Graham Higman proved that a finitely generated group can be embedded in a finitely presented group if and only if it has a recursively enumerable set of defining relations. The analogue for lattice-ordered groups is considered here. Clearly, the finitely generated lattice-ordered groups that can be $\ell$-embedded in finitely presented lattice-ordered groups must have recursively enumerable sets of defining relations. The converse direction is proved for a special class of lattice-ordered groups.

THEOREM. Every finitely generated Abelian lattice-ordered group that has finite rank and a recursively enumerable set of defining relations can be$\ell$-embedded in a finitely presented lattice-ordered group.

If $\xi$ is a real number, let $D(\xi)$ be the Abelian rank 2 group $\Z^2$ with order $(m,n)>0$ if and only if $m+n\xi>0$.

COROLLARY. $D(\xi)$can be$\ell$-embedded in a finitely presented lattice-ordered group if and only if$\xi$is a recursive real number.

Thus an algebraic characterisation of recursive real numbers is obtained. In particular, $\pi$ is ‘$\ell$-algebraic’ in that it can be captured by finitely many relations in this language.

The paper continues the investigation of the Hecke structure of spaces of half-integral weight cusp forms ${S_{k + 1/2}}(4N, \chi)$, where $k$ and $N$ are positive integers with $N$ odd, and $\chi$ is an even quadratic Dirichlet character modulo $4N$. In the Hecke decomposition of these spaces, contributions are determined arising from newforms which are quadratic twists of newforms at lower levels. Combining this result with its counterpart in a previous paper regarding non-twists gives this paper's main result: necessary and sufficient conditions under which a given newform has equivalent cusp forms in ${S_{k + 1/2}}(4N, \chi)$. The result reformulates, in classical terms, the representation-theoretic conditions given by Flicker and Waldspurger. The conditions involve easily-verified information about the primes dividing the level of the newform, and about the behavior of the newform under certain quadratic twists and Atkin–Lehner involutions. The theorem is applied to give explicit examples of twisted newforms having no equivalent half-integral weight cusp forms in any space ${S_{k + 1/2}}(4N, \chi)$ as above.

Goldston and Montgomery proved that the strong pair correlation conjecture and two second moments of primes in short intervals are equivalent to each other under the Riemann hypothesis. The paper obtains the second main terms for each of the above, and it is shown that they are almost equivalent to each other.

Inspired by the work of Bloch and Kato in [2], David Burns constructed several ‘equivariant Tamagawa invariants’ associated to motives of number fields. These invariants lie in relative $K$-groups of group-rings of Galois groups, and in [3] Burns gave several conjectures (see Conjecture 3.1) about their values. In this paper I shall verify Burns' conjecture concerning the invariant $T\Omega^{\rm loc}( N/{\bf Q},1)$ for some families of quaternion extensions $N/{\bf Q}$. Using the results of [9] I intend in a subsequent paper to verify Burns' conjecture for those families of quaternion fields which are not covered here.

Let $k$ be a field of characteristic zero, $f(X,Y), g(X,Y)\,{\in}\,k[X,Y]$, $g(X,Y)\,{\notin}\,(X,Y)$ and $d\,{:=} g(X,Y)\,{\partial}/ {\partial X}+f(X,Y)\,{\partial}/{\partial Y}$. A connection is established between the $d$-simplicity of the local ring $k[X,Y]_{(X,Y)}$ and the transcendency of the solution in $tk[[t]]$ of the algebraic differential equation $g(t,y(t))\,{\cdot}\, ({\partial}/{\partial t})y(t)=f(t,y(t))$. This connection is used to obtain some interesting results in the theory of the formal power series and to construct new examples of differentially simple rings.

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.

Let $A$ be a finite-dimensional division algebra containing a base field $k$ in its center $F$. $A$ is defined over a subfield $F_0$ if there exists an $F_0$-algebra $A_0$ such that $A = A_0 \bigotimes_{F_0} F$. The following are shown. (i) In many cases $A$ can be defined over a rational extension of $k$. (ii) If $A$ has odd degree $n \geq 5$, then $A$ is defined over a field $F_0$ of transcendence degree $\leq {\frac{1}{2}}(n-1)(n-2)$ over $k$. (iii) If $A$ is a $\mathbb{Z}/m \times \mathbb{Z}/2$-crossed product for some $m \geq 2$ (and in particular, if $A$ is any algebra of degree 4) then $A$ is Brauer equivalent to a tensor product of two symbol algebras. Consequently, ${\rm M}_m(A)$ can be defined over a field $F_0$ such that ${\rm trdeg}_k(F_0) \leq 4$. (iv) If $A$ has degree 4 then the trace form of $A$ can be defined over a field $F_0$ of transcendence degree $\leq 4$. (In (i), (iii) and (iv) it is assumed that the center of $A$ contains certain roots of unity.)

Further properties of a group $\Gamma$ introduced by the first author in 1980 (sometimes called the first Grigorchuk group) are established. These are notably that any two infinite finitely generated subgroups of $\Gamma$ have isomorphic subgroups of finite index and that the generalized word problem for $\Gamma$ is soluble.

Acyclic groups of low dimension are considered. To indicate the results simply, let $G^{\prime}$ be the nontrivial perfect commutator subgroup of a finitely presentable group $G$. Then $\mathrm{def}(G)\leq1$. When $\mathrm{def}(G)=1$, $G^{\prime}$ is acyclic provided that it has no integral homology in dimensions above 2 (a sufficient condition for this is that $G^{\prime}$ be finitely generated); moreover, $G/G^{\prime}$ is then $Z$ or $Z^{2}$. Natural examples are the groups of knots and links with Alexander polynomial 1. A further construction is given, based on knots in $S^{2}\times S^{1}$. In these geometric examples, $G^{\prime}$ cannot be finitely generated; in general, it cannot be finitely presentable. When $G$ is a 3-manifold group it fails to be acyclic; on the other hand, if $G^{\prime}$ is finitely generated it has finite index in the group of a $\mathbb{Q}$-homology 3-sphere.

In this study of the behaviour of the number of conjugacy classes in finite $p$-groups using pro-$p$ groups, the conjugacy growth function $r_n(G)= \max\{r(G/N)\,{|}\,N\triangleleft_o G,|G\,{:}\,N|=n\}$ is introduced. It is proved that there are no infinite pro-$p$ groups of linear conjugacy growth (that is, there is no $c$ such that $r_n(G)\le c\log_2 n$ for all $n>1$) and it is shown that many known pro-$p$ groups are of exponential conjugacy growth (that is, there exists $\epsilon\,{>}\,0$ such that $r_n(G)\,{\ge}\,n^\epsilon$ for infinitely many values of $n$).

For a holomorphic function $f$ in the open unit disc $D$, the $N$th partial sum of its Taylor series with center $\zeta \in D$ is denoted by $S_N(f,\zeta)(z)=$${\sum\nolimits^N_{n=0}}({{f^{(n)}(\zeta)}/n!})(z-\zeta)^n$. Generically, all functions $f$ in $H(D)$ satisfy the following. For every compact set $K\subset\Bbb C$ with $K{\cap}\,D=\varnothing$ and $K^c$ connected and every polynomial $h$, there exists a sequence of positive integers $\{\lambda_n\}^{\infty}_{n=1}$ such that, for every $l \in \{ 0,1,2, \ldots \}$, \[ \sup_{z \in K}\Big\vert {{\partial^l}\over {\partial z^l}} S_{\lambda_n}(f,0)(z)-h^{(l)}(z)\Big\vert \,{\to}\,0 \quad {\rm as} \; n\,{\to}\,{+}\,\,\infty.\]

It is proved that the duality map $\langle\,,\rangle:(\ell^\infty,\hbox{weak})\times((\ell^\infty)^*,\hbox{weak}^* )\longrightarrow {\bf R}$ is not Borel. More generally, the evaluation $e:(C(K),\wk)\times K\longrightarrow{\bf R}$, $e(f,x) = f(x)$, is not Borel for any function space $C(K)$ on a compact $F$-space. It is also shown that a non-coincidence of norm-Borel and weak-Borel sets in a function space does not imply that the duality map is non-Borel.

Let $\phi$ be a bounded linear functional on $A$, where $A$ is a commutative Banach algebra, then the bilinear functional $\skew5\check{\phi}$ is defined as $\skew5\check{\phi} (a,b)\,{=}\,\phi (ab)-\phi (a) \phi (b)$ for each $a$ and $b$ in $A$. If the norm of $\skew5\check{\phi}$ is small then $\phi$ is approximately multiplicative, and it is of interest whether or not $\|\skew5\check{\phi}\|$ being small implies that $\phi$ is near to a multiplicative functional. If this property holds for a commutative Banach algebra then $A$ is an AMNM algebra (approximately multiplicative functionals are near multiplicative functionals). The main result of the paper shows that ${\text{C}}^N {[0,1]}^M$ (the complex valued functions defined on ${[0,1]}^M$ with all $N$th order partial derivatives continuous) is AMNM. It is also shown that a similar proof can be applied to certain Lipschitz algebras.

Let $A$ be an infinite-dimensional $C^*$-algebra. It is proved that every nonempty relatively weakly open subset of the closed unit ball $B_A$ of $A$ has diameter equal to 2. This implies that $B_A$ is not dentable, and that there is not any point of continuity for the identity mapping $(B_A,{\rm weak)\,{\longrightarrow}\,(B_A,{\rm norm})$.

The flag curvature of a Finsler metric is called a Riemannian quantity because it is an extension of sectional curvature in Riemannian geometry. In Finsler geometry, there are several non-Riemannian quantities such as the (mean) Cartan torsion, the (mean) Landsberg curvature and the S-curvature, which all vanish for Riemannian metrics. It is important to understand the geometric meanings of these quantities. In the paper, Finsler metrics of scalar curvature (that is, the flag curvature is a scalar function on the slit tangent bundle) are studied and the flag curvature is partially determined when certain non-Riemannian quantities are isotropic. Using the obtained formula for the flag curvature, locally projectively flat Randers metrics with isotropic S-curvature are classified.

A formula of Walczak is applied to two situations in differential geometry. Holomorphic distributions on Kähler manifolds are studied, and it is shown how the formula simplifies to a Bochner type formula, which is particularly useful in the study of integrable distributions. Then the Walczak formula is applied in the context of harmonic morphisms, where it provides a means of investigating the vertical Laplacian of the dilation. It is shown that, under some additional conditions on the map and the domain, the $p$-energy is infinite for $p$ sufficiently large.

On a manifold with polynomial volume growth satisfying Gaussian upper bounds of the heat kernel, a simple characterization of the matching lower bounds is given in terms of a certain Sobolev inequality. The method also works in the case of so-called sub-Gaussian or sub-diffusive heat kernels estimates, which are typical of fractals. Together with previously known results, this yields a new characterization of the full upper and lower Gaussian or sub-Gaussian heat kernel estimates.