Fix an abelian group $\Gamma $ and an injective endomorphism $F\colon \Gamma \to \Gamma $ . Improving on the results of [2], new characterizations are here obtained for the existence of spanning sets, F-automaticity, and F-sparsity. The model theoretic status of these sets is also investigated, culminating with a combinatorial description of the F-sparse sets that are stable in $(\Gamma ,+)$ , and a proof that the expansion of $(\Gamma ,+)$ by any F-sparse set is NIP. These methods are also used to show for prime $p\ge 7$ that the expansion of $(\mathbb {F}_p[t],+)$ by multiplication restricted to $t^{\mathbb {N}}$ is NIP.

Non-n-ampleness as denned by Pillay [20] and Evans [5] is preserved under analysability. Generalizing this to a more general notion of Σ-ampleness, this gives an immediate proof for all simple theories of a weakened version of the Canonical Base Property (CBP) proven by Chatzidakis [4] for types of finite SU-rank. This is then applied to the special case of groups.

Let U be a connected noetherian scheme of finite étale cohomological dimension in which every finite set of points is contained in an affine open subscheme. Suppose that α is a class in H2(Uét,ℂm)tors. For each positive integer m, the K-theory of α-twisted sheaves is used to identify obstructions to α being representable by an Azumaya algebra of rank m2. The étale index of α, denoted eti(α), is the least positive integer such that all the obstructions vanish. Let per(α) be the order of α in H2(Uét,ℂm)tors. Methods from stable homotopy theory give an upper bound on the étale index that depends on the period of α and the étale cohomological dimension of U; this bound is expressed in terms of the exponents of the stable homotopy groups of spheres and the exponents of the stable homotopy groups of B(ℤ/(per(α))). As a corollary, if U is the spectrum of a field of finite cohomological dimension d, then , where [] is the integer part of , whenever per(α) is divided neither by the characteristic of k nor by any primes that are small relative to d.

Cosmogenic 10Be and 26Al measurements from bedrock exposures in East Antarctica provide indications of how long the rock surface has been free from glacial cover. Samples from the crests of Zakharoff Ridge and Mount Harding, two typical nunataks in the Grove Mountains, show minimum 10Be ages of 2.00 ± 0.22 and 2.30 ± 0.26 Ma, respectively. These ages suggest that the crests were above the ice sheet at least since the Plio–Pleistocene boundary. Adopting a ‘reasonable’ erosion rate of 5–10 cm Ma-1 increases the exposure ages of these two samples to extend into the mid-Pliocene. The bedrock exposure ages steadily decrease with decreasing elevation on the two nunataks, which indicates ~200 m decrease of the ice sheet in the Grove Mountains since mid-Pliocene time. Seven higher elevation samples exhibit a simple exposure history, which indicates that the ice sheet in the Grove Mountains decreased only ~100 m over a period as long as 1–2 Ma. This suggests that the East Antarctic Ice Sheet (EAIS) was relatively stable during the Pliocene warm interval. Five lower elevation samples suggest a complex exposure history, and indicate that the maximum subsequent increase of the EAIS was only 100 m higher than the present ice surface. Considering the uncertainties, their total initial exposure and subsequent burial time could be later than mid-Pliocene, which may not conflict with the stable mid-Pliocene scenario.

A type analysable in one-based types in a simple theory is itself one-based.

Based on recurrence equation theory and relative error (rather than absolute error) analysis, the concept and criterion for the stability of a recurrence equation are clarified. A family of recursions, called congruent recursions, is proved to be strongly stable in evaluating its non-negative solutions. A type of strongly unstable recursion is identified. The recursive formula discussed by Panjer (1981) is proved to be strongly stable in evaluating the compound Poisson and the compound Negative Binomial (including Geometric) distributions. For the compound Binomial distribution, the recursion is shown to be unstable. A simple method to cope with this instability is proposed. Many other recursions are reviewed. Illustrative numerical examples are given.

The paper investigates stochastic processes directed by a randomized time process. A new family of directing processes called Hougaard processes is introduced. Monotonicity properties preserved under subordination, and dependence among processes directed by a common randomized time are studied. Results for processes subordinated to Poisson and stable processes are presented. Potential applications to shock models and threshold models are also discussed. Only Markov processes are considered.

Suppose λ is an isolated eigenvalue of the (bounded linear) operator T on the Banach space X and the algebraic multiplicity of λ is finite. Let Tn be a sequence of operators on X that converge to T pointwise, that is, Tnx → Tx for every x ∈ X. If ‖(T − Tn)Tn‖ and ‖Tn(T − Tn)‖ converge to 0 then Tn is strongly stable at λ.

Twelve cultivares were grown along with released high yielding hybrid CSH-1 (susceptible to stem-borer) to investigate their relative resistance to stem-borer at three locations. Results obtained have established that the 12 cultivars possess moderate levels of resistance to stem-borer. The damage suffered by them was significantly lower than CSH-1. Further, four cultivars viz. E601, E 602, E 603 and E 604 suffered less damage.

The stability of the resistance was worked out by following the model of Eberhart and Russell (1966) [Crop Sci. 6, 36–40]. The means of various cultivars were calculated and regression coefficients (bi) were tested against unity to locate most adaptable cultivar for borer damage. The cultivar × environment interactions for leaf injury and stem tunnelling (two parameters used for assessing stem-borer damage) were significant and in most of the cases were due to linear response of different cultivars. Cultivar E 601 proved to be most promising and stable for resistance to stem-borer.

The notion of self-decomposability for -valued random variables as introduced by Steutel and van Harn [10] and its generalization by van Harn, Steutel and Vervaat [5], are used to study the limiting behaviour of continuous-time Markov branching processes with immigration. This behaviour provides analogues to the behaviour of sequences of random variables obeying a certain difference equation as studied by Vervaat [12] and their continuous-time counterpart considered by Wolfe [13]. An application in queueing theory is indicated. Furthermore, discrete-state analogues are given for results on stability in the processes studied by Wolfe, and for results on self-decomposability in supercritical branching processes by Yamazato [14].