Polynômes d'Euler et Fractions Continues de Stieltjes-Rogers

It is well-known that the Euler polynomials E2n(x) with n 0 can be expressed as a polynomial Hn(x(x – 1)) of x(x – 1). We extend Hn(u) to formal power series for n < 0 and prove several properties of the coefficients appearing in these polynomials or series, which generalize some recent results, independently obtained by Hammersley [7] and Horadam [8], and answer a question of Kreweras [9]. We also deduce several continued fraction expansions for the generating function of Euler polynomials, some of these formulae had been published by Stieltjes [14] and by Rogers [12] without proof. These formulae generalize our earlier results concerning Genocchi numbers, Euler numbers and Springer numbers [5, 4].     

Betti Numbers of Small Covers and Their Two-Fold Coverings

We compute the Betti numbers of two-fold coverings of small covers with some special properties; in particular, we use the results of Davis and Januszkiewicz on the cohomology of small covers. It turned out that their proof contains some gap that we describe in detail and fill in.     

A characterization of the Â-genus as a linear combination of Pontrjagin numbers

We show in this short note that if a rational linear combination of Pontrjagin numbers vanishes on all simply-connected 4k-dimensional closed connected and oriented spin manifolds admitting a Riemannian metric whose Ricci curvature is nonnegative and not identically zero, then this linear combination must be a multiple of the Â-genus, which improves a result of Gromov and Lawson. Our proof combines an idea of Atiyah and Hirzebruch and the celebrated Calabi–Yau theorem.     

On Inequalities of the Landau–Kolmogorov–Hörmander Type on a Segment and Real Straight Line

We prove inequalities of the Landau–Kolmogorov–Hörmander type for the uniform norms (on some subinterval) of positive and negative parts of intermediate derivatives of functions defined on a finite interval. By using the limit transition, we obtain a new proof or the well-known Hörmander result.     

Superfilters, Ramsey theory, and van der Waerden's Theorem

Superfilters are generalizations of ultrafilters, and capture the underlying concept in Ramsey-theoretic theorems such as van der Waerden's Theorem. We establish several properties of superfilters, which generalize both Ramsey's Theorem and its variants for ultrafilters on the natural numbers. We use them to confirm a conjecture of Kočinac and Di Maio, which is a generalization of a Ramsey-theoretic result of Scheepers, concerning selections from open covers. Following Bergelson and Hindman's 1989 Theorem, we present a new simultaneous generalization of the theorems of Ramsey, van der Waerden, Schur, Folkman–Rado–Sanders, Rado, and others, where the colored sets can be much smaller than the full set of natural numbers.     

Hodge Integrals and Hurwitz Numbers via Virtual Localization

We give another proof of Ekedahl, Lando, Shapiro, and Vainshtein's remarkable formula expressing Hurwitz numbers (counting covers of P¹ with specified simple branch points, and specified branching over one other point) in terms of Hodge integrals. Our proof uses virtual localization on the moduli space of stable maps. We describe how the proof could be simplified by the proper algebro-geometric definition of a 'relative space'. Such a space has recently been defined by J. Li.     

Generalization of the Gale–Ryser Theorem

We prove an inequality for the Kostka–Foulkes polynomials K_λ,μ(q) and give a criteria for the existence of a unique configuration of the given type (λ, μ). As a corollary, we obtain a nontrivial lower bound for the Kostka numbers which is a generalization the Gale–Ryser theorem on an existence of a (0,1)-matrix with given sums of rows and columns. A new proof of the Berenstein–Zelevinsky weight-multiplicity-one criteria is given.     

論Понтрягин示性類(Ⅳ)

<正> 前言 在前一文中,我們曾應用Steenrod冪以證明一個可微分閉流形上法示性類的拓撲不變性.本文的目的則在提供一個不同的方法,不假助於Steenrod冪以證明同一事實.同樣的方法,亦可用於Stiefel-Whitney示性類,而由此獲得這些類的拓撲不變性的一個不同證明,而在證明中避免用到Steenrod     

Spin^{\rm c} -manifolds with Pin(2)-action

We study -manifolds with Pin(2)-action. The main tool is a vanishing theorem for certain indices of twisted -Dirac operators. This theorem is used to show that the Witten genus vanishes on such manifolds provided the first Chern class and the first Pontrjagin class are torsion. We apply the vanishing theorem to cohomology complex projective spaces and give partial evidence for a conjecture of Petrie. For example we prove that the total Pontrjagin class of a cohomology with -action has standard form if the first Pontrjagin class has standard form. We also determine the intersection form of certain 4-manifolds with Pin(2)-action. Received: 26 June 1998     

Bilinear Singular Integral Operators,Smooth Atoms and Molecules

We present a bilinear T1 theorem in the context of Triebel–Lizorkin spaces. The proof uses atomic decomposition techniques and some a priori L ^\infty-estimates for the action of bilinear Calderón–Zygmund operators.     

A Simpler Proof of the Excluded Minor Theorem for Higher Surfaces

We give a simple proof of the fact (which follows from the Robertson–Seymour theory) that a graph which is minimal of genusgcannot contain a subdivision of a large grid. Combining this with the tree-width theorem and the quasi-wellordering of graphs of bounded tree-width in the Robertson–Seymour theory, we obtain a simpler proof of the generalized Kuratowski theorem for each fixed surface. The proof requires no previous knowledge of graph embeddings.     

Pontrjagin numbers and pseudo-riemannian geometry

Summary An integral formula for the Pontrjagin numbers of a compact orientable real 4k dimensional differentiable manifold which has a pseudo-Riemannian metric is derived. This formula allows the Pontrjagin numbers to be expressed in terms of the index, or signature, of the differentiable manifold. The application of these formulae to the four dimensional Lorentzian manifolds of the general theory of relativity is discussed. A corresponding formula for the Chern numbers of a complex differentiable manifold with a Hermitian metric is also given.     

A new topological proof of the Abhyankar–Moh theorem

We present a new simple proof of the famous theorem of Abhyankar, Moh and Suzuki about rational curves in a plane. This proof relies on the Poincaré–Hopf theorem. This work was supported by Polish KBN Grant No 2 P03A 041 15Mathematics Subject Classification (2002):14E25.     

A New Proof of the Assmus–Mattson Theorem for Non-Binary Codes

C. Bachoc gave a new proof of theAssmus–Mattson theorem for linear binary codes using harmonicweight enumerators which she defined B. We give a new proof ofthe Assmus–Mattson theorem for linear codes over any finitefield using similar methods.     

Stokes Cocycle and Differential Galois Groups

The classification of germs of ordinary linear differential systems with meromorphic coefficients at 0 under convergent gauge transformations and fixed normal form is essentially given by the non-Abelian 1-cohomology set of Malgrange–Sibuya. (Germs themselves are actually classified by a quotient of this set.) It is known that there exists a natural isomorphism h between a unipotent Lie group (called the Stokes group) and the 1-cohomology set of Malgrange–Sibuya; the inverse map which consists of choosing, in each cohomology class, a special cocycle called a Stokes cocycle is proved to be natural and constructive. We survey here the definition of the Stokes cocycle and give a combinatorial proof for the bijectivity of h. We state some consequences of this result, such as Ramis, density theorem in linear differential Galois theory; we note that such a proof based on the Stokes cocycle theorem and the Tannakian theory does not require any theory of (multi-)summation.     

Normal characteristic numbers

This paper computes the greatest common divisor of certain normal Pontrjagin numbers and Chern numbers. These calculations are of interest in studying singularities.

     

Curvature identities derived from the integral formula for the first Pontrjagin number

We give an integral formula for the first Pontrjagin number of a compact almost Hermitian surface and derive curvature identities from the integral formula based on the fundamental fact that the first Pontrjagin number in the deRham cohomology group is a topological invariant. Further, we provide some applications of the identities.     

Algebraic number fields

This survey of the theory of algebraic numbers covers material abstracted in theReferativnyi Zhurnal Matematika during the period 1975–1980. The survey focused mainly on the arithmetic of Abelian and cyclotomic fields.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 22, pp. 117–204, 1984.     

Weighted inversion numbers,restricted growth functions,and standard young tableaux

We give a family of weighted inversion numbers with the same generating function which interpolate between the inversion number and MacMahon's major index. Foata's bijection is obtained in a natural way from a simple involution. An alternative proof uses q-difference equations which yield some new results. We obtain a new generating function for restricted growth functions and two q-analogs of a formula for the number of standard Young tableaux of a given shape. While the first really goes back to MacMahon, the second uses one of our weighted inversion numbers and appears to be new.     

Instanton moduli as a novel map from tori to K3-surfaces

A map is constructed from the moduli of hyper-Kähler tori to hyper-Kähler K3 surfaces which does not coincide with the Kummer map. The map takes a torus to the moduli space of SO(3) connections on a bundle with nontrivial first Stiefel-Whitney class and first Pontrjagin class equal to –4. This map is shown to intersect the Kummer moduli and also certain subvarieties of singular K3 surfaces. Our map is shown to satisfy the local Torelli theorem, and the K3-surfaces in its image are shown to carry a natural metric which is Calabi-Yau.