Evaluation of Generalized Error Function via Fast-Converging Power Series

A generalized form of the error function, Gp(x)=pΓ(1/p)∫0xe−tpdt, which is directly associated with the gamma function, is evaluated for arbitrary real values of p>1and 0x≤+∞by employing a fast-converging power series expansion developed in resolving the so-called Grandi’s paradox. Comparisons with accurate tabulated values for well-known cases such as the error function are presented using the expansions truncated at various orders.

The Bivariate Normal Integral via Owen’s T Function as a Modified Euler’s Arctangent Series

The Owen’s T function is presented in four new ways, one of them as a series similar to the Euler’s arctangent series divided by 2π, which is its majorant series. All possibilities enable numerically stable and fast convergent computation of the bivariate normal integral with simple recursion. When tested computation on a random sample of one million parameter triplets with uniformly distributed components and using double precision arithmetic, the maximum absolute error was 3.45 × 10-16. In additional testing, focusing on cases with correlation coefficients close to one in absolute value, when the computation may be very sensitive to small rounding errors, the accuracy was retained. In rare potentially critical cases, a simple adjustment to the computation procedure was performed—one potentially critical computation was replaced with two equivalent non-critical ones. All new series are suitable for vector and high-precision computation, assuming they are supplemented with appropriate efficient and accurate computation of the arctangent and standard normal cumulative distribution functions. They are implemented by the R package Phi2rho, available on CRAN. Its functions allow vector arguments and are ready to work with the Rmpfr package, which enables the use of arbitrary precision instead of double precision numbers. A special test with up to 1024-bit precision computation is also presented.

Non Degeneration of Fibonacci Series, Pascal’s Elements and Hex Series

Generally Fibonacci series and Lucas series are the same, they converge to golden ratio. After I read Fibonacci series, I thought, is there or are there any series which converges to golden ratio. Because of that I explored the inter relations of Fibonacci series when I was intent on Fibonacci series in my difference parallelogram. In which, I found there is no degeneration on Fibonacci series. In my thought, Pascal triangle seemed like a lower triangular matrix, so I tried to find the inverse for that. In inverse form, there is no change against original form of Pascal elements matrix. One day I played with ring magnets, which forms hexagonal shapes. Number of rings which forms Hexagonal shape gives Hex series. In this paper, I give the general formula for generating various types of Fibonacci series and its non-degeneration, how Pascal elements maintain its identities and which shapes formed by hex numbers by difference and matrices.

Powell's optimal identification of material constants of thin-walled box girders based on Fibonacci series search method

A dynamic Bayesian error function of material constants of the structure is developed for thin-walled curve box girders. Combined with the automatic search scheme with an optimal step length for the one-dimensional Fibonacci series, Powell＇s optimization theory is used to perform the stochastic identification of material constants of the thin-walled curve box. Then, the steps in the parameter identification are presented. Powell＇s identification procedure for material constants of the thin-walled curve box is compiled, in which the mechanical analysis of the thin-walled curve box is completed based on the finite curve strip element （FCSE） method. Some classical examples show that Powell＇s identification is numerically stable and convergent, indicating that the present method and the compiled procedure are correct and reliable. During the parameter iterative processes, Powell＇s theory is irrelevant with the calculation of the FCSE partial differentiation, which proves the high computation efficiency of the studied methods. The stochastic performances of the system parameters and responses axe simultaneously considered in the dynamic Bayesian error function. The one-dimensional optimization problem of the optimal step length is solved by adopting the Fibonacci series search method without the need of determining the region, in which the optimized step length lies.

基于1DCNN和D-S多信息融合的光伏系统直流母线串联电弧故障检测

直流母线是光伏系统输出能源的主干道,由于长期曝晒、风化等作用,电缆、连接器等组件劣化,光伏系统直流母线中发生电弧的可能性急剧上升,极易引发火灾、触电等事故。在光伏系统中,串联电弧故障将使回路电流下降,传统的过流保护无法识别。因此,本文提出基于深度学习和证据理论(D-S)的方法来识别串联电弧故障,该方法基于并联电容器电流和电压信号,采用一维卷积神经网络(1DCNN)对检测数据进行电弧识别;在此基础上将基于单个传感数据的识别结果作为证据,运用D-S多信息合成法则计算得到信度分配,最后利用决策规则判断是否发生串联电弧故障。搭建多参数可调模型获取数据进行测试,结果表明:使用1DCNN识别方法,基于并联电容器电流和电压信号的串联电弧识别准确率分别为97.19%和94.98%,而基于1DCNN和D-S多信息融合的光伏系统直流串联电弧故障检测的识别准确率可提升至99%以上。

Anomalies identification of Earth’s rotation rate time series (2012-2017) for possible correlation with strong earthquakes occurrence

The analysis of the Earth’s rotation rate time series,from January 1,2012 till December 31,2017,is performed using two different time series analysis methods,both based on signal decomposition joined with forecasting approach.Anomalies in the time series are detected making the comparison between the raw signal and the forecasting one at the 95% confidence interval.The two methods show consistent results and the best is selected according to the evaluation of the prediction uncertainty.Both methods highlight correlations between detected anomalies in the Earth’s rotation rate time series and the world’s earthquakes occurrence with magnitude≥7 and/or number of events≥150 per day,within a time interval of ±10 days from each earthquake event.This study brings an innovation in the analysis of such time series and helps to better understand the extent of this relationship.

Classification of Short Time Series in Early Parkinson’s Disease With Deep Learning of Fuzzy Recurrence Plots

There are many techniques using sensors and wearable devices for detecting and monitoring patients with Parkinson’s disease(PD).A recent development is the utilization of human interaction with computer keyboards for analyzing and identifying motor signs in the early stages of the disease.Current designs for classification of time series of computer-key hold durations recorded from healthy control and PD subjects require the time series of length to be considerably long.With an attempt to avoid discomfort to participants in performing long physical tasks for data recording,this paper introduces the use of fuzzy recurrence plots of very short time series as input data for the machine training and classification with long short-term memory(LSTM)neural networks.Being an original approach that is able to both significantly increase the feature dimensions and provides the property of deterministic dynamical systems of very short time series for information processing carried out by an LSTM layer architecture,fuzzy recurrence plots provide promising results and outperform the direct input of the time series for the classification of healthy control and early PD subjects.

APPLICATIONS OF HYPERGEOMETRIC SUMMATION THEOREMS OF KUMMER AND DIXON INVOLVING DOUBLE SERIES

Using series iteration techniques identities and apply each of these identities in we derive a number of general double series order to deduce several hypergeometric reduction formulas involving the Srivastava-Daoust double hypergeometric function. The results presented in this article are based essentially upon the hypergeometric summation theorems of Kummer and Dixon.

ON THE CONDITIONAL VERSION OFKOLMOGOROV'S THREE-SERIES THEOREM

This paper studies the conditional version of Kolmogorov’s three-series theorem, and gets a new extention form of the conditional version. The results here present us an answer to the question when (or where) the conditional version also provide necessary conditions for convergence in dependent cases. Furthermore, some new sufficient conditions are obtained.

Extended Wiener Measure by Nonstandard Analysis for Financial Time Series

We propose a new approach to construct an extended Wiener measure using nonstandard analysis by E. Nelson. For the new definition we construct non-standardized convolution of probability measure for independent random variables. As an application, we consider a simple calculation of financial time series.

R/S method for evaluation of pollutant time series in environmental quality assessment

The significance of the fluctuation and randomness of the time series of each pollutant in environmental quality assessment is described for the first time in this paper. A comparative study was made of three different computing methods： the same starting point method, the striding averaging method, and the stagger phase averaging method. All of them can be used to calculate the Hurst index, which quantifies fluctuation and randomness. This study used real water quality data from Shazhu monitoring station on Taihu Lake in Wuxi, Jiangsu Province. The results show that, of the three methods, the stagger phase averaging method is best for calculating the Hurst index of a pollutant time series from the perspective of statistical regularity.

Forecasting Short Time Series with Missing Data by Means of Energy Associated to Series

In this work an algorithm to predict short times series with missing data by means energy associated of series using artificial neural networks (ANN) is presented. In order to give the prediction one step ahead, a comparison between this and previous work that involves a similar approach to test short time series with uncertainties on their data, indicates that a linear smoothing is a well approximation in order to employ a method for uncompleted datasets. Moreover, in function of the long- or short-term stochastic dependence of the short time series considered, the training process modifies the number of patterns and iterations in the topology according to a heuristic law, where the Hurst parameter H is related with the short times series, of which they are considered as a path of the fractional Brownian motion. The results are evaluated on high roughness time series from solutions of the Mackey-Glass Equation (MG) and cumulative monthly historical rainfall data from San Agustin, Cordoba. A comparison with ANN nonlinear filters is shown in order to see a better performance of the outcomes when the information is taken from geographical point observation.

基于RST算法提取乌什M_(S)7.1地震地表温度异常

为探讨RST算法在新疆地区地震热红外异常提取中的适用性,以乌什M_(S)7.1地震为例,构建研究区MODIS地表温度数据集,并对数据集进行逐像元时间序列趋势去除后,利用RST算法提取乌什M_(S)7.1地震地表温度异常信息。结果表明:乌什M_(S)7.1地震地表温度异常的变化过程分为“出现—消失—再出现—扩大—达到最大—消失”6个阶段;乌什M_(S)7.1地震的地表温度异常区域大致沿构造带的走向分布;乌什M_(S)7.1地震发生在地表温度异常面积出现最大值后的第99天。利用MODIS数据结合RST算法提取地震热红外异常在新疆地区相关研究中具有一定的可行性。

Discrete Convolution Associated with Fractional Cosine and Sine Series

Fractional sine series(FRSS)and fractional cosine series(FRCS)are the discrete form of the fractional cosine transform(FRCT)and fractional sine transform(FRST).The recent stud-ies have shown that discrete convolution is widely used in optics,signal processing and applied mathematics.In this paper,firstly,the definitions of fractional sine series(FRSS)and fractional co-sine series(FRCS)are presented.Secondly,the discrete convolution operations and convolution theorems for fractional sine and cosine series are given.The relationship of two convolution opera-tions is presented.Lastly,the discrete Young’s type inequality is established.The proposed theory plays an important role in digital filtering and the solution of differential and integral equations.

Solution of Delay Differential Equations Using a Modified Power Series Method

This paper presents a Modified Power Series Method (MPSM) for the solution of delay differential equations. Unlike the traditional power series method which is applied to solve only linear differential equations, this new approach is applicable to both linear and nonlinear problems. The method produces a system of algebraic equations which is solved to determine the coefficients in the trial solution. The method provides the solution in form of a rapid convergent series. The obtained results for numerical examples demonstrate the reliability and efficiency of the method.

Hurst’s Memory for Chaotic, Tree Ring, and SOI Series

Hurst’s memory that roots in early work of the British hydrologist H.E. Hurst remains an open problem in stochastic hydrology. Today, the Hurst analysis is widely used for the hydrological studies for the memory and characteristics of time series and many methodologies have been developed for the analysis. So, there are many different techniques for the estimation of the Hurst exponent (H). However, the techniques can produce different characteristics for the persistence of a time series each other. This study uses several techniques such as adjusted range, rescaled range (RR) analysis, modified rescaled range (MRR) analysis, 1/f power spectral density analysis, Maximum Likelihood Estimation (MLE), detrended fluctuations analysis (DFA), and aggregated variance time (AVT) method for the Hurst exponent estimation. The generated time series from chaos and stochastic systems are analyzed for the comparative study of the techniques. Then, this study discusses the advantages and disadvantages of the techniques and also the limitations of them. We found that DFA is the most appropriate technique for the Hurst exponent estimation for both the short term memory and long term memory. We analyze the SOI (Southern Oscillations Index) and 6 tree-ring series for USA sites by means of DFA and the BDS statistic is used for nonlinearity test of the series. From the results, we found that SOI series is nonlinear time series which has a long term memory of H = 0.92. Contrary to earlier work, all the tree ring series are not random from our analysis. A certain tree ring series show a long term memory of H = 0.97 and nonlinear property. Therefore, we can say that the SOI series has the properties of long memory and nonlinearity and tree ring series could also show long memory and non-linearity.

A Statistical Method for Determining the Fractal Dimension of Time Series

In this paper,we present a new method for determining the fractal dimension of time series and the algorithm of H index(Hurst index).

An Exponential Series Method for the Solution of Free Convection Boundary Layer Flow in a Saturated Porous Medium

Third order nonlinear ordinary differential equation, subject to appropriate boundary conditions, arising in fluid mechanics is solved exactly using more suggestive schemes- Dirichlet series and method of stretching variables. These methods have advantages over pure numerical methods in obtaining derived quantities accurately for various values of the parameters involved at a stretch and are valid in a much larger domain compared with classical numerical schemes.

Feynman Perturbation Series for the Morse Potential

In this paper we give an alternative treatment of the Schrodinger equation with the Morse potential, which based on the exact summation of the Feynman perturbation series in its original form. Using Fourier transform we establish a recurrence equation between terms of the perturbation series. Finally, by the inverse Fourier transform and some technical tools of the ordinary differential equations of the second order, we can compute the exact sum of the perturbation series which is the Green’s function of the problem.

ZTE Rolls Out a Series of Industry's First TD-SCDMA Handset Products at PT/WIRELESS & NETWORKS COMM CHINA 2007

ZTE Corporation (ZTE), a leading global provider of telecommunications equipment and network solutions, unveiled