New Advances in Differential Geometry and Optimizations on Manifolds

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: closed (31 August 2022) | Viewed by 9638

Special Issue Editors

Department of Mathematics-Informatics, Faculty of Applied Sciences, University Politehnica of Bucharest, Splaiul Independentei 313, 060042 Bucharest, Romania
Interests: differential geometry; optimizations on Riemannian manifolds; magnetic dynamical systems; geometric dynamics; multi-time optimal control
Special Issues, Collections and Topics in MDPI journals
Department of Mathematics, University of Haifa, Mount Carmel, Haifa 3498838, Israel
Interests: smooth manifold; submanifold; foliation; metric structure; curvature; tensor
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Mathematics has created this Special Issue to form collections of papers on “New Advances in Differential Geometry and Optimizations on Manifolds”. The aim is to build a community of authors and readers to discuss the latest research and develop new ideas and research directions. This Special Issue focuses on the interface of two traditional disciplines: differential geometry and optimization. We notice a surge in interest of such topics at recent conferences: The XIV International Conference of Differential Geometry and Dynamical Systems (DGDS-2020), 27–29 August 2020, University Politehnica of Bucharest, Bucharest, Romania; ICDGA 2021: 15. International Conference on Differential Geometry and Applications, 7–8 October 2021, New York, United States. The topics of the Special Issue are addressed by many doctoral schools around the world:  University Politehnica of Bucharest, Stanford University, Universita degli Studi di Bergamo, Universidad de Granada, etc.

Prof. Constantin Udriste
Prof. Dr. Vladimir Rovenski
Guest Editors

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Keywords

  • Riccati PDEs for flatness
  • Geometric nonlinear PDEs
  • Submanifolds
  • Adapted metrics and connections
  • Contact structures
  • Geometric dynamical systems
  • Optimization methods on manifolds
  • Constrained optimization
  • Dynamic optimization
  • Variational calculus
  • Optimal control
  • Optimal bounds and extremal manifolds
  • Operation research
  • Multivariate approximation
  • Symmetries

Published Papers (6 papers)

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Research

12 pages, 304 KiB  
Article
Solutions for Multitime Reaction–Diffusion PDE
by Cristian Ghiu and Constantin Udriste
Mathematics 2022, 10(19), 3623; https://0-doi-org.brum.beds.ac.uk/10.3390/math10193623 - 03 Oct 2022
Viewed by 921
Abstract
A previous paper by our research group introduced the nonlinear multitime reaction–diffusion PDE (with oblique derivative) as a generalized version of the single-time model. This paper states and uses some hypotheses that allow the finding of some important explicit families of the exact [...] Read more.
A previous paper by our research group introduced the nonlinear multitime reaction–diffusion PDE (with oblique derivative) as a generalized version of the single-time model. This paper states and uses some hypotheses that allow the finding of some important explicit families of the exact solutions for multitime reaction–diffusion PDEs of any dimension that have a multitemporal directional derivative term. Some direct methods for determining the exact solutions of nonlinear PDEs from mathematical physics are presented. In the single-time case, our methods present many advantages in comparison with other known approaches. Particularly, we obtained classes of ODEs and classes of PDEs whose solutions generate solutions of the multitime reaction–diffusion PDE. Full article
(This article belongs to the Special Issue New Advances in Differential Geometry and Optimizations on Manifolds)
11 pages, 285 KiB  
Article
General Relativistic Space-Time with η1-Einstein Metrics
by Yanlin Li, Fatemah Mofarreh, Santu Dey, Soumendu Roy and Akram Ali
Mathematics 2022, 10(14), 2530; https://0-doi-org.brum.beds.ac.uk/10.3390/math10142530 - 21 Jul 2022
Cited by 21 | Viewed by 1451
Abstract
The present research paper consists of the study of an η1-Einstein soliton in general relativistic space-time with a torse-forming potential vector field. Besides this, we try to evaluate the characterization of the metrics when the space-time with a semi-symmetric energy-momentum tensor [...] Read more.
The present research paper consists of the study of an η1-Einstein soliton in general relativistic space-time with a torse-forming potential vector field. Besides this, we try to evaluate the characterization of the metrics when the space-time with a semi-symmetric energy-momentum tensor admits an η1-Einstein soliton, whose potential vector field is torse-forming. In adition, certain curvature conditions on the space-time that admit an η1-Einstein soliton are explored and build up the importance of the Laplace equation on the space-time in terms of η1-Einstein soliton. Lastly, we have given some physical accomplishment with the connection of dust fluid, dark fluid and radiation era in general relativistic space-time admitting an η1-Einstein soliton. Full article
(This article belongs to the Special Issue New Advances in Differential Geometry and Optimizations on Manifolds)
10 pages, 259 KiB  
Article
Explicit Information Geometric Calculations of the Canonical Divergence of a Curve
by Domenico Felice and Carlo Cafaro
Mathematics 2022, 10(9), 1452; https://0-doi-org.brum.beds.ac.uk/10.3390/math10091452 - 26 Apr 2022
Viewed by 1138
Abstract
Information geometry concerns the study of a dual structure (g,,*) upon a smooth manifold M. Such a geometry is totally encoded within a potential function usually referred to as a divergence or contrast function of [...] Read more.
Information geometry concerns the study of a dual structure (g,,*) upon a smooth manifold M. Such a geometry is totally encoded within a potential function usually referred to as a divergence or contrast function of (g,,*). Even though infinitely many divergences induce on M the same dual structure, when the manifold is dually flat, a canonical divergence is well defined and was originally introduced by Amari and Nagaoka. In this pedagogical paper, we present explicit non-trivial differential geometry-based proofs concerning the canonical divergence for a special type of dually flat manifold represented by an arbitrary 1-dimensional path γ. Highlighting the geometric structure of such a particular canonical divergence, our study could suggest a way to select a general canonical divergence by using the information from a general dual structure in a minimal way. Full article
(This article belongs to the Special Issue New Advances in Differential Geometry and Optimizations on Manifolds)
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22 pages, 1033 KiB  
Article
Riemannian Formulation of Pontryagin’s Maximum Principle for the Optimal Control of Robotic Manipulators
by Juan Antonio Rojas-Quintero, François Dubois and Hedy César Ramírez-de-Ávila
Mathematics 2022, 10(7), 1117; https://0-doi-org.brum.beds.ac.uk/10.3390/math10071117 - 30 Mar 2022
Cited by 2 | Viewed by 1950
Abstract
In this work, we consider robotic systems for which the mass tensor is identified to be the metric in a Riemannian manifold. Cost functional invariance is achieved by constructing it with the identified metric. Optimal control evolution is revealed in the form of [...] Read more.
In this work, we consider robotic systems for which the mass tensor is identified to be the metric in a Riemannian manifold. Cost functional invariance is achieved by constructing it with the identified metric. Optimal control evolution is revealed in the form of a covariant second-order ordinary differential equation featuring the Riemann curvature tensor that constrains the control variable. In Pontryagin’s framework of the maximum principle, the cost functional has a direct impact on the system Hamiltonian. It is regarded as the performance index, and optimal control variables are affected by this fundamental choice. In the present context of cost functional invariance, we show that the adjoint variables are the first-order representation of the second-order control variable evolution equation. It is also shown that adding supplementary invariant terms to the cost functional does not modify the basic structure of the optimal control covariant evolution equation. Numerical trials show that the proposed invariant cost functionals, as compared to their non-invariant versions, lead to lower joint power consumption and narrower joint angular amplitudes during motion. With our formulation, the differential equations solver gains stability and operates dramatically faster when compared to examples where cost functional invariance is not considered. Full article
(This article belongs to the Special Issue New Advances in Differential Geometry and Optimizations on Manifolds)
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15 pages, 295 KiB  
Article
Classification of Holomorphic Functions as Pólya Vector Fields via Differential Geometry
by Lucian-Miti Ionescu, Cristina-Liliana Pripoae and Gabriel-Teodor Pripoae
Mathematics 2021, 9(16), 1890; https://0-doi-org.brum.beds.ac.uk/10.3390/math9161890 - 09 Aug 2021
Cited by 2 | Viewed by 1476
Abstract
We review Pólya vector fields associated to holomorphic functions as an important pedagogical tool for making the complex integral understandable to the students, briefly mentioning its use in other dimensions. Techniques of differential geometry are then used to refine the study of holomorphic [...] Read more.
We review Pólya vector fields associated to holomorphic functions as an important pedagogical tool for making the complex integral understandable to the students, briefly mentioning its use in other dimensions. Techniques of differential geometry are then used to refine the study of holomorphic functions from a metric (Riemannian), affine differential or differential viewpoint. We prove that the only nontrivial holomorphic functions, whose Pólya vector field is torse-forming in the cannonical geometry of the plane, are the special Möbius transformations of the form f(z)=b(z+d)1. We define and characterize several types of affine connections, related to the parallelism of Pólya vector fields. We suggest a program for the classification of holomorphic functions, via these connections, based on the various indices of nullity of their curvature and torsion tensor fields. Full article
(This article belongs to the Special Issue New Advances in Differential Geometry and Optimizations on Manifolds)
14 pages, 278 KiB  
Article
Invariant Geometric Curvilinear Optimization with Restricted Evolution Dynamics
by Andreea Bejenaru
Mathematics 2021, 9(8), 802; https://0-doi-org.brum.beds.ac.uk/10.3390/math9080802 - 07 Apr 2021
Viewed by 1047
Abstract
This paper begins with a geometric statement of constraint optimization problems, which include both equality and inequality-type restrictions. The cost to optimize is a curvilinear functional defined by a given differential one-form, while the optimal state to be determined is a differential curve [...] Read more.
This paper begins with a geometric statement of constraint optimization problems, which include both equality and inequality-type restrictions. The cost to optimize is a curvilinear functional defined by a given differential one-form, while the optimal state to be determined is a differential curve connecting two given points, among all the curves satisfying some given primal feasibility conditions. The resulting outcome is an invariant curvilinear Fritz–John maximum principle. Afterward, this result is approached by means of parametric equations. The classical single-time Pontryagin maximum principle for curvilinear cost functionals is revealed as a consequence. Full article
(This article belongs to the Special Issue New Advances in Differential Geometry and Optimizations on Manifolds)
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