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General introduction
Published in Adedeji B. Badiru, Handbook of Industrial and Systems Engineering, 2013
Exponential function series expansion ez=expz=1+z1!z22!+z33!+…(z=x+iy)
Basic Mathematical Calculations for Data Analytics
Published in Adedeji B. Badiru, Data Analytics, 2020
Exponential function series expansion ez=expz=1+z1!z22!+z33!+…(z=x+iy)
An Alternative Class of Orthogonal Functions
Published in Anish Deb, Suchismita Ghosh, Power Electronic Systems, 2017
A special orthogonal function series known as the slant function series was introduced by Enomoto and Shibata [36] for image transmission analysis. These functions have a finite but a large number of possible states as shown in Figure 2.9. These functions are similar to the general form of the sequency ordering of Walsh functions. The superiority of this function set lies in its transform characteristics that permit a compaction of the image energy to only a few transformed samples. Thus, the efficiency of image data transmission in this form is improved. As expected, the slant functions are related to the Walsh functions through the algebra of transformation matrices.
A new finite strip formulation based on Carrera unified formulation for the free vibration analysis of composite laminates
Published in Mechanics of Advanced Materials and Structures, 2022
Behnam Daraei, Saeed Shojaee, Saleh Hamzehei-Javaran
in Eq. (5) can be written as: where is the nth of the basic function series which satisfy the end conditions (at and ). Here, trigonometric functions are used as basic functions because of their valuable orthogonal properties [34]. For instance, corresponding to both simply supported (S-S) and both clamped (C-C) end conditions in and can be considered as: in which where r is the number of terms of basic functions.
Technical Note: The Anti-plane Scattering of SH Waves by the Non-circular Cavity in an Infinite Strip
Published in Journal of Earthquake Engineering, 2022
The incident guided wave and the scattering guided wave are both the compatible SH waves in the strip-shaped medium so that the stress free condition on the plane of the upper and lower boundary surfaces and is satisfied, and as a solution of the scattering problem, it and satisfy the stress free condition on the surface of the circular cavity boundary . The factors of the wave function series of the shot-guided wave are set to obtain Equation (18), which is a function of the angle variable , and the diagonal variable is expanded to (19) by Fourier series, where and are the Fourier factors of the Q term, respectively (20) and (21).
Global exponential stability on anti-periodic solutions in proportional delayed HIHNNs
Published in Journal of Experimental & Theoretical Artificial Intelligence, 2021
Motivated by the discussion above, the main purpose of this paper is by applying the uniform convergence theory of function series to establish the existence of anti-periodic solutions of proportional delayed HIHNNs. The contribution of this paper includes the following four aspects: (1) the proportional delayed high-order inertial Hopfield neural networks involving time-varying coefficients are proposed. (2) A suitable Lyapunov function is applied to study the anti-periodicity of the addressed system. (3) A simple criterion which guarantees the global exponential stability of anti-periodic solutions of HIHNNs is established without using reduced-ordera method. (4) The results of this paper are completely new and the methods used in this paper are different from those used in (Ke & Miao, 2017, 2013; Xu & Zhang, 2015). In particular, the effectiveness and advantages of the obtained results are demonstrated by a numerical example.