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.

基于格林函数法研究分数阶Bessel-Gaussian光束近场传输

为揭示FBG(fractional-order Bessel-Gaussian)光束在近场传输时的光强和相位演化情况,基于光传播的独立性和叠加性原理,通过引入虚光源点,并结合格林函数法,严格推导了FBG光束的非傍轴积分表达式,并进一步利用非傍轴下的各阶修正解,精确计算了FBG光束轴上的光强分布.计算结果表明:对于分数阶Bessel-Gaussian光束的远场传输,利用经典的傍轴近似理论可以得到准确的计算结果,但在近场传输时,傍轴近似理论的计算结果误差很大,只有利用非傍轴修正解才能精确研究FBG光束的近场光学传输.研究结果为将FBG光束更好地应用于近场光学奠定必要的理论基础.

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.

Gaussian PI Controller Network Classifier for Grid-Connected Renewable Energy System

Multi-port converters are considered as exceeding earlier period decade owing to function in a combination of different energy sources in a single processing unit.Renewable energy sources are playing a significant role in the modern energy system with rapid development.In renewable sources like fuel combustion and solar energy,the generated voltages change due to their environmental changes.To develop energy resources,electric power generation involved huge awareness.The power and output voltages are plays important role in our work but it not considered in the existing system.For considering the power and voltage,Gaussian PI Controller-Maxpooling Deep Convolutional Neural Network Classifier(GPIC-MDCNNC)Model is introduced for the grid-connected renewable energy system.The input information is collected from two input sources.After that,input layer transfer information to hidden layer 1 fuzzy PI is employed for controlling voltage in GPIC-MDCNNC Model.Hidden layer 1 is transferred to hidden layer 2.Gaussian activation is employed for determining the output voltage with help of the controller.At last,the output layer offers the last value in GPIC-MDCNNC Model.The designed method was confirmed using one and multiple sources by stable and unpredictable input voltages.GPIC-MDCNNC Model increases the performance of grid-connected renewable energy systems by enhanced voltage value compared with state-of-the-art works.The control technique using GPIC-MDCNNC Model increases the dynamics of hybrid energy systems connected to the grid.

基于RUL和SVs-GFF的云服务器老化预测方法

针对云服务器中存在软件老化现象,将造成系统性能衰退与可靠性下降问题,借鉴剩余使用寿命(Remaining useful life,RUL)概念,提出基于支持向量和高斯函数拟合(Support vectors and Gaussian function fitting,SVs-GFF)的老化预测方法.首先,提取云服务器老化数据的统计特征指标,并采用支持向量回归(Support vector regression,SVR)对统计特征指标进行数据稀疏化处理,得到支持向量(Support vectors,SVs)序列数据;然后,建立基于密度聚类的高斯函数拟合(Gaussian function fitting,GFF)模型,对不同核函数下的支持向量序列数据进行老化曲线拟合,并采用Fréchet距离优化算法选取最优老化曲线;最后,基于最优老化曲线,评估系统到达老化阈值前的RUL,以预测系统何时发生老化.在OpenStack云服务器4个老化数据集上的实验结果表明,基于RUL和SVs-GFF的云服务器老化预测方法与传统预测方法相比,具有更高的预测精度和更快的收敛速度.

高斯函数拟合的地磁日变规律分析及改正

地磁数据的采集精度直接影响着地磁匹配定位的准确度,地磁日变改正是获取高精度地磁数据的重要前提.在远海以及一些远离台站有效范围的偏远区域设立日变站相对困难,而且保证日变站持续稳定的工作对日变站的设立提出了更高要求.日变改正步骤复杂而繁琐,大多通过多台站联合改正或监测相应时刻日变站的变化实现,对于区域地磁数据采集过程的日变改正研究相对较少.针对这些问题,文章顾及地磁静日变化存在一定周期性的特点,基于高斯拟合函数对单点位多个太阳日下的地磁场变化规律进行分析,并对区域多点的地磁场数据进行日变拟合及改正分析.试验结果表明:在一定区域范围内高斯函数拟合得到的点位磁场的静日变化能够满足地磁日变改正精度,拟合误差在几个nT以内,中误差也不超过10 nT;区域点位实验中,将高斯函数拟合获得的日变改正值与常规方法获得的日变改正值进行对比,其中82.3%的点位日变改正值的差值在±5 nT以内.因此,基于高斯函数拟合的地磁日变改正分析能够根据观测时间获得相应区域的日变改正估值,实现日变改正,获得高精度地磁数据,为地磁匹配定位提供了基础.

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.

基于U-Net和CNN深度学习求取地质属性建模变差函数参数

在油气藏地质属性建模中,变差函数的求取尤为关键,一般是通过拟合实验变差函数求取变程、方位角、基台值等参数的方式获得,但当研究区内样本点数量过少时,实验变差函数拟合效果较差,从而影响属性建模质量。为了克服传统方法的不足,最大限度地利用空间数据,文中提出一种基于U-Net和CNN网络求取变差函数参数的新方法:以孔隙度属性建模为例,首先选用球状模型并利用序贯高斯模拟(SGS)算法模拟生成多组孔隙度模型,以所得孔隙度平面模型抽取的数据点为基准构成样本集;然后采用U-Net进行孔隙度模型重构,保证孔隙度分布的空间相关性;最后利用CNN对样本集进行深度学习,从而建立求取变差函数的模型。实际应用表明,利用所提方法取得的主变程方位角与通过实验变差函数拟合求取的方位角仅相差1.52°,与沉积微相展布方向一致,得到的主次变程与实验变差函数非常契合,证明求取的变差函数结果可靠。同时,该方法简化了地质建模工作流程,减少了求取实验变差函数主观性,降低了研究区内样本点数量的局限性,为变差函数的预测提供了一种新的思路。

一种变步长的零吸引归一化自适应滤波算法

为了使最小均方算法对高斯噪声环境和稀疏系统具有更好的收敛速度和稳态误差,提出了一种变步长的零吸引归一化最小均方算法。该算法将改进的Versoria函数的稀疏感知范数与归一化最小均方算法相结合后,引入了一种新的类高斯函数的变步长策略,解决了固定步长条件下算法收敛速度较慢、跟踪性能较差的问题。从理论层面分析了所提算法的收敛性,并基于MATLAB平台讨论了改进的类高斯步长函数中各参数对算法性能的影响。最后将所提算法与其他同类算法应用于不同信噪比条件下的高斯噪声环境以及稀疏环境中进行未知系统辨识实验,仿真结果表明,所提算法具有更快的收敛速度、更好的跟踪能力以及较小的稳态误差。

基于福井函数方法的芳杂环化合物亲电取代反应位点预测及图形可视化教学应用

基于福井函数方法,利用Gaussian和Multiwfn软件预测了常见芳杂环化合物及取代芳杂环化合物的亲电取代反应位点,使之以可视化图像形式展现并应用于基础有机化学的教学实践。

大气湍流下部分相干涡旋光束经粗糙目标回波特性

利用广义Huygens-Fresnel原理和随机高斯粗糙面模型,研究部分相干高斯-谢尔模型涡旋光束的回波特性,通过特殊积分推导出部分相干涡旋光束经随机粗糙面散射后到达接收端的交叉谱密度函数和光强解析式,并进行仿真分析。结果表明:随着拓扑荷数、束腰半径和相干长度的增加,光强自相关函数曲线下降的越来越陡峭;在强湍流环境下,自相关函数衰减越来越严重;内尺度会影响光强自相关函数,并且随着内尺度的增大而衰减严重,外尺度对其影响作用不大;随着粗糙目标相干长度的增大,光强自相关函数的半峰宽值逐渐趋增大。

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.