Radius of Star-Likeness for Certain Subclasses of Analytic Functions

In this paper, we investigate a normalized analytic (symmetric under rotation) function, f, in an open unit disk that satisfies the condition ℜfzgz>0, for some analytic function, g, with ℜz+1−2nzgz>0,∀n∈N. We calculate the radius constants for different classes of analytic functions, including, for example, for the class of star-like functions connected with the exponential functions, i.e., the lemniscate of Bernoulli, the sine function, cardioid functions, the sine hyperbolic inverse function, the Nephroid function, cosine function and parabolic star-like functions. The results obtained are sharp.

Fractals as Julia Sets of Complex Sine Function via Fixed Point Iterations

We explore some new variants of the Julia set by developing the escape criteria for a function sin(zn)+az+c, where a,c∈C, n≥2, and z is a complex variable, utilizing four distinct fixed point iterative methods. Furthermore, we examine the impact of parameters on the deviation of dynamics, color, and appearance of fractals. Some of these fractals represent the stunning art on glass, and Rangoli (made in different parts of India, especially during the festive season) which are useful in interior decoration. Some fractals are similar to beautiful objects found in our surroundings like flowers (to be specific Hibiscus and Catharanthus Roseus), and ants.

Comparative assessments of strain measures for nonlinear analysis of truss structures at large deformations

PurposeIn this paper the use of classical strain measures in analysis of trusses at finite deformations will be discussed. The results will be compared to the ones acquired using a novel strain measure based on the Hyperbolic Sine function. Through the evaluation of results, algebraic development and graph analysis, the properties of the Hyperbolic Sine strain measure will be examined.Design/methodology/approachThrough graph plotting, comparisons between the novel strain measure and the classic ones will be made. The formulae for the implementation of the Hyperbolic Sine strain measure into a positional finite element method are developed. Four engineering applications are presented and comparisons between results obtained using all strain measures studied are made.FindingsThe proposed strain measure, Hyperbolic Sine, has objectivity and symmetry. The linear constitutive model formed by the Hyperbolic Sine strain and its conjugated stress presents an increasing stiffness, both in compression and tension, a behavior that can be useful in the modeling of several materials.Research limitations/implicationsThe structural analysis performed on the four examples of trusses in this article did not consider the variation of the cross-sectional area of the elements or the buckling phenomenon, moreover, only elastic behavior is considered.Originality/valueThe present article proposes the use of a novel strain measure family, based on the Hyperbolic Sine function and suitable for structural applications. Mathematical expressions for the use of the Hyperbolic Sine strain measure are established following the energetic concepts of the positional formulation of the finite element method.

Using Sine Function-Based Nonlinear Feedback to Control Water Tank Level

This manuscript addresses the feasibility and significance of using a sine function to modify the system error of a normal linear feedback control to achieve more efficient capabilities in terms of energy-saving. The associated mathematic modeling and assessment were demonstrated by presenting a case analysis on the capabilities of controlling water level for a single tank. The principle of robust control and the theories and detailed algorithm of Lyapunov stability were applied to assess the result derived by novel nonlinear feedback in the form of sine function for optimizing the robustness of the PID (Proportional–Integral–Derivative) controller and economizing energy. Two control simulations are compared: nonlinear feedback control using a sine function and conventional fuzzy control. The results reveal that using the nonlinear feedback controller, a reduction of up to 32.9% of the average controlled quantity is achieved, and the performance index is improved by 24.0% with satisfactory robustness. The proposed nonlinear feedback control using a sine function provides simplicity, convenient implementation, and energy efficiency.

Maclaurin's series expansions of real powers of inverse (hyperbolic) cosine and sine functions with applications

In the paper, by means of the Faa di Bruno formula, with the help of explicit formulas for special values of the Bell polynomials of the second kind with respect to a specific sequence, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes Maclaurin's series expansions for real powers of the inverse cosine (sine) function and the inverse hyperbolic cosine (sine) function. By applying different series expansions for the square of the inverse cosine function, the author not only finds infinite series representations of the circular constant Pi and its square, but also derives two combinatorial identities involving central binomial coefficients.

Dynamical Behaviors of Sinusoidal Nanocomposite Plates Subjected to Thermomechanical Loads

In this paper, a new plate structure has been found with the change of profile according to the sine function which we temporarily call as the sinusoidal plate. The classical plate theory and Galerkin’s technique have been utilized in estimating the nonlinear vibration behavior of the new non-rectangular plates reinforced by functionally graded (FG) graphene nanoplatelets (GNPs) resting on the Kerr foundation. The FG-GNP plates were assumed to have two horizontal variable edges according to the sine function. Four different configurations of the FG-GNP plates based on the number of cycles of sine function were analyzed. The material characteristics of the GNPs were evaluated in terms of two models called the Halpin–Tsai micromechanical model and the rule of mixtures. First, to verify this method, the natural frequencies of new non-rectangular plates made of metal were compared with those obtained by the Finite Element Method (FEM). Then, the numerical outcomes are validated by comparing with the previous papers for rectangular FGM/GNP plates — a special case of this structure. Furthermore, the impacts of the thermal environment, geometrical parameters, and the elastic foundation on the dynamical responses are scrutinized by the 2D/3D graphical results and coded in Wolfram-Mathematica. The results of this work proved that the introduced approach has the advantages of being fast, having high accuracy, and involving uncomplicated calculation.

Robust adaptive filtering algorithms based on (inverse)hyperbolic sine function

Recently, adaptive filtering algorithms were designed using hyperbolic functions, such as hyperbolic cosine and tangent function. However, most of those algorithms have few parameters that need to be set, and the adaptive estimation accuracy and convergence performance can be improved further. More importantly, the hyperbolic sine function has not been discussed. In this paper, a family of adaptive filtering algorithms is proposed using hyperbolic sine function (HSF) and inverse hyperbolic sine function (IHSF) function. Specifically, development of a robust adaptive filtering algorithm based on HSF, and extend the HSF algorithm to another novel adaptive filtering algorithm based on IHSF; then continue to analyze the computational complexity for HSF and IHSF; finally, validation of the analyses and superiority of the proposed algorithm via simulations. The HSF and IHSF algorithms can attain superior steady-state performance and stronger robustness in impulsive interference than several existing algorithms for different system identification scenarios, under Gaussian noise and impulsive interference, demonstrate the superior performance achieved by HSF and IHSF over existing adaptive filtering algorithms with different hyperbolic functions.

Periodic variability of the z = 2.0 quasar QSO B1312+7837

We report here the first results from a 15-yr long variability monitoring of the z = 2.0 quasar QSO B1312+7837. It shows luminosity changes with a period P ∼ 6.13 yr (P ∼ 2.04 yr at rest frame) and an amplitude of ∼0.2 mag, superimposed on a gradual dimming at a rate of ∼0.55 mag per 100 yrs. Two false periods associated with power peaks in the data windowing function were discarded. The measured period is confirmed with a bootstrapping Monte Carlo simulation. A damped random walk model yields a better fit to the data than a sine-function model, but at the cost of employing some high frequency variations which are typically not seen in quasars. We consider the possible mechanisms driving this variability, and conclude that orbital motion of two supermassive black holes – result from a recent galaxy merger – is a possible explanation.

New Bounds for the Sine Function and Tangent Function

Using the power series expansion technique, this paper established two new inequalities for the sine function and tangent function bounded by the functions x2sin(λx)/(λx)α and x2tan(μx)/(μx)β. These results are better than the ones in the previous literature.

Fourth Toeplitz Determinants for Starlike Functions Defined by Using the Sine Function

In this article, we aim to study the upper bounds of the fourth Toeplitz determinant T 4 2 for the function class S s ∗ , which are connected with the sine function.