On two problems in extension theory

In this note we introduce the concept of a quasi-finite complex. Next, we show that for a given countable simplicial complex L the following conditions are equivalent:

•

L is quasi-finite.

•

There exists a [L]-invertible mapping of a metrizable compactum X with e-dimX?[L] onto the Hilbert cube.Finally, we construct an example of a quasi-finite complex L such that its extension type [L] does not contain a finitely dominated complex.

     

Norm continuity of weakly continuous mappings into Banach spaces

Let T be the class of Banach spaces E for which every weakly continuous mapping from an α-favorable space to E is norm continuous at the points of a dense subset. We show that:

•

T contains all weakly Lindelöf Banach spaces;

•

l^∞∉T, which brings clarity to a concern expressed by Haydon ([R. Haydon, Baire trees, bad norms and the Namioka property, Mathematika 42 (1995) 30-42], pp. 30-31) about the need of additional set-theoretical assumptions for this conclusion. Also, (l^∞/c₀)∉T.

•

T is stable under weak homeomorphisms;

•

E∈T iff every quasi-continuous mapping from a complete metric space to (E,weak) is densely norm continuous;

•

E∈T iff every quasi-continuous mapping from a complete metric space to (E,weak) is weakly continuous at some point.

     

Intersection cohomology of the circle actions

A classical result says that a free action of the circle S¹ on a topological space X is geometrically classified by the orbit space B and by a cohomological class e∈H²(B,Z), the Euler class. When the action is not free we have a difficult open question:

(Π)

“Is the space X determined by the orbit space B and the Euler class?”

The main result of this work is a step towards the understanding of the above question in the category of unfolded pseudomanifolds. We prove that the orbit space B and the Euler class determine:

•

the intersection cohomology of X,

•

the real homotopy type of X.

     

The Mukai pairing—II: the Hochschild-Kostant-Rosenberg isomorphism

We continue the study of the Hochschild structure of a smooth space that we began in our previous paper, examining implications of the Hochschild-Kostant-Rosenberg theorem. The main contributions of the present paper are:

•

we introduce a generalization of the usual notions of Mukai vector and Mukai pairing on differential forms that applies to arbitrary manifolds;

•

we give a proof of the fact that the natural Chern character map K₀(X)→HH₀(X) becomes, after the HKR isomorphism, the usual one ; and

•

we present a conjecture that relates the Hochschild and harmonic structures of a smooth space, similar in spirit to the Tsygan formality conjecture.

     

Strong zero-dimensionality of hyperspaces

For a space X, X² denotes the collection of all non-empty closed sets of X with the Vietoris topology, and K(X) denotes the collection of all non-empty compact sets of X with the subspace topology of X². The following are known:

•

ω² is not normal, where ω denotes the discrete space of countably infinite cardinality.

•

For every non-zero ordinal γ with the usual order topology, K(γ) is normal iff whenever cf γ is uncountable.

In this paper, we will prove:

(1)

ω² is strongly zero-dimensional.

(2)

K(γ) is strongly zero-dimensional, for every non-zero ordinal γ.

In (2), we use the technique of elementary submodels.     

Banach spaces with the Daugavet property, and the centralizer

We introduce representable Banach spaces, and prove that the class R of such spaces satisfies the following properties:

(1)

Every member of R has the Daugavet property.

(2)

It Y is a member of R, then, for every Banach space X, both the space L(X,Y) (of all bounded linear operators from X to Y) and the complete injective tensor product lie in R.

(3)

If K is a perfect compact Hausdorff topological space, then, for every Banach space Y, and for most vector space topologies τ on Y, the space C(K,(Y,τ)) (of all Y-valued τ-continuous functions on K) is a member of R.

(4)

If K is a perfect compact Hausdorff topological space, then, for every Banach space Y, most C(K,Y)-superspaces (in the sense of [V. Kadets, N. Kalton, D. Werner, Remarks on rich subspaces of Banach spaces, Studia Math. 159 (2003) 195-206]) are members of R.

(5)

All dual Banach spaces without minimal M-summands are members of R.

     

On some contributions of John Horváth to the theory of distributions

Our main task is a presentation of J. Horváth's results concerning

•

singular and hypersingular integral operators,

•

the analytic continuation of distribution-valued meromorphic functions, and

•

a general definition of the convolution of distributions.

At some instances minor supplements to his results are given.     

The Fréchet-Urysohn property of Pixley-Roy hyperspaces

Let F[X] be the Pixley-Roy hyperspace of a regular space X. In this paper, we prove the following theorem.

Theorem. For a space X, the following are equivalent:

(1)

F[X]is a k-space;

(2)

F[X]is sequential;

(3)

F[X]is Fréchet-Urysohn;

(4)

Every finite power of X is Fréchet-Urysohn for finite sets;

(5)

Every finite power ofF[X]is Fréchet-Urysohn for finite sets.

As an application, we improve a metrization theorem onF[X].     

D-spaces and thick covers

The following results are obtained.

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An open neighbornet U of X has a closed discrete kernel if X has an almost thick cover by countably U-close sets.

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Every hereditarily thickly covered space is aD and linearly D.

-

Every t-metrizable space is a D-space.

-

X is a D-space if X has a cover {Xα:α<λ} by D-subspaces such that, for each β<λ, the set ?{Xα:α<β} is closed.

     

Finite Eulerian posets which are binomial, Sheffer or triangular

In this paper we study finite Eulerian posets which are binomial, Sheffer or triangular. These important classes of posets are related to the theory of generating functions and to geometry. The results of this paper are organized as follows:

•

We completely determine the structure of Eulerian binomial posets and, as a conclusion, we are able to classify factorial functions of Eulerian binomial posets.

•

We give an almost complete classification of factorial functions of Eulerian Sheffer posets by dividing the original question into several cases.

•

In most cases above, we completely determine the structure of Eulerian Sheffer posets, a result stronger than just classifying factorial functions of these Eulerian Sheffer posets.

We also study Eulerian triangular posets. This paper answers questions posed by R. Ehrenborg and M. Readdy. This research is also motivated by the work of R. Stanley about recognizing the boolean lattice by looking at smaller intervals.     

A Lagrangian heuristic algorithm for a real-world train timetabling problem

The train timetabling problem (TTP) aims at determining an optimal timetable for a set of trains which does not violate track capacities and satisfies some operational constraints.In this paper, we describe the design of a train timetabling system that takes into account several additional constraints that arise in real-world applications. In particular, we address the following issues:

•

Manual block signaling for managing a train on a track segment between two consecutive stations.

•

Station capacities, i.e., maximum number of trains that can be present in a station at the same time.

•

Prescribed timetable for a subset of the trains, which is imposed when some of the trains are already scheduled on the railway line and additional trains are to be inserted.

•

Maintenance operations that keep a track segment occupied for a given period.

We show how to incorporate these additional constraints into a mathematical model for a basic version of the problem, and into the resulting Lagrangian heuristic. Computational results on real-world instances from Rete Ferroviaria Italiana (RFI), the Italian railway infrastructure management company, are presented.     

The signature operator at 2

It is well known that the signature operator on a manifold defines a K-homology class which is an orientation after inverting 2. Here we address the following puzzle: What is this class localized at 2, and what special properties does it have? Our answers include the following:

•

the K-homology class Δ_M of the signature operator is a bordism invariant;

•

the reduction mod 8 of the K-homology class of the signature operator is an oriented homotopy invariant;

•

the reduction mod 16 of the K-homology class of the signature operator is not an oriented homotopy invariant.

     

Bounds on sets with few distances

We derive a new estimate of the size of finite sets of points in metric spaces with few distances. The following applications are considered:

•

we improve the Ray-Chaudhuri-Wilson bound of the size of uniform intersecting families of subsets;

•

we refine the bound of Delsarte-Goethals-Seidel on the maximum size of spherical sets with few distances;

•

we prove a new bound on codes with few distances in the Hamming space, improving an earlier result of Delsarte.

We also find the size of maximal binary codes and maximal constant-weight codes of small length with 2 and 3 distances.     

Complementably universal Banach spaces, II

The two main results are:

A.

If a Banach space X is complementably universal for all subspaces of c₀ which have the bounded approximation property, then X^∗ is non-separable (and hence X does not embed into c₀).

B.

There is no separable Banach space X such that every compact operator (between Banach spaces) factors through X.

Theorem B solves a problem that dates from the 1970s.     

Krasnosel’skii type fixed point theorems under weak topology features

In this paper we prove the following Krasnosel’skii type fixed point theorem: Let M be a nonempty bounded closed convex subset of a Banach space X. Suppose that A:M→X and B:X→X are two weakly sequentially continuous mappings satisfying:

(i)

AM is relatively weakly compact;

(ii)

B is a strict contraction;

(iii)

.

Then A+B has at least one fixed point in M.This result is then used to obtain some new fixed point theorems for the sum of a weakly compact and a nonexpansive mapping. The results presented in this paper encompass several earlier ones in the literature.     

The topology of continua that are approximated by disjoint subcontinua

Suppose that is a collection of disjoint subcontinua of continuum X such that limi→∞dH(Yi,X)=0 where dH is the Hausdorff metric. Then the following are true:

(1)

X is non-Suslinean.

(2)

If each Yi is chainable and X is finitely cyclic, then X is indecomposable or the union of 2 indecomposable subcontinua.

(3)

If X is G-like, then X is indecomposable.

(4)

If all lie in the same ray and X is finitely cyclic, then X is indecomposable.

     

Galois representations modulo p and cohomology of Hilbert modular varieties

The aim of this paper is to extend some arithmetic results on elliptic modular forms to the case of Hilbert modular forms. Among these results let us mention:

•

control of the image of Galois representations modulo p,

•

Hida's congruence criterion outside an explicit set of primes,

•

freeness of the integral cohomology of a Hilbert modular variety over certain local components of the Hecke algebra and Gorenstein property of these local algebras.

We study the arithmetic properties of Hilbert modular forms by studying their modulo p Galois representations and our main tool is the action of inertia groups at primes above p. In order to determine this action, we compute the Hodge-Tate (resp. Fontaine-Laffaille) weights of the p-adic (resp. modulo p) étale cohomology of the Hilbert modular variety. The cohomological part of our paper is inspired by the work of Mokrane, Polo and Tilouine on the cohomology of Siegel modular varieties and builds upon geometric constructions of Tilouine and the author.     

A new generation of applied probability textbooks

This article reviews the following books:

•

S. Asmussen, Applied Probability and Queues, second ed., Springer, Berlin, 2003, ISBN 0-387-00211-1, xii+438pp., EUR 85.55.

•

H. Chen, D. Yao, Fundamentals of Queueing Networks, Springer, Berlin, 2003, ISBN 0-387-95166-0, xviii+405pp., EUR 74,95.

•

W. Whitt, Stochastic-Process Limits, Springer, Berlin, 2002, ISBN 0-387-95358-2, xxiv+602pp., EUR 106,95.

     

A canonical form for nonderogatory matrices under unitary similarity

A square matrix is nonderogatory if its Jordan blocks have distinct eigenvalues. We give canonical forms for

•

nonderogatory complex matrices up to unitary similarity, and

•

pairs of complex matrices up to similarity, in which one matrix has distinct eigenvalues.

The types of these canonical forms are given by undirected and, respectively, directed graphs with no undirected cycles.