Asymptotic Scheme Research Articles

Calculations of Molecular Properties in Hybrid Coupled-Cluster and Molecular Mechanics Approach

We report benchmark calculations obtained with our new coupled-cluster singles and doubles (CCSD) code for calculating the first- and second-order molecular properties. This code can be easily incorporated into combined [Valiev, M.; Kowalski, K. J. Chem. Phys. 2006, 125, 211101] classical molecular mechanics (MM) and ab initio coupled-cluster (CC) calculations using NWChem, enabling us to study molecular properties in a realistic environment. To test this methodology, we discuss the results of calculations of dipole moments and static polarizabilities for the Cl2O system in the CCl4 solution using the CCSD (CC with singles and doubles) linear response approach. We also discuss the application of the asymptotic extrapolation scheme (AES) [Kowalski, K.; Valiev, M. J. Phys. Chem. A 2006, 110, 13106] in reducing the numerical cost of CCSD calculations.

Rayleigh waves guided by topography

It would not be far-fetched to say that the work of Lord Rayleigh on surface guided waves has had fundamental and far-reaching effects upon modern life and many things we take for granted today, stretching from mobile phones through to the study of earthquakes. Many of these things take advantage of surface waves that are topographically guided thereby allowing energy to be carried in specific directions along some topography of the surface. Much of the emphasis has so far been placed on devices which are essentially thin rectangular ridge-like defects upon the surface. Indeed, experiments with these so-called SAW devices have demonstrated the presence of trapped waves. It remains to provide any convincing theory that captures this trapping effect for surface waves—this is our aim. We seek to resolve the issue of whether topography can support a trapped wave, whose energy is localized to within some vicinity of the topography, and to explain physically how trapping occurs. The trapping is first addressed by developing an asymptotic scheme that exploits a small parameter associated with the surface topography. We then provide numerical evidence to support results obtained from the asymptotic scheme; however, no rigorous proof of existence is presented.

Interfacial viscous ship waves near the cusp lines

The interfacial waves due to a steady Oseenlet in a system of two semi-infinite immiscible fluids of different densities are investigated analytically. The two-fluid system which consists of an upper inviscid and a lower viscous fluid is assumed to be incompressible, homogenous and stable. The interfacial elevation is given by a double Fourier integral, which involves a generic amplitude function, a complex dispersion function and an elementary phase function. With the aid of four kinds of expansion of the phase function, the asymptotic representations of the far-field wave profiles are explicitly derived for four regions as the azimuth angle is less than, less than but close to, at, and greater than but close to, the maximal angle of the V-wedge for Kelvin ship waves. Graphical representations of wave profiles show the consistency of four asymptotic schemes. A combination of the Stokes and Scorer methods of stationary phase gives uniformly valid results over the panoramic region.

An asymptotic preserving relaxation scheme for a moment model of radiative transfer

A new relaxation scheme is exhibited to approximate the weak solutions of the M 1 model to simulate radiative transfer. This numerical method uses better wavespeed approximations than the current schemes. In addition, it is proved to satisfy all the required stability properties and the asymptotic preserving property. To cite this article: C. Berthon et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Multiple currents in charged continua

It is argued that continuum realizations of distributions of collisionless charged particles should accommodate a dynamically evolving number of electric currents even if the continuum is composed of only one species of particle, such as electrons. A model is proposed that self-consistently describes the interaction of such a continuum and its electromagnetic field. It is formulated using a Lagrangian approach and employs a ‘folded’ flow map to describe the bulk particle motion. An asymptotic perturbation scheme is developed to analyse ultra-relativistic multi-component current configurations. The model is fully relativistic and is formulated over Minkowski spacetime using intrinsic tensor field theory.

On the 75th Birthday of V. M. Zolotarev

February 27, 2007, was the 75th birthday of the eminent mathematician, Professor Vladimir Mikhailovich Zolotarev. In 1949, Vladimir entered the faculty of mechanics and mathematics of Moscow State University. As his specialization field he chose probability theory and began his studies under the supervision of Eugene Borisovich Dynkin. After graduating from the university he was recommended to graduate studies, where his advisor was Andrei Nikolaevich Kolmogorov. Other distinguished mathematicians also have had a potent effect on Zolotarev's mathematical talent. Later, he mentions more than once not only his teachers E. B. Dynkin and A. N. Kolmogorov, but also B. V. Gnedenko and Yu. V. Linnik. In his graduate studies, Vladimir begins to study the properties of stable distributions. He continues to be interested in this theme even today. At first he was dealing with the stable distributions in the scheme of summation of independent identically distributed random variables. Later, he extended the concept of...

Two-dimensional bin packing with one-dimensional resource augmentation

The two-dimensional bin packing problem is a generalization of the classical bin packing problem and is defined as follows. Given a collection of rectangles specified by their width and height, pack these into a minimum number of square bins of unit size. Recently, the problem was proved to be APX-hard even in the asymptotic case, i.e. when the optimum solutions require a large number of bins [N. Bansal, J. Correa, C. Kenyon, M. Sviridenko, Bin packing in multiple dimensions: Inapproximability results and approximation schemes, Math. Oper. Res. 31 (1) (2006) 31–49]. On the positive side, there exists a polynomial time algorithm that uses OPT bins whose sides have length ( 1 + ϵ ) , where OPT denotes the number of unit sized bins used by the optimum solution [N. Bansal, J. Correa, C. Kenyon, M. Sviridenko, Bin packing in multiple dimensions: Inapproximability results and approximation schemes, Math. Oper. Res. 31 (1) (2006) 31–49]. A natural question that remains is the approximability of the problem when we are allowed to relax the size of the unit bins in only one dimension. In this paper, we show that there exists an asymptotic polynomial time approximation scheme for packing rectangles into bins of size 1 × ( 1 + ϵ ) .

Asymptotic Extrapolation Scheme for Large-Scale Calculations with Hybrid Coupled Cluster and Molecular Dynamics Simulations

In this paper we discuss a simple extrapolation scheme based on the asymptotic behavior of the electronic energies considered as functions of cutoff factor for orbital energies corresponding to virtual orbitals. The performance of this approach is illustrated in the context of large-scale dynamic simulations for excitation energies of the cytosine molecule in its native DNA environment. We demonstrate that the extrapolation errors are significantly smaller than the excitation-energy fluctuations, due to the fluctuating environment.

Rayleigh waves guided by topography

We consider wave propagation along the surface of an elastic half-space, whose surface is flat except for a straight, infinite length, ridge or trench that does not vary in its cross-section. We seek to resolve the issue of whether such a perturbed surface can support a trapped wave, whose energy is localized to within some vicinity of the defect, and explain physically how this trapping occurs. First, the trapping is addressed by developing an asymptotic scheme, which exploits a small parameter associated with the surface variation, to perturb about the base state of a flat half-space (which supports a surface wave, as demonstrated by Lord Rayleigh in 1885). We then provide convincing numerical evidence to support the results obtained from the asymptotic scheme; however, no rigorous proof of existence is presented.

A Spectral Domain Integral Equation Method Utilizing Analytically Derived Characteristic Basis Functions for the Scattering From Large Faceted Objects

A novel technique, based on a spectral domain integral equation method with analytically derived characteristic basis functions, is introduced in this paper. It enables us to treat scattering problems from electrically large faceted bodies in a numerically rigorous and computationally efficient manner, in terms of both time and memory. The analytically derived characteristic basis functions include certain desirable features of the asymptotic schemes and are defined on subdomains that can be electrically large, not being bound to the typical discretization size of the conventional method of moments. By properly weighting through a Galerkin procedure the resulting electric field integral equation, the problem is reduced to a matrix equation having dimensions that do not depend on the size of the scatterer but only on its shape. Electrically large problems can be handled in a computationally efficient manner by using the proposed method since the associated matrix size is relatively small; moreover, all the reduced matrix elements are calculated in the spectral domain without evaluating any convolution products

A general asymptotic scheme for inference under order restrictions

Limit distributions for the greatest convex minorant and its derivative are considered for a general class of stochastic processes including partial sum processes and empirical processes, for independent, weakly dependent and long range dependent data. The results are applied to isotonic regression, isotonic regression after kernel smoothing, estimation of convex regression functions, and estimation of monotone and convex density functions. Various pointwise limit distributions are obtained, and the rate of convergence depends on the self similarity properties and on the rate of convergence of the processes considered.

Efficient evaluation of highly oscillatory acoustic scattering surface integrals

We consider the approximation of some highly oscillatory weakly singular surface integrals, arising from boundary integral methods with smooth global basis functions for solving problems of high frequency acoustic scattering by three-dimensional convex obstacles, described globally in spherical coordinates. As the frequency of the incident wave increases, the performance of standard quadrature schemes deteriorates. Naive application of asymptotic schemes also fails due to the weak singularity. We propose here a new scheme based on a combination of an asymptotic approach and exact treatment of singularities in an appropriate coordinate system. For the case of a spherical scatterer we demonstrate via error analysis and numerical results that, provided the observation point is sufficiently far from the shadow boundary, a high level of accuracy can be achieved with a minimal computational cost.

Asymptotic adaptive bipartite entanglement-distillation protocol

We present a new asymptotic bipartite entanglement distillation protocol that outperforms all existing asymptotic schemes. This protocol is based on the breeding protocol with the incorporation of two-way classical communication. Like breeding, the protocol starts with an infinite number of copies of a Bell-diagonal mixed state. Breeding can be carried out as successive stages of partial information extraction, yielding the same result: one bit of information is gained at the cost (measurement) of one pure Bell state pair (ebit). The basic principle of our protocol is at every stage to replace measurements on ebits by measurements on a finite number of copies, whenever there are two equiprobable outcomes. In that case, the entropy of the global state is reduced by more than one bit. Therefore, every such replacement results in an improvement of the protocol. We explain how our protocol is organized as to have as many replacements as possible. The yield is then calculated for Werner states.

Alternative approximations of the bias and MSE of the IV estimator under weak identification with an application to bias correction

Abstract We provide analytical formulae for the asymptotic bias (ABIAS) and mean-squared error (AMSE) of the IV estimator, and obtain approximations thereof based on an asymptotic scheme which essentially requires the expectation of the first stage F-statistic to converge to a finite (possibly small) positive limit as the number of instruments approaches infinity. Our analytical formulae can be viewed as generalizing the bias and MSE results of [Richardson and Wu 1971. A note on the comparison of ordinary and two-stage least squares estimators. Econometrica 39, 973–982] to the case with nonnormal errors and stochastic instruments. Our approximations are shown to compare favorably with approximations due to [Morimune 1983. Approximate distributions of k -class estimators when the degree of overidentifiability is large compared with the sample size. Econometrica 51, 821–841] and [Donald and Newey 2001. Choosing the number of instruments. Econometrica 69, 1161–1191], particularly when the instruments are weak. We also construct consistent estimators for the ABIAS and AMSE, and we use these to further construct a number of bias corrected OLS and IV estimators, the properties of which are examined both analytically and via a series of Monte Carlo experiments.

A continuum model for flows of cholesteric liquid crystal polymers and permeation flows

We present a hydrodynamic theory for flows of cholesteric liquid crystal polymers (CLCPs) following the continuum mechanics formulation of McMillan’s second order tensor theory for liquid crystals [E.H. MacMillan, Ph.D. thesis, A Theory of Anisotropic Fluid Department of Mechanics, University of Minnesota, 1987], in which anisotropic distortional elastic coefficients are prescribed as functions of the local nematic order. We explore two classes of spatially inhomogeneous equilibrium solutions associated with the variation of the uniaxial order parameter and the helical director pattern. In the latter solution, we show that the chirality effectively couples the distortional elasticity and the cholesteric pitch to impact on the local nematic order and vice versa. When the model is used to study permeation flows of weakly sheared cholesteric liquid crystal polymers, we find that secondary flows are strongly coupled to the primary flow and the director dynamics. At the first order of the coarse-grained asymptotic scheme, the flow and orientational director dynamics dominate while the dynamics of the orientation order parameters show up at the next order. The leading order solutions exhibit boundary layers of thickness in the order of O ( q − 1 ) , where q is the dimensionless wave number in the base cholesteric equilibrium. In plane Couette flows, the velocity components are nearly constants outside the boundary layers and the director angle varies linearly across the shear cell while the apparent viscosity scales like O ( q ) ; whereas in weak Poiseuille flows, the angle is quadratic outside the boundary layers and velocity components barely change while the apparent viscosity scales like O ( q 2 ) .

On the asymptotic solution of the Maxey-Riley equation

The Maxey-Riley [Phys. Fluids 26, 883 (1983)] particle equation of motion is considered without the history term and for an asymptotically small Stokes number. The equation admits a globally attractive invariant manifold identified as the Eulerian particle velocity field asymptotically close to the unperturbed fluid velocity field, thus suppressing the inconsequential initial transients. A recursive asymptotic scheme is obtained for the calculation of the invariant manifold in any order of accuracy. The dimension of the particle equation on the invariant manifold is reduced by half, which considerably facilitates the analysis of its motion in physical space. Structural stability theory provides comprehensive qualitative description of the particle motion.

Space isotropy and weak equivalence principle in a scalar theory of gravity

We consider a preferred-frame bimetric theory in which the scalar gravitational field both influences the metric and has direct dynamical effects. A modified version ("v2") is built, by assuming now a locally-isotropic dilation of physically measured distances, as compared with distances evaluated with the Euclidean space metric. The dynamical equations stay unchanged: they are based on a consistent formulation of Newton's second law in a curved space-time. To obtain a local conservation equation for energy with the new metric, the equation for the scalar field is modified: now its l.h.s. is the flat wave operator. Fluid dynamics is formulated and the asymptotic scheme of post-Newtonian approximation is adapted to v2. The latter also explains the gravitational effects on light rays, as did the former version (v1). The violation of the weak equivalence principle found for gravitationally-active bodies at the point-particle limit, which discarded v1, is proved to not exist in v2. Thus that violation was indeed due to the anisotropy of the space metric assumed in v1.

Approximation Algorithms for Extensible Bin Packing

In a variation of bin packing called extensible bin packing, the number of bins to use is specified as part of the input, and bins may be extended to hold more than the usual unit capacity. The cost of a bin is one if it is not extended, and the size if it is extended. The goal is to pack a set of items of given sizes with minimum cost. Adapting ideas in [7, 8, ?], we give a fully polynomial asymptotic approximation scheme (FPTAAS) for extensible bin packing. We note that under a different scaling the problem could not admit an FPTAAS unless P = NP.

Bin Packing in Multiple Dimensions: Inapproximability Results and Approximation Schemes

We study the following packing problem: Given a collection of d-dimensional rectangles of specified sizes, pack them into the minimum number of unit cubes. We show that unlike the one-dimensional case, the two-dimensional packing problem cannot have an asymptotic polynomial time approximation scheme (APTAS), unless P = NP. On the positive side, we give an APTAS for the special case of packing d-dimensional cubes into the minimum number of unit cubes. Second, we give a polynomial time algorithm for packing arbitrary two-dimensional rectangles into at most OPT square bins with sides of length 1 + ε, where OPT denotes the minimum number of unit bins required to pack these rectangles. Interestingly, this result has no additive constant term, i.e., is not an asymptotic result. As a corollary, we obtain the first approximation scheme for the problem of placing a collection of rectangles in a minimum-area encasing rectangle.

Gaussian fluctuations for random walks in random mixing environments

We consider a class of ballistic, multidimensional random walks in random environments where the environment satisfies appropriate mixing conditions. Continuing our previous work [2] for the law of large numbers, we prove here that the fluctuations are Gaussian when the environment is Gibbsian satisfying the “strong mixing condition” of Dobrushin and Shlosman and the mixing rate is large enough to balance moments of some random times depending on the path. Under appropriate assumptions the annealed Central Limit Theorem (CLT) applies in both nonnestling and nestling cases, and trivially in the case of finite-dependent environments with “strong enough bias”. Our proof makes use of the asymptotic regeneration scheme introduced in [2]. When the environment is only weakly mixing, we can only prove that if the fluctuations are diffusive then they are necessarily Gaussian.