In this paper we give two different proofs that the flat cover conjecture is true: that is, every module has a flat cover. The two proofs are of completely different nature, and, we hope, will have different applications. The first of the two proofs (due to the third author) is essentially an application of the work of P. Eklof and J. Trlifaj (work which is more set-theoretic). The second proof (due to the first two authors) is more direct, and has a model-theoretic flavour.

The entities A, B, X, Y in the title are operators, by which we mean either linear transformations on a finite-dimensional vector space (matrices) or bounded (= continuous) linear transformations on a Banach space. (All scalars will be complex numbers.) The definitions and statements below are valid in both the finite-dimensional and the infinite-dimensional cases, unless the contrary is stated.

For f a meromorphic function on the plane domain D and a ∈ [Copf ], let Ēf(a) = {z ∈ D[ratio ]f(z) = a}. Let [Fscr ] be a family of meromorphic functions on D, all of whose zeros are of multiplicity at least k. If there exist b ≠ 0 and h > 0 such that for every f ∈ [Fscr ], Ēf(0) = Ēf(k)(b) and 0 < [mid ]f(k+1)(z)[mid ] [les ] h whenever z ∈ Ēf(0), then [Fscr ] is a normal family on D. The case Ēf(0) = Ø is a celebrated result of Gu [5].

We describe a general construction of a module A from a given module B such that Ext(B, A) = 0, and we apply it to answer several questions on splitters, cotorsion theories and saturated rings.

This paper encompasses a motley collection of ideas from several areas of mathematics, including, in no particular order, random walks, the Picard group, exchange rate networks, chip-firing games, cohomology, and the conductance of an electrical network. The linking threads are the discrete Laplacian on a graph and the solution of the associated Dirichlet problem. Thirty years ago, this subject was dismissed by many as a trivial specialisation of cohomology theory, but it has now been shown to have hidden depths. Plumbing these depths leads to new theoretical advances, many of which throw light on the diverse applications of the theory.

Commutative rings form a very special subclass of rings, which shows quite different behaviour from the general case. For example, in a (non-trivial) commutative ring, the absence of zero-divisors is sufficient as well as necessary for the existence of a field of fractions, whereas for general rings, another infinite set of conditions is needed to characterize subrings of skew fields. This suggests the study of a class of rings which includes all commutative rings as well as all integral domains: reversible rings, where a ring is called reversible if ab = 0 implies ba = 0. It turns out that this condition helps to simplify other ring conditions, as we shall see in Section 2, although most of these results are at a somewhat superficial level. We therefore introduce a more technical notion, full reversibility, in Section 3, and show that this is the precise condition for the least matrix ideal to be proper and consist entirely of non-full matrices. Further, we show in Section 4 that a fully reversible ring is embeddable in a skew field if and only if it is an integral domain.

In what follows, all rings are associative, with a unit element 1 which is preserved by ring homomorphisms, inherited by subrings and acts unitally on modules.

I am grateful to V. de O. Ferreira for his comments, in particular the suggestion of using the notion of unit-stable rings.

We derive a sharp, localized version of elliptic type gradient estimates for positive solutions (bounded or not) to the heat equation. These estimates are related to the Cheng–Yau estimate for the Laplace equation and Hamilton's estimate for bounded solutions to the heat equation on compact manifolds. As applications, we generalize Yau's celebrated Liouville theorem for positive harmonic functions to positive ancient (including eternal) solutions of the heat equation, under certain growth conditions. Surprisingly this Liouville theorem for the heat equation does not hold even in ${\mathbb R}^n$ without such a condition. We also prove a sharpened long-time gradient estimate for the log of the heat kernel on noncompact manifolds.

Jensen's operator inequality and Jensen's trace inequality for real functions defined on an interval are established in what might be called their definitive versions. This is accomplished by the introduction of genuine non-commutative convex combinations of operators, as opposed to the contractions considered in earlier versions of the theory by the authors, and by Brown and Kosaki. As a consequence, one no longer needs to impose conditions on the interval of definition. It is shown how this relates to the pinching inequality of Davis, and how Jensen's trace inequality generalizes to C*-algebras.

The purpose of this work is to show that the fractional maximal operator has somewhat unexpected regularity properties. The main result shows that the fractional maximal operator maps $L^p$-spaces boundedly into certain first-order Sobolev spaces. It is also proved that the fractional maximal operator preserves first-order Sobolev spaces. This extends known results for the Hardy–Littlewood maximal operator.

We give a condition for an exact functor between triangulated categories to be an equivalence. Applications to Fourier–Mukai transforms are discussed. In particular, we obtain a large number of such transforms for K3 surfaces.

In this paper, we prove the existence of an unbounded sequence of weak solutions under an appropriate oscillating behaviour of φ for a Neumann problem of the type

The purpose of this note is to provide a new proof for the explicit formulas of the heat kernel on hyperbolic space. By definition, the hyperbolic space IHn is a (unique) simply connected complete n-dimensional Riemannian manifold with a constant negative sectional curvature −1.

A description is given of methods that have been used to analyze the spectrum of non-self-adjoint differential operators, emphasizing the differences from the self-adjoint theory. It transpires that even in cases in which the eigenfunctions can be determined explicitly, they often do not form a basis; this is closely related to a high degree of instability of the eigenvalues under small perturbations of the operator.

A counterexample has been constructed to show that a conjectured global solution structure for bifurcation of non-trivial solutions from a simple eigenvalue of the linearization at zero really can occur. In addition, new results and counterexamples have been obtained for bifurcation from an eigenvalue of geometric multiplicity 1 and odd algebraic multiplicity.

1. Introduction 514

2. Eigenvalues and eigenfunctions 516

2.1 Spectral asymptotics 516

2.2 The isoperimetric problem 518

2.3 The Rellich inequality 519

2.4 Numerical studies of the rectangular plate 519

2.5 The ground state 521

2.6 Stability properties 522

2.7 The ideal column 523

3. Solving elliptic equations 524

3.1 One-dimensional results 524

3.2 Constant-coefficient operators 524

3.3 Green functions of bounded regions 525

3.4 The Poisson problem 526

3.5 The Dirichlet problem 526

3.6 Conical points 527

4. The semigroup e−Ht 528

4.1 Elliptic systems 528

4.2 Extensions to Lp 529

4.3 Generalized Schrödinger operators 530

4.4 Kato class potentials 530

5. Heat kernel bounds 531

5.1 General properties of heat kernels 532

5.2 Gaussian upper bounds 532

5.3 Sharp constants 533

5.4 Complex time bounds 534

5.5 Pointwise lower bounds 535

6. Lp spectral theory 536

6.1 Functional calculus 536

6.2 Helffer–Sjöstrand calculus 537

6.3 Spectral independence 538

7. Non-self-adjoint operators 538

7.1 Second-order operators 538

7.2 Lp multiplier theory 539

7.3 H∞ functional calculus 540

7.4 Applications of the H∞ functional calculus 540

References 541

We construct several examples of positive definite functions, and use the positive definite matrices arising from them to derive several inequalities for norms of operators.

Free entropy is the analogue of entropy in free probability theory. The paper is a survey of free entropy, its applications to von Neumann algebras and its connections to random matrix theory, as well as a discussion of open problems and of a basic variational problem, connected to random multimatrix models.

This paper introduces baby Verma modules for symplectic reflection algebras of complex reflection groups at parameter $t=0$ (the so-called rational Cherednik algebras at parameter $t=0$), and presents their most basic properties. Baby Verma modules are then used to answer several problems posed by Etingof and Ginzburg, and to give an elementary proof of a theorem of Finkelberg and Ginzburg.

A subset $A$ of the integers is said to be sum-free if there do not exist elements $x,y,z \,{\in}\, A$ with $x \,{+}\, y \,{=}\, z$. It is shown that the number of sum-free subsets of $\{1,\ldots,N\}$ is $O(2^{N/2})$, confirming a well-known conjecture of Cameron and Erdős.

Let $G\subseteq \mathbb{C}^{2}$ be the open symmetrized bidisc, namely $G= \{(\lambda_{1}+\lambda_{2},\lambda_{1}\lambda_{2}):|\lambda_{1}|<1,|\lambda_{2}|<1\}$. In this paper, a proof is given that $G$ is not biholomorphic to any convex domain in $\mathbb{C}^{2}$. By combining this result with earlier work of Agler and Young, the author shows that $G$ is a bounded domain on which the Carathéodory distance and the Kobayashi distance coincide, but which is not biholomorphic to a convex set.