Ramification of valuations

In this paper we develop a very explicit theory of ramification of general valuations in algebraic function fields. In characteristic zero and arbitrary dimension, we obtain the strongest possible generalization of the classical ramification theory of local Dedekind domains. We further develop a ramification theory of algebraic functions fields of dimension two in positive characteristic. We prove that local monomialization and simultaneous resolution hold under very mild assumptions, and give pathological examples.     

Evaluation codes and plane valuations

We apply tools coming from singularity theory, as Hamburger–Noether expansions, and from valuation theory, as generating sequences, to explicitly describe order functions given by valuations of 2-dimensional function fields. We show that these order functions are simple when their ordered domains are isomorphic to the value semigroup algebra of the corresponding valuation. Otherwise, we provide parametric equations to compute them. In the first case, we construct, for each order function, families of error correcting codes which can be decodified by the Berlekamp–Massey–Sakata algorithm and we give bounds for their minimum distance depending on minimal sets of generators for the above value semigroup.     

Sections of convex bodies and splitting problem for selections

Let and be any convex-valued lower semicontinuous mappings and let be any linear surjection. The splitting problem is the problem of representation of any continuous selection f of the composite mapping L(F₁;F₂) in the form f=L(f₁;f₂), where f₁ and f₂ are some continuous selections of F₁ and F₂, respectively. We prove that the splitting problem always admits an approximate solution with fi being an ε-selection (Theorem 2.1). We also propose a special case of finding exact splittings, whose occurrence is stable with respect to continuous variations of the data (Theorem 3.1) and we show that, in general, exact splittings do not exist even for the finite-dimensional range.     

The expected convex hull trimmed regions of a sample

Given a data set in the multivariate Euclidean space, we study regions of central points built by averaging all their subsets with a fixed number of elements. The averaging of these sets is performed by appropriately scaling the Minkowski or elementwise summation of their convex hulls. The volume of such central regions is proposed as a multivariate scatter estimate and a circular sequence algorithm to compute the central regions of a bivariate data set is described. Extended version of the conference paper Cascos (2006) presented at the 17th Conference of the European Regional Section of the International Association for Statistical Computing (COMPSTAT 2006, Rome, Italy, August 28–September 1, 2006). Supported by the Spanish Ministry of Education and Science under grant MTM2005-02254.     

Lattice Polytopes Associated to Certain Demazure Modules of sl n + 1

Let w be an element of the Weyl group of sl _{n + 1}. We prove that for a certain class of elements w (which includes the longest element w₀ of the Weyl group), there exist a lattice polytope R ^l(w) , for each fundamental weight _i of sl _{n + 1}, such that for any dominant weight = _{i = 1} ⁿ a _{i i} , the number of lattice points in the Minkowski sum _w = _{i = 1} ⁿ a _{i i} ^w is equal to the dimension of the Demazure module E _w (). We also define a linear map A ^w : R ^l(w) P _Z R where P denotes the weight lattice, such that char E _w () = e e^–A(x) where the sum runs through the lattice points x of ^w .     

Blowing-ups and valuations on surfaces

In this paper, we show that all the nontrivial valuations on surfaces can be given by the infinite sequences of blowing-ups, and give the process of blowing-ups.     

极限的一个定理及其应用

给出了一个极限定理,能较好地解决一类特殊“和式”的极限问题.同时,利用对数函数的性质,又能够用来解决一些“积式”的极限.     

Convolution Estimates and Model Surfaces of Low Codimension

For k≥d/2 we give examples of measures on k-surfaces in ℝ ^d . These measures satisfy convolution estimates which are nearly optimal. The author was supported in part by NSF grant DMS-0552041.     

On the convolution and subordination of convex functions

In this work we present some new results on convolution and subordination in geometric function theory. We prove that the class of convex functions of order α is closed under convolution with a prestarlike function of the same order. Using this, we prove that subordination under the convex function order α is preserved under convolution with a prestarlike function of the same order. Moreover, we find a subordinating factor sequence for the class of convex functions. The work deals with several ideas and techniques used in geometric function theory, contained in the book Convolutions in Geometric Function Theory by Ruscheweyh (1982).     

Convolution products of probability measures on completely simple semigroups

Convolution products of probability measures are considered in the context of completely simple semigroups. Given a sequence of measures (μn)⊂Prob(S), where S is a finite completely simple semigroup, results are proven which (1) relate the supports of the measures in the sequence to the supports of the tail limit measures, and (2) determine necessary and sufficient conditions for convergence of the convolution products in the case of rectangular groups. An example showing how the theory can be used to analyze the convergence behavior of non-homogeneous Markov chains is included.     

卷积算子的交换子

利用Fourier变换估计和对算子的局部性质做精细的分析,建立与一类卷积算子交换子相关的一些算子的LP有界性结果．作为这些结果的应用,得到了与球平均算子交换子相关的离散极大算子以及一类低维集上奇异积分算子交换子等的LP有界性．     

两个分段函数卷积的数值算法

本文推广了文［１］中卷积的数值计算方法，使之适用于更广的范围，并用Ｃ语言编程举例说明．     

Convolution with measures on flat curves in low dimensions

We prove Lp→Lq convolution estimates for the affine arclength measure on certain flat curves in Rd when d∈{2,3,4}. For d=2,3, we also establish certain related Lorentz space estimates.     

二进小波变换的卷积定理

本文研究二进小波变换在信号处理中的应用.首先证明了两个满足容许性条件和规范性条件的二进小波的卷积和相关仍满足容许性和规范性条件,然后证明了二进小波变换的卷积定理和相关性定理,最后给出数值例子说明二进小波变换的卷积定理在加噪信号重构中的优越性.     

Convergence Acceleration of Taylor Sections by Convolution

Taylor sections S _n (f) of an entire function f often provide easy computable polynomial approximants of f . However, while the rate of convergence of (S _n (f)) _n is nearly optimal on circles around the origin, this is no longer true for other plane sets as, for example, real compact intervals. The aim of this paper is to construct for certain families of (entire) functions sequences of polynomial approximants which are computable with essentially the same effort as Taylor sections and which have a better rate of convergence on some parts of the plane. The resulting method may be applied, for example, to (modified) Bessel functions, to confluent hypergeometric functions, or to parabolic cylinder functions. October 2, 1997. Date revised: March 12, 1998. Date accepted: April 28, 1998.     

Convolution functions that are nowhere differentiable

In 1951 V. Jarník constructed two continuous functions whose Volterra convolution is nowhere differentiable. We generalize Jarník?s results by proving that the set of such functions is maximal lineable. This would shed some light on a question posed in 1973 on the structure of the set of continuous functions whose Volterra convolution is nowhere differentiable.     

Convolution on homogeneous groups

Let G be a homogeneous group with homogeneous dimension Q, and let So denote the space of Schwartz functions on G with all moments vanishing. Let be the usual Euclidean Fourier transform. For j∈R, we let be the space of J, smooth away from 0, satisfying |α^∂J(ξ)|?Cβ|ξ|^j−|β|, where both |ξ| and |β| are taken in the homogeneous sense. We characterize , and show that as elements of . If j₁,j₂,j₁+j₂>−Q, one can replace So, by S, S^′ in this result. A key ingredient of our proof is a lemma from the fundamental wavelet paper from 1985 by Frazier and Jawerth [4]. We believe that, in turn, our result will be useful in the theory of wavelets on homogeneous groups.     

Weak subordination for convex univalent harmonic functions

For two complex-valued harmonic functions f and F defined in the open unit disk Δ with f(0)=F(0)=0, we say f is weakly subordinate to F if f(Δ)⊂F(Δ). Furthermore, if we let E be a possibly infinite interval, a function with f(⋅,t) harmonic in Δ and f(0,t)=0 for each t∈E is said to be a weak subordination chain if f(Δ,t₁)⊂f(Δ,t₂) whenever t₁,t₂∈E and t₁<t₂. In this paper, we construct a weak subordination chain of convex univalent harmonic functions using a harmonic de la Vallée Poussin mean and a modified form of Pommerenke's criterion for a subordination chain of analytic functions.