On a generalization of the Hadwiger–Nelson problem

For a field F and a quadratic form Q defined on an n-dimensional vector space V over F, let QG _Q , called the quadratic graph associated to Q, be the graph with the vertex set V where vertices u,w ∈ V form an edge if and only if Q(v ? w) = 1. Quadratic graphs can be viewed as natural generalizations of the unit-distance graph featuring in the famous Hadwiger–Nelson problem. In the present paper, we will prove that for a local field F of characteristic zero, the Borel chromatic number of QG _Q is infinite if and only if Q represents zero non-trivially over F. The proof employs a recent spectral bound for the Borel chromatic number of Cayley graphs, combined with an analysis of certain oscillatory integrals over local fields. As an application, we will also answer a variant of question 525 proposed in the 22nd British Combinatorics Conference 2009 [6].     

On the Rubin–Stark Conjecture for a special class of CM extensions of totally real number fields

We use Greithers recent results on Brumers Conjecture to prove Rubins integral version of Starks Conjecture, up to a power of 2, for an infinite class of CM extensions of totally real number fields, called nice extensions under the assumption that a certain Iwasawa –invariant vanishes. As a consequence and under the same assumption, we show that the Brumer–Stark Conjecture is true for nice extensions, up to a power of 2.Mathematical Subject Classification (1991): 11R42, 11R58, 11R27Research on this project was partially supported by NSF grants DMS–9801267 and DMS–0196340.in final form: 16 July 2003     

On the Mordell–Weil rank of a surface fibration

Abstract

In this article, we give a new upper bound for the Mordell–Weil rank of a surface fibration and we prove an analog result in positive characteristic.     

On a conjecture of the Randi? index

The Randi? index of a graph G is defined as , where d(u) is the degree of vertex u and the summation goes over all pairs of adjacent vertices u, v. A conjecture on R(G) for connected graph G is as follows: R(G)≥r(G)−1, where r(G) denotes the radius of G. We proved that the conjecture is true for biregular graphs, connected graphs with order n≤10 and tricyclic graphs.     

On dominant contractions and a generalization of the zero–two law

Zaharopol proved the following result: let \({T,S:L^1(X,{\mathcal{F}},\mu)\to L^1(X, {\mathcal{F}},\mu)}\) be two positive contractions such that T ≤ S. If \({\|S-T\| <1 }\) then \({\left\|S^n-T^n\right\| <1 }\) for all \({n\in\mathbb{N}}\). In the present paper we generalize this result to multi-parameter contractions acting on L ¹. As an application of that result we prove a generalization of the “zero–two” law.     

On the Kashiwara–Vergne conjecture

Let G be a connected Lie group, with Lie algebra . In 1977, Duflo constructed a homomorphism of -modules , which restricts to an algebra isomorphism on invariants. Kashiwara and Vergne (1978) proposed a conjecture on the Campbell-Hausdorff series, which (among other things) extends the Duflo theorem to germs of bi-invariant distributions on the Lie group G. The main results of the present paper are as follows. (1) Using a recent result of Torossian (2002), we establish the Kashiwara–Vergne conjecture for any Lie group G. (2) We give a reformulation of the Kashiwara–Vergne property in terms of Lie algebra cohomology. As a direct corollary, one obtains the algebra isomorphism , as well as a more general statement for distributions.     

On a conjecture of ōshima

The set of homotopy classes of self maps of a compact, connected Lie group G is a group by the pointwise multiplication which we denote by H(G), and it is known to be nilpotent. ōshima [H. ōshima, Self homotopy group of the exceptional Lie group G₂, J. Math. Kyoto Univ. 40 (1) (2000) 177-184] conjectured: if G is simple, then H(G) is nilpotent of class ?rankG. We show this is true for PU(p) which is the first high rank example.     

On a problem of Yau regarding a higher dimensional generalization of the Cohn–Vossen inequality

We show that a problem asked by Yau (Open problems in geometry. Chern–a great geometer of the twentieth century, pp. 275–319, 1992) cannot be true in general. The counterexamples are constructed based on the recent work of Wu and Zheng (Examples of positively curved complete Kähler manifolds. Geometry and Analysis, vol. 17, pp. 517–542, 2010).     

On one conjecture in the theory of isochronous Liénard systems

New normal forms are obtained for the center as well as isochronous center of holomorphic Liénard systems.     

New generalization of the two-dimensional Bernfeld–Haddock conjecture and its proof

In this paper, we investigate a class of systems of delay differential equations. These systems have important practical applications and also are a two-dimensional generalization of the Bernfeld–Haddock conjecture. It is shown that each bounded solution of the systems tends to a constant vector under a desirable condition. Our results improve some corresponding ones already known and, in particular, give a proof of the Bernfeld–Haddock conjecture.     

On the coincidence of the factor and Gondran–Minoux rank functions of matrices over a semiring

We consider the rank functions of matrices over semirings, functions that generalize the classical notion of the rank of a matrix over a field. We study semirings over which the factor and Gondran–Minoux ranks of any matrix coincide. It is shown that every semiring satisfying that condition is a subsemiring of a field. We provide an example of an integral domain over which the factor and Gondran–Minoux ranks are different.     

On a conjecture of Füredi

Füredi conjectured that the Boolean lattice $2^{\left[n\right]}$ can be partitioned into $\left(\genfrac{}{}{00}{}{n}{\lfloor n/2\rfloor }\right)$ chains such that the size of any two differs in at most one. In this paper, we prove that there is an absolute constant $\alpha \approx 0.8482$ with the following property: for every $ϵ>0$, if $n$ is sufficiently large, the Boolean lattice $2^{\left[n\right]}$ has a chain partition into $\left(\genfrac{}{}{00}{}{n}{\lfloor n/2\rfloor }\right)$ chains, each of them of size between $(\alpha -ϵ)\sqrt{n}$ and $O(\sqrt{n}/ϵ)$.We deduce this result by looking at the more general setup of unimodal normalized matching posets. We prove that a unimodal normalized matching poset $P$ of width $w$ has a chain partition into $w$ chains, each of size at most $\frac{2\left|P\right|}{w}+5$, and it has a chain partition into $w$ chains, where each chain has size at least $\frac{\left|P\right|}{2w}-\frac{1}{2}$.     

On a mild generalization of the Schönemann irreducibility criterion

We state a mild generalization of the classical Schönemann irreducibility criterion in ?[x] and provide an elementary proof.     

On a generalization of the Dvoretzky–Wald–Wolfowitz theorem with an application to a robust optimization problem

A generalization of the Dvoretzky–Wald–Wolfowitz theorem to the case of conditional expectations is provided assuming that the σ-field on the state space has no conditional atoms. Neither continuity nor compactness assumptions are made on the model. An application to a robust (maxmin) optimization problem is given.     

On the Brezis–Nirenberg problem on S3, and a conjecture of Bandle–Benguria

We consider the following Brezis–Nirenberg problem on ${\mathbf{S}}^3$

$?{\mathrm{\Delta }}_{{\mathbf{S}}^3}u=\lambda u+u^5\text{in}D,u>0\text{in}D\text{and}u=0\text{on }?D,$

where D is a geodesic ball on ${\mathbf{S}}^3$ with geodesic radius ${\theta }_1$, and ${\mathrm{\Delta }}_{{\mathbf{S}}^3}$ is the Laplace–Beltrami operator on ${\mathbf{S}}^3$. We prove that for any $\lambda <?\frac{3}{4}$ and for every ${\theta }_1<\pi$ with $\pi ?{\theta }_1$ sufficiently small (depending on λ), there exists bubbling solution to the above problem. This solves a conjecture raised by Bandle and Benguria [J. Differential Equations 178 (2002) 264–279] and Brezis and Peletier [C. R. Acad. Sci. Paris, Ser. I 339 (2004) 291–394]. To cite this article: W. Chen, J. Wei, C. R. Acad. Sci. Paris, Ser. I 341 (2005).