Finite dimensional convex structures II: the invariants

Various relations between the dimension and the classical invariants of a topological convex structure have been obtained, leading to an equivalence between Helly's and Carathéodory's theorem, and to the closedness of the hull of compact sets in finite-dimensional convexities. It is also shown that the Radon number of an n-dimensional binary convexity is in most cases equal to the Radon number of the n-cube, and a natural condition is presented under which the invariants are equal to dimension plus one.     

Planarity and duality of finite and infinite graphs

We present a short proof of the following theorems simultaneously: Kuratowski's theorem, Fary's theorem, and the theorem of Tutte that every 3-connected planar graph has a convex representation. We stress the importance of Kuratowski's theorem by showing how it implies a result of Tutte on planar representations with prescribed vertices on the same facial cycle as well as the planarity criteria of Whitney, MacLane, Tutte, and Fournier (in the case of Whitney's theorem and MacLane's theorem this has already been done by Tutte). In connection with Tutte's planarity criterion in terms of non-separating cycles we give a short proof of the result of Tutte that the induced non-separating cycles in a 3-connected graph generate the cycle space. We consider each of the above-mentioned planarity criteria for infinite graphs. Specifically, we prove that Tutte's condition in terms of overlap graphs is equivalent to Kuratowski's condition, we characterize completely the infinite graphs satisfying MacLane's condition and we prove that the 3-connected locally finite ones have convex representations. We investigate when an infinite graph has a dual graph and we settle this problem completely in the locally finite case. We show by examples that Tutte's criterion involving non-separating cycles has no immediate extension to infinite graphs, but we present some analogues of that criterion for special classes of infinite graphs.     

Fixed point theorems for precomplete numberings

In the context of his theory of numberings, Ershov showed that Kleene's recursion theorem holds for any precomplete numbering. We discuss various generalizations of this result. Among other things, we show that Arslanov's completeness criterion also holds for every precomplete numbering, and we discuss the relation with Visser's ADN theorem, as well as the uniformity or nonuniformity of the various fixed point theorems. Finally, we base numberings on partial combinatory algebras and prove a generalization of Ershov's theorem in this context.     

Hindman's theorem and groups

We prove a theorem which limits the possible uncountable generalizations of Hindman's theorem.     

A canonical partition relation for finite subsets of ω

We prove a canonical partition relation for finite subsets of ω that generalizes Hindman's theorem in much the same way that the Erdös-Rado canonical partition relation generalizes Ramsey's theorem. As an application of this we establish a generalized pigeon-hole principle for infinite dimensional vector spaces over the two element field.     

Homogeneous continua in 2-manifolds

In 1960 R.H. Bing [2] proved that every homogeneous plane continuum that contains an arc is a simple closed curve. At that time Bing [2, p. 228] asked if every 1-dimensional homogeneous continuum that contains an arc and lies on a 2-manifold is a simple closed curve. We prove that no 2-manifold contains uncountably many disjoint triods. We use this theorem and decomposition theorems of F.B. Jones [10] and H.C. Wiser [19] to answer Bing's question in the affirmative. We also prove that every homogeneous indecomposable continuum in a 2-manifold can be embedded in the plane. It follows from this result and another theorem of Wiser [20] that every homogeneous continuum that is properly contained in an orientable 2-manifold is planar.     

Ramsey's theorem with sums or unions

Using Hindman's theorem as a strong pigeonhole principle, we prove strengthened versions of Ramsey's theorem and of various generalizations of Ramsey's theorem due to Nash-Williams, Galvin and Prikry, and Silver.     

Extensions of Liouville theorems

The method of deriving Liouville's theorem for subharmonic functions in the plane from the corresponding Hadamard three-circles theorem is extended to a more general and abstract setting. Two extensions of Liouville's theorem for vector-valued holomorphic functions of several complex variables are also mentioned.     

The colorful Helly theorem and colorful resolutions of ideals

We demonstrate that the topological Helly theorem and the algebraic Auslander-Buchsbaum theorem may be viewed as different versions of the same phenomenon. Using this correspondence we show how the colorful Helly theorem of I. Barany and its generalizations by G. Kalai and R. Meshulam translate to the algebraic side. Our main results are algebraic generalizations of these translations, which in particular give a syzygetic version of Helly’s theorem.     

Positivity of intersection multiplicity over a two-dimensional base

We prove the positivity of Serre's Intersection Multiplicity for regular local rings that are essentially smooth over a two-dimensional, regular base. Afterward, we apply this result to prove a transversality theorem for unramified regular local rings via a local analysis on the blowup.     

Dynamic Picone's identity and its applications

We prove some Picone-type identities and inequalities for a class of first-order nonlinear dynamic systems and derive various weighted inequalities of Wirtinger type and Hardy type on time scales. As applications we study oscillatory and related properties of these systems including Reid's roundabout theorem on disconjugacy, Sturm's separation and comparison theorems, as well as a variational method in the oscillation theory.     

Burkholder-Gundy-Davis inequality in martingale Hardy spaces with variable exponent

In this article, by extending classical Dellacherie's theorem on stochastic sequences to variable exponent spaces, we prove that the famous Burkholder-Gundy-Davis inequality holds for martingales in variable exponent Hardy spaces. We also obtain the variable exponent analogues of several martingale inequalities in classical theory, including convexity lemma, Chevalier's inequality and the equivalence of two kinds of martingale spaces with predictable control. Moreover, under the regular condition on σ-algebra sequence we prove the equivalence between five kinds of variable exponent martingale Hardy spaces.     

Carathéodory-type Theorems à la Bárány

We generalise the famous Helly–Lovász theorem leading to a generalisation of the Bárány–Carathéodory theorem for oriented matroids in dimension ≤3. We also provide a non-metric proof of the latter colourful theorem for arbitrary dimensions and explore some generalisations in dimension 2.     

A simple theorem on 3-connectivity

It is well known that for any element of a connected matroid, either the deletion or the contraction of that element preserves connectivity. We prove a simple and natural generalization to 3-connected matroids. This result is used to prove Seymour's generalization of a theorem of Kelmans.     

Nowhere-zero 6-flows

We prove that every graph with no isthmus has a nowhere-zero 6-flow, that is, a circulation in which the value of the flow through each edge is one of ±1, ±2,…, ±5. This improves Jaeger's 8-flow theorem, and approaches Tutte's 5-flow conjecture.     

On constructivity and the Rosser property: a closer look at some Gödelean proofs

The proofs of Kleene, Chaitin and Boolos for Gödel's First Incompleteness Theorem are studied from the perspectives of constructivity and the Rosser property. A proof of the incompleteness theorem has the Rosser property when the independence of the true but unprovable sentence can be shown by assuming only the (simple) consistency of the theory. It is known that Gödel's own proof for his incompleteness theorem does not have the Rosser property, and we show that neither do Kleene's or Boolos' proofs. However, we show that a variant of Chaitin's proof can have the Rosser property. The proofs of Gödel, Rosser and Kleene are constructive in the sense that they explicitly construct, by algorithmic ways, the independent sentence(s) from the theory. We show that the proofs of Chaitin and Boolos are not constructive, and they prove only the mere existence of the independent sentences.     

Several theorems of combinatorial geometry

A set is said to be H-convex if it can be represented by an intersection of a family of closed half-spaces whose outer normals belong to a given subset of the set H of the unit sphereS ⁿ⁻¹⊂R. On the basis of Helly’s theorem for H-convex sets recently obtained by us, we prove in this note certain extensions of Blaschke’s theorem (on the radius of an inscribed sphere) and of several other well-known theorems of combinatorial geometry. Translated from Matematicheskie Zametki, Vol. 21, No. 1, pp. 117–124, January, 1977.     

On partitions of a partially ordered set

Using linear programming we prove a generalization of Greene and Kleitman's generalization of Dilworth's theorem on the decomposition of a partially ordered set into chains.     

An operator-valued Lyapunov theorem

We generalize Lyapunov's convexity theorem for classical (scalar-valued) measures to quantum (operator-valued) measures. In particular, we show that the range of a nonatomic quantum probability measure is a weak^?-closed convex set of quantum effects (positive operators bounded above by the identity operator) under a sufficient condition on the non-injectivity of integration. To prove the operator-valued version of Lyapunov's theorem, we must first define the notions of essentially bounded, essential support, and essential range for quantum random variables (Borel measurable functions from a set to the bounded linear operators acting on a Hilbert space).     

Hurwitz' theorem implies Rouché's theorem

It is well known that Hurwitz's theorem is easily proved from Rouché's theorem. We show that conversely, Rouché's theorem is readily proved from Hurwitz' theorem. Since Hurwitz' theorem is easily proved from the formula giving the number of roots of an analytic function, our result thus gives also a simple proof of Rouché's theorem.