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20 pages, 380 KB

Open AccessReview

Integral Formulas for Almost Product Manifolds and Foliations

by Vladimir Rovenski

Mathematics 2022, 10(19), 3645; https://doi.org/10.3390/math10193645 - 5 Oct 2022

Cited by 2 | Viewed by 1818

Integral formulas are powerful tools used to obtain global results in geometry and analysis. The integral formulas for almost multi-product manifolds, foliations and multiply twisted products of Riemannian, metric-affine and sub-Riemannian manifolds, to which this review paper is devoted, are useful for studying such problems as (i) the existence and characterization of foliations with a given geometric property, such as being totally geodesic, minimal or totally umbilical; (ii) prescribing the generalized mean curvatures of the leaves of a foliation; (iii) minimizing volume-like functionals defined for tensors on foliated manifolds. We start from the series of integral formulas for codimension one foliations of Riemannian and metric-affine manifolds, and then we consider integral formulas for regular and singular foliations of arbitrary codimension. In the second part of the article, we represent integral formulas with the mixed scalar curvature of an almost multi-product structure on Riemannian and metric-affine manifolds, give applications to hypersurfaces of space forms with

k = 2, 3

distinct principal curvatures of constant multiplicities and then discuss integral formulas for foliations or distributions on sub-Riemannian manifolds. Full article

(This article belongs to the Section E: Applied Mathematics)

23 pages, 2336 KB

Open AccessArticle

FPGA-Based Implementation of an Optimization Algorithm to Maximize the Productivity of a Microbial Electrolysis Cell

by José de Jesús Colín-Robles, Ixbalank Torres-Zúñiga, Mario A. Ibarra-Manzano and Víctor Alcaraz-González

Processes 2021, 9(7), 1111; https://doi.org/10.3390/pr9071111 - 25 Jun 2021

Cited by 7 | Viewed by 3490

Abstract

In this work, the design of the hardware architecture to implement an algorithm for optimizing the Hydrogen Productivity Rate (HPR) in a Microbial Electrolysis Cell (MEC) is presented. The HPR in the MEC is maximized by the golden section search algorithm in conjunction with a super-twisting controller. The development of the digital architecture in the implementation step of the optimization algorithm was developed in the Very High Description Language (VHDL) and synthesized in a Field Programmable Gate Array (FPGA). Numerical simulations demonstrated the feasibility of the proposed optimization strategy embedded in an FPGA Cyclone II. Results showed that only

21 %

of the total logic elements,

5.19 %

of dedicated logic registers, and

64 %

of the total eight-bits multipliers of the FPGA were used. On the other hand, the estimated power consumption required by the FPGA-embedded optimization algorithm was only 146 mW. Full article

(This article belongs to the Special Issue Mathematical Modeling and Control of Bioprocesses)

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11 pages, 284 KB

Open AccessArticle

On a Metric Affine Manifold with Several Orthogonal Complementary Distributions

by Vladimir Rovenski and Sergey E. Stepanov

Mathematics 2021, 9(3), 229; https://doi.org/10.3390/math9030229 - 25 Jan 2021

Cited by 2 | Viewed by 2075

Abstract

A Riemannian manifold endowed with

k > 2

A Riemannian manifold endowed with

k > 2

orthogonal complementary distributions (called here an almost multi-product structure) appears in such topics as multiply twisted or warped products and the webs or nets composed of orthogonal foliations. In this article, we define the mixed scalar curvature of an almost multi-product structure endowed with a linear connection, and represent this kind of curvature using fundamental tensors of distributions and the divergence of a geometrically interesting vector field. Using this formula, we prove decomposition and non-existence theorems and integral formulas that generalize results (for

k = 2

) on almost product manifolds with the Levi-Civita connection. Some of our results are illustrated by examples with statistical and semi-symmetric connections. Full article

(This article belongs to the Special Issue Differential Geometry: Theory and Applications)

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