scholarly journals Sharpening and Generalizations of Shafer's Inequality for the Arc Tangent Function

2009 ◽  
Vol 2009 (1) ◽  
pp. 930294 ◽  
Author(s):  
Feng Qi ◽  
Shi-Qin Zhang ◽  
Bai-Ni Guo
Keyword(s):  
Tangent Function ◽  
Shafer’S Inequality

scholarly journals The monotonicity of ratios involving arc tangent function with applications

2019 ◽  
Vol 17 (1) ◽  
pp. 1450-1467
Author(s):  
Zhen-Hang Yang ◽  
King-Fung Tin ◽  
Qin Gao
Keyword(s):  
Accurate Estimate ◽  
Tangent Function ◽  
Relative Errors

Abstract In this paper, we investigate the monotonicity of the functions $$\begin{array}{} \begin{split}{} \displaystyle x &\mapsto &\frac{1}{x}\left( 1-a+\sqrt{\frac{2}{3}ax^{2}+a^{2}}\right) \arctan x, \\ x &\mapsto &\frac{1}{x}\left( \frac{4}{\pi ^{2}}+\sqrt{\frac{4}{\pi ^{2}}% x^{2}+a}\right) \arctan x \end{split} \end{array}$$ on (0, ∞) for a > 0, which not only gives relative errors of known bounds with quadratic for arctan x, but also yields some new accurate bounds. Moreover, the known bounds are extended and a more accurate estimate for arctan x is presented.

An improved nonlinear innovation-based parameter identification algorithm for ship models

2021 ◽  
pp. 1-9
Author(s):  
Baigang Zhao ◽  
Xianku Zhang
Keyword(s):  
Parameter Identification ◽  
Test Data ◽  
Stochastic Gradient ◽  
Least Square ◽  
Gradient Algorithm ◽  
Model Parameters ◽  
Identification Algorithm ◽  
Process Innovations ◽  
Stochastic Gradient Algorithm ◽  
Tangent Function

Abstract To solve the problem of identifying ship model parameters quickly and accurately with the least test data, this paper proposes a nonlinear innovation parameter identification algorithm for ship models. This is based on a nonlinear arc tangent function that can process innovations on the basis of an original stochastic gradient algorithm. A simulation was carried out on the ship Yu Peng using 26 sets of test data to compare the parameter identification capability of a least square algorithm, the original stochastic gradient algorithm and the improved stochastic gradient algorithm. The results indicate that the improved algorithm enhances the accuracy of the parameter identification by about 12% when compared with the least squares algorithm. The effectiveness of the algorithm was further verified by a simulation of the ship Yu Kun. The results confirm the algorithm's capacity to rapidly produce highly accurate parameter identification on the basis of relatively small datasets. The approach can be extended to other parameter identification systems where only a small amount of test data is available.

scholarly journals Teaching Materials Development and Meeting the Needs of the Subject: A Sample Application

2018 ◽  
Vol 11 (8) ◽  
pp. 27 ◽  
Author(s):  
Abdullah Aydin ◽  
Cahit Aytekin
Keyword(s):  
Teaching Effectiveness ◽  
Real Life ◽  
Teaching Material ◽  
Teaching Materials ◽  
Mathematics Textbook ◽  
Waste Products ◽  
Material Development ◽  
Bridge Model ◽  
Tangent Function ◽  
The Subject

It has been determined that the drawings, photographs and pictures related to the subject of the continuity of the tangent function on page 68 of the Ministry of National Education’s twelfth-grade mathematics textbook contradict principles 1, 7 and 10 of Yanpar’s (2007) teaching material development principles. According to these principles, teaching materials should: i) be simple, plain, and understandable, ii) reflect real life as much as possible, and iii) be easy to develop or revise, if necessary. This study aims to develop a portable tangent bridge model to meet the needs of the subject of the continuity of the tangent function. With this aim: i) teaching with the analogies model in the design of the teaching material, ii) “this is my project” format in the development and iii) Yanpar’s (2007) principles were considered. The design of the model lasted 14 weeks. At the end of the study, a portable tangent bridge model from waste products was designed and developed. This model is thought to contribute to the teaching effectiveness of teachers (Shulman, 1987) with content knowledge alongside with pedagogical knowledge (Shulman, 1986). With this contribution, the needs of the subject as described by Taba (1962) and Tyler (1949) will be met. This model will also serve as an example of meeting the needs of the subjects of knowledge and its product, technology, as highlighted by Cahit Arf (Terzioglu &Yilmaz, 2006).

scholarly journals Continued Fractions Related to $(t,q)$-Tangents and Variants

10.37236/2014 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
General Version ◽  
Continued Fraction Expansion ◽  
Special Cases ◽  
Tangent Function

For the $q$-tangent function introduced by Foata and Han (this volume) we provide the continued fraction expansion, by creative guessing and a routine verification. Then an even more recent $q$-tangent function due to Cieslinski is also expanded. Lastly, a general version is considered that contains both versions as special cases.

scholarly journals Real-Time Prediction of the COVID-19 Epidemic in Thailand using Simple Model-Free Method and Time Series Regression Model

2021 ◽  
Vol 18 (14) ◽  
Author(s):  
Rati WONGSATHAN
Keyword(s):  
Genetic Algorithm ◽  
Time Series ◽  
Regression Model ◽  
Regression Models ◽  
Gaussian Function ◽  
Logistic Function ◽  
Hyperbolic Tangent ◽  
Time Series Regression ◽  
Model Free ◽  
Tangent Function

The novel coronavirus 2019 (COVID-19) pandemic was declared a global health crisis. The real-time accurate and predictive model of the number of infected cases could help inform the government of providing medical assistance and public health decision-making. This work is to model the ongoing COVID-19 spread in Thailand during the 1st and 2nd phases of the pandemic using the simple but powerful method based on the model-free and time series regression models. By employing the curve fitting, the model-free method using the logistic function, hyperbolic tangent function, and Gaussian function was applied to predict the number of newly infected patients and accumulate the total number of cases, including peak and viral cessation (ending) date. Alternatively, with a significant time-lag of historical data input, the regression model predicts those parameters from 1-day-ahead to 1-month-ahead. To obtain optimal prediction models, the parameters of the model-free method are fine-tuned through the genetic algorithm, whereas the generalized least squares update the parameters of the regression model. Assuming the future trend continues to follow the past pattern, the expected total number of patients is approximately 2,689 - 3,000 cases. The estimated viral cessation dates are May 2, 2020 (using Gaussian function), May 4, 2020 (using a hyperbolic function), and June 5, 2020 (using a logistic function), whereas the peak time occurred on April 5, 2020. Moreover, the model-free method performs well for long-term prediction, whereas the regression model is suitable for short-term prediction. Furthermore, the performances of the regression models yield a highly accurate forecast with lower RMSE and higher R2 up to 1-week-ahead. HIGHLIGHTS COVID-19 model for Thailand during the first and second phases of the epidemic The model-free method using the logistic function, hyperbolic tangent function, and Gaussian function  applied to predict the basic measures of the outbreak Regression model predicts those measures from one-day-ahead to one-month-ahead The parameters of the model-free method are fine-tuned through the genetic algorithm  GRAPHICAL ABSTRACT

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