Inthis note, we present a thorough investigation of convolution operators that are naturally associated to vector measures. We characterize those convolution operators that are weakly compact and compact on Ll(G) and C(G) as well as those that are p summing, (1 ≤ p ≤ ∞) and nuclear on C(G).

We construct an algebraic almost closed 1-form with zero scheme not expressible (even locally) as the critical locus of a holomorphic function on a non-singular variety. The result answers a question of Behrend–Fantechi. We correct here an error in our paper (D. Maulik, R Pandharipande and R. P. Thomas, Curves on K3 surfaces and modular forms, J. Topol.3 (2010) 937–996. arXiv:1001.2719v3), where an incorrect construction with the same claimed properties was proposed.

Let Δ denote the Laplace operator acting on the space L2(Г/H) of automorphic functions with respect to a congruence group Г, square integrable over the fundamental domain F=Г/H. It is known that Δ has a point spectrum

with (Weyl's law)

and it has a purely continuous spectrum on [¼,∞) of finite multiplicity equal to the number of inequivalent cusps. The eigenpacket of the continuous spectrum is formed by the Eisenstein series Ea(z, s) on s = ½+it where a ranges over inequivalent cusps. The eigenfunctions ui(z) with positive eigenvalues are Maass cusp forms.

Harte (1982, Math. Z.179, 431–436) initiated the study of Fredholm theory relative to a unital homomorphism T: A → B between unital Banach algebras A and B based on the following notions: an element a ∈ A is called Fredholm if 0 is not in the spectrum of Ta, while a is Weyl (Browder) if there exist (commuting) elements b and c in A with a = b + c such that 0 is not in the spectrum of b and c is in the null space of T. We introduce and investigate the concepts of r-Fredholm, r-Weyl and r-Browder elements, where 0 in these definitions is replaced by the spectral radii of a and b, respectively.

Let G, H and K be finite groups such that K acts on both G and H. The action of K on G and H induces an action of K on their tensor product G ⊗ H, and we shall denote the K-stable subgroup of G ⊗ H by (G ⊗ H)K. In section 1 of this note we shall obtain necessary and sufficient conditions for (G ⊗ H)K = G ⊗ H. The importance of this result is that the direct product of G and H has Schur multiplier M(G × H) isomorphic to M(G) × M(H) × (G ⊗ H); moreover K: acts on M(G × H), and M(G × H)K is one of the terms contained in a fundamental exact sequence concerning the Schur multiplier of the semidirect product of K and G × H (see [3, (2.2.10) and (2.2.5)] for details). Indeed in section 2 we shall assume that G is abelian and use the fact that M(G) ≅ G ∧ G to find necessary and sufficient conditions for M(G)K = M(G).

We give some extensions of classical results of Kellogg and Warschawski to a class of quasiconformal (q.c.) mappings. Among the other results we prove that a q.c. mapping f, between two planar domains with smooth C1,α boundaries, together with its inverse mapping f−1, is C1,α up to the boundary if and only if the Beltrami coefficient μf is uniformly α Hölder continuous (0 < α < 1).

Let G by any given group. A homomorphic mapping μ of a subgroup A of G onto a second subgroup B of G, where A and B need not be distinct, is called a partial endomorphism of G. When μ is defined on the whole of G, that is when A = G, we call μ a total endomorphism of G; or simply an endomorphism of G.

A partial (or total) endomorphism μ* of a supergroup G* of G is said to extend (or continue) μ if μ* is defined on a supergroup A* of A, that is, μ* is defined for at least the elements for which μ. is defined, and moreover μ* coincides with μ on A.

In this paper, we study geometric structures on 2-dimensional simplicial complexes. In particular, we consider hyperbolic structures and measured foliations on these simplicial complexes. We describe the spaces of such structures and we relate the two resulting spaces in a manner which is analogous to Thurston's compactification of the Teichmüller space of a surface.

A multicurve${\mathcal{C}}$ in a closed orientable surface Sg of genus g is defined to be a finite collection of disjoint non-isotopic essential simple closed curves. A left-handed Dehn twist$t_{\mathcal{C}}$about${\mathcal{C}}$ is the product of left-handed Dehn twists about the individual curves in ${\mathcal{C}}$. In this paper, we derive necessary and sufficient conditions for the existence of a root of $t_{\mathcal{C}}$ in the mapping class group Mod(Sg). Using these conditions, we obtain combinatorial data that correspond to roots, and use it to determine upper bounds on the degree of a root. As an application of our theory, we classify all such roots up to conjugacy in Mod(S4). Finally, we establish that no such root can lie in the level m congruence subgroup of Mod(Sg), for m ≥ 3.

Let $\Delta$ be one of the dual polar spaces $DW(5,q)$ or $DH(5,q^2)$. We consider a class of subspaces of $\Delta$, each member of which carries the structure of a near hexagon, and classify all these subspaces. Using this classification, we determine all hyperplanes of $DW(5,q)$ without ovoidal quads.

It is well known that, provided that the indexing set I is not too large, the product

of a family of bornological locally convex topological vector spaces Eαis bornological. Products of bornological spaces were first studied by Mackey [3]. He reduced the problem to the study of R1, showing that this space is bornological if and only if I satisfies a certain condition, related to a problem in measure theory posed by Ulam [5]. We shall therefore call it the Mackey-Ulam condition on I. A similar study of the spaces R1 is to be found in the paper [4] by Simons; see also [1, Ch. IV, §6, exercise 3].

In a recent paper Cooke [1] obtained a solution of the integral equation

by using the identity

and the technique, first used by Copson, of interchanging the orders of integration and hence reducing the problem to that of the successive solution of two Abel integral equations. It is also shown in [1] that the above identity can also be used to solve the dual series equations

The kernel in equation (1) is a particular member of a general class of kernels which the author [6] has shown to be such that the resulting integral equation is directly soluble by using Copson's technique. The particular example of equation (1) is given in [6] and the identity of equation (2) was used by the author [7] to obtain the solution of equation (3).

In the theory of spectral (and prespectral) operators in a Banach space or in a locally convex topological vector space the countable additivity (in some topology) of a resolution of the identity of the operator is a standing assumption. One might wonder why. Even if one cannot completely agree with the opinion of Diestel and Uhl ([6, p. 32]) stating that “countable additivity [of a set function] is often more of a hindrance than a help”, it might be interesting to study which portions of the theory of (pre)spectral operators and in which form extend to the more general situation described below.

1. Introduction: Suppose the group G is generated by subnormal subgroups H and K, and that A, B are normal subgroups of finite index in H, Krespectively. The following question has been asked by J. C. Lennox: Under what circumstances is the subgroupJ = (A, B) subnormal in G? In particular, it is of interest to know when J has finite index in G, for, if this is the case, we may factor out by the normal core of J in G and apply Wielandt's theorem on joins of subnormal subgroups of finite groups [11] to deduce that J is subnormal in G. Here we prove the following result.

Number fields such as described in the title play a rôle in the study of Artin L-functions and automorphic forms for the groups SL2 over rings of integers in quadratic extensions of ℚ. They are also of some interest on their own. We have not found many examples in the literature. Lang [4] mentions an unramified A5-extension of a real quadratic number field which is due to E. Artin.

In Schinzel [1] the following interesting question is asked: does there exist an absolute constant K such that every trinomial in ℚ[x] has a factor irreducible in ℚ[x] which has at most K terms? The only known result appears to be that of Mrs. H. Smyczek, given in the above paper, that if K exists, then K ≥ 6. We here extend this bound to K ≥ 8 by exhibiting a trinomial in ℤ[x] which splits into the product of two irreducible factors, each having 8 terms.

Let

be the rath cyclotomic polynomial, and denote by An the absolute value of the largest coefficient of Fn(x).Schur proved that

and Emma Lehmer [5] showed that An>cn1/3 for infinitely many n; in fact she proved that n can be chosen as the product of three distinct primes. I proved [3] that there exists a positive constant q such that, for infinitely many n

and Bateman [1] proved very simply that, for every ∈>0 and all n>no(∈),

We generalise statements known about Springer fibres associated to nilpotents with two Jordan blocks to Spaltenstein varieties. We study the geometry of generalised irreducible components (i.e. Bialynicki-Birula cells) and their pairwise intersections. In particular, we develop a graphical calculus that encodes their structure as iterated fibre bundles with ℂℙ1 as base spaces, and compute their cohomology. At the end, we present a connection with coloured cobordisms generalising the construction of Khovanov (M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101(3) (2000), 359–426) and Stroppel (C. Stroppel, Parabolic category O, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology, Compositio Mathematica145(4) (2009), 954–992).

The trace on matrix rings, along with the augmentation map and Kaplansky trace on group rings, are some of the many examples of linear functions on algebras that vanish on all commutators. We generalize and unify these examples by studying traces on (contracted) semigroup rings over commutative rings. We show that every such ring admits a minimal trace (i.e., one that vanishes only on sums of commutators), classify all minimal traces on these rings, and give applications to various classes of semigroup rings and quotients thereof. We then study traces on Leavitt path algebras (which are quotients of contracted semigroup rings), where we describe all linear traces in terms of central maps on graph inverse semigroups and, under mild assumptions, those Leavitt path algebras that admit faithful traces.

In this paper we study a seemingly unnoticed property of bases in a Hilbert space that falls in the general area of constructing new bases from old, yet is quite atypical of others in this regard. Namely, if $\{x_n\}$ is any normalized basis for a Hilbert space $ H $ and $ \{\,f_n\}$ the associated basis of coefficient functionals, then the sequence $\{x_n+f_n\} $ is again a basis for $ H $. The unusual aspect of this observation is that the basis $ \{x_n+f_n\} $ obtained in this way from $\{x_n\} $ and $ \{\,f_n\} $ need not be equivalent to either, in contrast to the standard techniques of constructing new bases from given ones by means of an isomorphism on $ H $. In this paper we study bases of this form and their relation to the component bases $ \{x_n\} $ and $ \{\,f_n\}$.