Some elementary results are given on restricted sums $2^{\wedge}{\cal A}$ and $3^{\wedge}{\cal A}$ of a subset ${\cal A}\subset {\bb Z}/n{\bb Z}$ in the case when the cardinality of ${\cal A}$ , is close to $n/2$ . In the last section, some of these results are used to derive some new values of a function related to the Erdős–Ginzburg–Ziv problem.

This paper deals with some character values of the symmetric group Sn as well as its double cover S˜n.

Let χλ(ρ) be the irreducible character of Sn, indexed by the partition λ and evaluated at the conjugacy class ρ. Comparing the character tables of S2 and S4, one observes that

for ρ = (2), 2ρ = (4) and ρ = (12), 2ρ = (22). A number of such observations lead to what we call Littlewood's multiple formula (Theorem 1.1). This formula appears in Littlewood's book [2]. We include a proof that is based on an ‘inflation’ of the variables in a Schur function. This is different from one given in [2], and we claim that it is more complete than the one given there.

Our main objective is to obtain the spin character version of Littlewood's multiple formula (Theorem 2.3).

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.

For any algebraic curve with unbounded branches in the plane, two numbers in [0, ∞] are defined that measure how closely to the curve it is possible to find infinitely many integer points. These numbers are shown to be 1 for almost all such curves. The state of the art in estimating these numbers for several classes of curves is also discussed.

The paper studies a generalized spherical function, or a generalized Whittaker model for generalized principal series representations of $G={\rm Sp}(2,{\bf R})$ induced from the Jacobi maximal parabolic subgroup $P_{\rm J}$ , which is called the Fourier–Jacobi type. In particular, a multiplicity theorem and an explicit formula via the Meijer $G$ -functions for this function are given.

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.

The paper computes the Hausdorff dimension of sets of real numbers which are close to infinitely many real algebraic integers of bounded degree. It also investigates the distribution of real algebraic integers of bounded degree, which are proved to be evenly spaced.

A uniform version of the Schanuel conjecture is discussed that has some model-theoretical motivation. This conjecture is assumed, and it is proved that any ‘non-obviously-contradictory’ system of equations in the form of exponential sums with real exponents has a solution.

Let $(R,{\frak m})$ be a local ring with prime ideals ${\frak p}$ and ${\frak q}$ such that $\sqrt{{\frak p}+{\frak q}}={\frak m}$ . If $R$ is regular and contains a field, and $\dim(R/{\frak p})+\dim(R/{\frak q})=\dim(R)$ , then it is proved that ${\frak p}^{(m)}\cap {\frak q}^{(n)}\subseteq {\frak m}^{m+n}$ for all positive integers $m$ and $n$ . This is proved using a generalization of Serre's Intersection Theorem which is applied to a hypersurface $R/fR$ . The generalization gives conditions that guarantee that Serre's bound on the intersection dimension $\dim(R/{\frak p})+\dim(R/{\frak q})\le \dim(R)$ holds when $R$ is nonregular.

An example is given which shows that the Denef–Loeser zeta function (usually called the topological zeta function) associated to a germ of a complex hypersurface singularity is not a topological invariant of the singularity. The idea is the following. Consider two germs of plane curves singularities with the same integral Seifert form but with different topological type and which have different topological zeta functions. Make a double suspension of these singularities (consider them in a 4-dimensional complex space). A theorem of M. Kervaire and J. Levine states that the topological type of these new hypersurface singularities is characterized by their integral Seifert form. Moreover the Seifert form of a suspension is equal (up to sign) to the original Seifert form. Hence these new singularities have the same topological type. By means of a double suspension formula the Denef–Loeser zeta functions are computed for the two 3-dimensional singularities and it is verified that they are not equal.

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 $J^*$ .

There is a one-to-one correspondence between nonsingular pencils of quadrics in an odd-dimensional projective space and nonsingular hyperelliptic curves. There is also a close relation between the moduli spaces of vector bundles of ranks 1, 2 on the hyperelliptic curve and spaces associated to the intersection of quadrics of the corresponding pencil. This has been studied extensively by many authors [1, 5, 8, 9, 11]. An interesting application of this was in the proof of the Verlinde formulae by Szenes [12]. This relationship was generalised to a relation between spaces related to singular pencils of quadrics associated to irreducible nodal curves and moduli of torsion-free sheaves on the nodal curves [2, 4]. The projective models of the moduli of vector bundles on smooth curves (given by this relationship) led to explicit representations of integrable systems of [6, 7]. Similar results on singular curves would provide important classes of special solutions. The pencils of quadrics considered in previous works had associated Segre symbols of type [1 … 1] and [2 … 2 1 … 1]. Kleiman (and Emma Previato) had raised the question: What happens in the case of Segre symbol [2 2 2]? In this case the associated curve Y has two rational components intersecting transversely at three points. Unlike in the previous cases, the compactified Jacobian has more than one component. Moreover the stability of line bundles and the compactification of the Jacobian depends on the choice of a polarisation on Y, making the problem more interesting and difficult.

It is proved that there exists a scheme that represents the functor of line modules over a graded algebra, and it is called the line scheme of the algebra. Its properties and its relationship to the point scheme are studied. If the line scheme of a quadratic, Auslander-regular algebra of global dimension 4 has dimension 1, then it determines the defining relations of the algebra.

Moreover, the following counter-intuitive result is proved. If the zero locus of the defining relations of a quadratic (not necessarily regular) algebra on four generators with six defining relations is finite, then it determines the defining relations of the algebra. Although this result is non-commutative in nature, its proof uses only commutative theory.

The structure of the line scheme and the point scheme of a 4-dimensional regular algebra is also used to determine basic incidence relations between line modules and point modules.

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.

The following autoduality theorem is proved for an integral projective curve $C$ in any characteristic. Given an invertible sheaf ${\cal L}$ of degree 1, form the corresponding Abel map $A_{\cal L}:C\longrightarrow \bar{J}$ , which maps $C$ into its compactified Jacobian, and form its pullback map $A^{\ast}_{\cal L}:{\rm Pic}^0_{\bar{J}}\longrightarrow J$ , which carries the connected component of $0$ in the Picard scheme back to the Jacobian. If $C$ has, at worst, points of multiplicity $2$ , then $A^{\ast}_{\cal L}$ is an isomorphism, and forming it commutes with specializing $C$ .

Much of the work in the paper is valid, more generally, for a family of curves with, at worst, points of embedding dimension $2$ . In this case, the determinant of cohomology is used to construct a right inverse to $A^{\ast}_{\cal L}$ . Then a scheme-theoretic version of the theorem of the cube is proved, generalizing Mumford's, and it is used to prove that $A^{\ast}_{\cal L}$ is independent of the choice of ${\cal L}$ . Finally, the autoduality theorem is proved. The presentation scheme is used to achieve an induction on the difference between the arithmetic and geometric genera; here, special properties of points of multiplicity $2$ are used.

Let K be a kernel on Rn, that is, K is a non-negative, unbounded L1 function that is radially symmetric and decreasing. We define the convolution K [midast ] F by

and note from Lp-capacity theory [11, Theorem 3] that, if F ∈ Lp, p > 1, then K [midast ] F exists as a finite Lebesgue integral outside a set A ⊂ Rn with CK,p(A) = 0. For a Borel set A,

where

We define the Poisson kernel for Rn+1+ = {(x, y) [ratio ] x ∈ Rn, y > 0} by

and set

Thus u is the Poisson integral of the potential f = K [midast ] F, and we write

We are concerned here with the limiting behaviour of such harmonic functions at boundary points of Rn+1+, and in particular with the tangential boundary behaviour of these functions, outside exceptional sets of capacity zero or Hausdorff content zero.

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.

At the regional conference held at the University of California, Irvine, in 1985 [24], Harald Upmeier posed three basic questions regarding derivations on JB*-triples:

(1) Are derivations automatically bounded?

(2) When are all bounded derivations inner?

(3) Can bounded derivations be approximated by inner derivations?

These three questions had all been answered in the binary cases. Question 1 was answered affirmatively by Sakai [17] for C*-algebras and by Upmeier [23] for JB-algebras. Question 2 was answered by Sakai [18] and Kadison [12] for von Neumann algebras and by Upmeier [23] for JW-algebras. Question 3 was answered by Upmeier [23] for JB-algebras, and it follows trivially from the Kadison–Sakai answer to question 2 in the case of C*-algebras.

In the ternary case, both question 1 and question 3 were answered by Barton and Friedman in [3] for complex JB*-triples. In this paper, we consider question 2 for real and complex JBW*-triples and question 1 and question 3 for real JB*-triples. A real or complex JB*-triple is said to have the inner derivation property if every derivation on it is inner. By pure algebra, every finite-dimensional JB*-triple has the inner derivation property. Our main results, Theorems 2, 3 and 4 and Corollaries 2 and 3 determine which of the infinite-dimensional real or complex Cartan factors have the inner derivation property.