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

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Purpose. Female runners are known to be at greater risk from chronic running injuries than age-matched males, although the exact mechanisms are often poorly understood. The aim of the current investigation was to determine if female recreational runners exhibit distinct limb and joint stiffness characteristics in relation to their male counterparts. Methods. Fourteen male and fourteen female runners ran over a force platform at 4.0 m · s -1 . Lower limb kinematics were collected using an eight-camera optoelectric motion capture system operating at 250 Hz. Measures of limb and joint stiffness were calculated as a function of limb length and joint moments divided by the extent of limb and joint excursion. All stiffness and joint moment parameters were normalized to body mass. Sex differences in normalized limb and knee and ankle joint stiffness were examined statistically using independent samples t tests. Results. The results indicate that normalized limb (male = 0.18 ± 0.07, female = 0.37 ± 0.10 kN · kg · m -1 ) and knee stiffness (male = 5.59 ± 2.02, female = 7.34 ± 1.78 Nm · kg · rad -1 ) were significantly greater in female runners. Conclusions. On the basis that normalized knee and limb stiffness were shown to be significantly greater in female runners, the findings from the current investigation may provide further insight into the aetiology of the distinct injury patterns observed between sexes.

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