cyclic permutation
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Author(s):  
DMITRI I. PANYUSHEV ◽  
OKSANA S. YAKIMOVA

AbstractLet 𝔮 be a finite-dimensional Lie algebra. The symmetric algebra (𝔮) is equipped with the standard Lie–Poisson bracket. In this paper, we elaborate on a surprising observation that one naturally associates the second compatible Poisson bracket on (𝔮) to any finite order automorphism ϑ of 𝔮. We study related Poisson-commutative subalgebras (𝔮; ϑ) of 𝒮(𝔮) and associated Lie algebra contractions of 𝔮. To obtain substantial results, we have to assume that 𝔮 = 𝔤 is semisimple. Then we can use Vinberg’s theory of ϑ-groups and the machinery of Invariant Theory.If 𝔤 = 𝔥⊕⋯⊕𝔥 (sum of k copies), where 𝔥 is simple, and ϑ is the cyclic permutation, then we prove that the corresponding Poisson-commutative subalgebra (𝔮; ϑ) is polynomial and maximal. Furthermore, we quantise this (𝔤; ϑ) using a Gaudin subalgebra in the enveloping algebra 𝒰(𝔤).


2020 ◽  
Author(s):  
James G. Saulsbury

AbstractThe analysis of patterns in comparative data has come to be dominated by least-squares regression, mainly as implemented in phylogenetic generalized least-squares (PGLS). This approach has two main drawbacks: it makes relatively restrictive assumptions about distributions and can only address questions about the conditional mean of one variable as a function of other variables. Here I introduce two new non-parametric constructs for the analysis of a broader range of comparative questions: phylogenetic permutation tests, based on cyclic permutations and permutations conserving phylogenetic signal. The cyclic permutation test, an extension of the restricted permutation test that performs exchanges by rotating nodes on the phylogeny, performs well within and outside the bounds where PGLS is applicable but can only be used for balanced trees. The signal-based permutation test has identical statistical properties and works with all trees. The statistical performance of these tests compares favorably with independent contrasts and surpasses that of a previously developed permutation test that exchanges closely related pairs of observations more frequently. Three case studies illustrate the use of phylogenetic permutations for quantile regression with non-normal and heteroscedastic data, testing hypotheses about morphospace occupation, and comparative problems in which the data points are not tips in the phylogeny.


Biometrika ◽  
2020 ◽  
Author(s):  
Lihua Lei ◽  
Peter J Bickel

Abstract We propose the cyclic permutation test to test general linear hypotheses for linear models. This test is nonrandomized and valid in finite samples with exact Type-I error α for an arbitrary fixed design matrix and arbitrary exchangeable errors, whenever 1 / α is an integer and n / p ≥ 1 / α – 1. The test applies the marginal rank test on 1 / α linear statistics of the outcome vector where the coefficient vectors are determined by solving a linear system such that the joint distribution of the linear statistics is invariant to a nonstandard cyclic permutation group under the null hypothesis. The power can be further enhanced by solving a secondary nonlinear travelling salesman problem, for which the genetic algorithm can find a reasonably good solution. We show that the Cyclic Permutation Test has comparable power with existing tests through extensive simulation studies. When testing for a single contrast of coefficients, an exact confidence interval can be obtained by inverting the test. Furthermore, we provide a selective yet extensive literature review of the century-long efforts on this problem, highlighting the novelty of our test.


Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Risong Li ◽  
Tianxiu Lu

In this paper, we study some chaotic properties of s-dimensional dynamical system of the form Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1, where ak∈Hk for any k∈1,2,…,s, s≥2 is an integer, and Hk is a compact subinterval of the real line ℝ=−∞,+∞ for any k∈1,2,…,s. Particularly, a necessary and sufficient condition for a cyclic permutation map Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1 to be LY-chaotic or h-chaotic or RT-chaotic or D-chaotic is obtained. Moreover, the LY-chaoticity, h-chaoticity, RT-chaoticity, and D-chaoticity of such a cyclic permutation map is explored. Also, we proved that the topological entropy hΨ of such a cyclic permutation map is the same as the topological entropy of each of the following maps: gj∘gj−1∘⋯∘g1l∘gs∘gs−1∘⋯∘gj+1, if j=1,…,s−1and gs∘gs−1∘⋯∘g1, and that Ψ is sensitive if and only if at least one of the coordinates maps of Ψs is sensitive.


2020 ◽  
Vol 31 (10) ◽  
pp. 2050080
Author(s):  
M. Bischoff

Using a result of Longo and Xu, we show that the anomaly arising from a cyclic permutation orbifold of order 3 of a holomorphic conformal net [Formula: see text] with central charge [Formula: see text] depends on the “gravitational anomaly” [Formula: see text]. In particular, the conjecture that holomorphic permutation orbifolds are non-anomalous and therefore a stronger conjecture of Müger about braided crossed [Formula: see text]-categories arising from permutation orbifolds of completely rational conformal nets are wrong. More generally, we show that cyclic permutations of order [Formula: see text] are non-anomalous if and only if [Formula: see text] or [Formula: see text]. We also show that all cyclic permutation gaugings of [Formula: see text] arise from conformal nets.


2020 ◽  
Vol 277 ◽  
pp. 172-179
Author(s):  
Mariusz Grech ◽  
Andrzej Kisielewicz

2020 ◽  
Vol 54 ◽  
pp. 2
Author(s):  
Golnaz Badkobeh ◽  
Pascal Ochem

We construct an infinite word w over the 5-letter alphabet such that for every factor f of w of length at least two, there exists a cyclic permutation of f that is not a factor of w. In other words, w does not contain a non-trivial conjugacy class. This proves the conjecture in Gamard et al. [Theoret. Comput. Sci. 726 (2018) 1–4].


2019 ◽  
Vol 35 (6) ◽  
pp. 1405-1432 ◽  
Author(s):  
Mariusz Grech ◽  
Andrzej Kisielewicz

Abstract In this paper we establish conditions for a permutation group generated by a single permutation to be an automorphism group of a graph. This solves the so called concrete version of König’s problem for the case of cyclic groups. We establish also similar conditions for the symmetry groups of other related structures: digraphs, supergraphs, and boolean functions.


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