It has been conjectured that equation

formula here

has only finitely many solutions. We observe that (x, y, n, q)=(3, 11, 5, 2), (7, 20, 4, 2) and (18, 7, 3, 3) are solutions of (1). These are the only known solutions and perhaps (1) has no other solution. Ljunggren [8] proved in 1943 that (1) with q=2 has no solution other than x=3, y=11, n=5 and x=7, y=20, n=4. Shorey and Tijdeman [16] confirmed the conjecture if x is fixed. Let z>1 be an integer. The main purpose of this paper is to show that (1) has no solution if x is restricted to the infinite set of squares z2 with z[ges ]32 and z∈{2, 3, 4, 8, 9, 16, 27}.

If Q is a finitely generated abelian group, k a field, V a finitely generated kQ-module Bieri and Groves have proved that [otimes ]mV is a finitely generated as a kQ-module with diagonal Q-action if and only if ∧iV is finitely generated as a kQ-module for all 1[les ]i[les ]m. We generalize this result by showing that if the mth exterior power of V or the mth symmetric power of V is finitely generated as a kQ-module so is the mth tensor power of V. Further we show the equivalence between the finite generation of symmetric and tensor powers in the case when the ground ring is a PID of characteristic 0.

We describe the construction of explicit representations of some of the exceptional covers of generic groups, that is, covers whose existence is not explained by the general theory of such groups.

We generalize a theorem of R. Thomas, which sometimes allows one to tell by inspection that a finitely presented group G is infinite. Groups to which his theorem applies have presentations with not too many more relators than generators, with at least some of the relators being proper powers. Our generalization provides lower bounds for the ranks of the abelianizations of certain normal subgroups of G in terms of their indices. We derive Thomas's theorem as a special case.

In this paper, we will prove that if A is a C*-algebra with an effective coaction ε by a compact quantum group, then the fixed point algebra and the reduced crossed product are Morita equivalent. As an application, we prove an imprimitivity type theorem for crossed products of coactions by discrete Kac C*-algebras.

Let (Vi, ti) be a pair consisting of a solid torus Vi and an unknotted arc ti properly imbedded in Vi for i=1 and 2. Taking the union of these pairs we obtain a pair (M, K) of the 3-dimensional sphere or a lens space M and a knot K. We call such a knot K a genus one 1-bridge knot. In this paper we determine when a genus one 1-bridge knot is the trivial knot or a cabled knot.

The Funk transform is the integral transform from the space of smooth even functions on the unit sphere S2⊂ℝ3 to itself defined by integration over great circles. One can regard this transform as a limit in a certain sense of the Penrose transform from [Copf ]ℙ2 to [Copf ]ℙ*ast;2. We exploit this viewpoint by developing a new proof of the bijectivity of the Funk transform which proceeds by considering the cohomology of a certain involutive (or formally integrable) structure on an intermediate space. This is the simplest example of what we hope will prove to be a general method of obtaining results in real integral geometry by means of complex holomorphic methods derived from the Penrose transform.

Let X be a complex nonsingular projective threefold of general type. Suppose the canonical system of X is composed of a pencil, i.e. dimΦ∼KX∼(X)=1. It is often important to understand birational invariants of X such as pg(X), q(X), h2(OX) and χ(OX) etc. In this paper, we mainly study the irregularity of X.

We may suppose that ∼KX∼ is free of base points. There is a natural fibration f[ratio ]X→C onto a nonsingular curve after the Stein factorization of Φ∼KX∼. Let F be a general fibre of f, then we know that F is a nonsingular projective surface of general type. Set b[ratio ]=g(C) and pg(F), q(F) for the respective invariants of F. The main result is the following theorem.

We obtain a formula for the essential supremum of the packing dimensions of the sections of sets parallel to a given subspace. This depends on a variant of packing dimension defined in terms of local projections of sets.

Suppose that H=H*[ges ]0 on L2(X, dx) and that e−Ht has an integral kernel K(t, x, y) which is a continuous function of all three variables. It follows from the fact that e−Ht is a non-negative self-adjoint operator that K(t, x, x)[ges ]0 for all t>0 and x∈X. Our main abstract results, Theorems 2 and 3, provide a positive lower bound on K(t, x, x) under suitable general hypotheses. As an application we obtain a explicit positive lower bound on K(t, x, y) when x is close enough to y and H is a higher order uniformly elliptic operator in divergence form acting in L2(RN, dx); see Theorem 6.

We emphasize that our results are not applicable to second order elliptic operators (except in one space dimension). For such operators much stronger lower bounds can be obtained by an application of the Harnack inequality. For higher order operators, however, we believe that our result is the first of its type which does not impose any continuity conditions on the highest order coefficients of the operators.

Existence and regularity results are obtained for the Cauchy problem for the Kadomtsev–Petviashvili–I equation by the inverse spectral method. The initial data are small Schwartz functions which are not assumed to satisfy the zero mass constraint ∫ℝdxq(x, y, 0)=0.

We obtain sharp rates of uniform convergence in the real inversion theorem for Laplace transforms. This entails giving estimates of the Kolmogorov distance between normalized Poisson mixtures and their mixing distribution. Our approach combines probabilistic representations in terms of gamma processes with approximation-theoretic techniques.

We introduce a quasitriangular Hopf algebra or ‘quantum group’ U(B), the double-bosonization, associated to every braided group B in the category of H-modules over a quasitriangular Hopf algebra H, such that B appears as the ‘positive root space’, H as the ‘Cartan subalgebra’ and the dual braided group B* as the ‘negative root space’ of U(B). The choice B=Uq(n+) recovers Lusztig's construction of Uq(g); other choices give more novel quantum groups. As an application, our construction provides a canonical way of building up quantum groups from smaller ones by repeatedly extending their positive and negative root spaces by linear braided groups; we explicitly construct Uq(sl3) from Uq(sl2) by this method, extending it by the quantum-braided plane. We provide a fundamental representation of U(B) in B. A projection from the quantum double, a theory of double biproducts and a Tannaka–Krein reconstruction point of view are also provided.