Volume 18, Issue 5

Our interest is in the generating function E N,β (I;ξ;w β ) for the probabilities E N,β (n;I;w β ) that in a matrix ensemble with unitary ( β = 2) or orthogonal ( β = 1) symmetry, characterized by the weight w β (λ) and having N eigenvalues, the interval I contains exactly n eigenvalues. Using a determinant formula for E N ,2 , a general quadratic identity is obtained which relates E N ,2 in the case I and w 2 ( x ) even to a product of generating functions E N ,2 with different I , w 2 (λ) and N , and for which the eigenvalues are positive. Also, generalizing some earlier calculations, the sum E N ,1 (2 n − 1; I ; w 1 ) + E N ,1 (2 n ; I ; w 1 ) for N even, I = (− t , t ) and w 1 an even classical weight is shown to equal E N /2,2 ( n ; (0, t 2 ); w 2 ) for w 2 related to w 1 . Implications of these identities are discussed.

Optimal embeddings for Orlicz-Sobolev spaces of any order into Orlicz spaces and into general rearrangement invariant spaces are established. Related Poincaré inequalities both for compactly supported functions and for arbitrary functions are also derived.

We study symplectic structures on characteristically nilpotent Lie algebras (CNLAs) by computing the cohomology space H 2 (𝔤, k ) for certain Lie algebras 𝔤. Among these Lie algebras are filiform CNLAs of dimension n ≤ 14. It turns out that there are many examples of CNLAs which admit a symplectic structure. A generalization of a sympletic structure is an affine structure on a Lie algebra.

Distributional and strong equalities in Colombeau algebra 𝒢 of generalized functions are compared. Also it is done for 𝒢 ∞ . Moreover a positive answer to a question of M. Oberguggenberger is given: A generalized function, invariant under all translations, is a generalized constant.

We define a family of groups with balanced presentations and prove that these groups correspond to spines (or, equivalently, to Heegaard diagrams) of a certain class of Seifert fibered 3-manifolds. These manifolds are constructed from triangulated 3-balls by identifying pairs of boundary faces via orientation-reversing homeomorphisms. Then we describe the manifolds as cyclic branched coverings of certain lens spaces when the groups are cyclically presented. Finally, we give explicit computations of the Casson-Walker-Lescop invariant and the Rohlin invariant for many manifolds in the above class.

The concept of thickening was systematically studied by C. T. C. Wall in [Wall C. T. C.: Classification problems in differential topology—IV. Thickenings. Topology 5 (1966), 73–94]. The suspension theorem of that paper is an exact sequence relating the n -dimensional thickenings of a finite complex to its ( n + 1)-dimensional ones. The object of this note is to fill in what we believe is a missing argument in the proof of that theorem.

Some necessary and sufficient conditions on rearrangement-invariant spaces A, B are stated, which provide compactness of the corresponding Sobolev embedding W A k (Ω) ↪ B (Ω) for any bounded open set Ω ⊂ ℝ n . It is shown that this property is equivalent to compactness of some integral operator of Hardy type acting from A to B . This enables the compactness problem to be connected with behaviour of the ratio of fundamental functions ϕ A ( t )/ϕ B ( t ) at zero. We also show that any optimal Sobolev embedding is not compact and extend this result to some other Sobolev embeddings “close” to optimal one.

We study the problem of extending rationally connected fibrations from an ample subvariety to the ambient variety, applying our results to particular rationally connected fibrations, i.e. Fano fibrations coming from numerically effective extremal contractions of the ample subvariety.