In this paper we establish function field versions of two classical conjectures on prime numbers. The first says that the number of primes in intervals (x,x+x^epsilon] is about x^epsilon/log x and the second says that the number of primes p1 and m\geq 3 if q is even and deg f' \leq 1. We show that this estimation fails in the neglected cases. Let \pi_q(k) be the number of monic prime polynomials of degree k with coefficients in the finite field with q elements \FF_q. For relatively prime polynomials f,D\in \FF_q[t] we prove that the number N' of monic prime polynomials g that are congruent to f modulo D and of degree k satisfies |N'-\pi_q(k)/\phi(D)|\leq c(k)\pi_q(k)q^{-1/2}/\phi(D), as long as 1\leq \deg D\leq k-3 (or \leq k-4 if p=2 and (f/D)' is constant). We also generalize these results to other factorization types.Comment: Changes in the introduction, accepted to Duk

Abstract. A finitely generated subgroup Γ of a real Lie group G is said to be Diophantine if there is β > 0 such that non-trivial elements in the word ball BΓ(n) centered at 1 ∈ Γ never approach the identity of G closer than |BΓ(n)| −β . A Lie group G is said to be Diophantine if for every k ≥ 1 a random k-tuple in G generates a Diophantine subgroup. Semi-simple Lie groups are conjectured to be Diophantine but very little is proven in this direction. We give a characterization of Diophantine nilpotent Lie groups in terms of the ideal of laws of their Lie algebra. In particular we show that nilpotent Lie groups of class at most 5, or derived length at most 2, as well as rational nilpotent Lie groups are Diophantine. We also find that there are non Diophantine nilpotent and solvable (non nilpotent) Lie groups.

Abstract. We study the general problem of extremality for metric diophantine approximation on submanifolds of matrices. We formulate a criterion for extremality in terms of a certain family of algebraic obstructions and show that it is sharp. In general the almost sure diophantine exponent of a submanifold is shown to depend only on its Zariski closure, and when the latter is defined over Q, we prove that the exponent is rational and give a method to effectively compute it. This method is applied to a number of cases of interest. In particular we prove that the diophantine exponent of rational nilpotent Lie groups exists and is a rational number, which we determine explicitly in terms of representation theoretic data.

Serum lipids were measured in children and their parents from 40 families in which the father had a myocardial infarction or coronary heart disease (CHD) before age 40 years. The relationship between physical activity and serum lipid concentrations in the children was also evaluated.Twenty six men had one or more abnormal lipid value (in mmol/1): total venous cholesterol (TVC) >6-24, triglycerides >2-55, low density lipoprotein cholesterol (LDL-C) >4-42, or high density lipoprotein cholesterol (HDL-C) <0 91. There were 15 spouses with significant hyperlipidaemia (values above). In the 107 children examined, TVC mean (SD) was 4*68 (1-17), triglycerides 1-4 (0.8), LDL-C 3 0 (1.0), and HDL-C 1-18 (0.28).Altogether 42% ofthe children had significant hyperlipidaemia. No significant correlation was found between the degree of physical activity of the children and their LDL-C and TVC concentrations. However, a significant positive correlation was found between the degree of physical activity and HDL-C and a significant negative one with triglyceride concentrations.It is concluded that screening the progeny of young CHD patients is highly productive in identifying young people at excessive risk for future CHD. The data also suggest that promoting high degrees of activity among these children may have a positive influence on risk factors for adult onset CHD. (Arch Dis Child 1995; 73: 62-65) Keywords: coronary heart disease, lipoprotein screening, parental history.Earlier reports have shown that children whose parents had a myocardial infarction at an early age may demonstrate serum lipoprotein abnormalities.' 2 Increased total cholesterol and low density lipoprotein cholesterol (LDL-C) concentrations are found to aggregate within certain families in which children exhibit hypercholesterolaemia.3 Increased cholesterol concentrations in childhood are often predictive of those in adult life.4 To detect children at high risk, in 1985, the National Institute of Health, USA, recommended a blood cholesterol test for children whose parents, grandparents, or first degree relatives had hypercholesterolaemia or premature coronary heart disease (CHD) (<50 years of age in men, <60 years of age in women).5 Many surveys examining the lipoproteins in the progeny of young men with CHD have followed this recommendation.' 2 6-8A relationship between physical activity and increased concentrations of high density lipoprotein cholesterol (HDL-C) in both young9 and middle aged men has been demonstrated.'0 However, similar studies performed in young children show conflicting data." 12The present investigation was undertaken to study the lipoprotein fractions in children whose fathers had myocardial infarction or CHD before the age of 40 years. The lipoprotein fractions were analysed in the parents as well. The relationship between physical activity and serum lipid concentrations in these children was also evaluated. Subjects and methodsChildren from 40 families in which the father had a myocardial infarction or CHD before the age of ...