The notion of stationary reflection is one of the most important notions of combinatorial set theory. Weak reflection, which is, as its name suggests, a weak version of stationary reflection, is investigated. The main result is that modulo a large cardinal assumption close to 2-hugeness, there can be a regular cardinal $\kappa$ such that the first cardinal weakly reflecting at $\kappa$ is the successor of a singular cardinal. This answers a question of Cummings, Džamonja and Shelah.

Finding polynomial solutions of Pell's equation is of interest as such solutions sometimes allow the fundamental units to be determined in an infinite class of real quadratic fields.

In the paper, for each triple of positive integers $(c, h, f)$ satisfying $c^2-fh^2 = 1$ , where $(c, h)$ are the smallest pair of integers satisfying this equation, several sets of polynomials $(c(t), h(t), f(t))$ that satisfy $c(t)^2-f(t)h(t)^2 = 1$ and $(c(0), h(0), f(0)) = (c, h, f)$ are derived. Moreover, it is shown that the pair $(c(t), h(t))$ constitute the fundamental polynomial solution to the Pell equation above.

The continued fraction expansion of $\sqrt{f(t)}$ is given in certain general cases (for example when the continued fraction expansion of $\sqrt{f}$ has odd period length, or has even period length, or has period length $\equiv 2 \mod 4$ and the middle quotient has a particular form, etc.). Some applications to the determination of the fundamental unit in real quadratic fields is also discussed.

It is proved that the simultaneous lattice packing and lattice covering constant of every three-dimensional centrally symmetric convex body is less than or equal to 7/4. Consequently, for any three-dimensional convex body K there is a corresponding lattice packing in which no extra translate of K can be added.

Given $q$ , a power of a prime $p$ , denote by $F$ the finite field ${\rm GF}(q)$ of order $q$ , and, for a given positive integer $n$ , by $E$ its extension ${\rm GF}(q^n)$ of degree $n$ . A primitive element of $E$ is a generator of the cyclic group $E^\ast$ . Additively too, the extension $E$ is cyclic when viewed as an $FG$ -module, $G$ being the Galois group of $E$ over $F$ . The classical form of this result – the normal basis theorem – is that there exists an element $\alpha \in E$ (an additive generator) whose conjugates $\{\alpha, \alpha^q, \ldots, \alpha^{q^{n-1}}\}$ form a basis of $E$ over $F; \alpha$ is a free element of $E$ over $F$ , and a basis like this is a normal basis over $F$ . The core result linking additive and multiplicative structure is that there exists $\alpha \in E$ , simultaneously primitive and free over $F$ . This yields a primitive normal basis over $F$ , all of whose members are primitive and free. Existence of such a basis for every extension was demonstrated by Lenstra and Schoof [5] (completing work by Carlitz [1, 2] and Davenport [4]).

Finite-dimensional algebras over an algebraically closed field are divided into two disjoint classes, called tame and wild respectively, by Drozd's tame and wild dichotomy (see [5] and [2]). A tame algebra, roughly speaking, has its n-dimensional indecomposable modules parametrized by finitely many one-parameter families, for all natural numbers n, but a wild algebra has more indecomposable modules and it is considered hopeless to classify them. In [2], Crawley-Boevey showed that all but finitely many n-dimensional indecomposable modules over a tame algebra are τ-invariant, for all natural numbers n, and conjectured that the converse would be true, where τ := DTr is the Auslander–Reiten translation (see [1]) and we call an indecomposable module X τ-invariant if X ≅ = τX.

The paper determines all permutation groups with a transitive minimal normal subgroup that have no fixed point free elements of prime order. All such groups are primitive and are wreath products in a product action involving $M_{11}$ in its action on 12 points. These groups are not 2-closed and so substantial progress is made towards asserting the truth of the polycirculant conjecture that every 2-closed transitive permutation group has a fixed point free element of prime order. All finite simple groups $T$ with a proper subgroup meeting every ${\rm Aut}(T)$ -conjugacy class of elements of $T$ of prime order are also determined.

Most of our notation is taken from James [7], where further details of the representation theory of the symmetric groups may be found; note, however, that we write functions on the left.

Let $n$ be a non-negative integer, and $\lambda$ a partition of $n$ . Say that two $\lambda$ -tableaux are row equivalent if one can be obtained from the other by permuting the entries within each row, and define column equivalence similarly. Let $\sim_{\rm row}$ and $\sim_{\rm col}$ denote these relations.

Let $K$ and $\mu$ be the self-similar set and the self-similar measure associated with an IFS (iterated function system) with probabilities $(S_i, p_i)_{i=1,\ldots,N}$ satisfying the open set condition. Let $\Sigma=\{1,\ldots,N\}^{\bb N}$ denote the full shift space and let $\pi : \Sigma \longrightarrow K$ denote the natural projection. The (symbolic) local dimension of $\mu$ at $\omega \in \Sigma$ is defined by $\lim_n (\log \mu K_{\omega\mid n}/\log\hbox{ diam }K_{\omega\mid n})$ , where $K_{\omega\mid n}=S_{\omega 1}\circ\ldots\circ S_{\omega_n}(K)$ for $\omega = (\omega_1, \omega_2,\ldots) \in \Sigma$ . A point $\omega$ for which the limit $\lim_n (\log \mu K_{\omega\mid n}/\log\hbox{ diam }K_{\omega\mid n})$ does not exist is called a divergence point. In almost all of the literature the limit $\lim_n (\log \mu K_{\omega\mid n}/\log\hbox{ diam }K_{\omega\mid n})$ is assumed to exist, and almost nothing is known about the set of divergence points. In the paper a detailed analysis is performed of the set of divergence points and it is shown that it has a surprisingly rich structure. For a sequence $(x_n)_n$ , let ${\sf A}(x_n)$ denote the set of accumulation points of $(x_n)_n$ . For an arbitrary subset $I$ of ${\bb R}$ , the Hausdorff and packing dimension of the set

\[ \left\{\omega\in\Sigma\left\vert {\sf A}\left(\frac{\log\mu K_{\omega\mid n}}{\log\hbox{ diam }K_{\omega\mid n}}\right)\right.=I\right\} \]

and related sets is computed. An interesting and surprising corollary to this result is that the set of divergence points is extremely ‘visible’; it can be partitioned into an uncountable family of pairwise disjoint sets each with full dimension.

In order to prove the above statements the theory of normal and non-normal points of a self-similar set is formulated and developed in detail. This theory extends the notion of normal and non-normal numbers to the setting of self-similar sets and has numerous applications to the study of the local properties of self-similar measures including a detailed study of the set of divergence points.

Recently, systematic studies of mappings of finite distortion have emerged as a key area in geometric function theory. The connection with deformations of elastic bodies and regularity of energy minimizers in the theory of nonlinear elasticity is perhaps a primary motivation for such studies, but there are many other applications as well, particularly in holomorphic dynamics and also in the study of first order degenerate elliptic systems, for instance the Beltrami systems we consider here.

In most cases, the third order approximation of a scalar Newtonian equation can lead to the Lyapunov stability of a periodic solution through the obtaining of a nonzero twist coefficient. Recently, Ortega obtained the twist property of a periodic solution when the second order coefficient does not change sign and the third one is negative under a crucial limitation to the rotation of the linearization equation. The paper finds that the best bound on the limitation of the rotations is $\theta^*_0=\arccos(-1/4)$.

Let $X,Y$ be Banach spaces. Of concern are the higher order abstract Cauchy problem $({\rm ACP}_n)$ in $X$ and its inhomogeneous version $({\rm IACP}_n)$ . A new operator family of bounded linear operators from $Y$ to $X$ is introduced, called an existence family for $({\rm ACP}_n)$ , so that the existence and continuous dependence on initial data of the solutions of $({\rm ACP}_n)$ and $({\rm IACP}_n)$ can be studied, and some basic results in a quite general setting can be obtained. A sufficient and necessary condition ensuring that $({\rm ACP}_n)$ possesses an exponentially bounded existence family, in terms of Laplace transforms, is presented. As a partner of the existence family, for $({\rm ACP}_n)$ , a uniqueness family of bounded linear operators on $X$ is defined to guarantee the uniqueness of solutions. These two operator families for $({\rm ACP}_n)$ are generalizations of the classical strongly continuous semigroups and sine operator functions, the $C$ -regularized semigroups and sine operator functions, the existence and uniqueness families for $(ACP_1)$ , and the $C$ -propagation families for $({\rm ACP}_n)$ . They have a special function in treating those ill-posed $({\rm ACP}_n)$ and $({\rm IACP}_n)$ whose coefficient operators lack commutativity.

A new non-oscillatory theory is presented for higher order delay and neutral equations. The results include both singular and nonsingular problems.

The paper deals with the estimates of the blow-up rate and the blow-up set of positive solutions to semilinear parabolic systems coupled in the equations and boundary conditions. The explicit upper and lower bounds of the blow-up rate are obtained. Moreover, it is proved that the blow-up set contains a single point x = 1 under the proper conditions on the initial data.

Let $f(z)$ be a transcendental meromorphic function. The paper investigates, using the hyperbolic metric, the relation between the forward orbit $P(f)$ of singularities of $f^{-1}$ and limit functions of iterations of $f$ in its Fatou components. It is mainly proved, among other things, that for a wandering domain $U$ , all the limit functions of $\{f^n\vert_U\}$ lie in the derived set of $P(f)$ and that if $f^{np}\vert_V\rightarrow q(n\rightarrow +\infty)$ for a Fatou component $V$ , then either $q$ is in the derived set of $S_p(f)$ or $f^p(q) = q$ . As applications of main theorems, some sufficient conditions of the non-existence of wandering domains and Baker domains are given.

The paper proves an almost-orthogonality principle for maximal operators over arbitrary sets of directions in ${\bb R}^2$. Namely, the Lp-bounds for an operator of this type are obtained from the corresponding Lp-bounds of the maximal functions associated to a certain partition of the set of directions, and from the particular structure of this partition. Applications to several types of maximal operators are provided.

For each $p$ in $[1, \infty)$ let ${\bf E}_p$ denote the closure of the region of holomorphy of the Ornstein–Uhlenbeck semigroup $\{{\cal H}_t : t >0\}$ on $L^p$ with respect to the Gaussian measure. Sharp weak type and strong type estimates are proved for the maximal operator $f \mapsto {{\cal H}^*}_pf=\sup\{\vert {\cal H}_zf\vert :z\in {\bf E}_p\}$ and for a class of related operators. As a consequence, a new and simpler proof of the weak type 1 estimate is given for the maximal operator associated to the Mehler kernel.

Let $1 < p <\infty, 0 < v < p^\prime$ , let $\Omega$ be a bounded domain in ${\bb R}^n$ , and denote by ${\rm id}_{\omega}$ the limiting compact embedding of the Besov space $B^{n/p}_{pp}({\bb R}^n)$ into the exponential Orlicz space $L_{\exp (t^v)}(\Omega)$ , mapping a function $f$ onto its restriction $f\vert_{\Omega}$ . In 1993 Triebel established, among others, two-sided estimates for the entropy numbers of ${\rm id}_{\omega}$ , which are even asymptotically optimal for ‘small’ $\nu$ . The aim of the paper is to improve the upper bounds in the case of ‘large’ $\nu$ , where Triebel's estimates are not yet sharp, thus making a further step towards the conjectured correct asymptotic behaviour.

Sufficient conditions are found for stochastic convolution integrals driven by a Wiener process in a Hilbert space to belong to the Orlicz space exp $L^2$ ; standard exponential tail estimates follow from these results. Proofs are based on the extrapolation theory and are rather simple.

A random walk that is certain to visit $(0, \infty)$ has associated with it, via a suitable $h$ -transform, a Markov chain called ‘random walk conditioned to stay positive’, which is defined properly below. In continuous time, if the random walk is replaced by Brownian motion then the analogous associated process is Bessel-3. Let $\phi(x) = \log\log x$ . The main result obtained in this paper, which is stated formally in Theorem 1, is that, when the random walk has zero mean and finite variance, the total time for which the random walk conditioned to stay positive is below $x$ ultimately lies between $Lx^2/\phi(x)$ and $Ux^2\phi(x)$ , for suitable (non-random) positive $L$ and finite $U$ , as $x$ goes to infinity. For Bessel-3, the best $L$ and $U$ are identified.