The Zassenhaus filtration,Massey products,and representations of profinite groups

We consider the p  -Zassenhaus filtration (_Gn)$(G_n)$ of a profinite group G  . Suppose that G=S/N$G=S/N$ for a free profinite group S and a normal subgroup N of S   contained in _Sn$S_n$. Under a cohomological assumption on the n-fold Massey products (which holds, e.g., if G has p  -cohomological dimension ≤ 1), we prove that _Gn+1$G_{n+1}$ is the intersection of all kernels of upper-triangular unipotent (n+1)$(n+1)$-dimensional representations of G   over _Fp${\mathbb{F}}_p$. This extends earlier results by Miná?, Spira, and the author on the structure of absolute Galois groups of fields.     

Power integral bases for Selmer-like number fields

The Selmer trinomials are the trinomials f(X)∈{Xn−X−1,Xn+X+1|n>1 is an integer} over Z. For these trinomials we show that the ideal C=(f(X),f^′(X))Z[X] has height two and contains the linear polynomial (n−1)X+n. We then give several necessary and sufficient conditions for D[X]/(f(X)D[X]) to be a regular ring, where f(X) is an arbitrary polynomial over a Dedekind domain D such that its ideal C has height two and contains a product of primitive linear polynomials. We next specialize to the Selmer-like trinomials bXn+cX+d and bXn+cXn−1+d over D and give several more such necessary and sufficient conditions (among them is that C is a radical ideal). We then specialize to the Selmer trinomials over Z and give quite a few more such conditions (among them is that the discriminant Disc(Xn−X−1)=±(nn−(1−n)ⁿ⁻¹) of Xn−X−1 is square-free (respectively Disc(Xn+X+1)=±(nn+(1−n)ⁿ⁻¹) of Xn+X+1 is square-free)). Finally, we show that nn+(1−n)ⁿ⁻¹ is never square-free when n≡2 (mod 3) and n>2, but, otherwise, both are very often (but not always) square-free.     

The Farahat-Higman ring of wreath products and Hilbert schemes

We study the structure constants of the class algebra of the wreath products Γ_n associated to an arbitrary finite group Γ with respect to the basis of conjugacy classes. We show that a suitable filtration on gives rise to the graded ring with non-negative integer structure constants independent of n (some of which are computed), which are then encoded in a Farahat-Higman ring . The real conjugacy classes of Γ come to play a distinguished role and are treated in detail in the case when Γ is a subgroup of . The above results provide new insight to the cohomology rings of Hilbert schemes of points on a quasi-projective surface X.     

A Berry-Esseen type bound in kernel density estimation for strong mixing censored samples

In this paper, we discuss the estimation of a density function based on censored data by the kernel smoothing method when the survival and the censoring times form a stationary α-mixing sequence. A Berry-Esseen type bound is derived for the kernel density estimator at a fixed point x. For practical purposes, a randomly weighted estimator of the density function is also constructed and investigated.     

Weak approximation on del Pezzo surfaces of degree 1

We study del Pezzo surfaces of degree 1 of the form

w²=z³+Ax⁶+By⁶     

Continuous weak selections for products

A weak selection on an infinite set X   is a function σ:[X]²→X$\sigma :{[X]}^2\to X$ such that σ({x,y})∈{x,y}$\sigma (\{x,y\})\in \{x,y\}$ for each {x,y}∈[X]²$\{x,y\}\in {[X]}^2$. A weak selection on a space is said to be continuous if it is a continuous function with respect to the Vietoris topology on [X]²${[X]}^2$ and the topology on X  . We study some topological consequences from the existence of a continuous weak selection on the product X×Y$X\times Y$ for the following particular cases:

(i)

Both X and Y are spaces with one non-isolated point.     

Classification of regular embeddings of hypercubes of odd dimension

By a regular embedding of a graph into a closed surface we mean a 2-cell embedding with the automorphism group acting regularly on flags. Recently, Kwon and Nedela [Non-existence of nonorientable regular embeddings of n-dimensional cubes, Discrete Math., to appear] showed that no regular embeddings of the n-dimensional cubes Q_n into nonorientable surfaces exist for any positive integer n>2. In 1997, Nedela and Škoviera [Regular maps from voltage assignments and exponent groups, European J. Combin. 18 (1997) 807-823] presented a construction giving for each solution of the congruence a regular embedding M_e of the hypercube Q_n into an orientable surface. It was conjectured that all regular embeddings of Q_n into orientable surfaces can be constructed in this way. This paper gives a classification of regular embeddings of hypercubes Q_n into orientable surfaces for n odd, proving affirmatively the conjecture of Nedela and Škoviera for every odd n.     

Perverse coherent sheaves and the geometry of special pieces in the unipotent variety

Let X be a scheme of finite type over a Noetherian base scheme S admitting a dualizing complex, and let U⊂X be an open set whose complement has codimension at least 2. We extend the Deligne-Bezrukavnikov theory of perverse coherent sheaves by showing that a coherent intermediate extension (or intersection cohomology) functor from perverse sheaves on U to perverse sheaves on X may be defined for a much broader class of perversities than has previously been known. We also introduce a derived category version of the coherent intermediate extension functor.Under suitable hypotheses, we introduce a construction (called “S₂-extension”) in terms of perverse coherent sheaves of algebras on X that takes a finite morphism to U and extends it in a canonical way to a finite morphism to X. In particular, this construction gives a canonical “S₂-ification” of appropriate X. The construction also has applications to the “Macaulayfication” problem, and it is particularly well-behaved when X is Gorenstein.Our main goal, however, is to address a conjecture of Lusztig on the geometry of special pieces (certain subvarieties of the unipotent variety of a reductive algebraic group). The conjecture asserts in part that each special piece is the quotient of some variety (previously unknown for the exceptional groups and in positive characteristic) by the action of a certain finite group. We use S₂-extension to give a uniform construction of the desired variety.     

On the independence of Heegner points associated to distinct quadratic imaginary fields

Let E/Q be an elliptic curve with no CM and a fixed modular parametrization and let be Heegner points attached to the rings of integers of distinct quadratic imaginary fields k₁,…,kr. We prove that if the odd parts of the class numbers of k₁,…,kr are larger than a constant C=C(E,ΦE) depending only on E and ΦE, then the points P₁,…,Pr are independent in .     

Large deviations for statistics of the Jacobi process

This paper aims to derive large deviations for statistics of the Jacobi process already conjectured by M. Zani in her thesis. To proceed, we write in a simpler way the Jacobi semi-group density. Being given by a bilinear sum involving Jacobi polynomials, it differs from Hermite and Laguerre cases by the quadratic form of its eigenvalues. Our attempt relies on subordinating the process using a suitable random time change. This gives a Mehler-type formula whence we recover the desired semi-group density. Once we do, an adaptation of Zani’s result [M. Zani, Large deviations for squared radial Ornstein–Uhlenbeck processes, Stochastic. Process. Appl. 102 (1) (2002) 25–42] to the non-steep case will provide the required large deviations principle.     

Centers for a 6-parameter family of polynomial vector fields of arbitrary degree

For all non-negative integers n₁,n₂,n₃,j₁,j₂ and j₃ with nk+jk>1 for k=1,2,3, (nk,jk)≠(nl,jl) if k≠l, j₃=n₃−1 and jk≠nk−1 for k=1,2, we study the center variety of the 6-parameter family of real planar polynomial vector given, in complex notation, by , where z=x+iy and A,B,C∈C\{0}.     

Vector fields on smooth threefolds vanishing on complete intersections

A result of J. Wahl shows that the existence of a vector field vanishing on an ample divisor of a projective normal variety X implies that X is a cone over this divisor. If X is smooth, X will be isomorphic to the n-dimensional projective space. This paper is a first attempt to generalize Wahl's theorem to higher codimensions: Given a complex smooth projective threefold X and a vector field on X vanishing on an irreducible and reduced curve which is the scheme theoretic intersection of two ample divisors, X is isomorphic to the 3-dimensional projective space or the 3-dimensional quadric. Received: 24 April 2001     

Infinitesimal bendings of high orders for homogeneous surfaces with positive curvature and a flat point

For any given n∈Z⁺ and k∈Z⁺, infinitesimal bendings of order n and of class Ck are constructed for a class of surfaces in R³. These surfaces are given as graphs of homogeneous polynomials P(x,y) and they have positive curvature except at the origin.     

Further examples of maximal curves

It is shown that the curve over F_q²ⁿ with n≥3 odd, that generalizes Serre’s curve y⁴+y=x³ over F₆₄, is also maximal. We also investigate a family of maximal curves over F_q²ⁿ and provide isomorphisms between these curves.     

Tail behavior of random products and stochastic exponentials

In this paper we study the distributional tail behavior of the solution to a linear stochastic differential equation driven by infinite variance α$\alpha$-stable Lévy motion. We show that the solution is regularly varying with index α$\alpha$. An important step in the proof is the study of a Poisson number of products of independent random variables with regularly varying tail. The study of these products merits its own interest because it involves interesting saddle-point approximation techniques.     

Discrete Morse theory for totally non-negative flag varieties

In a seminal 1994 paper Lusztig (1994) [26], Lusztig extended the theory of total positivity by introducing the totally non-negative part (G/P)_?0 of an arbitrary (generalized, partial) flag variety G/P. He referred to this space as a “remarkable polyhedral subspace”, and conjectured a decomposition into cells, which was subsequently proven by the first author Rietsch (1998) [33]. In Williams (2007) [40] the second author made the concrete conjecture that this cell decomposed space is the next best thing to a polyhedron, by conjecturing it to be a regular CW complex that is homeomorphic to a closed ball. In this article we use discrete Morse theory to prove this conjecture up to homotopy-equivalence. Explicitly, we prove that the boundaries of the cells are homotopic to spheres, and the closures of cells are contractible. The latter part generalizes a result of Lusztig's (1998) [28], that (G/P)_?0 - the closure of the top-dimensional cell - is contractible. Concerning our result on the boundaries of cells, even the special case that the boundary of the top-dimensional cell (G/P)_>0 is homotopic to a sphere, is new for all G/P other than projective space.     

A classification of CO spaces which are continuous images of compact ordered spaces

A topological space X is called a CO space, if every closed subset of X is homeomorphic to some clopen subset of X. Every ordinal with its order topology is a CO space. This work gives a complete classification of CO spaces which are continuous images of compact ordered spaces.     

Tests for multiple regression based on simplicial depth

A general approach for developing distribution free tests for general linear models based on simplicial depth is applied to multiple regression. The tests are based on the asymptotic distribution of the simplicial regression depth, which depends only on the distribution law of the vector product of regressor variables. Based on this formula, the spectral decomposition and thus the asymptotic distribution is derived for multiple regression through the origin and multiple regression with Cauchy distributed explanatory variables. The errors may be heteroscedastic and the concrete form of the error distribution does not need to be known. Moreover, the asymptotic distribution for multiple regression with intercept does not depend on the location and scale of the explanatory variables. A simulation study suggests that the tests can be applied also to normal distributed explanatory variables. An application on multiple regression for shape analysis of fishes demonstrates the applicability of the new tests and in particular their outlier robustness.     

Limit theorems in the Fourier transform method for the estimation of multivariate volatility

In this paper, we prove some limit theorems for the Fourier estimator of multivariate volatility proposed by Malliavin and Mancino (2002, 2009) [14] and [15]. In a general framework of discrete time observations we establish the convergence of the estimator and some associated central limit theorems with explicit asymptotic variance. In particular, our results show that this estimator is consistent for synchronous data, but possibly biased for non-synchronous observations. Moreover, from our general central limit theorem, we deduce that the estimator can be efficient in the case of a synchronous regular sampling. In the non-synchronous sampling case, the expression of the asymptotic variance is in general less tractable. We study this case more precisely through the example of an alternate sampling.