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Combinatorics
Published in Sriraman Sridharan, R. Balakrishnan, Foundations of Discrete Mathematics with Algorithms and Programming, 2019
Sriraman Sridharan, R. Balakrishnan
In this chapter on combinatorics, we start from the elementary rules of counting, then study permutations and combinations, binomial coefficients, binomial theorem, multinomial coefficients, multinomial theorem, Stirling numbers of the first and the second kind, Bell numbers, the Principle of Inclusion and Exclusion (simple and weighted versions), some applications of the Principle of Inclusion and Exclusion to number theory and the theory of permanents, generating function techniques and recurrence relations, Bernoulli numbers, Catalan numbers, and an algorithm for generating all the subsets of a given finite set.
Multi-surface phase-shifting algorithm using the window function fitted by the nonlinear least squares method
Published in Journal of Modern Optics, 2022
Lin Chang, Sergiy Valyukh, Tingting He, Zhu Chen, Yingjie Yu
Before fitting , we need to study the distribution characteristics of its values, which have not been deeply analyzed [17–22]. For a polynomial composed of variables with power n, according to the multinomial-theorem [23], the distribution of its coefficients can be expressed as: By analyzing the distribution of the coefficients, it can be concluded that the form of these coefficients is similar to the Gaussian distribution, which can be seen in Figure 2. Besides, considering the multinomial-theorem and Eqs. (7)-(8), the window length can be given as Z = Zmax×N-Zmax+1.
On Secure QoS-based NOMA Networks with Multiple Antennas and Eavesdroppers over Nakagami-m Fading
Published in IETE Journal of Research, 2022
Tam Nguyen Kieu, Duc-Dung Tran, Dac-Binh Ha, Miroslav Voznak
Based on (1), under Nakagami-m fading, the cumulative distribution function (CDF) of has the following form [11]: By using binomial expansion [19, Eq. 1.111], and multinomial theorem, is rewritten as where , , , and .