Dynamics of the a-map over residually finite Dedekind domains and applications

Qureshi, Claudio; Reis, Lucas

doi:10.1016/j.jnt.2019.03.018

Cited by 5 publications

(6 citation statements)

References 20 publications

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“…Basics on functional graphs. We use the same terminology as in [13] and [16]. It is a well known fact that the connected components of functional graphs are composed of a cycle and each vertex of this cycle is the root of a tree (the direction of the edges is from leaves to the root).…”

Section: Preliminariesmentioning

confidence: 99%

“…Trees associated with non-increasing sequences as above are called elementary trees, see [16] for more details on elementary trees. Figure 1 shows the inductive process to contruct T v for a 4-term sequence.…”

Section: Given Rooted Trees Tmentioning

confidence: 99%

“…Elementary trees appear in a wide range of context when we describe the dynamics of different classes of maps, see for example [5,10,13,14,15,16,20]. They will play a key role also in our description of the functional graph associated with the power map over flower groups.…”

Section: Given Rooted Trees Tmentioning

confidence: 99%

“…When G is cyclic, the graph G(ϕ t /G) is completely described in [13]. The latter can be extended to abelian groups, mainly using the results of [16] as we show in Section 2.3. For non abelian groups, only results for special families of groups are known.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2 we introduce the basic notation and operations on graphs and review the structure theorem for the functional graph of the power map on cyclic groups. Then, using results from [16], we show how to extend this structure theorem for finite abelian groups. In the last part of this section we provide some general results on the functional graph of the power map on arbitrary finite groups that is further used to prove our main results.…”

Section: Introductionmentioning

confidence: 99%

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On the functional graph of the power map over finite groups

Qureshi¹,

Reis²

2021

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In this paper we study the description of the digraph associated with the power map over finite groups. Our main motivation comes from the nice description of such digraphs in the case of cyclic groups. In particular, we derive results on abelian groups, and also on flower groups, which are introduced in this paper. The class of flower groups includes many non abelian groups such as dihedral and generalized quaternion groups, and the projective general linear group of order two over a finite field. In particular, we provide improvements on past works.

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Section: Preliminariesmentioning

confidence: 99%

Section: Given Rooted Trees Tmentioning

confidence: 99%

Section: Given Rooted Trees Tmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the functional graph of the power map over finite groups

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Dynamics of polynomial maps over finite fields

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ReLiCADA: Reservoir Computing Using Linear Cellular Automata design algorithm

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In this paper, we present a novel algorithm to optimize the design of Reservoir Computing using Cellular Automata models for time series applications. Besides selecting the models’ hyperparameters, the proposed algorithm particularly solves the open problem of Linear Cellular Automaton rule selection. The selection method pre-selects only a few promising candidate rules out of an exponentially growing rule space. When applied to relevant benchmark datasets, the selected rules achieve low errors, with the best rules being among the top 5% of the overall rule space. The algorithm was developed based on mathematical analysis of Linear Cellular Automaton properties and is backed by almost one million experiments, adding up to a computational runtime of nearly one year. Comparisons to other state-of-the-art time series models show that the proposed Reservoir Computing using Cellular Automata models have lower computational complexity and, at the same time, achieve lower errors. Hence, our approach reduces the time needed for training and hyperparameter optimization by up to several orders of magnitude.

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Dynamics of the a-map over residually finite Dedekind domains and applications

Abstract: Let D be a residually finite Dedekind domain, a ∈ D be a nonzero element and n be a nonzero ideal of D. In this paper we describe the dynamics of the map x → ax over the quotient ring D/n. We further present some applications of our main result.

Cited by 5 publications

References 20 publications

On the functional graph of the power map over finite groups

On the functional graph of the power map over finite groups

Dynamics of polynomial maps over finite fields

ReLiCADA: Reservoir Computing Using Linear Cellular Automata design algorithm

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