On the evaluation of generalized exponential integral functions

This paper deals with generalized exponential integral (GEI) functions arising in the study of anisotropic scattering in a multidimensional media. These functions are represented as finite linear combinations of basic GEIs introduced in this work. The recurrence relations are derived for the linear combination coefficients and basic GEIs. Numerical results are also given. The extensive test calculations show that the work proposed in this algorithm is the most efficient practical computations of GEI functions.     

Determination of Moisture Content in Vegetative Cultivated Plants Using Millimeter-Wave Spectroscopy for the Tasks of Increasing Plant Productivity

Technical Physics - Genetic and breeding study shows inefficiency of the analysis of the genotype-environment interaction (GEI) in plants at a molecular level, since the GEI effect completely...     

Convergence of observables on quantum logics

We define two types of convergence for observables on a quantum logic which we call M-weak and uniform M-weak convergence. These convergence modes correspond to weak convergence of probability measures. They are motivated by the idea that two (in general unbounded) observables are close if bounded functions of them are close. We show that M-weak and uniform M-weak convergence generalize strong resolvent and norm resolvent convergence for self-adjoint operators on a Hilbert space. Also, these types of convergence strengthen the weak operator convergence and operator norm convergence of bounded self-adjoint operators on a Hilbert space. Finally, we consider spectral perturbation by showing that the spectra of approximating observables approach the spectrum of the limit in a certain sense.     

The classical field limit of scattering theory for non-relativistic many-boson systems. I

We study the classical field limit of non-relativistic many-boson theories in space dimensionn≧3. When ?→0, the correlation functions, which are the averages of products of bounded functions of field operators at different times taken in suitable states, converge to the corresponding functions of the appropriate solutions of the classical field equation, and the quantum fluctuations are described by the equation obtained by linearizing the field equation around the classical solution. These properties were proved by Hepp [6] for suitably regular potentials and in finite time intervals. Using a general theory of existence of global solutions and a general scattering theory for the classical equation, we extend these results in two directions: (1) we consider more singular potentials, (2) more important, we prove that for dispersive classical solutions, the ?→0 limit is uniform in time in an appropriate representation of the field operators. As a consequence we obtain the convergence of suitable matrix elements of the wave operators and, if asymptotic completeness holds, of theS-matrix.     

Asymptotic of Grazing Collisions and Particle Approximation for the Kac Equation without Cutoff

The subject of this article is the Kac equation without cutoff. We first show that in the asymptotic of grazing collisions, the Kac equation can be approximated by a Fokker-Planck equation. The convergence is uniform in time and we give an explicit rate of convergence. Next, we replace the small collisions by a small diffusion term in order to approximate the solution of the Kac equation and study the resulting error. We finally build a system of stochastic particles undergoing collisions and diffusion, that we can easily simulate, which approximates the solution of the Kac equation without cutoff. We give some estimates on the rate of convergence.     

Variational-integral perturbation corrections of some lower excited states for hydrogen atoms in magnetic fields

A variational-integral perturbation method(VIPM) is established by combining the variational perturbation with the integral perturbation.The first-order corrected wave functions are constructed,and the second-order energy corrections for the ground state and several lower excited states are calculated by applying the VIPM to the hydrogen atom in a strong uniform magnetic field.Our calculations demonstrated that the energy calculated by the VIPM only shows a negative value,which indicates that the VIPM method is more accurate than the other methods.Our study indicated that the VIPM can not only increase the accuracy of the results but also keep the convergence of the wave functions.     

Coupled bending-bending vibrations of pretwisted tapered cantilever beams treated by the Reissner method

Theoretical natural frequencies and mode shapes of the first four coupled modes of a uniform pretwisted cantilever blade and the first five coupled flexural frequencies of pretwisted tapered blading are determined by using the Reissner method. The shape functions for the bending moments and deflections are developed in series form and with these used in the dynamic Reissner functional, the frequency equation is obtained by minimizing it through the Ritz process. A convergence study made in the case of the pretwisted uniform blade indicates that there appears to be a quicker convergence of the natural frequencies and that a five-term solution yields a set of results that are in good agreement with the theoretical and experimental values of other authors, available in the literature. The mode shapes obtained from the present analysis are compared with those from an earlier investigation and the effect of ignoring the shear deflection and rotary inertia in the analysis is discussed. The effects of breadth taper and depth taper on the vibration characteristics of pretwisted cantilever blading are discussed from the results obtained in the present limited study and it is observed that an extensive investigation appears to be necessary to draw positive conclusions covering wide ranges of pretwisted blade parameters.     

Some properties of mode coupling equations

In this paper it is proven that some classes of mode coupling equations for correlation functions have a unique solution which exhibits all the standard properties like causality and stability. This is done by demonstrating the uniform convergence of an iteration procedure, which was previously used for a numerical solution of these equations.     

Regularity and decay of lattice Green's functions

In the paper we study a class of lattice, covariant Laplace operators with external gauge fields. We prove that these operators are positive and that their Green's functions decay exponentially. They also have regularity properties similar to continuous space Green's functions. All the bounds are uniform in the lattice spacing.     

Convergence to the Time Average by Stochastic Regularization

In Ergodic Theory it is natural to consider the pointwise convergence of finite time averages of functions with respect to the flow of dynamical systems. Since the pointwise convergence is too weak for applications to Hamiltonian Perturbation Theory, requiring differentiability, we first introduce regularized averages obtained through a stochastic perturbation of an integrable Hamiltonian flow, and then we provide detailed estimates. In particular, for a special vanishing limit of the stochastic perturbation, we obtain convergence even in a Sobolev norm taking into account the derivatives.     

Continuity of Eigenfunctions of Uniquely Ergodic Dynamical Systems and Intensity of Bragg Peaks

We study uniquely ergodic dynamical systems over locally compact, sigma-compact Abelian groups. We characterize uniform convergence in Wiener/Wintner type ergodic theorems in terms of continuity of the limit. Our results generalize and unify earlier results of Robinson and Assani respectively. We then turn to diffraction of quasicrystals and show how the Bragg peaks can be calculated via a Wiener/Wintner type result. Combining these results we prove a version of what is sometimes known as the Bombieri/Taylor conjecture. Finally, we discuss various examples including deformed model sets, percolation models, random displacement models, and linearly repetitive systems.     

Construction of the two-dimensional sine-Gordon model for β<8π

We present a rigorous renormalization group construction of the two-dimensional massless and massive quantum sine-Gordon models in finite volume for the range 0<<8. We prove analyticity in the coupling constant , which implies the convergence of perturbation theory. The field correlation functions and their generating functional are analyzed and shown to have the short distance asymptotics of the free field theory. In the massive case the bounds are uniform in volume and we also obtain uniform estimates on the long distance decay of correlations.Research supported by NSF Grant PHY-9001178Research supported by the Natural Sciences and Engineering Research Council of Canada     

Multiscale finite element methods for high-contrast problems using local spectral basis functions

In this paper we study multiscale finite element methods (MsFEMs) using spectral multiscale basis functions that are designed for high-contrast problems. Multiscale basis functions are constructed using eigenvectors of a carefully selected local spectral problem. This local spectral problem strongly depends on the choice of initial partition of unity functions. The resulting space enriches the initial multiscale space using eigenvectors of local spectral problem. The eigenvectors corresponding to small, asymptotically vanishing, eigenvalues detect important features of the solutions that are not captured by initial multiscale basis functions. Multiscale basis functions are constructed such that they span these eigenfunctions that correspond to small, asymptotically vanishing, eigenvalues. We present a convergence study that shows that the convergence rate (in energy norm) is proportional to (H/Λ₁)^1/2, where Λ₁ is proportional to the minimum of the eigenvalues that the corresponding eigenvectors are not included in the coarse space. Thus, we would like to reach to a larger eigenvalue with a smaller coarse space. This is accomplished with a careful choice of initial multiscale basis functions and the setup of the eigenvalue problems. Numerical results are presented to back-up our theoretical results and to show higher accuracy of MsFEMs with spectral multiscale basis functions. We also present a hierarchical construction of the eigenvectors that provides CPU savings.     

Steady-profile fingering flows in Marangoni driven thin films

We present experimental and computational results indicating the existence of finite-amplitude fingering solutions in a flow of a thin film of a viscous fluid driven by thermally induced Marangoni stresses. Using carefully controlled experiments, spatially periodic perturbations to the contact line of an initially uniform thin film flow are shown to lead to the development of steady-profile two-dimensional traveling wave fingers. Using an infrared laser and scanning mirror, we impose thermal perturbations with a known wavelength to an initially uniform advancing fluid front. As the front advances in the experiment, we observe convergence to fingers with the initially prescribed wavelength. Experiments and numerical computations show that this family of solutions arises from a subcritical bifurcation.     

Convergence of Migdal-Kadanoff iterations: A simple and general proof

A short proof is presented showing convergence of Migdal-Kadanoff iterations for a large class of functions on compact connected Lie groups. From our estimate of the rate of convergence, we deduce lower bounds for string tension and mass gap in hierarchical four-dimensional lattice gauge and twodimensional spin models.     

Rates of Convergence in the CLT¶for Two-Dimensional Dispersive Billiards

We consider billiards in the two-dimensional torus with convex obstacles. Central Limit Theorems have been established for regular functions for the billiard transformation in [2], [1] and [14]. We are interested here in the problem of the rate of convergence. In this paper, we establish a rate in (for any α > 0) for the billiard transformation, by adapting the proof of [7, 6, 8]. In our proof, we use a strong decorrelation result obtained by the method developped in [14] for the study of general hyperbolic systems. Moreover, we establish a rate of convergence in in the Central Limit Theorem for the billiard flow. Received: 23 April 2001 / Accepted: 3 August 2001     

Limitations of variational methods for hydrogenic systems

Theoretical methods that give good approximations of the behaviours of hydrogen-like impurity states in semiconductors in uniform external magnetic fields are by now well established. With variational methods in mind the sizes of the determinants that give sufficiently accurate energies greatly depend on the quality of the wave functions chosen as these affect the rapidity of the series convergence. The recently developed hybrid functions enable hydrogenic energies to be calculated up ton=7 using fairly small determinants. However, it has emerged that there is a restricted region of variational parameters which maintains the numerical symmetry of the hamiltonian matrix elements for each group of states. Although each such parameter regime appears to narrow as the magnetic quantum number increases, the accuracies of the calculated eigenvalues remain satisfactory.     

On Higher Order Pyramidal Finite Elements

In this paper we first prove a theorem on the nonexistence of pyramidal polynomial basis functions. Then we present a new symmetric composite pyramidal finite element which yields a better convergence than the nonsymmetric one. It has fourteen degrees of freedom and its basis functions are incomplete piecewise triquadratic polynomials. The space of ansatz functions contains all quadratic functions on each of four sub-tetrahedra that form a given pyramidal element.     

Distances in spaces of physical models: partition functions versus spectra

We study the relation between convergence of partition functions (seen as general Dirichlet series) and convergence of spectra and their multiplicities. We describe applications to convergence in physical models, e.g., related to topology change and averaging in cosmology.     

A macro traffic flow model with probability distribution function

In this paper, we introduce three probability distribution functions into the dynamic equation and propose a macro traffic flow model to investigate the impacts of the probability distribution functions on the evolutions of traffic flow under three typical states (i.e., uniform flow, shock, rarefaction wave, and small perturbation). The numerical results indicate that the probability distribution functions do not change the density and speed distributions of uniform flow, produce a two-layer shock but have no prominent effects on rarefaction wave, and have little effect on small perturbation.