Gaussian Functions,Γ-Functions and Wavelets

The relations between Gaussian function and Γ function is revealed first at one dimensional situation. Then, the Fourier transformation of n dimensional Gaussian function is deduced by a lemma. Following the train of thought in one dimensional situation, the relation between n dimensional Gaussian function and Γ function is given. By these, the possibility of arbitrary derivative of an n dimensional Gaussian function being a mother wavelet is indicated. The result will take some enlightening role in exploring the internal relations between Gaussian function and Γ function as well as in finding high dimensional mother wavelets.

Simulation of γ spectrum-shifting based on the parameter adjustment of Gaussian function space

Based on the statistical characteristics of energy spectrum and the features of spectrum-shifting in spectrometry,the parameter adjustment method of Gaussian function space was applied in the simulation of spectrum-shifting.The transient characteristics of energy spectrum were described by the Gaussian function space,and then the Gaussian function space was transferred by parameter adjustment method.Furthermore,the spectrum-shifting in measurement of energy spectrum was simulated.The applied example shows that the parameters can be adjusted flexibly by this method to meet the various requirements in simulation of energy spectrum-shifting.This method was one parameterized simulation method with good performance for the practical application.

Berezin Quantization of Gaussian Functions Depending by a Quantum and Compression Parameter

The aim of this work is to study the Berezin quantization of a Gaussian state. The result is another Gaussian state that depends on a quantum parameter α, which describes the relationship between the classical and quantum vision. The compression parameter λ>0 is associated to the harmonic oscillator semigroup.

RBF neural network based on q-Gaussian function in function approximation

To enhance the generalization performance of radial basis function （RBF） neural networks, an RBF neural network based on a q-Gaussian function is proposed. A q-Gaussian function is chosen as the radial basis function of the RBF neural network, and a particle swarm optimization algorithm is employed to select the parameters of the network. The non-extensive entropic index q is encoded in the particle and adjusted adaptively in the evolutionary process of population. Simulation results of the function approximation indicate that an RBF neural network based on q-Gaussian function achieves the best generalization performance.

Study of hadrons using the Gaussian functional method in the O(4) linear σ model

We study properties of hadrons in the O（4） linear σ model, where we take into account fluctuations of mesons around their mean field values using the Gaussian functional （GF） method. In the GF method we calculate dressed σ and π masses, where we include the effect of fluctuations of mesons to find a better ground state wave function than the mean field approximation. Then we solve the Bethe-Salpeter equations and calculate physical σ and π masses. We recover the Nambu-Goldstone theorem for the physical pion mass to be zero in the chiral limit. The σ meson is a strongly correlated meson-meson state, and seems to have a two meson composite structure. We calculate σ and π masses as functions of temperature for both the chiral limit and explicit chiral symmetry breaking case. We get similar behaviors for the physical σ and π masses as the case of the mean field approximation, but the coupling constants are much larger than the values of the case of the mean field approximation.

Audio-Visual Underdetermined Blind Source Separation Algorithm Based on Gaussian Potential Function

Most existing algorithms for the underdetermined blind source separation(UBSS) problem are two-stage algorithm, i.e., mixing parameters estimation and sources estimation. In the mixing parameters estimation, the previously proposed traditional clustering algorithms are sensitive to the initializations of the mixing parameters. To reduce the sensitiveness to the initialization, we propose a new algorithm for the UBSS problem based on anechoic speech mixtures by employing the visual information, i.e., the interaural time difference(ITD) and the interaural level difference(ILD), as the initializations of the mixing parameters. In our algorithm, the video signals are utilized to estimate the distances between microphones and sources, and then the estimations of the ITD and ILD can be obtained. With the sparsity assumption in the time-frequency domain, the Gaussian potential function algorithm is utilized to estimate the mixing parameters by using the ITDs and ILDs as the initializations of the mixing parameters. And the time-frequency masking is used to recover the sources by evaluating the various ITDs and ILDs. Experimental results demonstrate the competitive performance of the proposed algorithm compared with the baseline algorithms.

Gaussian Radial Basis Function interpolation in vertical deformation analysis

In many deformation analyses,the partial derivatives at the interpolated scattered data points are required.In this paper,the Gaussian Radial Basis Functions(GRBF)is proposed for the interpolation and differentiation of the scattered data in the vertical deformation analysis.For the optimal selection of the shape parameter,which is crucial in the GRBF interpolation,two methods are used:the Power Gaussian Radial Basis Functions(PGRBF)and Leave One Out Cross Validation(LOOCV)(LGRBF).We compared the PGRBF and LGRBF to the traditional interpolation methods such as the Finite Element Method(FEM),polynomials,Moving Least Squares(MLS),and the usual GRBF in both the simulated and actual Interferometric Synthetic Aperture Radar(InSAR)data.The estimated results showed that the surface interpolation accuracy was greatly improved by LGRBF and PGRBF methods in comparison withFEM,polynomial,and MLS methods.Finally,LGRBF and PGRBF interpolation methods are used to compute invariant vertical deformation parameters,i.e.,changes in Gaussian and mean Curvatures in the Groningen area in the North of Netherlands.

Seismic interferometry and estimation of the Green's function using Gaussian beams

This study investigates seismic interferometry in which the Green＇s function is estimated between two receiv- ers by cross-correlation and integration over sources. For smoothly varying source strengths, the dominant contributions of the correlation integral come from the stationary phase directions in the forward and backward directions from the alignment of the two receivers. Gaussian beams can be used to evaluate the correlation integral and concentrate the amplitudes in a vicinity of the stationary phase regions instead of completely relying on phase interference. Several numerical examples are shown to illustrate how this process works. The use of Gaussian beams for the evaluation of the correlation integral results in stable estimates, and also provides physical insight into the estimation of the Green＇s function based on seismic interferometry.

On Development of Fuzzy Controller: The Case of Gaussian and Triangular Membership Functions

In recent years, the use of Fuzzy set theory has been popularised for handling overlap domains in control engineering but this has mostly been within the context of triangular membership functions. In actual practice however, such domains are hardly triangular and in fact for most engineering applications the membership functions are usually Gaussian and sometimes cosine. In an earlier paper, we derived explicit Fourier series expressions for systematic and dynamic computation of grade of membership in the overlap and non-overlap regions of triangular Fuzzy sets. In another paper, we extended the methodology to cover cases of cosine, exponential and Gaussian Fuzzy sets by presenting explicit Fourier series representation for encoding fuzziness in the overlap and non-overlap domains of Fuzzy sets. This current paper presents the development of a “Fuzzy Controller” device, which incorporates the formal mathematical representation for computing grade of membership of Gaussian and triangular Fuzzy sets. It is shown that triangular approximation of Gaussian membership function in Fuzzy control can lead to wrong linguistic classification which may have adverse effects on operational and control decisions. The development of the Fuzzy controller demonstrates that the proposed technique can indeed be incorporated in engineering systems for dynamic and systematic computation of grade of membership in the overlap and non-overlap regions of Fuzzy sets;and thus provides a basis for the design of embedded Fuzzy controller for mission critical applications.

The Wigner distribution functions of coherent and partially coherent Bessel-Gaussian beams

Based on the integral representation of the Bessel functions and the generating function of the Tricomi function, an analytical expression of the Wigner distribution function （WDF） for a coherent or partially coherent Bessel Gaussian beam is presented. The reduced two-dimensional WDFs are also demonstrated graphically, which reveals the dependence of the reduced WDFs on the beam parameters.

Wigner function and the entanglement of a quantized Bessel-Gaussian vortex state of a quantized radiation field

A new kind of quantum non-Gaussian state with a vortex structure, termed a Bessel-Gaussian vortex state, is constructed, which is an eigenstate of the sum of squared annihilation operators a2 ＋ b2. The Wigner function of the quantum vortex state is derived and exhibits negativity which is an indication of nonclassicality. It is also found that a quantized vortex state is always in entanglement. And a scheme for generating such quantized vortex states is proposed.

Analytical Evaluation of Non-Elementary Integrals Involving Some Exponential, Hyperbolic and Trigonometric Elementary Functions and Derivation of New Probability Measures Generalizing the Gamma-Type and Gaussian-Type Distributions

The non-elementary integrals involving elementary exponential, hyperbolic and trigonometric functions, and where α,η and β are real or complex constants are evaluated in terms of the confluent hypergeometric function 1F1 and the hypergeometric function 1F2. The hyperbolic and Euler identities are used to derive some identities involving exponential, hyperbolic, trigonometric functions and the hypergeometric functions 1F1 and 1F2. Having evaluated, these non-elementary integrals, some new probability measures generalizing the gamma-type and Gaussian distributions are also obtained. The obtained generalized probability distributions may, for example, allow to perform better statistical tests than those already known (e.g. chi-square (x2) statistical tests and other statistical tests constructed based on the central limit theorem (CLT)), while avoiding the use of computational approximations (or methods) which are in general expensive and associated with numerical errors.

利用Gaussian型RBF网络进行函数逼近的构造性估计

前馈人工神经网络有着极其广泛的应用 ,如何估计隐层神经元数及相应的逼近误差 ,一直是确定前馈网络结构的难点和关键。 RBF网络是一种最重要的前馈神经网络 ,本文给出了利用 Gaussian型 RBF网络逼近连续函数或 Lebesgue-可积函数时的构造性的隐层单元数显式估算式及相应的显式逼近误差估算式。文中的结论也易于推广到离散样本的情形。这些结论对于提高 Guassian型 RBF在实际应用时的计算精度和减少计算量具有一定的指导意义。

基于Gaussian-支持向量回归机的高速公路短时交通量预测

针对高速公路短时交通量的实时性、波动性和非线性的特点,将支持向量回归机(SVR)应用于高速公路短时交通量预测,并采用Gaussian损失函数来代替ε-不敏感损失函数,对原始序列进行降噪处理,为了更好的优选SVR模型参数,采用遗传算法(GA)进行参数优选,建立了基于GA优化的GA-Gaussian-SVR高速公路短时交通量预测模型,将本路段前几个时段交通量、天气因素和出行日期作为影响因素输入,结合实例进行了仿真预测.结果表明该方法可有效应用于高速公路短时交通量预测.

一种基于Gaussian函数的双向选择径向基函数神经网络算法

径向基函数神经网络是一类重要的神经网络算法。本文对现有的径向基函数神经网络算法进行了总结分析,将现有算法分为前向选择和后向选择两类。在分析各自优缺点的基础上从提高神经网络泛化能力的角度提出了一种新的基于Gaussian函数的双向选择径向基函数神经网络算法——BSRBF,从数理角度研究了神经元选择的基本技术方法,并对算法的基本思想和具体步骤进行了阐述。最后,用一个实验对比验证了双向选择算法的有效性。

Gaussian型RBF神经网络的函数逼近仿真研究

为了研究Gaussian型RBF神经网络的逼近能力,首先介绍了Gaussian型RBF神经网络的结构和算法,然后在MATLAB7.0环境下,编程建立了Gaussian型RBF神经网络和BP神经网络,并以具体的非线性函数为例,分别用两种神经网络对其进行逼近.仿真结果表明,相对于传统BP神经网络而言,Gaussian型RBF神经网络对于非线性函数的逼近精度更高、收敛速度更快,具有良好的逼近能力,为解决非线性函数的逼近问题提供了良好的解决手段.

基于格林函数法的奇型Mathieu-Gaussian光束

根据光束传播的独立性和叠加性原理,引入了一组能够产生第一类(2n+2阶)奇型Mathieu-Gaussian光束的虚光源点.利用虚源点技术和格林函数法,计算得到第一类奇型Mathieu-Gaussian光束的严格解析积分表达式.利用该表达式得到了轴上光场分布的积分解析解.以三阶非旁轴修正为例,得到了第一类奇型Mathieu-Gaussian光束保留到三阶非旁轴修正项的轴上光场分布精确解.

Fuzzy adaptive learning control network with sigmoid membership function

To get simpler operation in modified fuzzy adaptive learning control network （FALCON） in some engineering application, sigmoid nonlinear function is employed as a substitute of traditional Gaussian membership function. For making the modified FALCON learning more efficient and stable, a simulated annealing （SA） learning coefficient is introduced into learning algorithm. At first, the basic concepts and main advantages of FALCON were briefly reviewed. Subsequently, the topological structure and nodes operation were illustrated; the gradient-descent learning algorithm with SA learning coefficient was derived; and the distinctions between the archetype and the modification were analyzed. Eventually, the significance and worthiness of the modified FALCON were validated by its application to probability prediction of anode effect in aluminium electrolysis cells.

基于各向异性Gaussian改进模型的卫星材料BRDF计算

对目标表面双向反射分布函数(BRDF)进行研究,在各向异性Gaussian模型的基础上引入波长因子,提出了改进的BRDF各向异性Gaussian计算模型。运用该模型对卫星太阳能板及包裹多层隔热材料进行了BRDF计算,结果表明该模型参数物理意义明确、简洁高效,能够有效地计算物体表面材料BRDF数值。

基于小波分析和Gaussian回归的急性低血压预测

急性低血压是危害病人健康的并发症之一,对急性低血压发生的提早预测,能够帮助医生对重症病人找到更好的医疗处理方案。提出了一个基于趋势分量的Gaussian函数拟合预测模型,即用小波多尺度分析提取出信号的趋势分量;再根据Gaussian回归模型对趋势分量进行函数拟合,得到的函数参数作为特征值,用支持向量机SVM对数据分类。Gaussian回归模型使用的是数据驱动,用系数来描述数据之间的关系。通过在较大病人数据集上实验得到了较好的效果。