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

passive stochastic approximation with constant step size and window width motivated by the desire to solve a steady state estimation and detection problem in chemical engineering and inspired by the recent progress in passive stochastic approximation recursive algorithms combining stochastic approximation and kernel estimation are studied and developed in this work the underlying problem can be stated as finding the roots f x 0 provided only noisy measurements y sub n f x sub n spl xi sub n are available where ef x spl xi sub n f x the main difficulty lies in that unlike the traditional approach in stochastic approximation the sequence x sub n is generated randomly and cannot be adjusted in accordance with our wish similar to those used in the decreasing step size algorithms another sequence z sub n is generated to approximate the roots of f x 0 some of the features of the algorithms include constant step size and constant window width and correlated random processes under fairly general conditions it is proven that a weak convergence result holds for an interpolated sequence of the iterates error bounds are obtained and a local limit theorem is also derived the algorithm is then applied to an estimation problem in chemical engineering simulation has shown promising results

one dimensional cellular automata characterization by the roughness exponent abstract cellular automata ca are discrete spatially homogeneous locally interacting dynamical systems of very simple construction but which exhibit a rich intrinsic behavior ca can even starting from disordered initial configurations evolve into ordered states with complex structures crystallized in its space time patterns in this paper we concentrate on the several wolfram qualitative classes of ca behavior in order to better characterize these classes we apply the roughness exponent method to the profiles generated by the spatiotemporal patterns of one dimensional elementary ca rules we find that this method can separate wolfram class iv from other ones

improvements of the weil bound for artin schreier curves for the artin schreier curve y q y f x defined over a finite field mathbb f q of q elements the celebrated weil bound for the number of mathbb f q r rational points can be sharp especially in super singular cases and when r is divisible in this paper we show how the weil bound can be significantly improved using ideas from moment l functions and katz s work on l adic monodromy calculations roughly speaking we show that in favorable cases which happens quite often one can remove an extra sqrt q factor in the error term

computation theory of cellular automata self organizing behaviour in cellular automata is discussed as a computational process formal language theory is used to extend dynamical systems theory descriptions of cellular automata the sets of configurations generated after a finite number of time steps of cellular automaton evolution are shown to form regular languages many examples are given the sizes of the minimal grammars for these languages provide measures of the complexities of the sets this complexity is usually found to be non decreasing with time the limit sets generated by some classes of cellular automata correspond to regular languages for other classes of cellular automata they appear to correspond to more complicated languages many properties of these sets are then formally non computable it is suggested that such undecidability is common in these and other dynamical systems

power law distributions in some random boolean networks the kauffman net is a dynamical system of logical variables receiving two random inputs and each randomly assigned a boolean function we show that the attractor and transient lengths exhibit scaleless behavior with power law distributions over up to 10 orders of magnitude in probability our results provide evidence for the existence of the edge of chaos as a distinct regime between the ordered and chaotic phases analogous to a critical point in statistical mechanics the power law distributions are robust to the changes in the composition of the transition rules and network dynamics

asymptotic behaviour of a class of stochastic approximation procedures letb be a real banach space considering stochastic approximation procedures of robbins monro and kiefer wolfowitz type one is led to recursion formulas of the form u k 1 u k k 1 gamma k u k k rho v k k eta t k and positive real numbers uk vk tkb valued random variables k random linear operatorsb b where in addition to some further assumptions k almost surely andtk t say almost surely letting y k n k 1 n n n be an array ofb valued random variables such that the partial sums sum limits k 1 m v k and sum limits k 1 m y k n do not differ too much uniformly inm 1 n we shall investigate the distance of u m m 1 n and u n m m 1 n where u k 1 left n right u k left n right k 1 gamma u k left n right k rho y k n k eta t fork 1 n 1 the proofs are based on a deterministic argument that enables us to present a unified approach for proving central limit theorems weak and almost sure invariance principles and variants of the bounded and the functional law of the iterated logarithm for the sequence un

using the renormalization group to classify boolean functions this paper argues that the renormalization group technique used to characterize phase transitions in condensed matter systems can be used to classify boolean functions a renormalization group transformation is presented that maps an arbitrary boolean function of n boolean variables to one of n 1 variables applying this transformation to a generic boolean function one whose output for each input is chosen randomly and independently to be one or zero with equal probability yields another generic boolean function moreover applying the transformation to some other functions known to be non generic such as boolean functions that can be written as polynomials of degree with n and functions that depend on composite variables such as the arithmetic sum of the inputs yields non generic results one can thus define different phases of boolean functions as classes of functions with different types of behavior upon repeated application of the renormalization transformation possible relationships between different phases of boolean functions and computational complexity classes studied in computer science are discussed

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

reversible boolean networks i distribution of cycle lengths abstract we consider a class of models describing the dynamics of n boolean variables where the time evolution of each depends on the values of k of the other variables previous work has considered models with dissipative dynamics here we consider time reversible models which necessarily have the property that every possible point in the state space is an element of one and only one cycle the orbits can be classified by their behavior under time reversal the orbits that transform into themselves under time reversal have properties quite different from those that do not in particular a significant fraction of latter type orbits have lengths enormously longer than orbits that are time reversal symmetric for large k and moderate n the vast majority of points in the state space are on one of the time reversal singlet orbits and a random hopping model gives an accurate description of orbit lengths however for any finite k the random hopping approximation fails qualitatively when n is large enough n 22k as in the dissipative case when k is large typical orbit lengths grow exponentially with n whereas for small enough k typical orbit lengths grow much more slowly with n the numerical data are consistent with the existence of a phase transition at which the average orbit length grows as a power of n at a value of k between 1 4 and 1 7 however in the reversible models the interplay between the discrete symmetry and quenched randomness can lead to enormous fluctuations of orbit lengths and other interesting features that are unique to the reversible case

critical phenomena in an one dimensional probabilistic cellular automation a one dimensional probabilistic cellular automaton that models a transition from elementary rule 4 to elementary rule 22 following wolfram s nomenclature scheme is studied here the evolution of the automaton follows rule 4 with probability 1 p and rule 22 with probability p in course of the transition the system shows two critical points a trivial pc1 0 and a nontrivial pc2 0 75 at which the relaxation time of the system is observed to diverge in the form of a power law p pc1 z1 and pc2 p z2 with z1 0 86 and z2 0 92 the point pc2 is also a point of phase transition with the density of occupied sites in the equilibrium state as the order parameter the order parameter goes to zero as n p pc2 s s 0 32 for p pc2 the possible cause of the observed behaviour is discussed

relevant elements magnetization and dynamical properties in kauffman networks a numerical study abstract this is the first of two papers about the structure of kauffman networks in this paper we define the relevant elements of random networks of automata following previous work by flyvbjerg j phys a 21 1988 l955 l960 and flyvbjerg and kjaer j phys a 21 1988 1695 1718 and we study numerically their probability distributions in the chaotic phase and on the critical line of the model a simple approximate argument predicts that their number scales as n on the critical line while it is linear with n in the chaotic phase and independent on system size in the frozen phase this argument is confirmed by numerical results the study of the relevant elements gives useful information about the properties of the attractors in critical networks where the pictures coming from either approximate computation methods or from simulations are not very clear

chaos and diffusion in deterministic cellular automata abstract it is shown that some of the deterministic one dimensional cellular automata studied recently by wolfram exhibit a kind of spontaneous symmetry breaking the associated kinks bloch walls perform annihilating diffusive walks for infinite lattices and large times in addition close analogies between these automata and strange attractors are exhibited by mapping the automata onto dynamical systems defined by iterations of the unit square

universality and complexity in cellular automata abstract cellular automata are discrete dynamical systems with simple construction but complex self organizing behaviour evidence is presented that all one dimensional cellular automata fall into four distinct universality classes characterizations of the structures generated in these classes are discussed three classes exhibit behaviour analogous to limit points limit cycles and chaotic attractors the fourth class is probably capable of universal computation so that properties of its infinite time behaviour are undecidable

scale free network with boolean dynamics as a function of connectivity in this work we analyze scale free networks with different power law spectra n k ensuremath sim k ensuremath ensuremath gamma under a boolean dynamic where the boolean rule that each node obeys is a function of its connectivity k this is done by using only two logical functions and and xor which are controlled by a parameter q using a damage spreading technique we show that the hamming distance and the number of 1 s exhibit power law behavior as a function of q the exponents appearing in the power laws depend on the value of ensuremath gamma

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

nonparametric sequential estimation of zeros and extrema of regression functions let x y x l y l x 2 y 2 cdots be independent identically distributed pairs of random variables and let m x e y x x be the regression curve of y on x the estimation of zeros and extrema of the regression curve via stochastic approximation methods is considered consistency results of some sequential procedures are presented and termination rules are defined providing fixed width confidence intervals for the parameters to be estimated

lyapunov exponents in random boolean networks a new order parameter approximation to random boolean networks rbn is introduced based on the concept of boolean derivative a statistical argument involving an annealed approximation is used allowing to measure the order parameter in terms of the statistical properties of a random matrix using the same formalism a lyapunov exponent is calculated allowing to provide the onset of damage spreading through the network and how sensitive it is to minimal perturbations finally the lyapunov exponents are obtained by means of dierent approximations through distance method and a discrete variant of the wolf s method for continuous systems c 2000 elsevier science b v all rights reserved

families of modular eigenforms this article is an expansion of the notes to a one hour lecture for an msri workshop on computational number theory the editors of mathematics of computation kindly asked us to submit these notes for publication and we are enormously pleased to do so our original audience did not consist of experts in the field of modular forms and we have tried to keep this article accessible to nonexperts we have made an experimental investigation of certain arithmetic conjec tures using macsyma this investigation requires a search for certain mod ular eigenforms of high weight these computations pose problems which we feel may be interesting on their own we are novices here and we seek advice from people more experienced in making computations of an analogous sort we are in fact deeply grateful to j f mestre who came to our aid and who vastly extended our computations using pari 1 mestre has graciously allowed us to present his computations in this article the kind of families we have in mind as in the title of this lecture has as its prototypical example the standard family of classical eisenstein series of level 1 and weight k for k 4 6 8 whose fourier expansions are given by oo

on bayesian supremum norm contraction rates building on ideas from castillo and nickl 6 a method is pro vided to study nonparametric bayesian posterior convergence rates when strong measures of distances such as the sup norm are considered in particular we show that likelihood methods can achieve optimal minimax sup norm rates in density estimation on the unit interval the introduced methodology is used to prove that commonly used families of prior distri butions on densities namely log density priors and dyadic random density histograms can indeed achieve optimal sup norm rates of convergence new results are also derived in the gaussian white noise model as a further il lustration of the presented techniques ams 2000 subject classications primary 62g20 secondary 62g05 62g07

complexity measures from interaction structures we evaluate information theoretic quantities that quantify complexity in terms of k text th order statistical dependences that cannot be reduced to interactions among k ensuremath 1 random variables using symbolic dynamics of coupled maps and cellular automata as model systems we demonstrate that these measures are able to identify complex dynamical regimes

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

the red queen s walk abstract we study a new type of walk with memory which might serve as a toy model for the behavior one must adopt to avoid exhaustion of resources and attraction of parasites and predators the walk takes place on a regular square lattice with periodic boundary conditions although the walk is completely deterministic it mimics a true self avoiding walk i e a random walk with weak autocorrelation this shows that the red queen effect can lead to aperiodic behavior in addition to the case of single walkers in a flat landscape we also study the cases of hilly landscapes and of several walkers performing simultaneous walks

intriguing properties of global structure in some classes of finite cellular automata abstract the simplest one dimensional cellular automata introduced recently by wolfram are analyzed particularly the finite cellular automata with two special evolution rules are studied non trivial correspondence in global structure is found between the state transition diagrams under the two rules correspondence of cycle structure basin volume of each cycle and structure of transient trees etc origin of the correspondence is explored on the basis of local properties of the evolution rules it is shown that this correspondence becomes exact in the thermodynamic limit

long range effects in an elementary cellular automaton we present evidence that one of the elementary one dimensional cellular automata in the sense of wolfram rule 22 in wolfram s notation involves very complex long range effects similar to a critical phenomenon this is in contrast to superficial evidence that would suggest that this rule leads to fairly simple behavior

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

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

adaptive density estimation from data with small measurement errors in this paper we study the problem of density estimation from data that contains small measurement errors the only assumption on these errors is that the maximal measurement error is bounded by some real number converging to zero for sample size tending to infinity in particular we do not assume that the measurement errors are independent with expectation zero we estimate the density by a standard kernel density estimate applied to data with measurement errors and derive a data driven method to choose its bandwidth we derive an adaptation result for this estimate and analyze the expected l 1 error of our density estimate depending on the smoothness of the density and the size of the maximal measurement error the results are applied in a density estimation problem in a simulation model where we show under suitable assumptions that the l 1 error of our newly proposed estimate converges to zero much faster than the l 1 error of the standard kernel density estimate if both are based on the same number of observations in the simulation model the performance of the method in case of finite sample size is analyzed using simulated data

monte carlo study of infinite dimensional cellular automata abstract the well known one dimensional elementary cellular automata are generalized to an infinite number of dimensions the generalization of the most interesting automaton in 1d rule 22 is studied in detail by means of monte carlo as well as enumeration methods it is shown that monte carlo data extrapolate very smoothly to infinite system size numerical deviation from mean field theory is analyzed

sequential fixed width confidence intervals for a nonparametric density function in this paper sequential fixed width confidence intervals for a nonparametric density function are derived these are formed from certain values of a kernel density estimator the efficiency of such procedures is measured in terms of the expected stopping time

the one dimensional exactly 1 cellular automaton replication periodicity and chaos from finite seeds in the exactly 1 cellular automaton also known as rule 22 every site of the one dimensional lattice is either in state 0 or in state 1 and a synchronous update rule dictates that a site is in state 1 next time if and only if it sees a single 1 in its three site neighborhood at the current time we analyze this rule started from finite seeds i e those initial configurations that have only finitely many 1 s three qualitatively different types of evolution are observed replication periodicity and chaos we focus on rigorous results assisted by algorithmic searches for the first two behaviors in particular we explain why replication is observed so frequently and present a method for collecting the smallest periodic seeds some empirical observations about chaotic seeds are also presented

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

on line identification of time varying systems by nonparametric techniques a nonparametric algorithm based on orthogonal series is proposed for real time identification of nonstationary systems it is proved that the algorithm converges in the mean square sense the convergence conditions are closely related to those in dupac 2

shtukas and jacobi sums we show how the analogues of jacobi sums in the context of function fields introduced and studied in t1 t2 t3 can be obtained from shtukas introduced and studied in d2 d3 m we apply this to obtain some results on the prime factorization of analogues of gauss sums and to prove an analogue of the gross koblitz formula for general function field generalizing the results in t2 for this purpose we also introduce and interpolate a new analogue of gamma function

tannakian duality for anderson drinfeld motives and algebraic independence of carlitz logarithms we develop a theory of tannakian galois groups for t motives and relate this to the theory of frobenius semilinear difference equations we show that the transcendence degree of the period matrix associated to a given t motive is equal to the dimension of its galois group using this result we prove that carlitz logarithms of algebraic functions that are linearly independent over the rational function field are algebraically independent

a new type of kinetic critical phenomenon the authors study a new critical phenomenon in an non thermal one dimensional lattice model it is characterised by the transition from stability to instability of kinks between ordered states below the critical point the kinks are stable and move by annihilating random walks above the critical point they are unstable against creation of kink antikink pairs the spontaneous production of pairs is assumed to be absent at the critical point p pcr the density of kinks decreases like n approximately talpha with alpha 0 27 or 0 08 above the critical point the density of kinks in the stationary state is approximately p pcr beta with beta 0 06 or 0 2 possible extensions to two or more dimensions and possible applications are discussed

on the reduction of rank one drinfel d modules the drinfeld modules of rank one associated to all elliptic curves over the finite fields if2 and if3 are computed in explicit form these examples illustrate the theory of the j invariant of such modules as developed by gekeler and dorman let f x e lfq x be a monic polynomial of odd degree n 3 and let k fq x be the hyperelliptic extension obtained by adjoining a root of 1 y2 f x if q is odd or 2 y axy a3y f x al a3 e fq not both zero if q is even to the field of rational functions fq x if q is odd we require that f x be square free which means that the affine plane curve defined by 1 has no singular points for q even we restrict f x and a1 a3 also by requiring that the affine curve defined by 2 be nonsingular in either case the affine coordinate ring a fq x i1y is integrally closed in k since n is odd the infinite place of fq x ramifies in k fq x let oo denote its unique extension to k and let k be the completion of k at oo fix vix e kco and let sgn k0 fq be the unique sign function for which sgn l v x 1 we choose y so that sgn y 1 the ring a is the ring of functions in k which are holomorphic away from oo let p be a sgn normalized rank one drinfeld a module defined over the algebraic closure of k then p is determined by its values px x aq3 o 2 2 n l n py y c1o c2 nc i nc where cn sgn y 1 and the coefficients a c1 c2 cnare elements of the hilbert class field h of a the degree hk h k is the class number of k let b be the integral closure of a in h one knows that the coefficients received july 25 1990 1980 mathematics subject classification 1985 revision primary 1 lg15 1 lg20 11 r58 partially supported by nsf grant dms 8702716 q 1991 american mathematical society 0025 5718 91 1 00 25 per page

the topological skeleton of cellular automaton dynamics abstract we have developed statistical techniques to study the structure of the state transition graphs of cellular automata with periodic boundary conditions in the limit of large system size we organize our results around the concept of a topological skeleton the topological skeleton is the set of physically relevant states covering this skeleton is a surface typically thin and dense which contains the bulk of the set of states states in the skeleton have some long histories states on the surface by contrast have only short histories they are reached only near the beginning of cellular automaton evolution we study in detail a sequence of rules which exhibit mostly skeletal to mostly surface structure

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

efficient estimation in dynamic conditional quantile models in this paper we consider the problem of semiparametric efficient estimation in conditional quantile models with time series data we construct an m estimator which achieves the semiparametric efficiency bound recently derived by komunjer and vuong forthcoming our efficient m estimator is obtained by minimizing an objective function which depends on a nonparametric estimator of the conditional distribution of the variable of interest rather than its density the estimator is new and not yet seen in the literature we illustrate its performance through a monte carlo experiment

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

classification of square lattice cellular automata abstract all 65 536 automata on the square lattice with nearest neighbor interactions ignoring the central spin were simulated we determined which percentage of them ended up in a fixed point with all spins parallel in other fixed points in up down oscillations with all spins parallel and in other oscillations of period two nine tenths of the rules did not fall into these classes we also checked for chaos i e how does the damage spread if one line of spins is reversed for 62 percent of the possible rules it spread over the lattice 31 percent had the damage remaining localized but nonzero and in 7 percent of all rules the damage healed out after some time these simulations were done with a speed of up to 120 updates per microsecond and cray xmp processor

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

long range correlations in chaotic cellular automata abstract one dimensional cellular automata of class 3 are studied as models for spatially extended chaotic systems their irreversible dynamics can generate weak but long range spatial correlations the one labelled 22 by wolfram is treated in an exemplary way its stationary state is approximated by a markov chain the regular grammar underlying the markov chain is chosen with care so that it includes a maximum amount of information on the stationary state the approximation can be performed without simulating the dynamics of the cellular automaton 22 and reproduces its long range correlations qualitatively their origin is explained intuitively and they are argued to be a frequent feature of cellular automata with more complicated rules

some more exact enumeration results for 1d cellular automata the author presents exact enumeration results for the elementary cellular automation rules 22 30 45 and 120 in wolfram s notation obtained by a new algorithm they improve previous results considerably they confirm the existence of hidden power behaved long range correlations in the patterns created by these rules

newton slopes for artin schreier witt towers we fix a monic polynomial f x in mathbb f q x over a finite field and consider the artin schreier witt tower defined by f x this is a tower of curves cdots rightarrow c m rightarrow c m 1 rightarrow cdots rightarrow c 0 mathbb a 1 with total galois group mathbb z p 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 form arithmetic progressions which are independent of the conductor of the character as a corollary we obtain a result on the behavior of the slopes of the eigencurve associated to the artin schreier witt tower analogous to the result of buzzard and kilford

p adic banach spaces and families of modular forms let p be a prime cp the completion of an algebraic closure of the p adic numbers qp and k a nite extension of qp contained in cp let v be the valuation on cp such that v p 1 and let be the absolute value on cp such that x p v x for x cp suppose n is a positive integer prime to p let x1 np denote the modular curve over k which represents elliptic curves with 1 np structure and let up be the hecke operator on modular forms on x1 np which takes a form with q expansion n anq n to the modular form with q expansion n anpq n a modular form f is said to have slope q if there is a polynomial r t over cp such that r up f 0 and such that the newton polygon of r t has only one side and its slope is

convergence rates in density estimation for data from infinite order moving average processes the effect of long range dependence in nonparametric probability density estimation is investigated under the assumption that the observed data are a sample from a stationary infinite order moving average process it is shown that to first order the mean integrated squared error mise of a kernel estimator for moving average data may be expanded as the sum of mise of the kernel estimator for a same sizerandom sample plus a term proportional to the variance of the moving average sample mean the latter term does not depend on bandwidth and so imposes a ceiling on the convergence rate of a kernel estimator regardless of how bandwidth is chosen this ceiling can be quite significant in the case of long range dependence we show thatall density estimators have the convergence rate ceiling possessed by kernel estimators

topological classification of cellular automata the author summarizes an extensive numerical study of basin and attractor sizes among the 88 distinct elementary cellular automata ca rules based on this study and on previous work in discretized dynamical systems he proposes a new classification of ca complementary to that of wolfram in which attractor globality is important with the use of fixed boundary conditions he finds global periodic attractors in ca for the first time not a single instance of attractor chaos is observed in this class of rules

transients in a one dimensional probabilistic cellular automaton abstract the characteristics of the distribution of transient times of a one dimensional probabilistic cellular automaton are studied by computer simulation the mean and width of the distribution are found to diverge by the same power law at each of the two critical points p c 1 0 and p c 2 0 75 of the model critical exponents obtained from finite size scaling indicate that p c 1 belongs to the universality class of directed random walks whereas p c 2 belongs to the universality class of directed percolation between the two critical points there exists a point of minimum transient length at p m 0 23 where the mean transient time scales logarithmically with the system size

effects of alternative connectivity on behavior of randomly constructed boolean networks random boolean networks rbns were first introduced and used to model regulatory gene networks by kauffman in 1969 and since then have been frequently used to model networks at various levels of biological organization the standard model has a characteristic topology in which the network has n genes with an average of k input connections selected at random we have used rbns to examine regulatory gene networks with four different topologies which are characterized by different rank distributions of output connections that vary from uniform to highly skewed among these is the topology of the standard model which is included for purposes of comparison and a topology with a power law rank distribution which is based on recent data for the regulatory gene network of the bacterium escherichia coli we also examine effects of bias in the distribution of boolean functions for the network the dynamical properties and mutual information of these networks depend not only on their size but also on their topology and boolean functions networks with the more uniform rank distributions exhibit longer lengths of attractors larger numbers of attractors and less mutual information networks with the more skewed rank distributions have complementary properties when viewed as biological decision making networks those with either the most uniform or the most skewed rank distributions have disadvantageous properties the intermediate rank distribution exhibited by the regulatory gene network of e coli avoids these disadvantages

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

two step estimation of semiparametric censored regression models root n consistent estimators of the regression coecients in the linear censored regression model under conditional quantile restrictions on the error terms were pro posed by powell journal of econometrics 25 1984 303 325 32 1986a 143 155 while those estimators have desirable asymptotic properties under weak regularity conditions simulation studies have shown these estimators to exhibit a small sam ple bias in the opposite direction of the least squares bias for censored data this paper introduces two step estimators for these models which minimize convex objec tive functions and are designed to overcome thisnite sample bias the paper gives regularity conditions under which the proposed two step estimators are consistent and asymptotically normal a monte carlo study compares thenite sample behavior of the proposed methods with their one step counterparts 2001 elsevier science s a all rights reserved

optimal and robust kernel algorithms for passive stochastic approximation the problem of estimating a root of an equation f x 0 is considered in the situation where the values of f x are measured with random errors at random points and the choice of these points cannot be controlled nonlinear modification of the recursive hardle nixdorf method is studied almost sure and mean square convergence is proved and the rate of convergence is estimated the optimal choice of parameters and of a kernel is presented it is shown that for the optimal procedure the lower bound for the accuracy of arbitrary methods of solving the problem is attained

reversible boolean networks ii phase transitions oscillations and local structures we continue our consideration of a class of models describing the reversible dynamics of n boolean variables each with k inputs we investigate in detail the behavior of the hamming distance as well as of the distribution of orbit lengths as n and k are varied we present numerical evidence for a phase transition in the behavior of the hamming distance at a critical value kc 1 62 and also an analytic theory that yields the exact bounds 1 5 k c 2 we also discuss the large oscillations that we observe in the hamming distance for k k c as a function of time as well as in the distribution of cycle lengths as a function of cycle length for moderate k both greater than and less than kc we propose that local structures or subsets of spins whose dynamics are not fully coupled to the other spins in the system play a crucial role in generating these oscillations the simplest of these structures are linear chains called linkages and rings called circuits we discuss the properties of the linkages in some detail and sketch the properties of circuits we argue that the observed oscillation phenomena can be largely understood in terms of these local structures 2001 published by elsevier science b v

l functions of symmetric products of the kloosterman sheaf over z the classical n variable kloosterman sums over the finite field f p give rise to a lisse overline bf q l sheaf kl n 1 on bf g m bf f p bf p 1 bf f p 0 infty which we call the kloosterman sheaf let l p g m f p sym k kl n 1 s be the l function of the k fold symmetric product of kl n 1 we construct an explicit virtual scheme x of finite type over spec z such that the p euler factor of the zeta function of x coincides with l p g m f p sym k kl n 1 s we also prove similar results for otimes k rm kl n 1 and bigwedge k rm kl n 1

the number and probability of canalizing functions canalizing functions have important applications in physics and biology for example they represent a mechanism capable of stabilizing chaotic behavior in boolean network models of discrete dynamical systems when comparing the class of canalizing functions to other classes of functions with respect to their evolutionary plausibility as emergent control rules in genetic regulatory systems it is informative to know the number of canalizing functions with a given number of input variables this is also important in the context of using the class of canalizing functions as a constraint during the inference of genetic networks from gene expression data to this end we derive an exact formula for the number of canalizing boolean functions of n variables we also derive a formula for the probability that a random boolean function is canalizing for any given bias p of taking the value 1 in addition we consider the number and probability of boolean functions that are canalizing for exactly k variables finally we provide an algorithm for randomly generating canalizing functions with a given bias p and any number of variables which is needed for monte carlo simulations of boolean networks

on methods of sieves and penalization we develop a general theory which provides a unified treatment for the asymptotic normality and efficiency of the maximum likelihood estimates mle s in parametric semiparametric and nonparametric models we find that the asymptotic behavior of substitution estimates for estimating smooth functionals are essentially governed by two indices the degree of smoothness of the functional and the local size of the underlying parameter space we show that when the local size of the parameter space is not very large the substitution standard nonsieve substitution sieve and substitution penalized mle s are asymptotically efficient in the fisher sense under certain stochastic equicontinuity conditions of the loglikelihood moreover when the convergence rate of the estimate is slow the degree of smoothness of the functional needs to compensate for the slowness of the rate in order to achieve efficiency when the size of the parameter space is very large the standard and penalized maximum likelihood procedures may be inefficient whereas the method of sieves may be able to overcome this difficulty this phenomenon is particularly manifested when the functional of interest is very smooth especially in the semiparametric case

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

estimating the information content of symbol sequences and efficient codes several variants of an algorithm for estimating shannon entropies of symbol sequences are presented they are all related to the lempel ziv algorithm 1976 1977 and to recent algorithms for estimating hausdorff dimensions the average storage and running times increase as n and nlog n respectively with the sequence length n these algorithms proceed basically by constructing efficient codes they seem to be the optimal algorithms for sequences with strong long range correlations e g natural languages an application to written english illustrates their use

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

phase transitions in an elementary probabilistic cellular automaton with memory abstract standard cellular automata ca are ahistoric memoryless i e the new state of a cell depends on the neighborhood configuration only at the preceding time step this article considers an extension to the standard framework of ca by implementing memory capabilities in cells thus in ca with memory while the update rules of the ca remain unaltered historic memory of all past iterations is retained by featuring each cell by a summary of all its past states a study is conducted of the effect of historic memory on a given side of the hypercube of elementary probabilistic ca which shows interesting phase transition phenomena

functional equations of l functions for symmetric products of the kloosterman sheaf we determine the arithmetic local monodromy at 0 and at of the kloosterman sheaf using local fourier transformations and laumon s stationary phase principle we then calculate e factors for symmetric products of the kloosterman sheaf using laumon s product formula we get functional equations of l functions for these symmetric products and prove a conjecture of evans on signs of constants of functional equations 2010 american mathematical society

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

period spaces for hodge structures in equal characteristic we develop the analog in equal positive characteristic of fontaine s theory for crystalline galois representations of a p adic eld in particular we describe the analog of fontaine s functor which assigns to a crystalline galois representation an isocrystal with a hodge ltration in equal characteristic the role of isocrystals and hodge ltrations is played by z isocrystals and hodge pink structures the latter were invented by pink our rst main result in this article is the analog of the colmez fontaine theorem that weakly admissible implies admissible next we construct period spaces for hodge pink structures on a xed z isocrystal these period spaces are analogs of the rapoport zink period spaces for fontaine s ltered isocrystals in mixed characteristic and likewise are rigid analytic spaces for our period spaces we prove the analog of a conjecture of rapoport and zink stating the existence of a universal local system on a berkovich open subspace of the period space as a consequence of weakly admissible implies admissible this berkovich open subspace contains every classical rigid analytic point of the period space as the principal tool to demonstrate these results we use the analog of kedlaya s slope filtration theorem which we also formulate and prove here

the dimensionality reduction principle for generalized additive models soit x y une paire de variables aleatoires telles que x x 1 x j s etende sur c 0 1 j la distribution conditionnelle de y etant donne x x est supposee appartenir a une famille exponentielle appropriee ayant pour parametre r f x denote la dependance de vis a vis de x et f l approximation additive de f ayant la log vraisemblance esperee maximum possible