A note on Property Np

Let M be a very ample line bundle on a smooth complex projective variety Y and let ϕ _M :Y→P(H ⁰(Y, M)^*) be the map associated to M; we are concerned with the problem to see whether the syzygies of ϕ _M (Y) give information on the syzygies of ϕ _M ^s (Y). In particular we prove that if Y is a smooth complex projective variety and M is a line bundle on Y satisfying Property N _p , then M ^s satisfies Property N _p if s≥p. Received: 11 June 1999 / Revised version: 22 November 1999     

有穷等差数列的一个性质及其应用

本文指出从有穷等差数列中可重复地任取两项求和时 ,不相等的和数的个数及每个和数出现的频数 ,据此可以解决一类古典概率问题 .     

On the Summing Property of the Sobolev Embedding Operators

We prove that the Sobolev embedding operator S _d,k,p : , where 1/s=1/p-k/d , is (v,1) -absolutely summing for appropriate v > 1 . The result is optimal for s 2 .     

On Semidirect Products and the Arithmetic Lifting Property

Let G be a finite group and let K be a hilbertian field. Manyfinite groups have been shown to satisfy the arithmetic liftingproperty over K, that is, every G-Galois extension of K arisesas a specialization of a geometric branched covering of theprojective line defined over K. The paper explores the situationwhen a semidirect product of two groups has this property. Inparticular, it shows that if H is a group that satisfies thearithmetic lifting property over K and A is a finite cyclicgroup then G = A H also satisfies the arithmetic lifting propertyassuming the orders of H and A are relatively prime and thecharacteristic of K does not divide the order of A. In thiscase, an arithmetic lifting for any AH-Galois extension of Kis explicitly constructed and the existence of the arithmeticlifting for any G-Galois extension is deduced. It is also shownthat if A is any abelian group, and H is the group with thearithmetic lifting property then AH satisfies the property aswell, with some assumptions on the ground field K. In the constructionproperties of Hilbert sets in hilbertian fields and spectralsequences in étale cohomology are used.     

Embedding of Biological Regulatory Networks and Property Preservation

In the course of understanding biological regulatory networks (BRN), scientists usually start by studying small BRNs that they believe to be of particular importance to represent a biological function, and then, embed them in a larger network. Such a reduction can lead to neglect relevant regulations and to study a network whose properties can be very different from the properties of this network viewed as a part of the whole. In this paper we study, from a logical point of view, on which conditions concerning both networks, properties can be inherited by BRNs from sub-BRNs. We give some conditions on the nature of the network embeddings ensuring that dynamic properties on the embedded sub-BRNs are preserved at the level of the whole BRN.     

Property insurance,quality and reservation premium: A note

The aim of this note is to investigate the effects of the introduction of a ‘quality’ variable in a model of non-life insurance demand. The framework, based on the model of Mossin (1968) with deductible, analyses the variations of the reservation premium when the quality is of the additive or the multiplicative type and the deductible is absolute or proportional.     

A note on the core

This note studies an exchange economy in which there are n traders and n “kinds” of commodities. Each trader has n utility functions corresponding to n “kinds” of commodities, respectively. Thus, a multiple non-transferable utility game can be derived from this exchange economy. It is shown that a sufficient condition for non-emptiness of the core of a multiple non-transferable utility game. The result is an extension of Scarf-Billera theorem.     

A note on the nucleolus

It was shown byKohlberg [1972] that the nucleolus can be obtained by solving a linear program of extremely large size (2 ⁿ ! constraints). We show here how this program can be reduced to a more tractable size (4 ⁿ constraints).     

A note on the shameful conjecture

Let $P_G\left(q\right)$ denote the chromatic polynomial of a graph $G$ on $n$ vertices. The ‘shameful conjecture’ due to Bartels and Welsh states that, $\frac{P_G\left(n\right)}{P_G(n-1)}\ge \frac{n^n}{{(n-1)}^n}\text{.}$ Let $\mu \left(G\right)$ denote the expected number of colors used in a uniformly random proper $n$-coloring of $G$. The above inequality can be interpreted as saying that $\mu \left(G\right)\ge \mu \left(O_n\right)$, where $O_n$ is the empty graph on $n$ nodes. This conjecture was proved by F.M. Dong, who in fact showed that, $\frac{P_G\left(q\right)}{P_G(q-1)}\ge \frac{q^n}{{(q-1)}^n}$ for all $q\ge n$. There are examples showing that this inequality is not true for all $q\ge 2$. In this paper, we show that the above inequality holds for all $q\ge 36D^{3/2}$, where $D$ is the largest degree of $G$. It is also shown that the above inequality holds true for all $q\ge 2$ when $G$ is a claw-free graph.     

A note on the Multiplier Conjecture

In order to identify multipliers of abelian (υ, k, λ)-difference sets the First and the Second Multiplier Theorem of Hall, Ryser and Chowla, resp. of Hall and Menon, need a divisor m of n = k − λ that is coprime to υ. Moreover, both theorems require that m > λ. The famous Multiplier Conjecture asserts that the restriction m > λ is not necessary. We present a generalization of the Second Multiplier Theorem where m is not necessarily coprime to υ. Here the requirement that m > λ generalizes to the condition m/(υ, m) > λ. This gives rise to a generalized Multiplier Conjecture which asserts that this condition is not necessary. We disprove this conjecture by showing that there exist counterexamples.     

A note on the exterior centralizer

The notion of the exterior centralizer C_G^ù(x){C_G^{^\wedge}(x)} of an element x of a group G is introduced in the present paper in order to improve some known results on the non-abelian tensor product of two groups. We study the structure of G by looking at that of C_G^ù(x){C_G^{^\wedge}(x)} and we find some bounds for the Schur multiplier M(G) of G.     

A note on the spectral theorem

We give necessary and sufficient conditions to prove a spectral theorem and a functional calculus for certain nonselfadjoint operators, H. Our method is non-perturbative: the conditions are given in terms of the resolvent (z-H)^–1. We give an example of an operator satisfying these conditions. This operator is not a spectral operator of scalar type. Its spectral projections are unbounded operators defined on a common dense domainD.This research was supported in part by Department of Energy Grant No. DE-AS05-80ER10711 and National Science Foundation Grant No. DMA-8312451.     

A note on the classical brachistochrone

Summary It is shown that the classical Bernoulli's brachistochrone problem and the brachistochrone problem in a central force field may be solved by the maximum principle of Pontryagin. According to the optimum control theory these problems are singular.

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Zusammenfassung In der Arbeit ist gezeigt, daß das klassische Bernoullische brachistochrone Problem und Brachistochronen im zentralen Schwerefeld durch das Pontryagin Maximum Prinzip gelöst werden können. Gemäss Theorie der optimalen Regelung sind diese Probleme singulär.

     

A note on the classical brachistochrone

It is shown that the classical Bernoulli's brachistochrone problem and the brachistochrone problem in a central force field may be solved by the maximum principle of Pontryagin. According to the optimum control theory these problems are singular.