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15 pages, 309 KiB

Open AccessArticle

Existence of Solutions for a Perturbed N-Laplacian Boundary Value Problem with Critical Growth

by Sheng Shi and Yang Yang

Axioms 2024, 13(11), 733; https://doi.org/10.3390/axioms13110733 - 23 Oct 2024

Viewed by 414

In this paper, we investigate a perturbed elliptic boundary value problem that exhibits critical growth characterized by a Trudinger–Moser-type inequality. Our primary focus is to establish the existence of two nontrivial solutions. This is achieved by employing a combination of the Trudinger–Moser-type inequality [...] Read more.

In this paper, we investigate a perturbed elliptic boundary value problem that exhibits critical growth characterized by a Trudinger–Moser-type inequality. Our primary focus is to establish the existence of two nontrivial solutions. This is achieved by employing a combination of the Trudinger–Moser-type inequality and a linking theorem based on the

Z_{2}

-cohomological index. The main feature and novelty of this paper lies in extending the equation to N-Laplacian boundary value problems utilizing the aforementioned methods. This extension not only broadens the applicability of these techniques but also enriches the research outcomes in the field of nonlinear analysis. Full article

(This article belongs to the Special Issue Differential Equations and Related Topics, 2nd Edition)

28 pages, 407 KiB

Open AccessArticle

Continuity Equation of Transverse Kähler Metrics on Sasakian Manifolds

by Yushuang Fan and Tao Zheng

Mathematics 2024, 12(19), 3132; https://doi.org/10.3390/math12193132 - 7 Oct 2024

Viewed by 538

Abstract

We introduce the continuity equation of transverse Kähler metrics on Sasakian manifolds and establish its interval of maximal existence. When the first basic Chern class is null (resp. negative), we prove that the solution of the (resp. normalized) continuity equation converges smoothly to [...] Read more.

We introduce the continuity equation of transverse Kähler metrics on Sasakian manifolds and establish its interval of maximal existence. When the first basic Chern class is null (resp. negative), we prove that the solution of the (resp. normalized) continuity equation converges smoothly to the unique

η

-Einstein metric in the basic Bott–Chern cohomological class of the initial transverse Kähler metric (resp. first basic Chern class). These results are the transverse version of the continuity equation of the Kähler metrics studied by La Nave and Tian, and also counterparts of the Sasaki–Ricci flow studied by Smoczyk, Wang, and Zhang. Full article

(This article belongs to the Special Issue Complex Analysis and Geometric Function Theory, 2nd Edition)

17 pages, 296 KiB

Open AccessArticle

Crossed Modules and Non-Abelian Extensions of Differential Leibniz Conformal Algebras

by Hui Wu, Shuangjian Guo and Xiaohui Zhang

Axioms 2024, 13(10), 685; https://doi.org/10.3390/axioms13100685 - 2 Oct 2024

Viewed by 380

Abstract

In this paper, we introduce two-term differential

L e i b_{\infty}

-conformal algebras and give characterizations of some particular classes of such two-term differential

L e i b_{\infty}

-conformal algebras. Furthermore, we discuss the classification of the non-Abelian extensions in terms [...] Read more.

In this paper, we introduce two-term differential

L e i b_{\infty}

-conformal algebras and give characterizations of some particular classes of such two-term differential

L e i b_{\infty}

-conformal algebras. Furthermore, we discuss the classification of the non-Abelian extensions in terms of non-Abelian cohomology groups. Finally, we explore the inducibility of pairs of automorphisms and derive the analog Wells exact sequences under the circumstance of differential Leibniz conformal algebras. Full article

(This article belongs to the Special Issue Computational Algebra, Coding Theory and Cryptography: Theory and Applications)

32 pages, 437 KiB

Open AccessArticle

The Dirac-Dolbeault Operator Approach to the Hodge Conjecture

by Simone Farinelli

Symmetry 2024, 16(10), 1291; https://doi.org/10.3390/sym16101291 - 1 Oct 2024

Viewed by 1726

Abstract

The Dirac-Dolbeault operator for a compact Kähler manifold is a special case of Dirac operator. The Green function for the Dirac Laplacian over a Riemannian manifold with boundary allows the expression of the values of the sections of the Dirac bundle in terms [...] Read more.

The Dirac-Dolbeault operator for a compact Kähler manifold is a special case of Dirac operator. The Green function for the Dirac Laplacian over a Riemannian manifold with boundary allows the expression of the values of the sections of the Dirac bundle in terms of the values on the boundary, extending the mean value theorem of harmonic analysis. Utilizing this representation and the Nash–Moser generalized inverse function theorem, we prove the existence of complex submanifolds of a complex projective manifold satisfying globally a certain partial differential equation under a certain injectivity assumption. Thereby, internal symmetries of Dolbeault and rational Hodge cohomologies play a crucial role. Next, we show the existence of complex submanifolds whose fundamental classes span the rational Hodge classes, proving the Hodge conjecture for complex projective manifolds. Full article

(This article belongs to the Section Mathematics)

4 pages, 210 KiB

Open AccessCorrection

Correction: Bychkov et al. The σ₋ Cohomology Analysis for Symmetric Higher-Spin Fields. Symmetry 2021, 13, 1498

by Alexey S. Bychkov, Kirill A. Ushakov and Mikhail A. Vasiliev

Symmetry 2024, 16(9), 1115; https://doi.org/10.3390/sym16091115 - 28 Aug 2024

Viewed by 330

Abstract

There was an error in the original publication [...] Full article

(This article belongs to the Section Physics)

7 pages, 241 KiB

Open AccessArticle

Geometry of Torsion Gerbes and Flat Twisted Vector Bundles

by Byungdo Park

Axioms 2024, 13(8), 504; https://doi.org/10.3390/axioms13080504 - 26 Jul 2024

Viewed by 449

Abstract

Gerbes and higher gerbes are geometric cocycles representing higher degree cohomology classes, and are attracting considerable interest in differential geometry and mathematical physics. We prove that a 2-gerbe has a torsion Dixmier–Douady class if and only if the gerbe has locally constant cocycle [...] Read more.

Gerbes and higher gerbes are geometric cocycles representing higher degree cohomology classes, and are attracting considerable interest in differential geometry and mathematical physics. We prove that a 2-gerbe has a torsion Dixmier–Douady class if and only if the gerbe has locally constant cocycle data. As an application, we give an alternative description of flat twisted vector bundles in terms of locally constant transition maps. These results generalize to n-gerbes for

n = 1

and

n \geq 3

, providing insights into the structure of higher gerbes and their applications to the geometry of twisted vector bundles. Full article

(This article belongs to the Special Issue Differential Geometry and Its Application, 2nd Edition)

17 pages, 310 KiB

Open AccessArticle

Cohomology and Crossed Modules of Modified Rota–Baxter Pre-Lie Algebras

by Fuyang Zhu and Wen Teng

Mathematics 2024, 12(14), 2260; https://doi.org/10.3390/math12142260 - 19 Jul 2024

Viewed by 749

Abstract

The goal of the present paper is to provide a cohomology theory and crossed modules of modified Rota–Baxter pre-Lie algebras. We introduce the notion of a modified Rota–Baxter pre-Lie algebra and its bimodule. We define a cohomology of modified Rota–Baxter pre-Lie algebras with [...] Read more.

The goal of the present paper is to provide a cohomology theory and crossed modules of modified Rota–Baxter pre-Lie algebras. We introduce the notion of a modified Rota–Baxter pre-Lie algebra and its bimodule. We define a cohomology of modified Rota–Baxter pre-Lie algebras with coefficients in a suitable bimodule. Furthermore, we study the infinitesimal deformations and abelian extensions of modified Rota–Baxter pre-Lie algebras and relate them with the second cohomology groups. Finally, we investigate skeletal and strict modified Rota–Baxter pre-Lie 2-algebras. We show that skeletal modified Rota–Baxter pre-Lie 2-algebras can be classified into the third cohomology group, and strict modified Rota–Baxter pre-Lie 2-algebras are equivalent to the crossed modules of modified Rota–Baxter pre-Lie algebras. Full article

(This article belongs to the Special Issue Advanced Research in Pure and Applied Algebra)

28 pages, 999 KiB

Open AccessArticle

Applications of Differential Geometry Linking Topological Bifurcations to Chaotic Flow Fields

by Peter D. Neilson and Megan D. Neilson

AppliedMath 2024, 4(2), 763-790; https://doi.org/10.3390/appliedmath4020041 - 15 Jun 2024

Viewed by 835

Abstract

At every point

p

on a smooth

n

-manifold M there exist

(n + 1)

skew-symmetric tensor spaces spanning differential

r

-forms

ω

with

r = 0, 1, \dots, n

. Because

d \circ d

is always zero where

d

[...] Read more.

At every point

p

on a smooth

n

-manifold M there exist

(n + 1)

skew-symmetric tensor spaces spanning differential

r

-forms

ω

with

r = 0, 1, \dots, n

. Because

d \circ d

is always zero where

d

is the exterior differential, it follows that every exact

r

-form (i.e.,

ω = d λ

where

λ

is an

(r - 1)

-form) is closed (i.e.,

d ω = 0

) but not every closed

r

-form is exact. This implies the existence of a third type of differential

r

-form that is closed but not exact. Such forms are called harmonic forms. Every smooth

n

-manifold has an underlying topological structure. Many different possible topological structures exist. What distinguishes one topological structure from another is the number of holes of various dimensions it possesses. De Rham’s theory of differential forms relates the presence of

r

-dimensional holes in the underlying topology of a smooth

n

-manifold

M

to the presence of harmonic

r

-form fields on the smooth manifold. A large amount of theory is required to understand de Rham’s theorem. In this paper we summarize the differential geometry that links holes in the underlying topology of a smooth manifold with harmonic fields on the manifold. We explore the application of de Rham’s theory to (i) visual, (ii) mechanical, (iii) electrical and (iv) fluid flow systems. In particular, we consider harmonic flow fields in the intracellular aqueous solution of biological cells and we propose, on mathematical grounds, a possible role of harmonic flow fields in the folding of protein polypeptide chains. Full article

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14 pages, 289 KiB

Open AccessArticle

Abelian Extensions of Modified λ-Differential Left-Symmetric Algebras and Crossed Modules

by Fuyang Zhu, Taijie You and Wen Teng

Axioms 2024, 13(6), 380; https://doi.org/10.3390/axioms13060380 - 4 Jun 2024

Viewed by 664

Abstract

In this paper, we define a cohomology theory of a modified

λ

-differential left-symmetric algebra. Moreover, we introduce the notion of modified

λ

-differential left-symmetric 2-algebras, which is the categorization of a modified

λ

-differential left-symmetric algebra. As applications of cohomology, we classify [...] Read more.

In this paper, we define a cohomology theory of a modified

λ

-differential left-symmetric algebra. Moreover, we introduce the notion of modified

λ

-differential left-symmetric 2-algebras, which is the categorization of a modified

λ

-differential left-symmetric algebra. As applications of cohomology, we classify linear deformations and abelian extensions of modified

λ

-differential left-symmetric algebras using the second cohomology group and classify skeletal modified

λ

-differential left-symmetric 2-algebra using the third cohomology group. Finally, we show that strict modified

λ

-differential left-symmetric 2-algebras are equivalent to crossed modules of modified

λ

-differential left-symmetric algebras. Full article

(This article belongs to the Section Algebra and Number Theory)

9 pages, 253 KiB

Open AccessArticle

The de Rham Cohomology Classes of Hemi-Slant Submanifolds in Locally Product Riemannian Manifolds

by Mustafa Gök and Erol Kılıç

Mathematics 2024, 12(11), 1730; https://doi.org/10.3390/math12111730 - 2 Jun 2024

Viewed by 469

Abstract

This paper aims to discuss the de Rham cohomology of hemi-slant submanifolds in locally product Riemannian manifolds. The integrability and geodesical invariance conditions of the distributions derived from the definition of a hemi-slant submanifold are given. The existence and non-triviality of de Rham [...] Read more.

This paper aims to discuss the de Rham cohomology of hemi-slant submanifolds in locally product Riemannian manifolds. The integrability and geodesical invariance conditions of the distributions derived from the definition of a hemi-slant submanifold are given. The existence and non-triviality of de Rham cohomology classes of hemi-slant submanifolds are investigated. Finally, an example is presented. Full article

(This article belongs to the Special Issue Differentiable Manifolds and Geometric Structures)

12 pages, 244 KiB

Open AccessArticle

A Classification of Compact Cohomogeneity One Locally Conformal Kähler Manifolds

by Daniel Guan

Mathematics 2024, 12(11), 1710; https://doi.org/10.3390/math12111710 - 30 May 2024

Viewed by 746

Abstract

In this paper, we apply a result of the classification of a compact cohomogeneity one Riemannian manifold with a compact Lie group G to obtain a classification of compact cohomogeneity one locally conformal Kähler manifolds. In particular, we prove that the compact complex [...] Read more.

In this paper, we apply a result of the classification of a compact cohomogeneity one Riemannian manifold with a compact Lie group G to obtain a classification of compact cohomogeneity one locally conformal Kähler manifolds. In particular, we prove that the compact complex manifold is a complex one-dimensional torus bundle over a projective rational homogeneous, or cohomogeneity one manifold except of a class of manifolds with a generalized Hopf surface bundle over a projective rational homogeneous space. Additionally, it is a homogeneous compact complex manifold under the complexification

G^{C}

of the given compact Lie group G under an extra condition that the related closed one form is cohomologous to zero on the generic G orbit. Moreover, the semi-simple part S of the Lie group action has hypersurface orbits, i.e., it is of cohomogeneity one with respect to the semi-simple Lie group S in that special case. Full article

15 pages, 347 KiB

Open AccessArticle

In Pursuit of BRST Symmetry and Observables in 4D Topological Gauge-Affine Gravity

by Oussama Abdelghafour Belarbi and Ahmed Meziane

Symmetry 2024, 16(5), 528; https://doi.org/10.3390/sym16050528 - 28 Apr 2024

Viewed by 823

Abstract

The realization of a BRST cohomology of the 4D topological gauge-affine gravity is established in terms of a superconnection formalism. The identification of fields in the quantized theory occurs directly as is usual in terms of superconnection and its supercurvature components with the [...] Read more.

The realization of a BRST cohomology of the 4D topological gauge-affine gravity is established in terms of a superconnection formalism. The identification of fields in the quantized theory occurs directly as is usual in terms of superconnection and its supercurvature components with the double covering of the general affine group

\bar{G A} (4, R)

. Then, by means of an appropriate decomposition of the metalinear double-covering group

\bar{S L} (5, R)

with respect to the general linear double-covering group

\bar{G L} (4, R)

, one can easily obtain the enlargements of the fields while remaining consistent with the BRST algebra. This leads to the descent equations, allowing us to build the observables of the theory by means of the BRST algebra constructed using a

\bar{s a} (5, R)

algebra-valued superconnection. In particular, we discuss the construction of topological invariants with torsion. Full article

(This article belongs to the Special Issue Symmetries in Gravity Research: Classical and Quantum)

19 pages, 310 KiB

Open AccessFeature PaperArticle

The Gauge Equation in Statistical Manifolds: An Approach through Spectral Sequences

by Michel Nguiffo Boyom and Stephane Puechmorel

Mathematics 2024, 12(8), 1177; https://doi.org/10.3390/math12081177 - 14 Apr 2024

Viewed by 1060

Abstract

The gauge equation is a generalization of the conjugacy relation for the Koszul connection to bundle morphisms that are not isomorphisms. The existence of nontrivial solution to this equation, especially when duality is imposed upon related connections, provides important information about the geometry [...] Read more.

The gauge equation is a generalization of the conjugacy relation for the Koszul connection to bundle morphisms that are not isomorphisms. The existence of nontrivial solution to this equation, especially when duality is imposed upon related connections, provides important information about the geometry of the manifolds under consideration. In this article, we use the gauge equation to introduce spectral sequences that are further specialized to Hessian structures. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Its Applications)

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17 pages, 282 KiB

Open AccessArticle

Maps on the Mirror Heisenberg–Virasoro Algebra

by Xuelian Guo, Ivan Kaygorodov and Liming Tang

Mathematics 2024, 12(6), 802; https://doi.org/10.3390/math12060802 - 8 Mar 2024

Viewed by 718

Abstract

Using the first cohomology from the mirror Heisenberg–Virasoro algebra to the twisted Heisenberg algebra (as the mirror Heisenberg–Virasoro algebra module), in this paper, we determined the derivations on the mirror Heisenberg–Virasoro algebra. Based on this result, we proved that any two-local derivation on [...] Read more.

Using the first cohomology from the mirror Heisenberg–Virasoro algebra to the twisted Heisenberg algebra (as the mirror Heisenberg–Virasoro algebra module), in this paper, we determined the derivations on the mirror Heisenberg–Virasoro algebra. Based on this result, we proved that any two-local derivation on the mirror Heisenberg–Virasoro algebra is a derivation. All half-derivations are described, and as corollaries, we have descriptions of transposed Poisson structures and local (two-local) half-derivations on the mirror Heisenberg–Virasoro algebra. Full article

(This article belongs to the Section Algebra, Geometry and Topology)

16 pages, 303 KiB

Open AccessArticle

Generalized Reynolds Operators on Hom-Lie Triple Systems

by Yunpeng Xiao, Wen Teng and Fengshan Long

Symmetry 2024, 16(3), 262; https://doi.org/10.3390/sym16030262 - 21 Feb 2024

Viewed by 1467

Abstract

In this paper, we first introduce the notion of generalized Reynolds operators on Hom-Lie triple systems associated to a representation and a 3-cocycle. Then, we develop a cohomology of generalized Reynolds operators on Hom-Lie triple systems. As applications, we use the first cohomology [...] Read more.

In this paper, we first introduce the notion of generalized Reynolds operators on Hom-Lie triple systems associated to a representation and a 3-cocycle. Then, we develop a cohomology of generalized Reynolds operators on Hom-Lie triple systems. As applications, we use the first cohomology group to classify linear deformations and we study the obstruction class of an extendable order n deformation. Finally, we introduce and investigate Hom-NS-Lie triple system as the underlying structure of generalized Reynolds operators on Hom-Lie triple systems. Full article

(This article belongs to the Section Mathematics)

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