimates in sobolev spaces for moving least square approximations the aim of this paper is to obtain error estimates for moving least square mls approximations in bbb r n we prove that under appropriate hypotheses on the weight function and the distribution of points the method produces optimal order error estimates in l infty and l2 for the approximations of the function and its first derivatives these estimates are important in the analysis of galerkin approximations based on the mls method in particular our results provide error estimates optimal in order and regularity for second order coercive problems

algebraic cycles and higher k theory here g x is the grothendieck group of coherent sheaves on x 13 gr refers to the graded group defined by the y filtration on g x cf kratzer 14 soult 20 and ch x is the chow group of codimension i algebraic cycles defined by fulton 9 the left hand isomorphism is a formal consequence of the existence of a i structure on g x while the existence of r is the central theme of the b f m rr theorem the main purpose of this paper is to define a theory of higher chow groups ch x n n 2 0 so as to obtain isomorphisms

a weak form rbf generated finite difference method abstract in this paper the idea of direct discretization via radial basis functions rbfs is applied on a local petrov galerkin test space of a partial differential equation pde this results to a weak based rbf generated finite difference rbf fd scheme that possesses some useful properties the error and stability issues are considered when the pde solution or the basis function has low smoothness the new method gives more accurate results than the already well established strong based collocation methods although the method uses a galerkin formulation it still remains meshless because not only the approximation process relies on scattered point layouts but also integrations are done over non connected independent and well shaped subdomains some applications to potential and elasticity problems on scattered data points support the theoretical analysis and show the efficiency of the proposed method

on the kernel of the reciprocity map of varieties over local fields this thesis is about higher dimensional class field theory of varieties over local and finite fields their abelianized etale fundamental group is compared with an attached class group via the reciprocity map like in the classical case of a local or global field the main results in this thesis are about the structure of the kernel of their reciprocity maps and criteria for the vanishing of it resp

artin tate motivic sheaves with finite coefficients over an algebraic variety we propose a construction of a tensor exact category fxm of artin tate motivic sheaves with finite coefficients z m over an algebraic variety x over a field k of characteristic prime to m in terms of etale sheaves of z m modules over x among the objects of fxm in addition to the tate motives z m j there are the cohomological relative motives with compact support mccm y x of varieties y quasi finite over x exact functors of inverse image with respect to morphisms of algebraic varieties and direct image with compact supports with respect to quasi finite morphisms of varieties y x act on the exact categories fxm assuming the existence of triangulated categories of motivic sheaves dm x z m over algebraic varieties x over k and a weak version of the six operations in these categories we identify fxm with the exact subcategory in dm x z m consisting of all the iterated extensions of the tate twists mccm y x j of the motives mccm y x an isomorphism of the z m modules ext between the tate motives z m j in the exact category fxm with the motivic cohomology modules predicted by the beilinson lichtenbaum etale descent conjecture recently proved by voevodsky rost et al holds for smooth varieties x over k if and only if the similar isomorphism holds for artin tate motives over fields containing k when k contains a primitive m root of unity the latter condition is equivalent to a certain koszulity hypothesis as shown in our previous paper positselski mixed artin tate motives with finite coefficients mosc math j 11 2011 317 402

on a conjecture of vorst let k be an infinite perfect field of positive characteristic such that strong resolution of singularities holds over k we prove that a localization of a d dimensional commutative k algebra r of finite type is kd 1 regular if and only if it is regular this partially affirms a conjecture of vorst

zeta functions of regular arithmetic schemes at s 0 lichtenbaum conjectured in 22 the existence of a weil etale cohomology in order to describe the vanishing order and the special value of the zeta function of an arithmetic scheme x at s 0 in terms of euler poincare characteristics assuming the conjectured finite generation of some motivic cohomology groups we construct such a cohomology theory for regular schemes proper over spec z in particular we compute uncon ditionally the right weil etale cohomology of number rings and projective spaces over number rings we state a precise version of lichtenbaum s con jecture which expresses the vanishing order resp the special value of the zeta function x s at s 0 as the rank resp the determinant of a single perfect complex of abelian groups r w c x z then we relate this conjecture to soule s conjecture and to the tamagawa number conjecture lichtenbaum s conjecture for projective spaces over the ring of integers of an abelian number field follows

the iwasawa main conjecture for semistable abelian varieties over function fields we prove the iwasawa main conjecture over the arithmetic mathbb z p extension for semistable abelian varieties over function fields of characteristic p 0

h 2 convergence of least squares kernel collocation methods the strong form asymmetric kernel based collocation method commonly referred to as the kansa method is easy to implement and hence is widely used for solving engineering problems and partial differential equations despite the lack of theoretical support the simple least squares ls formulation on the other hand makes the study of its solvability and convergence rather nontrivial in this paper we focus on general second order linear elliptic differential equations in omega subset mathbb r d under dirichlet boundary conditions with kernels that reproduce h m omega and some smoothness assumptions on the solution we provide conditions for a constrained ls method and a class of weighted ls algorithms to be convergent theoretically for max 2 lceil d 1 2 rceil leq nu leq m we identify some h nu omega convergent ls formulations that have an optimal error behavior like h m nu for d leq3 the proposed methods are optimal in h 2 omega we demonstrate the effect

milnor k theory of rings higher chow groups and applications if r is a smooth semi local algebra of geometric type over an infinite field we prove that the milnor k group kmn r surjects onto the higher chow group chn r n for all n 0 our proof shows moreover that there is an algorithmic way to represent any admissible cycle in chn r n modulo equivalence as a linear combination of symbolic elements defined as graphs of units in r as a byproduct we get a new and entirely geometric proof of results of gabber kato and rost related to the gersten resolution for the milnor k sheaf furthermore it is also shown that in the semi local pid case we have under some mild assumptions an isomorphism some applications are also given

error estimates for scattered data interpolation on spheres we study sobolev type estimates for the approximation order resulting from using strictly positive definite kernels to do interpolation on the n sphere the interpolation knots are scattered our approach partly follows the general theory of golomb and weinberger and related estimates these error estimates are then based on series expansions of smooth functions in terms of spherical harmonics the markov inequality for spherical harmonics is essential to our analysis and is used in order to find lower bounds for certain sampling operators on spaces of spherical harmonics

the k theory of fields in characteristic p we show that for a field k of characteristic p h i k n is uniquely p divisible for i n we use higher chow groups as our definition of motivic cohomology this implies that the natural map k n m k k n k from milnor k theory to quillen k theory is an isomorphism up to uniquely p divisible groups and that k n m k and k n k are p torsion free as a consequence one can calculate the k theory mod p of smooth varieties over perfect fields of characteristic p in terms of cohomology of logarithmic de rham witt sheaves for example k n x p r 0 for n dimx another consequence is gersten s conjecture with finite coefficients for smooth varieties over discrete valuation rings with residue characteristic p as the last consequence bloch s cycle complexes localized at p satisfy all beilinson lichtenbaum milne axioms for motivic complexes except possibly the vanishing conjecture

motivic periods and grothendieck arithmetic invariants abstract we construct a period regulator for motivic cohomology of an algebraic scheme over a subfield of the complex numbers for the field of algebraic numbers we formulate a period conjecture for motivic cohomology by saying that this period regulator is surjective showing that a suitable betti de rham realization of 1 motives is fully faithful we can verify this period conjecture in several cases the divisibility properties of motivic cohomology imply that our conjecture is a neat generalization of the classical grothendieck period conjecture for algebraic cycles on smooth and proper schemes these divisibility properties are treated in an appendix by b kahn extending previous work of bloch and colliot thelene raskind

isogeometric residual minimization igrm for non stationary stokes and navier stokes problems abstract we show that it is possible to obtain a linear computational cost fem based solver for non stationary stokes and navier stokes equations our method employs a technique developed by guermond and minev 2011 which consists of singular perturbation plus a splitting scheme while the time integration schemes are implicit we use finite elements to discretize the spatial counterparts at each time step we solve a pde having weak derivatives in one direction only which allows for the linear computational cost at the expense of handling strong second order derivatives of the previous time step solution on the right hand side of these pdes this motivates the use of smooth functions such as b splines for high reynolds numbers some of these pdes become unstable to deal robustly with these instabilities we propose to use a residual minimization technique we test our method on problems having manufactured solutions as well as on the cavity flow problem

convergence of unsymmetric kernel based meshless collocation methods this paper proves convergence of variations of the unsymmetric kernel based collocation method introduced by kansa in 1986 since then this method has been very successfully used in many applications though it may theoretically fail in special situations and though it had no error bound or convergence proof up to now thus it is necessary to add assumptions or to make modifications our modifications prevent numerical failure by dropping strict collocation and allow a rigorous mathematical analysis proving error bounds and convergence rates these rates improve with the smoothness of the solution the domain and the kernel providing the trial spaces but they are currently not yet optimal and deserve refinement they are based on rates of approximation to the residuals by nonstationary meshless kernel based trial spaces and they are independent of the type of differential operator the results are applicable to large classes of linear problems in strong form provided that there is a smooth solution and the test and trial discretizations are chosen with some care our analysis does not require assumptions like ellipticity and it can be extended to ill posed problems

the approximation power of moving least squares a general method for near best approximations to functionals on r d using scattered data information is discussed the method is actually the moving least squares method presented by the backus gilbert approach it is shown that the method works very well for interpolation smoothing and derivatives approximations for the interpolation problem this approach gives mclain s method the method is near best in the sense that the local error is bounded in terms of the error of a local best polynomial approximation the interpolation approximation in r d is shown to be a c function and an approximation order result is proven for quasi uniform sets of data points

on unsymmetric collocation by radial basis functions solving partial differential equations by collocation with radial basis functions can be efficiently done by a technique first proposed by kansa in 1990 it rewrites the problem as a generalized interpolation problem and the solution is obtained by solving a possibly large linear system the method has been used successfully in a variety of applications but a proof of nonsingularity of the linear system was still missing this paper shows that a general proof of this fact is impossible however numerical evidence shows that cases of singularity are rare and have to be constructed with quite some effort

motivic cohomology over dedekind rings we study properties of bloch s higher chow groups on smooth varieties over dedekind rings we prove the vanishing of open image in new window for i n and the existence of a gersten resolution for open image in new window if the residue characteristic is p we also show that the bloch kato conjecture implies the beilinson lichtenbaum conjecture open image in new window an identification open image in new window for m invertible and a gersten resolution with arbitrary finite coefficients over a complete discrete valuation ring of mixed characteristic 0 p we construct a map from motivic cohomology to syntomic cohomology which is a quasi isomorphism provided the bloch kato conjecture holds

hasse principles for higher dimensional fields for rather general excellent schemes x k kato deflned complexes of gerstenbloch ogus type involving the galois cohomology groups of all residue flelds of x for arithmetically interesting schemes i e schemes over global or local flelds or over their rings of integers he developed a web of conjectures on some of these complexes generalizing the fundamental exact sequence of brauer groups for a global fleld he proved these conjectures for low dimensions we prove kato s conjecture with inflnite coe cients over number flelds in particular it gives a hasse principle for function flelds f over a number fleld k involving the corresponding function flelds fv over the completions kv of k we get a conditional result over global flelds k of positive characteristic assuming resolution of singularities this is unconditional for dim x 3 due to recent results on resolution there are also applications to other cases considered by kato

approximate interpolation with applications to selecting smoothing parameters in this paper we study the global behavior of a function that is known to be small at a given discrete data set such a function might be interpreted as the error function between an unknown function and a given approximant we will show that a small error on the discrete data set leads under mild assumptions automatically to a small error on a larger region we will apply these results to spline smoothing and show that a specific a priori choice of the smoothing parameter is possible and leads to the same approximation order as the classical interpolant this has also a surprising application in stabilizing the interpolation process by splines and positive definite kernels

on some invariants of fields of characteristic p 0 abstract let k be a field of characteristic p 0 k n the k vector space of n differential forms and k n k n d k n 1 the artin schreier operator given by x d x 1 x 1 d x n x n x p x d x 1 x 1 d x n x n let h p n 1 k coker and define p k min m h p m 1 k 0 we show that for any finite extension l k p k p l p k 1

on the equivariant tamagawa number conjecture for tate motives let l be a finite abelian extension of and let k be any subfield of l for each integer r with r 0 we prove the equivariant tamagawa number conjecture for the pair h0 spec l r gal l k

unsymmetric meshless methods for operator equations a general framework for proving error bounds and convergence of a large class of unsymmetric meshless numerical methods for solving well posed linear operator equations is presented the results provide optimal convergence rates if the test and trial spaces satisfy a stability condition operators need not be elliptic and the problems can be posed in weak or strong form without changing the theory non stationary kernel based trial and test spaces are shown to fit into the framework disregarding the operator equation as a special case unsymmetric meshless kernel based methods solving weakly posed problems with distributional data are treated in some detail this provides a foundation of certain variations of the meshless local petrov galerkin technique of s n atluri and collaborators

preconditioning landweber iteration in hilbert scales in this paper we investigate convergence of landweber iteration in hilbert scales for linear and nonlinear inverse problems as opposed to the usual application of hilbert scales in the framework of regularization methods we focus here on the case s 0 which for tikhonov regularization corresponds to regularization in a weaker norm in this case the hilbert scale operator l 2 s appearing in the iteration acts as a preconditioner which significantly reduces the number of iterations needed to match an appropriate stopping criterion additionally we carry out our analysis under significantly relaxed conditions i e we only require inlinemediaobject not available see fulltext instead of inlinemediaobject not available see fulltext which is the usual condition for regularization in hilbert scales the assumptions needed for our analysis are verified for several examples and numerical results are presented illustrating the theoretical ones

rost nilpotence and higher unramified cohomology abstract we develop an approach to proving the rost nilpotence principle involving higher unramified cohomology along with a decomposition of the diagonal technique we use this to prove the principle for certain threefolds over a perfect field

theoremes de dualite pour les corps de fonctions sur des corps locaux superieurs let k be the function field of a smooth projective curve x over a higher dimensional local field k we define tate shafarevich groups of a commutative group scheme via cohomology classes locally trivial at each completion of k coming from a closed point of x we establish duality theorems between tate shafarevich groups for finite groups schemes and for tori

gersten s conjecture for thek theory with torsion coefficients of a discrete valuation ring between quillen k groups induced by the obvious inclusion functors are all zero in 6 quillen proved this conjecture if r is essentially of finite type over a field in the case of discrete valuation rings this conjecture reduces to conjecture let r be a discrete valuation ring with residue field k and field of fractions f then for all q z 0 the transfer map which is defined since every k vector space is of finite projective dimension as an r module k k k r is zero equivalently the map r ec f induced by the ring homomorphism r p f is injective a third equivalent formulation is that the boundary map

discrete least squares radial basis functions approximations abstract we consider discrete least squares methods using radial basis functions a general l2 tikhonov regularization with w 2 m penalty is considered we provide error estimates that are comparable to kernel based interpolation in cases which the function it is approximating is within and is outside of the native space of the kernel our proven theories concern the denseness condition of collocation points and selection of regularization parameters in particular the unregularized least squares method is shown to have w 2 convergence for d 2 on smooth domain r d for any properly regularized least squares method the same convergence estimates hold for a large range of 0 these results are extended to the case of noisy data numerical demonstrations are provided to verify the theoretical results in terms of applications we also apply the proposed method to solve a heat equation whose initial condition has huge oscillation in the domain

on motivic cohomology with z l coefficients in this paper we prove the conjecture of bloch and kato which relates milnor s k theory of a field with its galois cohomology as well as the related comparisons results for motivic cohomology with finite coefficients in the nisnevich and etale topologies

a unified stability analysis of meshless particle methods a uni ed stability analysis of meshless methods with eulerian and lagrangian kernels is presented three types of instabilities were identi ed in one dimension an instability due to rank de ciency a tensile instability and a material instability which is also found in continua the stability of particle methods with eulerian and lagrangian kernels is markedly di erent lagrangian kernels do not exhibit the tensile instability in both kernels the instability due to rank de ciency can be suppressed by stress points in two dimensions the stabilizing e ect of stress points is dependent on their locations it was found that the best approach to stable particle discretizations is to use lagrangian kernels with stress points the stability of the least squares stabilization was also studied copyright 2000 john wiley sons ltd

uber die lokale zetafunktion von shimuravarietaten monodromiefiltration und verschwindende zyklen in ungleicher charakteristik

murthy s conjecture on 0 cycles we show that the levine weibel chow group of 0 cycles mathrm ch d a of a reduced affine algebra a of dimension d ge 2 over an algebraically closed field is torsion free among several applications it implies an affirmative solution to an old conjecture of murthy in classical k theory

logarithmic hodge witt sheaves on normal crossing varieties in this paper we define two kinds homological and cohomological of etale logarithmic hodge witt sheaves on normal crossing varieties over a perfect field of positive characteristic and discuss some fundamental properties in particular puity and duality

p adic deformation of algebraic cycle classes we study the p adic deformation properties of algebraic cycle classes modulo rational equivalence we show that the crystalline chern character of a vector bundle on the closed fibre lies in a certain part of the hodge filtration if and only if rationally the class of the vector bundle lifts to a formal pro class in k theory on the p adic scheme

realisation l adique des motifs triangules geometriques ii for geometrical triangulated motives with rational coefficients over a ground field of characteristic zero which is embeddable into mathbb c huber j algebraic geom 9 4 755 799 2000 j algebraic geom 13 1 195 207 2004 has constructed a realization functor with values in the category of mixed realization of huber mixed motives and their realization in derived categories lecture notes in mathematics vol 1604 springer berlin 1995 in this sequel to ivorra doc math 12 607 671 2007 we prove that the l adic realization functor obtained in theorem 4 3 of ivorra doc math 12 607 671 2007 is the same up to a canonical isomorphism as the l adic component of a huber s construction in this way ivorra in doc math 12 607 671 2007 might be viewed as an integral generalization to all noetherian separed schemes of the work huber in j algebraic geom 9 4 755 799 2000 j algebraic geom 13 1 195 207 2004 as far as the l adic setting is concerned we also prove a comparison theorem with the classical l adic cycle class map over a perfect field using a naive motivic cycle class map

the order of higher brauer groups let x be a smooth projective variety of dimension 2r over a finite field mathbb f we prove that for every prime ell ne mathrm char mathbb f the order of the non divisible quotient of the ell primary torsion group of the higher brauer group mathrm br r x ell mathrm nd is a square number

factorization of p adic rankin l series we prove that the p adic l series of the tensor square of a p ordinary modular form factors as the product of the symmetric square p adic l series of the form and a kubota leopoldt p adic l series this establishes a generalization of a conjecture of citro greenberg s exceptional zero conjecture for the adjoint follows as a corollary of our theorem our method of proof follows that of gross who proved a factorization result for the katz p adic l series associated to the restriction of a dirichlet character whereas gross s method is based on comparing circular units with elliptic units our method is based on comparing these same circular units with a new family of units called beilinson flach units that we construct the beilinson flach units are constructed using bloch s intersection theory of higher chow groups applied to products of modular curves we relate these units to special values of classical and p adic l functions using work of beilinson as generalized by lei loeffler zerbes in the archimedean case and bertolini darmon rotger as generalized by kings loeffler zerbes in the p adic case central to our method are two compatibility theorems regarding bloch s intersection pairing and the classical and p adic beilinson regulators defined on higher chow groups

rost nilpotence and etale motivic cohomology abstract a smooth projective scheme x over a field k is said to satisfy the rost nilpotence principle if any endomorphism of x in the category of chow motives that vanishes on an extension of the base field k is nilpotent we show that an etale motivic analogue of the rost nilpotence principle holds for all smooth projective schemes over a perfect field this provides a new approach to the question of rost nilpotence and allows us to obtain an elegant proof of rost nilpotence for surfaces as well as for birationally ruled threefolds over a field of characteristic 0

error bounds for kernel based numerical differentiation the literature on meshless methods shows that kernel based numerical differentiation formulae are robust and provide high accuracy at low cost this paper analyzes the error of such formulas using the new technique of growth functions it allows to bypass certain technical assumptions that were needed to prove the standard error bounds on interpolants and their derivatives since differentiation formulas based on polynomials also have error bounds in terms of growth functions we have a convenient way to compare kernel based and polynomial based formulas it follows that kernel based formulas are comparable in accuracy to the best possible polynomial based formulas a variety of examples is provided

the geisser levine method revisited and algebraic cycles over a finite field we reformulate part of the arguments of t geisser and m levine relating motivic cohomology with finite coefficients to truncated etale cohomology with finite coefficients 9 10 this reformulation amounts to a uniqueness theorem for motivic cohomology and shows that the geisser levine method can be applied generally to compare motivic cohomology with other types of cohomology theories we apply this to prove an equivalence between conjectures of tate and beilinson on cycles in characteristic p and a vanishing conjecture for continuous etale cohomology

duality via cycle complexes we show that bloch s complex of relative zero cycles can be used as a dualizing complex over perfect fields and number rings this leads to duality theorems for torsion sheaves on arbitrary separated schemes of finite type over algebraically closed fields finite fields local fields of mixed characteristic and rings of integers in number rings generalizing results which so far have only been known for smooth schemes or in low dimensions and unifying the p adic and l adic theory as an application we generalize rojtman s theorem to normal projective schemes

a least squares radial basis function partition of unity method for solving pdes recently collocation based radial basis function rbf partition of unity methods pums for solving partial differential equations have been formulated and investigated numerically and theoretically when combined with stable evaluation methods such as the rbf qr method high order convergence rates can be achieved and sustained under refinement however some numerical issues remain the method is sensitive to the node layout and condition numbers increase with the refinement level here we propose a modified formulation based on least squares approximation we show that the sensitivity to node layout is removed and that conditioning can be controlled through oversampling we derive theoretical error estimates both for the collocation and least squares rbf pums numerical experiments are performed for the poisson equation in two and three space dimensions for regular and irregular geometries the convergence experiments confirm the theoretical estimates and the least squares formulation is shown to

a nonlinear discretization theory abstract this paper extends for the first time schaback s linear discretization theory to nonlinear operator equations relying heavily on the methods in bohmer s 2010 book there is no restriction to elliptic problems or to symmetric numerical methods like galerkin techniques trial spaces can be arbitrary including spectral and meshless methods but have to approximate the solution well and testing can be weak or strong on the downside stability is not easy to prove for special applications and numerical methods have to be formulated as optimization problems results of this discretization theory cover error bounds and convergence rates some numerical examples are added for illustration

an approximation theorist s view on solving operator equations with special attention to trefftz mfs mps and drm methods abstract when an approximation theorist looks at well posed pde problems or operator equations and standard solution algorithms like finite elements rayleigh ritz or trefftz techniques methods of fundamental or particular solutions and their combinations they boil down to approximation problems and stability issues these two can be handled by approximation theory and this paper shows how with special applications to the aforementioned algorithms the intention is that the approximation theorist s viewpoint is helpful for readers who are somewhat away from that subject

the bloch kato conjecture and a theorem of suslin voevodsky we give a new proof of the theorem of suslin voevodsky which shows that the bloch kato conjecture implies a portion of the beilinsonlichtenbaum conjectures our proof does not rely on resolution of singularities and thereby extends the suslin voevodsky theorem to positive characteristic

addendum to values of zeta functions of varieties over finite fields the original article it amer j math bf 108 1986 no 2 297 360 expressed the special values of the zeta function of a variety over a finite field in terms of the widehat bbb z cohomology of the variety as the article was being completed september 1983 lichtenbaum conjectured the existence of complexes of sheaves bbb z r extending the sequence bbb z bbb g m 1 ldots the complexes given by bloch s higher chow groups are known to satisfy most of the axioms for bbb z r using lichtenbaum s weil e tale topology we can now give a beautiful restatement of the main theorem of the original article in terms of bbb z cohomology groups

decidability and complexity analysis by basic paramodulation it is shown that for sets of horn clauses saturated underbasic paramodulationthe word and unifiability problems are in np and the number of minimal unifiers is simply exponential i for horn sets saturated w r t a special ordering under the more restrictive inference rule ofbasic superposition the word and unifiability problems are still decidable and unification is finitary ii these two results are applied to the following languages forshallowpresentations equations with variables at depth at most one we show that the closure under paramodulation can be computed in polynomial time applying result i it follows that shallow unifiability is in np which is optimal since unifiability in ground theories is already np hard the shallow word problem is even shown to be polynomial generalizing shallow theories to the horn case we obtain two versions of a language we callcatalog a natural extension of datalog to include functions and equality the closure under paramodulation is finite for catalog sets hence i still applies for catalog sets s the decidability of the full first order theory of t f sis shown as well finally we definestandard theories which include and significantly extendshallowtheories standard presentations can be finitely closed under superposition and result ii applies thus obtaining a new fundamental class with decidable word and unifiability problems and where unification is finitary

comparison of dualizing complexes we prove that there is a map from bloch s cycle complex to kato s complex of milnor k theory which induces a quasi isomorphism from etale sheafified cycle complex to the gersten complex of logarithmic de rham witt sheaves next we show that the truncation of bloch s cycle complex at 3 is quasi isomorphic to spiess dualizing complex

error analysis of the reproducing kernel particle method interest in meshfree or meshless methods has grown rapidly in recent years in solving boundary value problems bvps arising in mechanics especially in dealing with difficult problems involving large deformation moving discontinuities etc in this paper we provide a theoretical analysis of the reproducing kernel particle method rkpm which belongs to the family of meshfree methods one goal of the paper is to set up a framework for error estimates of rkpm we introduce the concept of a regular family of particle distributions and derive optimal order error estimates for rkp interpolants on a regular family of particle distributions the interpolation error estimates can be used to yield error estimates for rkp solutions of bvps

extrinsic meshless collocation methods for pdes on manifolds we proposed ways to implement meshless collocation methods extrinsically for solving elliptic pdes on smooth closed connected and complete riemannian manifolds with arbitrary codimensions our m

all well posed problems have uniformly stable and convergent discretizations this paper considers a large class of linear operator equations including linear boundary value problems for partial differential equations and treats them as linear recovery problems for functions from their data well posedness of the problem means that this recovery is continuous discretization recovers restricted trial functions from restricted test data and it is well posed or stable if this restricted recovery is continuous after defining a general framework for these notions this paper proves that all well posed linear problems have stable and refinable computational discretizations with a stability that is determined by the well posedness of the problem and independent of the computational discretization provided that sufficiently many test data are used the solutions of discretized problems converge when enlarging the trial spaces and the convergence rate is determined by how well the data of the function solving the analytic problem can be approximated by the data of the trial functions this allows new and very simple proofs of convergence rates for generalized finite elements symmetric and unsymmetric kansa type collocation and other meshfree methods like meshless local petrov galerkin techniques it is also shown that for a fixed trial space weak formulations have a slightly better convergence rate than strong formulations but at the expense of numerical integration since convergence rates are reduced to those coming from approximation theory and since trial spaces are arbitrary this also covers various spectral and pseudospectral methods all of this is illustrated by examples

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

weil etale cohomology over finite fields we calculate the derived functors r for the base change from the weil etale site to the etale site for a variety over a finite field for smooth and proper varieties we apply this to express tate s conjecture and lichtenbaum s conjecture on special values of functions in terms of weil etale cohomology of the motivic complex n

lower bounds for the minima of certain symplectic and unitary group lattices we obtain lower bounds for the minimal norms in several families of integral lattices associated to the weil representations of finite symplectic and unitary groups ultimately these bounds come from elkies bounds for a certain family of mordell weil lattices of elliptic curves and from the minima of the barnes wall lattices

doubly stochastic radial basis function methods abstract we propose a doubly stochastic radial basis function dsrbf method for function recoveries instead of a constant we treat the rbf shape parameters as stochastic variables whose distribution were determined by a stochastic leave one out cross validation loocv estimation a careful operation count is provided in order to determine the ranges of all the parameters in our methods the overhead cost for setting up the proposed dsrbf method is o n 2 for function recovery problems with n basis numerical experiments confirm that the proposed method not only outperforms constant shape parameter formulation in terms of accuracy with comparable computational cost but also the optimal loocv formulation in terms of both accuracy and computational cost

etale duality for constructible sheaves on arithmetic schemes in this note we relate the following three topics for arithmetic schemes a general duality for etale constructible torsion sheaves a theory of etale homology and the arithmetic complexes of gersten bloch ogus type defined by k kato kct in brief there is an absolute duality using certain dualizing sheaves on these schemes we describe and characterize the dualizing sheaves to some extent relate them to symbol maps define etale homology via the dualizing sheaves and show that the niveau spectral sequence for the latter constructed by the method of bloch and ogus bo leads to the complexes defined by kato some of these relations may have been expected by experts and some have been used implicitly in the literature although we do not know any explicit reference for statements or proofs moreover the main results are used in a crucial way in a paper by two of us js so a major aim is to fill a gap in the literature and a special emphasis is on precise formulations including the determination of signs but the general picture developed here may be of interest itself

improved error bounds for scattered data interpolation by radial basis functions if additional smoothness requirements and boundary conditions are met the well known approximation orders of scattered data interpolants by radial functions can roughly be doubled

the de rham witt and z p cohomologies of an algebraic variety abstract we prove that for a smooth complete variety x over a perfect field h i x z p r hom d c b r 1 r w x r i where h i x z p r lim n h i r x et n r amer j math 108 2 1986 297 360 w x is the de rham witt complex on x ann scient ec num sup 12 1979b 501 661 and d c b r is the triangulated category of coherent complexes over the raynaud ring inst hautes etuder sci publ math 57 1983 73 212

convergence analysis of general spectral methods abstract if a spectral numerical method for solving ordinary or partial differential equations is written as a biinfinite linear system b z a with a map z l 2 l 2 that has a continuous inverse this paper shows that one can discretize the biinfinite system in such a way that the resulting finite linear system b z a is uniquely solvable and is unconditionally stable i e the stability can be made to depend on z only not on the discretization convergence rates of finite approximations b of b then carry over to convergence rates of finite approximations a of a spectral convergence is a special case some examples are added for illustration

syntacticness cycle syntacticness and shallow theories abstract solving equations in the free algebra t f x i e unification uses the two rules s t s t decomposition and s x x occur check these two rules are not correct in quotients of t f x by a finitely generated congruence e following c kirchner we first define classes of equational theories called syntactic and cycle syntactic respectively for which it is possible to derive some rules replacing the two above then we show that these abstract classes are relevant all shallow theories i e theories which can be generated by equations in which variables occur at depth at most one are both syntactic and cycle syntactic moreover the new set of unification rules is terminating which proves that unification is decidable and finitary in shallow theories we give still further extensions if the set of equivalence classes is infinite a problem which turns out to be decidable in shallow theories then shallow theories fulfill colmerauer s independence of disequations principle a conjunction of n disequations is solvable iff each disequation alone is solvable this allows us to derive quantifier elimination rules it turns out that these rules do terminate for shallow theories hence the first order theory of the quotient algebra t f e is decidable when f is finite and e is shallow this extends mal cev results on the classes of permutative locally free algebras that are completely axiomatizable

permutative rewriting and unification permutative rewriting provides a way of analyzing deduction modulo a theory defined by leaf permutative equations our analysis naturally leads to the definition of the class of unify stable axiom sets in order to enforce a simple reduction strategy we then give a uniform unification algorithm modulo theories e axiomatized this way we prove that it computes complete sets of unifiers of simply exponential cardinality and that the e unification decision problem belongs to np