The main aim of this paper is to provide two new proofs of Ramanujan's cubic transformation formula for 2F1(1/3, 2/3; 1; z) (see (1·8) below). For our first proof, we have to develop Ramanujan's elliptic functions in the theory of signature 3 using a different approach from that given in a recent paper by Berndt, Bhargava and Garvan. For our second proof, we use two of Goursat's formulas.

A classical reciprocity formula for Gauss sums due to Cauchy, Dirichlet, and Kronecker states that

formula here

where a, b are nonzero integers and w∈ℚ such that ab+2aw∈2ℤ. For a detailed proof and historical background, see [4, chapter IX]. Various versions of formula (1) have been extensively studied in connection with transformation properties of theta functions. A version of (1) for multivariate Gauss sums was first obtained by A. Krazer [10] in 1912, see also [2, 5, 11, 14]. Krazer's formula generalizes the case w=0 of (1) via replacing one of the numbers a, b by an integer quadratic form of several variables.

Recently, Florian Deloup [6] found a new and most beautiful reciprocity formula for multivariate Gauss sums. His formula is a far reaching generalization of Krazer's result. Roughly speaking, Deloup replaces both numbers a, b with quadratic forms. Deloup involves Wu classes of quadratic forms which allows him to remove the evenness condition appearing in Krazer's formulation. However, Deloup's formula covers only the cases w=0 and w=b/2 of (1).

In this paper we establish a more general reciprocity for Gauss sums including the Krazer and Deloup formulas and formula (1) in its full generality. Our reciprocity law involves two quadratic forms and a so-called rational Wu class, as defined below.

The original proofs of Cauchy, Dirichlet, Kronecker and Krazer are analytical and involve a study of a limit of a transformation formula for theta-functions. Deloup's proof goes by a reduction to the case w=b/2 of (1) based on a careful study of Witt groups of quadratic forms. Our proof is more direct and uses only (a generalization of) the van der Blij computation of Gauss sums via signatures of integer quadratic forms. In particular, our argument provides a new proof of (1).

Let A be a commutative ring. We denote by a standard A-algebra a commutative graded A-algebra U=[oplus ]n[ges ]0Un with U0=A and such that U is generated as an A-algebra by the elements of U1. Take x a (possibly infinite) set of generators of the A-module U1. Let V=A[t] be the polynomial ring with as many variables t (of degree one) as x has elements and let f[ratio ]V→U be the graded free presentation of U induced by the x. For n[ges ]2, we will call the module of effective n-relations the A-module E(U)n= ker fn/V1· ker fn. The minimum positive integer r[ges ]1 such that the effective n-relations are zero for all n[ges ]r+1 is known to be an invariant of U. It is called the relation type of U and is denoted by rt(U). For an ideal I of A, we define E(I)n= E([Rscr ](I))n and rt(I)=rt([Rscr ](I)), where [Rscr ](I)= [oplus ]n[ges ]0Intn⊂A [t] is the Rees algebra of I.

In this paper we give two descriptions of the A-module of effective n-relations. In terms of André–Quillen homology we have that E(U)n= H1(A, U, A)n (see 2·3). It turns out that this module does not depend on the chosen [x]. In terms of Koszul homology we prove that E(U)n= H1([x], U)n (see 2·4). Using these characterizations, we show later some properties on the module of effective n-relations and the relation type of a graded algebra. Our approach has connections with several earlier works on the subject (see [2, 5–7, 9, 10, 13, 14]).

In a recent paper [7], Erdmann has calculated Ext1G between Weyl modules for SL2. In this paper we generalize this result to solve the corresponding problem for quantum GL2 as defined by Dipper and Donkin in [2]. We also show how our result also holds for the Manin quantization. To apply the methods of [7], it is necessary to determine the block structure of quantum GL2, so the first main result of this paper is a description of this, derived from the analysis of the subcomodule structure of the symmetric powers in [10].

After an initial section of generalities, the next section consists of the determination of the block structure. We also need a quantum analogue of two short exact sequences from [11], which we give in the following section. With these results, the argument now follows much as in [7]; we consider the infinitesimal case, and then use the Lyndon–Hochschild–Serre spectral sequence to obtain the desired result. Finally we show how the result also holds for the Manin quantization.

It should be noted that the result here uses the classical case, so is not independent of that in [7]. The only real difference in the arguments used occurs in Lemma 4·8 where the original methods do not generalize, so we use a more direct argument. There is also an unfortunate typographical error in the statement of the main result in [7].

We attack the conjecture that the only spherical classes in the homology of Q0S0 are Hopf invariant one and Kervaire invariant one elements. We do this by computing products in the E2-term of the unstable Adams spectral sequence converging to π∗(Q0S0), using results about the Dickson algebra and by studying the Lannes–Zarati homomorphism.

Let [Mfr ]g be the moduli space of smooth curves C of genus g[ges ]4 over the complex field [Copf ] and, as in [4], denote by [Mfr ](n)g⊂[Mfr ]g the locus of points representing curves having n vanishing theta-null, i.e. n points of order two in Sing (Θ).

I. V. Dolgachev proved the rationality of [Mfr ](1)4 (see [4, p. 13]) and of [Mfr ](2)5 (see [4, p. 11]). The aim of this paper is to prove the

Theorem. [Mfr ](1)tis rational.

In [8] we studied equivariant bifurcation problems with a symmetry group acting on parameters, from the point of view of singularity theory. We followed the now classical theory originated by Damon [5], using the ideas presented in [5, 13, 14]. We adapted general results about unfoldings, the algebraic characterization of finite determinacy, and the recognition problems, to multiparameter bifurcation problems f(x, λ)=0 with ‘diagonal’ symmetry on both the state variables and on the bifurcation parameters. More precisely, such bifurcation problems satisfy the condition f(γx, γλ)=γf(x, λ) for all γ∈Γ, where Γ is a compact Lie group

In this paper we attack the same problem from a different angle: the path formulation. This idea can be traced back to the first papers of Mather [17] and Martinet [15, 16]. It was used explicitly in Golubitsky and Schaeffer [12] (see also their earlier paper [11]) as a way of relating bifurcation problems in one state variable without symmetry to a miniversal unfolding in the sense of catastrophe theory. At that time the techniques of singularity theory were not powerful enough to handle the full power of the idea efficiently – either in theory or in computational practice. This is why the path formulation was abandoned in favour of contact equivalence with distinguished parameters, as developed in Golubitsky and Schaeffer [12]. Considerable progress has been made since then; for example Montaldi and Mond [19] use the path formulation to apply the idea of [Kscr ]V-equivalence introduced by Damon [6] to equivariant bifurcation theory. Bridges and Furter [3] studied equivariant gradient bifurcation problems using the path formulation, and defined an equivalence relation in the space of paths and their unfoldings that respects contact equivalence of the gradients. Here we describe an algebraic approach to the path formulation that has the advantage of organizing the classification of normal forms. Moreover, it minimizes the calculation involved in obtaining the normal forms (compare with the classical framework in Furter et al. [8]). The geometric approach to the path formulation using [Kscr ]V-equivalence is still open in the context of a symmetry group acting diagonally on parameters.

We give an explicit description of the construction of inverse limits in the category of locales and show a localic version of Steenrod's theorem without the Axiom of Choice; as an application we generalize the classic result by Steenrod.

Let

formula here

be the complete elliptic integral of the first kind. Recently, S.-L. Qiu and M. K. Vamanamurthy have proved for all r∈(0, 1):

formula here

with r′=√1−r2. In this paper we establish that the converse inequality

formula here

holds for all r∈(0, 1). Moreover, we show that the constant factors ¼ and π/(4 log (2))−1 in (a 1) and (a 2) are best possible.

We study the global behaviour of Gaussian curvature K and normal curvature K⊥ of zero mean curvature spacelike surfaces (stationary surfaces) in a four-dimensional Lorentzian space form L4(c). In particular, we show that the only complete stationary surfaces in Minkowski space E41 with K[ges ]0 are those with K≡0≡K⊥ and we give an explicit description of them. More general results are obtained for stationary surfaces in L4(c). We also discuss applications to Willmore surfaces in both Lorentzian and Riemannian three spaces. We give new examples of complete stationary surfaces in E41 with finite total curvature.

In this paper we prove that the conjectures of Frisch and Parisi and Arneodo et al. (called the multifractal formalism for functions) may fail for some non-homogeneous selfsimilar functions on ℝ2. In these cases, we compute the correct spectrum of singularities and we show how the multifractal formalism must be modified.

If each of E and F is a real Banach space, H a compact Hausdorff space, C(H, E) the Banach space (sup norm ∥·∥∞) of continuous E-valued functions defined on H, L: C(H, E)→F a continuous linear transformation (=operator) with representing measure m, [sum ] the σ-algebra of Borel subsets of H and m˜(A) the semivariation of m on A∈[sum ], then m maps [sum ] into B(E, F**), the Banach space of all operators from E into F** (= the bidual of F), ∥L∥=m˜(H) and L(f)=∫ fdm. The reader may consult [9] or [6] for a detailed discussion of the Riesz representation theorem in this setting.

This paper deals with an initial value problem of a neutral functional-differential equation with variable time delay. The effect of the neutral term on the existence and uniqueness of solutions is investigated. A comparison theorem is given and is applied to a functional Riccati equation. The asymptotic behaviour of solutions of a linear perturbation of the generalized pantograph equation is studied.