Cotangent Function Research Articles

Euler sine product and the continued fraction of π

The Euler sine product and the continued fraction of $\pi$ are discussed in this article. Some of the infinite series for cotangent and its derivative are obtained by implementing the concept of Euler sine product and some of the standard series are derived as the immediate consequence of the main results. Furthermore, the continued fraction for odd powers of $\pi$ similar to the expression of $\pi$ derived by Brouncker is presented in this article. Meanwhile, an expression relating the Basel's constant and the cotangent function is obtained as follows: \begin{equation*} \frac{\coth{r}}{2}-\frac{1}{2r}=\sum_{n\in\mathbb{N}}\frac{2^{2n}}{2(2n)!}B_{2n}r^{2n-1}. \end{equation*}

Deriving the new cotangent Fréchet distribution with real data analysis

This study introduces the family of the novel ”cot-generator” distributions based on the cotangent function. The construction of the probability density function based on the cot-generator and its corresponding moments, reliability indicators, and order statistics are derived. Regarding inferential procedures, the maximum likelihood estimation method is implemented to estimate the model parameters in a non-closed form. The novel Cot-Fréchet distribution (NCFD) is then focused on a specific class member, constructed using the Fréchet distribution as the baseline. The estimators’ bias, variance, and mean square error (MSE) efficiency are examined and evaluated through Monte Carlo simulation experiments. The novel model is also applied to some real data sets to assess its efficacy and is compared with other competing distributions.

Construction of optical solitary wave solutions and their propagation for Kuralay system using tanh-coth and energy balance method

In this article, we study the complex coupled Kuralay system (CKS), which have many applications in different field such as ferromagnetic material and fiber optics. We implement an analytical method, namely, the tanh–coth method to considered system to formulate a system of algebraic equations. Solving the obtained system, we acquire different traveling wave solutions that are expressed in term of hyperbolic tangent functions and hyperbolic cotangent functions. Also, we employ the energy balance technique to the selected model to sort out the behavior of traveling wave solutions. By this method, we obtain periodic wave solution that are expressed in term of cosine function. In order to show the graphical presentation of the acquired solitary waves solutions, 3D and 2D curve are created by selecting well-suited parametric values via Mathematica.

Extrapolation and Interpolation of Information in Computer Science

Proposed method is dealing with multi-dimensional data modeling, extrapolation and interpolation using the set of high dimensional feature vectors. Identification of handwriting, signature, faces or fingerprints need data modeling and each model of the pattern is built by a choice of characteristic key points and multi-dimensional modeling functions. Novel modeling via nodes combination and parameter γ as N-dimensional function enables data parameterization and interpolation for feature vectors. Multi-dimensional data is modelled and interpolated via different functions for each feature: polynomial, sine, cosine, tangent, cotangent, logarithm, exponent, arc sin, arc cos, arc tan, arc cot or power function.

MCDM technique using single-valued neutrosophic trigonometric weighted aggregation operators

Motivated based on the trigonometric t-norm and t-conorm, the aims of this article are to present the trigonometric t-norm and t-conorm operational laws of SvNNs and then to propose the SvNN trigonometric weighted average and geometric aggregation operators for the modelling of a multiple criteria decision making (MCDM) technique in an inconsistent and indeterminate circumstance. To realize the aims, this paper first proposes the trigonometric t-norm and t-conorm operational laws of SvNNs, which contain the hybrid operations of the tangent and arctangent functions and the cotangent and inverse cotangent functions, and presents the SvNN trigonometric weighted average and geometric operators and their properties. Next, a MCDM technique is proposed in view of the presented two aggregation operators in the circumstance of SvNNs. In the end, an actual case of the choice issue of slope treatment schemes is provided to indicate the practicability and effectivity of the proposed MCDM technique.

Similarity measures of double-valued neutrosophic sets based on cotangent function and applications to football players’ competitive ability evaluation

At present, China’s football sports are relatively backward in training theory and practice. If you want to break out of Asia and enter the world, and gain a firm foothold in the international football arena, you must use a scientific and realistic attitude, absorb the successful experience of advanced football countries, reflect on our training concepts and practices, and deeply study the training laws of football, in order to find a way suitable for our development. Athletes’ competitive ability is the core issue of sports training. The failure of our football level is directly related to our systematic understanding of athletes’ competitive ability. This problem has led to the separation of our training practice from the actual competition, making training unable to meet the needs of the competition. Only by solving this problem, can we improve the level of football in China. The football players’ competitive ability evaluation is affirmed as multiple attribute decision making (MADM). In such paper, motivated by the idea of cotangent similarity measure (CSM), the CSMs are extended to DVNSs and four CSMs are created under DVNSs. Then, two weighted CSMs are built for MADM under DVNSs. Finally, a numerical example for Football players’ competitive ability evaluation is affirmed and some comparative algorithms are produced to affirm the built method.

SOME TRIGONOMETRIC SIMILARITY MEASURES OF COMPLEX FUZZY SETS WITH APPLICATION

Similarity measures of fuzzy sets are applied to compare the closeness among fuzzy sets. These measures have numerous applications in pattern recognition, image processing, texture synthesis, medical diagnosis, etc. However, in many cases of pattern recognition, digital image processing, signal processing, and so forth, the similarity measures of the fuzzy sets are not appropriate due to the presence of dual information of an object, such as amplitude term and phase term. In these cases, similarity measures of complex fuzzy sets are the most suitable for measuring proximity between objects with two-dimensional information. In the present paper, we propose some trigonometric similarity measures of the complex fuzzy sets involving similarity measures based on the sine, tangent, cosine, and cotangent functions. Furthermore, in many situations in real life, the weight of an attribute plays an important role in making the right decisions using similarity measures. So in this paper, we also consider the weighted trigonometric similarity measures of the complex fuzzy sets, namely, the weighted similarity measures based on the sine, tangent, cosine, and cotangent functions. Some properties of the similarity measures and the weighted similarity measures are discussed. We also apply our proposed methods to the pattern recognition problem and compare them with existing methods to show the validity and effectiveness of our proposed methods.

Adaptive finite-time fault-tolerant control for the full-state-constrained robotic manipulator with novel given performance

Most robot manipulators usually operate under different nonlinearities, such as state constraints, time delays, and actuator input saturation faults. This paper proposes an adaptive finite-time fault-tolerant controller for the full-state-constrained robotic manipulator with novel prescribed performance and time-varying delays. Firstly, a novel prescribed performance function consisting of hyperbolic tangent and hyperbolic cotangent functions is constructed to the full-state performance constraints of the robotic manipulator. Such a design can circumvent the demands for accurate initial conditions of the state variables and guarantee the convergence of the state errors in a prescribed time. Then, based on the finite-time principle, a finite-time command filter and a compensation mechanism with fractional-power terms are used to bypass the “explosion of complexity” and further compensate for the filter errors within a finite time. Moreover, a proper Lyapunov–Krasovskii functional is devised within the controller design to compensate for the time-varying delays. And the radial basis function neural networks are used to estimate the other nonlinearities including actuator saturation faults, and external perturbations. Finally, simulation results exhibit that the controller designed in this paper has a faster convergence speed and minor state errors.

Cosine Similarity Measures of (m, n)-Rung Orthopair Fuzzy Sets and Their Applications in Plant Leaf Disease Classification

A fuzzy set is a powerful tool to handle uncertainty and ambiguity, and generally, the notions of symmetry and similarity are also exhibited in the fuzzy set theory. The class of (m, n)-rung orthopair fuzzy sets through two universes are more flexible and efficient than the q-rung orthopair fuzzy sets when discussing the symmetry and similarity between multiple objects. This research article comprehensively investigates ten similarity measures that employ cosine and cotangent functions for comparing (m, n)-rung orthopair fuzzy sets, which are a superclass of q-rung orthopair fuzzy sets. Moreover, the proposed weighted similarity measures are applied to real-world problems in building material analysis. A comparative analysis is conducted between the proposed measures and the existing cosine and cotangent measures of q-rung orthopair fuzzy sets, showing that the proposed measures are more efficient than existing ones. Additionally, a numerical example demonstrates the practical and scientific applications of these similarity measures in classifying plant leaf diseases. The sensitivity analysis shows that the existing measures cannot be applied to (m, n)-fuzzy data for distinct values of m and n. The results are supported by graphical interpretations, further illustrating the efficacy of the proposed measures.

Determinantal Expressions, Identities, Concavity, Maclaurin Power Series Expansions for van der Pol Numbers, Bernoulli Numbers, and Cotangent

In this paper, basing on the generating function for the van der Pol numbers, utilizing the Maclaurin power series expansion and two power series expressions of a function involving the cotangent function, and by virtue of the Wronski formula and a derivative formula for the ratio of two differentiable functions, the authors derive four determinantal expressions for the van der Pol numbers, discover two identities for the Bernoulli numbers and the van der Pol numbers, prove the increasing property and concavity of a function involving the cotangent function, and establish two alternative Maclaurin power series expansions of a function involving the cotangent function. The coefficients of the Maclaurin power series expansions are expressed in terms of specific Hessenberg determinants whose elements contain the Bernoulli numbers and binomial coefficients.

A Cotangent Fractional Derivative with the Application

In this work, we present a new type of fractional derivatives (FD) involving exponential cotangent function in their kernels called Riemann–Liouville Dσ,γ and Caputo cotangent fractional derivatives CDσ,γ, respectively, and their corresponding integral Iσ,γ. The advantage of the new fractional derivatives is that they achieve a semi-group property, and we have special cases; if γ=1 we obtain the Riemann–Liouville FD (RL-FD), Caputo FD (C-FD), and Riemann–Liouville fractional integral (RL-FI). We give some theorems and lemmas, and we give solutions to linear cotangent fractional differential equations using the Laplace transform of the Dσ,γ, CDσ,γ and Iσ,γ. Finally, we give the application of this new type on the SIR model. This new type of fractional calculus can help other researchers who still work on the actual subject.

A half-discrete Hilbert-type inequality in the whole plane with the constant factor related to a cotangent function

In this work, by the introduction of some parameters, a new half-discrete kernel function in the whole plane is defined, which involves both the homogeneous and the nonhomogeneous cases. By employing some techniques of real analysis, especially the method of a weight function, a new half-discrete Hilbert-type inequality with the new kernel function, as well as its equivalent Hardy-type inequalities are established. Moreover, it is proved that the constant factors of the newly obtained inequalities are the best possible. Finally, assigning special values to the parameters, some new half-discrete Hilbert-type inequalities with special kernels are presented at the end of the paper.

INEQUALITIES INVOLVING GENERALIZED TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS

Trigonometric and Hyperbolic inequalities, which have been obtained by C. Huygens[5], D. S. Mitrinovic[10] and many more, have attracted attention of several researchers. We oer several renements and generalization of few trigonometric and hyperbolic inequalities involving tangent function, cotangent function, sine function, secant function and cosecant function. The established results are obtained with the aid of the Schwab-Borchardt mean.

The Exponentiated Cotangent Generalized Distributions: Characteristics and Applications Patients of Chemotherapy Treatments Data

Over the course of the last several decades, many algebraic generalised families and classes of statistical distributions have been developed. The purpose of this research is to construct a brand new cotangent exponentiated generalized and generator of distributions that have support on the real line. After that, two novel families of distributions incorporating the cotangent function are proposed: one called the cotangent exponentiated generalised (CE-G) family, and the other called the logistic cotangent exponentiated generalised (LCE-G) family. A comprehensive analysis of the mathematical and structural properties of the recently suggested G-family as well as a Burr-based novel model (LCEB) is presented here. The maximum likelihood approach is used in Monte-Carlo simulation studies to estimate model parameters and evaluate performance. This is done using the maximum likelihood method. The statistical analysis on the survival and waiting times data sets is carried out, and the outcomes confirm the competence, superiority, and utility of the suggested generator, G-family, and novel distribution in comparison to similar and competing Burr-based models that are already well-known.

Two identities and closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kind

Abstract In this article, the authors present two identities involving products of the Bernoulli numbers, provide four alternative proofs for these two identities, derive two closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kind, and supply simple proofs of series expansions of (hyperbolic) cosecant and cotangent functions.

The Method of Probabilistic Nodes Combination in Biomechanics

Proposed method, called Probabilistic Nodes Combination (PNC), is the method of 2D curve modeling and handwriting identification by using the set of key points. Nodes are treated as characteristic points of signature or handwriting for modeling and writer recognition. Identification of handwritten letters or symbols need modeling and the model of each individual symbol or character is built by a choice of probability distribution function and nodes combination. PNC modeling via nodes combination and parameter γ as probability distribution function enables curve parameterization and interpolation for each specific letter or symbol. Two-dimensional curve is modeled and interpolated via nodes combination and different functions as continuous probability distribution functions: polynomial, sine, cosine, tangent, cotangent, logarithm, exponent, arc sin, arc cos, arc tan, arc cot or power function.

(Invited) High-Performing Fibre Batteries at a Scalable Production

Fibre batteries are pioneering as flexible power solutions because they can be woven into textiles. Textile batteries offer a convenient and seamless way to power future wearable electronics. However, current fibre batteries are only centimetres long because longer fibre batteries are difficult to produce and they are thought to have significantly higher internal resistances that compromise electrochemical performance. Here, we show our discovery that internal resistance of fibre battery has a hyperbolic cotangent function relationship with fibre length, where it firstly decreases before leveling off as length increases. Systematic studies confirm this is true for different fibre batteries. We thus produce metres of high-performing fibre lithium-ion batteries via an optimised industrial process. Our mass-produced fibre batteries had high certified energy density based on the total weight of a lithium-cobalt-oxide/graphite full battery including packaging. Over 80% capacity can be maintained after bending for 100,000 cycles, indicating high flexibility. We show fibre batteries woven into safe and washable textile by industrial rapier loom can wirelessly charge a cellphone or power a health management jacket integrated with fibre sensors and textile display.

The Method of Probabilistic Nodes Combination in Handwritten Character Reconstruction

Proposed method, called Probabilistic Nodes Combination (PNC), is the method of 2D curve modeling and handwriting identification by using the set of key points. Nodes are treated as characteristic points of signature or handwriting for modeling and writer recognition. Identification of handwritten letters or symbols need modeling and the model of each individual symbol or character is built by a choice of probability distribution function and nodes combination. PNC modeling via nodes combination and parameter γ as probability distribution function enables curve parameterization and interpolation for each specific letter or symbol. Two-dimensional curve is modeled and interpolated via nodes combination and different functions as continuous probability distribution functions: polynomial, sine, cosine, tangent, cotangent, logarithm, exponent, arc sin, arc cos, arc tan, arc cot or power function.

Calculation of the value of the functions of the complex variable with by an interval argument, we will design in the hyperbolic form

Information about the interval numbers presented in the classical form, the CENTER-RADIUS system and in the hyperbolic form is given. Rules for the transition from one of the forms of representation of interval numbers to others are proposed. Information is given on complex interval numbers, the real and imaginary parts of which are presented in hyperbolic form. The rules for performing basic arithmetic operations with these numbers and the calculation of interval values of power, exponential, logarithmic functions, direct and inverse trigonometric functions, direct and inverse hyperbolic functions are described. For functions of a complex variable, information about their real and imaginary parts is given. The list of functions corresponds to the functions of a complex variable included in the EXCEL system. Relationships are obtained for determining the real and imaginary parts of the secant, cosecant, tangent and cotangent functions for circular trigonometric and hyperbolic functions, which were absent in the most common reference literature. It is shown that the operations of multiplication, division and raising to an integer power are most appropriate to perform with complex interval numbers, which are defined in hyperbolic form. The operation of calculating the root of degree n from an interval complex number presented in hyperbolic form is most expediently performed using the CENTER-RADIUS system in combination with the hyperbolic form of representing the interval number. Relationships are obtained that make it possible to obtain a function of an interval complex variable equivalent to the original one and suitable for further work with complex functions and numbers presented in hyperbolic form and in the CENTER-RADIUS system. Examples illustrating the application of the proposed technique are given.

Correlation analysis and prediction of PEM fuel cell voltage during start-stop operation based on real-world driving data

The accurate prediction of fuel cell voltage is critical for fuel cell health management, which directly affects the performance and cost of fuel cell vehicles. This paper proposes analysis and prediction approaches for the start-stop voltage of fuel cell based on data from a proton exchange membrane fuel cell (PEMFC) bus operated under real traffic conditions in Chengdu, China. First, three correlative analysis methods are used to identify the correlation between the PEMFC voltage and other variables during start-stop operation. Then, the original sequence of PEMFC voltage at start-stop is processed using the grey model, residual grey model, logarithmic-power function grey model, and cotangent function grey model. Four grey prediction models are combined and optimized based on a backpropagation neural network (BPNN). Finally, an organic grey BPNN model (OGNNM) is established to predict the PEMFC voltage during start-stop operation and analyze its accuracy. Results indicate that the PEMFC voltage changes sharply during start-stop, which is closely connected to the current, hydrogen gas tank pressure, and PEMFC stack temperature. In addition, the proposed OGNNM has higher prediction precision than the other data-driven models that provides a basis for the control of fuel cell hybrid systems.