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Articles 34711 - 34740 of 302422

Full-Text Articles in Physical Sciences and Mathematics

Impact Of Grazing Management On The Productivity Of Cold Temperate Grasslands Of Southern Patagonia - A Critical Assessment, A. F. Cibils, M. B. Coughenour Jan 2022

Impact Of Grazing Management On The Productivity Of Cold Temperate Grasslands Of Southern Patagonia - A Critical Assessment, A. F. Cibils, M. B. Coughenour

IGC Proceedings (1993-2023)

The contention that productivity of cold temperate grasslands of southern Patagonia have been greatly affected by domestic herbivore grazing in the past is fairly widespread. The consequences of spatiotemporal interactions between grazing and abiotic factors on the overall productivity of such systems, however, are still poorly understood. Predictions of both short- and long-term impacts of grazing management strategies on grassland productivity require a better understanding of these interactions. In this paper we make a critical assessment of the impact of grazing management on long-term sustainability of the grass steppes of southern Patagonia (Argentina). We discuss this issue in the context …

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Productivity Of Winter And Spring Native Improved Vegetation Under Different Grazing Stocking Rates On Basaltic Soil, M. Bemhaja Jan 2022

Productivity Of Winter And Spring Native Improved Vegetation Under Different Grazing Stocking Rates On Basaltic Soil, M. Bemhaja

IGC Proceedings (1993-2023)

Improved grassland with Lotus corniculatus, Trifolium repens and annual phosphorus fertilization, was evaluated during winter and spring for five consecutive years under grazing at three different stocking rates on medium basaltic soil. Specific contribution of Lotus and Trifolium reached 67% of grassland community in the first year. Stipa setigera and Poa lanigera increased their contribution from 0.8 to 17% in the fourth year of grazing experiment. Highest average value of crude protein was 19% during the second year and was related to legumes contribution. Forage availability was always up to 1.5 ton MS/ha at the three stocking treatments. Average daily …

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Effects Of Cattle Grazing On A Perennial Legume, Bitumen Trefoil (Bituminaria Bituminosa (L.) Stirton), In A Mediterranean Grassland, M. Gutman, M. Sternberg, A. Perevoltsky Jan 2022

Effects Of Cattle Grazing On A Perennial Legume, Bitumen Trefoil (Bituminaria Bituminosa (L.) Stirton), In A Mediterranean Grassland, M. Gutman, M. Sternberg, A. Perevoltsky

IGC Proceedings (1993-2023)

A six-year study was conducted in a Mediterranean grassland in north-eastern Israel to investigate the effects of cattle grazing management on a perennial legume, Bitumen Trefoil (Bituminaria bituminosa (L.) Stirton). The relationship between grazing pressure and its relative plant cover was studied in the context of inter-annual variation in rainfall. Treatments included manipulations of stocking rates (moderate, heavy and very heavy) and of grazing regimes (continuous vs. seasonal), in a factorial design. The results showed that inter-seasonal rainfall variation was a dominant factor in the expression of plant cover changes of this species. Grazing showed no significant effect on …

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A Fast Simulation Method For Ship Target Sar Signal Echo, Yuan Fei, Jianhong Li, Yin Hao, Yuhao Wang, Hong Sheng Jan 2022

A Fast Simulation Method For Ship Target Sar Signal Echo, Yuan Fei, Jianhong Li, Yin Hao, Yuhao Wang, Hong Sheng

Journal of System Simulation

Abstract: In order to meet the application requirements of synthetic aperture radar (SAR) in ocean remote sensing, a fast simulation method for the ship target SAR echo generation is presented, which carries out the accurate electromagnetic modeling to the important ship targets. “Four paths" model is used to calculate the complex echo between the ship target and sea surface, and the facet model is used to model the sea surface backscattering. After the two parts of echoes being synthesized, the SAR echo of whole scene is gotten, and the echo is processed by spot SAR imaging processing algorithm to verify …

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Study On Bidirectional Coupling Of Human Thermal Comfort Parameters And Cabin Thermal Environment, Jue Qu, Dayan Wang, Wang Wei, Sina Dang Jan 2022

Study On Bidirectional Coupling Of Human Thermal Comfort Parameters And Cabin Thermal Environment, Jue Qu, Dayan Wang, Wang Wei, Sina Dang

Journal of System Simulation

Abstract: At present, for the existing cockpit heat system, are studied more the airflow tissue parameters of the thermal environment and the human heat regulation is taken into account less, which leads to the low accuracy of simulation result evaluating the human thermal comfort. Through CFD (computational fluid dynamics) method, energy equation, RANS (reynolds-average navier-stokes) equation, and N-S(navier-stokes) equation, by combining the human temperature distribution with the cabin thermal environment parameters, the cabin heat system model based on the human heat regulation is established. The model considers the interacting influence of the human thermal regulation and the thermal environmental airflow …

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Visual Analysis Of Cross-Domain Association Of Time-Series Data, Beibei Han, Yingmei Wei, Yujie Fang, Shanshan Wan Jan 2022

Visual Analysis Of Cross-Domain Association Of Time-Series Data, Beibei Han, Yingmei Wei, Yujie Fang, Shanshan Wan

Journal of System Simulation

Abstract: Time series data is the important research object of data mining. The current visual analysis technology of time series data rarely conducts the cross-domain correlation. The development and evolution of statistical time series data in the spatio-temporal domain and the text theme data in the cognitive domain cannot be simulitaneously supervised in a unified view. The user's visual analysis process based on cross-domain time series data is abstracted and a visual analysis process model is proposed. A multi-view collaborative cross-domain correlation visual analysis tool is designed on the basis of the model. The case study of epidemic time …

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Wsn Clustering Routing Protocol For Bridge Structure Health Monitoring, Li Gang, Caixia Zhang, Shaolin Hu, Xiangdong Wang, Guo Jing Jan 2022

Wsn Clustering Routing Protocol For Bridge Structure Health Monitoring, Li Gang, Caixia Zhang, Shaolin Hu, Xiangdong Wang, Guo Jing

Journal of System Simulation

Abstract: In the specific application of bridge structure health monitoring (BSHM), clustering based only on the geographic location of nodes or using a single-hop strategy to complete inter-cluster routing may cause the unstability of the entire wireless sensor networks(WSN). For WSN in BSHM scenario, the concept of "energy distribution" is proposed, and an energy balance clustering routing protocol energy balance protocol(EBP) is designed. The second clustering, the high-energy areas in WSN bear more energy consumption, and a multi hop strategy based on region division is designed to control the number of forwarding hops. The simulation results show that, compared with …

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Optimization Of Household Electricity Consumption Period Based On Improved Multi-Objective Particle Swarm Optimization, Xiuying Yan, Miaomiao Dang Jan 2022

Optimization Of Household Electricity Consumption Period Based On Improved Multi-Objective Particle Swarm Optimization, Xiuying Yan, Miaomiao Dang

Journal of System Simulation

Abstract: Aiming at the household power load scheduling optimization, three objectives of the cost of electricity, satisfaction and user-side fluctuation degree are taken into comprehensive account. An improved adaptive weight multi-objective particle swarm optimization (IAW-MOPSO) algorithm is proposed to realize the scheduling optimization of household power load. The local improvement ability and global search ability of particle swarm optimization are balanced by updating the inertia weight of particle fitness value. The simulation results of five groups show that the proposed optimization strategy reduces the electricity charge by 29%, ensures the stability of electricity consumption in the peak period, and …

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Uncertainty Simulation Method Based On Deep Bayesian Networks Learning, Nie Kai, Kejun Zeng, Qinghai Meng Jan 2022

Uncertainty Simulation Method Based On Deep Bayesian Networks Learning, Nie Kai, Kejun Zeng, Qinghai Meng

Journal of System Simulation

Abstract: There are lots of uncertain elements in battlefields situation assessment and the uncertainty simulation would enhance the ability of situation assessment. A deep variational autoencoder bayesian networks (BN) model with memory module is proposed aiming at the problem of being unable to represent the uncertainties exactly caused by the various combat objects and more uncertain elements. Based on the deep BN learning, the situation assessment model is designed from the deep generative model. The principle of deep generative model mixing with the memory module is discussed and the leaning and reasoning process of the model is explained. The proposed …

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Optimization And Prediction For Multi-Robot Combination Maximum Coverage Area, Yutong Wang, Shiwei Ma, Yuanrui Yang, Chaoyu Chen Jan 2022

Optimization And Prediction For Multi-Robot Combination Maximum Coverage Area, Yutong Wang, Shiwei Ma, Yuanrui Yang, Chaoyu Chen

Journal of System Simulation

Abstract: Aiming at the optimal control of the multi-robot combination maximum coverage area, based on the intensity radial attenuation disc model and following the superposition principle, a method for estimating, optimizing and predicting the effective coverage area of the multi-robot combination is proposed. The Monte Carlo method is used to estimate the effective coverage area of the robot combination, and the multiple population genetic algorithm is used to obtain the maximum effective coverage area of the combination, and the support vector machine regression is used to predict the relationship between the number of robots and the maximum effective coverage area. …

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Modeling And Simulation Of Emergency Medical Resources Allocation In Shanghai During Covid-19, Changjia Fan, Yanqiu Du, Liang Di, Hu Kai, Jiayan Huang Jan 2022

Modeling And Simulation Of Emergency Medical Resources Allocation In Shanghai During Covid-19, Changjia Fan, Yanqiu Du, Liang Di, Hu Kai, Jiayan Huang

Journal of System Simulation

Abstract: Modeling and simulating on the allocation of emergency medical resources in Shanghai with COVID-19 is carried out. Based on the SEIR model of infectious diseases, combined with the process of outpatients visiting and inpatients treatment, a SEIOWHR(susceptible-exposed-infected-outpatients- waiting to hospitalized-hospitalized-removed) system dynamics model is established. If the Wuhan epidemic occurred in Shanghai, based on the model, the amount of emergency medical resources needed, the gap time of medical resources and the disease progression of patients who are waiting to hospitalized under the different supply of medical resources is simulated, and the key factors in the allocation of medical …

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Adaptive Optimization In Feature-Based Slam Visual Odometry, Yanan Yu, Dunhuang Shi, Chunjie Hua Jan 2022

Adaptive Optimization In Feature-Based Slam Visual Odometry, Yanan Yu, Dunhuang Shi, Chunjie Hua

Journal of System Simulation

Abstract: Aiming to reduce the impact of dynamic environments on simultaneous localization and mapping (SLAM) of mobile robots, an adaptive optimization method in a feature-based visual odometry is proposed. The method helps to improve the invariance of image feature in illumination changing situation and to extract features effectively in areas where the texture information is not sufficient to make contributions to feature matching. Meanwhile, down sampling is applied to establish image pyramids and each scaled image is divided into cells based on a defined rule. Illumination adaptive nonlinear adjustments for each cell are applied to increase the image details, and …

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Simulation Of Rocket Exhaust Plumes Recognition Based On Dynamic Time Warping, Liu Hao, Hongxia Mao, Zhihe Xiao, Liu Zheng Jan 2022

Simulation Of Rocket Exhaust Plumes Recognition Based On Dynamic Time Warping, Liu Hao, Hongxia Mao, Zhihe Xiao, Liu Zheng

Journal of System Simulation

Abstract: By analyzing the infrared radiation characteristics of the rocket exhaust plumes and summarizing the changing law of the radiant intensity sequence, an improved recognition algorithm based on Dynamic Time Warping algorithm is proposed. In order to improve the effect of sequence shape similarity measurement, the distance matrix and matching path are calculated by the derivative sequence, and the distance is recalculated according to the matching path and radiant intensity value. The problem of path matching affected by the non-uniformity of observation sequence length is solved to a certain extent by using the prefix and suffix relaxation factors. The simulation …

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Planning And Analysis On Uav Trajectory Based On Pce Method, Sijie Zeng, Yan Liang, Xiaojun Duan Jan 2022

Planning And Analysis On Uav Trajectory Based On Pce Method, Sijie Zeng, Yan Liang, Xiaojun Duan

Journal of System Simulation

Abstract: Focusing on the uncertainty in the UAV trajectory planning, combined with the artificial potential energy method, a UAV trajectory planning method based on polynomial chaos expansion (PCE), which can also efficiently obtain the optimal parameters of the model based on artificial potential field method is proposed. The PCE proxy model is established, and the stochastic collocation method is used to quickly solve the problem, so as to avoid the insufficient computing resources. Through the Sobol sensitivity analysis, the calculation overhead of the uncertainty parameters in the trajectory planning model is reduced. Cases of UAV trajectory planning prove the effectiveness …

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Autonomous Vehicle Path Tracking Control System Based On Energy Optimization, Xiaolong Wu, Fugen Xia, Chen Jing, Xu Jia Jan 2022

Autonomous Vehicle Path Tracking Control System Based On Energy Optimization, Xiaolong Wu, Fugen Xia, Chen Jing, Xu Jia

Journal of System Simulation

Abstract: Powertrain control is important to the dynamic performance and economy of driverless cars and a path following control strategy based on energy optimization is proposed. The control strategy includes two parts. The nonlinear model predictive control is used in the upper controller to calculate the required power parameters and front wheel angle. The lower-level controller is designed based on the optimal value of motor energy consumption which ensure the motor being always running at the optimal state of efficiency. In addition, the continuously variable transmission (CVT) is dynamically adjusted according to the motor state to meet the vehicle power …

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Agent- Based Research On Power Absorption Simulation Analysis Of Renewable Energy, Zhang Luan, Zhengjun Luo, Dequn Zhou Jan 2022

Agent- Based Research On Power Absorption Simulation Analysis Of Renewable Energy, Zhang Luan, Zhengjun Luo, Dequn Zhou

Journal of System Simulation

Abstract: Aiming at the “three abandonment”, a guarantee mechanism for the consumption of renewable energy power is proposed in our country. In order to stimulate the consumption of renewable energy power, a multi-agent simulation method is used to analyze the transaction behavior and interaction of market players, and the key factors affecting the consumption of renewable energy power is analyzed to simulated the consumption of renewable energy and the evolution of the number of active consumers. The results show that the subscribed green certificate can directly promote the consumption of renewable energy power, and it is necessary to comprehensively consider …

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Approximate Solution Of Generalized Modified B-Equation By Optimal Auxiliary Function Method, Aatif Ali, Laiq Zada, Rashid Nawaz Jan 2022

Approximate Solution Of Generalized Modified B-Equation By Optimal Auxiliary Function Method, Aatif Ali, Laiq Zada, Rashid Nawaz

International Journal of Emerging Multidisciplinaries: Mathematics

In this study, the implantation of a new semi-analytical method called the optimal auxiliary function method (OAFM) has been extended to partial differential equations. The adopted method was tested upon for approximate solution of generalized modified b-equation. The first-order numerical solution obtained by OAFM has been compared with the variational homotopy perturbation method (VHPM). The method possesses the auxiliary function and control parameters which can be easily handled during simulation of the nonlinear problem. From the numerical and graphical results, we concluded the method is very effective and easy to implement for the nonlinear PDEs.

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Inequalities For Riemann - Stieltjes Integral, Mohammad Alomari Jan 2022

Inequalities For Riemann - Stieltjes Integral, Mohammad Alomari

International Journal of Emerging Multidisciplinaries: Mathematics

Two new inequalities for Riemann--Stieltjes integral are introduced for functions of bounded $p$-variation and H\"{o}lder continuous integrators.

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Engine Wear Fault Diagnosis Based On Supervised Kernel Entropy Component Analysis, Zhichao Zhu, Dinghui Wu, Yuanchang Yue Jan 2022

Engine Wear Fault Diagnosis Based On Supervised Kernel Entropy Component Analysis, Zhichao Zhu, Dinghui Wu, Yuanchang Yue

Journal of System Simulation

Abstract: Focus on the influence of environment on engine operation, which leads to a large amount of redundant information and nonlinear structure in oil spectral data that affects the engine fault diagnosis results, the feature extraction method of SKECA (supervised kernel entropy component analysis) is proposed. A supervised learning algorithm is adopted on the basis of Kernel Entropy Component Analysis, which extracts the inherent geometric features of oil spectrum data to make the extracted fault features include the discriminative information. GA (genetic algorithm) is used to find parameters to optimize the results of feature extraction, and SVM (support vector machine) …

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Research And Implementation Of On-Board Human Factors Collaborative Simulation System, Changqing Yin, Tianran Tan, Jianmin Wang Jan 2022

Research And Implementation Of On-Board Human Factors Collaborative Simulation System, Changqing Yin, Tianran Tan, Jianmin Wang

Journal of System Simulation

Abstract: In order to improve the multi-vehicle evaluation and follow the trend of the Internet of Vehicles in modern vehicle-road collaborative system, an on-board human factors collaborative simulation system is built. Distributed theory is used to establish a joint platform for vehicle simulation, and an on-board human factors co-simulation system is built and MQTT network is used to optimize the data distribution mechanism and data collection mechanism. The simulation experiment shows that the built on-board human factor co-simulation platform reduces the coupling between the participating experimental subsystems and the main system, and has good robustness and flexibility. The distributed data …

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Machine Learning Approach To Simulate Soil Co2 Fluxes Under Cropping Systems, Toby A. Adjuik, Sarah C. Davis Jan 2022

Machine Learning Approach To Simulate Soil Co2 Fluxes Under Cropping Systems, Toby A. Adjuik, Sarah C. Davis

Biosystems and Agricultural Engineering Graduate Research

With the growing number of datasets to describe greenhouse gas (GHG) emissions, there is an opportunity to develop novel predictive models that require neither the expense nor time required to make direct field measurements. This study evaluates the potential for machine learning (ML) approaches to predict soil GHG emissions without the biogeochemical expertise that is required to use many current models for simulating soil GHGs. There are ample data from field measurements now publicly available to test new modeling approaches. The objective of this paper was to develop and evaluate machine learning (ML) models using field data (soil temperature, soil …

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Bernstein And Bernstein-Like Inequalities For Modulation Spaces, Sa'ud Al-Sa'di, Salti Samarah Jan 2022

Bernstein And Bernstein-Like Inequalities For Modulation Spaces, Sa'ud Al-Sa'di, Salti Samarah

International Journal of Emerging Multidisciplinaries: Mathematics

It is shown that the modulation spaces Mwp can be characterized by the approximation behavior of their elements using Local Fourier bases. In analogy to the Local Fourier bases, we show that the modulation spaces can also be characterized by the approximation behavior of their elements using Gabor frames. We derive direct and inverse approximation theorems that describe the best approximation by linear combinations of N terms of a given function using its modulates and translates.

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Fault Tolerant Control And Simulation Of Quadrotor Based On Adaptive Observer, Zhao Jing, Wang Peng, Xiaoqian Ding, Guoping Jiang, Fengyu Xu, Yanfei Sun Jan 2022

Fault Tolerant Control And Simulation Of Quadrotor Based On Adaptive Observer, Zhao Jing, Wang Peng, Xiaoqian Ding, Guoping Jiang, Fengyu Xu, Yanfei Sun

Journal of System Simulation

Abstract: Focusing on the actuator fault of quadrotor, an integral backstepping sliding mode combined with adaptive observer is proposed to ensure the safety and reliability of the quadrotor. A dynamic model of the quadrotor with actuator fault are established. An adaptive observer is proposed to observe the state and estimate the actual value of the fault. The attitude fault tolerant controller and position controllers are designed by the method of integral backstepping combined with the sliding mode control to complete the trajectory tracking of attitude and position. The simulation results show that the control strategy can quickly and accurately track …

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Novel Techniques For Solving Goursat Partial Differential Equations In The Linear And Nonlinear Regime, Tahir Naseem Jan 2022

Novel Techniques For Solving Goursat Partial Differential Equations In The Linear And Nonlinear Regime, Tahir Naseem

International Journal of Emerging Multidisciplinaries: Mathematics

The Goursat problem, which is related to hyperbolic partial differential equations, occurs in a variety of branches of physics and engineering. We studied the solution of the Goursat partial differential equation utilizing the reduced differential transform (RDT) and Adomian decomposition (AD) techniques in this inquiry. The problem's analytical solution is found in series form, which converges to exact solutions. The approaches' reliability and efficiency were evaluated using the Goursat problems (linear and non-linear). Additionally, the accuracy of the findings obtained demonstrates the reduced differential approach's superiority over the Adomian decomposition method and other numerical methods previously applied to the Goursat …

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Autoregressive Distributed Lag Transformation For Exchange Rate And Trade Balance, Muhammad Asif, Shamsul Haq Jan 2022

Autoregressive Distributed Lag Transformation For Exchange Rate And Trade Balance, Muhammad Asif, Shamsul Haq

International Journal of Emerging Multidisciplinaries: Mathematics

In this study the prime objective is to initiate autoregressive distributed lag transformation for exchange rate and trade balance to avoid the possible existence of multicollinearity among the explanatory variables and to analyze the variation in real exchange rate and its impact on trade balance in Pakistan. The Koyck model was used for studying the immediate impact and long run relationship between the variables by using time series annual data ranging from 1981 to 2021. The result showed that there is an inverse association between the trade balance and the exchange rate. It is evident from the results that the …

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A New Companion Inequality Of Ostrowski's Type With Applications To P.D.F.'S, Izharul Haq Jan 2022

A New Companion Inequality Of Ostrowski's Type With Applications To P.D.F.'S, Izharul Haq

International Journal of Emerging Multidisciplinaries: Mathematics

In this paper, variant inequalities of Ostrowski’s type for absolutely continuous mappings whose derivatives are monotonic, belongs to L1[a, b], Lq[a, b], (q > 1) and L∞[a, b] are established.

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Examining The Behavior Of A Solid Particle Interacting With Circular Obstacles In An Incompressible Flow, Kamran Usman, Muhammad Yaqoob, Kainat Komal Kayani, Muhammad Shahid Jan 2022

Examining The Behavior Of A Solid Particle Interacting With Circular Obstacles In An Incompressible Flow, Kamran Usman, Muhammad Yaqoob, Kainat Komal Kayani, Muhammad Shahid

International Journal of Emerging Multidisciplinaries: Mathematics

We have examined the effects on fluid and particle motion due to solid particles passing around circular obstacles in particulate flows. Particle interaction with internal obstacles, outer boundary and with the fluid is inspected. Eulerian approach using a fixed computational mesh is used across which the solid particles move freely in fluid. Treatment of fluid and particle interaction inside the whole domain is carried using Fictitious boundary method (FBM). A collision model is presented to handle particle-cylinder interactions. The particulate flow is computed using multigrid finite element solver FEATFLOW. Numerical experiments are performed considering different particle positions and different alignment …

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Modeling And Analysis Of Fractional Tb Model With Atangana-Baleanu Derivative, Aatif Ali, Saeed Islam, Quaid Iqbal, Huma Gul, Muhammad Nafees Jan 2022

Modeling And Analysis Of Fractional Tb Model With Atangana-Baleanu Derivative, Aatif Ali, Saeed Islam, Quaid Iqbal, Huma Gul, Muhammad Nafees

International Journal of Emerging Multidisciplinaries: Mathematics

In recent years Atangana and Baleanu proposed a new fractional derivative with non-singular and non-local kernel, this paper formulate a fragmentary request numerical TB model with AtanganaBaleanu derivative (AB derivative). We figured the basic reproduction number ( R0 ) and assessment of boundary dependent on genuine information of Khyber Pakhtunkhwa Pakistan, Initially we present the fundamental properties of the model, the existence and uniqueness of the model is proved using fixed point theory. At last, the model is tackled mathematically through Adams-Bashforth Moulton technique. The mathematical results for the extended model of the elements of Tuberculosis is shown graphically to …

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A Stimulus Generalization Of Double Integral Inequalities By The Way Of Fractional Integrals, Muhammad Muddassar, Muhammad Iqbal, Tahira Jabeen, Ghulam Haider Jan 2022

A Stimulus Generalization Of Double Integral Inequalities By The Way Of Fractional Integrals, Muhammad Muddassar, Muhammad Iqbal, Tahira Jabeen, Ghulam Haider

International Journal of Emerging Multidisciplinaries: Mathematics

This study presents some of the latest results annexed to Hermite Hadamard inequality by utilizing double integrals by the way of Riemann-Liouville fractional integrals. Another aim of this article is to generalize some of the recent developments on Hermite Hadamard's type inequalities.

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Analysis Of Non-Isothermal Mhd Thin Film Flow From Moving Belt With Slip And Variable Viscosity, Shamsul Haq, Aamir Shahzad, Muhammad Ilyas Jan 2022

Analysis Of Non-Isothermal Mhd Thin Film Flow From Moving Belt With Slip And Variable Viscosity, Shamsul Haq, Aamir Shahzad, Muhammad Ilyas

International Journal of Emerging Multidisciplinaries: Mathematics

This paper aims the study of electrically conducting Newtonian fluid flow and heat transfer considering the slip at the moving belt with temperature dependent viscosity. A domain decomposition method (ADM) is employed to solve the non-linear system of equations. Explicit expressions are obtained for velocity profile and temperature distribution. Effect of variable viscosity parameter, slip, Hartmann number, Brinkmann number and Stoke number are discussed and graphically shown.

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