a note on a class of observability problems for pdes abstract the question of observability arises naturally in the analysis of control problems if the solution of a pde initial boundary value problem is known to be zero in a part of the domain does this guarantee it is zero everywhere the most popular techniques to establish such results are based on local unique continuation results holmgren s theorem or carleman estimates the purpose of this note is to draw attention to a class of problems where the observed region is bounded by characteristics and local unique continuation fails nevertheless observability may hold a problem of this nature arose in recent work by the author on control of viscoelastic flows m renardy are viscoelastic flows under control or out of control system control lett 54 2005 1183 1193 in this note we first analyze a simple example which shares the same essential features specifically we consider the problem u x t u for spatially periodic solutions we show that observability holds for data given on the line x 0 we shall show however that there is no observability estimate we shall then show how the methods used in the more elementary example can be extended to the case of viscoelastic shear flows

arithmetic duality theorems for 1 motives over function fields in this paper we obtain a poitou tate exact sequence for finite and flat group schemes over a global function field in addition we extend the duality theorems for 1 motives over number fields obtained by d harari and t szamuely to the function field case

the winding angle distribution of an ordinary random walk the winding angle about a selected point of a random walk quantifies what is perhaps the simplest manifestation of entanglement a phenomenon of great importance in the study of polymers the distribution of winding angles is studied here for ordinary or non self avoiding walks new results highlight the crucial influence of the exclusion of a finite region about the selected point on the winding angle distribution the results of this analysis are compared with simulations of random walks on a lattice the agreement between the two is very good there are however small and as yet unexplained discrepancies

on albanese torsors and the elementary obstruction we show that the elementary obstruction to the existence of 0 cycles of degree 1 on an arbitrary variety x over an arbitrary field can be expressed in terms of the albanese 1 motives associated with dense open subsets of x arithmetic applications are given

some controllability results for linear viscoelastic fluids we analyze the controllability properties of systems which provide a description at first approximation of a kind of viscoelastic fluid we consider linear maxwell fluids first we establish the large time approximate finite dimensional controllability of the system with distributed or boundary controls supported by arbitrary small sets then we prove the large time exact controllability of fluids of the same kind with controls supported by suitable large sets the proofs of these results rely on classical arguments in particular the approximate controllability result is implied by appropriate unique continuation properties while exact controllability is a consequence of observability inverse inequalities we also discuss questions concerning the controllability of viscoelastic fluids and some related open problems

are viscoelastic flows under control or out of control we discuss the controllability of viscoelastic shear flows both linear and nonlinear results are reviewed an interesting aspect of the problem is that the nonlinear problem introduces new stress components which are automatically zero in the linear case it is therefore a challenging problem to determine to what extent these stresses are controllable

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

extended picard complexes and linear algebraic groups for a smooth geometrically integral variety x over a field k of characteristic 0 we introduce and investigate the extended picard complex upic x it is a certain complex of galois modules of length 2 whose zeroth cohomology is k x k and whose first cohomology is pic x where k is a fixed algebraic closure of k and x is obtained from x by extension of scalars to k when x is a k torsor of a connected linear k group g we compute upic x upic g in the derived category in terms of the algebraic fundamental group 1 g as an application we compute the elementary obstruction for such x

approximate controllability results for viscoelastic flows with infinitely many relaxation modes abstract we prove results on approximate controllability for linear viscoelastic flows with a localized distributed control in the momentum balance equation the constitutive law is a multimode maxwell or jeffreys model with an infinite number of relaxation modes

on the control of viscoelastic jeffreys fluids abstract this paper is devoted to analyzing the control of vicoelastic fluids of the jeffreys kind also known as oldroyd models we will present the interesting problems with special emphasis in the difficulties that they involve then we will consider appropriate linear approximations and we will establish some partial approximate finite dimensional controllability results in an arbitrarily small time with distributed or boundary controls supported by arbitrarily small sets the proofs rely on some specific unique continuation properties which are implied by the structure of the solutions

on abelian birational sections for a smooth and geometrically irreducible variety x over a field k the quotient of the absolute galois group gk x by the commutator subgroup of g k x projects onto gk we investigate the sections of this projection we show that such sections correspond to infinite divisions of the elementary obstruction of colliot thelene and sansuc if k is a number field and the tate shafarevich group of the picard variety of x is finite then such sections exist if and only if the elementary obstruction vanishes for curves this condition also amounts to the existence of divisors of degree 1 finally we show that the vanishing of the elementary obstruction is not preserved by extensions of scalars

the elementary obstruction and homogeneous spaces let k be a field of characteristic zero and let k be an algebraic closure of k for a geometrically integral variety x over k we write k x for the function field of x x k k if x has a smooth k point the natural embedding of multiplicative groups k k x admits a galois equivariant retraction in the first part of this article equivalent conditions to the existence of such a retraction are given over local and then over global fields those conditions are expressed in terms of the brauer group of x in the second part of the article we restrict attention to varieties that are homogeneous spaces of connected but otherwise arbitrary algebraic groups with connected geometric stabilizers for k local or global and for such a variety x in many situations but not all the existence of a galois equivariant retraction to k k x ensures the existence of a k rational point on x for homogeneous spaces of linear algebraic groups the technique also handles the case where k is the function field of a complex surface resume soient k un corps de caracteristique nulle et k une cloture algebrique de k pour une k variete x geometriquement integre on note k x le corps des fonctions de x x k k si x possede un k point lisse le plongement naturel de groupes multiplicatifs k k x admet une retraction equivariante pour l action du groupe de galois de k sur k dans la premiere partie de l article sur les corps locaux puis sur les corps globaux on donne des conditions equivalentes a l existence d une telle retraction equivariante ces conditions s expriment en terme du groupe de brauer de la variete x dans la seconde partie de l article on considere le cas des espaces homogenes de groupes algebriques connexes non necessairement lineaires avec groupes d isotropie duke mathematical journal vol 141 no 2 c 2008 received 6 november 2006 revision received 23 april 2007 2000 mathematics subject classification primary 14g05 11g99 12g05 secondary 11e72 14f22 14k15

une formule pour le groupe de brauer algebrique dʼun torseur abstract given a homogeneous space x of a connected algebraic group g with connected stabilizers over a field k of characteristic zero we construct a complex of galois modules of length 3 and a canonical isomorphism between a hypercohomology group of this complex and an explicit subgroup of the brauer group of x this result is obtained as a consequence of a formula describing the algebraic brauer group of a torsor and it generalizes recent results by borovoi and van hamel by considering non linear groups g and by taking into account some transcendental elements in the brauer group of x

local global principles for 1 motives building upon our arithmetic duality theorems for 1 motives we prove that the manin obstruction related to a finite subquotient x of the brauer group is the only obstruction to the hasse principle for rational points on torsors under semiabelian varieties over a number field assuming the finiteness of the tate shafarevich group of the abelian quotient this theorem answers a question by skorobogatov in the semiabelian case and is a key ingredient of recent work on the elementary obstruction for homogeneous spaces over number fields we also establish a cassels tate type dual exact sequence for 1 motives and give an application to weak approximation

notes on the two dimensional fractional brownian motion we study the two dimensional fractional brownian motion with hurst parameter h 1 2 in particular we show using stochastic calculus that this process admits a skew product decomposition and deduce from this representation some asymptotic properties of the motion

galois sections for abelianized fundamental groups given a smooth projective curve x of genus at least 2 over a number field k grothendieck s section conjecture predicts that the canonical projection from the etale fundamental group of x onto the absolute galois group of k has a section if and only if the curve has a rational point we show that there exist curves where the above map has a section over each completion of k but not over k in the appendix victor flynn gives explicit examples in genus 2 our result is a consequence of a more general investigation of the existence of sections for the projection of the etale fundamental group with abelianized geometric part onto the galois group we also point out the relation to the elementary obstruction of colliot thelene and sansuc

integral points for multi norm tori we construct a finite subgroup of brauer manin obstruction for detecting the existence of integral points on integral models of principle homogeneous spaces of multi norm tori several explicit examples are provided

winding number of fractional brownian motion we find the exact winding number distribution of riemann liouville fractional brownian motion for large times in two dimensions using the propagator of a free particle the distribution is similar to the brownian motion case and is of cauchy type in addition we find the winding number distribution of the fractal time process i e the time fractional fokker planck equation in the presence of a finite size winding center

fractional langevin equation we investigate fractional brownian motion with a microscopic random matrix model and introduce a fractional langevin equation we use the latter to study both subdiffusion and superdiffusion of a free particle coupled to a fractal heat bath we further compare fractional brownian motion with the fractal time process the respective mean square displacements of these two forms of anomalous diffusion exhibit the same power law behavior here we show that their lowest moments are actually all identical except the second moment of the velocity this provides a simple criterion that enable us to distinguish these two non markovian processes