Every 4k-edge-connected graph is Weakly 3k-linked

We prove:

1. Fork ≥ 2 andα = 0, 1, every (4k + 2α)-edge-connected graph is weakly (3k + 2α)-linked.
2. IfG is ak-edge-connected graph (k ≥ 2),s, t are vertices andf is an edge, then there exists a pathP betweens andt such thatf ? E(P) andG ? E(P) ? f is (k ? 2)-edge-connected, whereE(P) denotes the edge set ofP.

     

Every simple 3-polytype with 34 vertices is Hamiltonian

The smallest known example of a simple 3-polytype with no Hamiltonian circuit has 38 vertices. By extending the methods of Okamura we prove that if a simple 3-polytype has 38 most 34 vertices then it is Hamiltonian.     

Every point is critical

We show that, for any compact Alexandrov surface S$S$ (without boundary) and any point y$y$ in S$S$, there exists a point x$x$ in S$S$ for which y$y$ is a critical point. Moreover, we prove that uniqueness characterizes the surfaces homeomorphic to the sphere among smooth orientable surfaces.     

Every 3-connected essentially 10-connected line graph is Hamilton-connected

Thomassen conjectured that every 4-connected line graph is Hamiltonian. Lai et al. (in 2006) [5] considered whether the high essential connectivity of the 3-connected line graphs can guarantee the existence of the Hamiltonian cycle in graphs and they showed that every 3-connected, essentially 11-connected line graph is Hamiltonian. In this note, we show that every 3-connected, essentially 10-connected line graph is Hamiltonian-connected. The result strengthens the known result mentioned above.     

Every 3-connected claw-free graph with domination number at most 3 is hamiltonian-connected

We prove that every 3-connected claw-free graph with domination number at most 3 is hamiltonian-connected. The result is sharp and it is inspired by a conjecture posed by Zheng, Broersma, Wang and Zhang in 2020.     

Every Proposition is a Counterfactual

I present and discuss two logical results. The first shows that a non-trivial counterfactual analysis exists for any contingent proposition that is false in at least two possible worlds. The second result identifies a set of conditions that are individually necessary and jointly sufficient for the success of a counterfactual analysis. I use these results to shed light on the question whether disposition ascribing propositions can be analyzed as Stalnaker-Lewis conditional propositions. The answer is that they can, but, in order for a counterfactual analysis to work, the antecedent and consequent must be related in a particular way, and David Lewis’s Time’s Arrow constraints on comparative world similarity must be relaxed. The upshot is that counterfactual analyses are easy to come by, in principle, even if not in practice. In that sense, it’s easy to be iffy.     

Every 3-connected,locally connected,claw-free graph is Hamilton-connected

A graph G is locally connected if the subgraph induced by the neighbourhood of each vertex is connected. We prove that a locally connected graph G of order p ≥ 4, containing no induced subgraph isomorphic to K_1,3, is Hamilton-connected if and only if G is 3-connected. © 1996 John Wiley & Sons, Inc.     

Every simple 3-polytope of order 32 or less is Hamiltonian

Using theorems of Butler, Goodey, and Okamura we show that every simple 3-polytope of order 32 or less is Hamiltonian.     

Every null-additive set is meager-additive

It is proved that every null-additive subset of^ω2 is meager-additive. Several characterizations of the null-additive subsets of^ω2 are given, as well as a characterization of the meager additive subsets of^ω2. Under CH, an uncountable null-additive subset of^ω2 is constructed. Publ. No. 445. First version written in April 1991. Partially supported by the Basic Research Fund of the Israel Academy of Sciences and the Edmund Landau Center for Research in Mathematical Analysis, supported by the Minerva Foundation (Germany).     

Every group is the automorphism group of a rank-3 matroid

Every group is the automorphism group of a rank-3 extension of a rank-3 Dowling geometry.Partially supported by The George Washington University UFF grant.Partially supported by the National Security Agency under grant MDA904-91-H-0030.     

Every scattered space is subcompact

We prove that every scattered space is hereditarily subcompact and any finite union of subcompact spaces is subcompact. It is a long-standing open problem whether every ?ech-complete space is subcompact. Moreover, it is not even known whether the complement of every countable subset of a compact space is subcompact. We prove that this is the case for linearly ordered compact spaces as well as for ω  -monolithic compact spaces. We also establish a general result for Tychonoff products of discrete spaces which implies that dense _Gδ$G_{\delta }$-subsets of Cantor cubes are subcompact.     

Every Banach Space is Reflexive

The title above is wrong, because the strong dual of a Banach space is too strong to assert that the natural correspondence between a space and its bidual is an isomorphism. However, for many applications it suffices to replace the norm on the first dual by the weak*-structure in order to solve the non-reflexiveness problem [1]. But in this way, only the original vector space is recovered by taking the second dual. In this work we introduce a suitable numerical structure on vector spaces such that Banach balls, or more precisely totally convex modules, arise naturally in duality, namely as a category of Eilenberg–Moore algebras. This numerical structure naturally overlies the weak*-topology on the algebraic dual, so the entire Banach space can be reconstructed as a second dual. Moreover, the isomorphism between the original space and its bidual is the unit of an adjunction between the two-dualisation functors. Notice that the weak*-topology is normable only if it lives on a finite dimensional space; in that case the original space is trivial as well, hence reflexive. So the overlying numerical structure should be something more general than a norm or a seminorm and thus approach theory [2, 3] enters the picture.     

Every Akivis Algebra is Linear

An Akivis algebra is a vector space V endowed with a skew-symmetric bilinear product [x,y] and a trilinear product A(x,y,z) that satisfy the identity

Every diassociative A-loop is Moufang

An A-loop is a loop in which every inner mapping is an automorphism. A problem which had been open since 1956 is settled by showing that every diassociative A-loop is Moufang.

     

Every surface is a leaf

FOR EACH integer n≥3, there are open, connected n-manifolds that cannot be realized as leaves in any compact, C⁰-foliated (n+1)-manifold [8] (for the C² case, cf. [10]). For n=2, the corresponding question has been thought to be hard. Here we give a proof, announced in [7], that all open surfaces are realizable. This answers a basic question posed in [14].In the following, L denotes an open, connected 2-manifold, M a closed, connected 3-manifold.     

Every transformation is bilaterally deterministic

Every ergodic transformationT with a finite generator α, has another finite generator β, which refines α, and is bilaterally deterministic, i.e. V_|i|>n T ⁱβ is the full σ-algebra for everyn.