Cameron–Praeger designs with parameters $t-(v,k,\lambda)$ are studied. Cameron and Praeger showed that in such designs, $t=2$ or 3. In 1989, Delandtsheer and Doyen proved that if $t=2$ then \[ v \leqslant \left(\left({k \atop 2}\right)-1\right)^2. \] In 2000, Mann and Tuan improved this equality and showed that if $t=3$ then \[ v \leqslant \left({k \atop 2}\right)+1. \] Three infinite families of Cameron–Praeger 3-designs for which this bound is met have been constructed by Mann and Tuan and by Sebille. The paper constructs infinitely many infinite families of such Cameron–Praeger 3-designs via a study of a divisibility problem for polynomials. Further, the construction generalizes the previous constructions.

Fix an integer $N\ge 2$. For a positive integer $n\in\Bbb N$, let $n=d_{0}(n)+d_{1}(n)N+d_{2}(n)N^{2}+\ldots+d_{\gamma(n)}(n)N^{\gamma(n)}$ where $d_{i}(n)\in\{0,1,2,\ldots,N-1\}$ and $d_{\gamma(n)}(n)\not=0$ denote the $N$-ary expansion of $n$. For a probability vector $\bold p=(p_{0},\ldots,p_{N-1})$ and $r>0$, the $r$ approximative discrete Besicovitch–Eggleston set $B_{r}(\bold p)$ is defined by $$B_{r}(\bold p) = \bigg\{n\in\Bbb N\,{|}\,\bigg\|\,\frac{|\{0\le k\le\gamma(n)\,{|}\,d_{k}(n)=i\}|}{\gamma(n)\,{+}\,1} - p_{i} \bigg| \le r \,\,\text{for all $i$} \bigg\}\,,$$ that is, $B_{r}(\bold p)$ is the set of positive integers $n$ such that the frequency of the digit $i$ in the $N$-ary expansion of $n$ differs from $p_{i}$ by less than $r$ for all $i\in\{0,1,2,\ldots,N-1\}$. Three natural fractional dimensions of subsets $E$ of ${\mathbb N}$ are defined, namely, the lower fractional dimension $\underline\dim(E)$, the upper fractional dimension $\overline\dim(E)$ and the exponent of convergence $\delta(E)$, and the dimensions of various subsets of ${\mathbb N}$ defined in terms of the frequencies of the digits in the $N$-ary expansion of the positive integers are studied. In particular, the dimensions of $B_{r}(\bold p)$ are computed (in the limit as $r\searrow 0$). Let ${\bold p}=(p_{0},\ldots,p_{N-1})$ be a probability vector. Then $$\lim_{r\searrow 0}\,\,\underline{\dim}(B_{r}({\bold p})) = \lim_{r\searrow 0}\,\,\overline{\dim}(B_{r}({\bold p})) = \lim_{r\searrow 0}\,\,\delta(B_{r}({\bold p})) = -\frac{\sum_{i}p_{i}\log p_{i}}{\log N}\,.$$ This result provides a natural discrete analogue of a classical result due to Besicovitch and Eggleston on the Hausdorff dimension of certain sets of non-normal numbers.

Several applications to the theory of normal numbers are given.

Convergent sums $T = {\sum}_{n=1}^{\infty} P(n)/Q(n)$ and $U = {\sum}_{n=1}^{\infty}(-1)^n P(n)/Q(n)$, where $P(X), Q(X) \in {\mathbb Q}[X]$, and $Q(X)$ has only simple rational roots, are investigated. Adhikari, Shorey and the authors have shown that $T$ and $U$ are either rational or transcendental. Simple necessary and sufficient conditions are formulated for the transcendence of $T$ and $U$ if the degree of $Q$ is 3 and 2, respectively.

Let $p$ be a prime and let $q = p^a$, where $a$ is a positive integer. Let $G = G({\mathbb{F}}_q)$ be a Chevalley group over $\mathbb{ F}_q$, with associated system of roots $\Phi$ and Weyl group $W$. Steinberg showed in 1957 that $G$ has an irreducible complex representation whose degree equals the $p$-part of $|G|$ [11]. This representation, now known as the Steinberg representation, has remarkable properties, which reflect the structure of $G$, and there have been many research papers devoted to its study. The module constructed in [11] is in fact a right ideal in the integral group ring $\mathbb{ Z}G$ of $G$, and is thus a $\mathbb{ Z}G$-lattice, which we propose to call the Steinberg lattice of $G$. It should be noted that lattices not integrally isomorphic to the Steinberg lattice may also afford the Steinberg representation, and such lattices may differ considerably in their properties compared with the Steinberg lattice.

The $\Sigma^3$-conjecture for metabelian groups is proved in the split extension case.

Let $T$ be an abelian group and $\lambda$ an uncountable regular cardinal. The question of whether there is a $\lambda$-universal group $U$ among all torsion-free abelian groups $G$ of cardinality less than or equal to $\lambda$ satisfying $\Ext\left(G,T\right)=0$ is considered. $U$ is said to be $\lambda$-universal for $T$ if, whenever a torsion-free abelian group $G$ of cardinality at most $\lambda$ satisfies $\Ext\left(G,T\right)=0$, there is an embedding of $G$ into $U$. For large classes of abelian groups $T$ and cardinals $\lambda$, it is shown that the answer is consistently no, that is to say, there is a model of ZFC in which, for pairs T and $\lambda$, there is no universal group. In particular, for $T$ torsion, this solves a problem by Kulikov.

It is proved that the measure algebra M(G) of a locally compact group G is Connes-amenable if and only if G is amenable.

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.

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.

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 $\Bbb H^n$.

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.

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.

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.

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.

It is shown that every special Lagrangian cone in $\C^3$ determines, and is determined by, a primitive harmonic surface in the 6-symmetric space ${\rm SU}_3/{\rm SO}_2$. For cones over tori, this allows the classification theory of harmonic tori to be used to describe the construction of all the corresponding special Lagrangian cones. A parameter count is given for the space of these, and some of the examples found recently by Joyce are put into this context.

The class of linearly recurrent Cantor systems contains the substitution subshifts and some odometers. For substitutionsubshifts, measure-theoretical and continuous eigenvalues are the same. It is natural to ask whether this rigidity property remains true for the class of linearly recurrent Cantor systems. Partial answers are given to this question.

Let $\Omega X$ be the space of Moore loops on a finite, $q$-connected, $n$-dimensional CW complex $X$, and let $R\subset\Q$ be a subring containing 1/2. Let $\rho\left(R\right)$ be the least non-invertible prime in $R$. For a graded $R$-module $M$ of finite type, let $FM = M / {\rm Torsion}\,M$. We show that the inclusion $P \subset FH_{*}\left(\Omega X;R\right)$ of the sub-Lie algebra of primitive elements induces an isomorphism of Hopf algebras $$UP \overset{\cong}{\longrightarrow} FH_{*}\left(\Omega X;R\right)},$$ provided that $\rho\left(R\right) \geqslant n/q$. Furthermore, the Hurewicz homomorphism induces an embedding of $F(\pi_{*}\left(\Omega X\right)\otimes R)$ in $P$, with $P/F(\pi_{*}\left(\Omega X\right)\otimes R)$ torsion. As a corollary, if $X$ is elliptic, then $FH_{*}\left(\Omega X;R\right)$ is a finitely generated $R$-algebra.