Relation of Some Known Functions in terms of Generalized Meijer G -Functions

The aim of this paper is to prove some identities in the form of generalized Meijer G -function. We prove the relation of some known functions such as exponential functions, sine and cosine functions, product of exponential and trigonometric functions, product of exponential and hyperbolic functions, binomial expansion, logarithmic function, and sine integral, with the generalized Meijer G -function. We also prove the product of modified Bessel function of first and second kind in the form of generalized Meijer G -function and solve an integral involving the product of modified Bessel functions.

A new Multi Sine-Cosine algorithm for unconstrained optimization problems

The Sine-Cosine algorithm (SCA) is a population-based metaheuristic algorithm utilizing sine and cosine functions to perform search. To enable the search process, SCA incorporates several search parameters. But sometimes, these parameters make the search in SCA vulnerable to local minima/maxima. To overcome this problem, a new Multi Sine-Cosine algorithm (MSCA) is proposed in this paper. MSCA utilizes multiple swarm clusters to diversify & intensify the search in-order to avoid the local minima/maxima problem. Secondly, during update MSCA also checks for better search clusters that offer convergence to global minima effectively. To assess its performance, we tested the MSCA on unimodal, multimodal and composite benchmark functions taken from the literature. Experimental results reveal that the MSCA is statistically superior with regards to convergence as compared to recent state-of-the-art metaheuristic algorithms, including the original SCA.

Stable DNA Sequence Over Close-Ending and Pairing Sequences Constraint

DNA computing is a new method based on molecular biotechnology to solve complex problems. The design of DNA sequences is a multi-objective optimization problem in DNA computing, whose objective is to obtain optimized sequences that satisfy multiple constraints to improve the quality of the sequences. However, the previous optimized DNA sequences reacted with each other, which reduced the number of DNA sequences that could be used for molecular hybridization in the solution and thus reduced the accuracy of DNA computing. In addition, a DNA sequence and its complement follow the principle of complementary pairing, and the sequence of base GC at both ends is more stable. To optimize the above problems, the constraints of Pairing Sequences Constraint (PSC) and Close-ending along with the Improved Chaos Whale (ICW) optimization algorithm were proposed to construct a DNA sequence set that satisfies the combination of constraints. The ICW optimization algorithm is added to a new predator–prey strategy and sine and cosine functions under the action of chaos. Compared with other algorithms, among the 23 benchmark functions, the new algorithm obtained the minimum value for one-third of the functions and two-thirds of the current minimum value. The DNA sequences satisfying the constraint combination obtained the minimum of fitness values and had stable and usable structures.

SOME FORMULAS OF FRACTIONAL TRIGONOMETRIC FUNCTIONS

This paper studies some properties of fractional trigonometric sine and cosine functions and we obtain the fractional Dirichlet kernel. The Mittag-Leffler function plays an important role in this article, and the results we obtained are the generalizations of formulas of the classical sine and cosine functions.

Finding the fractional value of 2pi

Rational order integral and derivative of a myriad of functions—ln(x); e^(ax) and t^(ax)—are known. Nevertheless, the investigation focuses on rational order integrals and derivatives of sine and cosine as these functions follow a cyclic order and determiningwhether properties of sine and cosine extend to their fractional integrals and derivativescould expand current applications of sine and cosine, which are extensively in engineeringand economics. For example, Mehdi and Majid outline in, "Applications of FractionalCalculus" how fractional integrals and derivatives come handy while modeling ultrasonicwave propagation in human cancellous bones and bettering edge detection technology. Thus, this explorative study investigates the properties of fractional derivatives and fractional integrals of standard sine and cosine functions. THe study also looks at how to generalize the definition of the 2pi using fractional calculus.

Controllability of stochastic nonlinear oscillating delay systems driven by the Rosenblatt distribution

In this paper, we study the controllability of second-order nonlinear stochastic delay systems driven by the Rosenblatt distributions in finite dimensional spaces. A set of sufficient conditions are established for controllability of nonlinear stochastic delay systems using fixed point theory, delayed sine and cosine matrices and delayed Grammian matrices. Furthermore, controllability results for second-order stochastic delay systems driven by Rosenblatt distributions via the representation of solution by delayed sine and cosine functions are presented. Finally, our theoretical results are illustrated through numerical simulation.

ESSENTIAL TRIGONOMTERY WITHOUT GEOMETRY PART II: GENERALIZATIONS AND APPLICATIONS

In a previous paper (Gresham et al. 2019), the properties, theorems, and identities of the sine and cosine functions were developed using only analytical methods and without geometric constructions. We follow those results and use them to develop generalizations of the key theorems of trigonometry, again using purely analytical methods. We conclude with a connection to the Law of Conservation of Energy in physics.

Elementos do processo evolutivo do conceito das funções seno e cosseno: contribuições para uma razão de ser na construção de um PEPElements of the evolutionary process of the concept of sine and cosine functions: contributions to a reason to be in the construction of a PEP

As funções seno e cosseno são consideradas como um dos conteúdos de difícil entendimento. Desse modo, fomos em busca na história, de elementos que nos permitissem compreender o processo evolutivo das funções seno e cosseno, e detectar possíveis incompletudes que possa ter influenciado o ensino e aprendizagem atual desses conceitos. Realizamos uma análise bibliográfica com o objetivo de construir nossa revisão histórica, e compreender fatores determinantes na evolução do conceito das funções seno e cosseno. E baseado nesses estudos, notamos - durante todo o processo evolutivo do campo da trigonometria - a existência de técnicas que venha evoluir com o passar dos tempos, mostrando uma ligação forte com a astronomia, bem como a existência da razão de ser social, em especial, no período da pré-história e Idade Antiga, pois a partir da Idade Média até a Idade Moderna começaram a prevalecer com maior ênfase as razões matemáticas. Esse fato pode ter influenciado na perda da razão ser social no ensino das funções seno e cosseno. Destarte, a partir dos resultados da análise, ressaltamos a importância em desenvolver um PEP que pode se apoiar em Questões Sócio-científicas-QSC, com intuito de resgatar o ensino das funções seno e cosseno com a razão de ser social integrando não apenas o conhecimento científico, mas conhecimentos éticos e sociais.