Cranks 0, Schoolboys 1

بررسی‌های ساختاری و مغناطیسی نانوذرات‌ پروسکایت LaMn1-xCoxO3 (0/1، 75/0، 5/0، 25/0، 0/0 = x)

In this work, the structural and magnetic properties of LaMn1-xCoxO3 (x = 0.00, 0.25, 0.50, 0.75, 1.00) are investigated. The structural characterization of compounds by X-ray powder diffraction is evidence for a rhombohedral structure (R -3c space group). Much less increase of the unit cell volume suggests that Co enters in lattice as Co+2 for x ≤ 0.5 while much more decrease of the unit cell ...

Predators and cranks.

Cranks and T -cores

New statistics on partitions (called cranks) are defined which combinatorially prove Ramanujan’s congruences for the partition function modulo 5, 7, 11, and 25. Explicit bijections are given for the equinumerous crank classes. The cranks are closely related to the t-core of a partition. Using q-series, some explicit formulas are given for the number of partitions which are t-cores. Some related...

Cranks – Really, the Final Problem

A survey of Ramanujan’s work on cranks in his lost notebook is given. We give evidence that Ramanujan was concentrating on cranks when he died, that is to say, the final problem on which Ramanujan worked was cranks – not mock theta functions.

0 0 2 1 − 2 − 1 1 0 1 1 − 1 − 1 0 0 0 1 − 1

This article is to contribute to the classification of finite-dimensional complex pointed Hopf algebras with Weyl groups of E6, E7, F4, G2. Many papers are about the classification of finite dimensional pointed Hopf algebras, for example, [AS98, AS02, AS00, AS05, He06, AHS08, AG03, AFZ, AZ07, Gr00, Fa07, AF06, AF07, ZZC, ZC]. In these research ones need the centralizers and character tables of ...