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).

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.

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.

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.

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.

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.

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.

Let g [ratio ] M → N be a quasiconformal harmonic diffeomorphism between noncompact Riemann surfaces M and N. In this paper we study the relation between the map g and the complex structures given on M and N. In the case when M and N are of finite analytic type we derive a precise estimate which relates the map g and the Teichmüller distance between complex structures given on M and N. As a corollary we derive a result that every two quasiconformally related finitely generated Kleinian groups are also related by a harmonic diffeomorphism. In addition, we study the question of whether every quasisymmetric selfmap of the unit circle has a quasiconformal harmonic extension to the unit disk. We give a partial answer to this problem. We show the existence of the harmonic quasiconformal extensions for a large class of quasisymmetric maps. In particular it is proved that all symmetric selfmaps of the unit circle have a unique quasiconformal harmonic extension to the unit disk.

Some almost sharp sufficient conditions of oscillation and nonoscillation are obtained for the superlinear delay differential equation

The paper studies the existence of multiple solutions to the following p-Laplacian type elliptic problem (p > 1):

where Ω is a bounded domain in ℝN(N [ges ] 1) with smooth boundary ∂Ω, and f(x, u) goes asymptotically in u to [mid ]u[mid ]p−2u at infinity. It is well known that this kind of nonlinear term creates some difficulties in the application of the mountain pass theorem because of the lack of an Ambrosetti–Rabinowitz type superlinear condition on f(x, u). An improved mountain pass theorem is used to prove that the above problem possesses multiple solutions under some natural conditions on f(x, u), and some known results are generalized.

The problem of studying the largest rearrangement invariant space of functions with almost everywhere convergent Fourier series is considered. A quasi-Banach space QA of functions is defined with almost everywhere convergent Fourier series that strictly contains Antonov and Soria's spaces ([Lscr ] log [Lscr ] log log log [Lscr ] and B*ϕ1([ ])).

A shadow of a subset A of ℝn is the image of A under a projection onto a hyperplane. Let C be a closed nonconvex set in ℝn such that the closures of all its shadows are convex. If, moreover, there are n independent directions such that the closures of the shadows of C in those directions are proper subsets of the respective hyperplanes then it is shown that C contains a copy of ℝn−2. Also for every closed convex set B ‘minimal imitations’ C of B are constructed, that is, closed subsets C of B that have the same shadows as B and that are minimal with respect to dimension.

The closed 3-manifolds of constant positive curvature were classified long ago by Seifert and Threlfall. Using well-known information about the orthogonal group O(4), their full isometry groups Isom(M) are calculated. It is determined which elliptic 3-manifolds admit Seifert fiberings that are invariant under all isometries, and it is verified that the inclusion of Isom(M) to Diff(M) is a bijection on path components.

For a discrete group G there are two well known completions. The first is the Malcev (or unipotent) completion. This is a prounipotent group [Uscr ], defined over ℚ, together with a homomorphism ψ : G → [Uscr ] that is universal among maps from G into prounipotent ℚ-groups. To construct [Uscr ], it suffices for us to consider the case where G is nilpotent; the general case is handled by taking the inverse limit of the Malcev completions of the G/ΓrG, where Γ[bull ]G denotes the lower central series of G. If G is abelian, then [Uscr ] = G [otimes ] ℚ. We review this construction in Section 2.

Suppose that A is a pointed CW-complex. The paper looks at how difficult it is to construct an A-cellular space B from copies of A by repeatedly taking homotopy colimits; this is determined by an ordinal number called the complexity of B. Studying the complexity leads to an iterative technique, based on resolutions, for constructing the A-cellular approximation CWA(X) of an arbitrary space X.

It is shown that the topological classification and the smooth classification are generically the same for certain families of plane curves in a semi-local case (that is, the double local case). The normal form of transversely joined two families of plane curves with second-order contact at the envelope is also given.

The paper gives a necessary and sufficient criterion on the Lévy measure that determines whether a Lévy process creeps. (The author says that a Lévy process creeps if it can continuously pass a fixed level.)