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Hybrid surfaces
Published in Trevor J. Cox, Peter D'Antonio, Acoustic Absorbers and Diffusers, 2016
Trevor J. Cox, Peter D'Antonio
Schroeder11 showed that a folding technique called the Chinese remainder theorem could be applied to phase grating diffusers based on polyphase sequences. D'Antonio1 used the same technique for a binary hybrid diffuser. The Chinese remainder theorem folds a 1D sequence into a 2D array and yet preserves the good autocorrelation and Fourier properties. To use this method, N and M must be coprime. By coprime, it is meant that the only common factor for the two numbers is 1.
Basics of Number Theory
Published in Sriraman Sridharan, R. Balakrishnan, Foundations of Discrete Mathematics with Algorithms and Programming, 2019
Sriraman Sridharan, R. Balakrishnan
Two numbers a and b are coprime or relatively prime or prime to each other, if gcd(a,b)=1 $ gcd(a,b)=1 $ .
The Restricted Congruences Toolbox
Published in Khodakhast Bibak, Restricted Congruences in Computing, 2020
In order to study the restricted congruences and their applications, we need a wide variety of tools and techniques from several areas. In this chapter, we introduce these tools. Throughout the book we let Zn={0,…,n−1}. Two integers are said to be coprime (relatively prime) if their greatest common divisor (gcd) is 1. The set of integers from Zn coprime to n form a group under multiplication modulo n, called the multiplicative group of integers modulo n, and denoted by Zn∗. We use (a1, …, ak) and lcm(a1,…,ak) to denote, respectively, the greatest common divisor and the least common multiple of integers a1, …, ak, and write 〈a1, …, ak〉 for an ordered k-tuple of integers. Also, for a∈Z∖{0} and a prime p, we use the notation pr|| a if pr| a and pr+1∤a. For a set X, we write x←X to denote that x is chosen uniformly at random from X. We also use 0 to denote the vector of all zeroes.
Causal polynomial pole assignment and stabilisation of SISO strictly causal linear discrete processes
Published in International Journal of Control, 2019
E. N. Rosenwasser, B. P. Lampe, W. Drewelow, T. Jeinsch
Since the polynomials a(z), b(z) are coprime, there exists a set of separations where α(z), β(z) are polynomials, and the fractions on the right side are irreducible. We can easily verify that any pair of polynomials (α(z), β(z)), configured in (17), defines a solution of Equation (11). The reverse is also true: any pair of polynomials as solution of Equation (11) builds a separation of form (17). Among all possible separations (17), there exists exactly one, in which the polynomials α(z) = α*(z), β(z) = β*(z) satisfy and, moreover, The so-constructed solution α*(z), β*(z) of Equation (11) is called β-minimal controller and it is denoted by the pair (α*(z), β*(z)).
Absolute phase recovery with multiple-wavelength fringes
Published in Journal of Modern Optics, 2023
Jiale Long, Jianmin Zhang, Jiangtao Xi, Yi Ding, Fujian Chen, Zihao Du
When are not coprime, there are two possible situations. First, let be the greatest common divisor (g.c.m.) of , we have where are positive integers coprime. Equation (9) can be reproduced as follows: Obviously, Equation (18) can be proved with the same approach as Equation (9). Thus Equation (18) will hold when , which is equivalent to .
An elementary proof of Niven's theorem via the tangent function
Published in International Journal of Mathematical Education in Science and Technology, 2021
Bonaventura Paolillo, Giovanni Vincenzi
If one of and is even and the other is odd, then is odd, and hence no prime divisor of can divide , because and are coprime. So that is a reduced fraction, and hence by position (1), we have . This is a contradiction.