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.

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.

Estimation of Hourly and Daily Global Solar Radiation at Clear Days Using an Approach Based on Modified Version of Gaussian Distribution

The performance of two models,Jam and Baig,based on the modified version of Gaussian distribution function in estimating the daily total of global solar radiation and its distribution through the hours of the day from sunrise to sunset al any clear day is evaluated with our own measured data in the period from June 1992 to May 1993 in Qena Egypt The results show a high relative deviation of calculated values from measured ones,especially for Jain model,in the most hours of the day,except for those near to local noon.This misfit behavior is quite obvious in the early morning and late afternoon A new approach has been proposed in this paper to estimate the daily and hourly global solar radiation This model performs with very high accuracy on the recorded data in our region.The validity of this approach was verified with new measurements in some clear days in June and August 1994.The resultant very low relative deviation of the calculated values of global solar radiation from the measured ones confirms the high performance of the approach proposed in this work

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.

Open World Recognition of Communication Jamming Signals

To improve the recognition ability of communication jamming signals,Siamese Neural Network-based Open World Recognition(SNNOWR)is proposed.The algorithm can recognize known jamming classes,detect new(unknown)jamming classes,and unsupervised cluseter new classes.The network of SNN-OWR is trained supervised with paired input data consisting of two samples from a known dataset.On the one hand,the network is required to have the ability to distinguish whether two samples are from the same class.On the other hand,the latent distribution of known class is forced to approach their own unique Gaussian distribution,which is prepared for the subsequent open set testing.During the test,the unknown class detection process based on Gaussian probability density function threshold is designed,and an unsupervised clustering algorithm of the unknown jamming is realized by using the prior knowledge of known classes.The simulation results show that when the jamming-to-noise ratio is more than 0d B,the accuracy of SNN-OWR algorithm for known jamming classes recognition,unknown jamming detection and unsupervised clustering of unknown jamming is about 95%.This indicates that the SNN-OWR algorithm can make the effect of the recognition of unknown jamming be almost the same as that of known jamming.

Identification of linear continuous-time system using wavelet modulating filters

An approach to identification of linear continuous-time system is studied with modulating functions. Based on wavelet analysis theory, the multi-resolution modulating functions are designed, and the corresponding filters have been analyzed. Using linear modulating filters, we can obtain an identification model that is parameterized directly in continuous-time model parameters. By applying the results from discrete-time model identification to the obtained identification model, a continuous-time estimation method is developed. Considering the accuracy of parameter estimates, an instrumental variable (Ⅳ) method is proposed, and the design of modulating integral filter is discussed. The relationship between the accuracy of identification and the parameter of modulating filter is investigated, and some points about designing Gaussian wavelet modulating function are outlined. Finally, a simulation study is also included to verify the theoretical results.

Detecting winter canola(Brassica napus) phenological stages using an improved shape-model method based on time-series UAV spectral data

Accurate information about phenological stages is essential for canola field management practices such as irrigation, fertilization, and harvesting. Previous studies in canola phenology monitoring focused mainly on the flowering stage, using its apparent structure features and colors. Additional phenological stages have been largely overlooked. The objective of this study was to improve a shape-model method(SMM) for extracting winter canola phenological stages from time-series top-of-canopy reflectance images collected by an unmanned aerial vehicle(UAV). The transformation equation of the SMM was refined to account for the multi-peak features of the temporal dynamics of three vegetation indices(VIs)(NDVI, EVI, and CI). An experiment with various seeding scenarios was conducted, including four different seeding dates and three seeding densities. Three mathematical functions: asymmetric Gaussian function(AGF), Fourier function, and double logistic function, were employed to fit timeseries vegetation indices to extract information about phenological stages. The refined SMM effectively estimated the phenological stages of canola, with a minimum root mean square error(RMSE) of 3.7 days for all phenological stages. The AGF function provided the best fitting performance, as it captured multiple peaks in the growth dynamics characteristics for all seeding date scenarios using four scaling parameters. For the three selected VIs, CIred-edgeachieved the greatest accuracy in estimating the phenological stage dates. This study demonstrates the high potential of the refined SMM for estimating winter canola phenology.

ON THE BOUNDS OF THE PERIMETER OF AN ELLIPSE

In this paper,we present new bounds for the perimeter of an ellipse in terms of harmonic,geometric,arithmetic and quadratic means;these new bounds represent improvements upon some previously known results.

Generalized Gr?tzsch ring function and generalized elliptic integrals

Abstract. In this paper, we study the quotient of hypergeometric functions μα （r） in the theory of Ramanujan＇s generalized modular equation for α ∈（0, 1/2]. Several new inequalities are given for this and related functions. Our main results complement and generalize some known results in the literature.

A RANDOM TRANSPORT-DIFFUSION EQUATION

In this paper we study the integral curve in a random vector field perturbed by white noise. It is related to a stochastic transport-diffusion equation. Under some conditions on the covariance function of the vector field, the solution of this stochastic partial differential equation is proved to have moments. The exact p-th moment is represented through integrals with respect to Brownian motions. The basic tool is Girsanov formula.

A Novel Method for Solving Ordinary Differential Equations with Artificial Neural Networks

This research work investigates the use of Artificial Neural Network (ANN) based on models for solving first and second order linear constant coefficient ordinary differential equations with initial conditions. In particular, we employ a feed-forward Multilayer Perceptron Neural Network (MLPNN), but bypass the standard back-propagation algorithm for updating the intrinsic weights. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the initial or boundary conditions and contains no adjustable parameters. The second part involves a feed-forward neural network to be trained to satisfy the differential equation. Numerous works have appeared in recent times regarding the solution of differential equations using ANN, however majority of these employed a single hidden layer perceptron model, incorporating a back-propagation algorithm for weight updation. For the homogeneous case, we assume a solution in exponential form and compute a polynomial approximation using statistical regression. From here we pick the unknown coefficients as the weights from input layer to hidden layer of the associated neural network trial solution. To get the weights from hidden layer to the output layer, we form algebraic equations incorporating the default sign of the differential equations. We then apply the Gaussian Radial Basis function (GRBF) approximation model to achieve our objective. The weights obtained in this manner need not be adjusted. We proceed to develop a Neural Network algorithm using MathCAD software, which enables us to slightly adjust the intrinsic biases. We compare the convergence and the accuracy of our results with analytic solutions, as well as well-known numerical methods and obtain satisfactory results for our example ODE problems.

An alternative approach to match field production data from unconventional gas-bearing systems

Nowadays,the unconventional gas-bearing system plays an increasingly important role in energy market.The performances of the current history-matching techniques are not satisfied when applied to such systems.To overcome this shortfall,an alternative approach was developed and applied to investigate production data from an unconventional gas-bearing system.In this approach,the fluid flow curve obtained from the field is the superposition of a series of Gaussian functions.An automatic computing program was developed in the MATLAB,and both gas and water field data collected from a vertical well in the Linxing Block,Ordos Basin,were used to present the data processing technique.In the reservoir study,the automatic computing program was applied to match the production data from a single coal seam,multiple coal seams and multiple vertically stacked reservoirs with favourable fitting results.Compared with previous approaches,the proposed approach yields better results for both gas and water production data and can calculate the contributions from different reservoirs.The start time of the extraction for each gas-containing unit can also be determined.The new approach can be applied to the field data prediction and designation for the well locations and patterns at the reservoir scale.

Binary Tomography Reconstruction with Limited-Data by a Convex Level-Set Method

This paper proposes a new level-set-based shape recovery approach that can be applied to a wide range of binary tomography reconstructions.In this technique,we derive generic evolution equations for shape reconstruction in terms of the underlying level-set parameters.We show that using the appropriate basis function to parameterize the level-set function results in an optimization problem with a small number of parameters,which overcomes many of the problems associated with the traditional level-set approach.More concretely,in this paper,we use Gaussian functions as a basis function placed at sparse grid points to represent the parametric level-set function and provide more flexibility in the binary representation of the reconstructed image.In addition,we suggest a convex optimization method that can overcome the problem of the local minimum of the cost function by successfully recovering the coefficients of the basis function.Finally,we illustrate the performance of the proposed method using synthetic images and real X-ray CT projection data.We show that the proposed reconstruction method compares favorably to various state-of-the-art reconstruction techniques for limited-data tomography,and it is also relatively stable in the presence of modest amounts of noise.Furthermore,the shape representation using a compact Gaussian radial basis function works well.

Improved Bacterial Foraging Optimization Algorithm Based on Fuzzy Control Rule Base

Manual construction of a rule base for a fuzzy system is the hard and time-consuming task that requires expert knowledge.In this paper we proposed a method based on improved bacterial foraging optimization（IBFO）,which simulates the foraging behavior of “E.coli” bacterium,to tune the Gaussian membership functions parameters of an improved Takagi-Sugeno-Kang fuzzy system（C-ITSKFS） rule base.To remove the defect of the low rate of convergence and prematurity,three modifications were produced to the standard bacterial foraging optimization（BFO）.As for the low accuracy of finding out all optimal solutions with multi-method functions,the IBFO was performed.In order to demonstrate the performance of the proposed IBFO,multiple comparisons were made among the BFO,particle swarm optimization（PSO）,and IBFO by MATLAB simulation.The simulation results show that the IBFO has a superior performance.

Research on Image Quality Enhancement Algorithm Using Hessian Matrix

The Hessian matrix has a wide range of applications in image processing,such as edge detection,feature point detection,etc.This paper proposes an image enhancement algorithm based on the Hessian matrix.First,the Hessian matrix is obtained by convolving the derivative of the Gaussian function.Then use the Hessian matrix to enhance the linear structure in the image.Experimental results show that the method proposed in this paper has strong robustness and accuracy.

东西伯利亚地区植被对气温和降水交互作用的定量化响应

Significant changes to the world’s climate over the past few decades have had an impact on the development of plants.Vegetation in high latitude regions,where the ecosystems are fragile,is susceptible to climate change.It is possible to better understand vegetation’s phenological response to climate change by examining these areas.Traditional studies have mainly investigated how a single meteorological factor affects changes in vegetation phenology through linear correlation analysis,which is insufficient for quantitatively revealing the effects of various climate factor interactions on changes in vegetation phenology.We used the asymmetric Gaussian method to fit the normalized difference vegetation index(NDVI)curve and then used the dynamic threshold method to extract the phenological parameters,including the start of the season(SOS),end of the season(EOS),and length of the season(LOS),of the vegetation in this study area in the Tundra-Tagar transitional zone in eastern and western Siberia from 2000 to 2017.The monthly temperature and precipitation data used in this study were obtained from the climate research unit(CRU)meteorological dataset.The degrees to which the changes in temperature and precipitation in the various months and their interactions affected the changes in the three phenological parameters were determined using the GeoDetector,and the results were explicable.The findings demonstrate that the EOS was more susceptible to climate change than the SOS.The vegetation phenology shift was best explained by the climate in March,April,and September,and the combined effect of the temperature and precipitation had a greater impact on the change in the vegetation phenology compared with the effects of the individual climate conditions.The results quantitatively show the degree of interaction between the variations in temperature and precipitation and their effects on the changes in the different phenological parameters in the various months.Understanding how various climatic variations effect phenology changes in plants at different times may be more intuitive.This research provides as a foundation for research on how global climate change affects ecosystems and the global carbon cycle.

A Fully Pipelined Probability Density Function Engine for Gaussian Copula Model

The Gaussian Copula Probability Density Function （PDF） plays an important role in the fields of finance, hydrological modeling, biomedical study, and texture retrieval. However, the existing schemes for evaluating the Gaussian Copula PDF are all computationally-demanding and generally the most time-consuming part in the corresponding applications. In this paper, we propose an FPGA-based design to accelerate the computation of the Gaussian Copula PDF. Specifically, the evaluation of the Gaussian Copula PDF is mapped into a fully-pipelined FPGA dataflow engine by using three optimization steps： transforming the calculation pattern, eliminating constant computations from hardware logic, and extending calculations to multiple pipelines. In the experiments on 10 typical large-scale data sets, our FPGA-based solution shows a maximum of 1870 times speedup over a well-tuned single- core CPU-based solution, and 610 times speedup over a well-optimized parallel quad-core CPU-based solution when processing two-dimensional data.

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.

东西伯利亚苔原-泰加林过渡带植被遥感物候时空特征及其对气温变化的响应

Phenology is an important indicator of climate change.Studying spatiotemporal variations in remote sensing phenology of vegetation can provide a basis for further analysis of global climate change.Based on time series data of MODIS-NDVI from 2000 to 2017,we extracted and analyzed four remote sensing phenological parameters of vegetation,including the Start of Season(SOS),the End of Season(EOS),the Middle of Season(MOS)and the Length of Season(LOS),in tundra-taiga transitional zone in the East Siberia,using asymmetric Gaussian function and dynamic threshold methods.Meanwhile,we analyzed the responses of the four phenological parameters to the temperature change based on the temperature change data from Climate Research Unit(CRU).The results show that:in regions south of 64°N,with the rise of temperature in April and May,the SOS in the corresponding area was 5-15 days ahead of schedule;in the area between 64°N and 72°N,with the rise of temperature in May and June,the SOS in the corresponding area was 10-25 days ahead of schedule;in the northernmost of the study area on the coast of the Arctic Ocean,with the drop of temperature in May and June,the SOS in the corresponding area was 15-25 days behind schedule;in the northwest of the study area in August and the southwest in September,with the drop of temperature,the EOS in the corresponding areas was 15-30 days ahead of schedule;in regions south of 67°N,with the rise of temperature in September and October,the EOS in the corresponding area was 5-30 days behind schedule;the change of the EOS in autumn was more sensitive to the change of the SOS in spring,because the smaller temperature fluctuation can cause the larger change of the EOS;the growth season of vegetation in the study area was generally moving forward,and the LOS in the northwest was shortened,while the LOS in the middle and south of the study area was prolonged.

Real-time comprehensive driving ability evaluation algorithm for intelligent assisted driving

To meet the needs of the human-machine co-driving decision problem in the intelligent assisted driving system for real-time comprehensive driving ability evaluation of drivers,this paper proposes a real-time comprehensive driving ability evaluation method that integrates driving skill,driving state,and driving style.Firstly,by analyzing the driving experiment data obtained based on the intelligent driving simulation platform(the experiment can effectively distinguish the driver's driving skills and avoid the interference of driving style),the feature values that significantly represent driving skills and driving state are selected,and the time correlation between driving state and driving skills is pointed out.Furthermore,the concept of relativity in comprehensive driving ability evaluation is further proposed.Under this concept,the natural driving trajectory dataset-HighD is used to establish the distribution map of feature values of the human driver group as the evaluation benchmark to realize the relative evaluation of driving skill and driving state.Similarly,HighD is used to establish a distribution map of human driver style feature values as an evaluation benchmark to achieve relative driving style evaluation.Finally,a comprehensive driving ability evaluation model with a“punishment”and“affirmation”mechanism is proposed.The experimental comparative analysis shows that the evaluation algorithm proposed in this paper can take into account the driver's driving skill,driving state,and driving style in the real-time comprehensive driving ability evaluation,and draw differential evaluation conclusions based on the“punishment”and“affirmation”mechanism model to achieve a comprehensive and objective evaluation of the driver's driving ability.It can meet the needs of human-machine shared driving decisions for driver's driving ability evaluation.