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Graph Algorithms I
Published in R. Balakrishnan, Sriraman Sridharan, Discrete Mathematics, 2019
R. Balakrishnan, Sriraman Sridharan
Let us illustrate this problem with an example (see graph of Figure 1.38). In the graph of Figure 1.38, the different tours and their corresponding costs are given in Table 1.13. In Table 1.13, there are three tours and the minimum tour is the tour number 2 whose cost is 19. Note that the tours (1, 2, 3, 4, 1) and the reverse tour (1, 4, 3, 2, 1) give the same cost. Hence such duplicated tours are not listed in the table. The following lemma gives the number of Hamiltonian tours in a complete graph Kn. The number of Hamiltonian tours in Kn is (n − 1)!/2 (by identifying the reverse tours, that is, for n = 3, (1, 2, 3, 1) = (1, 3, 2, 1), because they have the same cost).Proof. Every tour of Kn defines a cyclic permutation and conversely every cyclic permutation of the set {1, 2…, n} defines a tour of Kn. But the number of cyclic permutation of the n set [n] is (n − 1)! (see [6]). By identifying a tour and its reverse tour (because they have the same total cost), the desired number is (n − 1)!/2.
Subfield codes and trace codes
Published in Jürgen Bierbrauer, Introduction to Coding Theory, 2016
It should now be clear when two codes are equivalent under permutations of coordinates. A highly interesting situation occurs if we find a permutation σ such that σ() = (in words: the image of our code under σ is the code itself). In this case we call σ an automorphism of . The automorphisms of form a subgroup, the stabilizer or automorphism group of . We saw a large example in Chapter 7. The generator matrix for the binary Golay code [23, 12, 7]2 given there has the special feature that all rows are cyclic shifts of the first row. In other words, the cyclic permutation σ : 1 ↦ 2, 2 ↦ 3,... 22 ↦ 23, 23 ↦ 1
Properties of the Tensors of Local and Non-Local Optical Response
Published in S. V. Popov, P Svirko Yu, N. I. Zheludev, Susceptibility Tensors for Nonlinear Optics, 2017
S. V. Popov, P Svirko Yu, N. I. Zheludev
We should point out here that the cyclic permutation of Cartesian indices in (2.24) is accompanied by a cyclic permutation of frequencies. Also, the frequency argument that the permutation transfers to position one, and the argument which is transferred from this position, have their signs reversed.
Numerical simulation of compressible flows by lattice Boltzmann method
Published in Numerical Heat Transfer, Part A: Applications, 2019
For two-dimensional problems presented in this article, the 16-velocity model, known as D2Q16 (see Figure 1), is used for spatial discretization that can be expressed as: where cyc indicates the cyclic permutation. The discrete form of Boltzmann equation with the Bhatanger–Gross–Krook approximation reads as [27]: where is the particle distribution function, is the relaxation time and is spatial direction. Furthermore, the equilibrium distribution function, in its discrete form is obtained as: where the relevant constant coefficients for are arranged in Table 1. The multiple-relaxation time (MRT) equation for compressible flow on 16-velocity components also reads as: where N is the number of discrete velocities and is kinetic moment space which is equal to and are the elements of the following matrix and they are defined as Eqs. (11) and (12).