Laws of large numbers for classes of functions

Let (X, , P) be a probability space and _n, n ≥ 1, a sequence of classes of measurable complex-valued functions on (X, , P). Under a weak metric entropy condition on _n and sup {g_∞: g _n}, Glivenko-Cantelli theorems are established for the classes _n with respect to the probability measure P; i.e., lim_{n → ∞} sup_{g n} ∫ g(dP_n − dP) = 0 a.s. The result is applied to kernel density estimation and a law of the logarithm is derived for the maximal deviation between a kernel density estimator and its expected value, improving upon and generalizing the recent results of W. Stute (Ann. Probab. 10 (1982), 414–422). This result is also used to derive improved rates of uniform convergence for the empirical characteristic function.     

Note on g-Hilbertian fields and nonrational curves¶with the Hilbert property

In a recent paper, Fried and Jarden prove the existence, for all integers g, of non-Hilbertian fields K which cannot be covered by a finite number of sets of the form ϕ (X(K)), where X is a curve of genus ≤g and ϕ is a rational function on X of degree ≥ 2. (If no bound is given on the genus we recover the notion of Hilbertian field.) This generalizes the case g=0, obtained previously by Corvaja and Zannier with a more elementary method. By a suitable modification of that method, we give here a new proof of the result of Fried and Jarden which avoids the use of deep group theoretical results. By a somewhat related construction we give an example of a curve X/Q of any prescribed genus and a Hilbertian field K⊂ˉQ such that X/K has the Hilbert property, i.e. the set of rational points X(K) is not thin. Received: 10 March 1998 / Revised version: 20 April 1998     

Ordered sets with small width and large jump number

The following result is proved in this note: For any positive integers w and t, if an ordered set P has jump number at least (t+1) ^w–1, then either the width of P is more than w, or P has a tower, i.e., a linear sum of pairs of noncomparable elements, of height more than t.     

Extending analyticK-subanalytic functions

Letg:U→ℝ (U open in ℝⁿ) be an analytic and K-subanalytic (i. e. definable in ℝ _an ^K , whereK, the field of exponents, is any subfield ofℝ) function. Then the set of points, denoted Σ, whereg does not admit an analytic extension is K-subanalytic andg can be extended analytically to a neighbourhood of Ū\∑. Partially supported by the European RTN Network RAAG (contract no. HPRN-CT-00271)     

Upper and lower bounds of the valence-functional

For the non-negative integerg let (M, g) denote the closed orientable 2-dimensional manifold of genusg. K-realizationsP of (M, g) are geometric cell-complexes inP with convex facets such that set (P) is homeomorphic toM. ForK-realizationsP of (M, g) and verticesv ofP, val (v,P) denotes the number of edges ofP incident withv and the weighted vertex-number Σ(val(v, P)-3) taken over all vertices ofP is called valence-valuev (P) ofP. The valence-functionalV, which is important for the determination of all possiblef-vectors ofK-realisations of (M, g), in connection with Eberhard's problem etc., is defined byV(g):=min[v(P)|P is aK-realization of (M,g)]. The aim of the note is to prove the inequality 2g+1≦V(g)≦3g+3 for every positive integerg.     

SPECIAL AUTOMORPHISMS WITH CONTINUOUS SPECTRUM

Abstract

In this paper conditions are given for the primitive automorphism of a cyclic KS approximation to have continuous spectrum. If T: X → X admits a cyclic KS approximation with speed o(1/n) it is then shown that for a dense set of measurable sets A € 𝔉, T^A: X^A → X^A is weakly mixing, i.e. has continuous spectrum. In particular it is shown that if T_α is an irrational rotation on the unit circle there exists an uncountable dense set of measurable sets for which (T^α)^A has continuous spectrum.     

On the greedy dimension of a partial order

This paper introduces a new concept of dimension for partially ordered sets. Dushnik and Miller in 1941 introduced the concept of dimension of a partial order P, as the minimum cardinality of a realizer, (i.e., a set of linear extensions of P whose intersection is P). Every poset has a greedy realizer (i.e., a realizer consisting of greedy linear extensions). We begin the study of the notion of greedy dimension of a poset and its relationship with the usual dimension by proving that equality holds for a wide class of posets including N-free posets, two-dimensional posets and distributive lattices.     

Maximizing the distance between center,centroid and characteristic set of a tree

Consider a tree P_n-g,g , n≥ 2, 1≤ g≤ n-1 on n vertices which is obtained from a path on [1,2,?…?,n-g] vertices by adding g pendant vertices to the pendant vertex n-g. We prove that over all trees on n?≥?5 vertices, the distance between center and characteristic set, centroid and characteristic set, and center and centroid is maximized by trees of the form P_n-g,g , 2?≤?g?≤?n-3. For n≥ 5, we also supply the precise location of the characteristic set of the tree P_n-g,g , 2?≤?g?≤?n-3.     

Strongly Amorphous Sets and Dual Dedekind Infinity

1. If A is strongly amorphous (i.e., all relations on A are definable), then its power set P(A) is dually Dedekind infinite, i. e., every function from P(A) onto P(A) is injective. 2. The class of “inexhaustible” sets is not closed under supersets unless AC holds.     

Faisceaux sans torsion et faisceaux quasi localement libres sur les courbes multiples primitives

This paper is devoted to the study of some coherent sheaves on non reduced curves that can be locally embedded in smooth surfaces. If Y is such a curve and n is its multiplicity, then there is a filtration C₁ = C ? C₂ ? … ? C_n = Y such that C is the reduced curve associated to Y, and for every P ∈ C, if z ∈ O_Y,P is an equation of C then (zⁱ) is the ideal of C_i in O_Y,P. A coherent sheaf on Y is called torsion free if it does not have any non zero subsheaf with finite support. We prove that torsion free sheaves are reflexive. We study then the quasi locally free sheaves, i.e., sheaves which are locally isomorphic to direct sums of the O_Ci.We define an invariant for these sheaves, the complete type, and prove the irreducibility of the set of sheaves of given complete type. We study the generic quasi locally free sheaves, with applications to the moduli spaces of stable sheaves on Y (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the topological type of anticonformal square roots of automorphisms of Riemann surfaces

Let X be a compact Riemann surface and f be a conformal automorphism of X of order n. An anticonformal square root of f is an anticonformal automorphism g of X such that g²=f. If g₁ and g₂ are two anticonformal square roots of f we study the problem of whether g₁ and g₂ have the same topological type, i. e., if there exists a homeomorphism h:X→X such that g₁=hg₂h⁻¹. We use techniques of noneuclidean crystallographic (NEC) groups and the topological classification of orientation reversing maps of finite period on surfaces given in [C1] and [Y]. Partially supported by DGICYT PB92-0716 and EC project CHRX-CT93-408     

A packing problem its application to Bose's families

We propose and study the following problem: given X ⊂ Z_n, construct a maximum packing of dev X (the development of X), i.e., a maximum set of pairwise disjoint translates of X. Such a packing is optimal when its size reaches the upper bound . In particular, it is perfect when its size is exactly equal to i.e. when it is a partition of Z_n. We apply the above problem for constructing Bose's families. A (q, k) Bose's family (BF) is a nonempty family F of subsets of the field GF(q) such that: (i) each member of F is a coset of the kth roots of unity for k odd (the union of a coset of the (k - 1)th roots of unity and zero for k even); (ii) the development of F, i.e., the incidence structure , is a semilinear space. A (q, k)-BF is optimal when its size reaches the upper bound . In particular, it is perfect when its size is exactly equal to ; in this case the (q, k)-BF is a (q, k, 1) difference family and its development is a linear space. If the set of (q, k)-BF's is not empty, there is a bijection preserving maximality, optimality, and perfectness between this set with the set of packings of dev X, where X is a suitable -subset of Z_n, for k odd, for k even. © 1996 John Wiley & Sons, Inc.     

Metrics on products of surfaces with non-positive sectional curvature

Let (S _i, g_i),i=1, 2 be two compact riemannian surfaces isometrically embedded in euclidean spaces. In this paper we show that ifM=S ₁×S₂,then for any functionF: M→R, the graph ofF, i.e. the manifold {(x, F(x)): x∈M}, does not have positive sectional curvature.     

Surfaces with pg=q=3

 In this paper it is proved that a complete algebraic surface of general type with p _g =q=3, without irrational pencil of genus bigger than one is birationally equivalent to the two-symmetric product of a curve of genus 3. This result completes the classification of the surfaces with p _g =q=3. The main tools are the Lefschetz theorem and the use of the paracanonical system on the surface. Received 14 August 2001 / Revised version: 17 January 2002     

On the maximum number of pairwise compatible euler cycles

B. Jackson [4] made the following conjecture: If G is an Eulerian graph with δ(G) ≥ 2k, then G has a set of 2k - 2 pairwise compatible Euler cycles (i.e., every pair of adjacent edges appears in at most one of these Euler cycles as a pair of consecutive edges). We verify this conjecture in the case where every circuit of G is a block of G.     

Certain 4-manifolds with non-negative sectional curvature

In this paper, we study certain compact 4-manifolds with non-negative sectional curvature K. If s is the scalar curvature and W. is the self-dual part of Weyl tensor, then it will be shown that there is no metric g on S ^• × S ^• with both (i) K > 0 and (ii) ÷ s − W _• ⩾ 0. We also investigate other aspects of 4-manifolds with non-negative sectional curvature. One of our results implies a theorem of Hamilton: “If a simply-connected, closed 4-manifold M^• admits a metric g of non-negative curvature operator, then M^• is one of S^•, ℂP^• and S^•×S^•”. Our method is different from Hamilton’s and is much simpler. A new version of the second variational formula for minimal surfaces in 4-manifolds is proved.      

Largest induced suborders satisfying the chain condition

For a finite ordered set P, let c(P) denote the cardinality of the largest subset Q such that the induced suborder on Q satisfies the Jordan-Dedekind chain condition (JDCC), i.e., every maximal chain in Q has the same cardinality. For positive integers n, let f(n) be the minimum of c(P) over all ordered sets P of cardinality n. We prove:      

Locally connected models for Julia sets

Let P be a polynomial with a connected Julia set J. We use continuum theory to show that it admits a finest monotone map φ onto a locally connected continuumJP_∼, i.e. a monotone map φ:J→JP_∼ such that for any other monotone map ψ:J→J^′ there exists a monotone map h with ψ=h°φ. Then we extend φ onto the complex plane C (keeping the same notation) and show that φ monotonically semiconjugates PC_| to a topological polynomialg:C→C. If P does not have Siegel or Cremer periodic points this gives an alternative proof of Kiwi's fundamental results on locally connected models of dynamics on the Julia sets, but the results hold for all polynomials with connected Julia sets. We also give a characterization and a useful sufficient condition for the map φ not to collapse all of J into a point.     

Molecules of a partially ordered set

It is shown that in a general partially ordered set (P, ≤) the notion of a molecule (i.e., a nonzero elementm ofP such that any two nonzero elements ofP which are ≤m have a nonzero lower bound) is in close analogy to the notion of an atom in a Boolean ring. Presented by R. P. Dilworth.     

Sets of nonnegative matrices with positive inhomogeneous products

Let X be a set of k×k matrices in which each element is nonnegative. For a positive integer n, let P(n) be an arbitrary product of n matrices from X, with any ordering and with repetitions permitted. Define X to be a primitive set if there is a positive integer n such that every P(n) is positive [i.e., every element of every P(n) is positive]. For any primitive set X of matrices, define the index g(X) to be the least positive n such that every P(n) is positive. We show that if X is a primitive set, then g(X)?2^k?2. Moreover, there exists a primitive set Y such that g(Y) = 2^k?2.