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The Real Number System
Published in Hemen Dutta, Pinnangudi N. Natarajan, Yeol Je Cho, Concise Introduction to Basic Real Analysis, 2019
Hemen Dutta, Pinnangudi N. Natarajan, Yeol Je Cho
The real numbers are frequently represented geometrically as points on a line, called the real line or the real axis. A point is chosen to represent 0 and another to represent 1, as shown in the following figure:
Fractional-order Bessel wavelet functions for solving variable order fractional optimal control problems with estimation error
Published in International Journal of Systems Science, 2020
Haniye Dehestani, Yadollah Ordokhani, Mohsen Razzaghi
In this section, we discuss the error analysis of the approximate functions in the Sobolev space. For this aim, the Sobolev norm of integer order in the interval in the real line is defined (Canuto et al., 2006) where The seminorm is defined as Canuto et al. (2006) Also, to obtain the convenient results, we define the following seminorm for and as Due to the above relation, we have where
Propagation of waves in fractal spaces
Published in Waves in Random and Complex Media, 2023
Rami Ahmad El-Nabulsi, Alireza Khalili Golmankhaneh
Let us consider fractal Lagrangian as follows The corresponding fractal Euler–Lagrange equation is with the following conditions Suppose the solution of Equation (37) be in the following form Then we have and It follows that Then we can write and In view of , we obtain and Using and fractal Fourier sine series, we get Let , then it leads Utilizing , we can write a smooth version as follows We plot in Figures 9–16, the variations of the fractal dimension on the solutions of the fractal Laplace equation for n = 1, 2. Note that in this section, if we choose which is the dimension of real-line we arrive at ordinary calculus.
A random weak ergodic property of infinite products of operators in metric spaces
Published in Optimization, 2019
Simeon Reich, Alexander J. Zaslavski
Let be a metric space and let denote the real line. We say that a mapping is a metric embedding of into X if for all real s and t. The image of under a metric embedding is called a metric line. The image of a real interval under such a mapping is called a metric segment.