the proper dissipative extensions of a dual pair let a and widetilde a be dissipative operators on a hilbert space mathcal h and let a widetilde a form a dual pair i e a subset widetilde a resp widetilde a subset a we present a method of determining the proper dissipative extensions widehat a of this dual pair i e a subset widehat a subset widetilde a provided that mathcal d a cap mathcal d widetilde a is dense in mathcal h applications to symmetric operators symmetric operators perturbed by a relatively bounded dissipative operator and more singular differential operators are discussed finally we investigate the stability of the numerical range of the different dissipative extensions

hasse principle for hermitian spaces over semi global fields abstract in a recent paper colliot thelene parimala and suresh conjectured that a local global principle holds for projective homogeneous spaces under connected linear algebraic groups over function fields of p adic curves in this paper we show that the conjecture is true for any linear algebraic group whose almost simple factors of its semisimple part are isogenous to unitary groups or special unitary groups of hermitian or skew hermitian spaces over central simple algebras with involutions the proof implements patching techniques of harbater hartmann and krashen as an application we obtain a springer type theorem for isotropy of hermitian spaces over odd degree extensions of function fields of p adic curves

the u invariant of the function fields of p adic curves the u invariant of a field is the maximum dimension of ansiotropic quadratic forms over the field it is an open question whether the u invariant of function fields of p aidc curves is 8 in this paper we answer this question in the affirmative for function fields of nondyadic p adic curves

fractality of deterministic diffusion in the nonhyperbolic climbing sine map abstract the nonlinear climbing sine map is a nonhyperbolic dynamical system exhibiting both normal and anomalous diffusion under variation of a control parameter we show that on a suitable coarse scale this map generates an oscillating parameter dependent diffusion coefficient similarly to hyperbolic maps whose asymptotic functional form can be understood in terms of simple random walk approximations on finer scales we find fractal hierarchies of normal and anomalous diffusive regions as functions of the control parameter by using a green kubo formula for diffusion the origin of these different regions is systematically traced back to strong dynamical correlations starting from the equations of motion of the map these correlations are formulated in terms of fractal generalized takagi functions obeying generalized de rham type functional recursion relations we finally analyze the measure of the normal and anomalous diffusive regions in the parameter space showing that in both cases it is positive and that for normal diffusion it increases by increasing the parameter value

dynamical crossover in deterministic diffusion we study diffusion in a one dimensional periodic array of scatterers modeled by a simple map the chaotic scattering process of the map can be changed by a control parameter and exhibits a dynamics analogous to a crisis in chaotic scattering we show that the associated strong backscattering induces a crossover between different asymptotic laws for the parameter dependent diffusion coefficient these laws are obtained from exact diffusion coefficient results and are supported by simple random walk models we conjecture that the main physical feature of this crossover is present in many other dynamical systems exhibiting nonequilibrium transport s1063 651x 97 50502 2 pacs number s 05 45 1b 05 60 1w

simple deterministic dynamical systems with fractal diffusion coefficients we analyze a simple model of deterministic diffusion the model consists of a one dimensional array of scatterers with moving point particles the particles move from one scatterer to the next according to a piecewise linear expanding deterministic map on unit intervals the microscopic chaotic scattering process of the map can be changed by a control parameter the macroscopic diffusion coefficient for the moving particles is well defined and depends upon the control parameter we calculate the diffusion coefficent and the largest eigenmodes of the system by using markov partitions and by solving the eigenvalue problems of respective topological transition matrices for different boundary conditions we find that the largest eigenmodes of the map match the ones of the simple phenomenological diffusion equation our main result is that the diffusion coefficient exhibits a fractal structure as a function of the control parameter we provide qualitative and quantitative arguments to explain features of this fractal structure

scale invariant motion in intermittent chaotic systems we investigate the dynamics generated from iterated maps and analyze the motion in terms of the probabilistic continuous time random walk ctrw approach two different ctrw models are considered i particles jump between sites turning points or ii particles move at a constant velocity between sites and choose a new direction at random for both models we study the mean squared displacement mathit r 2 t and the propagator p r t the probability to be at location r at time t having started at the origin at t 0 iterated maps are used to generate both dispersive and enhanced diffusion and the results are analyzed using the ctrw framework and scaling arguments for the case of dispersive motion we discuss the problem of the stationary state and point out its relevance

local global principles for norm one tori over semi global fields let k be a complete discretely valued field with the residue field kappa let f be the function field of a smooth projective geometrically integral curve over k and mathscr x be a regular proper model of f such that the reduced special fibre x is a union of regular curves with normal crossings suppose that the graph associated to mathscr x is a tree e g f k t let l f be a galois extension of degree n such that n is coprime to char kappa suppose that kappa is an algebraically closed field or a finite field containing a primitive n mathrm th root of unity then we show that the local global principle holds for the norm one torus associated to the extension l f with respect to discrete valuations on f i e an element in f times is a norm from the extension l f if and only if it is a norm from the extensions l otimes f f nu f nu for all discrete valuations nu of f

phase diffusion in a chaotic pendulum the rate of expansion of the phase coordinate for a harmonically driven pendulum is considered the mean squared displacement is found to grow as a linear function of time during chaotic motion indicating deterministic diffusion the diffusion coefficient can be significantly influenced by the proximity of a window containing a periodic solution we find that diffusion associated with intermittent chaos can be described in terms of an interleaving of the diffusion properties of the separate modes taking part in the intermittency copyright ital 1996 the american physical society

negative and nonlinear response in an exactly solved dynamical model of particle transport we consider a simple model of particle transport on the line defined by a dynamical map f satisfying f x 1 1 f x for all x and f x ax b for x 1 and therein mainly to the part a 1 we study along the lines of previous investigations r klages and j r dorfman phys rev lett 74 387 1995 on the restricted symmetric b 0 one parameter version of the present model the parameter dependence of the transport properties i e not only of the diffusion coefficient d a b but this time also of the current j a b a major difference however is that this time an important tool for such a study has been available in the form of exact expressions for j and d obtained recently by one of the authors these expressions allow for a quite efficient numerical implementation which is important because the functions encountered typically have a fractal character the main results of our present preliminary survey of the parameter plane of the model are presented in several plots of these functions j a b and d a b and in an over all chart displaying in the parameter plane in principle all possibly relevant information on the system including e g the dynamical phase diagram as well as by way of illustration values of some topological invariants kneading numbers which according to the formulas determine the singularity structure of j a b and d a b what we regard as our most significant findings are 1 nonlinear response the parameter dependence of these transport properties is throughout the ergodic part of the parameter plane i e outside the infinitely many arnol d tongues fractally nonlinear 2 negative response inside certain regions with an apparently fractal boundary the current j and the bias b have opposite signs

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

applications of patching to quadratic forms and central simple algebras this paper provides applications of patching to quadratic forms and central simple algebras over function fields of curves over henselian valued fields in particular we use a patching approach to reprove and generalize a recent result of parimala and suresh in preprint arxiv 0708 3128 2007 on the u invariant of p adic function fields p 2 the strategy relies on a local global principle for homogeneous spaces for rational algebraic groups combined with local computations

normal and anomalous diffusions in web map driven by a constant external force abstract we consider the conservative web map x n 1 y n y n 1 x n k sin y n where is the external constant force for a range of values of k with 0 the diffusion is found to be anomalous zaslavsky et al chaos 7 1997 159 due to the presence of accelerator mode islands we investigate the effect of on anomalous both dispersive and enhanced and normal diffusions for four fixed values of k we show the occurrence of transition from anomalous to normal and vice versa processes by varying the value of from zero we characterize the anomalous and normal dynamics using the exponents describing the evolution of mean square displacement and describing the evolution of variance of short time moving distance velocity spectrum kurtosis and propagator the values of the exponents and are found to depend strongly on the external force we show that for anomalous diffusion the probability distributions of the variables x and y are levy

an expanded phillips theory and its application to differential operators abstract a new characterization of maximal accretive extensions of accretive operators is presented it is based on expanded spaces of the abstract boundary spaces developed by r s phillips and is therefore called the expanded phillips theory in particular when the operator is symmetric a concrete realization of the expanded spaces is given in applications of the expanded phillips theory to ordinary differential equations we characterize all maximal accretive extensions of singular symmetric ordinary differential operators of order 2 n

on the u invariant of function fields of curves over complete discretely valued fields abstract let k be a complete discretely valued field with residue field if char k 0 char 2 and 2 d we prove that there exists an integer n depending on d such that the u invariant of any function field in one variable over k is bounded by n the method of proof is via introducing the notion of uniform boundedness for the p torsion of the brauer group of a field and relating the uniform boundedness of the 2 torsion of the brauer group to the finiteness of the u invariant we prove that the 2 torsion of the brauer group of function fields in one variable over k is uniformly bounded

on the behaviour of brauer p dimensions under finitely generated field extensions abstract the present paper shows that if q p or q 0 where p is the set of prime numbers then there exist characteristic q fields e q k k n of brauer dimension brd e q k k and infinite absolute brauer p dimensions abrd p e q k for all p p not dividing q 2 q this ensures that brd p f q k p q 2 q for every finitely generated transcendental extension f q k e q k we also prove that each sequence a p b p p p satisfying the conditions a 2 b 2 and 0 b p a p equals the sequence abrd p e brd p e p p for a field e of characteristic zero

simple maps with fractal diffusion coefficients we consider chains of one dimensional piecewise linear chaotic maps with uniform slope we study the diffusive behavior of an initially nonuniform distribution of points as a function of the slope of the map by solving the frobenius perron equation for markov partition values of the slope we relate the diffusion coefficient to eigenvalues of the topological transition matrix the diffusion coefficient obtained shows a fractal structure as a function of the slope of the map this result may be typical for a wide class of maps such as two dimensional sawtooth maps

classification theorems for hermitian forms the rost kernel and hasse principle over fields with cd2 k 3 abstract in this paper we prove classification theorems for hermitian forms over some central simple algebras with involution over a field k with cd 2 k 3 we apply these results to show the triviality of the kernel of the rost invariant for the classical algebraic groups associated to such hermitian forms over k we also deduce a hasse principle for algebraic groups defined over function fields of curves over p adic fields thus proving a conjecture due to colliot thelene parimala suresh for a large class of groups

on the extension problem for singular accretive differential operators on considere la description explicite des conditions aux limites pour tous les operateurs differentiels accretifs maximaux possibles deductibles d une expression differentielle formelle donnee

noncyclic division algebras over fields of brauer dimension one abstract let k be a complete discretely valued field of rank one with residue field q p it is well known that period equals index in br k we prove that when p 2 there exist noncyclic k division algebras of every 2 power degree divisible by four otherwise every k division algebra is cyclic this gives the first published example of a field whose brauer dimension and cyclic length are not equal

colliot thelene s conjecture and finiteness of u invariants let k be a number field and f a function field in one variable over k we prove that the ramification of a p torsion element in br f on a regular proper model over the ring of integers in k can be split in an extension of degree p 3 using this result we show that colliot thelene s conjecture on 0 cycles of degree 1 implies finiteness for the u invariant of the function field of a curve over a totally imaginary number field and period index bounds for the brauer groups of arbitrary fields of transcendence degree 1 over the rational numbers

on the extension problem for accretive differential operators abstract let t0 denote the minimal operator corresponding to the formal differential expression y x w x 1 j 0n 1 j pn j x y j j x i in l2 i under the assumption that t0 is an accretive operator in l2 i a complete description of all of the maximal accretive extensions of t0 via explicit boundary conditions is given for a wide class of differential expressions

statistical approach to nonhyperbolic chaotic systems spectral properties of the evolution operator for probability densities are obtained for unimodal maps for which all periodic orbits are unstable and the lyapunov exponent calculated from the first iterate of the critical point converges to a positive constant the method is applied to the logistic map both for parameter values at which finite markov partitions can be found as well as for more typical parameter values a universal behavior is found for the spectral gap in the period doubling inverse cascade of chaotic band merging bifurcations full agreement with numerical simulation is obtained textcopyright 1996 the american physical society

period index bounds for arithmetic threefolds the standard period index conjecture for brauer groups of p adic surfaces s predicts that mathrm ind alpha mathrm per alpha 3 for every alpha in mathrm br mathbf q p s using gabber s theory of prime to ell alterations and the deformation theory of twisted sheaves we prove that mathrm ind alpha mathrm per alpha 4 for alpha of period prime to 6p giving the first uniform period index bounds over such fields

local global principles for torsors over arithmetic curves we consider local global principles for torsors under linear algebraic groups over function fields of curves over complete discretely valued fields the obstruction to such a principle is a version of the tate shafarevich group and for groups with rational components we compute it explicitly and show that it is finite this yields necessary and sufficient conditions for local global principles to hold our results rely on first obtaining a mayer vietoris sequence for galois cohomology and then showing that torsors can be patched we also give new applications to quadratic forms and central simple algebras

period index and u invariant questions for function fields over complete discretely valued fields let k be a complete discretely valued field with residue field and f the function field of a curve over k let p be the characteristic of and l a prime not equal to p if the brauer l dimensions of all finite extensions of are bounded by d and the brauer l dimensions of all extensions of of transcendence degree at most 1 are bounded by d 1 then it is known that the brauer l dimension of f is at most d 2 lieblich in j reine angew math 659 1 41 2011 saltman in j ramanujan math soc 12 25 47 1997 harbater et al in invent math 178 231 263 2009 in this paper we give a bound for the brauer p dimension of f in terms of the p rank of as an application we show that if is a perfect field of characteristic 2 then any quadratic form over f in at least 9 variables is isotropic this leads to the fact that every element in h 3 f mathbb z 2 mathbb z is a symbol if is a finite field of characteristic 2 u f 8 is a result of heath brown leep heath brown in compos math 146 271 287 2010 leep in j reine angew math 2013 to appear