The topological types of some irregular Kähler surfaces

The topological type of generalized Kummer surfaces is described in terms of sphere bundles over Riemann surfaces and the complex projective plane. Explicit examples of sets of pairwise non-diffeomorphic K?hler surfaces of the same topological type are given. Received: 5 January 2000     

A note on the approximate amenability of semigroup algebras

It is known that the bicyclic semigroup S ₁ is an amenable inverse semigroup. In this note we show that the convolution semigroup algebra ℓ ¹(S ₁) is not approximately amenable.     

Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras

We construct an explicit isomorphism between blocks of cyclotomic Hecke algebras and (sign-modified) cyclotomic Khovanov-Lauda algebras in type A. These isomorphisms connect the categorification conjecture of Khovanov and Lauda to Ariki’s categorification theorem. The Khovanov-Lauda algebras are naturally graded, which allows us to exhibit a non-trivial ℤ-grading on blocks of cyclotomic Hecke algebras, including symmetric groups in positive characteristic.     

The centers of spin symmetric group algebras and Catalan numbers

Generalizing the work of Farahat-Higman on symmetric groups, we describe the structures of the even centers of integral spin symmetric group superalgebras, which lead to universal algebras termed as the spin FH-algebras. A connection between the odd Jucys-Murphy elements and the Catalan numbers is developed and then used to determine the algebra generators of the spin FH-algebras and of the even centers .     

Completely splittable representations of affine Hecke-Clifford algebras

We classify and construct irreducible completely splittable representations of affine and finite Hecke-Clifford algebras over an algebraically closed field of characteristic not equal to 2.     

The study of an infinite class of dendrimer nanostars by topological index approaches

A topological index for a molecular graph G is a numeric quantity invariant under automorphisms of G. A dendrimer is an artificially manufactured or synthesized molecule built up from branched units called monomers. In this article an infinite class of dendrimer nanostars is investigated under three topological indices containing PI, Szeged and edge Szeged.     

Comparison semigroups and algebras of transformations

We characterize algebras of transformations on a set under the operations of composition and the pointwise switching function defined as follows: (f,g)[h,k](x)=h(x) if f(x)=g(x), and k(x) otherwise. The resulting algebras are both semigroups and comparison algebras in the sense of Kennison. The same characterization holds for partial transformations under composition and a suitable generalisation of the quaternary operation in which agreement of f,g includes cases where neither is defined. When a zero element is added (modelling the empty function), the resulting signature is rich enough to encompass many operations on semigroups of partial transformations previously considered, including set difference and intersection, restrictive product, and a functional analog of union. When an identity element is also added (modelling the identity function), further domain-related operations can be captured.     

The projection onto the center of an ℓ-algebra

In a uniformly complete -algebra with unite>0 the principal band {e} ^dd generated bye is a projection band. We prove that the formula of A. R. Schep for the projection onto this band which holds in the -algebra of order bounded operators on a Dedekind complete vector lattice, in general defines a projection onto the commutant of {e} ^dd . We present some examples to show that the commutant may be strictly larger than the band {e} ^dd , and also conditions which guarantee equality.This paper was written during the fall semester of 1993 while the first author visited the Department of Mathematics of the University of Leiden as a guest of Dr. C. B. Huijsmans, to whom the express his gratitude for the hospitality offered to him.     

The Poincaré series of divisorial valuations in the plane defines the topology of the set of divisors

With a plane curve singularity one associates a multi-index filtration on the ring of germs of functions of two variables defined by the orders of a function on irreducible components of the curve. The Poincaré series of this filtration turns out to coincide with the Alexander polynomial of the curve germ. For a finite set of divisorial valuations on the ring corresponding to some components of the exceptional divisor of a modification of the plane, in a previous paper there was obtained a formula for the Poincaré series of the corresponding multi-index filtration similar to the one associated with plane germs. Here we show that the Poincaré series of a set of divisorial valuations on the ring of germs of functions of two variables defines “the topology of the set of the divisors” in the sense that it defines the minimal resolution of this set up to combinatorial equivalence. For the plane curve singularity case, we also give a somewhat simpler proof of the statement by Yamamoto which shows that the Alexander polynomial is equivalent to the embedded topology.     

The Intrinsic Diameter of the Surface of a Parallelepiped

In the paper we obtain an explicit formula for the intrinsic diameter of the surface of a rectangular parallelepiped in 3-dimensional Euclidean space. As a consequence, we prove that an parallelepiped with relation for its edge lengths has maximal surface area among all rectangular parallelepipeds with given intrinsic diameter.     

On some algebras diagonalized by M-bases of ℓ2

We study reflexive algebrasA whose invariant lattices LatA are generated by M-bases of ². Examples are given whereA differs from ( being the rank one subalgebra ofA), and where together with the identity I is not strongly dense inA. For M-bases in a special class, we characterize the cases when they are strong, and also when the identity I is the ultraweak limit of a sequence of contractions in . We show that this holds provided that I is approximable by compact operators inA at any two points of ². We show that the spaceA+^* (where is the annihilator of ) is ultraweakly dense in (²), and characterize the M-bases in this class for which the sum is direct. We give a class of automorphisms ofA which are strongly continuous but not spatial.     

The product formula for the topological degree of strict γ-contractions

Leray and Schauder [8] have defined a topological degree for mappings in Banach spaces of the form I-f, where f is compact. A product formula for this degree, i.e. a formula for the degree of the composed map (I-f)(I-g), has been given by Nagumo [10]. In recent years, this degree has been extended to mappings I-f, where f is a strict -contraction or -condensing with respect to a rather general measure of noncompactness . It has been shown that all the properties of the Leray-Schauder degree are still valid (see [11] and [12]), while the validity of the product formula remained an open question. Nussbaum [11] has announced this formula for the special case of ₁ -spaces and mappings f that are strict - and ß-contractions, simultaneously. Fenske [5] has shown that this formula holds in every Banach space provided f is a differentiable strict -contraction. In this note we shall establish this formula for an arbitrary Banach space and an arbitrary strict -contraction.     

The arithmetic-geometric scaling spectrum for continued fractions

To compare continued fraction digits with the denominators of the corresponding approximants we introduce the arithmetic-geometric scaling. We will completely determine its multifractal spectrum by means of a number-theoretical free-energy function and show that the Hausdorff dimension of sets consisting of irrationals with the same scaling exponent coincides with the Legendre transform of this free-energy function. Furthermore, we identify the asymptotic of the local behaviour of the spectrum at the right boundary point and discuss a connection to the set of irrationals with continued-fraction digits exceeding a given number which tends to infinity.     

The Method of Virtual Components in the Multivariate Setting

We describe the so-called method of virtual components for tight wavelet framelets to increase their approximation order and vanishing moments in the multivariate setting. Two examples of the virtual components for tight wavelet frames based on bivariate box splines on three or four direction mesh are given. As a byproduct, a new construction of tight wavelet frames based on box splines under the quincunx dilation matrix is presented.     

The topological type of the α-sections of convex sets

An α=(α₁,…,αk)(0?αi?1) section of a family {K₁,…,Kk} of convex bodies in Rd is a transversal halfspace H⁺ for which Vold(Ki∩H⁺)=αi⋅Vold(Ki) for every 1?i?k. Our main result is that for any well-separated family of strictly convex sets, the space of α-sections is diffeomorphic to Sd−k.