products of languages with counter abstract it is well known that varieties of rational languages are in one to one correspondence with varieties of finite monoids this correspondence often extends to operations on languages and on monoids we investigate the special case of the product of languages with counter and describe the associated operations on monoids and varieties

convergence of large deviation estimators we study the convergence of statistical estimators used in the estimation of large deviation functions describing the fluctuations of equilibrium nonequilibrium and manmade stochastic systems we give conditions for the convergence of these estimators with sample size based on the boundedness or unboundedness of the quantity sampled and discuss how statistical errors should be defined in different parts of the convergence region our results shed light on previous reports of phase transitions in the statistics of free energy estimators and establish a general framework for reliably estimating large deviation functions from simulation and experimental data and identifying parameter regions where this estimation converges

mesures d independance lineaire de logarithmes dans un groupe algebrique commutatif this work falls within the theory of linear forms in logarithms over a connected and commutative algebraic group defined over the field of algebraic numbers overline mathbb q let g be such a group let w be a hyperplane of the tangent space at the origin of g defined over overline mathbb q and u be a complex point of this tangent space such that the image of u by the exponential map of the lie group g is an algebraic point then we obtain a lower bound for the distance between u and w which improves the results known before and which is in particular the best possible for the height of the hyperplane w the proof rests on baker s method and hirata s reduction as well as a new arithmetic argument chudnovsky s process of variable change which enables us to give a precise estimate of the ultrametric norms of some algebraic numbers built during the proof

on special l values of t modules we show that taelman s conjecture on special l values of ander son t modules holds for a large class of t modules this class contains all mixed a finite and uniformizable t modules whose hodge pink weights are at least 1 as a consequence we deduce various log algebraicity identities for tensor powers of the carlitz module generalizing the work of anderson thakur

arithmetic of function field units we introduce the module of stark units associated to a drinfeld module defined over the ring of integers of a global function field we investigate the arithmetic properties of these objects and as an application we prove a discrete analogue for taelman s class modules of certain conjectures formulated by greenberg for cyclotomic fields

a herbrand ribet theorem for function fields we prove a function field analogue of the herbrand ribet theorem on cyclotomic number fields the herbrand ribet theorem can be interpreted as a result about cohomology with p coefficients over the splitting field of p and in our analogue both occurrences of p are replaced with the mathfrak p torsion scheme of the carlitz module for a prime mathfrak p in f q t

infinite iteration of matrix semigroups ii structure theorem for arbitrary semigroups up to aperiodic morphism where 0 is defined by a a 8 a a here intuitively a is easily globally computed because concatenation of two strings is a transparent multiplication and a a i s surely special however here 0 need not be line the unsolvability of the word problem for semigroups is one way to state this we shall formulate it differently later in fact using the 25 0021 8693 86 3 00

rare behavior of growth processes via umbrella sampling of trajectories we compute probability distributions of trajectory observables for reversible and irreversible growth processes these results reveal a correspondence between reversible and irreversible processes at particular points in parameter space in terms of their typical and atypical trajectories thus key features of growth processes can be insensitive to the precise form of the rate constants used to generate them recalling the insensitivity to microscopic details of certain equilibrium behavior we obtained these results using a sampling method inspired by the s ensemble large deviation formalism that amounts to umbrella sampling in trajectory space the method is a simple variant of existing approaches and applies to ensembles of trajectories controlled by the total number of events it can be used to determine large deviation rate functions for trajectory observables in or out of equilibrium

values of certain l series in positive characteristic we introduce a class of deformations of the values of the goss zeta function we prove with the use of the theory of deformations of vectorial modular forms as well as with other techniques a formula for their value at 1 and some arithmetic properties of values at other positive integers our formulas involve anderson and thakur s function we discuss how our formulas may be used to investigate the existence of a kind of functional equation for the goss zeta function

a dirichlet unit theorem for drinfeld modules we show that the module of integral points on a drinfeld module satisfies an analogue of dirichlet s unit theorem despite its failure to be finitely generated as a consequence we obtain a construction of a canonical finitely generated sub module of the module of integral points we use the results to give a precise formulation of a conjectural analogue of the class number formula

arithmetic of characteristic p special l values recently the second author has associated a finite fq t module h to the carlitz module over a finite extension of fq t this module is an analogue of the ideal class group of a number field in this paper we study the galois module structure of this module h for cyclotomic extensions of fq t we obtain function field analogues of some classical results on cyclotomic number fields such as the p adic class number formula and a theorem of mazur and wiles about the fitting ideal of ideal class groups we also relate the galois module h to anderson s module of circular units and give a negative answer to anderson s kummer vandiver type conjecture these results are based on a kind of equivariant class number formula which refines the second author s class number formula for the carlitz module

special l values of drinfeld modules we state and prove a formula for a certain value of the goss l function of a drinfeld module this gives characteristic p valued function eld analogues of the class number formula and of the birch and swinnerton dyer conjecture the formula and its proof are presented in an entirely selfcontained fashion

two algebraic approaches to variants of the concatenation product we extend an existing approach of the bideterministic concatenation product of languages aiming at the study of three other variants unambiguous left deterministic and right deterministic such an approach is based on monoid expansions the proofs are purely algebraic and use another approach based on properties on the kernel category of a monoid relational morphism without going through the languages this gives a unified fashion to deal with all these variants and allows us to better understand the connections between these two approaches finally we show that local finiteness of an m variety is transferred to the m varieties corresponding to these variants and apply the general results to the m variety of idempotent and commutative monoids

algebraic tools for the concatenation product this paper is a contribution to the algebraic study of the concatenation product in the first part of the paper we extend to the ordered case standard algebraic tools related to the concatenation product like the schutzenberger product and the relational morphisms we show in a precise way how the ordered schutzenberger product corresponds to polynomial operations on languages in the second part of the paper we apply these results to establish a bridge between the three standard concatenation hierarchies namely the straubing therien hierarchy the brzozowski s or dot depth hierarchy and the group hierarchy

actions wreath products of c varieties and concatenation product the framework of c varieties introduced by the third author extends the scope of eilenberg s variety theory to new classes of languages in this paper we first define c varieties of actions which are closely related to automata and prove their equivalence with the original definition of c varieties of stamps next we complete the study of the wreath product initiated by esik and ito by extending its definition to c varieties in two different ways which are proved to be equivalent we also state an extension of the wreath product principle a standard tool of language theory finally our main result generalizes to c varieties the algebraic characterization of the closure under product of a variety of languages

iwasawa main conjecture for the carlitz cyclotomic extension and applications we prove an iwasawa main conjecture for the class group of the mathfrak p cyclotomic extension mathcal f of the function field mathbb f q theta mathfrak p is a prime of mathbb f q theta showing that its fitting ideal is generated by a stickelberger element we use this and a link between the stickelberger element and a mathfrak p adic l function to prove a close analog of the ferrero washington theorem for mathcal f and to provide information on the mathfrak p adic valuations of the bernoulli goss numbers beta j i e on the values of the carlitz goss zeta function at negative integers

stark units in positive characteristic we show that the module of stark units associated to a sign normalized rank one drinfeld module can be obtained from anderson s equivariant a harmonic series we apply this to obtain a class formula a la taelman and to prove a several variable log algebraicity theorem generalizing anderson s log algebraicity theorem we also give another proof of anderson s log algebraicity theorem using shtukas and obtain various results concerning the module of stark units for drinfeld modules of arbitrary rank

gauss sums for fq t the purpose of this paper is to define gauss sums taking values in function fields of one variable over a finite field and to prove analogues of various classical and recent results these results include stickelberger s theorem the hasse davenport theorem weil s theorem on jacobi sums as hecke characters and the gross koblitz theorem for comparison the reader may consult g k and references given there