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11 pages, 2323 KiB

Open AccessArticle

Investigation of Buckling Behaviors in Carbon Nanorings Using the Chebyshev–Ritz Method

by Xiaobo Wang, Guowen Kuang, Hongmei Tian, Zhibin Shao, Ning Dong, Tao Lin and Li Huang

Nanomaterials 2024, 14(23), 1868; https://doi.org/10.3390/nano14231868 - 21 Nov 2024

Viewed by 997

Carbon nanorings (CNRs) serve as an ideal quantum system for novel electronic and magnetic properties. Although extensive theoretical studies utilizing molecular dynamics (MD) simulations have investigated the formation and structural characteristics of CNRs, systematically analyzing their properties across various toric sizes remains challenging due to the inherent complexity of this system. In this study, we introduce a novel finite element method, the Chebyshev–Ritz method, as an alternative approach to investigating the structural properties of CNRs. Previous MD simulations demonstrated that stable CNRs adopt a regular buckled shape at specific toric sizes. By meticulously selecting mechanical parameters, we observe that the critical deformation of a CNR with 50 repeated units, as determined by the Chebyshev–Ritz method, aligns with an MD simulation presenting a buckling number of 14. Additionally, the implementation of the Chebyshev–Ritz method with a constant mechanical parameter for 50 repeated units reveals a structural transition at varying toric sizes, leading to the stabilization of buckling numbers 13, 14, and 15. This structural transition across different buckling modes has also been corroborated by MD simulations. Our approach offers a reliable and accurate means of examining the structural properties of large-scale nanomaterials and paves the way for further exploration in nanoscale mechanics. Full article

(This article belongs to the Special Issue Structural Modeling and Theoretical Study of Low-Dimensional Materials)

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11 pages, 284 KiB

Open AccessArticle

Minimal Systems of Binomial Generators for the Ideals of Certain Monomial Curves

by Manuel B. Branco, Isabel Colaço and Ignacio Ojeda

Mathematics 2021, 9(24), 3204; https://doi.org/10.3390/math9243204 - 11 Dec 2021

Cited by 4 | Viewed by 2301

Abstract

Let

a, b

and

n > 1

be three positive integers such that a and

\sum_{j = 0}^{n - 1} b^{j}

are relatively prime. In this paper, we prove that the toric ideal I associated to the submonoid of [...] Read more.

Let

a, b

and

n > 1

be three positive integers such that a and

\sum_{j = 0}^{n - 1} b^{j}

are relatively prime. In this paper, we prove that the toric ideal I associated to the submonoid of

N

generated by

{\sum_{j = 0}^{n - 1} b^{j}} \cup {\sum_{j = 0}^{n - 1} b^{j} + a \sum_{j = 0}^{i - 2} b^{j} ∣ i = 2, \dots, n}

is determinantal. Moreover, we prove that for

n > 3

, the ideal I has a unique minimal system of generators if and only if

a < b - 1

. Full article

(This article belongs to the Special Issue Combinatorics and Computation in Commutative Algebra)

12 pages, 310 KiB

Open AccessFeature PaperArticle

Toric Rings and Ideals of Stable Set Polytopes

by Kazunori Matsuda, Hidefumi Ohsugi and Kazuki Shibata

Mathematics 2019, 7(7), 613; https://doi.org/10.3390/math7070613 - 10 Jul 2019

Cited by 9 | Viewed by 3082

Abstract

In the present paper, we study the normality of the toric rings of stable set polytopes, generators of toric ideals of stable set polytopes, and their Gröbner bases via the notion of edge polytopes of finite nonsimple graphs and the results on their toric ideals. In particular, we give a criterion for the normality of the toric ring of the stable set polytope and a graph-theoretical characterization of the set of generators of the toric ideal of the stable set polytope for a graph of stability number two. As an application, we provide an infinite family of stable set polytopes whose toric ideal is generated by quadratic binomials and has no quadratic Gröbner bases. Full article

(This article belongs to the Special Issue Current Trends on Monomial and Binomial Ideals)

31 pages, 538 KiB

Open AccessArticle

Morphisms and Order Ideals of Toric Posets

by Matthew Macauley

Mathematics 2016, 4(2), 39; https://doi.org/10.3390/math4020039 - 4 Jun 2016

Viewed by 4643

Abstract

Toric posets are in some sense a natural “cyclic” version of finite posets in that they capture the fundamental features of a partial order but without the notion of minimal or maximal elements. They can be thought of combinatorially as equivalence classes of acyclic orientations under the equivalence relation generated by converting sources into sinks, or geometrically as chambers of toric graphic hyperplane arrangements. In this paper, we define toric intervals and toric order-preserving maps, which lead to toric analogues of poset morphisms and order ideals. We develop this theory, discuss some fundamental differences between the toric and ordinary cases, and outline some areas for future research. Additionally, we provide a connection to cyclic reducibility and conjugacy in Coxeter groups. Full article

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