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.

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.

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 $N$.

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 $P_2(X)\,{=}\,P_2(Y)$.

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

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.

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 $\lambda$.

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.

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.

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].

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.

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.

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.

Necessary and sufficient conditions are given for a Banach space operator with the single-valued extension property to satisfy Weyl's theorem and $a$-Weyl's theorem. It is shown that if $T$ or $T^{\ast}$ has the single-valued extension property and $T$ is transaloid, then Weyl's theorem holds for $f(T)$ for every $f\in H(\sigma (T))$. When $T^{\ast}$ has the single-valued extension property, $T$ is transaloid and $T$ is $a$-isoloid, then <formula form="inline" disc="math" id="ffm012"><formtex notation="AMSTeX">$a$-Weyl's theorem holds for $f(T)$ for every $f\in H(\sigma(T))$. It is also proved that if $T$ or $T^{\ast}$ has the single-valued extension property, then the spectral mapping theorem holds for the Weyl spectrum and for the essential approximate point spectrum.

A connection and a curving on a bundle gerbe associated with lifting a structure group of a principal bundle to a central extension are constructed. The construction is based on certain structures on the bundle, that is, connections and bundle splittings. The Deligne cohomology class of the lifting bundle gerbe with the connection and with the curving coincides with the obstruction class of the lifting problem with these structures.

The paper describes some qualitative properties of minimizers on a manifold $\mathcal M$ endowed with a discontinuous metric. The discontinuity occurs on a hypersurface $\Sigma$ disconnecting $\mathcal M$. Denote by $\Omega_1$ and $\Omega_2$ the open subsets of $\mathcal M$ such that $\mathcal M\setminus\Sigma=\Omega_1\cup\Omega_2$. Assume that $\overline\Omega_1$ and $\overline\Omega_2$ are endowed with metrics $\left\langle\cdot,\cdot\right\rangle_{\left(1\right)}$ and $\left\langle\cdot,\cdot\right\rangle_{\left(2\right)}$, respectively, such that $\overline\Omega_i (i=1, 2)$ is convex or concave. The existence of a minimizer of the length functional on curves joining two given points of $\mathcal M$ is proved. The qualitative properties obtained allows the refraction law in a very general situation to be described.