The definable fundamental group of a definable set in an $o$ -minimal expansion of a field is computed. This is achieved by proving the relevant case of the $o$ -minimal van Kampen theorem. This result is applied to show that if the geometrical realization of a simplicial complex over an $o$ -minimal expansion of a field is a definable manifold of dimension not 4, then its geometrical realization over the reals is a topological manifold.

On page 45 of his lost notebook, Ramanujan recorded two asymptotic formulas for two continued fractions involving the Riemann zeta-function and Dirichlet $L$ -functions. The paper proves a more general theorem and derives Ramanujan's claims as a corollary of the theorem.

A square-ordered field, also called a Hilbert field of type (A), is understood to be an ordered field all of whose positive elements are squares. The problem of classifying, up to isomorphism, all $4$ -dimensional quadratic division algebras over a square-ordered field $k$ is shown to be equivalent to the problem of finding normal forms for all pairs $(X, Y)$ of $3 \times 3$ matrices over $k, X$ being antisymmetric and $Y$ being positive definite, under simultaneous conjugation by ${\rm SO}_3(k)$ . A solution is derived for the subproblem of this matrix pair problem defined by requiring $Y+Y^t$ to be orthogonally diagonalizable. The classifying list is given in terms of a $9$ -parameter family of configurations in $k^3$ , formed by a pair of points and an ellipsoid in normal position.

Each $4$ -dimensional quadratic division algebra $A$ over a square-ordered field $k$ is shown to determine, uniquely up to sign, a self-adjoint linear endomorphism $\alpha$ of its purely imaginary hyperplane. Calling A diagonalizable in case $\alpha$ is orthogonally diagonalizable, the achieved solution of the matrix pair subproblem yields a full classification of all diagonalizable $4$ -dimensional quadratic division $k$ -algebras. This generalizes earlier results of both Hefendehl-Hebeker who classified, over Hilbert fields, those $4$ -dimensional quadratic division algebras having infinite automorphism group, and Dieterich, who achieved a full classification of all real $4$ -dimensional quadratic division algebras.

Finally, the paper describes explicitly how Hefendehl-Hebeker's classifying list, given in terms of a $4$ -parameter family of pairs of definite $3 \times 3$ matrices over $k$ , embeds into the classifying list of configurations. The image of this embedding turns out to coincide with the sublist of the list formed by all non-generic configurations.

An approach of Swinnerton-Dyer is extended to obtain new upper bounds for the number of lattice points close to a smooth curve. One consequence of these bounds is a new asymptotic result for the distribution of squarefull numbers in short intervals.

Let $B = k[x_1, \ldots, x_n]$ be a polynomial ring over a field $k$ , and let $A$ be a quotient ring of $B$ by a homogeneous ideal $J$ . Let $\frak{m}$ denote the maximal graded ideal of $A$ . Then the Rees algebra $R = A[{\frak{m}} t]$ also has a presentation as a quotient ring of the polynomial ring $k[x_1, \ldots, x_n, y_1, \ldots, y_n]$ by a homogeneous ideal $J^*$ . For instance, if $A = k[x_1, \ldots, x_n]$ , then \[ R \cong k[x_1, \ldots, x_n, y_1, \ldots, y_n]/(x_i y_j - x_j y_i\mid i,j = 1, \ldots, n). \] In this paper we want to compare the homological properties of the homogeneous ideals $J$ and $J^*$ .

Let $G$ be a finite group scheme operating on an algebraic variety $X$ , both defined over an algebraically closed field $k$ . The paper first investigates the properties of the quotient morphism $X\longrightarrow X/G$ over the open subset of $X$ consisting of points whose stabilizers have maximal index in $G$ . Given a $G$ -linearized coherent sheaf on $X$ , it describes similarly an open subset of $X$ over which the invariants in the sheaf behave nicely in some way. The points in $X$ with linearly reductive stabilizers are characterized in representation theoretic terms. It is shown that the set of such points is nonempty if and only if the field of rational functions $k(X)$ is an injective $G$ -module. Applications of these results to the invariants of a restricted Lie algebra ${\frak g}$ operating on the function ring $k[X]$ by derivations are considered in the final section. Furthermore, conditions are found ensuring that the ring $k[X]^{\frak g}$ is generated over the subring of $p$ th powers in $k[X]$ , where $p={\rm char}\,k>0$ , by a given system of invariant functions and is a locally complete intersection.

Motivated by, but independent of, some recent work in quantum stochastic calculus, a theory of differential and integral calculus is developed which is intrinsic to the universal enveloping algebra of a Lie algebra whose Lie bracket is obtained by taking commutators in an associative algebra. The differential map satisfies a generalisation of Leibniz' formula called the Leibniz–Itô formula, which involves the associative multiplication. There is an analogue of the Taylor–Maclaurin expansion. Through passing to formal power series, a theory of product integrals is developed; such integrals are characterised by a group-like property with respect to the coproduct.

First, the strong Stone–Weierstrass conjecture for postliminal ${\rm JB}^*$ -triples is proved. This is combined with the recent classification of ${\rm JB}^*$ -triples $A$ for which $\partial_e(A^*_1)$ is ${\rm weak}^*$ dense in $A^*_1$ , and the weak Stone–Weierstrass theorem for ${\rm JB}^*$ -triples is obtained. If $B$ is a ${\rm JB}^*$ -subtriple of a ${\rm JB}^*$ -triple $A$ such that $B$ separates $\overline{\partial_e(A^*_1)}\cup\{0\}$ , then $B=A$ .

A linear autonomous system of differential equations $\dot{x}=Ax$ can be transformed to its Jordan normal form, that is, the transformed system is in block diagonal form and the blocks correspond to different eigenvalues. This result is generalized to arbitrary nonautonomous linear systems $\dot{x}=A(t)x$ with a locally integrable matrix function $A:{\bb R}\longrightarrow {\bb R}^{N\times N}$ .

It is determined which Bloch-type conditions on a function $f\in \bigcap_{0 < p <\infty}H^p$ ensure that $f\in {\rm BMOA}$ .

Let $F$ be a germ of a holomorphic function at $0$ in ${\bb C}^{n+1}$ , having $0$ as a critical point not necessarily isolated, and let $\tilde{X}:= \sum^n_{j=0} X^j(\partial/\partial z_j)$ be a germ of a holomorphic vector field at $0$ in ${\bb C}^{n+1}$ with an isolated zero at $0$ , and tangent to $V := F^{-1}(0)$ . Consider the ${\cal O}_{V,0}$ -complex obtained by contracting the germs of Kähler differential forms of $V$ at $0$ \renewcommand{\theequation}{0.\arabic{equation}} \begin{equation} \Omega^i_{V,0}:=\frac{\Omega^i_{{\bb C}^{n+1},0}}{F\Omega^i_{{\bb C}^{n+1},0}+dF\wedge{\Omega^{i-1}}_{{\bb C}^{n+1}},0} \end{equation} with the vector field <formula form="inline" disc="math" id="frm14"><formtex notation="AMSTeX"> $X:=\tilde{X}|_V$ on $V$ : \begin{equation} 0\longleftarrow {\cal O}_{V,0} {\buildrel X\over\longleftarrow}\,\Omega_{V,0}^1\,{\buildrel X\over\longleftarrow}\, \cdots \,{\buildrel X\over\longleftarrow}\, \Omega_{V,0}^n\, {\buildrel X\over\longleftarrow}\, \Omega_{V,0}^{n+1}\longleftarrow 0. \end{equation}

Periodic neutral functional differential equations are considered. Sufficient conditions for existence, uniqueness and global attractivity of periodic solutions are established by combining the theory of monotone semiflows generated by neutral functional differential equations and Krasnosel'skii's fixed-point theorem. These results are applied to a concrete neutral functional differential equation that can model single-species growth, the spread of epidemics, and the dynamics of capital stocks in a periodic environment.

The paper first gives sufficient conditions on the critical points and the Schwarzian derivative of a real rational function $R$ such that the Julia set of $R$ is $\bar{{\bb C}}$ . Further, it is shown that under mild conditions on another real rational function $\tilde{R}$ with possibly non-empty Fatou set, the Julia set of $\tilde{R} \circ R$ is the whole Riemann sphere again. Then families of rational functions are given whose Julia set is $\bar{{\bb C}}$ and whose critical points are not necessarily preperiodic. Concrete examples were previously available only for the preperiodic case. Finally, it is demonstrated that the methods presented also apply to the construction of polynomials whose Julia sets are dendrites and whose critical points in the Julia set are not necessarily preperiodic.

A central limit theorem for a generalized Harper operator on a crystal lattice is obtained. As the limit, the continuous semigroup of a uniform magnetic Schrödinger operator is captured on a vector space equipped with a special Euclidean structure. The standard realization of the crystal lattice is a key to the Euclidean structure and a linear vector potential on the Euclidean space from combinatorial data of the generalized Harper operator.

A partial differential operator associated with natural oscillations of an incompressible fluid in the neighbourhood of an elliptical flow is considered. The differentiation is only taken with respect to the angular variable, and thus the operator becomes a family of ordinary differential operators parametrized by the radial variable. It is shown that the spectra of these ordinary differential operators completely determine the spectrum of the given operator which turns out to have a kind of skeleton structure.

It is shown that a weakly closed subspace ${\cal S}$ of a nest algebra ${\cal A}$ is closed under conjugation by invertible elements in ${\cal A}$ , that is, $a^{-1}{\cal S}\,a={\cal S}$ if and only if ${\cal S}$ is a Lie ideal. A similar result holds for not-necessarily-closed subspaces of algebras of infinite multiplicity. An explicit characterisation of weakly closed Lie ideals in a nest algebra is given.