families of a optimal block designs for comparing test treatments with a control a optimal designs for comparing each of v test treatments simultaneously with a control in b blocks of size k each are considered it is shown that several families of bib designs in the test treatments augmented by t replications of a control in each block are a optimal in particular these designs with t 1 are optimal whenever k 2 2 1 v s k 1 2 irrespective of the number of blocks this includes bib designs associated with finite projective and euclidean geometries

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

the de rham isomorphism for drinfeld modules over tate algebras abstract introduced by angles pellarin and tavares ribeiro drinfeld modules over tate algebras are closely connected to anderson log algebraicity identities pellarin l series and taelman class modules in the present paper we define the de rham map for drinfeld modules over tate algebras and we prove that it is an isomorphism under natural hypotheses as part of this investigation we determine further criteria for the uniformizability and rigid analytic triviality of drinfeld modules over tate algebras

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

on identities for zeta values in tate algebras zeta values in tate algebras were introduced by pellarin in 2012 they are generalizations of carlitz s zeta values and play an increasingly important role in function field arithmetic in this paper we prove a conjecture of pellarin on identities for these zeta values the proof is based on arithmetic properties of carlitz s zeta values and an explicit formula for bernoulli type polynomials attached to pellarin s zeta values

the classification of kleinian surface groups ii the ending lamination conjecture thurston s ending lamination conjecture states that a hyperbolic 3manifold n with nitely generated fundamental group is uniquely determined by its topological type and its end invariants in this paper we prove this conjecture for kleinian surface groups the general case when n has incompressible ends relative to its cusps follows readily the main ingredient is a uniformly bilipschitz model for the quotient of h 3 by a kleinian surface group

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

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

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

levy statistics in a hamiltonian system enhanced diffusion in a hamiltonian system is studied in terms of the continuous time random walk formulation for l evy walks the previous l evy walk scheme is extended i to include interruptions by periods of temporal localization and ii to describe motion in two dimensions we analyze a case of conservative motion in a two dimensional periodic potential numerical calculations of the mean squared displacements and the propagators for intermediate energies are consistent with the l evy walk description

universal gauss thakur sums and l series we prove that the maximal abelian extension tamely ramified at infinity of the rational function field over mathbb f q is generated by the values at the points in the algebraic closure of mathbb f q of the higher derivatives of the so called anderson and thakur function omega we deduce a similar property for the special values of the higher derivatives of a new kind of l series introduced by the second author

an analytical approximation to the diffusion coefficient in overdamped multidimensional systems abstract an analytical approximation for the mobility of an overdamped particle in a periodic multi dimensional system is presented attention is focused on two dimensions quasi 2d approximation in the most generic case of a 2d coupled periodic potential in a rectangular lattice and of a position dependent friction the approximation is derived in the framework of the linear response theory by fixing the value of one coordinate and solving the problem of diffusion along the other coordinate as strictly 1d this is expected to be essentially correct if all the most relevant diffusion paths are straight lines two different specific applications have been considered diffusion in a square egg carton potential and diffusion in absence of potential in a 2d channel with unsurmountable periodic walls exact results are available in literature in the latter case and are obtained in the first case by solving the smoluchowski equation matrix continued fraction method comparisons with the quasi 2d approximation show that the agreement is excellent for the egg carton potential but far less satisfying for migration in the 2d periodically shaped channel characterized by important diffusion paths not being straight lines

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