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Special Issue "Geometric Methods and their Applications"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: 28 February 2021.

Special Issue Editors

Prof. Paul Popescu

Guest Editor
Department of Applied Mathematics, University of Craiova, 200585 Craiova, Romania
Interests: geometry of differential equations on manifolds; geometric theory of foliations; Finsler and Lagrange geometry; geometric algebra; geometric methods in mechanics
Assoc. Prof. Marcela Popescu

Guest Editor
Department of Applied Mathematics, University of Craiova, 200585 Craiova, Romania
Interests: geometry of submanifolds; Finsler and Lagrange geometry; geometric methods in mechanics

Special Issue Information

Dear Colleagues,

The aim of our Special Issue, titled Geometric Methods and their Applications, is to bring together outstanding theoretical contributions using geometric methods from various mathematical and physical research areas, with real-world applications.

Geometric methods retain the force they have long had in new and old domains, which would seem to be exhausted. For example:

- Lagrangians and Hamiltonians can reinterpret the classical analysis on manifolds.

- The use of foliations can give new perspectives to classical mechanics.

- Classical generalizations using different types of algebroids or groupoids can make the transfer to the study of singular structures using classical methods for regular structures.

- Some methods of discretization have a discrete setting, but some combine the discrete with the continuous, such as Veselov-type discretizations of tangent spaces.

- Symmetries, the Klein basis of geometry, or Noether’s base of physics can be involved in varied forms in geometry and physics, related to various settings.

The purpose of this Special Issue is to include works containing new and significant original results in the topics specified above, but also a limited number of exceptional survey papers. We will select and accept only high-quality papers, written and organized impeccably, including significant examples and applications.

The research topics include, but are not limited to, the following:

•  First- and higher-order Finslerians, Lagrangians, and Hamiltonians;
•  Geometric theory of foliations;
•  Nonholonomic spaces;
•  Geometric symmetries;
•  Geometric methods and differential equations;
•  Induced structures on submanifolds;
•  Algebroids, groupoids and generalizations;
•  Discretization methods in geometry.

Prof. Paul Popescu
Assoc. Prof. Marcela Popescu
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, . Manuscripts can be submitted until the deadline. All papers will be 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. Mathematics 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 1200 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.

Published Papers (4 papers)

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Research

Open AccessArticle
On Singular Distributions With Statistical Structure
by , and
Mathematics 2020, 8(10), 1825; - 17 Oct 2020
Abstract
In this paper, we extend our previous study regarding a Riemannian manifold endowed with a singular (or regular) distribution, generalizing Bochner’s technique and a statistical structure. Following the construction of an almost Lie algebroid, we define the central concept of the paper: The [...] Read more.
In this paper, we extend our previous study regarding a Riemannian manifold endowed with a singular (or regular) distribution, generalizing Bochner’s technique and a statistical structure. Following the construction of an almost Lie algebroid, we define the central concept of the paper: The Weitzenböck type curvature operator on tensors, prove the Bochner–Weitzenböck type formula and obtain some vanishing results about the null space of the Hodge type Laplacian on a distribution. Full article
(This article belongs to the Special Issue Geometric Methods and their Applications)
Open AccessArticle
Quaternionic Product of Equilateral Hyperbolas and Some Extensions
by and
Mathematics 2020, 8(10), 1686; - 01 Oct 2020
Abstract
This note concerns a product of equilateral hyperbolas induced by the quaternionic product considered in a projective manner. Several properties of this composition law are derived and, in this way, we arrive at some special numbers as roots or powers of unit. Using [...] Read more.
This note concerns a product of equilateral hyperbolas induced by the quaternionic product considered in a projective manner. Several properties of this composition law are derived and, in this way, we arrive at some special numbers as roots or powers of unit. Using the algebra of octonions, we extend this product to oriented equilateral hyperbolas and to pairs of equilateral hyperbolas. Using an inversion we extend this product to Bernoulli lemniscates and q-lemniscates. Finally, we extend this product to a set of conics. Three applications of the given products are proposed. Full article
(This article belongs to the Special Issue Geometric Methods and their Applications)
Open AccessArticle
Riemannian Structures on Z 2 n -Manifolds
by and
Mathematics 2020, 8(9), 1469; - 01 Sep 2020
Abstract
Very loosely, $Z2n$-manifolds are ‘manifolds’ with $Z2n$-graded coordinates and their sign rule is determined by the scalar product of their $Z2n$-degrees. A little more carefully, such objects can be understood within a sheaf-theoretical framework, [...] Read more.
Very loosely, $Z2n$-manifolds are ‘manifolds’ with $Z2n$-graded coordinates and their sign rule is determined by the scalar product of their $Z2n$-degrees. A little more carefully, such objects can be understood within a sheaf-theoretical framework, just as supermanifolds can, but with subtle differences. In this paper, we examine the notion of a Riemannian $Z2n$-manifold, i.e., a $Z2n$-manifold equipped with a Riemannian metric that may carry non-zero $Z2n$-degree. We show that the basic notions and tenets of Riemannian geometry directly generalize to the setting of $Z2n$-geometry. For example, the Fundamental Theorem holds in this higher graded setting. We point out the similarities and differences with Riemannian supergeometry. Full article
(This article belongs to the Special Issue Geometric Methods and their Applications)
Open AccessArticle
Influence of Geometric Equations in Mixed Problem of Porous Micromorphic Bodies with Microtemperature
by and
Mathematics 2020, 8(8), 1386; - 18 Aug 2020
Abstract
The study of the mixed initial-boundary value problem, corresponding to the thermoelasticity of porous micromorphic materials under the influence of microtemperatures, represents the main objective of this article. Achieving qualitative results on the existence, uniqueness and continuous dependence on the initial data and [...] Read more.
The study of the mixed initial-boundary value problem, corresponding to the thermoelasticity of porous micromorphic materials under the influence of microtemperatures, represents the main objective of this article. Achieving qualitative results on the existence, uniqueness and continuous dependence on the initial data and loads, of the solution of the mixed problem, implies a new perspective of approaching these topics, imposed by the large number of unknowns, which increases the complexity of equations and conditions that characterize the thermoelastic porous micromorphic materials with microtemperatures. The use of the semigroup theory of operators represents the optimal solution for deducing these results, the theory being adaptable to the requirements of the demonstrations, the mixed problem turning into a problem of Cauchy type, with regards to an equation of evolution on a Hilbert space, chosen appropriately. Full article
(This article belongs to the Special Issue Geometric Methods and their Applications)