dwork s conjecture on unit root zeta functions annals of mathematics 150 1999 867 927 arxiv math 9911270v1 math nt 1 nov 1999 dwork s conjecture on unit root zeta functions by daqing wan 1 introduction in this article we introduce a systematic new method to investigate the conjectural p adic meromorphic continuation of professor bernard dwork s unit root zeta function attached to an ordinary family of algebraic varieties defined over a finite field of characteristic p after his pioneer p adic investigation of the weil conjectures on the zeta function of an algebraic variety over a finite field dwork went on to study the p adic analytic variation of a family of such zeta functions when the variety moves through an algebraic family in the course of doing so he was led to a new zeta function called the unit root zeta function which goes beyond the reach of the existing theory he conjectured 8 that such a unit root zeta function is p adic meromorphic everywhere these unit root zeta functions contain important arithmetic information about a family of algebraic varieties they are truly p adic in nature and are transcendental functions sometimes seeming quite mysterious in fact no single nontrivial example has been proved to be true about this conjecture other than the trivial overconvergent or log convergent case for which dwork s classical p adic theory already applies see 6 11 26 for various attempts in this article we introduce a systematic new method to study such unit root zeta functions our method can be used to prove the conjecture in the case when the involved unit root f crystal has rank one in particular this settles the first nontrivial case the rank one unit root f crystal coming from the family of higher dimensional kloosterman sums our method further allows us to understand reasonably well about analytic variation of an arithmetic family of such rank one unit root zeta functions motivated by the gouv ea mazur conjecture about dimension variation of classical and p adic modular forms we shall introduce another this work was partially supported by nsf the author wishes to thank p deligne and n katz for valuable discussions at one stage of this work during 1993 and 1994 the author would also like to thank s sperber for his careful reading of the manuscript and for his many detailed comments which led to a significant improvement of the exposition

adaptive grid generation from harmonic maps on reimannian manifolds abstract in this paper we describe a new method for generation of solution adaptive grids based on harmonic maps on riemannian manifolds the reliability of the method is assured by an existence and uniqueness theorem for one to one maps between multidimensional multiconnected domains we formulate an adaptive riemannian metric consistent with this theorem several examples demonstrating application of the developed procedure are provided

the completion of fuzzy metric spaces sherwood z showed that every menger space with continuous t norm has a completion which is unique up to isometry since fuzzy metric spaces resemble in some respects probabilistic metric spaces it is to be expected that at least some fuzzy metric spaces have a completion the purpose of this paper is to prove that we need the following definitions for the definition and properties of a fuzzy metric space see also 11

some new sequence spaces of fuzzy numbers defined by orlicz functions using a fuzzy metric using orlicz functions and a fuzzy metric some new sequence spaces are introduced in this paper using the concept that the distance between two points in a fuzzy metric space is a non negative upper semicontinuous normal and convex fuzzy number we study some basic properties of the new sequence spaces of fuzzy numbers also we investigate the relations between these spaces

higher rank case of dwork s conjecture journal of the american mathematical society volume 00 number 0 xxxx xxxx pages 000 000 s 0894 0347 xx 0000 0 arxiv math 0005309v1 math nt 9 may 2000 higher rank case of dwork s conjecture daqing wan dedicated to the memory of bernard dwork 1 introduction in this series of two papers we prove the p adic meromorphic continuation of the pure slope l functions arising from the slope decomposition of an overconvergent f crystal as conjectured by dwork 6 more precisely we prove a suitable extension of dwork s conjecture in our more general setting of modules see section 2 for precise definitions of the various notions used in this introduction our main result is the following theorem theorem 1 1 let x be a smooth affine variety defined over a finite field f q of characteristic p 0 let m be a finite rank overconvergent module over x f q then for each rational number s the pure slope s l function l s t attached to is p adic meromorphic everywhere the proof of this theorem will be completed in two papers in the present higher rank paper we introduce a reduction approach which reduces theorem 1 1 to the special case when the slope s s 0 part of has rank one and the base space x is the simplest affine space a n this part is essentially algebraic it depends on monsky s trace formula grothendieck s specialization theorem the hodge newton decomposition and katz s isogeny theorem in our next paper 23 we will handle the rank one case over the affine space a n the rank one case is very much analytic in nature and forces us to work in a more difficult infinite rank setting generalizing and improving the limiting approach introduced in 19 dwork s conjecture grew out of his attempt to understand the p adic analytic variation of the pure pieces of the zeta function of a variety when the variety moves through an algebraic family to give an important geometric example let us con sider the case that f y x is a smooth and proper morphism over f q with a smooth and proper lifting to characteristic zero berthelot s result 1 says that the relative crystalline cohomology r i f crys z p modulo torsion is an overconvergent f crystal m i over x applying theorem 1 1 we conclude that the pure l functions arising from these geometric overconvergent f crystals m i are p adic meromorphic in particular this implies the existence of an exact p adic formula for geometric p adic character sums and a suitable p adic equi distribution theorem for the roots of zeta functions for more detailed arithmetic motivations and further open prob lems see the expository papers 20 21 1991 mathematics subject classification primary 11g40 11s40 secondary 11m41 14g15 key words and phrases l functions p adic meromorphic continuation modules this work was partially supported by nsf c american mathematical society

absolute irreducibility of polynomials via newton polytopes abstract a multivariable polynomial is associated with a polytope called its newton polytope a polynomial is absolutely irreducible if its newton polytope is indecomposable in the sense of minkowski sum of polytopes two general constructions of indecomposable polytopes are given and they give many simple irreducibility criteria including the well known eisenstein criterion polynomials from these criteria are over any field and have the property of remaining absolutely irreducible when their coefficients are modified arbitrarily in the field but keeping a certain collection of them nonzero

the l functions of witt coverings results on l functions of artin schreier coverings by dwork bombieri and adolphson sperber are generalized to l functions of witt coverings

slopes for higher rank artin schreier witt towers author s ren rufei wan daqing xiao liang yu myungjun abstract we fix a monic polynomial bar f x in mathbb f q x over a finite field of characteristic p and consider the mathbb z p ell artin schreier witt tower defined by bar f x this is a tower of curves cdots to c m to c m 1 to cdots to c 0 mathbb a 1 whose galois group is canonically isomorphic to mathbb z p ell the degree ell unramified extension of mathbb z p which is abstractly isomorphic to mathbb z p ell as a topological group we study the newton slopes of zeta functions of this tower of curves this reduces to the study of the newton slopes of l functions associated to characters of the galois group of this tower we prove that when the conductor of the character is large enough the newton slopes of the l function asymptotically form a finite union of arithmetic progressions as a corollary we prove the spectral halo property of the spectral variety associated to the mathbb z p ell artin schreier witt tower this extends the main result in dwx from rank one case ell 1 to the higher rank case ell geq 1

on a conjecture of igusa ii in his 1978 tata lecture notes igusa conjectured the validity of a strong uniformity in the decay of complete exponential sums modulo powers of a prime number and determined by a homogeneous polynomial we prove this property for absolutely reduced ternary forms

pure anderson motives and abelian τ sheaves pure t motives were introduced by g anderson as higher dimensional generalizations of drinfeld modules and as the appropriate analogs of abelian varieties in the arithmetic of function fields in order to construct moduli spaces for pure t motives the second author has previously introduced the concept of abelian sheaf in this article we clarify the relation between pure t motives and abelian sheaves we obtain an equivalence of the respective quasi isogeny categories furthermore we develop the elementary theory of both structures regarding morphisms isogenies tate modules and local shtukas the later are the analogs of p divisible groups

a two fluid spectral element method abstract a spectral element formulation for two fluid flows is presented which uses arbitrary lagrangian eulerian mesh movement with an unstructured mesh the main advantage of this approach is that the spectral elements provide a high order of accuracy while the unstructured mesh allows geometric flexibility the accuracy of this approach is demonstrated for several relatively simple configurations with and without moving interfaces another unique feature of the formulation as compared to previous single fluid spectral element methods is that a fully implicit time advancement scheme is used this is made possible by the development of a spectral multigrid iteration which solves the implicit governing equations in a fixed number of iterations insensitive to both the polynomial degree of the spectral elements and the number of elements in the mesh by using this approach solutions for both stiff unsteady problems and steady state problems can be obtained efficiently

clustering by a fuzzy metric in this paper we consider the clustering problem in which a distance between any two elements is given by a fuzzy metric our clustering criterion is to minimize the total of the average within cluster sum of the fuzzy metrics and an optimal clustering is concretely sought by applying an approach by cuts of fuzzy numbers also a numerical example is given to illustrate our theoretical results

devising discontinuous galerkin methods for non linear hyperbolic conversation laws abstract in this paper we give a simple introduction to the devising of discontinuous galerkin dg methods for nonlinear hyperbolic conservation laws these methods have recently made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications the dg methods which are extensions of finite volume methods incorporate into a finite element framework the notions of approximate riemann solvers numerical fluxes and slope limiters coined during the remarkable development of the high resolution finite difference and finite volume methods for nonlinear hyperbolic conservation laws we start by stressing the fact that nonlinear hyperbolic conservation laws are usually obtained from well posed problems by neglecting terms modeling nondominant features of the model which nevertheless are essential in crucial small parts of the domain as a consequence the resulting problem becomes ill posed the main difficulty in devising numerical schemes for these conservation laws is thus how to re introduce the neglected physical information in order to approximate the physically relevant solution usually called the entropy solution for the classical case of the entropy solution of the nonlinear hyperbolic scalar conservation law we show how to carry out this process for two prototypical dg methods the first dg method is the so called shock capturing dg method which does not use slope limiters and is implicit the second is the runge kutta dg method which is an explicit method that does not employ a shock capturing term but uses a slope limiter instead we then focus on the runge kutta dg methods and show how to obtain a key stability property which holds independently of the accuracy of the scheme and of the nonlinearity of the conservation law we also show some computational results

exponential sums and newton polyhedra cohomology and estimates the basic objects of this study are exponential sums on a variety defined over a finite field fq q pa p char fq as we have remarked in some earlier articles 1 2 we find it more natural to begin with exponential sums on the torus gm n extend via the usual toric decomposition of an to exponential sums on affine n space and finally proceed via a standard character argument 4 to exponential sums on an affine variety defined over fq while this is the natural order of the work what we do in fact in the first part of this article is to combine the first two steps and deal with exponential sums on varieties v of the form v gm r x as r s n let f e fq xl xn xl xr be an arbitrary regular function on v then f is a sum of monomials and as such has a well defined newton polyhedron a ff at infinity this is the convex closure in rn of the lattice points which occur as exponents of the terms of f together with the origin we have indicated in our previous work 1 2 the description of some of the invariants of the associated l function in terms of properties of this polyhedron for example in 1 we showed how bounds for the degree and total degree of the l function associated with a general exponential sum on v can be expressed in terms of the volumes of a f and the intersections of a f and the various coordinate spaces in the present article assuming f is nondegenerate and commode with respect to af f we show these estimates are sharp our methods are p adic and are based on the work of dwork 11 12 our main accomplishment from which our other results follow is the extension of dwork s cohomology theory from smooth projective hypersurfaces in characteristic p to a general class of exponential sums given f regular on v we construct a complex of p adic banach spaces on which frobenius acts the alternating product of characteristic polynomials of frobenius describes the associated l function in fact when f is nondegenerate and commode with respect to a f the complex is acyclic in dimensions other than 0 and the characteristic

numerical solution of the quasilinear poisson equation in a nonuniform triangle mesh a finite difference method using a nonuniform triangle mesh is described for the numerical solution of the nonlinear two dimensional poisson equation s 0 where is a function of or its derivatives sis a function of position and or its normal derivative is specified on the boundary the finite difference equations are solved by successive overrelaxation the triangle mesh which is constructed numerically by solving laplace s equation is easily adapted to nonrectangular boundaries and interfaces examples of numerical results are given for the magnetostatic problem with iron and other possible applications are mentioned

meromorphic continuation of l functions of p adic representations x det i td x b xqd x l b xq b x where x is the teichmiiller lifting of zt and d x denotes the degree of x over fq this infinite product is a well defined power series with coefficients in rp because there are only finitely many closed points of a given degree if v fq is an affine variety contained in an then one defines l b v t to be the product in equation 1 1 except that we now restrict x to run only over the closed points of v fq using a standard reduction argument and dwork s

sequences in a fuzzy metric space in this paper we introduce the concept of bounded sequences in a fuzzy metric space x d min max and show that every convergent sequence in x is bounded we also discuss the space of convergent sequences in a complete fuzzy metric space x d min max

discretization of three dimensional free surface flows and moving boundary problems via elliptic grid methods based on variational principles abstract a new boundary fitted technique to describe free surface and moving boundary problems is presented we have extended the 2d elliptic grid generator developed by dimakopoulos and tsamopoulos 2003 19 and further advanced by chatzidai et al 2009 18 to 3d geometries the set of equations arises from the fulfillment of the variational principles established by brackbill and saltzman 1982 21 and refined by christodoulou and scriven 1992 22 these account for both smoothness and orthogonality of the grid lines of tessellated physical domains the elliptic grid equations are accompanied by new boundary constraints and conditions which are based either on the equidistribution of the nodes on boundary surfaces or on the existing 2d quasi elliptic grid methodologies the capabilities of the proposed algorithm are first demonstrated in tests with analytically described complex surfaces the sequence in which these tests are presented is chosen to help the reader build up experience on the best choice of the elliptic grid parameters subsequently the mesh equations are coupled with the navier stokes equations in order to reveal the full potential of the proposed methodology in free surface flows more specifically the problem of gas assisted injection in ducts of circular and square cross sections is examined where the fluid domain experiences extreme deformations finally the flow mesh solver is used to calculate the equilibrium shapes of static menisci in capillary tubes

on fuzzy metric spaces abstract in this paper we introduce the concept of a fuzzy metric space the distance between two points in a fuzzy metric space is a non negative upper semicontinuous normal and convex fuzzy number properties of fuzzy metric spaces are studied and some fixed point theorems are proved

exponential sums on ⁿ ii we prove a vanishing theorem for the p adic cohomology of exponential sums on a n in particular we obtain new classes of exponential sums on an that have a single nonvanishing p adic cohomology group the dimension of this cohomology group equals a sum of milnor numbers

variation of mixed hodge structure i around 1970 griffiths introduced the notion of a variation of hodge structure on a complex manifold s see 17 w 2 it constitutes the axiomatization of the features possessed by the local systems of cohomology associated to a family of compact k ihler manifolds esp smooth projective varieties parametrized by s these are for a variation of hodge structure u of weight m 0 a locally constant sheaf u of q vector spaces or u of real vector spaces 1 hodge filtration horizontality a decreasing filtration o r of gs by holomorphic subbundles such that the evident connection v on takes o p into q 2 hodge decompositions if gs denotes the sheaf of functions on s and if

rank one case of dwork s conjecture journal of the american mathematical society volume 00 number 0 xxxx xxxx pages 000 000 s 0894 0347 xx 0000 0 arxiv math 0005308v1 math nt 9 may 2000 rank one case of dwork s conjecture daqing wan 1 introduction in the higher rank paper 17 we reduced dwork s conjecture from higher rank case over any smooth affine variety x to the rank one case over the simplest affine space a n in the present paper we finish our proof by proving the rank one case of dwork s conjecture over the affine space a n which is called the key lemma in 17 the key lemma had already been proved in 16 in the special case when the frobenius lifting is the simplest q th power map x x q thus the aim of the present paper is to treat the general frobenius lifting case our method here is an improvement of the limiting method in 16 it allows us to move one step further and obtain some explicit information about the zeros and poles of the unit root l function as in 16 to handle the rank one case we are forced to work in the more difficult infinite rank setting see section 2 for precise definitions of the various basic infinite rank notions let f q denote the finite field of characteristic p 0 our main result of this paper is the following theorem theorem 1 1 let be a nuclear overconvergent module over the affine n space a n f q ordinary at the slope zero side let unit be the unit root slope zero part of assume that unit has rank one let be another nuclear overconvergent module over a n f q then for each integer k the l function l k unit t is p adic meromorphic furthermore the family l k t of l functions unit parametrized by integers k in each residue class modulo q 1 is a strong family of meromorphic functions with respect to the p adic topology of k a finite rank module is automatically nuclear thus theorem 1 1 includes the key lemma of 17 over a n as a special case the basic ideas of the present paper are the same as the limiting approach in 16 the details are however quite different in the simplest q th power frobenius lifting case one has the fundamental dwork trace formula available which is completely explicit for uniform estimates this makes it easy to extend the dwork trace formula to infinite rank setting it also makes it possible to see the various analytic subtleties involved in a concrete case as a result we were able to prove analytically optimal results in 16 for a general frobenius lifting even over the simplest affine n space a n as we shall work in this paper one has to use the much more difficult monsky trace formula which is a generalization of dwork s trace formula thus the first task of this paper is to extend the monsky trace formula to infinite rank setting and to make it sufficiently 1991 mathematics subject classification primary 11g40 11s40 secondary 11m41 14g15 key words and phrases l functions fredholm determinants p adic meromorphic continua tion nuclear modules and banach modules this work was partially supported by nsf c american mathematical society

a quasi elliptic transformation for moving boundary problems with large anisotropic deformations we have developed a quasi elliptic set of equations for generating a discretization mesh that optimally conforms to an entire domain that undergoes large deformations in primarily one direction we have applied this method to the axisymmetric problem of the transient displacement of a viscous liquid by a high pressure gas the liquid initially fills completely a tube the diameter of which may be constant or change smoothly suddenly or periodically along its finite length key ingredients for the success of the proposed transformation are limiting the orthogonality requirements on the mesh and employing an improved node distribution function along the deforming boundary the retained orthogonal term along with the penalty method for the imposition of the boundary conditions overcome the inherent restrictions of a conformal transformation producing meshes of high quality this term also eliminates the discontinuous slopes of the coordinate lines that are normal to the free surface these usually arise due to the harmonic transformation around highly deforming surfaces the mathematical anisotropy in the mesh generating equations directly corresponds to the physical one when the initially straight and normal to the axis of symmetry air fluid interface at the tube entrance develops mostly along the axial direction generating a long open bubble moreover the generalized node distribution imposes an optimal discretization of the bubble surface itself combining this scheme with the mixed finite element method produces a powerful tool with extended robustness and accuracy in order to reduce the computational cost the resulting set of equations is solved with a non linear gauss seidel technique and a variable time step specific applications of the proposed scheme are presented

application of a variational method for generating adaptive grids the application of variational methods for generating adaptive grids is not as straightforward as one is led to believe proper scaling suitable weight functions and appropriate clustering on boundaries must be employed to obtain a satisfactory grid this work which is based on the framework developed by brackbill and saltzman provides simple methods for determining scaling and investigates possible options for selecting the weight function and clustering points on the boundaries the concepts developed here are applied to two two dimensional problems a model problem based on burger s equation which contains two length scales and transonic flow past airfoils using the euler equations

the distribution of the zeros of the goss zeta function for a mathbb f _2 x y y 2 y x 3 x 1 let f be a global function field over a finite constant field and infty a place of f the ring a of functions regular away from infty in f is a dedekind domain for such a goss defined a zeta function which is a continuous function from mathbb z p to the ring of entire power series with coefficients in the completion f infty of f at infty he asks what one can say about the distribution of the zeros of the entire function at any parameter of mathbb z p in the simplest case a is the polynomial ring in one variable over a finite field here the question was settled completely by j sheats after previous work by j diaz vargas b poonen and d wan for any parameter in mathbb z p the zeros of the power series have pairwise different valuations and they lie in f infty in the present article we completely determine the distribution of zeros for the simplest case different from polynomial rings namely a mathbb f 2 x y y 2 y x 3 x 1 this a has class number 1 it is the affine coordinate ring of a supersingular elliptic curve and the place infty is mathbb f 2 rational the answer is slightly different from the above case of polynomial rings for arbitrary a such that infty is a rational place of f we describe a pattern in the distribution of zeros which we observed in some computational experiments finally we present some precise conjectures on the fields of rationality of these zeroes for one particular hyperelliptic a of genus 2

uniformizable families of t motives abelian t modules and the dual notion of t motives were introduced by anderson as a generalization of drinfeld modules for such anderson defined and studied the important concept of uniformizability it is an interesting question and the main objective of the present article to see how uniformizability behaves in families since uniformizability is an analytic notion we have to work with families over a rigid analytic base we provide many basic results and in fact a large part of this article concentrates on laying foundations for studying the above question building on these we obtain a generalization of a uniformizability criterion of anderson and among other things we establish that the locus of uniformizability is berkovich open

finite dimensional fuzzy normed linear space abstract the concept of a fuzzy normed linear space is introduced it is proved that in a finite dimensional fuzzy normed linear space fuzzy norms are the same upto fuzzy equivalence finite dimensional fuzzy subspaces of a fuzzy normed linear space are proved to be necessarily complete fuzzy normed linear spaces

applications and generalizations of variational methods for generating adaptive meshes abstract generating computation meshes for irregular regions have been of interest to a lot of people in many areas of research for a long time one technique that has met with sucess over the long run has been to generate methods using an elliptic equation or a system of elliptic equations the technique in its simplest form uses a system of laplace equations which are solved by direct or iterative methods as people gained more experience with this method source terms were added to the laplace equations to gain additional control of the mesh in variable coefficients of the derivatives were added for further flexibility in this paper the authors work with a method that systematically generates a set of elliptic equations without having to explicitly perturb a set of laplace equations with source terms and variable coefficients this technique uses the variational methods often associated with elliptic equations following this introduction they briefly discuss the variational formulation in two dimensional cartesian geometry then the formulation will be generalized to three dimensions next several three dimensional test problems will be shown after displaying these three dimensional test results the author wil the exhibit an application of the mesh generation technique in two dimensions this application involves generating an adaptive mesh for a supersonic flow past a step in a wind tunnel author

a direct comparison between volume and surface tracking methods with a boundary fitted coordinate transformation and third order upwinding a volume tracking vt and a front tracking ft algorithm are implemented and compared for locating the interface between two immiscible incompressible newtonian fluids in a tube with a periodically varying circular cross section initially the fluids are stationary and stratified in an axisymmetric arrangement so that one is around the axis of the tube core fluid and the other one surrounds it annular fluid a constant pressure gradient sets them in motion with both vt and ft a boundary fitted coordinate transformation is applied and appropriate modifications are made to adopt either method in this geometry the surface tension force is approximated using the continuous surface force method all terms appearing in the continuity and momentum equations are approximated using centered finite differences in space and one sided forward finite differences in time in each time step the incompressibility condition is enforced by a transformed poisson equation which is linear in pressure this equation is solved by either direct lu decomposition or a multigrid iterative solver when the two fluids have the same density the former method is about 3 5 times faster but when they do not the multigrid solver is as much as 10 times faster than the lu decomposition when the interface does not break and the reynolds number remains small the accuracy and rates of convergence of vt and ft are comparable the well known failure of centered finite differences arises as the reynolds number increases and leads to non physical oscillations in the interface and failure of both methods to converge with mesh refinement these problems are resolved and computations with reynolds as large as 500 converged by approximating the convective terms in the momentum equations by third order upwind differences using lagrangian polynomials when the volume of the core fluid or the weber number decrease increasing the importance of interfacial tension and leading to breakup of the interface forming a drop of core fluid the ft method converges faster with mesh refinement than the vt method and upwinding may be required finally examining the generation of spurious currents around a stationary bubble in the tube for ohnesorge numbers between 0 1 and 10 it is found that the maximum velocity remains approximately the same in spite mesh refinements when vt is applied whereas it is of the same order of magnitude for the coarsest mesh and monotonically decreases with mesh refinement when ft is applied

global l functions over function fields let c be a smooth projective curve over a finite field k with a marked point and let a be the ring of regular functions on c r furthermore let c be the completion of an algebraic closure of the function field k of c at a place above in this situation goss attaches a global l function to any family of drinfeld a modules via an infinite euler product on the domain s c zp cf 11 as in the classical situation it converges on some half plane of s a similar procedure yields for any closed point v of c and any family m of a motives a global lfunction l v m s which converges on a half plane of a suitably defined domain sv 12 8 the theme of the current article is to derive some consequences for such global l functions from the theory of crystals over function fields introduced by r pink and the current author in 2 this theory encompasses a motives as defined by anderson 1 indeed any family of a motives on a scheme x is represented by an a crystal on x furthermore for any compactifiable morphism f y x there is a functor direct image with compact support from a crystals on y to a crystals on x at least in the derived context for such f a trace formula is given in 2 for l functions of crystals it is this trace formula and the relations between goss global l functions and l functions of a crystals which we will mainly exploit slightly generalizing goss definition we obtain for any a scheme x of finite type over k any a crystal f on x and any closed point v of c a vadic l function l v f s as an euler product which converges on some half plane of sv cf definition 2 8 our principal goal is to prove that any such function has a meromorphic essentially algebraic continuation to all of sv in the sense of 12 8 along the way we will prove some interesting

mesh deformation using the biharmonic operator the use of the biharmonic operator for deforming a mesh in an arbitrary lagrangian eulerian simulation is investigated the biharmonic operator has the advantage that two conditions can be specified on each boundary of the mesh this allows both the position and the normal mesh spacing along a boundary to be controlled which is important for two fluid interfaces and periodic boundaries at these boundaries we can simultaneously fix the position of the boundary and ensure that the normal mesh spacing is continuous across the boundary in addition results for deforming surfaces show that greater surface deformation can be tolerated when using biharmonic equations compared to approaches using second order partial differential equations a final advantage is that with the biharmonic operator the integrity of a grid in a moving boundary layer can be preserved as the boundary moves the main disadvantage of the approach is its increased computational expense

discretization of free surface flows and other moving boundary problems a system of elliptic partial differential equations and boundary conditions has been developed for generating boundary fitted element discretizations of two dimensional free and moving boundary problems terms in the differential equations are scaled for dimensional homogeneity and adjustable weighting of orthogonality smoothness and concentration of the coordinate mesh they govern grid points become finite element nodes mapped isoparametrically or subparametrically from a simple or patched computational domain concentration terms contain control functions and parameters that influence node spacing along each coordinate independently overall control is by patchwise parameters and functions successful selection of these to follow deforming flow regions is straightforward and is illustrated by analysis of steady and transient slide coating flows

l adic gkz hypergeometric sheaves and exponential sums abstract to a torus action on a complex vector space gelfand kapranov and zelevinsky introduce a system of differential equations called the gkz hypergeometric system its solutions are gkz hypergeometric functions we study the l adic counterpart of the gkz hypergeometric system which we call the l adic gkz hypergeometric sheaf it is an object in the derived category of l adic sheaves on the affine space over a finite field traces of frobenius on stalks of this object at rational points of the affine space define the hypergeometric functions over the finite field introduced by gelfand and graev we prove that the l adic gkz hypergeometric sheaf is perverse calculate its rank and prove that it is irreducible under the non resonance condition we also study the weight filtration of the gkz hypergeometric sheaf determine its lisse locus and apply our result to the study of weights of twisted exponential sums

e 1 degeneration of the irregular hodge filtration for a regular function f on a smooth complex quasi projective variety j d yu introduced in 35 a filtration the irregular hodge filtration on the de rham complex with twisted differential dc df extending a definition of deligne in the case of curves in this article we show the degeneration ate1 of the spectral sequence attached to the irregular hodge filtration by using the method of 26 we also make explicit the relation with a complex intro duced by m kontsevich and give details on his proof of the corresponding e1 degeneration by reduction to characteristicp when the pole divisor of the function is reduced with normal crossings in appendix e m saito gives a different proof of thee1 degeneration

dimension variation of classical and p adic modular forms a quadratic bound is obtained for a conjecture of gouvea mazur on arithmetic variation of dimensions of classical and p adic modular forms

p adic variation of l functions of one variable exponential sums i for a polynomial f x in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of degree d 3 let inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i be the l function of the exponential sum of f mod p let np inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i denote the newton polygon of inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 04i let hp inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 05i denote the hodge polygon of inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 06i which is the lower convex hull in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 07i of the points n n n 1 2 d for 0 n d 1 let inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 08i be the space of degree d monic polynomials parameterized by their coefficients let gnp inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 09i inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 10i np inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 11i be the lowest newton polygon over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 12i if exists we prove that for p large enough gnp inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 13i exists and we give an explicit formula for it we also prove that there is a zariski dense open subset inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 14i defined over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 15i in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 16i such that for inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 17i and for p large enough we have np inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 18i gnp inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 19i furthermore as p goes to infinity their limit exists and is equal to hp inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 20i finally we prove analogous results for the space of polynomials f x x d ax with one parameter in particular for any nonzero a inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 21i we show that lim p np x d ax inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 22i hp inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 23i

a newton raphson pseudo solid domain mapping technique for free and moving boundary problems an implicit pseudo solid domain mapping technique is described that facilitates finite element analysis of free and moving boundary problems the technique is based on an implicit full newton strategy free of restrictions on mesh structure this leads to many advantages over existing domain mapping techniques the fully coupled approach using newton s method is particularly effective for problems with strong coupling between the internal bulk physics and the governing physics at unknown free boundary locations it is also useful when the distinguishing conditions which constrain the free boundary shape provide only an implicit dependence on the boundary location unstructured meshes allow for efficient resolution of internal and boundary layers and other regions of strong local variations in the solution and they also reduce the amount of user interaction required to define a problem since the meshes may be generated automatically the technique is readily applied to steady or transient problems in complex geometries of two and three dimensions examples are shown that include free and moving boundary problems from solidification and capillary hydrodynamics

finite element method for time dependent incompressible free surface flow we present a finite element method for time dependent incompressible free surface fluid flow problems described by the navier stokes equations the elements chosen have dimensions in both space and time and the resulting system of equations is block tridiagonal and lends itself to solution by standard techniques in the present article we restrict our attention to two dimensional problems although three dimensional problems may be solved by a straightforward generalization the method is essentially an implicit time stepping technique and therefore stable even for relatively large time steps with this choice of elements the method is completely adaptive to the changing nature of the solution an iterative procedure is used to find the position of the free surface this procedure is found to be rapidly convergent determining accurately the shape of the free surface within a few iterations numerical results are given for the problem of entrainment of fluid by a vertically moving plate which has applications to the chemical engineering problems of the free coating of metals we also consider the problem of circulation flow in a rectangular channel

on the elliptic mesh generation in domains containing multiple inclusions and undergoing large deformations we present an improved method to generate a sequence of structured meshes even when the physical domain contains deforming inclusions this method belongs to the class of arbitrary lagrangian eulerian ale methods for solving moving boundary problems its tools are either a separate mappings of the domain boundaries and enforcing the node distribution on lines emanating from singular points or b domain decomposition and separate mappings of each subdomain using suitable coordinate systems the latter is shown to be more versatile and general in both cases a set of elliptic equations is used to generate the grid extending in this way the method advanced by dimakopoulos and tsamopoulos y dimakopoulos j a tsamopoulos a quasi elliptic transformation for moving boundary problems with large anisotropic deformations j comput phys 192 2003 494 522 we shall present examples where this earlier method and all other mesh generating methods which are based on a conformal mapping or solving a quasi elliptic set of pdes fail to produce an acceptable mesh and accurate solutions in such geometries furthermore in contrast to other methods appropriate boundary conditions and constraints such as orthogonality of specific mesh lines and prespecified node distributions on them can be easily implemented along a specific part of the domain or its boundary hence no attractive terms at specific corners or singular points are needed to increase the mesh resolution around the moving interfaces while keeping low the memory requirements and the computational time a local mesh refinement technique has been incorporated as well the method is demonstrated in two challenging examples where no remeshing is required in spite of the large domain deformations in the first one the transient growth of two bubbles embedded in a viscoelastic filament undergoing stretching in the axial direction is examined while in the second one the linear and non linear dynamics of two bubbles in a viscous medium are determined in an acoustic field the large elasticity of the filament in the first case or the large inertia in the second case coupled with the externally induced large deformations of the liquid domain requires the accurate calculation which is achieved by the method we propose herein the governing equations are solved using the finite element galerkin method with appropriate modifications to solve the hyperbolic constitutive equation of a viscoelastic fluid these are coupled with an implicit euler method for time integration or with arnoldi s algorithm for normal mode analysis

a front tracking method for viscous incompressible multi fluid flows abstract a method to simulate unsteady multi fluid flows in which a sharp interface or a front separates incompressible fluids of different density and viscosity is described the flow field is discretized by a conservative finite difference approximation on a stationary grid and the interface is explicitly represented by a separate unstructured grid that moves through the stationary grid since the interface deforms continuously it is necessary to restructure its grid as the calculations proceed in addition to keeping the density and viscosity stratification sharp the tracked interface provides a natural way to include surface tension effects both two and three dimensional full numerical simulations of bubble motion are presented

l functions of symmetric powers of kloosterman sums unit root l functions and p adic estimates the l function of symmetric powers of classical kloosterman sums is a polynomial whose degree is now known as well as the complex absolute values of the roots in this paper we provide estimates for the p adic absolute values of these roots our method is indirect we first develop a dwork type p adic cohomology theory for the two variable infinite symmetric power l function associated to the kloosterman family and then study p adic estimates of the eigenvalues of frobenius a continuity argument then provides the desired p adic estimates

zeta functions and oscillatory integrals for meromorphic functions abstract in the 70 s igusa developed a uniform theory for local zeta functions and oscillatory integrals attached to polynomials with coefficients in a local field of characteristic zero in the present article this theory is extended to the case of rational functions or more generally meromorphic functions f g with coefficients in a local field of characteristic zero this generalization is far from being straightforward due to the fact that several new geometric phenomena appear also the oscillatory integrals have two different asymptotic expansions the usual one when the norm of the parameter tends to infinity and another one when the norm of the parameter tends to zero the first asymptotic expansion is controlled by the poles with negative real parts of all the twisted local zeta functions associated to the meromorphic functions f g c for certain special values c the second expansion is controlled by the poles with positive real parts of all the twisted local zeta functions associated to f g

newton polygons of zeta functions and l functions in this article we give a systematic treatment of newton polygons of exponential sums the newton polygon is a nice way to describe p adic values of the zeroes or poles of zeta functions and l functions our main objective is to show that the adolphson sperber conjecture 12 which asserts that under a simple condition the generic newton polygon of l functions coincides with its lower bound is false in its full form but true in a slightly weaker form we also show that the full form is true in various important special cases for example we show that for a generic projective hypersurface of degree d the newton polygon of the interesting part of the zeta function coincides with its lower bound the hodge polygon this gives a p adic proof of a recent theorem of illusie conjectured by dwork and mazur for more examples let us consider the family of affine hypersurfaces of degree d or the family of affine hypersurfaces defined by polynomials f xi x7n of degree di with respect to xi 1 i n where the di are fixed positive integers then for all large prime numbers p the generic newton polygon for the zeta functions of each of the two families of hypersurfaces coincides with its lower bound we obtain our main results namely several decomposition theorems using certain maximizing functions from linear programming our work suggests a possible connection between newton polygons and the resolution of singularities of toric varieties let p be a prime q pa and let fq be the finite field of q elements and fqm its extension of degree m fix a nontrivial additive character qp of fp for any laurent polynomial f xi xn e fq xl xj1 x xi1 we form

an algorithm for the simulation of transient viscoelastic flows with free surfaces we propose a numerical procedure for solving a class of transient viscoelastic flows with free surfaces it is based on a galerkin finite element technique on deforming elements combined with a predictor corrector scheme the method is applied to the analysis of jet breakup caused by capillary forces non linear effects known to experimentalists are predicted and a detailed comparison with asymptotic results is carried out

adjoint based optimal variable stiffness mesh deformation strategy based on bi elliptic equations there are many recent advances in mesh deformation methods for computational fluid dynamics simulation in deforming geometries we present a method of constructing dynamic mesh around deforming objects by solving the bi elliptic equation an extension of the biharmonic equation we show that introducing a stiffness coefficient fielda x in the bi elliptic equation can enable mesh deformation for very large boundary movements an indicator of the mesh quality is constructed as an objective function of a numerical optimization procedure to find the best stiffness coefficient field a x the optimization is efficiently solved using steepest descent along adjoint based integrated sobolev gradients a multi scenario optimization procedure is performed to calculate the optimal stiffness coefficient fielda x for a priori unpredictable boundary movements simulation of fluid structure interaction problems requires solving the governing equation of the unsteady flow inside a deforming physical domain whose geometry is determined by the deformation of the structure in numerical methods for solving such problems arbitrary lagrangian eulerian ale 1 method is one of the most popular types of approaches it formulates the governing equation in a coordinate system that moves with the boundary and discretizes the deforming domain with a deformable body conforming mesh ale methods are capable of using anisotropic mesh to efficiently resolve boundary layers however these methods requires an algorithm to deform the mesh so that it conforms to the boundary movement while maintaining good mesh quality there has been many recent advances in algorithms for computing mesh deformation in the context of computational fluid dynamics simulations in moving domains two of the most popular categories of such algorithms are interpolation based methods and structure analogy methods interpolation based methods determines the position or displacement of mesh points in the interior of the domain by interpolating the position or displacement of the mesh points on the boundary of the domain an example of interpolation based methods is the radial basis function interpolation method 23 structural analogy methods include methods based on structural models of the discrete computational mesh and methods based on modeling the computational domain as a continuous elastic media recent developments in discrete mesh structural model includes the torsional spring method 45 recent developments in the continuous elasticity model include the biharmonic equation method 6 and the quasi elliptic transformation method 7 in this paper we present a mesh deformation method based on continuous elasticity model our contribution is introducing a variable stiffness coefficient in t he biharmonic elastic model and developing a method to efficiently obtain the optimal stiffness coefficient d istribution we show that the variable stiffness coefficient algorithm can produce a good deformed mesh whilethe original scheme with constant coefficient produces unacceptable mesh to obtain a suitable stiffness coefficient we define a simple global mesh

functions of sheaves and drinfeld modules in this paper we apply dwork s p adic methods to study the meromorphic continuation and rationality of various l functions arising from 7r adic galois representations drinfeld modules and p sheaves as a consequence we prove some conjectures of goss about the rationality of the local l function and the meromorphic continuation of the global l function attached to a drinfeld module let fq be a finite field of q elements with characteristic p let 7r be a prime of the polynomial ring a fq t let a be the completion of the ring a at 7r this is an analogue of the classical ring 7p of p adic integers let x be an irreducible algebraic variety defined over fq and let 7r x be the arithmetic fundamental group of x ifq with respect to some base point the group 7r x may be regarded as the galois groiip of a separable closure of the function field of x fq modulo the inertia groups at the closed points of x lq suppose now that we are given a continuous 7r adic representation

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100