For an artin algebra $\Lambda$, cotorsion pairs are studied for the category ${\rm mod}\Lambda$ of finitely presented $\Lambda$-modules and for the category ${\rm Mod}\Lambda$ of all $\Lambda$-modules. It is shown that every cotorsion pair for ${\rm mod}\Lambda$ induces a cotorsion pair for ${\rm Mod}\Lambda$. This has some interesting applications, even for the category of finitely presented modules. Another theme of the paper is the interplay between cotorsion and torsion pairs. This leads to a conjecture which is an analogue of the telescope conjecture in stable homotopy theory.

A minimax principle is derived for the eigenvalues in the spectral gap of a possibly non-semibounded self-adjoint operator. It allows the nth eigenvalue of the Dirac operator with Coulomb potential from below to be bound by the nth eigenvalue of a semibounded Hamiltonian which is of interest in the context of stability of matter. As a second application it is shown that the Dirac operator with suitable non-positive potential has at least as many discrete eigenvalues as the Schrödinger operator with the same potential.

The paper studies the existence of multiple solutions to the following p-Laplacian type elliptic problem (p > 1):

where Ω is a bounded domain in ℝN(N [ges ] 1) with smooth boundary ∂Ω, and f(x, u) goes asymptotically in u to [mid ]u[mid ]p−2u at infinity. It is well known that this kind of nonlinear term creates some difficulties in the application of the mountain pass theorem because of the lack of an Ambrosetti–Rabinowitz type superlinear condition on f(x, u). An improved mountain pass theorem is used to prove that the above problem possesses multiple solutions under some natural conditions on f(x, u), and some known results are generalized.

Several important cases of vector bundles with extra structure (such as Higgs bundles and triples) may be regarded as examples of twisted representations of a finite quiver in the category of sheaves of modules on a variety/manifold/ringed space. It is shown that the category of such representations is an abelian category with enough injectives by the construction of an explicit injective resolution. Using this explicit resolution, a long exact sequence is found that computes the Ext groups in this new category in terms of the Ext groups in the old category. The quiver formulation is directly reflected in the form of the long exact sequence. It is also shown that under suitable circumstances, the Ext groups are isomorphic to certain hypercohomology groups.

An explicit asymptotic formula is obtained for the norms of the spectral projections of the non-self-adjoint harmonic oscillator $H$. It is deduced that the spectral expansion of $\rme^{-Ht}$ is norm convergent if and only if $t$ is greater than a certain explicit positive constant.

A transitive finite permutation group is called elusive if it contains no nontrivial semiregular subgroup. The purpose of the paper is to collect known information about elusive groups. The main results are recursive constructions of elusive permutation groups, using various product operations and affine group constructions. A brief historical introduction and a survey of known elusive groups are also included. In a sequel, Giudici has determined all the quasiprimitive elusive groups.

Part of the motivation for studying this class of groups was a conjecture due to Marušič, Jordan and Klin asserting that there is no elusive 2-closed permutation group. It is shown that the constructions given will not build counterexamples to this conjecture.

Cellular algebras have recently been introduced by Graham and Lehrer [5, 6] as a convenient axiomatization of all of the following algebras, each of them containing information on certain classical algebraic or finite groups: group algebras of symmetric groups in any characteristic, Hecke algebras of type A or B (or more generally, Ariki Koike algebras), Brauer algebras, Temperley–Lieb algebras, (q-)Schur algebras, and so on. The problem of determining a parameter set for, or even constructing bases of simple modules, is in this way reduced (but of course not solved in general) to questions of linear algebra.

The present paper has two aims. First, we make explicit an inductive construction of cellular algebras which has as input data of linear algebra, and which in fact produces all cellular algebras (but no other ones). This is what we call ‘inflation’. This construction also exhibits close relations between several of the above algebras, as can be seen from the computations in [6]. Among the consequences of the construction is a natural way of generalizing Hochschild cohomology. Another consequence is the construction of certain idempotents which is used in the second part of the paper.

The second aim is to study Morita equivalences of cellular algebras. Since the input of many of the constructions of representation theory of finite-dimensional algebras is a basic algebra, it is useful to know whether any finite-dimensional cellular algebra is Morita equivalent to a basic one by a Morita equivalence that preserves the cellular structure. It turns out that the answer is ‘yes’ if the underlying field has characteristic other than 2. However, there are counterexamples in the case of characteristic 2, or more generally for any ring in which 2 is not invertible. This also tells us that the notion of ‘cellular’ cannot be defined only in terms of the module category. However, in any characteristic we find some useful Morita equivalences which are compatible with cellular structures.

Let ${\mathcal H}_{q}(d)$ be the Iwahori–Hecke algebra of the symmetric group, where $q$ is a primitive $l$th root of unity. Using results from the cohomology of quantum groups and recent results about the Schur functor and adjoint Schur functor, it is proved that, contrary to expectations, for $l \geq 4$ the multiplicities in a Specht or dual Specht module filtration of an ${\mathcal H}_q(d)$-module are well defined. A cohomological criterion is given for when an ${\mathcal H}_{q}(d)$-module has such a filtration. Finally, these results are used to give a new construction of Young modules that is analogous to the Donkin–Ringel construction of tilting modules. As a corollary, certain decomposition numbers can be equated with extensions between Specht modules. Setting $q = 1$, results are obtained for the symmetric group in characteristic $p \geq 5$. These results are false in general for $p = 2$ or $3$.

Let G be a finite group and k be an algebraically closed field of prime characteristic. Corresponding to each closed homogeneous subvariety W of the maximal ideal spectrum of H*(G, k) we construct (usually infinite-dimensional) kG-modules E(W) and F(W) which are idempotent in the sense that E(W) and F(W) are isomorphic (up to projective summands) to E(W) [otimes ] E(W) and F(W) [otimes ] F(W) respectively. We study the properties of these modules, and as an application we use them to describe natural direct sum decompositions of modules in quotient categories.

The paper gives a necessary and sufficient criterion on the Lévy measure that determines whether a Lévy process creeps. (The author says that a Lévy process creeps if it can continuously pass a fixed level.)

We use generalized power series to construct algebraically a nonstandard model of the theory of the real field with exponentiation. This model enables us to show the undefinability of the zeta function and certain non-elementary and improper integrals. We also use this model to answer a question of Hardy by showing that the compositional inverse to the function (log x) (log log x) is not asymptotic as x→+∞ to a composition of semialgebraic functions, log and exp.

Given $q$ , a power of a prime $p$ , denote by $F$ the finite field ${\rm GF}(q)$ of order $q$ , and, for a given positive integer $n$ , by $E$ its extension ${\rm GF}(q^n)$ of degree $n$ . A primitive element of $E$ is a generator of the cyclic group $E^\ast$ . Additively too, the extension $E$ is cyclic when viewed as an $FG$ -module, $G$ being the Galois group of $E$ over $F$ . The classical form of this result – the normal basis theorem – is that there exists an element $\alpha \in E$ (an additive generator) whose conjugates $\{\alpha, \alpha^q, \ldots, \alpha^{q^{n-1}}\}$ form a basis of $E$ over $F; \alpha$ is a free element of $E$ over $F$ , and a basis like this is a normal basis over $F$ . The core result linking additive and multiplicative structure is that there exists $\alpha \in E$ , simultaneously primitive and free over $F$ . This yields a primitive normal basis over $F$ , all of whose members are primitive and free. Existence of such a basis for every extension was demonstrated by Lenstra and Schoof [5] (completing work by Carlitz [1, 2] and Davenport [4]).

an approximation theorem in inhomogeneous orlicz–sobolev spaces is proved which allows a second-order parabolic equation in orlicz spaces to be solved. a trace result is also given which shows that the solutions are continuous with respect to time.

In this paper we consider the form of the eigenvalue counting function ρ for Laplacians on p.c.f. self-similar sets, a class of self-similar fractal spaces. It is known that on a p.c.f. self-similar set K the function ρ(x)=O(xds/2) as x→∞, for some ds>0. We show that if there exist localized eigenfunctions (that is, a non-zero eigenfunction which vanishes on some open subset of the space) and K satisfies some additional conditions (‘the lattice case’) then ρ(x)x−ds/2 does not converge as x→∞. We next establish a number of sufficient conditions for the existence of a localized eigenfunction in terms of the symmetries of the space K. In particular, we show that any nested fractal with more than two essential fixed points has localized eigenfunctions.

One-parameter overpartition-theoretic analogues are given of two classical families of partition identities: Andrews' combinatorial generalization of the Gollnitz–Gordon identities and a theorem of Andrews and Santos on partitions with attached odd parts. Geometric counterparts arising from multiple $q$-series identities are also discussed. These involve representations of overpartitions in terms of generalized Frobenius partitions.

We develop a deformation theory for k-parameter families of pointed marked graphs with fixed fundamental group Fn. Applications include a simple geometric proof of stability of the rational homology of Aut(Fn), computations of the rational homology in small dimensions, proofs that various natural complexes of free factorizations of Fn are highly connected, and an improvement on the stability range for the integral homology of Aut(Fn).

In this paper we shall prove that the measure algebra $M(G)$ of a locally compact group $G$ is amenable as a Banach algebra if and only if $G$ is discrete and amenable as a group. Our contribution is to resolve a conjecture by proving that $M(G)$ is not amenable in the case where the group $G$ is not discrete. Indeed, we shall prove a much stronger result: the measure algebra of a non-discrete, locally compact group has a non-zero, continuous point derivation at a certain character on the algebra.

The question of when a $\phi$ -derivation on a Banach algebra has quasinilpotent values, and how this question is related to the noncommutative Singer–Wermer conjecture, is discussed.

The closure of the periodic points of rational maps over a non-archimedean field is studied. An analogue of Montel's theorem over non-archimedean fields is first proved. Then, it is shown that the (nonempty) Julia set of a rational map over a non-archimedean field is contained in the closure of the periodic points.

On page 212 of his lost notebook, Ramanujan defined a new class invariant λn and constructed a table of values for λn. The paper constructs a new class of series for 1/π associated with λn. The new method also yields a new proof of the Borweins' general series for 1/π belonging to Ramanujan's ‘theory of q2’.