On the global Hadamard parametrix in QFT and the signed squared geodesic distance defined in domains larger than convex normal neighbourhoods

We consider the global Hadamard condition and the notion of Hadamard parametrix whose use is pervasive in algebraic QFT in curved spacetime (see references in the main text). We point out the existence of a technical problem in the literature concerning well-definedness of the global Hadamard parametrix in normal neighbourhoods of Cauchy surfaces. We discuss in particular the definition of the (signed) geodesic distance $$\sigma $$ σ and related structures in an open neighbourhood of the diagonal of $$M\times M$$ M × M larger than $$U\times U$$ U × U , for a normal convex neighbourhood U, where (M, g) is a Riemannian or Lorentzian (smooth Hausdorff paracompact) manifold. We eventually propose a quite natural solution which slightly changes the original definition by Kay and Wald and relies upon some non-trivial consequences of the paracompactness property. The proposed re-formulation is in agreement with Radzikowski’s microlocal version of the Hadamard condition.

From the Strong Differential to Italian Domination in Graphs

Given a graph G and a subset of vertices $$D\subseteq V(G)$$ D ⊆ V ( G ) , the external neighbourhood of D is defined as $$N_e(D)=\{u\in V(G){\setminus } D:\, N(u)\cap D\ne \varnothing \}$$ N e ( D ) = { u ∈ V ( G ) \ D : N ( u ) ∩ D ≠ ∅ } , where N(u) denotes the open neighbourhood of u. Now, given a subset $$D\subseteq V(G)$$ D ⊆ V ( G ) and a vertex $$v\in D$$ v ∈ D , the external private neighbourhood of v with respect to D is defined to be $$\mathrm{epn}(v,D)=\{u\in V(G){\setminus } D: \, N(u)\cap D=\{v\}\}.$$ epn ( v , D ) = { u ∈ V ( G ) \ D : N ( u ) ∩ D = { v } } . The strong differential of a set $$D\subseteq V(G)$$ D ⊆ V ( G ) is defined as $$\partial _s(D)=|N_e(D)|-|D_w|,$$ ∂ s ( D ) = | N e ( D ) | - | D w | , where $$D_w=\{v\in D:\, \mathrm{epn}(v,D)\ne \varnothing \}$$ D w = { v ∈ D : epn ( v , D ) ≠ ∅ } . In this paper, we focus on the study of the strong differential of a graph, which is defined as $$\begin{aligned} \partial _s(G)=\max \{\partial _s(D):\, D\subseteq V(G)\}. \end{aligned}$$ ∂ s ( G ) = max { ∂ s ( D ) : D ⊆ V ( G ) } . Among other results, we obtain general bounds on $$\partial _s(G)$$ ∂ s ( G ) and we prove a Gallai-type theorem, which states that $$\partial _s(G)+\gamma _{_I}(G)=\mathrm{n}(G)$$ ∂ s ( G ) + γ I ( G ) = n ( G ) , where $$\gamma _{_I}G)$$ γ I G ) denotes the Italian domination number of G. Therefore, we can see the theory of strong differential in graphs as a new approach to the theory of Italian domination. One of the advantages of this approach is that it allows us to study the Italian domination number without the use of functions. As we can expect, we derive new results on the Italian domination number of a graph.

ON LOCALLY-BALANCED 2-PARTITIONS OF BIPARTITE GRAPHS

A \emph{$2$-partition of a graph $G$} is a function $f:V(G)\rightarrow \{0,1\}$. A $2$-partition $f$ of a graph $G$ is a \emph{locally-balanced with an open neighborhood}, if for every $v\in V(G)$, $\left\vert \vert \{u\in N_{G}(v)\colon\,f(u)=0\}\vert - \vert \{u\in N_{G}(v)\colon\,f(u)=1\}\vert \right\vert\leq 1$. A bipartite graph is \emph{$(a,b)$-biregular} if all vertices in one part have degree $a$ and all vertices in the other part have degree $b$. In this paper we prove that the problem of deciding, if a given graph has a locally-balanced $2$-partition with an open neighborhood is $NP$-complete even for $(3,8)$-biregular bipartite graphs. We also prove that a $(2,2k+1)$-biregular bipartite graph has a locally-balanced $2$-partition with an open neighbourhood if and only if it has no cycle of length $2 \pmod{4}$. Next, we prove that if $G$ is a subcubic bipartite graph that has no cycle of length $2 \pmod{4}$, then $G$ has a locally-balanced $2$-partition with an open neighbourhood. Finally, we show that all doubly convex bipartite graphs have a locally-balanced $2$-partition with an open neighbourhood.

Generalised Iwasawa invariants and the growth of class numbers

We study the generalised Iwasawa invariants of {\mathbb{Z}_{p}^{d}}-extensions of a fixed number field K. Based on an inequality between ranks of finitely generated torsion {\mathbb{Z}_{p}[\kern-2.133957pt[T_{1},\dots,T_{d}]\kern-2.133957pt]}-modules and their corresponding elementary modules, we prove that these invariants are locally maximal with respect to a suitable topology on the set of {\mathbb{Z}_{p}^{d}}-extensions of K, i.e., that the generalised Iwasawa invariants of a {\mathbb{Z}_{p}^{d}}-extension {\mathbb{K}} of K bound the invariants of all {\mathbb{Z}_{p}^{d}}-extensions of K in an open neighbourhood of {\mathbb{K}}. Moreover, we prove an asymptotic growth formula for the class numbers of the intermediate fields in certain {\mathbb{Z}_{p}^{2}}-extensions, which improves former results of Cuoco and Monsky. We also briefly discuss the impact of generalised Iwasawa invariants on the global boundedness of Iwasawa λ-invariants.

Total Weak Roman Domination in Graphs

Given a graph G = ( V , E ) , a function f : V → { 0 , 1 , 2 , ⋯ } is said to be a total dominating function if ∑ u ∈ N ( v ) f ( u ) > 0 for every v ∈ V , where N ( v ) denotes the open neighbourhood of v. Let V i = { x ∈ V : f ( x ) = i } . We say that a function f : V → { 0 , 1 , 2 } is a total weak Roman dominating function if f is a total dominating function and for every vertex v ∈ V 0 there exists u ∈ N ( v ) ∩ ( V 1 ∪ V 2 ) such that the function f ′ , defined by f ′ ( v ) = 1 , f ′ ( u ) = f ( u ) - 1 and f ′ ( x ) = f ( x ) whenever x ∈ V ∖ { u , v } , is a total dominating function as well. The weight of a function f is defined to be w ( f ) = ∑ v ∈ V f ( v ) . In this article, we introduce the study of the total weak Roman domination number of a graph G, denoted by γ t r ( G ) , which is defined to be the minimum weight among all total weak Roman dominating functions on G. We show the close relationship that exists between this novel parameter and other domination parameters of a graph. Furthermore, we obtain general bounds on γ t r ( G ) and, for some particular families of graphs, we obtain closed formulae. Finally, we show that the problem of computing the total weak Roman domination number of a graph is NP-hard.

An answer to Furstenberg’s problem on topological disjointness

In this paper we give an answer to Furstenberg’s problem on topological disjointness. Namely, we show that a transitive system $(X,T)$ is disjoint from all minimal systems if and only if $(X,T)$ is weakly mixing and there is some countable dense subset $D$ of $X$ such that for any minimal system $(Y,S)$, any point $y\in Y$ and any open neighbourhood $V$ of $y$, and for any non-empty open subset $U\subset X$, there is $x\in D\cap U$ such that $\{n\in \mathbb{Z}_{+}:T^{n}x\in U,S^{n}y\in V\}$ is syndetic. Some characterization for the general case is also given. By way of application we show that if a transitive system $(X,T)$ is disjoint from all minimal systems, then so are $(X^{n},T^{(n)})$ and $(X,T^{n})$ for any $n\in \mathbb{N}$. It turns out that a transitive system $(X,T)$ is disjoint from all minimal systems if and only if the hyperspace system $(K(X),T_{K})$ is disjoint from all minimal systems.

Traceability of locally hamiltonian and locally traceable graphs

International audience If $\mathcal{P}$ is a given graph property, we say that a graph $G$ is <i>locally</i> $\mathcal{P}$ if $\langle N(v) \rangle$ has property $\mathcal{P}$ for every $v \in V(G)$ where $\langle N(v) \rangle$ is the induced graph on the open neighbourhood of the vertex $v$. Pareek and Skupien (C. M. Pareek and Z. Skupien , On the smallest non-Hamiltonian locally Hamiltonian graph, J. Univ. Kuwait (Sci.), 10:9 - 17, 1983) posed the following two questions. <b>Question 1</b> Is 9 the smallest order of a connected nontraceable locally traceable graph? <b>Question 2</b> Is 14 the smallest order of a connected nontraceable locally hamiltonian graph? We answer the second question in the affirmative, but show that the correct number for the first question is 10. We develop a technique to construct connected locally hamiltonian and locally traceable graphs that are not traceable. We use this technique to construct such graphs with various prescribed properties.

Bifurcation at isolated singular points of the Hadamard derivative

For Banach spaces X and Y, we consider bifurcation from the line of trivial solutions for the equation F (λ, u) = 0, where F : ℝ × X → Y with F (λ, 0) = 0 for all λ ∈ ℝ. The focus is on the situation where F (λ, ·) is only Hadamard differentiable at 0 and Lipschitz continuous on some open neighbourhood of 0, without requiring any Fréchet differentiability. Applications of the results obtained here to some problems involving nonlinear elliptic equations on ℝN, where Fréchet differentiability is not available, are presented in some related papers, which shed light on the relevance of our hypotheses.

ISOMETRIES AND DISCRETE ISOMETRY SUBGROUPS OF HYPERBOLIC SPACES

Let n be the n-dimensional hyperbolic space with n ≥ 2. Suppose that G is a discrete, sense-preserving subgroup of Isomn, the isometry group of n. Let p be the projection map from n to the quotient space M = n/G. The first goal of this paper is to prove that for any a ∈ ∂n (the sphere at infinity of n), there exists an open neighbourhood U of a in n ∪ ∂ n such that p is an isometry on U ∩ n if and only if a ∈ oΩ(G) (the domain of proper discontinuity of G). This is a generalization of the main result discussed in the work by Y. D. Kim (A theorem on discrete, torsion free subgroups of Isomn, Geometriae Dedicata109 (2004), 51–57). The second goal is to obtain a new characterization for the elements of Isomn by using a class of hyperbolic geometric objects: hyperbolic isosceles right triangles. The proof is based on a geometric approach.