Coefficient estimates for certain a new subclass of univalent functions associated with quotient of cosine hyperbolic and exponential functions

It is well known that the first stage of mine-to-mill optimization is rock fragmentation by blasting. The degree of rock fragmentation can be expressed in terms of average grain (X50) size and size distribution. There are approaches in which exponential functions are used to estimate the size distribution of the pile that will be formed before blasting. The most common of these exponential functions used to estimate the average grain size is the Kuz–Ram and KCO functions. The exponential functions provide a curve from 0% to 100% using the mean grain size (X50), characteristic size (XC), and uniformity index (n) parameters. This distribution curve can make predictions in the range of fine grains and coarse grains outside the acceptable error limits in some cases. In this article, the usability of the hyperbolic tangent function, which is symmetrical at origin, in the estimation of the size distribution as an alternative to the exponential distribution functions used in almost all estimation models is investigated. As with exponential functions, the hyperbolic tangent function can express the aggregated size distribution as a percentage with reference to the variables X50 and XC. It has been shown that the hyperbolic tangent function provides 99% accuracy to the distribution of fine grains and coarse grains of the pile formed as a result of blasting data for the characteristic size (XC) parameter and the uniformity index (n).

녹색 미세조류인 Chlorella sp.를 사용하여 광도에 따른 미세조류의 비성정속도와 광합성을 측정하였고, 여러 가지 kinetic model을 이용해 실측된 자료에 가장 적합한 kinetic model을 모색하였다. Chlorella sp.는 광독립영양조건 (photoautotroph condition)과 27, 54, 81, 140, 281 <TEX>${\mu}mol/m^2{\cdot}s$</TEX>의 광도 하에서 배양되었으며 kinetic model을 사용하여 P-I 반응선도와 <TEX>${\mu}-I$</TEX> 반응선도를 구하였다. Chlorella sp.의 비성장속도와 광합성은 81 <TEX>${\mu}mol/m^2{\cdot}s$</TEX>에서 0.35/day, 0.94 <TEX>$mgO_2/g{\cdot}h$</TEX>로 가장 높았다. 실측된 자료를 기질속도론 (Tamiya), 광포화모형(hyperbolic tangent, 포아송 함수)과 두가지 광저해모형(Steel 지수함수, Andrew 기질저해함수)을 이용하여 그게 따른 계수를 추정하고 비교 분석하였다. <TEX>${\mu}-I$</TEX> 반응선도에서는 포화모형의 일종인 hyperbolic tangent 모형이 적합하였으며, P-I 반응선도에서는 광저해가 현저히 관찰되어 Steele 지수함수의 적합도가 우수하였다. We investigated which is the best kinetic model for expression of microalgal growth and photosynthesis by irradiance in green microalgae, marine Chlorella sp. is cultivated under 5 light intensities (27, 54, 81, 140, and 281 <TEX>${\mu}mol/m^2{\cdot}s$</TEX>). The maximum of specific growth rate and photosynthesis at light intensity of 81 <TEX>${\mu}mol/m^2{\cdot}s$</TEX> were observed as 0.35/day and 0.94 <TEX>$mg0_2/g{\cdot}h$</TEX>, respectively. To describe the effect of irradiance, we adapted both types for kinetic model fitting, saturation and inhibition models, Tamiya's modified Michaelis- Menten equation, hyperbolic tangent and Poisson function as saturation model, Steele's exponential and Andrew's function as the other and made the specific growth rate- irradiance response curve and photosynthesis- irradiance response curve. In case of microalgal growth, one of the saturation models, the hyperbolic tangent function was the most suitable, otherwise, the Steele's exponential function, one of the inhibition models, was suitable for photosynthesis.

ABSTRACTBackground: Smokers exhibit an unusually high willingness to forgo a delayed reward of greater magnitude to receive a smaller, more immediate reward. The functional form of such “delay discounting” behavior is central to the discounting-based operationalization of impulsivity, and has implications for theories regarding the basis of steep discounting among smokers and treatment approaches to addiction. Objectives: We examined the discounting behavior of current smokers, ex-smokers, and never-smokers to determine whether the functional form of discounting differs between these groups. Methods: Participants completed a 27-item delay discounting questionnaire (25). We used finite mixture modeling in analyzing data as the product of two simultaneous data-generating processes (DGPs), a hyperbolic function and an exponential function, and compared results to a quasi-hyperbolic (QH) approximation, in a relatively large sample (n = 1205). Results: Consistent with prior reports, current smokers exhibited steeper discounting relative to never-smokers across exponential, hyperbolic, and QH models. A mixture model provided significant support for exponential and hyperbolic discounting in the data, and both accounted for roughly 50% of the participants’ choices. This mixture model showed a statistically significantly better fit to the data than the exponential, hyperbolic, or QH functions alone. Contrary to the prevailing view, current smokers were not more likely to discount hyperbolically than nonsmokers, and, thus, were not more prone to time-inconsistent discounting. Conclusions: The results inform the interpretation of steep discounting among smokers and suggest that treatment approaches could be tailored to the type of discounting behavior that smokers exhibit.

In the traditional text-book presentation and evaluation of logarithmic, exponential, circular and hyperbolic functions there is a lack of uniformity of method, and the demands on the analytical powers of the average student are considerable, if not excessive It has been shown in The Mathematical Gazette of June, 1926, that logarithmic and exponential functions may readily be evaluated by ab initio arithmetical methods. Incidentally it may be noted that the theory of the method is independent of the theory of indices. The definition of loge a is virtually and leads to a simple assignment of meaning to irrational and imaginary powers of a number.

The complex Ginzburg-Landau model appears in the mathematical description of wave propagation in nonlinear optics. In this paper, the fractional complex Ginzburg-Landau model is investigated using the generalized exponential rational function method. The Kerr law and parabolic law are considered to discuss the nonlinearity of the proposed model. The fractional effects are also included using a novel local fractional derivative of order α . Many novel solutions containing trigonometric functions, hyperbolic functions, and exponential functions are acquired using the generalized exponential rational function method. The 3D-surface graphs, 2D-contour graphs, density graphs, and 2D-line graphs of some retrieved solutions are plotted using Maple software. A variety of exact traveling wave solutions are reported including dark, bright, and kink soliton solutions. The nature of the optical solitons is demonstrated through the graphical representations of the acquired solutions for variation in the fractional order of derivative. It is hoped that the acquired solutions will aid in understanding the dynamics of the various physical phenomena and dynamical processes governed by the considered model.

It is pointed out in this paper that Lomax's hyperbolic function is a special case of both Compound Gamma and Compound Weibull distributions, and both of these distributions provide better models for Lomax's business failure data than his hyperbolic and exponential functions. Since his exponential function fails to yield a valid distribution function, a necessary condition is established to remedy this drawback. In the light of this result, his exponential function is modified in several ways. It is further shown that a natural complement of Lomax's exponential function does not suffer from this drawback.

The present work continues the study for the superstability and solution of the Pexider type functional equation , which is the mixed functional equation represented by sum of the sine, cosine, tangent, hyperbolic trigonometric, and exponential functions. The stability of the cosine (d'Alembert) functional equation and the Wilson equation was researched by many authors: Baker [7], Badora [5], Kannappan [14], Kim ([16, 19]), and Fassi, etc [11]. The stability of the sine type equations was researched by Cholewa [10], Kim ([18], [20]). The stability of the difference type equation for the above equation was studied by Kim ([21], [22]). In this paper, we investigate the superstability of the sine functional equation and the Wilson equation from the Pexider type difference functional equation , which is the mixed equation represented by the sine, cosine, tangent, hyperbolic trigonometric functions, and exponential functions. Also, we obtain additionally that the Wilson equation and the cosine functional eqaution in the obtained results can be represented by the composition of a homomorphism. In here, the domain (G; +) of functions is a noncommutative semigroup (or 2-divisible Abelian group), and A is an unital commutative normed algebra with unit 1A. The obtained results can be applied and expanded to the stability for the difference type's functional equation which consists of the (hyperbolic) secant, cosecant, logarithmic functions.

Novel closed-form analytical solutions and modulation instability spectrum induced by the Salerno equation describing nonlinear discrete electrical lattice via symbolic computation

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.

The prime goal of this paper is to establish sharp lower and upper bounds for useful functions such as the exponential functions, with a focus on exp(−x^2), the trigonometric functions (cosine and sine) and the hyperbolic functions (cosine and sine). The bounds obtained for hyperbolic cosine are very sharp. New proofs, refinements as well as new results are offered. Some graphical and numerical results illustrate the findings.

In the present work, we introduce the function representing a rapidly convergent power series which extends the well-known confluent hypergeometric function \(_1F_1[z]\) as well as the integral function \( f(z) = \sum \nolimits _{n=1}^\infty \frac{z^n}{n!^n} \) considered by Sikkema (Differential operators and equations, P. Noordhoff N. V., Djakarta, 1953). We introduce the corresponding differential operators and obtain infinite order differential equations, for which these new special functions are the eigen functions. First we establish some properties, as the order zero of these entire (integral) functions, integral representations, differential equations involving a kind of hyper-Bessel type operators of infinite order. Then we emphasize on the special cases, especially the corresponding analogues of the exponential, circular and hyperbolic functions, called here as \({\ell }\)-H exponential function, \({\ell }\)-H circular and \({\ell }\)-H hyperbolic functions. At the end, the graphs of these functions are plotted using the Maple software.

The Atangana–Baleanu fractional integral and multiplier transformations are two functions successfully used separately in many recently published studies. They were previously combined and the resulting function was applied for obtaining interesting new results concerning the theories of differential subordination and fuzzy differential subordination. In the present investigation, a new approach is taken by using the operator previously introduced by applying the Atangana–Baleanu fractional integral to a multiplier transformation for introducing a new subclass of analytic functions. Using the methods familiar to geometric function theory, certain geometrical properties of the newly introduced class are obtained such as coefficient estimates, distortion theorems, closure theorems, neighborhoods and the radii of starlikeness, convexity, and close-to-convexity of functions belonging to the class. This class may have symmetric or assymetric properties. The results could prove interesting for future studies due to the new applications of the operator and because the univalence properties of the new subclass of functions could inspire further investigations having it as the main focus.

This paper presents a novel extension of trigonometric and hyperbolic functions based on the W α , β , ν -function recently introduced in the literature [Droghei. Properties of the multi-index special function W ( α ¯ , ν ¯ ) ( z ) . Fract Calc Appl Anal. 2023;26(5):2057–2068. doi: 10.1007/s13540-023-00197-6]. The classical factorial is generalized to offer greater flexibility for applications in advanced mathematical fields such as combinatorics, special functions, and complex analysis. Using the W α , β , ν -exponential function, we derive new expressions for generalized trigonometric and hyperbolic functions, including sine, cosine, hyperbolic sine, and hyperbolic cosine. These functions provide exact solutions for non-trivial fractional differential equations with variable coefficients. We discuss some relevant examples of applications to solve linear and nonlinear equations by using operational methods and separation of variables.

According to the wave power rule, the second derivative of a function with respect to the variable t is equal to negative n times the function raised to the power of 2n-1. Solving the ordinary differential equations numerically results in waves appearing in the figures. The ordinary differential equation is very simple; however, waves, including the regular amplitude and period, are drawn in the figure. In this study, the function for obtaining the wave is called the leaf function. Based on the leaf function, the exact solutions for the undamped and unforced Duffing equations are presented. In the ordinary differential equation, in the positive region of the variable, the second derivative becomes negative. Therefore, in the case that the curves vary with the time under the condition x(t)>0, the gradient constantly decreases as time increases. That is, the tangential vector on the curve of the graph changes from the upper right direction to the lower right direction as time increases. On the other hand, in the negative region of the variable, the second derivative becomes positive. The gradient constantly increases as time decreases. That is, the tangent vector on the curve changes from the lower right direction to the upper right direction as time increases. Since the behavior occurring in the positive region of the variable and the behavior occurring in the negative region of the variable alternately occur in regular intervals, waves appear by these interactions. In this paper, I present seven types of damped and divergence exact solutions by combining trigonometric functions, hyperbolic functions, hyperbolic leaf functions, leaf functions, and exponential functions. In each type, I show the derivation method and numerical examples, as well as describe the features of the waveform.

After bringing together various results mentioned before, in this chapter we introduce the quaternion hyperbolic functions, whose study will require us to master a new situation. Since the quaternion exponential function agrees with the real and complex exponential function of real and complex arguments, it follows that the quaternion hyperbolic functions also agree with their usual counterparts for real and complex input. This allows us to discuss some important hyperbolic identities and the existence of infinitely many zeros of the quaternion sine and cosine hyperbolic functions, and to solve equations involving hyperbolic functions. A remarkable result of the theory exhibits the deep connection between the hyperbolic and trigonometric functions discussed in the previous chapter. We hope that material presented in this part will make this beautiful topic accessible to the reader.KeywordsTrigonometric FunctionCosine FunctionHyperbolic FunctionPrevious ChapterComplex Exponential FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.