Exposure of pea plants (Pisum sativum L. ‘Little Marvel’) to vapors of S-(2,3-dichloroallyl)diisopropylthiocarbamate (diallate) increased the subsequent foliar uptake of 14C-labeled herbicides. Increased amounts of 14C-activity were found in the tissues of diallate-treated plants as compared to the controls when herbicides were applied to the leaves as aqueous solutions or embedded in agar blocks. The herbicides were (2,4-dichlorophenoxy)acetic acid-1-14C (2,4-D), uniformly ring-14C 2-chloro-4-(ethylamino)-6-(isopropylamino)-s-triazine (atrazine),trichloroacetic acid-2-14C (TCA), and ethylene-14C 6,7-dihydrodipyrido[1,2-a:2′,1′-c]pyrazinediium ion (diquat). The increased movement of herbicides into diallate-treated peas was primarily attributed to a reduction of epicuticular lipids on leaves of the treated plants.

Consumption of almonds has been shown to be associated with a decreased risk of CHD, which may be related to their fatty acid (FA) composition. However, the effect of almond consumption on the serum FA composition is not known. Therefore, in the present study, we investigated whether almond consumption would alter the serum FA profile and risk of CHD, as calculated using Framingham's 10-year risk score, in a dose-dependent manner in hyperlipidaemic individuals when compared with a higher-carbohydrate control group using dietary interventions incorporating almonds. A total of twenty-seven hyperlipidaemic individuals consumed three isoenergetic (mean 1770 kJ/d) supplements during three 1-month dietary phases: (1) full-dose almonds (50–100 g/d); (2) half-dose almonds with half-dose muffins; (3) full-dose muffins. Fasting blood samples were obtained at weeks 0 and 4 for the determination of FA concentrations. Almond intake (g/d) was found to be inversely associated with the estimated Framingham 10-year CHD risk score (P= 0·026). In both the half-dose and full-dose almond groups, the proportions of oleic acid (OA) and MUFA in the TAG fraction (half-almond: OA P= 0·003; MUFA P= 0·004; full-almond: OA P< 0·001; MUFA P< 0·001) and in the NEFA fraction (half-almond: OA P= 0·01; MUFA P= 0·04; full-almond: OA P= 0·12; MUFA P= 0·06) increased. The estimated Framingham 10-year CHD risk score was inversely associated with the percentage change of OA (P= 0·011) and MUFA (P= 0·016) content in the TAG fraction. The proportions of MUFA in the TAG and NEFA fractions were positively associated with changes in HDL-cholesterol concentrations. Similarly, the estimated Framingham 10-year CHD risk score was inversely associated with the percentage change of OA (P= 0·069) and MUFA content in the NEFA fraction (P= 0·009). In conclusion, the results of the present study indicate that almond consumption increases OA and MUFA content in serum TAG and NEFA fractions, which are inversely associated with CHD lipid risk factors and overall estimated 10-year CHD risk.

The effect of diet v. statins on LDL particle size as a risk factor for CVD has not been examined. We compared, in the same subjects, the impact of a dietary portfolio of cholesterol-lowering foods and a statin on LDL size electrophoretic characteristics. Thirty-four hyperlipidaemic subjects completed three 1-month treatments as outpatients in random order: a very-low saturated fat diet (control); the same diet with 20 mg lovastatin; a dietary portfolio high in plant sterols (1 g/4200 kJ), soya proteins (21·4 g/4200 kJ), soluble fibres (9·8 g/4200 kJ) and almonds (14 g/4200 kJ). LDL electrophoretic characteristics were measured by non-denaturing polyacrylamide gradient gel electrophoresis of fasting plasma at 0, 2 and 4 weeks of each treatment. The reductions in plasma LDL-cholesterol levels with the dietary portfolio and with statins were comparable and were largely attributable to reductions in the estimated concentration of cholesterol within the smallest subclass of LDL (portfolio − 0·69 (se 0·10) mmol/l, statin − 0·99 (se 0·10) mmol/l). These were significantly greater (P < 0·01) than changes observed after the control diet ( − 0·17 (se 0·08) mmol/l). Finally, baseline C-reactive protein levels were a significant predictor of the LDL size responsiveness to the dietary portfolio but not to the other treatments. The dietary portfolio, like the statin treatment, had only minor effects on several features of the LDL size phenotype, but the pronounced reduction in cholesterol levels within the small LDL fraction may provide additional cardiovascular benefit over the traditional low-fat diet of National Cholesterol Education Program Step II.

Studies conducted in the last 20 years have led to the identification of small dense LDL as an important risk factor for CVD. Consumption of plant sterols, soyabean proteins, viscous fibre and nuts are known to modulate the risk of CVD favourably through their cholesterol (Chol)-lowering properties, both independently and more recently in combination. Nevertheless, their combined impact on the LDL particle size phenotype has never been tested. In the present study, we assessed the effect of incorporating concurrently plant sterols (1 g/4·2 MJ), soyabean protein (23 g/4·2 MJ), viscous fibre (9 g/4·2 MJ) and almonds (15 g/4·2 MJ) into a diet very low in saturated fat in twelve patients with mildly elevated plasma LDL-Chol levels. Fasting blood lipids were obtained at the start of the study and at 2-week intervals during the 4-week study. The diet-induced reduction in plasma LDL-Chol of 30·0 (SE 3·0) % (P<0·0001) was attributed to concurrent reductions in the serum Chol concentrations of large (>26.0 nm−30 (SE 8) %, P<0·001), medium (25·5–26·0 nm−29 (SE 3) %, P<0·001) and small (<25·5 nm − 21 (SD 6) %, P<0·01) LDL particles, with near maximal reductions seen by week 2. These results indicate that foods and dietary components advocated for their potential to reduce the risk of CVD are effective in reducing serum concentrations of all LDL fractions including small dense LDL, thus potentially further contributing to an overall lower risk of CVD.

This paper is the third in a series with the general title ‘How to look at the five-dimensional shape space Σ4,3. Parts I and II were concerned with distributions and diffusions respectively. Here we ‘look at’ geodesics.

A remarkable distribution found by Mardia and Dryden for the shape of a random triangle in ℝ2 whose vertices are displaced from their initial positions by independent identical symmetrical gaussian perturbations is re-derived via stochastic calculus.

After some reminiscences about the founding of the Applied Probability journals the author discusses three open problems concerned respectively with:

(i) the prehistory of applied probability

(ii) Poisson–Delauney Simplexes

(iii) voids and pancakes in the universe.

This paper is concerned with the shape-density for a random triangle whose vertices are randomly labelled and i.i.d.-uniform in a compact convex polygon K. In earlier work we have already shown that there is a network of curves (the singular tessellation T(K)) across which suffers discontinuities of form. In two papers which will appear in parallel with this, Hui-lin Le finds explicit formulae for (i) the form of within the basic tile T0 of T(K), and (ii) the jump-functions which link the local forms of on either side of any curve separating two tiles. Here we exploit these calculations to find in the most general case. We describe the geometry of T(K), we examine the real-analytic structure of within a tile, and we establish by analytic continuation an explicit formula giving in an arbitrary tile T as the sum of the basic-tile function and the members of a finite sequence of jump-functions along a ‘stepping-stone' tile-to-tile route from T0 to T. Finally we comment on some of the problems that arise in the use of this formula in studies relating to the applications in archaeology and astronomy.

A survey is given of the attitude to and use of the concepts of deficient, perfect, and abundant number from the time of Nicomachus of Gerasa to that of David Anakht and Alcuin of York. Alcuin's ‘Letter to Dafnin', the source of an anecdote frequently mentioned in mathematical texts, is included as an appendix.

The statistics of deficient, perfect, and abundant numbers over the range 1–50000 are studied and presented graphically in several novel ways and compared with the work of Davenport and others on the distribution-law for the values of z = σ (n)/(2n).

Some queries are raised concerning observations which ancient writers might have been expected to make; for example, did they notice that about one half of all the even numbers are abundant?

Let n points in the plane be generated by some specified random mechanism and suppose that N(∊) of the resulting triads form triangles with largest angle ≧ π – ∊. The main object of the paper is to obtain asymptotic formulae for and Var (N(∊)) when ∊ ↓ 0, and to solve the associated data-analytic problem of testing whether an empirical set of n points should be considered to contain too many such ∊-blunt triads in the situation where the generating mechanism is unknown and where all that can be said about the tolerance ∊ is that it must be allowed to take values anywhere in a given interval (T0, T1) (0 < T0 < T1). This problem is solved by the introduction of a plot to be called the pontogram and by the introduction of simulation-based significance tests constructed by random lateral perturbations of the data.

After a general review of symmetric reversibility for countable-state continuous-time Markov chains the author shows that the birth-death-and-immigration process is symmetrically reversible and further that it remains so even when the description of the present state is refined to include a list of the sizes of all ‘families’ alive at the epoch in question. This result can be useful in genealogy because the operational direction of time there is the negative one. In view of the symmetric reversibility, some of the questions which face the genealogist can be answered without further calculation by quoting known results for the process with the usual (‘forward’ instead of ‘backward’) direction of time.

Further topics discussed include social mobility matrices, surname statistics, and Colin Rogers' ‘problem of the Spruces’.

1. Introduction. In a recent contribution to these Proceedings Alladi Rama-krishnan (23) has discussed a number of problems which arise when the development of a cascade shower of cosmic rays is considered from the standpoint of the theory of stochastic processes. As has often been remarked (see, for example, the important study by Niels Arley (1)), there is a close formal analogy between such physical phenomena and the growth of biological populations. In particular, if the distance of penetration (t) is identified with the time, and the energy (E) of a particle in the shower is replaced by the age (x) of an individual in the population, there emerges an obvious analogy between stochastic fluctuations in the energy spectrum of the shower and similar fluctuations in the age distribution of the population.