Until recently, the known symmetric nets were class-regular and therefore satisfied a certain geometric condition that defines the class of nets known as tactical symmetric nets. Thus the known symmetric nets were tactical. This paper gives a general method of construction for non-tactical symmetric nets, which are therefore not class-regular. It is not known whether there are tactical symmetric nets which are not class regular.

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.

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.

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.

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.

It is proved that the word problem of the direct product of two free groups of rank 2 can be recognised by a 2-tape real-time but not by a 1-tape real-time Turing machine. It is also proved that the Baumslag–Solitar groups $B$(1, $r$) have the 5-tape real-time word problem for all $r\neq 0$.

Motivated by Cremona and Mazur's notion of visibility of elements in Shafarevich–Tate groups [6, 27], there have been a number of recent works which test its compatibility with the Birch and Swinnerton–Dyer conjecture and the Bloch–Kato conjecture. These conjectures provide formulas for the orders of Shafarevich–Tate groups in terms of values of $L$-functions. For example, one may see recent work of Agashe, Dummigan, Stein and Watkins [1, 2, 10, 11]. In their examples, they find that the presence of visible elements agrees with the expected divisibility properties of the relevant $L$-values.

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.

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.

Estimates are given for the exponential sum $\sum_{x=1}^p \exp(2\pi i f(x)/p)$, $p$ a prime and $f$ a nonzero integer polynomial, of interest in cases where the Weil bound is worse than trivial. The results extend those of Konyagin for monomials to a general polynomial. Such bounds readily yield estimates for the corresponding polynomial Waring problem mod $p$, namely the smallest $\gamma$ such that $f(x_1)+\cdots +f(x_{\gamma})\equiv N$ (mod $p$) is solvable in integers for any $N$.

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]).

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

The purpose of the paper is to illustrate how vanishing theorems can be used to give effective criteria for a generically finite morphism $f\,{:}\,X\,{\longrightarrow}\,Y$ of smooth complex projective algebraic varieties to be birational. In particular, as a consequence of a non-vanishing theorem of Kollár, it is shown that if $Y$ is of general type and has generically large algebraic fundamental group, then $f$ is birational if and only if $P_2(X)\,{=}\,P_2(Y)$.

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.

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.

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.

A construction of the Monster simple group is described implicitly as $196882\,{\times}\,196882$ matrices over the field of 3 elements.

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.

According to a 1904 conjecture of Dickson [2] unless prevented by congruence conditions, any finite collection of linear forms in $\Z[x]$ with positive leading coefficients infinitely often simultaneously represent primes. For the forms $x,x+2$ this includes the conjectured infinitude of prime-pairs.