ramanujan s ternary quadratic form do not seem to obey any simple law following i kaplansky we call a non negative integer n eligible for a ternary form f x y z if there are no congruence conditions prohibiting f from representing n by the classical theory of quadratic forms it is well known that any given genus of positive definite ternary quadratic forms represents every eligible integer consequently if a genus consists of a single class with representative f x y z then f represents every eligible integer in the case of ramanujan s form this only implies that an eligible integer one not of the form 4 16 6 is represented by 1 or 2

moment analysis for localization in random schrodinger operators we study localization effects of disorder on the spectral and dynamical properties of schrodinger operators with random potentials the new results include exponentially decaying bounds on the transition amplitude and related projection kernels including in the mean these are derived through the analysis of fractional moments of the resolvent which are finite due to the resonance diffusing effects of the disorder the main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the lifshitz krein spectral shift associated with the local potential terms the difficulty is avoided here through the use of a weak l 1 estimate concerning the boundary value distribution of resolvents of maximally dissipative operators combined with standard tools of relative compactness theory

surface states and spectra let d 1 let h 0 be the discrete laplacian on the hilbert space l 2 d 1 with a dirichlet boundary condition and let v be a potential supported on the boundary d 1 we introduce the notions of surface states and surface spectrum of the operator h h 0 v and explore their properties our main result is that if the potential v is random and if the disorder is either large or small enough then in dimension two h has no surface spectrum on h 0 with probability one to prove this result we combine aizenman molchanov theory with techniques of scattering theory

new characterizations of the region of complete localization for random schrodinger operators we study the region of complete localization in a class of random operators which includes random schrodinger operators with anderson type potentials and classical wave operators in random media as well as the anderson tight binding model we establish new characterizations or criteria for this region of complete localization given either by the decay of eigenfunction correlations or by the decay of fermi projections these are necessary and sufficient conditions for the random operator to exhibit complete localization in this energy region using the first type of characterization we prove that in the region of complete localization the random operator has eigenvalues with finite multiplicity

the l2 decay for the 2d co rotation fene dumbbell model of polymeric flows abstract in this paper we mainly study the long time behavior of solutions to the finite extensible nonlinear elastic fene dumbbell model with dimension two in the co rotation case firstly we obtain the l 2 decay rate of the velocity of the 2d co rotation fene model is 1 t 1 2 with small data then by virtue of the littlewood paley theory we can remove the small condition our obtained sharp result improves considerably the recent results in 8 14

local existence for the fene dumbbell model of polymeric fluids we study the well posedness of a multi scale model of polymeric fluids the microscopic model is the kinetic theory of the finitely extensible nonlinear elastic fene dumbbell model the macroscopic model is the incompressible non newton fluids with polymer stress computed via the kramers expression the boundary condition of the fene type fokker planck equation is proved to be unnecessary by the singularity on the boundary other main results are the local existence uniqueness and regularity theorems for the fene model in certain parameter range

localization bounds for multiparticle systems we consider the spectral and dynamical properties of quantum systems of n particles on the lattice mathbb z d of arbitrary dimension with a hamiltonian which in addition to the kinetic term includes a random potential with iid values at the lattice sites and a finite range interaction two basic parameters of the model are the strength of the disorder and the strength of the interparticle interaction it is established here that for all n there are regimes of high disorder and or weak enough interactions for which the system exhibits spectral and dynamical localization the localization is expressed through bounds on the transition amplitudes which are uniform in time and decay exponentially in the hausdorff distance in the configuration space the results are derived through the analysis of fractional moments of the n particle green function and related bounds on the eigenfunction correlators

a new proof of localization in the anderson tight binding model we give a new proof of exponential localization in the anderson tight binding model which uses many ideas of the frohlich martinelli scoppola and spencer proof but is technically simpler particularly the probabilistic estimates

well posedness for the fene dumbbell model of polymeric flows we prove local and global well posedness for the fene dumbbell model for a very general class of potentials indeed in prior local or global well posedness results conditions on the strength of the singularity or on the parameter b were made here we give a proof in the general case we also prove global existence results if the data is small or if we restrict to the co rotational model in dimension 2 2008 wiley periodicals inc

many body localization in the droplet spectrum of the random xxz quantum spin chain abstract we study many body localization properties of the disordered xxz spin chain in the ising phase disorder is introduced via a random magnetic field in the z direction we prove a strong form of dynamical exponential clustering for eigenstates in the droplet spectrum for any pair of local observables separated by a distance l the sum of the associated correlators over these states decays exponentially in l in expectation this exponential clustering persists under the time evolution in the droplet spectrum our result applies to the large disorder regime as well as to the strong ising phase at fixed disorder with bounds independent of the support of the observables

hardy littlewood varieties and semisimple groups we are interested in counting integer and rational points in affine algebraic varieties also under congruence conditions we introduce the notions of a strongly hardy littlewood variety and a relatively hardy littlewood variety in terms of counting rational points satisfying congruence conditions the definition of a strongly hardy littlewood variety is given in such a way that varieties for which the hardy littlewood circle method is applicable are strongly hardy littlewood

existence of solution for a micro macro model of polymeric fluid the fene model abstract we analyse a non linear micro macro model of polymeric fluids in the case of a shear flow more precisely we consider the fene dumbbell model which models polymers by nonlinear springs accounting for the finite extensibility of the polymer chain we prove the existence of a unique solution to the stochastic differential equation which rules the evolution of a representative polymer in the flow and next deduce a local in time existence and uniqueness result on the system coupling the stochastic differential equation and the momentum equation on the fluid

global existence of weak solutions to the fene dumbbell model of polymeric flows systems coupling fluids and polymers are of great interest in many branches of sciences one of the most classical models to describe them is the fene finite extensible nonlinear elastic dumbbell model we prove global existence of weak solutions to the fene dumbbell model of polymeric flows the main difficulty is the passage to the limit in a nonlinear term that has no obvious compactness properties the proof uses many weak convergence techniques in particular it is based on the control of the propagation of strong convergence of some well chosen quantity by studying a transport equation for its defect measure in addition this quantity controls a rescaled defect measure of the gradient of the velocity

localization of surface spectra we study spectral properties of the discrete laplacian h on the half space with random boundary condition the v n are independent random variables on a probability space and is the coupling constant it is known that if the v n have densities then on the interval 2 d 1 2 d 1 h 0 the spectrum of the dirichlet laplacian the spectrum of h is p a s absolutely continuous for all jl1 here we show that if the random potential p satisfies the assumption of aizenman molchanov am then there are constants d and d such that for d the spectrum of h outside h 0 is p a s pure point with exponentially decaying eigenfunctions

anderson localization for random schrodinger operators with long range interactions we prove pure point spectrum with exponentially decaying eigenfunctions at all band edges for schrodinger operators with a periodic potential plus a random potential of the form v w x q i w f x i where f decays at infinity like x m for m 4d resp m 3d depending on the regularity of f the random variables q i are supposed to be independent and identically distributed we assume that their distribution has a bounded density of compact support

lieb robinson bounds and the exponential clustering theorem we give a lieb robinson bound for the group velocity of a large class of discrete quantum systems which can be used to prove that a non vanishing spectral gap implies exponential clustering in the ground state of such systems

bootstrap multiscale analysis and localization in random media we introduce an enhanced multiscale analysis that yields subexponentially decaying probabilities for bad events for quantum and classical waves in random media we obtain exponential decay for the resolvent of the corresponding random operators in boxes of side l with probability higher than 1 e l for any 0 1 the starting hypothesis for the enhanced multiscale analysis only requires the verification of polynomial decay of the finite volume resolvent at some sufficiently large scale with probability bigger than 1 d is the dimension note that from the same starting hypothesis we get conclusions that are valid for any 0 1 this is achieved by the repeated use of a bootstrap argument as an application we use a generalized eigenfunction expansion to obtain strong dynamical localization of any order in the hilbert schmidt norm and better estimates on the behavior of the eigenfunctions

lower bounds of rates of decay for solutions to the navier stokes equations where a p 3 2 p 1 and the constant c depends on the l2 and lnorms of the initial data u0 this paper deals with the more subtle problem of deriving lower bounds on the energy decay rates we show that for a certain class of initial data the solutions u x t to the 2d and 3d navier stokes equations admit an algebraic lower bound on the energy decay specifically there are two cases to consider in the first case the average of the initial data f u0 dx is nonzero this case was treated in the earlier paper 5 where it was established that

strong approximation for certain quadric fibrations with compact fibers abstract by developing the representation theory of quadratic diophantine equations we prove strong approximation with brauer manin obstruction for certain quadratic fibrations where each fiber is compact