Aut(X, σ) and Aut(X/σ) are not isomorphic in the category of surfaces with nodes

A symmetric Riemann surface is a pair (X,?σ) where X is a Riemann surface and?σ?is an anticonformal involution. We denote by Aut(X,?σ) the subgroup of Aut(X) defined by the automorphisms commuting with σ. There is a natural isomorphism between Aut(X,?σ) and Aut(X/σ). In this article we shall show that this isomorphism does not stand if X is a Riemann surface with nodes.     

Definability of Geometric Properties in Algebraically Closed Fields

We prove that there exists no sentence F of the language of rings with an extra binary predicat I₂ satisfying the following property: for every definable set X ? ?², X is connected if and only if (?, X) ? F, where I₂ is interpreted by X. We conjecture that the same result holds for closed subset of ?². We prove some results motivated by this conjecture.     

Factorization of Integer-Valued Polynomials with Square-Free Denominator

We describe an algorithm to compute the different factorizations of a given image primitive integer-valued polynomial f(X) = g(X)/d ∈ ?[X], where g ∈ ?[X] and d ∈ ? is square-free, assuming that the factorizations of g(X) in ?[X] and d in ? are known. We translate this problem into a combinatorial one.     

Deformations of trianalytic subvarieties of hyperkähler manifolds

Let M be a compact complex manifold equipped with a hyperk?hler metric, and X be a closed complex analytic subvariety of M. In alg-geom 9403006, we proved that X is trianalytic (i.e., complex analytic with respect to all complex structures induced by the hyperk?hler structure), provided that M is generic in its deformation class. Here we study the complex analytic deformations of trianalytic subvarieties. We prove that all deformations of X are trianalytic and naturally isomorphic to X as complex analytic varieties. We show that this isomorphism is compatible with the metric induced from M. Also, we prove that the Douady space of complex analytic deformations of X in M is equipped with a natural hyperk?hler structure.     

On von neumann's variation of the weierstrass-stone theorem

Let X be a compact HausdorfF space and let D(X) be the set of all continuous real-valued functions f defined on X and such that 0 ≤ f(x) ≤ 1, for all x ? X. The set D(X) is equipped with the uniform topology. We characterize the uniform closure of subsets A ? D(X) containing 0 and 1 and ?ψ + (1 ? ?)η, whenever they contain ?, ψ and η     

On relatively weakly almost Lindelöf subsets

Summary A subset Y of a space X is weakly almost Lindel?f in X if for every open cover U of X, there exists a countable subfamily V of U such that Y ⊆ comp (∩V ). We investigate the relationship between relatively weakly almost Lindel?f subsets and relatively almost Lindel?f subsets, and also study various properties of relatively weakly almost Lindel?f subsets.     

1 -convex manifolds are p- k?hler

We study here K?hler-type properties of 1-convex manifolds, using the duality between forms and compactly supported currents, and some properties of the Aeppli groups of (q-convex manifolds. We prove that, when the exceptional setS of the l-convex manifoldX has dimensionk, X is p-K?hler for everyp > k, and isk-K?hler if and only if “the fundamental class” ofS does not vanish. There are classical examples whereX is notk-K?hler even with a smoothS, but we prove that this cannot happen if2k ≥n = dimX, nor for suitable neighborhoods of S; in particular,X is always balanced (i.e.,(n - 1)-Kahler). Partially supported by MIUR research funds.     

On extensions of countable filterbases to ultrafilters and ultrafilter compactness

We show in the Zermelo-Fraenkel set theory ZF without the axiom of choice:

1. Given an infinite set X, the Stone space S(X) is ultrafilter compact.

2. For every infinite set X, every countable filterbase of X extends to an ultra-filter i? for every infinite set X, S(X) is countably compact.

3. ω has a free ultrafilter i? every countable, ultrafilter compact space is countably compact.

   We also show the following:

4. There are a permutation model 𝒩 and a set X ∈ 𝒩 such that X has no free ultrafilters and S(X) is not compact but S(X) is countably compact and every countable filterbase of X extends to an ultrafilter.

5. It is relatively consistent with ZF that every countable filterbase of ω extends to an ultrafilter but there exists a countable filterbase of ? which does not extend to an ultrafilter. Hence, it is relatively consistent with ZF that ? has free ultrafilters but there exists a countable filterbase of ? which does not extend to an ultrafilter.

     

Wirtinger numbers and holomorphic symplectic immersions

For any subvariety of a compact holomorphic symplectic K?hler manifold, we define the symplectic Wirtinger number W(X). We show that W(X) \leqslant 1,W(X) \leqslant 1, and the equality is reached if and only if the subvariety X ì MX \subset M is trianalytic, i.e. compatible with the hyperk?hler structure on M. For a sequence X₁ ? X₂ ? ?X_n ? MX_1 \to X_2 \to \ldots X_n \to M of immersions of simple holomorphic symplectic manifolds, we show that W( X₁ ) \leqslant W( X₂ ) \leqslant ?\leqslant W( X_n ).W\left( {X_1 } \right) \leqslant W\left( {X_2 } \right) \leqslant \ldots \leqslant W\left( {X_n } \right).     

The infinite duration lying oracle game

In the lying oracle game, a Player places bets on the outcomes of a sequence of coin flips. A second player, the Oracle, informs the Player what the outcome of each coin flip is, but may at times lie. We analyse this game when the duration of the game is infinite, where the Oracle's ability to lie or be truthful is specified by a set of lie patterns X, known to both players. By equipping X with Lebesgue measure, we prove that for any ε?>?0, the Player has a strategy that yields an expected fortune of at least 1/λ⁺(X)???ε, and that the Oracle has a strategy that limits the Player's expected fortune to at most 1/λ^?(X), where λ⁺(X) and λ^?(X) are the outer and inner Lebesgue measure of X, respectively.     

Open-Invariant Measures and the Covering Number of Sets

A result of J. Mycielski says that on every metric space (X, ?) with a non-empty compact thick set C ? X there exists a regular open-invariant Borel measure μ with μ(C) = 1. μ is called open-invariant if μ(A) = μ(B) for open isometric sets A, B ? X. We relate this result to the notion of a Hewitt-Stromberg measure and give a new independent existence proof for such an open-invariant measure μ on a compact metric space (X, ?). This proof works by induction, the well-known metric outer measure construction of Caratheodory-Hausdorff and a new property of the covering number N(X, q) of X.     

Arrangements of n Points whose Incident-Line-Numbers are at most n/2

We consider a set X of n noncollinear points in the Euclidean plane, and the set of lines spanned by X, where n is an integer with n ≥ 3. Let t(X) be the maximum number of lines incident with a point of X. We consider the problem of finding a set X of n noncollinear points in the Euclidean plane with t(X) ￡ ?n/2 ?{t(X) \le \lfloor n/2 \rfloor}, for every integer n ≥ 8. In this paper, we settle the problem for every integer n except n = 12k + 11 (k ≥ 4). The latter case remains open.     

Dunford-Pettis like properties on tensor products

In this paper we study equivalent formulations of the DP^? P_p (1 < p < ∞). We show that X has the DP^? P_p if and only if every weakly-p-Cauchy sequence in X is a limited subset of X. We give su?cient conditions on Banach spaces X and Y so that the projective tensor product X ?_π Y, the dual (X ?_? Y)^? of their injective tensor product, and the bidual (X ?_π Y)^?? of their projective tensor product, do not have the DP P_p, 1 < p < ∞. We also show that in some cases, the projective and the injective tensor products of two spaces do not have the DP^? P_p, 1 < p < ∞.     

Fonctions convexes et semimartingales

If ? is the difference of two convex functions on R ⁿ and X is a semimartingale, then ?(X) is also a semimartingale. We study the converse in this paper.     

An approximation theorem for sections of reflexive sheaves

 We show that, if X is a Stein manifold and D ? X an open set (not necessarily Stein) such that the restriction map has dense image, then, for any reflexive coherent analytic sheaf ℱ on X, the map has dense image, too. We also characterize the reflexivity of a torsion-free coherent sheaf on complex manifolds in terms of absolute gap sheaves or Kontinuit?tssatz. Received: 14 September 2001 / Revised version: 29 January 2002     

Real Hypersurfaces Having Many Pseudo-hyperplanes

Abstract

Let nand dbe natural integers satisfying n ≥ 3 and d ≥ 10. Let Xbe an irreducible real hypersurface Xin ? ⁿ of degree dhaving many pseudo-hyperplanes. Suppose that Xis not a projective cone. We show that the arrangement ? of all d ? 2 pseudo-hyperplanes of Xis trivial, i.e., there is a real projective linear subspace Lof ? ⁿ (?) of dimension n ? 2 such that L ? Hfor all H ∈ ?. As a consequence, the normalization of Xis fibered over ?¹in quadrics. Both statements are in sharp contrast with the case n = 2; the first statement also shows that there is no Brusotti-type result for hypersurfaces in ? ⁿ , for n ≥ 3.     

The Bourgain property and convex hulls

Let (Ω, Σ, μ) be a complete probability space and let X be a Banach space. We consider the following problem: Given a function f: Ω → X for which there is a norming set B ? B_{X *} such that Z_f,B = {x * ○ f: x * ∈ B } is uniformly integrable and has the Bourgain property, does it follow that f is Birkhoff integrable? It turns out that this question is equivalent to the following one: Given a pointwise bounded family ?? ? ?^Ω with the Bourgain property, does its convex hull co(??) have the Bourgain property? With the help of an example of D. H. Fremlin, we make clear that both questions have negative answer in general. We prove that a function f: Ω → X is scalarly measurable provided that there is a norming set B ? B_{X *} such that Z_f,B has the Bourgain property. As an application we show that the first problem has positive solution in several cases, for instance: (i) when B_{X *} is weak* separable; (ii) under Martin's axiom, for functions defined on [0, 1] with values in a Banach space with density character smaller than the continuum. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)