Construction of pseudo-isometries for treelike hyperbolic 3-manifolds of infinite volume

We introduce a family of rigid hyperbolic 3-manifolds of infinite volume with possibly infinitely many ends: the treelike manifolds. These manifolds generalize a family of constructive non compact surfaces – the equational surfaces – for which the homeomorphism problem is decidable. The proof of rigidity relies firstly on Thurston's theorem of compactness of the Teichmüller space of acylindrical compact 3-manifolds, and secondly, on Sullivan's rigidity theorem. To cite this article: O. Ly, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

On spectral analysis in locally compact motion groups

Beurling's theorem in spectral analysis of bounded functions on the real line is generalized to a class of locally compact motion groups and to Heisenberg groups.     

Interpolation and prediction problems for connected compact abelian groups

Extensions of the Nehari theorem and of the Sarason commutation theorem are given for compact abelian groups whose dual have a complete linear order compatible with the group structure. As a special case a version of the classical interpolation theorem due to Carathéodory — Féjer is obtained.For these groups an extension of the Helson — Szegö theorem and integral representations for positive definite generalized Toeplitz kernels are given.Partially supported by the CDCH of the Universidad Central de Venezuela, and by CONICIT grant G-97000668.     

An operator not satisfying Lomonosov's hypothesis

An example is presented of a Hilbert space operator such that no non-scalar operator that commutes with it commutes with a non-zero compact operator. This shows that Lomonosov's invariant subspace theorem does not apply to every operator.     

Characterisation of unbounded multiplier maps

This paper generalizes Wendel's theorem to cover an arbitrary closed linear map in L₁(G), G a locally compact group.     

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.     

Helly-type theorems for pseudoline arrangements in P2

We prove duals of Radon's theorem, Helly's theorem, Carathéodory's theorem, and Kirchberger's theorem for arrangements of pseudolines in the real projective plane, which generalize the original versions of those theorems for plane configurations of points. We also prove a topological generalization of the pseudoline-dual of Helly's theorem.     

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.     

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.     

Limiting variants of Krasnoselʼskiĭʼs compact interpolation theorem

We illustrate how limiting variants of Krasnosel?ski??s compact interpolation theorem may be obtained. As a special case of our results it follows that compactness is preserved on certain Lorentz–Zygmund spaces close to the end-point Lebesgue spaces.     

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.     

Hypergraphs and Whitney's theorem on edge-isomorphisms of graphs

This paper presents a new proof of Whitney's theorem on edge-isomorphisms of graphs and extends Whitney's theorem to hypergraphs. Whitney's theorem asserts that any two edge-isomorphic graphs of order at least 5 have their edge-isomorphism induced by a node-isomorphism isomorphism. Previous results of Gardner and of Berge and Rado are used.     

Ideals in the disk algebra

In this article, we determine a necessary and sufficient condition for a sum of closed ideals in the disk algebra to be closed. Essentially, the condition is that the generating inner functions must not simultaneously vanish in the sense used in the corona theorem. The proof uses interpolation properties of the disk algebra and Detraz' generalization of the corona theorem.     

Spectra of singular measures as multipliers on Lp

Stein's theorem on the interpolation of a family of operators between two analytic spaces is generalized, both to a multiply connected domain and to an interpolation between more than two spaces. The theorem is then applied to get setwise upper bounds for spectra of convolution operators on L^p of the circle. In particular the spectra of operators given by convolution by Cantor-Lebesgue-type measures on L^p are determined. The same is done for certain Riesz products. These results are used to derive a result on translation-invariant subspaces of L^p of the circle.     

Analytic proofs of a network feasibility theorem and a theorem of Fulkerson

We provide an elementary proff of Fulkerson's theorem which gives the permutation matrices as extreme points of a certain unbounded convex polyhedron. An adaptation of the proof also establishes an analogous feasibility theorem for network flows which has Fulkerson's theorem as a corollary.     

A short proof of Kneser's conjecture

In the paper a short proof is given for Kneser's conjecture. The proof is based on Borsuk's theorem and on a theorem of Gale.     

On the extension of von Neumann-Aumann's theorem

We give a measurable selection theorem which generalizes von Neumann-Aumann's theorem when the domain of definition is an abstract measurable space and the range space is a Suslin space.As application we give a measurable implicit function theorem and a parametrized version of Choquet's theorem on integral representation.     

A generalization of Ramsey's theorem for regular trees

An analog of Ramsey's theorem for regular trees is proved. The original theorem is a special case of it. It is shown that the denumerable case of Ramsey's theorem has no analog for binary trees.     

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.     

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.