T he achievement of Karl Mahlburg in this issue of PNAS (1) adds a lustrous chapter to a unique mathematical object: the crank. In 1944, the crank was first hinted at by Freeman Dyson (2), then an undergraduate at Cambridge University. He had written an article, titled Some Guesses in the Theory of Partitions, for Eureka, the undergraduate mathematics journal of Cambridge. Dyson discovered the ...

This paper describes a method for defining the maximum torque which can be produced at a joint from isovelocity torque measurements on an individual. The method is applied to an elite male gymnast in order to calculate subject specific joint torque parameters for the knee joint. Isovelocity knee extension torque data were collected for the gymnast using a two repetition concentric-eccentric pro...

Let N(≤ m,n) denote the number of partitions of n with rank not greater than m, and let M(≤ m,n) denote the number of partitions of n with crank not greater than m. Bringmann and Mahlburg observed that N(≤ m,n) ≤ M(≤ m,n) ≤ N(≤ m + 1, n) for m < 0 and 1 ≤ n ≤ 100. They also pointed out that these inequalities can be restated as the existence of a re-ordering τn on the set of partitions of n suc...

BACKGROUND The reported accuracy of clinical tests for superior labral anterior posterior lesions is extremely variable. Pooling results from multiple studies of higher quality is necessary to establish the best clinical tests to use. HYPOTHESIS Certain clinical tests are superior to others for diagnosing the presence or absence of a superior labral anterior posterior lesion. STUDY DESIGN M...

Andrews’ spt-function can be written as the difference between the second symmetrized crank and rank moment functions. Using the machinery of Bailey pairs a combinatorial interpretation is given for the difference between higher order symmetrized crank and rank moment functions. This implies an inequality between crank and rank moments that was only known previously for sufficiently large n and...

In this paper we present a posteriori error estimators for the approximate solutions of linear parabolic equations. We consider discretizations of the problem by discontinuous Galerkin method in time corresponding to variant Crank-Nicolson schemes and continuous Galerkin method in space. Especially, £nite element spaces are permitted to change at different time levels. Exploiting Crank-Nicolson...

The Burgers’ equation is a simplified form of the Navier-Stokes equations that very well represents their non-linear features. In this paper, numerical methods of the Adomian decomposition and the Modified Crank – Nicholson, used for solving the one-dimensional Burgers’ equation, have been compared. These numerical methods have also been compared with the analytical method. In contrast to...

The aim of this study was to evaluate the impact of a novel crank system on laboratory time-trial cycling performance. The Rotor system makes each pedal independent from the other so that the cranks are no longer fixed at 180°. Twelve male competitive but non-elite cyclists (mean ± s: 35 ± 7 yr, Wmax = 363 ± 38 W, VO2peak = 4.5 ± 0.3 L·min(-1)) completed 6-weeks of their normal training using e...

It is shown that A-acceptable and, more generally, A()-acceptablerational approximations of bounded analytic semigroups in Banach space are stable. The result applies, in particular, to the Crank-Nicolson method.