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17 pages, 618 KiB

Open AccessArticle

Multiplicative Renormalization of Stochastic Differential Equations for the Abelian Sandpile Model

by Dimitri Volchenkov

Dynamics 2024, 4(1), 40-56; https://doi.org/10.3390/dynamics4010003 - 4 Jan 2024

Cited by 3 | Viewed by 1849

The long-term, large-scale behavior in a problem of stochastic nonlinear dynamics corresponding to the Abelian sandpile model is studied with the use of the quantum-field theory renormalization group approach. We prove the multiplicative renormalization of the model including an infinite number of coupling parameters, calculate an infinite number of renormalization constants, identify a plane of fixed points in the infinite dimensional space of coupling parameters, discuss their stability and critical scaling in the model, and formulate a simple law relating the asymptotic size of an avalanche to a model exponent quantifying the time-scale separation between the slow energy injection and fast avalanche relaxation processes. Full article

(This article belongs to the Special Issue Recent Advances in Dynamic Phenomena)

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19 pages, 355 KiB

Open AccessArticle

Transposition Regular TA-Groupoids and Their Structures

by Xiaogang An and Xiaohong Zhang

Axioms 2022, 11(8), 378; https://doi.org/10.3390/axioms11080378 - 30 Jul 2022

Cited by 1 | Viewed by 2040

Abstract

Tarski associative groupoid (TA-groupoid) is a kind of non-associative groupoid satisfying Tarski associative law. In this paper, the new notions of transposition regular TA-groupoid are proposed and their properties and structural characteristics are studied by using band and quasi-separativity. In particular, the following conclusions are strictly proved: (1) every left transposition regular TA-groupoid is a semigroup; (2) every left transposition regular TA-groupoid is the disjoint union of sub Abelian groups; and (3) a finite TA-groupoid with quasi-separativity and a finite left transposition regular TA-groupoid are equivalent. Full article

(This article belongs to the Section Logic)

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21 pages, 6937 KiB

Open AccessArticle

A Variety of New Traveling Wave Packets and Conservation Laws to the Nonlinear Low-Pass Electrical Transmission Lines via Lie Analysis

by Muhammad Bilal Riaz, Jan Awrejcewicz, Adil Jhangeer and Muhammad Junaid-U-Rehman

Fractal Fract. 2021, 5(4), 170; https://doi.org/10.3390/fractalfract5040170 - 18 Oct 2021

Cited by 19 | Viewed by 2213

Abstract

This research is based on computing the new wave packets and conserved quantities to the nonlinear low-pass electrical transmission lines (NLETLs) via the group-theoretic method. By using the group-theoretic technique, we analyse the NLETLs and compute infinitesimal generators. The resulting equations concede two-dimensional Lie algebra. Then, we have to find the commutation relation of the entire vector field and observe that the obtained generators make an abelian algebra. The optimal system is computed by using the entire vector field and using the concept of abelian algebra. With the help of an optimal system, NLETLs convert into nonlinear ODE. The modified Khater method (MKM) is used to find the wave packets by using the resulting ODEs for a supposed model. To represent the physical importance of the considered model, some

3 D

2 D

, and density diagrams of acquired results are plotted by using Mathematica under the suitable choice of involving parameter values. Furthermore, all derived results were verified by putting them back into the assumed equation with the aid of Maple software. Further, the conservation laws of NLETLs are computed by the multiplier method. Full article

(This article belongs to the Special Issue Recent Advances in Computational Physics with Fractional Application)

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20 pages, 325 KiB

Open AccessArticle

A Group Law on the Projective Plane with Applications in Public Key Cryptography

by Raúl Durán Díaz, Luis Hernández Encinas and Jaime Muñoz Masqué

Mathematics 2020, 8(5), 734; https://doi.org/10.3390/math8050734 - 7 May 2020

Cited by 1 | Viewed by 2799

Abstract

In the context of new threats to Public Key Cryptography arising from a growing computational power both in classic and in quantum worlds, we present a new group law defined on a subset of the projective plane

F P^{2}

over an arbitrary [...] Read more.

F P^{2}

over an arbitrary field

F

, which lends itself to applications in Public Key Cryptography and turns out to be more efficient in terms of computational resources. In particular, we give explicitly the number of base field operations needed to perform the mentioned group law. Based on it, we present a Diffie-Hellman-like key agreement protocol. We analyze the computational difficulty of solving the mathematical problem underlying the proposed Abelian group law and we prove that the security of our proposal is equivalent to the discrete logarithm problem in the multiplicative group of the cubic extension of the finite field considered. We present an experimental setup in order to show real computation times along a comparison with the group operation in the group of points of an elliptic curve. Based on current state-of-the-art algorithms, we provide parameter ranges suitable for real world applications. Finally, we present a promising variant of the proposed group law, by moving from the base field

F

to the ring

Z / p q Z

, and we explain how the security becomes enhanced, though at the cost of a longer key length. Full article

(This article belongs to the Special Issue Mathematics Cryptography and Information Security)

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