In the social sciences, networks are used to represent relationships between social actors, be they individuals or aggregates. The structural importance of these actors is assessed in terms of centrality indices which are commonly defined as graph invariants. Many such indices have been proposed, but there is no unifying theory of centrality. Previous attempts at axiomatic characterization have been focused on particular indices, and the conceptual frameworks that have been proposed alternatively do not lend themselves to mathematical treatment.

We show that standard centrality indices, although seemingly distinct, can in fact be expressed in a common framework based on path algebras. Since, as a consequence, all of these indices preserve the neighbourhood-inclusion pre-order, the latter provides a conceptually clear criterion for the definition of centrality indices.

Consider a system of polynomials in many variables over the ring of integers of a number field  $K$ . We prove an asymptotic formula for the number of integral zeros of this system in homogeneously expanding boxes. As a consequence, any smooth and geometrically integral variety $X\subseteq \mathbb{P}_{K}^{m}$ satisfies the Hasse principle, weak approximation, and the Manin–Peyre conjecture if only its dimension is large enough compared to its degree. This generalizes work of Skinner, who considered the case where all polynomials have the same degree, and recent work of Browning and Heath-Brown, who considered the case where $K=\mathbb{Q}$ . Our main tool is Skinner’s number field version of the Hardy–Littlewood circle method. As a by-product, we point out and correct an error in Skinner’s treatment of the singular integral.

We generalize to higher dimensions the Bavard–Ghys construction of the hyperbolic metric on the space of polygons with fixed directions of edges. The space of convex $d$ -dimensional polyhedra with fixed directions of facet normals has a decomposition into type cones that correspond to different combinatorial types of polyhedra. This decomposition is a subfan of the secondary fan of a vector configuration and can be analyzed with the help of Gale diagrams. We construct a family of quadratic forms on each of the type cones using the theory of mixed volumes. The Alexandrov–Fenchel inequalities ensure that these forms have exactly one positive eigenvalue. This introduces a piecewise hyperbolic structure on the space of similarity classes of polyhedra with fixed directions of facet normals. We show that some of the dihedral angles on the boundary of the resulting cone-manifold are equal to $\unicode[STIX]{x1D70B}/2$ .

We say that a planar set $A$ has the Kakeya property if there exist two different positions of $A$ such that $A$ can be continuously moved from the first position to the second within a set of arbitrarily small area. We prove that if $A$ is closed and has the Kakeya property, then the union of the non-trivial connected components of $A$ can be covered by a null set which is either the union of parallel lines or the union of concentric circles. In particular, if $A$ is closed, connected and has the Kakeya property, then $A$ can be covered by a line or a circle.

In the 1970s, Feldman and Moore classified separably acting von Neumann algebras containing Cartan maximal abelian self-adjoint subalgebras (MASAs) using measured equivalence relations and 2-cocycles on such equivalence relations. In this paper we give a new classification in terms of extensions of inverse semigroups. Our approach is more algebraic in character and less point-based than that of Feldman and Moore. As an application, we give a restatement of the spectral theorem for bimodules in terms of subsets of inverse semigroups. We also show how our viewpoint leads naturally to a description of maximal subdiagonal algebras.

We give necessary and sufficient conditions for the Lp -well-posedness of the second-order degenerate differential equations with finite delay

with periodic boundary conditions (Mu)(0) = (Mu)(2π), (Mu)′(0) = (Mu)′(2π). Here A and M are closed operators on a complex Banach space X satisfying D(A) ⊂ D(M), α ∈ ℂ is fixed, F is a bounded linear operator from Lp ([−2π, 0],X) into X, and ut is given by ut (s) = u(t + s) when s ∈ [−2π, 0].

Let A = C(X) ⊗ K(H), where X is a compact Hausdorff space and K(H) is the algebra of compact operators on a separable infinite-dimensional Hilbert space. Let A s be the algebra of strong*-continuous functions from X to K(H). Then A s /A is the inner corona algebra of A. We show that if X has no isolated points, then A s /A is an essential ideal of the corona algebra of A, and Prim(A s /A), the primitive ideal space of A s /A, is not weakly Lindelof. If X is also first countable, then there is a natural injection from the power set of X to the lattice of closed ideals of A s /A. If X = βℕ\ℕ and the continuum hypothesis (CH) is assumed, then the corona algebra of A is a proper subalgebra of the multiplier algebra of A s /A. Several of the results are obtained in the more general setting of C 0(X)-algebras.

We say a graph is (Q n ,Q m )-saturated if it is a maximal Q m -free subgraph of the n-dimensional hypercube Q n . A graph is said to be (Q n ,Q m )-semi-saturated if it is a subgraph of Q n and adding any edge forms a new copy of Q m . The minimum number of edges a (Q n ,Q m )-saturated graph (respectively (Q n ,Q m )-semi-saturated graph) can have is denoted by sat(Q n ,Q m ) (respectively s-sat(Q n ,Q m )). We prove that

$$\begin{linenomath}\lim_{n\to\infty}\ffrac{\sat(Q_n,Q_m)}{e(Q_n)}=0,\end{linenomath}$$

$$\begin{linenomath}\ssat(Q_n,Q_m)\geq \ffrac{m+1}{2}\cdot 2^n,\end{linenomath}$$

A construction of Alon yields a sequence of highly pseudorandom triangle-free graphs with edge density significantly higher than one might expect from comparison with random graphs. We give an alternative construction for such graphs.

In this paper, we present a distributed Proportional-Integral (PI) strategy with self-tuning adaptive gains for reaching asymptotic consensus in networks of non-identical linear agents under constant disturbances. Alternative adaptive strategies are presented, based on global or local measures of the agents' disagreement. The proposed approaches are validated on a representative numerical example. Preliminary analytical results further confirm the viability of the self-tuning strategies.

Let G be an abelian group of cardinality n, where hcf(n, 6) = 1, and let A be a random subset of G. Form a graph ΓA on vertex set G by joining x to y if and only if x + y ∈ A. Then, with high probability as n → ∞, the chromatic number χ(ΓA ) is at most $(1 + o(1))\tfrac{n}{2\log_2 n}$ . This is asymptotically sharp when G = ℤ/nℤ, n prime.

We consider a fully practical finite element approximation of the Cahn–Hilliard–Stokes system:

$$\begin{align*}\gamma \tfrac{\partial u}{\partial t} + \beta v \cdot \nabla u -\nabla \cdot \left(\nablaw\right) & = 0 \,, \quadw= -\gamma \Delta u + \gamma ^{-1} \Psi ' (u) - \tfrac12 \alpha c'(\cdot,u)| \nabla \phi |^2\,, \\\nabla \cdot (c(\cdot,u) \nabla \phi) & = 0\,,\quad\begin{cases}-\Delta v + \nabla p = \varsigma w \nabla u,\\\nabla \cdot v = 0, \end{cases}\end{align*}$$

$\mathbb{R}_{>0}$

$\mathbb{R}_{\geq0}$

Let $F$ be a unramified finite extension of $\mathbb{Q}_{p}$ and $\overline{\unicode[STIX]{x1D70C}}$ be an irreducible mod $p$ two-dimensional representation of the absolute Galois group of $F$. The aim of this article is the explicit computation of the Kisin variety parameterizing the Breuil–Kisin modules associated to certain families of potentially Barsotti–Tate deformations of $\overline{\unicode[STIX]{x1D70C}}$. We prove that this variety is a finite union of products of $\mathbb{P}^{1}$. Moreover, it appears as an explicit closed connected subvariety of $(\mathbb{P}^{1})^{[F:\mathbb{Q}_{p}]}$. We define a stratification of the Kisin variety by locally closed subschemes and explain how the Kisin variety equipped with its stratification may help in determining the ring of Barsotti–Tate deformations of $\overline{\unicode[STIX]{x1D70C}}$.

For a given graph G of minimum degree at least k, let Gp denote the random spanning subgraph of G obtained by retaining each edge independently with probability p = p(k). We prove that if p ⩾ (logk + loglogk + ωk (1))/k, where ωk (1) is any function tending to infinity with k, then Gp asymptotically almost surely contains a cycle of length at least k + 1. When we take G to be the complete graph on k + 1 vertices, our theorem coincides with the classic result on the threshold probability for the existence of a Hamilton cycle in the binomial random graph.

We study a free boundary problem for the Fisher-KPP equation: u t = u xx + f(u) (g(t) < x < h(t)) with free boundary conditions h′(t) = −u x (t, h(t)) − α and g′(t) = −u x (t, g(t)) + β for 0 < β < α. This problem can model the spreading of a biological or chemical species, where free boundaries represent the spreading fronts of the species. We investigate the asymptotic behaviour of bounded solutions. There are two parameters α0 and α* with 0 < α0 < α* which play key roles in the dynamics. More precisely, (i) in case 0 < β < α0 and 0 < α < α*, we obtain a trichotomy result: (i-1) spreading, i.e., h(t) − g(t) → +∞ and u(t, ⋅ + ct) → 1 with c ∈ (c L , c R ), where c L and c R are the asymptotic spreading speed of g(t) and h(t), respectively, (c R > 0 > c L when 0 < β < α < α0; c R = 0 >c L when 0 < β < α = α0; 0 > c R > c L when α0 < α < α* and 0 < β < α0); (i-2) vanishing, i.e., limt→Th(t) = limt→T g(t) and limt→T u(t, x) = 0, where T is some positive constant; (i-3) transition, i.e., g(t) → −∞, h(t) → −∞, 0 < limt→∞[h(t) − g(t)] < +∞ and u(t, x) → V*(x − c*t) with c* < 0, where V*(x − c*t) is a travelling wave with compact support and which satisfies the free boundary conditions. (ii) in case β ≥ α0 or α ≥ α*, vanishing happens for any solution.

This paper is concerned with the asymptotic behaviour of the lifespan of solutions for a semilinear heat equation with initial datum λφ(x) in hyperbolic space. The growth rates for both λ → 0 and λ → ∞ are determined.

We prove that triangular configurations are plentiful in large subsets of Cartesian squares of finite quasirandom groups from classes having the quasirandom ultraproduct property, for example the class of finite simple groups. This is deduced from a strong double recurrence theorem for two commuting measure-preserving actions of a minimally almost periodic (not necessarily amenable or locally compact) group on a (not necessarily separable) probability space.