On sait associer à certaines structures de Poisson sur ℝn, de 1-jet nul en 0, des actions de ℝ2 sur ℝn, données par le ‘rotationnel’ de leur partie quadratique et un autre champ de vecteurs. Lorsque ces actions sont ‘non résonantes’ et ‘hyperboliques’, on montre que ces structures sont ‘quadratisables’, en ce sens qu'il existe des coordonnées dans lesquelles, elles sont quadratiques. Dans le cas de la dimension 3, nos résultats mènent à la ‘non-dégénérescence’ générique des structures de Poisson quadratiques à rotationnels inversibles.

We can associate with some Poisson structures defined on ℝn with a zero 1-jet at zero, actions from ℝ2 on ℝn, given by the ‘curl’ of their quadratic part and another vector field. Assuming that those actions are ‘hyperbolics’ and without ‘resonances’, we give a normal form for those structures. On ℝ3, we prove that every quadratic Poisson structure with invertible curl, is generically ‘non degenerate’.

Let u be a solution of the differential equation u″+Ru=0, where R is rational. Newton's method of finding the zeros of u consists of iterating the function f(z) =z−u(z)/u′(z). With suitable hypotheses on R and u, it is shown that the iterates of f converge on an open dense subset of the plane if they converge for the zeros of R. The proof is based on the iteration theory of meromorphic functions, and in particular on the result that, if the family of K-quasiconformal deformations of a meromorphic function f depends on only finitely many parameters, then every cycle of Baker domains of f contains a singularity of f−1. This result, together with classical results of Hille concerning the asymptotic behaviour of solutions of the above differential equations, is also used to study their value distribution. For example, it is shown that, if R is a rational function which satisfies R(z)∼amzm as z→∞ and has only k distinct zeros where k<(m+2)/2, then δ(0, u)[les ]k/(m+2−k)<1.

The positivity of the partial sums of the series of Legendre polynomials is a classical result of Fejér. The paper presents corresponding results for both even and odd Legendre polynomials with best possible leading constants. These results have applications to demonstrating positivity of Cotes numbers at Jacobi abscissas.

Here we develop estimates for Fourier coefficients of Siegel cusp forms. First we consider the case of Siegel modular forms for the full modular group $\Gamma_g$ having small weight (that is, weight k = g + 1). Afterwards the case of a certain subgroup $\Gamma_{g,0}(N)$ of $\Gamma_g$ is considered.

A characterization is given of the ‘special linear groups’ T(Ψ, W) as linear groups generated by a non-degenerate class Σ of abstract root groups such that the elements of A ∈ Σ are transvections.

The right-invariant Riemannian metric on simplex shape spaces in fact makes them particular Riemannian symmetric spaces of non-compact type. In the paper, the general properties of such symmetric spaces are made explicit for simplex shape spaces. In particular, a global matrix coordinate representation is suggested, with respect to which several geometric features, important for shape analysis, have simple and easily computable expressions. As a typical application, it is shown how to locate the Fréchet means of a class of probability measures on the simplex shape spaces, a result analogous to that for Kendall's shape spaces.

Let K be a field and let V be a vector space of dimension 2m over K. Let ∧V denote the exterior algebra of V and ∧kV its kth exterior power for 0[les ]k[les ]2m. Let f be a non-degenerate alternating bilinear form defined on V×V. The symplectic group Sp2m(K) is the group of all isometries of f and it acts as a group of vector space automorphisms on ∧kV. In the case that K is algebraically closed and 1[les ]k[les ]m, it is known that ∧kV contains a composition factor corresponding to the fundamental weight ωk of a root system of type Cm. We shall refer to the irreducible module for Sp2m(K) given by this composition factor as a fundamental module.

Let M be a manifold with conical ends. (For precise definitions see the next section; we only mention here that the cross-section [Kscr ] can have a nonempty boundary.) We study the scattering for the Laplace operator on M. The first question that we are interested in is the structure of the absolute scattering matrix [Sscr ](s). If M is a compact perturbation of ℝn, then it is well-known that [Sscr ](s) is a smooth perturbation of the antipodal map on a sphere, that is,

formula here

On the other hand, if M is a manifold with a scattering metric (see [8] for the exact definition), it has been proved in [9] that [Sscr ](s) is a Fourier integral operator on [Kscr ], of order 0, associated to the canonical diffeomorphism given by the geodesic flow at distance π. In our case it is possible to prove that [Sscr ](s) is in fact equal to the wave operator at a time t = π plus C∞ terms. See Theorem 3.1 for the precise formulation. This result is not too difficult and is obtained using only the separation of variables and the asymptotics of the Bessel functions.

Our second result is deeper and concerns the scattering phase p(s) (the logarithm of the determinant of the (relative) scattering matrix).

We say that a finite abelian group does not have the Rédei property if it can be expressed as a direct product of two of its subsets such that both subsets contain the identity element and both subsets span the whole group. It will be shown that only a small fraction of the finite abelian groups can have the Rédei property. For groups of odd order an explicit list of the possible exceptions is compiled.

It is shown that, if Q is a finitely generated abelian group, a finitely generated Q-group A is m-tame if and only if the mth tensor power of the augmentation ideal of ℤA is finitely generated over Am [rtimes ] Q, where Q acts diagonally on both Am and the tensor power. It is proved that quotients of metabelian groups of type FP3 are again of type FP3, and a necessary condition is found for a split extension of abelian-by-(nilpotent of class two) groups to be of type FP2. A conjecture is formulated that generalises the FPm-Conjecture for metabelian groups, and it is shown that one of the implications holds in the prime characteristic case.

Let $\Omega \subset \R^2$ be a bounded Lipschitz domain and let \[ F: \Omega \times {\mathbb R}_{+}^{2 \times 2} \longrightarrow {\mathbb R} \] be a Carathèodory integrand such that $F\left(x, \cdot\right)$ is polyconvex for ${\mathcal L}^2$-a.e. $x \in \Omega$. Moreover assume that $F$ is bounded from below and satisfies the condition $F\left(x, \xi\right) \searrow \infty$ as $\det \xi \searrow 0$ for ${\mathcal L}^2$-a.e. $x \in \Omega$. The paper describes the effect of domain topology on the existence and multiplicity of strong local minimizers of the functional \[ {\mathbb F} \left[u\right] := \int_{\Omega } F\left(x, \nabla u\left(x\right)\right) \, dx,\] where the map $u$ lies in the Sobolev space $W_{\rm {id}}^{1,p} (\Omega, {\mathbb R}^2)$ with $p \geqslant 2$ and satisfies the pointwise condition $\det \nabla u\left(x\right) >0$ for ${\mathcal L}^2$-a.e. $x \in \Omega$. The question is settled by establishing that ${\mathbb F}\left[\cdot\right]$ admits a set of strong local minimizers on {W^{1,p}_{\rm{id}}(\Omega, {\mathbb R}^2)} that can be indexed by the group $\Pn \oplus \Z^n$, the direct sum of Artin's pure braid group on $n$ strings and $n$ copies of the infinite cyclic group. The dependence on the domain topology is through the number of holes $n$ in $\Omega$ and the different mechanisms that give rise to such local minimizers are fully exploited by this particular representation.

Let t(G, q) denote the smallest integer t such that the vertex set V of the graph G can be partitioned into t classes V(G)=[Cup ]ti=1Vi such that the number of edges in the induced subgraph 〈Vi〉 for 1[les ]i[les ]t, is divisible by q. Using an algebraic theorem due to Baker and Schmidt we prove that if q is a prime power then t(G, q) can be computed and a corresponding partition can be presented in a polynomial time.

It is proved that the dual of a Banach space with the Mazur intersection property is almost weak* Asplund. Analogously, the predual of a dual space with the weak* Mazur intersection property is almost Asplund. Through the use of these arguments, it is found that, in particular, almost all (in the Baire sense) equivalent norms on [lscr ]1(Γ) and [lscr ]∞(Γ) are Fréchet differentiable on a dense Gδ subset. Necessary conditions for Mazur intersection properties in terms of convex sets satisfying a Krein–Milman type condition are also discussed. It is also shown that, if a Banach space has the Mazur intersection property, then every subspace of countable codimension can be equivalently renormed to satisfy this property.

For the nonlinear recurrence relation \[ \alpha_2 = \alpha_1(1 - \alpha_1), \quad \alpha_{k+1} = k^2 \alpha_k + \sum\limits_{m=2}^{k-1} \alpha_m \alpha_{k+1-m}, \quad k\geqslant 2, \] it is proved that the limit \[ p_{\infty}(\alpha_1) = \lim_{k \rightarrow \infty} \alpha_k /[(k-1)!]^2 \] exists and defines an entire function of $\alpha_2 = \alpha_1(1-\alpha_1)$ .

Upper and lower bounds are established for the entropy numbers of certain diagonal operators between Banach sequence spaces. These diagonal operators are isomorphisms between the spaces considered in the paper and weighted sequence spaces considered by Leopold so that the entropy numbers in question coincide with those considered by Leopold. The results in the paper improve the previous results in at least two ways. The estimates in the paper are ‘almost’ sharp in the sense that the upper and lower estimates differ only by logarithmic factors for a much wider range of parameters. Moreover, all the upper estimates are improvements on the previous ones, the improvement being quite significant in some cases.

It is proved that there is a certain degree of independence between stationary reflection phenomena at different cofinalities.

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 paper examines blow-up phenomena for the inequality \renewcommand{\theequation}{*} \begin{equation} u_t - Lu - \vert u \vert^{q-1} u \leqslant v_t - Lv - \vert v \vert^{q-1}v \end{equation} in the half-space ${\bb R}^1_+ \times {\bb R}^n,\; n \geqslant 1$ , where $L$ is a linear second-order partial differential operator in divergence form.

The paper studies weak solutions of (*) that belong only locally to the corresponding Sobolev spaces in the half-space ${\bb R}^1_+ \times {\bb R}^n$ . It also requires no conditions for the behavior of solutions of (*) on the hyperplane $t = 0$ .

The existence of critical blow-up exponents is obtained for solutions of (*) as a special case of a comparison principle for the corresponding solutions of (*). For example, the well-known Fujita result is a consequence of the comparison principle.

The approach developed in the paper is directly applicable to the study of analogous problems involving nonlinear differential operators. Its elliptic analogue has been recently developed by the authors.

An n-affine manifold is a differentiable manifold endowed with an atlas whose transition functions are locally affine transformations of ℝn . The paper studies affine dynamic, that is, affine endomorphisms of affine manifolds. The main goal is to relate the dynamical viewpoint to the completeness problem. In particular, it is shown that the Markus conjecture is true when dim Aff(M, [dtri ]) > dim M − 2.

Let f(z) and g(z) be transcendental entire functions. Fuchs and Song proved that if (f(z); g(z)) parametrizes some complex algebraic curve, then f and g must have a transcendental common right factor. The paper proves this result by a different method that also allows a similar result to be proved for some transcendental curves. This result is then used to solve some factorization problems of entire functions.