A d.c.e. (2-computably enumerable) degree d is pseudo-isolated if d itself is non-isolated (in the sense that no computably enumerable (c.e.) degree below d can bound the c.e. degrees below d) and there is a d.c.e. degree \textbf{b} < \textbf{d} bounding all c.e. degrees below d. We prove in this paper that the pseudo-isolated degrees are densely distributed in the c.e. degrees.

In an earlier paper (see Proc. London Math. Soc. (3) 84 (2002) 257–288) we showed that an irreducible integral binary cubic form $f(x, y)$ attains infinitely many prime values, providing that it has no fixed prime divisor. We now extend this result by showing that $f(m, n)$ still attains infinitely many prime values if $m$ and $n$ are restricted by arbitrary congruence conditions, providing that there is still no fixed prime divisor.

Two immediate consequences for the solvability of diagonal cubic Diophantine equations are given.

For an arbitrary field $F$ we study the double scissors congruence groups $A_n(F)$ generated by admissible pairs of simplices in the projective $n$-space ${\mathbb P}_F^n$ over $F$. Let $(L, M)$ be an admissible pair of simplices whose faces are defined as hyperplanes in ${\mathbb P}_F^n$ such that they do not have common faces of the same dimension. When $L$ and $M$ are in general positions we define a linearly constructible motivic perverse sheaf producing a motivic cohomology whose Betti realization is the relative cohomology $H^{\ast}({\mathbb P}_F^n\setminus L, M\setminus L; {\mathbb Q})$. The motivic cohomology provides a simple explanation of why the generic part of $A_\bullet$ forms a Hopf algebra with well-defined coproduct. When $n = 2$ we define explicitly the coproduct on $A_2$ which simplifies the approach of Beilinson et al. We also complete the same task for $A_3$ and $A_4$ which enables us to develop further results in another paper connecting Aomoto trilogarithms to the classical ones.

Using Grothendieck's semicontinuity theorem for half-exact functors, we derive two semicontinuity results on Hochschild cohomology. We apply these to show that the first Hochschild cohomogy group of the mesh algebra of a translation quiver over a domain vanishes if and only if the translation quiver is simply connected. We then establish an exact sequence relating the first Hochschild cohomology group of an algebra to that of the endomorphism algebra of a module and apply it to study the first Hochschild cohomology group of an Auslander algebra. Our main result shows that for a finite-dimensional and representation-finite algebra algebra $A$ over an algebraically closed field with Auslander algebra $\Lambda$ the following conditions are equivalent:

(1) $A$ admits no outer derivation;

(2) $\Lambda$ admits no outer derivations;

(3) $A$ is simply connected;

(4) $\Lambda$ is strongly simply connected.

The Jucys–Murphy elements for wreath products $\Gamma_n = \Gamma \wr S_n $ associated to any finite group $\Gamma$ are introduced and they play an important role in our study of the connections between class algebras of $\Gamma_n$ for all $n$ and vertex algebras. We construct an action of (a variant of) the $\mathcal W_{1 + \infty}$-algebra acting irreducibly on the direct sum ${\mathcal R}_\Gamma$ of the class algebras of $\Gamma_n$ for all $n$ in a group-theoretic manner. We establish various relations between convolution operators using JM elements and Heisenberg algebra operators acting on ${\mathcal R}_\Gamma$. As applications, we obtain two distinct sets of algebra generators for the class algebra of $\Gamma_n$ and establish various stability results concerning products of normalized conjugacy classes of $\Gamma_n$ and the power sums of Jucys--Murphy elements, etc. We introduce a stable algebra which encodes the class algebra structures of $\Gamma_n$ for all $n$, whose structure constants are shown to be non-negative integers. In the symmetric group case (that is, when $\Gamma$ is trivial), we recover and strengthen in a uniform approach various results of Lascoux and Thibon, Kerov and Olshanski, and Farahat and Higman, etc.

We make the category $\textbf{BGrb}_M$ of bundle gerbes on a manifold $M$ into a $2$-category by providing $2$-cells in the form of transformations of bundle gerbe morphisms. This description of $\textbf{BGrb}_M$ as a $2$-category is used to define the notion of a bundle $2$-gerbe. To every bundle $2$-gerbe on $M$ is associated a class in $H^4(M ; \mathbb{Z})$. We define the notion of a bundle $2$-gerbe connection and show how this leads to a closed, integral, differential $4$-form on $M$ which represents the image in real cohomology of the class in $H^4(M ; \mathbb{Z})$. Some examples of bundle $2$-gerbes are discussed, including the bundle $2$-gerbe associated to a principal $G$ bundle $P \to M$. It is shown that the class in $H^4(M ; \mathbb{Z})$ associated to this bundle $2$-gerbe coincides with the first Pontryagin class of $P$: this example was previously considered from the point of view of $2$-gerbes by Brylinski and McLaughlin.

We consider an arbitrary Riemann surface $X$, possibly of infinite hyperbolic area. The Liouville measure of the hyperbolic metric defines a measure on the space $G(\tilde{X})$ of geodesics of the universal covering $\tilde{X}$ of $X$. As we vary the Riemann surface structure, this gives an embedding from the Teichmüller space of $X$ into the Fréchet space of Hölder distributions on $G(\tilde{X})$. We show that the embedding is continuously differentiable. In particular, we obtain an explicit integral representation of the tangent map.

Let $A$ and $B$ be C*-algebras, with $A$ separable and $B$$\sigma$-unital and stable. It is shown that there are natural isomorphisms

$$ E(A,B) = KK(SA,Q(B)) = [SA,Q(B) \otimes K], $$

where $SA=C_0(0, 1) \otimes A$, $[\cdot, \cdot]$ denotes the set of homotopy classes of *-homomorphisms, $Q(B) = M(B) / B$ is the generalized Calkin algebra, and $K$ denotes the C*-algebra of compact operators of an infinite-dimensional separable Hilbert space.

Let $\ast$ denote convolution and let $\delta_x$ denote the Dirac measure at a point $x$. A function in $L^2({\mathbb{R}})$ is called a difference of order $1$ if it is of the form $ g- \delta_{x} \ast g$ for some $x \in {\mathbb{R}}$ and $g \in L^2 ({\mathbb{R}})$. Also, a difference of order $2$ is a function of the form $ g- 2^{-1}( \delta_{x} \ast g + \delta_{-x} \ast g)$ for some $x \in {\mathbb{R}}$ and $g \in L^2 ({\mathbb{R}})$. In fact, the concept of a ‘difference of order $s$’ may be defined in a similar manner for each $s > 0$. If $\widehat{f}$ denotes the Fourier transform of $f$, it is known that a function $f$ in $L^2 ({\mathbb{R}})$ is a finite sum of differences of order $s$ if and only if $\int_{-\infty}^{\infty} |\widehat {f}(x)|^2 |x|^{-2s} dx < \infty$, and the vector space of all such functions is denoted by ${\mathcal{D}}_{s} (L^2 ({\mathbb{R}}) )$. Every function in ${\mathcal{D}}_{s} (L^2 ({\mathbb{R}}) )$ is a sum of ${\mathrm{int}} (2s) + 1$ differences of order $s$, where ${\mathrm{int}} (t)$ denotes the integer part of $t$. Thus, every function in ${\mathcal{D}}_{1} (L^2 ({\mathbb{R}}) )$ is a sum of three first order differences, but it was proved in 1994 that there is a function in ${\mathcal{D}}_1 (L^2 ({\mathbb{R}}) )$ which is never the sum of two first order differences. This complemented, for the group ${\mathbb{R}}$, the corresponding result for first order differences obtained by Meisters and Schmidt in 1972 for the circle group. The results show that there is a function in $L^2 ({\mathbb{R}})$ such that, for each $s \ge 1/2$, this function is a sum of ${\mathrm{int}} (2s) + 1$ differences of order $s$ but it is never the sum of ${\mathrm{int}} (2s)$ differences of order $s$. The proof depends upon extending to higher dimensions the following result in two dimensions obtained by Schmidt in 1972 in connection with Heilbronn's problem: if $x_1, \ldots, x_n$ are points in the unit square,

$$\sum_{1 \le i < j \le n} |x_i - x_j|^{-2} \ge 200^{-1} n^2 \ln n.$$

Following on from the work of Meisters and Schmidt, this work further develops a connection between certain estimates in combinatorial geometry and some questions of sharpness in harmonic analysis.

For a dense $G_\delta$-set of parameters, the irrational rotation algebra is shown to contain infinitely many C*-subalgebras satisfying the following properties. Each subalgebra is isomorphic to a direct sum of two matrix algebras of the same (perfect square) dimension; the Fourier transform maps each summand onto the other; the corresponding unit projection is approximately central; the compressions of the canonical generators of the irrational rotation algebra are approximately contained in the subalgebra.

Let $H$ be the discrete Schrödinger operator

$$Hu(n):=u(n-1)+u(n+1)+v(n)u(n),\quad u(0)=0$$

acting on $l^2 (\mathbb{Z}^+)$, where the potential $v$ is real-valued and $v(n) \to 0$ as $n \to \infty$. Let <formula form="inline" disc="math" id="frm007"><formtex notation="AMSTeX">$P$ be the orthogonal projection onto a closed linear subspace $\Lambda \subset l^2 (\mathbb{Z}^+)$. In a recent paper E. B. Davies defines the second order spectrum ${\rm Spec}_2(H, \Lambda)$ of $H$ relative to $\Lambda$ as the set of $z \in \mathbb{C}$ such that the restriction to $\Lambda$ of the operator $P(H - z)^2P$ is not invertible within the space $\Lambda$. The purpose of this article is to investigate properties of ${\rm Spec}_2(H, \Lambda)$ when $\Lambda$ is large but finite dimensional. We explore in particular the connection between this set and the spectrum of $H$. Our main result provides sharp bounds in terms of the potential $v$ for the asymptotic behaviour of ${\rm Spec}_2(H, \Lambda)$ as $\Lambda$ increases towards $l^2(\mathbb{Z}^+)$.