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.

The points which converge to $\infty$ under iteration of the maps $z\mapsto\lambda\exp(z)$ for $\lambda \in \mathbb{C} \backslash \{0\}$ are investigated. A complete classification of such ‘escaping points’ is given: they are organized in the form of differentiable curves called rays which are diffeomorphic to open intervals, together with the endpoints of certain (but not all) of these rays. Every escaping point is either on a ray or the endpoint (landing point) of a ray. This answers a special case of a question of Eremenko. The combinatorics of occurring rays, and which of them land at escaping points, are described exactly. It turns out that this answer does not depend on the parameter $\lambda$.

It is also shown that the union of all the rays has Hausdorff dimension 1, while the endpoints alone have Hausdorff dimension 2. This generalizes results of Karpińska for specific choices of $\lambda$.

A tangent process of a random process X at a point z is defined to be the limit in distribution of some sequence of scaled enlargements of X about z. The main result of the paper is that a tangent process must be self-similar with stationary increments, at almost all points z where the tangent process is essentially unique. The consequences for tangent processes of certain classes of process are examined, including stable processes and processes with independent increments where unique tangent processes are Lévy processes.

Singly-periodic monogenic cotangent and cosecant functions are important to Clifford analysis because they are the building blocks of the Bergman and Szegö reproducing kernels for strip domains, that is, rectangular domains with a single bounded dimension. The paper establishes a wide range of explicit formulas for these functions, in terms of derivatives, of one-dimensional integrals, and of Fourier and plane wave multidimensional integrals. These results indicate how the elementary trigonometric functions $\cot(z)$, $\csc(z)$ and $\csc^2(z)$ are ramified into different entities when the setting is switched from complex analytic to Clifford monogenic.

It is shown that the natural action of the absolute Galois group on the ideals defining principal congruence subgroups of certain nonarithmetic Fuchsian triangle groups is compatible with its action on the algebraic curves that these congruence groups uniformize.

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 $\theta^*_0=\arccos(-1/4)$.

The pressure function $p(t)$ of a non-recurrent map is real analytic on some interval ($0,t_\ast$) with $t_\ast$ strictly greater than the dimension of the Julia set. The proof is an adaptation of the well known tower techniques to the complex dynamics situation. In general, $p(t)$ need not be analytic on the whole positive axis.

Let $\Bbb H^n$ denote the (2n+1)-dimensional Heisenberg group. Given an operator-valued function $M$, define the operator $T_{M}$ by $(T_{M}f)\hat{\vphantom{f}}=\skew4\hat{f} M$ with ‘$\hat{}$’ denoting the Fourier transform. Hörmander-type sufficient conditions are determined on $M$ for the $H^p$-boundedness, $p\le 1$, of the operator $T_{M}$ on $\Bbb H^n$.

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.

By means of generalized Riccati transformation techniques and generalized exponential functions, some oscillation criteria are given for the nonlinear dynamic equation \[ (p(t)x^{\Delta} (t))^{\Delta}+q(t)(f\circ x^{\sigma})=0 \] on time scales. The results are also applied to linear and nonlinear dynamic equations with damping, and some sufficient conditions are obtained for the oscillation of all solutions.

A holomorphic function defined in the unit disk $\Delta = \{z: |z|<1\}$ belongs to the MacLane class ${\cal A}$ if each point $\zeta$ of a dense subset of $\p \Delta$ is the endpoint of a curve $\gamma_{\zeta}$ (with

$\gamma_{\zeta}\setminus \zeta \subset \Delta)$ such that $f(z)$ tends to a limit (perhaps $\infty$) as $z \to \zeta$ on $\gamma_\zeta$. The classical Fatou theorem ensures that $f \in {\cal A}$ when $f$ is bounded. G. R. MacLane introduced ${\cal A}$ in [5], where he proved that $f \in {\cal A}$ if there is a set $E$ dense in $\p \Delta$ with \begin{equation}\int^1_0 (1-r) \log^+|f(re^{i\theta})| \, dr<\infty\qquad(\theta \in E).\end{equation}

For example, if $f$ is the modular function and $M(r)=\max_{|z|=r}|f(z)|$ its maximum modulus, then

$$\log M(r)\leq \log\frac{1}{1-r} +O(1),$$

so that (1.1) applies. An ample discussion of ${\cal A}$ is in [4, Chapter 10].

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

New results concerning the so-called Weiss conjecture on admissible operators for bounded analytic semigroups are given. Let \[ \left(T_t\right)_{t\geqslant 0} \] be a bounded analytic semigroup with generator $-A$ on some Banach space $X$. It is proved that if $A^{1/2}$ is admissible for $A$, that is, if there is an estimate \[ \int_{0}^{\infty^{\vphantom{-1}}}\|A^{1/2}e^{-tA}x\|^2\, dt\leqslant M^2\|x\|^2,\] then any continuous mapping $C : D\left(A\right)\longrightarrow Y$ valued in a Banach space $Y$ is admissible for $A$ provided that there is an estimate \[ \|\left(-{\rm Re}\left({\lambda}\right)\right)^{1/2}C\left(\lambda -A\right)^{-1}\|\leqslant K \] for $\lambda\in\mathbb{C}$, ${\rm Re}\left({\lambda}\right)<0$. This holds in particular if \[ \left(T_t\right)_{t\geqslant 0}\] is a contractive (analytic) semigroup on Hilbert space. In the converse direction, it is shown that this may happen for a bounded analytic semigroup on Hilbert space that is not similar to a contractive one. Applications in non-Hilbertian Banach spaces are also given.

If the sum of a series of positive numbers is not convergent and the series satisfies certain regularity conditions, a probability measure can be found whose support contains no interval, yet whose Fourier series decreases faster than the given series.

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 $A$ be a Banach algebra and let $p$ and $q$ be two positive integers. We show that if

$A$ has a left bounded sequential approximate identity $(e_n)_{n\ge1}$ such that ${\rm lim}\,{\rm inf}_{n\to+\infty}\|e^p_n-e^{p+q}_n\| < ({p \over {p+q}})^{p\over q}{q\over{p+q}}$ then

$A$ has a left-bounded sequential identity $(f_n)_{n\ge1}$ such that $f^2_n = f_n$ for $n\ge1$. A simple example shows that the constant $({p\over {p+q}})^{p\over q}{q\over{p+q}}$ is best possible.

This result is based on some algebraic or integral formulae which associate an idempotent to elements of a Banach algebra satisfying some inequalities involving polynomials or entire functions.

Given a non-atomic, finite and complete measure space $(\Omega,\Sigma,\mu)$ and a Banach space $X$, the modulus of continuity for a vector measure $F$ is defined as the function $\omega_F(t) = \sup_{\mu(E)\le t} |F|(E)$ and the space ${V}^{p,q}(X)$ of vector measures such that $t^{-1/p^\prime}\wf (t)\in L^q((0,\mu(\Omega)],dt/t)$ is introduced. It is shown that $V^{p,q}(X)$ contains isometrically $\lpq(X)$ and that $L^{p,q}(X)=V^{p,q}(X)$ if and only if $X$ has the Radon–Nikodym property. It is also proved that ${V}^{p,q}(X)$ coincides with the space of cone absolutely summing operators from $L^{p^\prime,q^\prime}$ into $X$ and the duality ${V}^{p,q}(X^*)= (L^{p^\prime,q^\prime}(X))^*$ where $1/p+1/p^\prime= 1/q+1/q^\prime=1$. Finally, $V^{p,q}(X)$ is identified with the interpolation space obtained by the real method $(V^{1}(X)$, $V^{\infty}(X))_{1/p',q}$. Spaces where the variation of $F$ is replaced by the semivariation are also considered.

T. Hosokawa, K. Izuchi and D. Zheng recently introduced the concept of asymptotic interpolating sequences (of type 1) in the unit disk for $H^\infty$(${\bb D}$). It is shown that these sequences coincide with sequences that are interpolating for the algebra $QA$. Also a characterization is given of the interpolating sequences of type $1$ for $H^\infty$(${\bb D}$), and asymptotic interpolating sequences in the spectrum of $H^\infty$(${\bb D}$) are studied. The existence of asymptotic interpolating sequences of type $1$ for $H^\infty(\Omega)$ on arbitrary domains is verified. It is shown that any asymptotic interpolating sequence in a uniform algebra eventually is interpolating.