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 a conjecture of the Randić index and the minimum degree of graphs

The Randi? index $R\left(G\right)$ of a graph $G$ is defined by $R\left(G\right)={\sum }_{uv}\frac{1}{\sqrt{d\left(u\right)d\left(v\right)}}$, where $d\left(u\right)$ is the degree of a vertex $u$ and the summation extends over all edges $uv$ of $G$. Delorme et al. (2002)  [6] put forward a conjecture concerning the minimum Randi? index among all$n$-vertex connected graphs with the minimum degree at least $k$. In this work, we show that the conjecture is true given the graph contains $k$ vertices of degree $n?1$. Further, it is true among $k$-trees.     

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 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).     

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 the logarithmic coefficients of Bazilevic? functions

The objective of the present paper is to study the logarithmic coefficients of Bazilevic? functions. We obtain the inequality ∣γ_n∣ ? An⁻¹logn (A is an absolute constant) which holds for Bazilevic? functions.     

Proof of Andrews' conjecture on a 4φ3 summation

We give a new proof of a ₄φ₃ summation due to G.E. Andrews and confirm another ₄φ₃ summation conjectured by him recently. Some variations of these two ₄φ₃ summations are also given.     

Curves on threefolds and a conjecture of Griffiths–Harris

We prove that any arithmetically Gorenstein curve on a smooth, general hypersurface of degree at least 6, is a complete intersection. This gives a characterisation of complete intersection curves on general type hypersurfaces in . We also verify that certain 1-cycles on a general quintic hypersurface are non-trivial elements of the Griffiths group.     

On the range of the derivative of Gâteaux-smooth functions on separable banach spaces

We prove that there exists a Lipschitz function froml ¹ into ℝ² which is Gateaux-differentiable at every point and such that for everyx, y εl ¹, the norm off′(x) −f′(y) is bigger than 1. On the other hand, for every Lipschitz and Gateaux-differentiable function from an arbitrary Banach spaceX into ℝ and for everyε > 0, there always exist two pointsx, y εX such that ‖f′(x) −f′(y)‖ is less thanε. We also construct, in every infinite dimensional separable Banach space, a real valued functionf onX, which is Gateaux-differentiable at every point, has bounded non-empty support, and with the properties thatf′ is norm to weak* continuous andf′(X) has an isolated pointa, and that necessarilya ε 0. This work has been initiated while the second-named author was visiting the University of Bordeaux. The second-named author is supported by grant AV 1019003, A1 019 205, GA CR 201 01 1198.     

On a conjecture of Hivert and Thiéry about Steenrod operators

We prove some results related to a conjecture of Hivert and Thiéry about the dimension of the space of q-harmonics (F. Hivert and N. Thiéry, 2004 [HT]). In the process we compute the actions of the involved operators on symmetric and alternating functions, which have some independent interest. We then use these computations to prove other results related to the same conjecture.     

On a conjecture of Erdős and Simonovits: Even cycles

Let F be a family of graphs. A graph is F-free if it contains no copy of a graph in F as a subgraph. A cornerstone of extremal graph theory is the study of the Turán number ex(n,F), the maximum number of edges in an F-free graph on n vertices. Define the Zarankiewicz number z(n,F) to be the maximum number of edges in an F-free bipartite graph on n vertices. Let C _k denote a cycle of length k, and let C _k denote the set of cycles C _?, where 3≤?≤k and ? and k have the same parity. Erd?s and Simonovits conjectured that for any family F consisting of bipartite graphs there exists an odd integer k such that ex(n,F ∪ C _k ) ～ z(n,F) — here we write f(n) ～ g(n) for functions f,g: ? → ? if lim _n→∞ f(n)/g(n)=1. They proved this when F ={C ₄} by showing that ex(n,{C ₄;C ₅})～z(n,C ₄). In this paper, we extend this result by showing that if ?∈{2,3,5} and k>2? is odd, then ex(n,C _2? ∪{C _k }) ～ z(n,C _2? ). Furthermore, if k>2?+2 is odd, then for infinitely many n we show that the extremal C _2? ∪{C _k }-free graphs are bipartite incidence graphs of generalized polygons. We observe that this exact result does not hold for any odd k<2?, and furthermore the asymptotic result does not hold when (?,k) is (3, 3), (5, 3) or (5, 5). Our proofs make use of pseudorandomness properties of nearly extremal graphs that are of independent interest.     

Proof of a conjecture of Zahariuta concerning a problem of Kolmogorov on the ε-entropy

We prove a conjecture of Zahariuta which itself solves a problem of Kolmogorov on the -entropy of some classes of analytic functions. For a given holomorphically convex compact subset K in a pseudoconvex domain D in Cⁿ, Zahariutas conjecture consists in approximating the relative extremal function u^*_K,D, uniformly on any compact subset of DK, by pluricomplex Green functions on D with logarithmic poles in the compact subset K.     

Some notes on a conjecture of Zygmund

In this paper, some classes of differentiation basis are investigated and several positive answers to a conjecture of Zygmund on differentiation of integrals are presented.