computational analysis of modeling error for the coupling of particle and continuum models by the arlequin method we propose in this paper a 1d model problem to study the convergence of surrogate approximations using an atomistic to continuum coupling method towards the solution of a full particle model the 1d problem consists of a collection of springs that exhibits a local defect materialized by a sudden change in the spring properties the surrogate model is obtained by the arlequin approach which introduces an overlap region in which the continuum and particle models are coupled together using lagrange multipliers the objective of the present work is to show via numerical experiments that the modeling error does indeed converge to zero as the distance of the overlap region from the defect and or its size are increased

traces and determinants of strongly stochastic operators periodic orbit theory allows calculations of long time properties of chaotic systems from traces dynamical zeta functions and spectral determinants of deterministic evolution operators which are in turn evaluated in terms of periodic orbits for the case of stochastic dynamics a direct numerical evaluation of the trace of an evolution operator is possible as a multidimensional integral techniques for evaluating such path integrals are discussed using as an example the logistic map f x lambdax 1 x with moderate to strong additive gaussian noise rapid convergence is demonstrated for all values of lambda with strong noise as well as at fixed lambda 5 for all noise levels

analysis of an energy based atomistic continuum approximation of a vacancy in the 2d triangular lattice we present an a priori error analysis of a practical energy based atomistic continuum coupling method a v shapeev multiscale model simul 9 3 905 932 2011 in two dimensions for finite range pair potential interactions in the presence of vacancy defects we establish first order consistency and stability of the method from which we obtain a priori error estimates in the h1 norm and the energy in terms of the mesh size and the smoothness of the atomistic solution in the continuum region from these error estimates we obtain heuristics for an optimal choice of the atomistic region and the finite element mesh as well as convergence rates in terms of the number of degrees of freedom our analytical predictions are supported by extensive numerical tests

a discrete carleson theorem along the primes with a restricted supremum consider the discrete maximal function acting on finitely supported functions on the integers begin aligned mathcal c lambda f n sup lambda in lambda left sum p in pm mathbb p f n p log p frac e 2 pi i lambda p p right end aligned where pm mathbb p pm p p text is a prime and lambda subset 0 1 we give sufficient conditions on lambda met by finite unions of lacunary sets for this to be a bounded sublinear operator on ell p mathbb z for frac 3 2 p 6 this result has as its precedent the work of bourgain and wierdl on the discrete maximal function along the primes begin aligned sup n left frac 1 n sum p in mathbb p p le n f n p log p right end aligned and mirek and trojan on maximally truncated singular integrals along the primes begin aligned sup n left sum p in pm mathbb p p le n f n p frac log p p right end aligned as well as the most recent work of mirek trojan and zorin kranich studying the larger variational variants of the above maximal operators the proof relies on bourgain s elegant multi frequency logarithmic lemma developed in his study of pointwise ergodic theorems along polynomial orbits combined with the results of hytonen and lacey on a vector valued version of carleson s theorem and those of oberlin seeger tao thiele and wright on a variational version of carleson s theorem

trace formulas for stochastic evolution operators weak noise perturbation theory periodic orbit theory is all effective tool for the analysis of classical and quantum chaotic systems in this paper we extend this approach to stochastic systems in particular to mappings with additive noise the theory is cast in the standard field theoretic formalism and weak noise perturbation theory written in terms of feynman diagrams the result is a stochastic analog of the next to leading h corrections to the gutzwiller trace formula with long time averages calculated from periodic orbits of the deterministic system the perturbative corrections are computed analytically and tested numerically on a simple 1 dimensional system

chaotic field theory a sketch abstract spatio temporally chaotic dynamics of a classical field can be described by means of an infinite hierarchy of its unstable spatio temporally periodic solutions the periodic orbit theory yields the global averages characterizing the chaotic dynamics as well as the starting semiclassical approximation to the quantum theory new methods for computing corrections to the semiclassical approximation are developed in particular a nonlinear field transformation yields the perturbative corrections in a form more compact than the feynman diagram expansions

stability instability and error of the force based quasicontinuum approximation due to their algorithmic simplicity and high accuracy force based model coupling techniques are popular tools in computational physics for example the force based quasicontinuum qcf approximation is the only known pointwise consistent quasicontinuum approximation for coupling a general atomistic model with a finite element continuum model in this paper we present a detailed stability and error analysis of this method our optimal order error estimates provide a theoretical justification for the high accuracy of the qcf approximation they clearly demonstrate that the computational efficiency of continuum modeling can be utilized without a significant loss of accuracy if defects are captured in the atomistic region the main challenge we need to overcome is the fact that the linearized qcf operator is typically not positive definite moreover we prove that no uniform inf sup stability condition holds for discrete versions of the w1 p w1 q duality pairing with 1 p 1 q 1 if 1 p however we were able to establish an inf sup stability condition for a discrete version of the w1 w1 1 duality pairing which leads to optimal order error estimates in a discrete w1 norm

discrete analogues in harmonic analysis i l 2 estimates for singular radon transforms this paper studies the discrete analogues of singular radon transforms we prove the 2 boundedness for those operators that are quasi translation invariant the approach used is related to the circle method of hardy and littlewood and requires multi dimensional extensions of weyl sums and gauss sums as well as variants that replace scalar sums by operator sums 1 introduction the purpose of this paper is to study discrete analogues of singular radon transforms their analysis will involve considerations originating in analytic number theory related to the circle method of hardy and littlewood and necessitates an extension of the notion of weyl sums and gauss sums this will require that we pass from the usual setting of scalar valued sums to certain operator analogues of these sums

dynamical entropy for systems with stochastic perturbation dynamics of deterministic systems perturbed by random additive noise is characterized quantitatively since for such systems the kolmogorov sinai ks entropy diverges if the diameter of the partition tends to zero we analyze the difference between the total entropy of a noisy system and the entropy of the noise itself we show that this quantity is finite and non negative and we call it the dynamical entropy of the noisy system in the weak noise limit this quantity is conjectured to tend to the ks entropy of the deterministic system in particular we consider one dimensional systems with noise described by a finite dimensional kernel for which the frobenius perron operator can be represented by a finite matrix

weak noise approach to the logistic map using a nonperturbative weak noise approach we investigate the interference of noise and chaos in simple 1d maps we replace the noise driven 1d map by an area preserving 2d map modelling the poincare sections of a conserved dynamical system with unbounded energy manifolds we analyze the properties of the 2d map and draw conclusions concerning the interference of noise on the nonlinear time evolution we apply this technique to the standard period doubling sequence in the logistic map from the 2d area preserving analogue we in addition to the usual period doubling sequence obtain a series of period doubled cycles which are elliptic in nature these cycles are spinning off the real axis at parameters values corresponding to the standard period doubling events

oscillation and variation for singular integrals in higher dimensions in this paper we continue our investigations of square function inequalities in harmonic analysis here we investigate oscillation and variation inequalities for singular integral operators in dimensions d 1 our estimates give quantitative information on the speed of convergence of truncations of a singular integral operator including upcrossing and a jump inequalities

hilbert integrals singular integrals and radon transforms i on introduit deux nouvelles classes d operateurs on etudie leur relation et on montre comment on peut les appliquer a l etude des problemes aux valeurs limites

asymptotic expansions for the escape rate of stochastically perturbed unimodal maps abstract the escape rate of a stochastic dynamical system can be found as an expansion in powers of the noise strength in the previous work the coefficients of such an expansion for a one dimensional map were fitted to a general form containing a few parameters these parameters were found to be related to the fractal structure of the repeller of the system the parameter the noise dimension remains to be interpreted this report presents new data for showing that the relation to the dimensions is more complicated than predicted in the earlier work and oscillates as a function of the map parameter in contrast to other dimension like quantities

theory based benchmarking of the blended force based quasicontinuum method we formulate an atomistic to continuum coupling method based on blending atomistic and continuum forces our precise choice of blending mechanism is informed by theoretical predictions we present a range of numerical experiments studying the accuracy of the scheme focusing in particular on its stability these experiments confirm and extend the theoretical predictions and demonstrate a superior accuracy of b qcf over energy based blending schemes

convergence analysis of a quasi continuum approximation for a two dimensional material without defects in many applications materials are modeled by a large number of particles or atoms where any particle can interact with any other the computational cost is very high since the number of atoms is huge recently much attention has been paid to a so called quasi continuum qc method which is a mixed atomistic continuum model the qc method uses an adaptive finite element framework to effectively integrate the majority of the atomistic degrees of freedom in regions where there is no serious defect however numerical analysis of this method is still in its infancy in this paper we will conduct a convergence analysis of the qc method in the case when there is no defect we will also remark on the case when the defect region is small the difference between our analysis and conventional analysis is that our exact atomistic solution is not a solution of a continuous partial differential equation but a discrete lattice scale solution which is not approximately related to any conventional partial differential equation

ell p mathbb z boundedness of discrete maximal functions along thin subsets of primes and pointwise ergodic theorems we establish the first pointwise ergodic theorems along thin sets of prime numbers a set with zero density with respect to the primes for instance we will be able to achieve this with the piatetski shapiro primes our methods will be robust enough to solve the ternary goldbach problem for some thin sets of primes

noisy one dimensional maps near a crisis i weak gaussian white and colored noise we study one dimensional single humped maps near the boundary crisis at fully developed chaos in the presence of additive weak gaussian white noise by means of a new perturbation like method the quasi invariant density is calculated from the invariant density at the crisis in the absence of noise in the precritical regime where the deterministic map may show periodic windows a necessary and sufficient condition for the validity of this method is derived from the quasi invariant density we determine the escape rate which has the form of a scaling law and compares excellently with results from numerical simulations we find that deterministic transient chaos is stabilized by weak noise whenever the maximum of the map is of orderz 1 finally we extend our method to more general maps near a boundary crisis and to multiplicative as well as colored weak gaussian noise within this extended class of noises and for single humped maps with any fixed orderz 0 of the maximum in the scaling law for the escape rate both the critical exponents and the scaling function are universal

calderon zygmund operators on amalgam spaces and in the discrete case abstract we prove the boundedness of calderon zygmund operators on weighted amalgam spaces l p l w q r n for 1 p q with muckenhoupt weights to do this we show the boundedness in the discrete case i e the boundedness on l w q z n we also investigate on l p l w r n as an application we consider an operator related to the navier stokes equation

noisy one dimensional maps near a crisis ii general uncorrelated weak noise the escape rate for one dimensional noisy maps near a crisis is investigated a previously introduced perturbation theory is extended to very general kinds of weak uncorrelated noise including multiplicative white noise as a special case for single humped maps near the boundary crisis at fully developed chaos an asymptotically exact scaling law for the rate is derived it predicts that transient chaos is stabilized by basically any noise of appropriate strength provided the maximum of the map is of sufficiently large order a simple heuristic explanation of this effect is given the escape rate is discussed in detail for noise distributions of levy dichotomous and exponential type in the latter case the rate is dominated by an exponentially leading arrhenius factor in the deep precritical regime however the preexponential factor may still depend more strongly than any power law on the noise strength

analysis of energy based blended quasi continuum approximations the development of patch test consistent quasi continuum energies for multidimensional crystalline solids modeled by many body potentials remains a challenge the original quasi continuum energy qce r miller and e tadmor model simul mater sci eng 17 2009 053001 has been implemented for many body potentials in two and three space dimensions but it is not patch test consistent we propose that by blending the atomistic and corresponding cauchy born continuum models of qce in an interfacial region with thickness of a small number k of blended atoms a general blended quasi continuum energy bqce can be developed with the potential to significantly improve the accuracy of qce near lattice instabilities such as dislocation formation and motion in this paper we give an error analysis of the blended quasi continuum energy bqce for a periodic one dimensional chain of atoms with next nearest neighbor interactions our analysis includes the optimization of the blending function for an improved convergence rate we show that the ell 2 strain error for the nonblended qce energy which has low order o varepsilon 1 2 where varepsilon is the atomistic length scale m dobson and m luskin siam j numer anal 47 2009 pp 2455 2475 p ming and j z yang multiscale model simul 7 2009 pp 1838 1875 can be reduced by a factor of k 3 2 for an optimized blending function where k is the number of atoms in the blending region the qce energy has been further shown to suffer from a o 1 error in the critical strain at which the lattice loses stability m dobson m luskin and c ortner j mech phys solids 58 2010 pp 1741 1757 we prove that the error in the critical strain of bqce can be reduced by a factor of k 2 for an optimized blending function thus demonstrating that the bqce energy for an optimized blending function has the potential to give an accurate approximation of the deformation near lattice instabilities such as crack growth

periodic orbit sum rules for billiards accelerating cycle expansions we show that the periodic orbit sums for two dimensional billiards satisfy an infinity of exact sum rules we demonstrate their utility by using the flow conservation sum rule to accelerate the convergence of cycle expansions for the overlapping three disc billiard the effectiveness of the approach is studied by applying the method on averages known explicitly by other sum rules the method is then applied to the lyapunov exponent

discrete radon transforms and applications to ergodic theory we prove lp boundedness of certain non translation invariant discrete maximal radon transforms and discrete singular radon transforms we also prove maximal pointwise and lp ergodic theorems for certain families of non commuting operators

a rule for quantizing chaos the authors find a real function delta e whose zeros approximate the quantum energy levels of a system with chaotic classical trajectories delta e is a finite sum over combinations of classical periodic orbits it is obtained from gutzwiller s infinite and divergent sum 1982 representing the spectral density in terms of periodic orbits by means of a resummation conjectured by analogy with a derivation of the riemann siegel formula for the riemann zeros they assess the practicality of the quantization condition

convergence of a force based hybrid method in three dimensions we study a force based hybrid method that couples an atomistic model with the cauchy born elasticity model we show that the proposed scheme converges to the solution of the atomistic model with second order accuracy since the ratio between lattice parameter and the characteristic length scale of the deformation tends to 0 convergence is established for the three dimensional system without defects with general finite range atomistic potential and simple lattice structure the proof is based on consistency and stability analysis general tools for stability analysis are developed in the framework opseudodifference operators in arbitrary dimensions 2012 wiley periodicals inc

a dynamical zeta function for pseudo riemannian foliations we investigate a generalization of geodesic random walks to pseudo riemannian foliations the main application we have in mind is to consider the logarithm of the associated zeta function as grand canonical partition function in a theory unifying aspects of general relativity quantum mechanics and dynamical systems

discrete maximal functions in higher dimensions and applications to ergodic theory we establish a higher dimensional counterpart of bourgain s pointwise ergodic theorem along an arbitrary integer valued polynomial mapping we achieve this by proving variational estimates v r on l p spaces for all 1

analysis of blended atomistic continuum hybrid methods we present a comprehensive error analysis of two prototypical atomistic to continuum coupling methods of blending type the energy based and the force based quasicontinuum methods our results are valid in two and three dimensions for finite range many body interactions e g eam type and in the presence of lattice defects we consider point defects and dislocations the two key ingredients in the analysis are 1 new force and energy consistency error estimates and 2 a new technique for proving energy norm stability of a c couplings that requires only the assumption that the exact atomistic solution is a stable equilibrium

entire fredholm determinants for evaluation of semiclassical and thermodynamical spectra proofs that fredholm determinants of transfer operators for hyperbolic flows are entire can be extended to a large new class of multiplicative evolution operators we construct such operators both for the gutzwiller semiclassical quantum mechanics and for classical thermodynamic formalism and introduce a new functional determinant which is expected to be entire for axiom a flows and whose zeros coincide with the zeros of the gutzwiller voros zeta function

estimates for singular convolution operators on the heisenberg group soit h n le groupe de heisenberg l operateur a co c h n donne par af f k s etend a un operateur borne de l p dans l p pour 1 p les operateurs ae co c donnes par aef f ke s etendent aux operateurs bornes sur l p pour 1 p avec des normes independantes de e

trace formula for noisy flows a trace formula is derived for the fokker planck equation associated with ito stochastic differential equations describing noisy time continuous nonlinear dynamical systems in the weak noise limit the trace formula provides estimations of the eigenvalues of the fokker planck operator on the basis of the pollicott ruelle resonances of the noiseless deterministic system which is assumed to be non bifurcating at first order in the noise amplitude the effect of noise on a periodic orbit is given in terms of the period and the derivative of the period with respect to the pseudo energy of the onsager machlup freidlin wentzell scheme