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Symmetry
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Asymptotic Methods in the Theory of Differential Equations and Mathematical...
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Asymptotic Methods in the Theory of Differential Equations and Mathematical Physics

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 May 2022) | Viewed by 8373

Special Issue Editors

Prof. Vladimir Nazaikinskii

E-Mail Website
Guest Editor
1. Ishlinsky Institute for Problems in Mechanics RAS, 119526 Moscow, Russia
2. Moscow Institute of Physics and Technology (National Research University), Dolgoprudny, 141701 Moscow Oblast, Russia
Interests: asymptotic methods in the theory of differential equations and mathematical physics; asymptotic methods in the statistics of many-particle systems; C * -algebras and noncommutative geometry; elliptic theory and index theory
Prof. Sergei Dobrokhotov

E-Mail Website
Guest Editor
1. Ishlinsky Institute for Problems in Mechanics RAS, 119526 Moscow, Russia
2. Moscow Institute of Physics and Technology (National Research University), Dolgoprudny, 14170Moscow Oblast, Russia
Interests: asymptotics; semiclassical and adiabatic approximations; waves; vortices; inhomogeneous media

Special Issue Information

Dear colleagues,

Asymptotic methods play an important role in the modern theory of differential equations, mathematical physics, and their applications in continuum mechanics, quantum mechanics, etc. One main aim of asymptotic methods is to provide descriptions and formulas that are sufficiently close to the exact ones and, at the same time, are efficient; that is, easy to analyze, interpret, and visualize. Of course, what is efficient and what is not depends on the tools available, and this is where modern technical computing systems like Wolfram Mathematica or MATLAB come in to make a breakthrough. With these systems, a combination of analytical and numerical approaches often works best, whereby, say, the numerical solution of ordinary differential equations of characteristics is used as an input for closed-form asymptotic expressions. This puts forward new challenges: new asymptotic formulas must be developed, and old ones must often be reworked with these computing systems in mind. This Special Issue intends to represent the state of the art in efficient asymptotics, mainly focusing on semiclassical (or geometric) asymptotics of rapidly varying solutions of wave and vortex type, boundary- and internal-layer asymptotics (including moving boundary layers), the adiabatic approximation, and asymptotics associated with homogenization (in spatial variables and in time).

Papers that employ the symmetry or asymmetry concept in their methodologies in the fields of Asymptotic Methods in the Theory of Differential Equations and Mathematical Physics are also welcomed.

Prof. Vladimir Nazaikinskii
Prof. Sergei Dobrokhotov
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • asymptotic methods
  • semiclassical asymptotics
  • geometric asymptotics
  • boundary layer
  • internal layer
  • adiabatic approximation
  • homogenization
  • analytical–numerical methods
  • efficient asymptotics
  • technical computing system
  • wolfram mathematica
  • MATLAB

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Published Papers (3 papers)

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Research

16 pages, 1196 KiB  
Article
Asymptotic Rules of Equilibrium Desingularization
by Yakov Krasnov and Iris Rabinowitz
Symmetry 2022, 14(10), 2186; https://doi.org/10.3390/sym14102186 - 18 Oct 2022
Cited by 1 | Viewed by 1296
Abstract
A local bifurcation analysis of a high-dimensional dynamical system dxdt=f(x) is performed using a good deformation of the polynomial mapping P:Cn→Cn. This theory is used to construct geometric [...] Read more.
A local bifurcation analysis of a high-dimensional dynamical system dxdt=f(x) is performed using a good deformation of the polynomial mapping P:Cn→Cn. This theory is used to construct geometric aspects of the resolution of multiple zeros of the polynomial vector field P(x). Asymptotic bifurcation rules are derived from Grothendieck’s theory of residuals. Following the Coxeter–Dynkin classification, the singularity graph is constructed. A detailed study of three types of multidimensional mappings with a large symmetry group has been carried out, namely: 1. A linear singularity (behaves similarly to a one-dimensional complex analysis theory); 2. The lattice singularity (generalized the linear and resembling regular crystal growth models); 3. The fan-shaped singularity (can be split radially like nuclear fission and fusion models). Full article
(This article belongs to the Special Issue Asymptotic Methods in the Theory of Differential Equations and Mathematical Physics)
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25 pages, 4757 KiB  
Article
RGB Image Encryption through Cellular Automata, S-Box and the Lorenz System
by Wassim Alexan, Mohamed ElBeltagy and Amr Aboshousha
Symmetry 2022, 14(3), 443; https://doi.org/10.3390/sym14030443 - 23 Feb 2022
Cited by 91 | Viewed by 4587
Abstract
The exponential growth in transmission of multimedia over the Internet and unsecured channels of communications is putting pressure on scientists and engineers to develop effective and efficient security schemes. In this paper, an image encryption scheme is proposed to help solve such a [...] Read more.
The exponential growth in transmission of multimedia over the Internet and unsecured channels of communications is putting pressure on scientists and engineers to develop effective and efficient security schemes. In this paper, an image encryption scheme is proposed to help solve such a problem. The proposed scheme is implemented over three stages. The first stage makes use of Rule 30 cellular automata to generate the first encryption key. The second stage utilizes a well-tested S-box, whose design involves a transformation, modular inverses, and permutation. Finally, the third stage employs a solution of the Lorenz system to generate the second encryption key. The aggregate effect of this 3-stage process insures the application of Shannon’s confusion and diffusion properties of a cryptographic system and enhances the security and robustness of the resulting encrypted images. Specifically, the use of the PRNG bitstreams from both of the cellular automata and the Lorenz system, as keys, combined with the S-box, results in the needed non-linearity and complexity inherent in well-encrypted images, which is sufficient to frustrate attackers. Performance evaluation is carried out with statistical and sensitivity analyses, to check for and demonstrate the security and robustness of the proposed scheme. On testing the resulting encrypted Lena image, the proposed scheme results in an MSE value of 8923.03, a PSNR value of 8.625 dB, an information entropy of 7.999, NPCR value of 99.627, and UACI value of 33.46. The proposed scheme is shown to encrypt images at an average rate of 0.61 Mbps. A comparative study with counterpart image encryption schemes from the literature is also presented to showcase the superior performance of the proposed scheme. Full article
(This article belongs to the Special Issue Asymptotic Methods in the Theory of Differential Equations and Mathematical Physics)
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9 pages, 248 KiB  
Article
Modeling 3D–1D Junction via Very-Weak Formulation
by Eduard Marušić-Paloka
Symmetry 2021, 13(5), 831; https://doi.org/10.3390/sym13050831 - 9 May 2021
Cited by 1 | Viewed by 1558
Abstract
We study the potential flow of an ideal fluid through a domain that consists of a reservoir and a pipe connected to it. The ratio of the pipe’s thickness and its length is considered as a small parameter. Using the rigorous asymptotic analysis [...] Read more.
We study the potential flow of an ideal fluid through a domain that consists of a reservoir and a pipe connected to it. The ratio of the pipe’s thickness and its length is considered as a small parameter. Using the rigorous asymptotic analysis with respect to that small parameter, we derive an effective model governing the the junction between a 1D and a 3D fluid domain. The obtained boundary-value problem has a measure boundary condition with Dirac mass concentrated in the junction point and is understood in the very-weak sense. Full article
(This article belongs to the Special Issue Asymptotic Methods in the Theory of Differential Equations and Mathematical Physics)
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