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Combinatorics
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
A cycle of length 1 is a fixed point, and so we know that the number of cycles of length 1 in a random permutation is expected to be one. An involution is a function that is its own inverse. That is, a function f:R→S such that f(f(x))=x for all x∈R. An involution is a permutation (since it must be both surjective and injective). It follows from the cycle representation that involutions are precisely those permutations that have no cycles of length greater than two.
In Many Circles. Permutations as Products of Cycles.
Published in Miklós Bóna, Combinatorics of Permutations, 2022
An involution is a permutation p so that p−1=p; in other words, p2 is the identity permutation. It is easy to see that this happens if and only if all cycles of p have length 1 or 2. Note that 1-cycles are often called fixed points.
Inverse problems for a multi-term time fractional evolution equation with an involution
Published in Inverse Problems in Science and Engineering, 2021
Asim Ilyas, Salman A. Malik, Summaya Saif
These solutions are illustrated in Figures 1 and 2 for T = 1. The effect of the order of the fractional derivative α is illustrated in Figure 1. It shows very small change in the solution profile, while its effect is more apparent in the graph of the source term. It shows that the solution is decreasing with increasing value of α. In Figure 2, graphs of the solution profile at different times and the source function are presented by fixing the fractional-order and the involution coefficient . The amplitude of the solution is growing over time, eventually reaching its maximum value. The final temperature in this example is perturbed to get the resulting noisy data . The source term is evaluated using the final data whereas the function recovered by using the noisy or perturbed data is represented by . The graphs of the solution are plotted by taking different orders of α and at time T = 1. Figure 3(a) shows the plot of and noisy data , whereas Figure 3(b) gives the plot of the solutions at .
Time-reversal symmetries in two-dimensional reversible partitioned cellular automata and their applications
Published in International Journal of Parallel, Emergent and Distributed Systems, 2022
Next, we show Lemma 3.9 stating that ESPCA-uvwxyz satisfying is T-symmetric under a certain simple involution. First, define a function as follows: for any . Next define an involution as follows. For all and : The involution gives the complement image of a configuration.
Fibonacci and Lucas Riordan arrays and construction of pseudo-involutions
Published in Applicable Analysis, 2021
Candice Marshall, Asamoah Nkwanta
Most of the Riordan matrices as well as the Lucas stochastic array presented in this paper are new and contain many new sequences of integers. The arrays were observed while studying certain algebraic properties of the Riordan group. Finding combinatorial interpretations of the arrays was not the focus of this paper. Thus, the arrays are open for combinatorial interpretations. Constructing a pseudo-involution starting with a modified Lucas generating function would be of interest. In addition, constructing pseudo-involutions with higher powers of would be of interest.