Ore-type and Dirac-type theorems for matroids

Let M be a connected binary matroid having no -minor. Let be a collection of cocircuits of M. We prove there is a circuit intersecting all cocircuits of if either one of two things hold:

(i) For any two disjoint cocircuits and in it holds that .

(ii) For any two disjoint cocircuits and in it holds that .

Part (ii) implies Ore's Theorem, a well-known theorem giving sufficient conditions for the existence of a hamilton cycle in a graph. As an application of part (i), it is shown that if M is a k-connected regular matroid and has cocircumference c^*2k, then there is a circuit which intersects each cocircuit of size c^*−k+2 or greater.We also extend a theorem of Dirac for graphs by showing that for any k-connected binary matroid M having no -minor, it holds that for any k cocircuits of M there is a circuit which intersects them.     

On consecutive happy numbers

Let e?1 and b?2 be integers. For a positive integer with 0?aj<b, define

     

Simultaneous Diophantine approximation on manifolds and Hausdorff dimension

Let M be an m-dimensional, C^k manifold in , for any , and for any τ>0 let

     

Braid groups of non-orientable surfaces and the Fadell-Neuwirth short exact sequence

Let M be a compact, connected non-orientable surface without boundary and of genus g?3. We investigate the pure braid groups P_n(M) of M, and in particular the possible splitting of the Fadell-Neuwirth short exact sequence

     

Sharp estimates for the non-centered maximal operator associated to Gaussian and other radial measures

In this work we investigate the integrability properties of the maximal operator M_μ, associated with a non-doubling measure μ defined on the Euclidean space , with special emphasis on the Gaussian and similar measures. Among other results we show for a wide class of radial and decreasing measures μ, that M_μ satisfies the modular inequality

     

Some starlikeness criterions for analytic functions

We determine a condition on M, α, λ and μ for which

     

Non-spectral self-affine measure problem on the plane domain

The self-affine measure μM,D corresponding to an expanding integer matrix

     

Le problème de Yamabe avec singularités

Let (Mn,g) be a compact riemannian manifold of dimension n?3. Under some assumptions, we prove that there exists a positive function φ solution of the Yamabe equation

     

Orthogonal exponentials on the generalized three-dimensional Sierpinski gasket

The self-affine measure μM,D corresponding to the expanding integer matrix

     

A mean-value theorem and its applications

For a function f defined in an interval I, satisfying the conditions ensuring the existence and uniqueness of the Lagrange mean L[f], we prove that there exists a unique two variable mean M[f] such that

     

The canonical Kaehler potential on the parameter space of the versal family of CR structures

Let (M,⁰T″) be a compact strongly pseudo convex CR structure with dim_RM=2n−1. Then, in [Michigan Math. J. 50 (2002) 517-549], the construction of the versal family of CR structures is settled. The purpose of this paper is to introduce a canonical Kaehler metric for the parameter space of this versal family if the CR structure admits a normal vector field and a non-vanishing CR n-form with the condition and

     

Large matchings from eigenvalues

We find lower bounds on the difference between the spectral radius λ₁ and the average degree of an irregular graph G of order n and size e. In particular, we show that, if n ? 4, then

     

Dissipative operators on Banach spaces

A bounded linear operator T on a Banach space is said to be dissipative if ‖etT‖?1 for all t?0. We show that if T is a dissipative operator on a Banach space, then:

(a)

.

(b)

If σ(T)∩iR is contained in [−iπ/2,iπ/2], then