A classical problem in analytic number theory is the distribution in short intervals of integers with a prescribed multiplicative structure, such as primes, almost-primes, k-free numbers and others. Recently, partly due to applications to cryptology, much attention has been received by the problem of the distribution in short intervals of integers without large prime factors, see Lenstra–Pila–Pomerance [3] and section 5 of the excellent survey by Hildebrand–Tenenbaum [1].

In this paper we deal with the distribution in short intervals of numbers representable as a product of a prime and integers from a given set [Sscr ], defined in terms of cardinality properties. Our results can be regarded as an extension of the above quoted results, and we will provide a comparison with such results by a specialization of the set [Sscr ].

Schur algebras are certain finite-dimensional algebras that completely control the polynomial representation theory of the general linear groups over an infinite field. Infinitesimal Schur algebras are truncated versions of the classical Schur algebras which control the polynomial representation theory of the Frobenius kernels of general linear groups. In this paper we use some elementary results on symmetric powers to classify the semisimple Schur algebras. We then classify the semisimple infinitesimal Schur algebras as well.

For a fixed finite group G, the numbers Ng of equivalence classes of orientation-preserving actions of G on closed orientable surfaces Σg of genus g can be encoded by a generating function [sum ]Ngzg. When equivalence is determined by the isomorphism class of the quotient orbifold Σg/G, we show that the generating function is rational. When equivalence is topological conjugacy, we examine the cases where G is abelian and show that the generating function is again rational in the cases where G is cyclic.

In this paper a new discreteness criterion for subgroups in SL(2, C) is established, which implies all known discreteness criteria by two-generator subgroup and is the best possible. And it is proved that every non-elementary and non-discrete subgroup of SL(2, C) has a non-elementary and non-discrete subgroup generated by two loxodromic elements.

Let Γ be a discrete group and p be a prime. One of the fundamental results in group cohomology is that H*(Γ, [ ]p) is a finitely generated [ ]p-algebra if Γ is a finite group [8, 24]. The purpose of this paper is to study the analogous question when Γ is no longer finite.

Recall that Γ is said to have finite virtual cohomological dimension (vcd) if there exists a finite index torion-free subgroup Γ′ of Γ such that Γ′ has finite cohomological dimension over ℤ [4]. By definition vcd Γ is the cohomological dimension of Γ′. It is easy to see that the mod p cohomology ring of a finite vcd-group does not have to be a finitely generated [ ]p-algebra in general. For instance, if Γ is a countably infinite free product of ℤ's, then H1(Γ, [ ]p) is not finite dimensional over [ ]p. The three most important classes of examples of finite vcd-groups in which the mod p cohomology ring is a finitely generated [ ]p-algebra are arithmetic groups [2], mapping class groups [9, 10] and outer automorphism groups of free groups [5]. In each of these examples, the proof of finite generation involves the construction of a specific Γ-complex with appropriate finiteness conditions. These constructions should be regarded as utilizing the geometry underlying these special classes of groups. In contrast, the result we prove will depend only on the algebraic structure of the group Γ.

For G a classical group of type Bn, Cn or Dn defined over an algebraically closed field (of characteristic other than two if G is a special orthogonal group), we show that the centralizer algebra of G on the space of endomorphisms of the module of traceless tensors Wr is naturally isomorphic to the group algebra of the symmetric group Sr if r[les ]n (for G of type Bn or Cn) or if r<n (for G of type Dn). This extends a characteristic zero result of Brauer and Weyl and recovers some of the related characteristic free results that De Concini and Strickland obtained for G of type Cn.

The usefulness of the Koszul complex in handling in an algebraic setting the two geometric notions of multiplicity and depth first became apparent with the work of Auslander and Buchsbaum [1] following a suggestion of Serre. Regarding the generators a1, …, an of the complex as a 1×n matrix first Eagon and Northcott [4] extended this work to a complex associated with an m×n matrix, then shortly afterwards a different extension was given by Buchsbaum and Rim [2, 3]. These two complexes are two of an infinite family [6] some of which inherit the depth sensitive property of the Koszul complex and all of which under a certain finiteness condition provide the same multiplicity as Euler–Poincaré characteristic [7].

These two properties prove useful in geometric applications, see for example Lago and Rodicio [7] for depth sensitivity and Kirby [8] for the characteristic as multiplicity of intersection. During the course of these developments it became clear that it was most appropriate to regard the complexes as being generated by linear forms. From this point of view it is natural to ask if the linearity of the forms is necessary. In the present note we begin a response to this question by extending the work of [6] to complexes associated with forms of arbitrary positive degree. In a sequel we shall similarly extend the results of multiplicity in [7].

Some thirty years ago Buchsbaum and Rim [1] extended the notion of multiplicity e(a1, …, an; E) for elements a1, …, an of a commutative ring R with identity and a Noetherian R-module E (≠0) with lengthR (E/[sum ]nt=1aiE) finite to give a multiplicity e((aij); E) associated with E and an m×n matrix (aij) over R satisfying a certain extended finiteness condition. One of their results states that for each of a set of m complexes depending on E, (aij) the Euler–Poincaré characteristic is a certain integer multiple of e((aij); E), at least when R is a local ring.

Some of these ideas were taken up in [4] where it is shown that when n[ges ]m−1, each of the complexes K((aij); E; t) with t∈ℤ introduced in [3] also have e((aij); E) as their Euler–Poincaré characteristic. With a slight change in viewpoint (aij) can be replaced by linear forms aj=[sum ]mi=1aijxi (j=1, …, n) of the graded polynomial ring R[x1, …, xm]; the complex K((aij); E; t) then becomes the component of degree t in a certain graded double complex

formula here

where K(a1, …, an; F) is the standard Koszul complex (see [4; section 2]). From this point of view the construction can be extended to allow the homogeneous polynomials a1, …, an to have any (possibly unequal) positive degrees [6]. The main aim of the present note is to extend similarly the results of [4] and to strengthen those results to give information on the vanishing of the multiplicity.

It is well known that totally geodesic Lagrangian submanifolds of a complex-space-form M˜n(4c) of constant holomorphic sectional curvature 4c are real-space-forms of constant sectional curvature c. In this paper we investigate and determine non-totally geodesic Lagrangian isometric immersions of real-space-forms of constant sectional curvature c into a complex-space-form M˜n(4c). In order to do so, associated with each twisted product decomposition of a real-space-form of the form f1I1×… ×fkIk×1Nn−k(c), we introduce a canonical 1-form, called the twistor form of the twisted product decomposition. Roughly speaking, our main result says that if the twistor form of such a twisted product decomposition of a simply-connected real-space-form of constant sectional curvature c is twisted closed, then it admits a ‘unique’ adapted Lagrangian isometric immersion into a complex-space-form M˜n(4c). Conversely, if L: Mn(c)→ M˜n(4c) is a non-totally geodesic Lagrangian isometric immersion of a real-space-form Mn(c) of constant sectional curvature c into a complex-space-form M˜n(4c), then Mn(c) admits an appropriate twisted product decomposition with twisted closed twistor form and, moreover, the Lagrangian immersion L is given by the corresponding adapted Lagrangian isometric immersion of the twisted product. In this paper we also provide explicit constructions of adapted Lagrangian isometric immersions of some natural twisted product decompositions of real-space-forms.

We show that the sequence 2n+n is a universal Hilbert sequence. That is, for each polynomial P(T, Y) irreducible in ℚ(T) [Y], the polynomial P(2n+n, Y) is irreducible in ℚ[Y] for all but finitely many n. This answers a question of M. Yasumoto. Other examples, like 2n+5n, are given. They are all obtained as special cases of a more general result which is proved from classical diophantine arguments.

For an infinite Bernoulli convolution whose support is a symmetric Cantor-type set, R. Kershner and A. Wintner [14] proved that the convolution is absolutely continuous or singular depending on whether the Lebesgue measure of the support is positive or zero respectively. Their proof employs the same construction which was developed originally by F. Hausdorff [11] in his celebrated paper, ‘Dimension und äusseres Mass’, where he introduced the notion of a fractional dimension. In this article we exploit this observation and prove a result which extends the theorem of Kershner and Wintner by considering Hausdorff instead of Lebesgue measure for the same class of infinite Bernoulli convolutions.

Let Bn denote the open unit ball in Cn. We write V to denote Lebesgue volume measure on Bn normalized so that V(Bn)=1. Fix −1<γ<∞ and let Vγ denote the measure given by dVγ(z)=cγ (1−[mid ]z[mid ]2)γdV(z), for z∈Bn, where cγ=Γ(n+γ+1)/ (n!Γ(γ+1)); then Vγ(Bn)=1. The weighted Bergman space A2,γ(Bn) is the space of all analytic functions in L2(Bn, dVγ). This is a closed linear subspace of L2(Bn, dVγ). Let Pγ denote the orthogonal projection of L2(Bn, dVγ) onto A2,γ(Bn). For a function f∈L∞(Bn) the Toeplitz operator Tf is defined on A2,γ(Bn) by Tfh=Pγ(fh), for h∈A2,γ(Bn). It is clear that Tf is bounded on A2,γ(Bn) with ∥Tf∥[les ]∥f∥∞. In this paper we will consider the question for which f∈L∞(Bn) the operator Tf is compact on A2,γ(Bn). Although a complete answer has been given by the author and D. Zheng (see the next section), the condition for compactness is somewhat unnatural. In this article we will give a more natural description for compactness of Toeplitz operators with sufficiently nice symbols. We will describe compactness in terms of behaviour of the so-called Berezin transform of the symbol, which has been useful in characterizing compactness of Toeplitz operators with positive symbols (see [5, 9]). Before we can define this Berezin transform we need to introduce more notation.

An orientation-preserving, recurrent homeomorphism of the two-sphere which is not the identity is shown to admit exactly two fixed points. A recurrent homeomorphism of a compact surface with negative Euler characteristic is periodic.

We consider the time evolution of an unbounded interface between a viscous fluid and a displacing fluid of negligible viscosity in a Hele-Shaw geometry. Using a conformal mapping transformation z=F(ζ, t) from the lower half ζ-plane to the flow domain Ω(t) in the z-plane, the dynamics is formulated in terms of an integro-differential equation for F on the real ζ axis. This formulation is similar to earlier results for radial and channel geometries. Using a certain explicit solution it is shown that the zero surface tension problem is ill-posed in the sense of Hadamard. Furthermore, certain earlier formal asymptotic results for small but non-zero surface tension [47], are rigorously established. This is achieved by relating the similarity reduction of the Harry–Dym equation to the Painlevé II equation, and by studying this equation in the complex plane. In particular, it is shown that there exists a unique solution of Painlevé II equation which satisfies the appropriate far-field algebraic decay condition in a certain sector Ŝ of the complex plane and which in Ŝ is free of singularities. This solution asymptotes to certain elliptic functions in other sectors of the complex plane and contains a set of denumerably infinite number of singularities that approach the boundary of Ŝ from the outside at large distances from the origin (in the similarity variable). This is consistent with earlier numerical results.