The Schubert Triangle Geometry on an Algebraic Surface

For a smooth complex projective surface, and for two families of curves with traditional singularities in it, we enumerate the pairs of curves in each family having two points of contact among them, thus generalizing the double contact formulae known or conjectured by Zeuthen and Schubert in the case of the complex projective plane. The technique we use to this purpose is a particular notion of triangle which can be defined in any smooth surface, thus potentially generalizing to arbitrary surfaces the Schubert technique of triangles.     

ON THE COMPUTATION OF TRACE FORMS OF ALGEBRAS WITH INVOLUTION

In this paper, we compute explicitely the isomorphism class of the trace of k-algebras with involution (of any kind) for some special base fields, especially for non formally real global fields, euclidean fields and the field of rational numbers.     

Bootstrapping complex functions

We formulate a new family of bootstrap algorithms suitable for learning non-Boolean functions from data. Within the Algorithmic Inference framework, the key idea is to consider a population of functions that are compatible with the observed sample. We generate items of this population from standard random seeds and reverse seed probabilities on the items. In this way we may compute in principle, and effectively achieve on paradigmatic examples, direct estimates and confidence intervals for any kind of complex function underlying the observed data according to any hypothesis on the randomness affecting the sample.     

Deformations of arithmetically Cohen-Macaulay subvarieties of pn

We show that every arithmetically Cohen-Macaulay two-codimensional subscheme ofP ⁿ can be deformed to a reduced union of two-codimensional linear subvarieties. This problem (classical for curves with the name of Zeuthen problem) was solved for curves by F.Gaeta.     

The real cohomology of virtually nilpotent groups

In this paper we present a method to compute the real cohomology of any finitely generated virtually nilpotent group. The main ingredient in our setup consists of a polynomial crystallographic action of this group. As any finitely generated virtually nilpotent group admits such an action (which can be constructed quite easily), the approach we present applies to all these groups. Our main result is an algorithmic way of computing these cohomology spaces. As a first application, we prove a kind of Poincaré duality (also in the nontorsion free case) and we derive explicit formulas in the virtually abelian case.

     

Orthogonal Helmholtz decomposition in arbitrary dimension using divergence-free and curl-free wavelets

We present tensor-product divergence-free and curl-free wavelets, and define associated projectors. These projectors enable the construction of an iterative algorithm to compute the Helmholtz decomposition of any vector field, in wavelet domain. This decomposition is localized in space, in contrast to the Helmholtz decomposition calculated by Fourier transform. Then we prove the convergence of the algorithm in dimension two for any kind of wavelets, and in larger dimension for the particular case of Shannon wavelets. We also present a modification of the algorithm by using quasi-isotropic divergence-free and curl-free wavelets. Finally, numerical tests show the validity of this approach for a large class of wavelets.     

Spectra of graphs attached to the space of melodies

For any integer n greater than or equal to two, two intimately related graphs on the vertices of the n-dimensional cube are introduced. All of their eigenvalues are found to be integers, and the largest and the smallest ones are also determined. As a byproduct, certain kind of generating function for their spectra is introduced and shown to be quite effective to compute the eigenvalues of some broader class of adjacency matrices of graphs.     

Restricted colored permutations and Chebyshev polynomials

Several authors have examined connections between restricted permutations and Chebyshev polynomials of the second kind. In this paper we prove analogues of these results for colored permutations. First we define a distinguished set of length two and length three patterns, which contains only 312 when just one color is used. Then we give a recursive procedure for computing the generating function for the colored permutations which avoid this distinguished set and any set of additional patterns, which we use to find a new set of signed permutations counted by the Catalan numbers and a new set of signed permutations counted by the large Schröder numbers. We go on to use this result to compute the generating functions for colored permutations which avoid our distinguished set and any layered permutation with three or fewer layers. We express these generating functions in terms of Chebyshev polynomials of the second kind and we show that they are special cases of generating functions for involutions which avoid 3412 and a layered permutation.     

An Analog Characterization of the Grzegorczyk Hierarchy

We study a restricted version of Shannon's general purpose analog computer in which we only allow the machine to solve linear differential equations. We show that if this computer is allowed to sense inequalities in a differentiable way, then it can compute exactly the elementary functions, the smallest known recursive class closed under time and space complexity. Furthermore, we show that if the machine has access to a function f(x) with a suitable growth as x goes to infinity, then it can compute functions on any given level of the Grzegorczyk hierarchy. More precisely, we show that the model contains exactly the nth level of the Grzegorczyk hierarchy if it is allowed to solve n−3 non-linear differential equations of a certain kind. Therefore, we claim that, at least in this region of the complexity hierarchy, there is a close connection between analog complexity classes, the dynamical systems that compute them, and classical sets of subrecursive functions.     

A Direct Splitting Method for Nonsmooth Variational Inequalities

We propose a direct splitting method for solving a nonsmooth variational inequality in Hilbert spaces. The weak convergence is established when the operator is the sum of two point-to-set and monotone operators. The proposed method is a natural extension of the incremental subgradient method for nondifferentiable optimization, which strongly explores the structure of the operator using projected subgradient-like techniques. The advantage of our method is that any nontrivial subproblem must be solved, like the evaluation of the resolvent operator. The necessity to compute proximal iterations is the main difficulty of other schemes for solving this kind of problem.     

Fundamental Groups of Triangle Geometries

利用Building理论获得一种计算某些三角几何的基本群的新的方法.这种方法能够容易地计算出无限多有限三角几何的基拓扑基本群.     

一类RNA二级结构的计数

多核苷酸的二级结构可视为一类顶点标号平面图，通常通过枚举每类RNA二级结构图的各种子图来计算其递推公式。本文作者给出了限制端环长度的RNA二级结构的递推公式，并运用隐式估计法计算它的渐近值。     

准时制下非核心企业最佳生产数学模型研究

通常,研究订单生产企业的生产计划求解模型往往只解决排产问题,制作的计划不能顾及各个方面的生产成本.其主要原因在于两个方面:一是,没有对由计划引发的成本作全面的考虑;二是,缺乏有效的成本表达方式,尤其是任意时间段内的各个产品的负荷总和以及能力波动费用的数学表达.研究了一种特殊的订单生产——准时制下非核心企业的生产,分析了计划范围内的成本,并分别就各种成本给出求解公式,尤其在能力波动费用的求解上,引进了符号函数,解决了在任意单位时间段内能力累计的数学表达以及能力波动费用的数学求解,在以上基础上给出了最佳生产的数学模型,给出了求解方法,讨论了最优解的存在条件,最后,讨论了在APS(A dvance P lan System)中的应用前景.     

Paralic Confinement: Models and Simulations

This paper deals with the mathematical modelling of confinement of paralic ecosystems. It is based on the recent paper (Frénod and Goubert in Ecol. Model. 200(1–2):139–148, 2007) that presents a modelling procedure in order to compute the confinement field of a lagoon. Here, we improve the existing model in order to account for tide oscillations in any kind of geometry such as non-rectangular lagoons with a non-flat bottom. The new model, that relies on PDEs rather than ODEs, is then implemented thanks to the finite element method. Numerical results confirm the feasibility of confinement studies thanks to the introduced model.     

The symplectic floer homology of the square knot and granny knots

Using a kind of Mayer-Vietoris principle for the symplectic Floer homology of knots, we compute the symplectic Floer homology of the square knot and granny knots. Partially supported by NSF Grant DMS 9626166     

Generators of some non-commutative stochastic processes

A fundamental result of Biane (Math Z 227:143–174, 1998) states that a process with freely independent increments has the Markov property, but that there are two kinds of free Lévy processes: the first kind has stationary increments, while the second kind has stationary transition operators. We show that a process of the first kind (with mean zero and finite variance) has the same transition operators as the free Brownian motion with appropriate initial conditions, while a process of the second kind has the same transition operators as a monotone Lévy process. We compute an explicit formula for the generators of these families of transition operators, in terms of singular integral operators, and prove that this formula holds on a fairly large domain. We also compute the generators for the $q$ -Brownian motion, and for the two-state free Brownian motions.     

On the spaltenstein-steinberg map for classical lie algebras

We compute the fibers of the canonical map from Weyl group elements to orbital varieties for a Lie algebra of type B,C, or D. We also compute one irreducible component of the associated variety of any simple highest weight module in any classical type, thereby sharpening a result of Joseph and partially proving a conjecture of Tanisaki.     

On a kind of curious binomial identity

As a generalization of Calkin's identity and its alternating form, we compute a kind of binomial identity involving some real number sequences and a partial sum of the binomial coefficients, from which many interesting identities follow.     

Fitting Quantiles: Doubling,HR, HQ,and HHH Distributions

Abstract

This article introduces a family of distributional shapes which is flexible in the sense that it contains skewed and symmetric laws as well as heavy-tailed and light-tailed laws. The proposed family is also practically convenient because it is easy to fit to a table of quantiles from any distribution. Inversely, for each of the distributional shapes it is trivial to compute quantiles for any desired probability, and it is possible to compute the corresponding densities.