Triangular approximation for Ising model and its application to Boltzmann machine

When we consider a problem in information processing, it is convenient to formulate the problem by using a random Ising model in statistical physics. However, a kind of computational difficulty arises in a case that the number of nodes becomes large. Hence approximation schemes such as a mean field approximation and a Bethe approximation have been used extensively for overcoming the difficulty. When frustration is essential in some problems, the Bethe approximation gives unfavorable results. In those problems, more advanced approximation schemes are needed beyond the Bethe approximation. In the present paper, we present explicitly the triangular approximation, which is the next approximation to the Bethe approximation. We apply the obtained approximation scheme to a Boltzmann machine in order to investigate the validity of the triangular approximation.     

Semiclassical theory of molecular collisions: Real and complex-valued classical trajectories for collinear atom Morse oscillator collisions

Real and complex-valued classical trajectories have been calculated for the collinear collision of an atom with a Morse oscillator. They are used in three semiclassical approximations for the transition probability: a Bessel uniform approximation, an Airy uniform approximation and a primitive semiclassical approximation. Comparison with exact quantum results shows that the Bessel uniform approximation is accurate even for near elastic collisions where the Airy and primitive approximations break down. The Airy and Bessel approximations agree quite closely for inelastic collisions however. The primitive semiclassical approximation is less accurate than either the Airy or Bessel approximation.     

基于P3近似的空间分辨漫反射研究

光源附近组织的空间分辨漫反射是近年来生物医学光子学领域的一个研究热点,其目的是发展一种能够测定活体生物组织光学参量的新技术。漫射近似理论研究光源附近组织的空间分辨漫反射具有很大局限性。P3近似理论考虑了相函数的三阶矩,能较准确地描述光源附近组织的光辐射分布。研究了基于P3近似的空间分辨漫反射,从输运理论的PN方程组出发,导出了P3近似方程组和P3近似的格林函数解;阐述了漫射近似与P1近似的关系,给出了外推边界条件下,准直光束近似后的P3近似漫反射率的完整表示,讨论了相函数二阶参量对P3近似漫反射的影响,并与漫射近似和蒙特卡罗模拟结果进行了比较,指出了P3近似的应用范围。     

The eikonal approximation revisited

Summary Certain problems in optical scattering are best understood when the more complicated exact scattering theory is replaced by an approximation. The Fraunhofer approximation is a well-known example. In the past ten years a considerable amount of work has been done in various disciplines towards assessing the usefulness of a new approximation referred to in the literature either as the eikonal approximation or as the high-energy approximation. The purpose of this paper is to provide a much needed review of this work and in addition to examine the historical evolution of this approximation which essentially started in optics when Bruns introduced the term eikonal in 1895. Part of this work was done when SKS was at Saha Institute of Nuclear Physics, Calcutta, and at Institute of Wetland Management and Ecological Design, Calcutta, India.     

Test for the cumulant approximation in the critical region

A cumulant approximation in wide use is tested, in the critical region, against a straightforward product-average decomposition approximation previously used in conjunction with a new integral representation for the Ising model. It is found that for the calculation of the transition temperature, the cumulant approximation is significantly inferior to the product-average approximation.     

Modelling of a displacive transformation in two-dimensional system within four-body approximation of Continuous Displacement Cluster Variation Method

Following our previous study on distorted to non-distorted displacive phase transformation in the two-dimensional square lattice based on the continuous displacement cluster variation method (CDCVM) within the Bragg–Williams (BW) approximation, we performed a higher order approximation, four-body approximation, in the entropy term and compared the results obtained by the two approximations. The transformation temperature decreases with the higher order approximation, which shares the common feature with conventional CVM studies on replacive transformations. The present study predicts the first-order transformation, which is markedly different from the previous study based on the BW approximation. Furthermore, by employing the four-body approximation, we are able to reproduce the o-type distribution of displaced atoms around a Bravais lattice point by changing the atomic interaction energy, which was by no means possible by the BW approximation.     

A high order multivariate approximation scheme for scattered data sets

We present a high order multivariate approximation scheme for scattered data sets. Each data point is represented as a Taylor series, and the high order derivatives in the Taylor series are treated as random variables. The approximation coefficients are then chosen to minimize an objective function at each point by solving an equality constrained least squares. The approximation is an interpolation when the data points are given as exact, or a nonlinear regression function when nonzero measurement errors are associated with the data points. Using this formulation, the gradient information on each data point can be used to significantly reduce the approximation error. All parameters of the approximation scheme can be computed automatically from the data points. An uncertainty bound of the approximation function is also produced by the scheme. Numerical experiments demonstrate that although this method is more computationally intensive than traditional methods, it produces more accurate approximation functions.     

The truncation of the Mori continued fraction for the spectral function in Heisenberg and XY models

We study the truncation of the Mori continued fraction for the two-spin spectral function of the XY model in various schemes suggested by the 3-pole approximation. These schemes are (i) a fully consistent version of the 3-pole approximation; (ii) 5-pole approximation (not fully consistent); and (iii) 5-pole approximate (fully consistent). We establish the equilivalence between the fully consistent 3-pole approximation and the Bennett-Martin approximation. Finally, we calculate the time-dependent spin-spin correlation function for the classical Heisenberg model in both versions of the 3-pole approximation and compare it with the exact spin-spin correlation function obtained by direct numerical calculation in one dimension.     

A note on the Brysk approximation of phase shifts

The Brysk approximation of phase shifts is derived in a manner differing from that used in the original study. The exact wave function is then replaced by its approximation, valid for small values ofr in a general integral expression for phase shifts. The approximation obtained is again the Brysk approximation. For the sake of simplicity, the scattering of a non-relativistic particle by a centrally symmetric potential is considered in all cases.     

任意阶标度分形格分抗与非正则格型标度方程

Carlson分形格电路是分抗的理想逼近情形,但仅具有负半阶运算性能,逼近效益随着电路节次数的增加逐渐降低.虽然可嵌套得到-1/2~n阶(n为大于或等于2的整数)分抗逼近电路,但结构复杂,无法实现任意分数阶运算.通过类比拓展Carlson分形格电路,获得具有高逼近效益的任意实数阶微积算子的分抗逼近电路——标度分形格分抗,并用非正则格型标度方程进行数学描述.分别探讨非正则格型标度方程的近似求解和真实解.通过调节电阻递进比α与电容递进比β的取值,可构造出具有任意运算阶的标度分形格分抗逼近电路.标度拓展极大地提高了标度分形格分抗电路的逼近效益.随着标度因子的增加,负半阶标度分形格分抗的逼近效益逐渐增大并明显高于Carlson分形格分抗.设计了基于五节Carlson分形格分抗与负半阶标度分形格分抗的半阶微分运算电路,并对周期三角波和周期方波信号进行半阶微分运算,实验测试结果与理论分析一致.     

Hybrid diffusion approximation in highly absorbing media and its effects of source approximation

A modified diffusion approximation model called the hybrid diffusion approximation that can be used for highly absorbing media is investigated. The analytic solution of the hybrid diffusion approximation for reflectance in two-source approximation and steady-state case with extrapolated boundary is obtained. The effects of source approximation on the analytic solution are investigated, and it is validated that two-source approximation in highly absorbing media to describe the optical properties of biological tissue is necessary. Monte Carlo simulation of recovering optical parameters from refiectant data is done with the use of this model. The errors of recovering μ＇s and μ＇s are smaller than 15% for the reduced albedo between 0.77 and 0.5 with the source-detector separation of 0.4-3 mm.     

APPROXIMATION THEORY OF OSCILLATING FACTOR SUPPRESSING AMPLITUDE IN QUANTUM PROCESS AND ITS APPLICATIONS

The oscillating factor suppressing amplitude approximation is proposed in this paper as a basic method to study the evolution of a quantum system. By making use of this method, not only the quantum adiabatic approximation is described again, But also both the higher-order corrections for the rotation wave approximation and the influences of other levels on two-level approximation can be analytically investigated with the explicit discussion for the quantitative conditions under which the two approximations hold.     

On the weak field approximation of the de Sitter gauge theory of gravity

The weak field approximation of a model of de Sitter gauge theory of gravity is studied in two cases. Without torsion and spin current, the model cannot give the right non-relativistic approximation unless the density is a constant. With small torsion, a satisfactory Newtonian approximation can be obtained.     

Association of evaluation methods of the effective permittivity of heterogeneous media on the basis of a generalized singular approximation

Various methods for evaluation of the effective permittivity of heterogeneous media, namely, the effective medium approximation (Bruggeman’s approximation), the Maxwell-Garnett approximation, Wiener’s bounds, and the Hashin-Shtrikman variational bounds (for effective static characteristics) are combined on the basis of a generalized singular approximation.     

Multiple scattering with a linearized propagator

Scattering by a many-body system is studied within the framework of the “fixed scatterer” approximation and the eikonal approximation formulated in terms of a linearized propagator. If properly treated, the “fixed scatterer” approximation is able to take into account the center-of-mass motion. We specifically study the linearized propagator proposed by Abarbanel and Itzykson. Although for potential scattering the above approximation is essentially equivalent to the Glauber eikonal approximation, its physical implications are quite different when applied to scattering by a composite system. The multiple-scattering series can generally no longer be simply expressed in terms of the individual on-shell scattering amplitudes, and the additivity of phase shifts is shown to break down for overlapping potentials. The implications for phenomenological calculations are discussed. Finally, the above approximation is explicitly applied to high-energy elastic nucleon-deuteron scattering and the results are compared with several variants of the Glauber multiple-scattering formalism.     

任意阶高运算恒定性分抗逼近电路—标度格型级联双口网络

标度拓展经典负半阶分抗逼近电路,可实现具有任意分数阶微积算子运算功能的分抗逼近电路,但牺牲了运算恒定性.从电路网络的角度分析具有恒定运算性能的负半阶Carlson分形格分抗逼近电路.根据标度分形格分抗逼近电路的等效无源双口网络,探讨该双口网络右侧端口的运算有效性,设计具有高运算恒定性的任意阶标度分形格分抗逼近电路.结合...     

多维梯度折射率介质内辐射传递的扩散近似

推导多维梯度折射率介质内稳态辐射传递的扩散近似方程.使用有限元法对扩散近似进行离散和求解,利用两个二维半透明介质的稳态辐射传递问题验证该扩散近似的精度及适用性.算例考虑介质为均匀折射率及梯度折射率两种情况.利用扩散近似分别求解辐射平衡时的边界热流、介质内温度场分布,并与辐射传递方程的求解结果进行对比分析.结果表明:介质折射率变化、散射特性、光学厚度及散射反照率均直接影响扩散近似的精度;在光学厚及强散射条件下,该扩散近似可以作为一种快速算法应用于梯度折射率介质稳态辐射传递的求解.     

Accuracy of the Time-Dependent Hartree–Fock Approximation for Uncorrelated Initial States

This article concerns the time-dependent Hartree–Fock (TDHF) approximation of single-particle dynamics in systems of interacting fermions. We find that the TDHF approximation is accurate when there are sufficiently many particles and the initial many-particle state is any Gibbs equilibrium state for noninteracting fermions (with Slater determinants as a special example). Assuming a bounded two-particle interaction, we obtain a bound on the error of the TDHF approximation, valid for short times. We further show that the error of the TDHF approximation vanishes at all times in the mean field limit.     

The ADM Hamiltonian at the postlinear approximation

The ADM Hamiltonian for a many-particle system is calculated up to the postlinear approximation, i.e., to the approximation that both the equations of motion for the particles and the equations of motion for the gravitational field in case of no-incoming radiation correctly result up to the postlinear approximation. The relation of this Hamiltonian to the ADM Hamiltonian obtained by a post-Newtonian approximation scheme which was applied up to the first radiation-reaction and radiation levels is discussed. From here the standard formulas for the mechanical angular momentum and energy losses as well as the radiated energy and angular momentum are deduced. Background logarithmic and logarithmic radiative terms are shown to be not present at our approximation if the condition of no-incoming radiation is fulfilled.     

标量衍射理论的非傍轴近似及其有效性

当光束束腰（或衍射孔孔径）可与波长相比拟或光束具有较大的发散角时,傍轴近似不再成立.在标量瑞利.索末菲衍射积分的基础上,进一步研究了衍射场的非傍轴近似解,并详细分析了解的有效性.以平面波圆孔衍射为例,对衍射场的精确解、非傍轴近似解以及菲涅耳近似解进行了详细的数值计算和比较研究.结果表明,非傍轴近似对微小孔衍射非常精确、有效.