scholarly journals Topological graph persistence

2020 ◽  
Vol 11 (1) ◽  
pp. 72-87
Author(s):  
Mattia G. Bergomi ◽  
Massimo Ferri ◽  
Lorenzo Zuffi
Keyword(s):  
Simplicial Complex ◽  
Independent Sets ◽  
Data Representation ◽  
Basic Tool ◽  
Topological Graph ◽  
Topological Information ◽  
One Dimensional ◽  
Low Dimensional ◽  
Dimensional Simplicial Complex

Abstract Graphs are a basic tool in modern data representation. The richness of the topological information contained in a graph goes far beyond its mere interpretation as a one-dimensional simplicial complex. We show how topological constructions can be used to gain information otherwise concealed by the low-dimensional nature of graphs. We do this by extending previous work in homological persistence, and proposing novel graph-theoretical constructions. Beyond cliques, we use independent sets, neighborhoods, enclaveless sets and a Ramsey-inspired extended persistence.

scholarly journals Local-to-global Urysohn width estimates

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Alexey Balitskiy ◽  
Aleksandr Berdnikov
Keyword(s):  
Riemannian Manifold ◽  
Simplicial Complex ◽  
Metric Space ◽  
Low Dimensional ◽  
Dimensional Simplicial Complex

Abstract The notion of the Urysohn d-width measures to what extent a metric space can be approximated by a d-dimensional simplicial complex. We investigate how local Urysohn width bounds on a Riemannian manifold affect its global width. We bound the 1-width of a Riemannian manifold in terms of its first homology and the supremal width of its unit balls. Answering a question of Larry Guth, we give examples of n-manifolds of considerable ( n - 1 ) {(n-1)} -width in which all unit balls have arbitrarily small 1-width. We also give examples of topologically simple manifolds that are locally nearly low-dimensional.

One-Dimensional Cell Complexes with Homeotopy Group Equal to Zero

1964 ◽  
Vol 16 ◽  
pp. 353-357 ◽  
Author(s):  
Louis V. Quintas
Keyword(s):  
Automorphism Group ◽  
Simplicial Complex ◽  
Topological Invariant ◽  
One Dimensional ◽  
Cell Complexes ◽  
Linear Graph ◽  
Group Of Homeomorphisms ◽  
Multiple Edges ◽  
Dimensional Cell ◽  
Dimensional Simplicial Complex

Let K denote a connected finite 1-dimensional cell complex (1, p. 95), G(K) its group of homeomorphisms, and D(K) the group of homeomorphisms of K which are isotopic to the identity. The group (K) = G(K)/D(K) is a topological invariant of K and is called the homeotopy group ofK (4). K may be thought of as a linear graph (connected finite 1- dimensional simplicial complex) extended to admit loops and multiple edges and (K) as the topological analogue of the automorphism group A(L), (the permutations of vertices which preserve edge incidence relations) of a linear graph L.

scholarly journals Atomic-scale one-dimensional materials identified from graph theory

2021 ◽  
Author(s):  
Shunning Li ◽  
Zhefeng Chen ◽  
Zhi Wang ◽  
Mouyi Weng ◽  
Jianyuan Li ◽  
...  
Keyword(s):  
Graph Theory ◽  
Structural Information ◽  
Functional Materials ◽  
Atomic Scale ◽  
Computational Techniques ◽  
Electronic Band ◽  
One Dimensional ◽  
Atomic Chains ◽  
Future Data ◽  
Low Dimensional

Abstract The past decades have witnessed an exponential growth in the discovery of functional materials, benefited from our unprecedented capabilities in characterizing their structure, chemistry, and morphology with the aid of advanced imaging, spectroscopic and computational techniques. Among these materials, atomic-scale low-dimensional compounds, as represented by the two-dimensional (2D) atomic layers, one-dimensional (1D) atomic chains and zero-dimensional (0D) atomic clusters, have long captivated scientific interest due to their unique topological motifs and exceptional properties. Their tremendous potentials in various applications make it a pressing urgency to establish a complete database of their structural information, especially for the underexplored 1D species. Here we apply graph theory in combination with first-principles high-throughput calculations to identify atomic-scale 1D materials that can be conceptually isolated from their parent bulk crystals. In total, two hundred and fifty 1D atomic chains are shown to be potentially exfoliable. We demonstrate how the lone electron pairs on cations interact with the p-orbitals of anions and hence stabilize their edge sites. Data analysis of the 2D and 1D materials also reveals the dependence of electronic band gap on the cationic percolation network determined by graph theory. The library of 1D compounds systematically identified in this work will pave the way for the predictive discovery of material systems for quantum engineering, and can serve as a source of stimuli for future data-driven design and understanding of functional materials with reduced dimensionality.

scholarly journals Hard Squares with Negative Activity and Rhombus Tilings of the Plane

10.37236/1093 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Jakob Jonsson
Keyword(s):  
Partition Function ◽  
Statistical Mechanics ◽  
Simplicial Complex ◽  
Transfer Matrix ◽  
Euler Characteristic ◽  
Independent Sets ◽  
Roots Of Unity ◽  
Hard Squares ◽  
Vertex Set

Let $S_{m,n}$ be the graph on the vertex set ${\Bbb Z}_m \times {\Bbb Z}_n$ in which there is an edge between $(a,b)$ and $(c,d)$ if and only if either $(a,b) = (c,d\pm 1)$ or $(a,b) = (c \pm 1,d)$ modulo $(m,n)$. We present a formula for the Euler characteristic of the simplicial complex $\Sigma_{m,n}$ of independent sets in $S_{m,n}$. In particular, we show that the unreduced Euler characteristic of $\Sigma_{m,n}$ vanishes whenever $m$ and $n$ are coprime, thereby settling a conjecture in statistical mechanics due to Fendley, Schoutens and van Eerten. For general $m$ and $n$, we relate the Euler characteristic of $\Sigma_{m,n}$ to certain periodic rhombus tilings of the plane. Using this correspondence, we settle another conjecture due to Fendley et al., which states that all roots of $\det (xI-T_m)$ are roots of unity, where $T_m$ is a certain transfer matrix associated to $\{\Sigma_{m,n} : n \ge 1\}$. In the language of statistical mechanics, the reduced Euler characteristic of $\Sigma_{m,n}$ coincides with minus the partition function of the corresponding hard square model with activity $-1$.

scholarly journals A Simple Alternating Direction Method for the Conic Trust Region Subproblem

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Honglan Zhu ◽  
Qin Ni
Keyword(s):  
Trust Region ◽  
Alternating Direction Method ◽  
New Method ◽  
Quadratic Model ◽  
Descent Direction ◽  
Orthogonal Direction ◽  
Trust Region Subproblem ◽  
One Dimensional ◽  
Alternating Direction ◽  
Low Dimensional

A simple alternating direction method is used to solve the conic trust region subproblem of unconstrained optimization. By use of the new method, the subproblem is solved by two steps in a descent direction and its orthogonal direction, the original conic trust domain subproblem into a one-dimensional subproblem and a low-dimensional quadratic model subproblem, both of which are very easy to solve. Then the global convergence of the method under some reasonable conditions is established. Numerical experiment shows that the new method seems simple and effective.

Low-dimensional projection approach for efficient sampling of molecular recognition and polymer aggregation

2020 ◽  
Vol 22 (13) ◽  
pp. 6953-6963
Author(s):  
Hiroya Nakata ◽  
Cheol Ho Choi
Keyword(s):  
Molecular Recognition ◽  
Force Field ◽  
Umbrella Sampling ◽  
Two Dimensional ◽  
Reactive Force ◽  
One Dimensional ◽  
Efficient Sampling ◽  
Projection Approach ◽  
The One ◽  
Low Dimensional

The one-dimensional projection (ODP) approach is extended to two-dimensional umbrella sampling (TDUS) and is applied to three different complex systems in combination with a reactive force field (ReaxFF).

Unusual properties of nanoscale ferroelectrics

2017 ◽  
Author(s):  
Huaxiang Fu
Keyword(s):  
Thin Films ◽  
Ferroelectric Phase ◽  
Structural Phase ◽  
Ferroelectric Thin Films ◽  
One Dimensional ◽  
Theoretical Approaches ◽  
Polarization Saturation ◽  
Low Dimensional ◽  
Ferroelectric Phase Transitions ◽  
Polarization Phase

This article describes the unusual properties of nanoscale ferroelectrics (FE), including widely tunable polarization and improved properties in strained ferroelectric thin films; polarization enhancement in superlattices; polarization saturation in ferroelectric thin films under very large inplane strains; occurrence of ferroelectric phase transitions in one-dimensional wires; existence of the toroidal structural phase in ferroelectric nanoparticles; and the symmetry-broken phase-transition path when one transforms a vortex phase into a polarization phase. The article first considers some of the critical questions on low-dimensional ferroelectricity before discussing the theoretical approaches used to determine the properties of ferroelectric nanostructures. It also looks at 2D ferroelectric structures such as surfaces, superlattices and thin films, along with 1D ferroelectric nanowires and ferroelectric nanoparticles.

scholarly journals Homogeneous one-dimensional Bose–Einstein condensate in the Bogoliubov’s regime

2016 ◽  
Vol 30 (22) ◽  
pp. 1650307 ◽  
Author(s):  
Elías Castellanos
Keyword(s):  
Ground State ◽  
Size Effects ◽  
Finite Size ◽  
Einstein Condensate ◽  
Finite Size Effects ◽  
One Dimensional ◽  
Bose Einstein Condensate ◽  
Ground State Properties ◽  
Low Dimensional ◽  
Bose Einstein

We analyze the corrections caused by finite size effects upon the ground state properties of a homogeneous one-dimensional (1D) Bose–Einstein condensate. We assume from the very beginning that the Bogoliubov’s formalism is valid and consequently, we show that in order to obtain a well-defined ground state properties, finite size effects of the system must be taken into account. Indeed, the formalism described in the present paper allows to recover the usual properties related to the ground state of a homogeneous 1D Bose–Einstein condensate but corrected by finite size effects of the system. Finally, this scenario allows us to analyze the sensitivity of the system when the Bogoliubov’s regime is valid and when finite size effects are present. These facts open the possibility to apply these ideas to more realistic scenarios, e.g. low-dimensional trapped Bose–Einstein condensates.

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