Let $a$ be a positive integer with $a > 1$. We show \[ [0;a,a^2,a^3,a^4,\ldots]=\frac{\sum\nolimits^{\infty}_{s=0}a^{-(s+1)^2}\prod\nolimits^s_{i=1}(a^{2i}-1)^{-1}}{\sum\nolimits^{\infty}_{s=0}a^{-s^2}\prod\nolimits^s_{i=1}(a^{2i}-1)^{-1}} \] and \[ [0;a,a,a^2,a^2,a^3,a^3,\ldots]=\frac{\sum\nolimits^{\infty}_{s=0}a^{-\frac{(s+1)(s+2)}{2}}\prod\nolimits^s_{i=1}(a^{i}-1)^{-1}}{\sum\nolimits^{\infty}_{s=0}a^{-\frac{s(s+1)}{2}}\prod\nolimits^s_{i=1}(a^{i}-1)^{-1}} \] A more general case is discussed.

Let $K$ be a convex body in ${\bb R}^n$ with volume $|K| = 1$. We choose $N$ points $x_1,\ldots, x_N$ independently and uniformly from $K$, and write $C(x_1,\ldots, x_N)$ for their convex hull. If $n(\log n)^2 \leqslant N \leqslant \exp(c_1n)$, we show that the expected volume radius \[ {\bb E}_{\frac{1}{n}}(K,N)=\int_K\cdots\int_K |C(x_1,\ldots, x_N)|^{\frac{1}{n}}dx_N\cdots dx_1 \] of this random $N$-tope can be estimated by \[ c_1\frac{\sqrt{\log(N/n)}}{\sqrt{n}}\leqslant{\bb E}_{1/n}(K,N)\leqslant c_2L_K\frac{\log(N/n)}{\sqrt{n}}, \] where $c_1, c_2$ are absolute positive constants and $L_K$ is the isotropic constant of $K$.

Philip Hall raised the following question which is stated in The Kourovka Notebook [12, p. 88]: is there a non-trivial group which is isomorphic with every proper extension of itself by itself? We will split the problem into two parts: we want to find non-commutative splitters, that are groups $G\ne 1$ with ${\rm Ext} (G, G) = 1$. The class of splitters fortunately is quite large so that extra properties can be added to $G$. We can consider groups $G$ with the following properties: there is a complete group $L$ with cartesian product $L^w \cong G, {\rm Hom}(L^w, S^w) = 0$($S_w$ the infinite symmetric group acting on $w$) and ${\rm End} (L, L) = {\rm Inn}\, L\cup\{0\}$. We will show that these properties ensure that $G$ is a splitter and hence obviously a Hall group in the above sense. Then we will apply a recent result from our joint paper [9] which also shows that such groups exist; in fact there is a class of Hall groups which is not a set.

The topology of a topological group $G$ is called an ${\cal L}_{\infty}$-topology if it can be represented as the intersection of a decreasing sequence of locally compact Hausdorff group topolgies on $G$. If ${\cal L}_1 < {\cal L}_2$ are two distinct ${\cal L}_{\infty}$-topologies on an Abelian group $G$, it is shown that the quotient of the corresponding character groups has cardinality ${\geqslant} 2^{\rm c}$. A conjecture in this sense announced by J. B. Reade in his paper [6] is thereby proved.

Let ${\rm q}$ be a Lie algebra and ξ a linear form on ${\rm q}$. Denote by ${\rm q}_{\xi}$ the set of all $x \in {\rm q}$ such that $\xi([{\rm q},x]) = 0$. In other words, ${\rm q}_{\xi}=\{x\in {\rm q} \mid({\rm ad}^*x)\cdot = 0\}$, where ${\rm ad}^*: {\rm q}\rightarrow {\rm g I}({\rm q}^*)$ is the coadjoint representation of ${\rm q}$. The index of ${\rm q}$, ind ${\rm q}$, is defined by \[ {\rm ind\, q} = \min_{\xi\in {\rm q}^*}\dim {\rm q}\xi. \]

Let $N$ be a prime number, and $X$ a curve that is intermediate to the cover $X_1(N)\rightarrow X_0(N)$. We study the automorphism group of $X$, and prove that in most cases it is generated by the Galois group of the cover $X\rightarrow X_0(N)$, and any lift of the Atkin–Lehner involution to $X$.

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.

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.

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 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.

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).