We discuss percolation and random walks in a class of homogeneous ultrametric spaces together with similarities and differences in ultrametric and Euclidean spaces. We briefly outline the role of these models in the study of interacting systems. Several open problems are presented.

An orbicular diorite from Fisher Lake, California, USA, contains multi-shelled, magmatic orbicules with branching and budding orthopyroxene crystals as well as feather and acicular plagioclase crystals that are oriented perpendicular to the growth horizon. Plagioclase and orthopyroxene show gradual, reverse compositional zoning along the long axes and normal zoning along the short axes. The reverse zoning varies from An87 to An93 and Mg68 to Mg74 over distances of 4 mm and 8 mm respectively. The close proximity of these two minerals makes it likely that only one mechanism is responsible for the reverse zoning. This zoning can be explained by using relevant temperature-composition diagrams and Gibbs free energy-composition plots. Under sudden and moderate undercoolings, which produce high growth but low nucleation rates, the difference in Gibbs free energy (ΔG) between the crystals and liquid is not initially maximized, i.e. initial compositions are not near-to-equilibrium. This results in crystal compositions that are closer to that of the bulk liquid than expected for crystallization under near-to-equilibrium conditions (i.e. very small ΔT). Over time, and under isothermal crystallization conditions, ΔG gradually increases to a maximum producing crystal compositions that also gradually attain near-to-equilibrium compositions. Subsequent to attaining these conditions, normal zoning occurs perpendicular to the crystal growth axes.

Nonsymmetric branching flow through a three-dimensional (3D) vessel is considered at medium-to-high flow rates. The branching is from one mother vessel to two or more daughter vessels downstream, with laminar steady or unsteady conditions assumed. The inherent 3D nonsymmetry is due to the branching shapes themselves, or the differences in the end pressures in the daughter vessels, or the incident velocity profiles in the mother. Computations based on lattice-Boltzmann methodology are described first. A subsequent analysis focuses on small 3D disturbances and increased Reynolds numbers. This reduces the 3D problem to a two-dimensional one at the outer wall in all pressure-driven cases. As well as having broader implications for feeding into a network of vessels, the findings enable predictions of how much swirling motion in the cross-plane is generated in a daughter vessel downstream of a 3D branch junction, and the significant alterations provoked locally in the shear stresses and pressures at the walls. Nonuniform incident wall-shear and unsteady effects are examined. A universal asymptotic form is found for the flux change into each daughter vessel in a 3D branching of arbitrary cross-section with a thin divider.

Common cocklebur, jimsonweed, and velvetleaf were grown with soybeans in the field to determine how soybean interference affects weed canopy architecture. Common cocklebur had more leaves within the soybean canopy than jimsonweed or velvetleaf. At the end of the season, common cocklebur leaf area was distributed evenly below and above the top of the soybean canopy, while nearly all the jimsonweed and velvetleaf leaf area was above the soybean canopy. Common cocklebur exhibited more shade tolerance than jimsonweed or velvetleaf by also maintaining leaves in the shade within the soybean canopy. Differences among these weeds in leaf distribution within the soybean canopy were not related to differences in abscission of the lower leaves but to a differential response of lower axillary buds to soybean shading. Growth from lower axillary buds in jimsonweed and velvetleaf was strongly inhibited by soybean interference, but interference had little effect on lower axillary bud growth in common cocklebur. Axillary bud growth in the lower canopies of both common cocklebur and soybeans, and their similarity in height, caused these plants to compete for the same aboveground niche. However, common cocklebur had more extensive axillary growth along the lower stem than soybeans, which may allow it to compete for resources in this niche more aggressively than soybeans. Velvetleaf and jimsonweed did not share the same aboveground niche with soybeans due to the placement of their leaves above rather than within the soybean canopy. The lower branching characteristics and apparent shade tolerance of common cocklebur may be important factors in the superior competitive ability with soybeans compared to jimsonweed and velvetleaf.

Field experiments were established at Columbus and near South Charleston, OH to determine the effects of giant ragweed population density on soybean yield and to characterize the development of giant ragweed grown in 76-cm soybean rows. An economic threshold was calculated for Ohio using a common treatment for giant ragweed control in soybean. A cost of $41/ha was estimated for a farmer to apply 0.56 kg/ha bentazon plus 0.28 kg/ha fomesafen plus COC (1.25% v/v). Assuming a soybean value of $0.22/kg, the cost of control was equivalent to 5.4 and 7.1% of the soybean yield in 1991 and 1992, respectively, which corresponded to the yield loss caused by 0.08 and 0.03 giant ragweed plants/m2. The competitiveness of giant ragweed can be at least partly attributed to its ability to initiate and maintain axillary leaves and branches within the shaded confines of the soybean canopy.

Burs of common cocklebur (Xanthium strumarium L.) obtained from Stoneville, Mississippi; Urbana, Illinois; and Lamberton, Minnesota, were used in greenhouse and growth chamber studies. The Mississippi and the Illinois strains were both classified as being in the “strumarium” complex of X. strumarium, whereas, the Minnesota strain was classified as being in the “hybrid” complex. The Minnesota common cocklebur developed branch leaves in the leaf axils of the main stem to a much greater extent than did the Mississippi and Illinois common cocklebur. The Mississippi common cocklebur was frequently more tolerant of postemergence applications of bentazon [3-isopropyl-1H-2,1,3-benzothiadiazin-4(3H)-one 2,2-dioxide] than were the Illinois and Minnesota common cockebur. The Mississippi and Illinois common cocklebur developed leaf necrosis when sprayed with an insecticide formulation containing malathion (o, o-dimethyl phosphorodithioate of mercaptosuccinate) and petroleum solvents; the Minnesota common cocklebur did not.

Kudzu (Pueraria lobata Ohwi # PUELO) was grown from seeds in controlled-environment chambers at 350, 675, or 1000 μl·1−1 CO2. Biomass and leaf area production, morphological characteristics, and growth analysis components were determined at 14, 24, 45, and 60 days after emergence. At 60 days, plants grown at 1000 μl·1−1 CO2 had 51% more biomass, 58% longer stems, and 50% more branches than plants grown at 350 μl·1−1 CO2. Plants grown at 675 μl·1−1 CO2 were intermediate. Growth analysis components indicated that CO2 enrichment increased growth by compounding effects due to increased net assimilation rates and increased leaf area duration. Relative growth rates were not significantly affected. The large CO2-induced increase in stem height versus stem diameter is in marked contrast to previously reported responses of woody erect growth forms. Possible ecological implications for competitive abilities are discussed.

Velvetleaf growth and canopy architecture were compared under a range of light conditions representative of competitive and noncompetitive environments typical of irrigated Mediterranean-type agroecosystems. Velvetleaf biomass and seed production exceeded those reported in the literature. Plants grown in full light produced 1,370 g dry weight and 44,200 seeds per plant and showed low relative variability. Velvetleaf grown with corn was reduced to 21 g dry weight and 349 seeds per plant, and had high relative variability for biomass and seed numbers. Velvetleaf grown with kidney bean, intraspecific neighbors, or under shadecloth had dry weights and seed numbers that were intermediate to plants grown in full light or with corn. Relative growth rate (RGR), net assimilation rate (NAR), and leaf area ratio (LAR) were assessed utilizing Richards functions, which were fitted to the primary biomass and leaf area data by weighted regression. RGR was highest for all plants early in the season, but declined later. Dynamics of NAR and LAR appeared to be correlated with increased self-shading, shading by neighbors, leaf age, and shedding of lower canopy leaves. Dynamics of specific leaf area corresponded with light availability such that the leaves exposed to full light were thicker than those exposed to shade. The branches of plants in all treatments had random azimuths and the foliage area density was concentrated along the perimeter of the plant's canopy. Velvetleaf increased the canopy radius through extensive branching when exposed to full sunlight. Leaf area distribution and branching patterns resulted in leaf area indices of less than 1.0. Leaves maintained a perpendicular angle to the sun throughout the day, but this depended on whether leaves received a consistent directional signal from the sun and not necessarily on whether they received a high-intensity signal. When shaded, the allocation of dry matter went primarily to the stem tissue, which increased the height rather than the girth of the plants. There was a 10- to 20-d delay for allocations to seed in the case of shaded plants relative to those grown in full sunlight. In brief, velvetleaf had a wide range of growth and canopy responses to a variety of light availabilities and it should have little difficulty in becoming fully established in the irrigated agroecosystems of Mediterranean-type regions.

The purpose of this article is to describe and analyze the reduction of unstressed vowels in Coratino, a language spoken in the Apulia region of Italy. Its vowel inventory includes seven vowels: /i, e, ε, a, ɔ, o, u/. All but /a/ are reduced to a schwa when they surface in unstressed positions. Furthermore, back and front vowels are not reduced in unstressed positions when they are adjacent to a labial consonant, or adjacent to a velar followed by a palatal. These vowels also remain non-reduced in word-initial position. Therefore, there are three contexts in which these vowels are not reduced (in stressed positions, adjacent to a consonant, and in word-initial positions). This article aims to reduce this disjunction.

In this paper we focus on a class of linear Hawkes processes with general immigrants. These are counting processes with shot-noise intensity, including self-excited and externally excited patterns. For such processes, we introduce the concept of the age pyramid which evolves according to immigration and births. The virtue of this approach that combines an intensity process definition and a branching representation is that the population age pyramid keeps track of all past events. This is used to compute new distribution properties for a class of Hawkes processes with general immigrants which generalize the popular exponential fertility function. The pathwise construction of the Hawkes process and its underlying population is also given.

Very little is known about how growth is distributed within thalli of fruticose lichens. To determine whether mature, non-apical regions of the beard lichen Ramalina usnea elongate, segments of the broadest axes were marked with thread and measured; thalli were then returned to the field to resume growth for about one year. Growth was evident in all segments measured and increments were roughly proportional to length, suggesting that elongation occurs throughout the thallus. New thallus axes (branches) were organized from buds that emerged from mature tissue, usually along pseudocyphellae and surfaces exposed by breakage. Axes were very fine at their extremities and possessed an inrolled apex from the time of bud emergence. No evidence of apical branching was observed. The thallus cortex consisted of very thick-walled fungal cells oriented longitudinally. In transverse section, cells gave the appearance of being grouped within fascicles, indicating that branch cells grow within the older cell wall material of their neighbours, as occurs in the cortex of Ramalina menziesii and the medullary cord of Usnea longissima. A continued accumulation of cell wall layers may be related to the mature tissue's capacity to sustain elongation growth.

We discuss the fourth-order thin film equation with a stable second-order diffusion term, in the context of a standard free-boundary problem with zero height, zero contact angle and zero-flux conditions imposed at an interface. For the first critical exponent where N ≥ 1 is the space dimension, there are continuous sets (branches) of source-type very singular self-similar solutions of the form For p ≠ p0, the set of very singular self-similar solutions is shown to be finite and consists of a countable family of branches (in the parameter p) of similarity profiles that originate at a sequence of critical exponents {pl, l ≥ 0}. At p = pl, these branches appear via a non-linear bifurcation mechanism from a countable set of second kind similarity solutions of the pure thin film equation Such solutions are detected by a combination of linear and non-linear ‘Hermitian spectral theory’, which allows the application of an analytical n-branching approach. In order to connect with the Cauchy problem in Part I, we identify the cauchy problem solutions as suitable ‘limits’ of the free-boundary problem solutions.

We consider the fourth-order thin film equation, with a stable second-order diffusion term. For the first critical exponent, where N ≥ 1 is the space dimension, the Cauchy problem is shown to admit countable continuous branches of source-type self-similar very singular solutions of the form These solutions are inherently oscillatory in nature and will be shown in Part II to be the limit of appropriate free-boundary problem solutions. For p ≠ p0, the set of very singular solutions is shown to be finite and to be consisting of a countable family of branches (in the parameter p) of similarity profiles that originate at a sequence of critical exponents {pl, l ≥ 0}. At p = pl, these branches appear via a non-linear bifurcation mechanism from a countable set of similarity solutions of the second kind of the pure thin film equation Such solutions are detected by the ‘Hermitian spectral theory’, which allows an analytical n-branching approach. As such, a continuous path as n → 0+ can be constructed from the eigenfunctions of the linear rescaled operator for n = 0, i.e. for the bi-harmonic equation ut = −Δ2u. Numerics are used, wherever appropriate, to support the analysis.

The Kesten-Stigum theorem for the one-type Galton-Watson process gives necessary and sufficient conditions for mean convergence of the martingale formed by the population size normed by its expectation. Here, the approach to this theorem pioneered by Lyons, Pemantle and Peres (1995) is extended to certain kinds of martingales defined for Galton-Watson processes with a general type space. Many examples satisfy stochastic domination conditions on the offspring distributions and suitable domination conditions combine nicely with general conditions for mean convergence to produce moment conditions, like the X log X condition of the Kesten-Stigum theorem. A general treatment of this phenomenon is given. The application of the approach to various branching processes is indicated. However, the main reason for developing the theory is to obtain martingale convergence results in a branching random walk that do not seem readily accessible with other techniques. These results, which are natural extensions of known results for martingales associated with binary branching Brownian motion, form the main application.

We define a stochastic process {Xn} based on partial sums of a sequence of integer-valued random variables (K0,K1,…). The process can be represented as an urn model, which is a natural generalization of a gambling model used in the first published exposition of the criticality theorem of the classical branching process. A special case of the process is also of interest in the context of a self-annihilating branching process. Our main result is that when (K1,K2,…) are independent and identically distributed, with mean a ∊ (1,∞), there exist constants {cn} with cn+1/cn → a as n → ∞ such that Xn/cn converges almost surely to a finite random variable which is positive on the event {Xn ↛ 0}. The result is extended to the case of exchangeable summands.

A transition probability kernel P(·,·) is said to be stochastically monotone if P(x, (–∞, y]) is non-increasing in x for every fixed y. A Markov chain is said to be stochastically monotone (SMMC) if its transition probability kernels are stochastically monotone. A new method for tackling the asymptotics of SMMC is given in terms of some limit variables {Wq}. In the temporally homogeneous case a cyclic pattern for {Wq} will describe the limit behaviour of suitably normed and centred processes. As a consequence, geometrically growing constants turn out to pertain to almost sure convergence. Some convergence criteria are given and applications to branching processes and diffusions are outlined.

The general framework of a Markov chain in a random environment is presented and the problem of determining extinction probabilities is discussed. An efficient method for determining absorption probabilities and criteria for certain absorption are presented in the case that the environmental process is a two-state Markov chain. These results are then applied to birth and death, queueing and branching chains in random environments.