Polynomial-Exponential Bounds for Some Trigonometric and Hyperbolic Functions

Recent advances in mathematical inequalities suggest that bounds of polynomial-exponential-type are appropriate for evaluating key trigonometric functions. In this paper, we innovate in this sense by establishing new and sharp bounds of the form (1−αx2)eβx2 for the trigonometric sinc and cosine functions. Our main result for the sinc function is a double inequality holding on the interval (0, π), while our main result for the cosine function is a double inequality holding on the interval (0, π/2). Comparable sharp results for hyperbolic functions are also obtained. The proofs are based on series expansions, inequalities on the Bernoulli numbers, and the monotone form of the l’Hospital rule. Some comparable bounds of the literature are improved. Examples of application via integral techniques are given.

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.

Multiplicative perturbations of local C-cosine functions

"We establish some left and right multiplicative perturbations of a local C-cosine function C(.) on a complex Banach space X with non-densely defined generator, which can be applied to obtain some new additive perturbation results concerning C(.)."

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.

On solutions of PDEs by using algebras

The components of complex differentiable functions define solutions for the Laplace’s equation. In this paper we generalize this result; for each PDE of the form $Au_{xx}+Bu_{xy}+Cu_{yy}=0$ we give an affine planar vector field $\varphi$ and an associative and commutative 2D algebra with unit $\mathbb A$, with respect to which the components of all functions of the form $\mathcal L\circ\varphi$ define solutions for this PDE, where $\mathcal L$ is differentiable in the sense of Lorch with respect to $\mathbb A$. In the same way, for each PDE of the form $Au_{xx}+Bu_{xy}+Cu_{yy}+Du_x+Eu_y+Fu=0$, the components of the exponential function $e^{\varphi}$ defined with respect to $\mathbb A$, define solutions for this PDE. In the case of PDEs of the form $Au_{xx}+Bu_{xy}+Cu_{yy}+Fu=0$, sine, cosine, hyperbolic sine, and hyperbolic cosine functions can be used instead of the exponential function. Also, solutions for two dependent variables $3^{\text{th}}$ order PDEs and a $4^{\text{th}}$ order PDE are constructed.

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.

Using an Optimization Algorithm to Detect Hidden Waveforms of Signals

Source signals often contain various hidden waveforms, which further provide precious information. Therefore, detecting and capturing these waveforms is very important. For signal decomposition (SD), discrete Fourier transform (DFT) and empirical mode decomposition (EMD) are two main tools. They both can easily decompose any source signal into different components. DFT is based on Cosine functions; EMD is based on a collection of intrinsic mode functions (IMFs). With the help of Cosine functions and IMFs respectively, DFT and EMD can extract additional information from sensed signals. However, due to a considerably finite frequency resolution, EMD easily causes frequency mixing. Although DFT has a larger frequency resolution than EMD, its resolution is also finite. To effectively detect and capture hidden waveforms, we use an optimization algorithm, differential evolution (DE), to decompose. The technique is called SD by DE (SDDE). In contrast, SDDE has an infinite frequency resolution, and hence it has the opportunity to exactly decompose. Our proposed SDDE approach is the first tool of directly applying an optimization algorithm to signal decomposition in which the main components of source signals can be determined. For source signals from four combinations of three periodic waves, our experimental results in the absence of noise show that the proposed SDDE approach can exactly or almost exactly determine their corresponding separate components. Even in the presence of white noise, our proposed SDDE approach is still able to determine the main components. However, DFT usually generates spurious main components; EMD cannot decompose well and is easily affected by white noise. According to the superior experimental performance, our proposed SDDE approach can be widely used in the future to explore various signals for more valuable information.