We study genus 5 curves C with a fixed point free involution. We give a geometrical (embedded) characterisation of these curves among all genus 5 curves: the points of the Prym variety associated to the involution give embeddings of the curve C in ${{\mathbb P}}_3$ so that C has infinitely many quadrisecant lines. Conversely, any genus 5 curve having such an embedding is endowed with a fixed point free involution and the embedding is given by a point of the Prym variety.

The work in this paper is part of an ongoing program to classify maximal orders on surfaces via their ramification data. Del Pezzo orders and ruled orders have already been classified by the authors and others. In this paper, we classify numerically Calabi–Yau orders which are the noncommutative analogues of minimal surfaces of Kodaira dimension zero.

Standard codeterminants for Donkin's symplectic Schur algebra $\bar{S}(n,r)$ are defined. It is shown that they form a basis of $\bar{S}(n,r)$. They are then used to give a purely combinatorial proof that $\bar{S}(n,r)$ is quasi-hereditary.

The paper investigates the partial regularity of the minimizers for quadratic functionals whose integrands have VMO coefficients in principal part and nonlinear terms that are Carathéodory functions. We use some majorizations for the functional, rather than the well known Euler equation associated to it.

The spectral gaps and thus the exponential rates of convergence to equilibrium are compared for ergodic one-dimensional diffusions on an interval. One of the results may be thought of as the diffusion analogue of a recent result for the spectral gap of one-dimensional Schrödinger operators. The similarities and differences between spectral gap results for diffusions and for Schrödinger operators are also discussed.

By using Morse theory, we study the existence and multiplicity of nontrivial solutions for a class of p-sublinear p-Laplacian elliptic equations.

On the basis of some new Liouville theorems, under suitable conditions, a priori estimates are obtained of positive solutions of the problem \[-\Delta _pu=\lambda u^{\alpha }-a(x)u^q\quad \mbox{in}\;\Omega,\qquad u|_{\partial \Omega }=0,\] where $\Omega \subset {\mathbb{R}}^N$ ($N\geq 2$) is a bounded smooth domain, $p>1$ and λ is a parameter, α, q are given constants such that $p-1<\alpha <p^*-1$, $\alpha <q$, $p^*=Np/(N-p)$ if $N > p$ and $p^*=\infty $ when $N\leq p$, and $a(x)$ is a continuous nonnegative function. Making use of the Leray–Schauder degree of a compact mapping and a priori estimates, the paper finds that the problem above possesses at least one positive solution. It also discusses the corresponding perturbed problem, where $a(x)$ is replaced by $a(x)+\epsilon$, $\epsilon>0$. The results are strikingly different from those obtained for the case $\alpha=p-1$.

The Cauchy problem is studied for the nonlinear equations with fractional power of the negative Laplacian \[ \left\{ \begin{array}{@{}r@{\,}c@{\,}l@{\qquad}l} u_{t}+( -\Delta ) ^{{\alpha}/{2}}u+u^{1+\sigma } &=&0, & x\in {\mathbf{R}}^{n},\text{ }t>0, \\[4pt] u( 0,x) &=& u_{0} ( x), &x\in {\mathbf{R}}^{n}, \end{array} \right. \label{A} \] where $\alpha \in ( 0,2)$, with critical $\sigma ={\alpha }/{ n}$ and sub-critical $\sigma \in ( 0,{\alpha }/{n}) $ powers of the nonlinearity. Let $u_{0}\,{\in}\, \mathbf{L}^{1,a}\cap \mathbf{L}^{\infty }\cap \mathbf{C}, $$u_{0}( x) \,{\geq}\, 0$ in $\mathbf{R}^{n},$$\theta \,{=}\,\int_{ \mathbf{R}^{n}}u_{0}( x) \,dx\,{>}\,0.$ The case of not small initial data is of interest. It is proved that the Cauchy problem has a unique global solution $u \in \mathbf{C}( [0,\infty); \mathbf{L}^{\infty }\cap \mathbf{L}^{1,a}\cap \mathbf{C})$ and the large time asymptotics are obtained.

For $(\l,a)\in \matbb{C}^*\times \mathbb{C}$, let $f_{\lambda,a}$ be the rational map defined by $f_{\lambda,a}(z) \,{=}\, \lambda z^2 {(az+1)/(z+a)}$. If $\alpha\in \mathbb{R}/\mathbb{Z}$ is a Brjuno number, we let ${\cal D}_\alpha$ be the set of parameters $(\lambda,a)$ such that $f_{\lambda,a}$ has a fixed Herman ring with rotation number $\alpha$ (we consider that $({\it e}^{2i\pi\alpha}{,}0)\,{\in}\, {\cal D}_\alpha$). Results obtained by McMullen and Sullivan imply that, for any $g\in {\cal D}_\alpha$, the connected component of ${\cal D}_\alpha\cap (\mathbb{C}^*\times(\mathbb{C}\setminus \{0,1\}))$ that contains g is isomorphic to a punctured disk.

We show that there is a holomorphic injection $\cal{F}_\alpha\,{:}\,\mathbb{D}\,{\longrightarrow}\, {\cal D}_\alpha$ such that $\cal{F}_\alpha(0) = ({\it e}^{2i\pi \alpha},0)$ and $\cal{F}_\alpha'(0)=(0,r_\alpha),$ where $r_\alpha$ is the conformal radius at 0 of the Siegel disk of the quadratic polynomial $z\longmapsto {\it e}^{2i\pi \alpha}z(1+z)$.

As a consequence, we show that for $a\in (0,1/3)$, if $f_{\l,a}$ has a fixed Herman ring with rotation number $\alpha$ and if $m_a$ is the modulus of the Herman ring, then, as $a\,{\rightarrow}\,0$, we have ${\it e}^{\pi m_a} \,{=} ({r_\alpha}/{a}) + {\cal O}(a).$

We finally explain how to adapt the results to the complex standard family $z\,{\longmapsto} \lambda z {\it e}^{({a}/{2})(z-1/z)}$.

We extend uncertainty principles which are valid for the Fourier transform to the setting of the ambiguity function. A general result is established for annihilating sets: strongly/weakly annihilating sets for the Fourier transform yield such sets for the ambiguity function, extending a result known for sets of finite measure. We apply this to sublevel sets of nondegenerate quadratic forms. Our main result is a sharp version of Beurling's uncertainty principle for the ambiguity function.

The natural geometric setting of quadrics commuting with a Hermitian surface of ${\rm PG}(3,q^2)$, q odd, is adopted and a hemisystem on the Hermitian surface ${\cal H}(3,q^2)$ admitting the group $\mathrm{{P}}\Omega^-(4,q)$ is constructed, yielding a partial quadrangle ${\rm PQ}((q-1)/2,q^2,(q-1)^2/2)$ and a strongly regular graph srg$((q^3+1)(q+1)/2,(q^2+1)(q-1)/2,(q-3)/2,(q-1)^2/2)$. For $q>3$, no partial quadrangle or strongly regular graph with these parameters was previously known, whereas when $q=3$, this is the Gewirtz graph. Thas conjectured that there are no hemisystems on ${\cal H}(3,q^2)$ for $q>3$, so these are counterexamples to his conjecture. Furthermore, a hemisystem on ${\cal H}(3,25)$ admitting $3.A_7.2$ is constructed. Finally, special sets (after Shult) and ovoids on ${\cal H}(3,q^2)$ are investigated.

We generalise group algebras to other algebraic objects with bounded Hilbert space representation theory; the generalised group algebras are called ‘host’ algebras. The main property of a host algebra is that its representation theory should be isomorphic (in the sense of the Gelfand–Raikov theorem) to a specified subset of representations of the algebraic object. Here we obtain both existence and uniqueness theorems for host algebras as well as general structure theorems for host algebras. Abstractly, this solves the question of when a set of Hilbert space representations is isomorphic to the representation theory of a C*-algebra. To make contact with harmonic analysis, we consider general convolution algebras associated to representation sets, and consider conditions for a convolution algebra to be a host algebra.

Given a directed graph, there exist a universal operator algebra and universal $C^*$-algebra associated to the directed graph. In this paper we give intrinsic constructions for these objects. We also provide an explicit construction for the maximal $C^*$-algebra of an operator algebra. We discuss uniqueness of the universal algebras for finite graphs, showing that for finite graphs the graph is an isomorphism invariant for the universal operator algebra of a directed graph. We show that the underlying undirected graph is a Banach algebra isomorphism invariant for the universal $C^*$-algebra of a directed graph.

The paper considers three questions about independent random points uniformly distributed in isotropic symmetric convex bodies $K, T_1,\ldots, T_s$. (a) Let $\varepsilon\,{\in}\, (0,1)$ and let $x_1,\ldots, x_N$ be chosen from K. Is it true that if $N\,{\geq}\, C(\varepsilon )n\log n$, then \[\left\| I-\frac{1}{NL_K^2}\sum_{i=1}^Nx_i\otimes x_i\right\|<\varepsilon\] with probability greater than $1\,{-}\,\varepsilon $? (b) Let $x_i$ be chosen from $T_i$. Is it true that the unconditional norm \[\|{\bf t}\|=\int_{T_1}\!{\ldots}\int_{T_s}\left\|\sum_{i=1}^st_ix_i\right\|_K\,dx_s\ldots dx_1\] is well comparable to the Euclidean norm in ${\mathbb R}^s$? (c) Let $x_1,\ldots, x_N$ be chosen from K. Let ${\mathbb E}\,(K,N):={\mathbb E}\,|{\rm conv}\{ x_1,\ldots, x_N\}|^{1/n}$ be the expected volume radius of their convex hull. Is it true that ${\mathbb E}\,(K,N)\,{\simeq}\, {\mathbb E}\,(B(n),N)$ for all N, where $B(n)$ is the Euclidean ball of volume 1?

It is proved that the answers to these questions are affirmative if there is a restriction to the class of unconditional convex bodies. The main tools come from recent work of Bobkov and Nazarov. Some observations about the general case are also included.

Necessary and sufficient conditions are given for linearly recurrent Cantor dynamical systems to have measurable and continuous eigenfunctions. Also an example of a linearly recurrent system with a nontrivial Kronecker factor and a trivial maximal equicontinuous factor is constructed explicitly.