In 1944, Freeman Dyson conjectured the existence of a "crank" function for partitions that would provide a combinatorial proof of Ramanujan's congruence modulo 11. Forty years later, Andrews and Garvan successfully found such a function and proved the celebrated result that the crank simultaneously "explains" the three Ramanujan congruences modulo 5, 7, and 11. This note announces the proof of ...

Higher moments of the partition rank and crank statistics have been studied for their connections to combinatorial objects such as Durfee symbols, as well as for their connections to harmonic Maass forms. This paper proves the first several cases of (and strengthens) a conjecture due to Garvan, which states that the moments of the crank function are always larger than the moments of the crank f...

The objective of this research was to use a pedal force decomposition approach to quantify the amount of negative muscular crank torque generated by a group of competitive cyclists across a range of pedaling rates. We hypothesized that negative muscular crank torque increases at high pedaling rates as a result of the activation dynamics associated with muscle force development and the need for ...

During lower limb exercise, blood flow through the resting upper limbs exhibits a change characterized by increased anterograde flow during systole, but also large increases in retrograde diastolic flow. One explanation for the retrograde flow is that increased sympathetic nervous system (SNS) tone and concomitant increased peripheral resistance generate a rebound during diastole. To examine wh...

For its first release in 2004, CRANK was shown to effectively detect and phase anomalous scatterers from single-wavelength anomalous diffraction data. Since then, CRANK has been significantly improved and many more structures can be built automatically with single- or multiple-wavelength anomalous diffraction or single isomorphous replacement with anomalous scattering data. Here, the new algori...

In the present investigation heat transfer character due to viscous dissipation and magnetic field applied to unsteady boundary layer fluid flow past a stretching sheet is investigated. The equations governing the unsteady boundary layer fluid flow are constructed and non-dimensionalized. The non-dimensional equations are discretized using implicit finite difference method of Crank-Nicolson typ...