Several results about the multiplicities of indecomposable injectives in the minimal injective resolution of a ring exist in the literature. Mostly these apply to universal enveloping algebras of finite dimensional solvable Lie algebras, and to Gorenstein noetherian PI local rings. These results are unified and extended to the much wider class of rings with Auslander dualizing complexes.

It is shown that for any meromorphic function $f$ the Julia set $J(f)$ has constant local upper and lower box dimensions, $\overline{d}(J(f))$ and $\underline{d}(J(f))$ respectively, near all points of $J(f)$ with at most two exceptions. Further, the packing dimension of the Julia set is equal to $\overline{d}(J(f))$. Using this result it is shown that, for any transcendental entire function $f$ in the class $B$ (that is, the class of functions such that the singularities of the inverse function are bounded), both the local upper box dimension and packing dimension of $J(f)$ are equal to 2. The approach is to show that the subset of the Julia set containing those points that escape to infinity as quickly as possible has local upper box dimension equal to 2.

It is shown that, for finitely generated residually soluble groups, a condition weaker than polynomial growth guarantees virtual nilpotence. Let $G$ be a residually soluble group having a finite generating set $X$, and suppose that the number $\ga_X(n)$ of elements of $G$ that are products of at most $n$ elements of $X\cup X^{-1}$ satisfies $\ga_X(n)\leqslant e^{\alpha(n)}$ for each $n$, where $\alpha(n)/e^{(1/2)(\ln n)^{1/2}}\to\infty$ as $n\to\infty$; then $G$ is virtually nilpotent.

When $T\,{:}\, X\,{\longrightarrow}\,X$ is a one-sided topologically mixing subshift of finite type and $\varphi\,{:}\,X\,{\longrightarrow}\,R$ is a continuous function, one can define the Ruelle operator ${\cal L}_\varphi\,{:}\,C(X)\,{\longrightarrow}\,C(X)$ on the space $C(X)$ of real-valued continuous functions on $X$. The dual operator ${\cal L}^*_\varphi$ always has a probability measure $\nu$ as an eigenvector corresponding to a positive eigenvalue (${\cal L}^*_\varphi\nu\,{=}\,\lambda\nu$ with $\lambda\,{>}\,0$). Necessary and sufficient conditions on such an eigenmeasure $\nu$ are obtained for $\varphi$ to belong to two important spaces of functions, $W(X,T)$ and ${\rm Bow} (X,T)$. For example, $\varphi\,{\in}\,{\rm Bow}(X,T)$ if and only if $\nu$ is a measure with a certain approximate product structure. This is used to apply results of Bradley to show that the natural extension of the unique equilibrium state $\mu_\varphi$ of $\varphi\,{\in}\,{\rm Bow}(X,T)$ has the weak Bernoulli property and hence is measure-theoretically isomorphic to a Bernoulli shift. It is also shown that the unique equilibrium state of a two-sided Bowen function has the weak Bernoulli property. The characterizations mentioned above are used in the case of $g$-measures to obtain results on the ‘reverse’ of a $g$-measure.

A method is presented to construct interpolation functions into the $2\times 2$ open spectral unit ball. For the spectral Nevanlinna–Pick problem, these functions are in some sense extremal, and the set of all these interpolation functions is enough to solve any interpolation problem, with solvable finite interpolation data. This fact is used to compute the complex geodesics for the symmetrized bidisc and for the spectral unit ball, and to solve completely the two-point interpolation problem for the two target sets.

Let $C$ be a genus 2 algebraic curve defined by an equation of the form $y^2\,{=}\,x(x^2\,{-}\,1)(x\,{-} a)(x\,{-}\,1/a)$. As is well known, the five accessory parameters for such an equation can all be expressed in terms of $a$ and the accessory parameter $\frak b$ corresponding to $a$. The main result of the paper is that if $a'\,{=}\,\sqrt{1\,{-}\,{a}^2}$, which in general yields a non-isomorphic curve $C'$, then $\acb'a'({a'}^2\,{-}\,1) {=}\,{-}\frac38\,{-}\,\acb\,a\,({a}^2\,{-}\,1)$.

This is proven by it being shown how the uniformizing function from the unit disk to $C'$ can be explicitly described in terms of the uniformizing function for $C$.

The main result ensures that the scalar problem $x''\,{=}\,f(x)$, $x(0)\,{=}\,x_0$, $x'(0)\,{=}\,x_1,$ has a nonconstant locally $W^{2,1}$ solution if and only if there exists a nontrivial interval $J$ such that $x_0 \in J$, $f \in L^1_{\rm loc}(J)$, $x_1^2+2\int_{x_0}^{y}{f(s)\,ds} > 0$ for almost all $y \in J$ and \[\frac{\max\{1,|f|\}}{ \sqrt{x_1^2+2\int_{x_0}^{\cdot}{f(s)\,ds}}} \in L^1_{\rm loc}(J).\] Necessary and sufficient conditions for local and global uniqueness and for existence of periodic solutions are also established.

The sector of analyticity of the Ornstein–Uhlenbeck semigroup is computed on the space $L^p_\mu \,{:=}\, L^p (\R^N; \mu)$ with respect to its invariant measure $\mu$. If $A\,{=}\,\Delta + Bx\cdot \nabla$ denotes the generator of the Ornstein–Uhlenbeck semigroup, then the angle $\theta_2$ of the sector of analyticity in $L^2_\mu$ is ${\pi}/{2}$ minus the spectral angle of $BQ_\infty, Q_\infty$ being the matrix determining the Gaussian measure $\mu$. The angle of analyticity in $L^p_\mu$ is then given by the formula \[\cot {\theta_p} = \frac{\sqrt{(p-2)^2+p^2(\cot\theta_2)^2}}{2\sqrt{p-1}}.\]

The convergent rates for bounded solutions of Dirichlet problems of quasilinear elliptic (possibly degenerate) equations in slab-like domains are derived in terms of the convergent rates of the boundary data and the coefficients of the operator. The equations considered include the prescribed mean curvature equation. The results are proved by constructing a family of local barrier functions based on the structure of the operator and the convergent rate of the boundary data. The construction of local barriers is inspired by early work due to Finn and Serrin that is related to the minimal surface equation.

Using critical point theory, some sufficient conditions are obtained for the existence of nonconstant $T$-periodic solutions of a class of second order self-adjoint difference equations.

A spectral inclusion theorem and a spectral mapping theorem are proved for the functional calculus for sectorial operators. Most proofs are generic, so that similar results can be obtained for other functional calculi. Applications are a new proof for the spectral mapping theorem for fractional powers and the identity $\sigma(\log A)\,{=}\,\log(\sigma(A)\ohne\{0\})$ for any injective sectorial operator $A$.

Let $\tau$ be a conjugation, alias a conjugate linear isometry of order 2, on a complex Banach space $X$ and let $X^{\tau}$ be the real form of $X$ of $\tau$-fixed points. In contrast to the Dunford–Pettis property, the alternative Dunford–Pettis property need not lift from $X^{\tau}$ to $X$. If $X$ is a C*-algebra it is shown that $X^{\tau}$ has the alternative Dunford–Pettis property if and only if $X$ does and an analogous result is shown when $X$ is the dual space of a C*-algebra. One consequence is that both Dunford–Pettis properties coincide on all real forms of C*-algebras.

Characterizations of subnormality due to Trent are extended to finite systems of operators, thus generalizing results of Gãvruţã and Suciu (subnormal pairs) and Binzar and Pãunescu (subnormal triplets). Vector and operator multidimensional moment problems with jointly subnormal solutions are characterized entirely in terms of the initial data. Bounded as well as unbounded solutions are taken into consideration.

It is shown that the invariant subspace of the Bergman space $A^p$ of the unit disk, generated by either an $A^p$-interpolating sequence or a singular inner function with a single point mass on the unit circle, is complemented in $A^p$.

Criteria are found which classify Schatten class membership for operators on the Hardy space in terms of their action on the reproducing kernels and their derivatives. The Hilbert–Schmidt conditions are shown to be both necessary and sufficient for arbitrary operators. For other Schatten classes, the criteria are shown to be either necessary or sufficient for arbitrary operators, depending upon the exponent of the class. However, using known results of Peller and Luecking, the criteria are shown to be both necessary and sufficient for Hankel operators and Carleson embeddings.

Extensions of the concept of asymptotic independence which are useful in the study of abelian C$^*$-algebras generated by the union of two of their C$^*$-subalgebras are examined. Non-trivial examples are also given in terms of classes of bounded continuous functions defined on a locally compact group.

Given a hyperbolic invariant set of a diffeomorphism on a surface, it is proved that, if the holonomies are sufficiently smooth, then the diffeomorphism on the hyperbolic invariant set is rigid in the sense that it is C$^{1+}$ conjugate to a hyperbolic affine model.

Multidirectional mean value inequalities provide estimates of the difference of the extremal value of a function on a given bounded set and its value at a given point in terms of its (sub)-gradient at some intermediate point. A generalization of such multidirectional mean value inequalities is derived by using new infinitesimal conditions for a weak r-growth of the lower semicontinuous function along approximate trajectories of differential inclusions. This new form of the multidirectional mean value inequality does not rely on the linear structure of the underlying space and removes a traditional assumption of lower boundedness on the function.

Certain linear operators from a Banach algebra $A$ into a Banach $A$-bimodule $X$, which are called approximately local derivations, are studied. It is shown that when $A$ is a ${\rm C^*}$-algebra, a Banach algebra generated by idempotents, a semisimple annihilator Banach algebra, or the group algebra of a SIN or a totally disconnected group, bounded approximately local derivations from $A$ into $X$ are derivations. This, in particular, extends a result of B. E. Johnson that ‘local derivations on ${\rm C^*}$-algebras are derivations’ and provides an alternative proof of it.

The covariogram of a convex body $K$ provides the volumes of the intersections of $K$ with all its possible translates. Matheron conjectured in 1986 that this information determines $K$ among all convex bodies, up to certain known ambiguities. It is proved that this is the case if $K\subset{\mathbb R}^2$ is not ${\rm C}^1$, or it is not strictly convex, or its boundary contains two arbitrarily small ${\rm C}^2$ open portions ‘on opposite sides’. Examples are also constructed that show that this conjecture is false in ${\mathbb R}^n$ for any $n\geq 4$.