We classify smooth $n$-dimensional varieties $X_n\subset{\bb P}^{2n+1}$ with one apparent double point and of degree $d \leqslant 2n + 4$, showing that these are only the smooth irreducible divisors of type (2,1), (0,2) and (1,2) on the Segre manifold ${\bb P}^1\times{\bb P}^n \subset {\bb P}^{2n+1}$, a 3-fold of degree 8 and two Mukai manifolds, the first one of dimension 4 and degree 12, the second one of dimension 6 and degree 16. We also prove that a linearly normal variety $X_n \subset{\bb P}^{2n+1}$ of degree $d \leqslant 2n+1$ and with ${\rm Sec}(X_n) = {\bb P}^{2n+1}$ is regular and simply connected, that it has one apparent double point and hence it is a divisor of type (2,1), (0,2) or (1,2) on the Segre manifold ${\bb P}^1\times{\bb P}^n\subset{\bb P}^{2n+1}$. To this aim we study linear systems of quadrics on projective space whose base locus is a smooth irreducible variety and we look for conditions assuring that they are (completely) subhomaloidal; we also show some new properties of varieties $X_n\subset{\bb P}^{2n+1}$ defined by quadratic equations and we study projections of such varieties from (subspaces of) the tangent space.

We prove that all generalized symmetric spaces of compact simple Lie groups are formal in the sense of Sullivan. Nevertheless, many of them, including all the non-symmetric flag manifolds, do not admit Riemannian metrics for which all products of harmonic forms are harmonic.

Let A be the generator of an analytic C0-semigroup on a Banach space X. We associate a closed operator ${\cal A}_{1}$ with A defined on Rad(X) and show that when X is a UMD-space, the Cauchy problem associated with A has maximal regularity if and only if the operator ${\cal A}_{1}{\rm g}$ generates an analytic C0-semigroup on Rad(X). This allows us to exploit known results on analytic C0-semigroups to study maximal regularity. Our results show that ${\cal R}$-boundedness is a local property for semigroups: an analytic C0-semigroup T of negative type is ${\cal R}$-bounded if and only if it is ${\cal R}$-bounded at z = 0. As applications, we give a perturbation result for positive semigroups. Finally, we show the following: when X is a UMD-space, T is an analytic C0-semigroup of negative type, then for every $f\in L^{p}(\RR_{+}; X)$, the mild solution of the corresponding inhomogeneous Cauchy problem with initial value 0 belongs to $W^{\theta,p}(\RR_{+}; X)$ for every $0<\theta < 1$.

The cohomology ring of the flag manifold associated to any compact Lie group has a simple description in terms of the Weyl group of $G$. The same algebraic procedure gives a ring associated to any complex re ection group, not necessarily a Weyl group, but much less is known about it. For example, in the case of Weyl groups (and the corresponding compact Lie groups), there are remarkably simple formulas for the degrees of Grassmannians and related projective varieties, due to Schubert and others. We find here a similar formula for the ‘degree’ of a ring associated to the complex re ection group $S_n**{\bf Z}/e$; in the case $e = 2$, this group is the Weyl group of the symplectic group and our formula specializes to the classical formula for the degree of the isotropic Grassmannian $Sp(2n)/U(n)$. We discuss the relation of our calculation to Hall–Littlewood polynomials at roots of unity.

The aim of this paper is to prove that the inverse of the normalization functor in the Dold–Kan correspondence $D:{\sf Ch}({\sf Ab})\rightarrow {\sf sAb}$ is an $E_{\infty}$-monoidal functor. This proves that generalized Eilenberg–MacLane spectra on differential graded commutative algebras are $E_{\infty}$-monoids in the category of $H{\bb Z}$-module spectra.

For $0<p<\infty$, the Bergman space $A^p$ consists of functions $f$ analytic in the unit disk $\Bbb D$ such that

$$\int_{\Bbb D}|f(z)|^p\,dA(z)<\infty,$$

where $dA(z)$ is the Lebesgue area measure on $\Bbb D.$ We say that a sequence of points $\{z_n\}$ in $\Bbb D$ is an $A^p$ zero set if some function $f\in A^p$ vanishes precisely on this set. Many interesting results about zero sets for Bergman spaces have been obtained by Horowitz [Ho1, Ho2], Korenblum [K] and Seip [S]. The reader is also referred to [HKZ]. The probabilistic approach to the study of $A^p$ zero sets was initiated by Leblanc [LeB] who proved a sufficient condition for a sequence $\{z_n\}$ to be an $A^2$ zero set almost certainly when the arguments of $z_n$ are chosen at random. His results have been improved and extended to $0<p\le 2$ by Bomash [Bo].

We exploit a recent approach to Brascamp–Lieb inequalities, due to Caffarelli [5], and reconsider earlier approaches to establish stochastic domination inequalities between Gaussian variables and random variables with density of the form $g\cdot h, g$ a Gaussian density and $h$ a log-concave or log-convex function. These extend to inequalities on random vectors via a classical result by Prékopa and Leindler and they complement the Brascamp–Lieb moment inequalities. Some applications to a class of Gibbs measures, the anharmonic crystals, are developed.

Let ${\cal A}$ be the mod 2 Steenrod algebra and $D_k$ the Dickson algebra of $k$ variables. We study the Lannes–Zarati homomorphisms \[ \varphi_k:{\rm Ext}_{{\cal A}}^{k, k+i}({\bb F}_2,{\bb F}_2)\rightarrow ({\bb F}_2\otimes_{{\cal A}}D_k)^{*}_{i}, \] which correspond to an associated graded of the Hurewicz map $H:\pi^s_*(S^0)\cong\pi_*(Q_0S^0)\rightarrow H_*(Q_0S^0)$. An algebraic version of the long-standing conjecture on spherical classes predicts that $\varphi_k = 0$ in positive stems, for $k > 2$. That the conjecture is no longer valid for $k = 1$ and $2$ is respectively an exposition of the existence of Hopf invariant one classes and Kervaire invariant one classes.

This conjecture has been proved for $k = 3$ by Hu'ng ‘9’. It has been shown that $\varphi_k$ vanishes on decomposable elements for $k > 2$ ‘14’ and on the image of Singer's algebraic transfer for $k > 2$ ‘9, 12’. In this paper, we establish the conjecture for $k = 4$. To this end, our main tools include (1) an explicit chain-level representation of $\varphi_k$ and (2) a squaring operation ${\rm Sq}^0$ on $({\bb F}_2\otimes_{{\cal A}}D_k)^*$, which commutes with the classical ${\rm Sq}^0$ on ${\rm Ext}_{{\cal A}}^k({\bb F}_2,{\bb F}_2)$ through the Lannes–Zarati homomorphism.

We consider a variational problem for steady axisymmetric vortex-rings, in which kinetic energy is maximised subject to prescribed impulse, with the ratio of vorticity to axial distance belonging to the class of rearrangements of a prescribed function, as proposed by Benjamin. We prove existence of maximisers in an extended constraint set, allowing some loss of vorticity. We then study a particular family of vortex-rings including Hill's spherical vortex, determining the precise range of impulse values for which the maximiser loses vorticity, and show that the maximisers are spherical when this happens.

We derive simple conditions, which are both necessary and sufficient, for the validity of sharp local embeddings $H^{\sigma,\alpha} X({\bf R}^{n}) \hookrightarrow Y({\bf {\Omega})$ and sharp global embeddings $H^{\sigma,\alpha}, X({\bf R}^{n}) \hookrightarrow Z({\bf R}^{n})$. Here $H^{\sigma,\alpha}$, stands for a Bessel potential operator involving the classical smoothness $\alpha$ and logarithmic smoothness $\alpha$, X, Y and Z are (generalized) Lorentz–Zygmund spaces, and $\Omega\subset {\bf R^{n}}$ is an open subset whose n-dimensional Lebesgue measure is finite. Our results extend those of [18] and improve them especially in the extreme cases when X is close to L1, or Y and Z are close to $L^{\infty}$, or when $\sigma=0$ and $\alpha > 0$.

It is shown that a spacetime with collisionless matter evolving from data on a compact Cauchy surface with hyperbolic symmetry can be globally covered by compact hypersurfaces on which the mean curvature is constant and by compact hypersurfaces on which the area radius is constant. Results for the related cases of spherical and plane symmetry are reviewed and extended. The prospects of using the global time coordinates obtained in this way to investigate the global geometry of the spacetimes concerned are discussed.

We give examples of acylindrical 3-manifolds such that every non-trivial closed curve on the boundary can appear as a boundary component of an immersed essential surface.

A theorem about local in time existence of spacelike foliations with prescribed mean curvature in cosmological spacetimes will be proved. The time function of the foliation is geometrically defined and fixes the diffeomorphism invariance inherent in general foliations of spacetimes. Moreover, in contrast to the situation of the more special constant mean curvature foliations, which play an important role in global analysis of spacetimes, this theorem overcomes the existence problem arising from topological restrictions for surfaces of constant mean curvature.

Let $f$ be an entire function. For $n \in {\bb N}$, let $f^n$ denote the $n$th iterate of $f$. The set \[ F(f)=\{z:(f^n)\hbox{ is normal in some neighbourhood of }z\} \] is the Fatou set or the set of normality and its complement $J(f)$ is the Julia set. If $U$ is a component of $F$$(f)$, then $f$$(U)$ lies in some component $V$ of $F$$(f)$. If $U_n\cap U_m=\phi$ for $n \ne m$ where $U_n$ denotes the component of $F$$(f)$ which contains $f^n(U)$, then $U$ is called a wandering domain, else $U$ is called a pre-periodic domain, and if $U_n = U$ for some $n \in {\bb N}$ then $U$ is called periodic domain.

It is known that for entire functions $f$ and $g$, $f$$(g)$ has wandering domain if and only if $f$$(g)$ has wandering domain. In this paper we discuss the existence of wandering domains of composite entire functions with regards to its factors $f$ and $g$.

We prove the following two theorems:

(I) Let $n \geqslant 1$ be a (fixed) integer. Then we have for $\theta \in (0, \pi)$: \[ \sum\limits^n_{k=1}\frac{\cos (k\theta)}{k}\leqslant -\log\left(\sin\left(\frac{\theta}{2}\right)\right)+\frac{\pi-\theta}{2}+\sigma_n, \] with the best possible constant $\sigma_n = \sum\nolimits^n_{k=1}(-1)^k/k$.

(II) For even integers $n \geqslant 2$ and for $\theta \in (0, \pi)$ we have \[ \sum\limits^n_{k=1}\frac{\sin(k\theta)}{k}\leqslant\alpha(\pi-\theta), \] with the best possible constant $\alpha = 0.66 395\ldots$.

Our results refine inequalities due to C. Hyltén-Cavallius ‘11’ and P. Turán ‘23’, respectively.

Let $E$ be a rearrangement invariant (r.i.) function space on ‘0, 1’. We consider the space $\Lambda({\cal R},E)$ of measurable functions $f$ such that $fg \in E$ for every a.e. converging series $g =\sum a_nr_n \in E$, where $(r_n)$ are the Rademacher functions. Curbera ‘4’ showed that, for a broad class of spaces $E$, the space $\Lambda({\cal R},E)$ is not order-isomorphic to a r.i. space. We study cases when $\Lambda({\cal R},E)$ is order-isomorphic to a r.i. space. We give conditions on $E$ so that $\Lambda({\cal R},E)$ is order-isomorphic to $L_{\infty}$. This includes certain classes of Lorentz and Marcinkiewicz spaces. We study further when $\Lambda({\cal R},E)$ is orderisomorphic to a r.i. space different from $L_{\infty}$. This occurs for the Orlicz spaces $E = L_{\Phi_q}$ with $\Phi_q(t)$ asymptotically equivalent to $\exp |t|^q-1$ and $0 < q < 2$.

We study the size of the range of the derivatives of a smooth function between Banach spaces. We establish conditions on a pair of Banach spaces $X$ and $Y$ to ensure the existence of a $C^p$ smooth (Fréchet smooth or a continuous Gâteaux smooth) function $f$ from $X$ onto $Y$ such that $f$ vanishes outside a bounded set and all the derivatives of $f$ are surjections. In particular we deduce the following results. For the Gâteaux case, when $X$ and $Y$ are separable and $X$ is infinite-dimensional, there exists a continuous Gâteaux smooth function $f$ from $X$ to $Y$, with bounded support, so that $f^{\prime}(X) = {\cal L}(X,Y)$. In the Fréchet case, we get that if a Banach space $X$ has a Fréchet smooth bump and ${\rm dens} X = {\rm dens} {\cal L}(X,Y)$, then there is a Fréchet smooth function $f: X \rightarrow Y$ with bounded support so that $f^{\prime}(X) = {\cal L}(X,Y)$. Moreover, we see that if $X$ has a $C^p$ smooth bump with bounded derivatives and ${\rm dens} X = {\rm dens} {\cal L}^m_s(X;Y)$ then there exists another $C^p$ smooth function $f: X \rightarrow Y$ so that $f^{(k)} (X) = {\cal L}^k_s(X;Y)$ for all $k = 0, 1,\ldots, m$. As an application, we show that every bounded starlike body on a separable Banach space $X$ with a (Fréchet or Gâteaux) smooth bump can be uniformly approximated by smooth bounded starlike bodies whose cones of tangent hyperplanes fill the dual space $X^*$. In the non-separable case, we prove that $X$ has such property if $X$ has smooth partitions of unity.

A left (right) approximate identity $(e_\alpha)$ for a normed algebra ${\cal A}$ is said to satisfy condition (U) if $e_{\alpha}\circ a({\rm resp.} a\circ e_{\alpha})$ converges to a uniformly for a in compact sets of ${\cal A}$. Let ${\cal A}$ be an operator algebra on a Banach space $X$, containing finite rank operators and being contained in the algebra of compact operators. It is shown that ${\cal A}$ has a left approximate identity satisfying condition (U) if and only if the identity operator on the space $X$ is approximable uniformly on compact sets of $X$ by operators in ${\cal A}$. The algebra ${\cal A}$ has a right approximate identity satisfying condition (U) if and only if the identity operator on the dual space $X^*$ is approximable uniformly on compact sets of $X^*$ by operators which are adjoints of the operators in ${\cal A}$. Moreover, ${\cal A}$ has a two-sided approximate identity satisfying condition (U) if it has a right approximate identity satisfying condition (U).