The Oriented and Flux-Weighted Current Density Stagnation Graph of LiH

A scheme is introduced to quantitatively analyze the magnetically induced molecular current density vector field $\mathbf{J}$. After determining the set of zero points of $\mathbf{J}$, which is called its {\em stagnation graph} (SG), the line integrals $\Phi_{\ell_i}=-\frac{1}{\mu_0} \int_{\ell_i} \mathbf{B}_\mathrm{ind}\cdot\mathrm{d}\mathbf{l}$ along all edges $\ell_i$ of the connected subset of the SG are determined. The edges $\ell_i$ are oriented such that all $\Phi_{\ell_i}$ are non-negative and they are weighted with $\Phi_{\ell_i}$. An oriented flux-weighted (current density) stagnation graph (OFW-SG) is obtained. Since $\mathbf{J}$ is in the exact theoretical limit divergence free and due to the topological characteristics of such vector fields the flux of all separate vortices and neighbouring vortex combinations can be determined by adding the weights of cyclic subsets of edges of the OFW-SG. The procedure is exemplified by the case of LiH for a perpendicular and weak homogeneous external magnetic field $\mathbf{B}$}

Some Geometric Properties of a Hyperbolic Univalent Function

In this paper, we analyze several aspects of a hyperbolic univalent function related to convexity properties, by assuming to be the univalent holomorphic function maps of the unit disk onto the hyperbolic convex region ( is an open connected subset of). This assumption leads to the coverage of some of the findings that are started by seeking a convex univalent function distortion property to provide an approximation of the inequality and confirm the form of the lower bound for . A further result was reached by combining the distortion and growth properties for increasing inequality . From the last result, we wanted to demonstrate the effect of the unit disk image on the condition of convexity estimation by proving the two inequalities of , and .

Phase transition of the 2-Choices dynamics on core–periphery networks

The 2-Choices dynamics is a process that models voting behavior on networks and works as follows: Each agent initially holds either opinion blue or red; then, in each round, each agent looks at two random neighbors and, if the two have the same opinion, the agent adopts it. We study its behavior on a class of networks with core–periphery structure. Assume that a densely-connected subset of agents, the core, holds a different opinion from the rest of the network, the periphery. We prove that, depending on the strength of the cut between core and periphery, a phase-transition phenomenon occurs: Either the core’s opinion rapidly spreads across the network, or a metastability phase takes place in which both opinions coexist for superpolynomial time. The interest of our result, which we also validate with extensive experiments on real networks, is twofold. First, it sheds light on the influence of the core on the rest of the network as a function of its connectivity toward the latter. Second, it is one of the first analytical results which shows a heterogeneous behavior of a simple dynamics as a function of structural parameters of the network.

Analysing the Vulnerability of Public Transport Networks

This paper analyses the robustness of specific public transport networks. Common attributes and which of them have more influence on the networks’ vulnerability are established. Initially, the structural properties of the networks in two graphical representations (L-Space and P-Space) are checked. Afterwards, the spread of problems (traffic jams, etc.) are simulated, employing a model based on a propagation and recovery mechanism, similar to those used in the epidemiological processes. Next, the size of the largest connected subset of stops of the network (giant component) is measured. What is shown is that the faults randomly happened at stops or links, also displaying that those that occurred in the highest weighted links spread slower than others. These others appear at stops with the largest level of betweenness, degree, or eigenvector centralities and PageRank. The modification of the giant component, when several stops and links are removed, proves that the removal of stops with the highest interactive betweenness, PageRank, and degree centralities has the most significant influence on the network’s integrity. Some equivalences in the degree, betweenness, PageRank, and eigenvector centrality parameters have been found. All networks show high modularity with values of index Q close to 1. The networks with the highest assortativity and lowest average number of stops are the ones which a passenger can use to travel directly to their destination, without any change. The Molloy–Reed parameter is higher than 2 in all networks, demonstrating that high integrity exists in them. All stops were characterized by low k-cores ≤3.

Generic rank of Betti map and unlikely intersections

Let $\mathcal {A} \rightarrow S$ be an abelian scheme over an irreducible variety over $\mathbb {C}$ of relative dimension $g$. For any simply-connected subset $\Delta$ of $S^{\mathrm {an}}$ one can define the Betti map from $\mathcal {A}_{\Delta }$ to $\mathbb {T}^{2g}$, the real torus of dimension $2g$, by identifying each closed fiber of $\mathcal {A}_{\Delta } \rightarrow \Delta$ with $\mathbb {T}^{2g}$ via the Betti homology. Computing the generic rank of the Betti map restricted to a subvariety $X$ of $\mathcal {A}$ is useful to study Diophantine problems, e.g. proving the geometric Bogomolov conjecture over char $0$ and studying the relative Manin–Mumford conjecture. In this paper we give a geometric criterion to detect this rank. As an application we show that it is maximal after taking a large fibered power (if $X$ satisfies some conditions); it is an important step to prove the bound for the number of rational points on curves (Dimitrov et al., Uniformity in Mordell–Lang for Curves, Preprint (2020), arXiv:2001.10276). Another application is to answer a question of André, Corvaja and Zannier and improve a result of Voisin. We also systematically study its link with the relative Manin–Mumford conjecture, reducing the latter to a simpler conjecture. Our tools are functional transcendence and unlikely intersections for mixed Shimura varieties.

Returning functions with closed graph are continuous

A function f : X → ℝ defined on a topological space X is called returning if for any point x ∈ X there exists a positive real number Mx such that for every path-connected subset Cx ⊂ X containing the point x and any y ∈ Cx ∖ {x} there exists a point z ∈ Cx ∖ {x, y} such that |f(z)| ≤ max{Mx, |f(y)|}. A topological space X is called path-inductive if a subset U ⊂ X is open if and only if for any path γ : [0, 1] → X the preimage γ–1(U) is open in [0, 1]. The class of path-inductive spaces includes all first-countable locally path-connected spaces and all sequential locally contractible spaces. We prove that a function f : X → ℝ defined on a path-inductive space X is continuous if and only if it is returning and has closed graph. This implies that a (weakly) Świątkowski function f : ℝ → ℝ is continuous if and only if it has closed graph, which answers a problem of Maliszewski, inscribed to Lviv Scottish Book.

On the spectrum of Cayley graphs

The set of eigenvalues of the adjacency matrix of a graph is called the spectrum of it. In the present paper, we introduce the spectrum of Cayley graphs of order pqr in terms of character table, where p,q,r are prime numbers. We also, stablish some properties of Cayley graphs of non-abelian groups with a normal symmetric connected subset.

New Computation of Resolving Connected Dominating Sets in Weighted Networks

In this paper we focus on the issue related to finding the resolving connected dominating sets (RCDSs) of a graph, denoted by G. The connected dominating set (CDS) is a connected subset of vertices of G selected to guarantee that all vertices in the graph are connected to vertices in the CDS. The connected dominating set with minimum cardinality, or minimum CDS (MCDS), is an adequate virtual backbone for information interchange in a network. When distinct vertices of G have also distinct representations with respect to a subset of vertices in the MCDS, it is said that the MCDS includes a resolving set (RS) of G. With this work, we explore different strategies to find the RCDS with minimum cardinality in complex networks where the vertices have different importances.

A Study of the Relationship between Convex Sets and Connected Sets

The principal objective of this research article is to explore the relationship between convex sets and connected sets. All convex sets are connected but in all cases connected sets are not convex. In the Maly theorem,let X be a Banach space, and let f:X → R be a (Fr´echet-)differentiable function. Then, for any closed convex subset C of X with nonempty interior,the image Df(C) of C by the differential Df of f is a connected subset of X∗ , where X∗ stands for thetopological dual space of X.The result does not hold true if C has an empty interior. There are counterexamples even with functions f of two variables. This article concludes that convexity cannot be replaced with the connectedness of C.

Convexity in complex networks

Metric graph properties lie in the heart of the analysis of complex networks, while in this paper we study their convexity through mathematical definition of a convex subgraph. A subgraph is convex if every geodesic path between the nodes of the subgraph lies entirely within the subgraph. According to our perception of convexity, convex network is such in which every connected subset of nodes induces a convex subgraph. We show that convexity is an inherent property of many networks that is not present in a random graph. Most convex are spatial infrastructure networks and social collaboration graphs due to their tree-like or clique-like structure, whereas the food web is the only network studied that is truly non-convex. Core–periphery networks are regionally convex as they can be divided into a non-convex core surrounded by a convex periphery. Random graphs, however, are only locally convex meaning that any connected subgraph of size smaller than the average geodesic distance between the nodes is almost certainly convex. We present different measures of network convexity and discuss its applications in the study of networks.