In an earlier paper [2], I gave explicit rules for calculating the Brauer group of a Del Pezzo surface of degree 3 or 4. Recent developments have made it desirable to extend these to the construction of the corresponding Azumaya algebras, so as to be able to calculate the associated Brauer–Manin obstruction to weak approximation or even to the Hasse principle. That is the main purpose of this note, which needs to be read in conjunction with [2]. The special case of diagonal cubic surfaces has already been treated in [1], though the process described there is not an algorithm in the strict sense; however, in numerical cases it is probably more efficient than the algorithm described in this note would be.

Suppose that f and g are transcendental entire functions of finite order, without wandering domains, and suppose that f∘g=g∘f. Then the Julia sets of f and g coincide.

We define Castelnuovo–Mumford regularity for graded modules over non-commutative graded algebras. Two fundamental commutative results are generalized to the non-commmutative case: a vanishing-theorem by Mumford, and a theorem on linear resolutions and syzygies by Eisenbud and Goto. The generalizations deal with sufficiently well-behaved algebras (i.e. so-called quantum polynomial algebras).

We go on to define Castelnuovo–Mumford regularity for sheaves on a non-commutative projective scheme, as defined by Artin. Again, a version of Mumford's vanishing-theorem is proved, and we use it to generalize a result by Martin, Migliore and Nollet, on degrees of generators of graded saturated ideals in polynomial algebras, to quantum polynomial algebras.

Finally, we generalize a practical result of Schenzel which determines the regularity of a module in terms of certain Tor-modules.

This paper asks: given a vector bundle ξ and a line bundle λ over the same base space, are λ[otimes ]ξ and ξ equivalent? We concentrate on real bundles ξ. Although the question is sensible in its own right, we explain in Section 2 our immediate motivation for studying it. In Section 3 we make some general comments about the question, the most significant being that under certain restrictions the answer depends on the stable class of ξ rather than on ξ itself (Proposition 3·4).

The rest of the paper tackles an interesting special case. To state the main result, let P(ℝn+1) denote n-dimensional real projective space, H the Hopf line bundle over it, and an+1 the order of the reduced Grothendieck group [wavy overbar]KO0(P(ℝn+1)).

The contact between a surface in Euclidean space and the family of spheres carries a good deal of useful geometrical information about the surface. The centres of those spheres having degenerate tangency with the surface (in Arnold's notation of type A2) form the focal set, and the set of points on the surface where there is a contact of type A3, the so-called ridge curve, is of some interest in the field of computer vision, as a robust feature of the surface. These ridges come together at umbilics, where the contact between the surface and the unique sphere of curvature is of type D. A dual notion, that of a subparabolic curve is also of some interest. In this paper we describe the way in which the ridge and subparabolic curves can evolve in a generic 1-parameter family of surfaces.

In this paper we investigate separation properties in the dual Ĝ of a connected, simply connected, nilpotent Lie group G. Following [4, 19], we are particularly interested in the question of when the group G is quasi-standard, in which case the group C*-algebra C*(G) may be represented as a continuous bundle of C*-algebras over a locally compact, Hausdorff, space such that the fibres are primitive throughout a dense subset. The same question for other classes of locally compact groups has been considered previously in [1, 5, 18]. Fundamental to the study of quasi-standardness is the relation of inseparability in Ĝ[ratio ]π∼σ in Ĝ if π and σ cannot be separated by disjoint open subsets of Ĝ. Thus we have been led naturally to consider also the set sep (Ĝ) of separated points in Ĝ (a point in a topological space is separated if it can be separated by disjoint open subsets from each point that is not in its closure).

The projective moduli variety [Sscr ][Uscr ]C(2) of semistable rank 2 vector bundles with trivial determinant on a smooth projective curve C comes with a natural morphism ϕ to the linear series [mid ]2×Θ[mid ] where Θ is the theta divisor on the Jacobian of C. Well- known results of Narasimhan and Ramanan say that ϕ is an isomorphism to P3 if C has genus 2 [16], and when C is nonhyperelliptic of genus 3 it is an isomorphism to a special Heisenberg-invariant quartic QC⊂P7 [18]. The present paper is an attempt to extend these results to higher genus.

The theory of variational bicomplexes was established at the end of the seventies by several authors [2, 17, 23, 26, 29–32]. The idea is that the operations which take a Lagrangian into its Euler–Lagrange morphism [9, 10, 12, 24] and an Euler–Lagrange morphism into its Helmholtz' conditions of local variationality [1–3, 7, 11, 13, 18, 27] are morphisms of a (long) exact sheaf sequence. This viewpoint overcomes several problems of Lagrangian formulations in mechanics and field theories [21, 28]. To avoid technical difficulties variational bicomplexes were formulated over the space of infinite jets of a fibred manifold. But in this formalism the information relative to the order of the jet where objects are defined is lost.

We refer to the recent formulation of variational bicomplexes on finite order jet spaces [13]. Here, a finite order variational sequence is obtained by quotienting the de Rham sequence on a finite order jet space with an intrinsically defined sub-sequence, whose choice is inspired by the calculus of variations. It is important to find an isomorphism of the quotient sequence with a sequence of sheaves of ‘concrete’ sections of some vector bundle. This task has already been faced locally [22, 25] and intrinsically [33] in the case of one independent variable.

In this paper, we give an intrinsic isomorphism of the variational sequence (in the general case of n independent variables) with a sequence which is made by sheaves of forms on a jet space of minimal order. This yields new natural solutions to problems like the minimal order Lagrangian corresponding to a locally variational Euler–Lagrange morphism and the search of variationally trivial Lagrangians. Moreover, we give a new intrinsic formulation of Helmholtz' local variationality conditions, proving the existence of a new intrinsic geometric object which, for an Euler–Lagrange morphism, plays a role analogous to that of the momentum of a Lagrangian.

We shall denote throughout by Ω⊂≠ℝd some conical region with vertex at 0. Let Σ=Σd={x∈ℝd; [mid ]x[mid ]=1} denote the unit sphere and by ΩΣ=Σ∩Ω. Let

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be the expression of the Laplacian in polar coordinates x=(r, σ) (r=[mid ]x[mid ], σ∈Σ) where Δr denote the corresponding r-dimensional spherical Laplacian. For d[ges ]2 we can always solve the first eigenfunction problem for ΩΣ

formula here

with Dirichlet boundary conditions. We shall assume throughout that ∂Ω is sufficiently regular to ensure that u is continuous up to the boundary and vanishes there. With α>0α(α+d−2)=λ the function u(x)=rαu(σ)= [mid ]x[mid ]αu(x/[mid ]x[mid ]) is then harmonic in Ω. It should be observed that α[ges ]1 if Ω is convex. If we further assume, as we shall do in this paper, that ∂Ω is Lipschitz, u(x) (x∈ℝd) is the unique, up tp multiplicative constant positive harmonic function in Ω that vanishes at the boundary ∂Ω This function is homogeneous and is called the réduite of Ω (cf. [1]). We shall denote throughout by α=degΩ the homogenity degree of u. If d=1 the réduite is u(x)≡Cx(x>0).

We shall denote by b(t)∈ℝd the standard Brownian motion in ℝd and by

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the first exit time from Ω. We shall then define the corresponding heat diffusion kernel and the corresponding ‘probability of life’

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