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19 pages, 657 KiB

Open AccessArticle

Study of Random Walk Invariants for Spiro-Ring Network Based on Laplacian Matrices

by Yasir Ahmad, Umar Ali, Daniele Ettore Otera and Xiang-Feng Pan

Mathematics 2024, 12(9), 1309; https://doi.org/10.3390/math12091309 - 25 Apr 2024

Viewed by 229

Abstract

The use of the global mean first-passage time (GMFPT) in random walks on networks has been widely explored in the field of statistical physics, both in theory and practical applications. The GMFPT is the estimated interval of time needed to reach a state [...] Read more.

The use of the global mean first-passage time (GMFPT) in random walks on networks has been widely explored in the field of statistical physics, both in theory and practical applications. The GMFPT is the estimated interval of time needed to reach a state j in a system from a starting state i. In contrast, there exists an intrinsic measure for a stochastic process, known as Kemeny’s constant, which is independent of the initial state. In the literature, it has been used as a measure of network efficiency. This article deals with a graph-spectrum-based method for finding both the GMFPT and Kemeny’s constant of random walks on spiro-ring networks (that are organic compounds with a particular graph structure). Furthermore, we calculate the Laplacian matrix for some specific spiro-ring networks using the decomposition theorem of Laplacian polynomials. Moreover, using the coefficients and roots of the resulting matrices, we establish some formulae for both GMFPT and Kemeny’s constant in these spiro-ring networks. Full article

(This article belongs to the Special Issue Geometry and Topology with Applications)

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13 pages, 297 KiB

Open AccessFeature PaperArticle

A Characterization of Procyclic Groups via Complete Exterior Degree

by Bernardo G. Rodrigues and Francesco G. Russo

Mathematics 2024, 12(7), 1018; https://doi.org/10.3390/math12071018 - 28 Mar 2024

Viewed by 435

Abstract

We describe the nonabelian exterior square

G \hat{\land} G

of a pro-p-group G (with p arbitrary prime) in terms of quotients of free pro-p-groups, providing a new method of construction of

G \hat{\land} G

and new structural [...] Read more.

We describe the nonabelian exterior square

G \hat{\land} G

of a pro-p-group G (with p arbitrary prime) in terms of quotients of free pro-p-groups, providing a new method of construction of

G \hat{\land} G

and new structural results for

G \hat{\land} G

. Then, we investigate a generalization of the probability that two randomly chosen elements of G commute: this notion is known as the “complete exterior degree” of a pro-p-group and we will use it to characterize procyclic groups. Among other things, we present a new formula, which simplifies the numerical aspects which are connected with the evaluation of the complete exterior degree. Full article

(This article belongs to the Special Issue Geometry and Topology with Applications)

6 pages, 284 KiB

Open AccessArticle

On the Space of G-Permutation Degree of Some Classes of Topological Spaces

by Ljubiša D. R. Kočinac, Farkhod G. Mukhamadiev and Anvar K. Sadullaev

Mathematics 2023, 11(22), 4624; https://doi.org/10.3390/math11224624 - 12 Nov 2023

Viewed by 521

Abstract

In this paper, we study the space of G-permutation degree of some classes of topological spaces and the properties of the functor

{SP}_{G}^{n}

of G-permutation degree. In particular, we prove: (a) If a topological space X is developable, then [...] Read more.

In this paper, we study the space of G-permutation degree of some classes of topological spaces and the properties of the functor

{SP}_{G}^{n}

of G-permutation degree. In particular, we prove: (a) If a topological space X is developable, then so is

{SP}_{G}^{n} X

; (b) If X is a Moore space, then so is

{SP}_{G}^{n} X

; (c) If a topological space X is an

M_{1}

-space, then so is

{SP}_{G}^{n} X

; (d) If a topological space X is an

M_{2}

-space, then so is

{SP}_{G}^{n} X

. Full article

(This article belongs to the Special Issue Geometry and Topology with Applications)

11 pages, 258 KiB

Open AccessArticle

Ricci Vector Fields

by Hanan Alohali and Sharief Deshmukh

Mathematics 2023, 11(22), 4622; https://doi.org/10.3390/math11224622 - 12 Nov 2023

Cited by 1 | Viewed by 549

Abstract

We introduce a special vector field

ω

on a Riemannian manifold

(N^{m}, g)

, such that the Lie derivative of the metric g with respect to

ω

is equal to

ρ R i c

, where

R i c

[...] Read more.

We introduce a special vector field

ω

on a Riemannian manifold

(N^{m}, g)

, such that the Lie derivative of the metric g with respect to

ω

is equal to

ρ R i c

, where

R i c

is the Ricci tensor of

(N^{m}, g)

and

ρ

is a smooth function on

N^{m}

. We call this vector field a

ρ

-Ricci vector field. We use the

ρ

-Ricci vector field on a Riemannian manifold

(N^{m}, g)

and find two characterizations of the m-sphere

S^{m} (α)

. In the first result, we show that an m-dimensional compact and connected Riemannian manifold

(N^{m}, g)

with nonzero scalar curvature admits a

ρ

-Ricci vector field

ω

such that

ρ

is a nonconstant function and the integral of

R i c (ω, ω)

has a suitable lower bound that is necessary and sufficient for

(N^{m}, g)

to be isometric to m-sphere

S^{m} (α)

. In the second result, we show that an m-dimensional complete and simply connected Riemannian manifold

(N^{m}, g)

of positive scalar curvature admits a

ρ

-Ricci vector field

ω

such that

ρ

is a nontrivial solution of the Fischer–Marsden equation and the squared length of the covariant derivative of

ω

has an appropriate upper bound, if and only if

(N^{m}, g)

is isometric to m-sphere

S^{m} (α)

. Full article

(This article belongs to the Special Issue Geometry and Topology with Applications)

12 pages, 274 KiB

Open AccessArticle

On c-Compactness in Topological and Bitopological Spaces

by Rehab Alharbi, Jamal Oudetallah, Mutaz Shatnawi and Iqbal M. Batiha

Mathematics 2023, 11(20), 4251; https://doi.org/10.3390/math11204251 - 11 Oct 2023

Viewed by 635

Abstract

The primary goal of this research is to initiate the pairwise c-compact concept in topological and bitopological spaces. This would make us to define the concept of c-compact space with some of its generalization, and present some necessary notions such as [...] Read more.

The primary goal of this research is to initiate the pairwise c-compact concept in topological and bitopological spaces. This would make us to define the concept of c-compact space with some of its generalization, and present some necessary notions such as the H-closed, the quasi compact and extremely disconnected compact spaces in topological and bitopological spaces. As a consequence, we derive numerous theoretical results that demonstrate the relations between c-separation axioms and the c-compact spaces. Full article

(This article belongs to the Special Issue Geometry and Topology with Applications)

12 pages, 390 KiB

Open AccessArticle

A Differential Relation of Metric Properties for Orientable Smooth Surfaces in ℝ³

by Sungmin Ryu

Mathematics 2023, 11(10), 2337; https://doi.org/10.3390/math11102337 - 17 May 2023

Cited by 2 | Viewed by 1196

Abstract

The Gauss–Bonnet formula finds applications in various fundamental fields. Global or local analysis on the basis of this formula is possible only in integral form since the Gauss–Bonnet formula depends on the choice of a simple region of an orientable smooth surface S [...] Read more.

The Gauss–Bonnet formula finds applications in various fundamental fields. Global or local analysis on the basis of this formula is possible only in integral form since the Gauss–Bonnet formula depends on the choice of a simple region of an orientable smooth surface S. The objective of the present paper is to construct a differential relation of the metric properties concerned at a point on S. Pointwise analysis on S is possible through the differential relation, which is expected to provide new geometrical insights into existing studies where the Gauss–Bonnet formula is applied in integral form. Full article

(This article belongs to the Special Issue Geometry and Topology with Applications)

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7 pages, 2101 KiB

Open AccessArticle

by Ema Jurkin and Marija Šimić Horvath

Mathematics 2023, 11(7), 1562; https://doi.org/10.3390/math11071562 - 23 Mar 2023

Viewed by 975

Abstract

The notion of the Gergonne point of a triangle in the Euclidean plane is very well known, and the study of them in the isotropic setting has already appeared earlier. In this paper, we give two generalizations of the Gergonne point of a [...] Read more.

The notion of the Gergonne point of a triangle in the Euclidean plane is very well known, and the study of them in the isotropic setting has already appeared earlier. In this paper, we give two generalizations of the Gergonne point of a triangle in the isotropic plane, and we study several curves related to them. The first generalization is based on the fact that for the triangle

A B C

and its contact triangle

A_{i} B_{i} C_{i}

, there is a pencil of circles such that each circle

k_{m}

from the pencil the lines

A A_{m}

,

B B_{m}

,

C C_{m}

is concurrent at a point

G_{m}

, where

A_{m}

,

B_{m}

,

C_{m}

are points on

k_{m}

parallel to

A_{i}, B_{i}, C_{i}

, respectively. To introduce the second generalization of the Gergonne point, we prove that for the triangle

A B C

, point I and three lines

q_{1}, q_{2}, q_{3}

through I there are two points

G_{1, 2}

such that for the points

Q_{1}, Q_{2}, Q_{3}

on

q_{1}, q_{2}, q_{3}

with

d (I, Q_{1}) = d (I, Q_{2}) = d (I, Q_{3})

, the lines

A Q_{1}, B Q_{2}

and

C Q_{3}

are concurrent at

G_{1, 2}

. We achieve these results by using the standardization of the triangle in the isotropic plane and simple analytical method. Full article

(This article belongs to the Special Issue Geometry and Topology with Applications)

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Review

Jump to: Research

10 pages, 247 KiB

Open AccessReview

On the Geometry and Topology of Discrete Groups: An Overview

by Renata Grimaldi

Mathematics 2024, 12(5), 766; https://doi.org/10.3390/math12050766 - 04 Mar 2024

Viewed by 594

Abstract

In this paper, we provide a brief introduction to the main notions of geometric group theory and of asymptotic topology of finitely generated groups. We will start by presenting the basis of discrete groups and of the topology at infinity, then we will [...] Read more.

In this paper, we provide a brief introduction to the main notions of geometric group theory and of asymptotic topology of finitely generated groups. We will start by presenting the basis of discrete groups and of the topology at infinity, then we will state some of the main theorems in these fields. Our aim is to give a sample of how the presence of a group action may affect the geometry of the underlying space and how in many cases topological methods may help the determine solutions of algebraic problems which may appear unrelated. Full article

(This article belongs to the Special Issue Geometry and Topology with Applications)

6 pages, 231 KiB

Open AccessFeature PaperReview

The Problems of Dimension Four, and Some Ramifications

by Valentin Poénaru

Mathematics 2023, 11(18), 3826; https://doi.org/10.3390/math11183826 - 06 Sep 2023

Viewed by 563

Abstract

In this short note, I present a very quick review of the peculiarities of dimension four in geometric topology. I consider, in particular, the role of geometric simple connectivity (which means handle decomposition without handles of index one) for both closed manifolds and [...] Read more.

In this short note, I present a very quick review of the peculiarities of dimension four in geometric topology. I consider, in particular, the role of geometric simple connectivity (which means handle decomposition without handles of index one) for both closed manifolds and open manifolds and for finitely presented groups, together with some of recent developments in geometric group theory. Full article

(This article belongs to the Special Issue Geometry and Topology with Applications)

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