Extended Legendre numbers of the first kind

The Legendre numbers are defined from the associated Legendre functions and three equivalent general forms are given for them. From one of these forms the Legendre numbers P ^m _n are extended to those of the forms P^‐m _n and P^‐m _‐n Several properties of these extended Legendre numbers are given and a combined table shows many properties common to all of the numbers.     

A generalization of triangular numbers

The triangular numbers are generalized in a natural way and a discovery‐type approach is used to find properties of these numbers. The numbers are connected to the Pascal numbers and generalized combinations that arise are connected to the Stirling numbers of the first kind.     

SQUID-based measuring systems

A program has been developed and initiated at the Indira Gandhi Centre for Atomic Research (IGCAR) for the utilization of SQUID sensors in various application areas. DC SQUID sensors based on Nb-AlO _x -Nb Josephson junctions have been designed and developed inhouse along with associated flux-locked loop (FLL) electronics. A compact low field SQUID magnetometer insertible in a liquid helium storage dewar has also been developed inhouse and is in use. Efforts to build a high field SQUID magnetometer, SQUID-DAC system, are in progress. A planar gradiometric DC SQUID sensor for non-destructive evaluation (NDE) application to be used in relatively unshielded environment has been designed and developed. An easily portable NDE cryostat with a small lift-off distance, to be used in external locations has been designed and tested. The magnetic field produced by a given two-dimensional current density distribution is inverted using the Fourier transform technique.     

Nanoelectromechanical systems

Nanoelectromechanical systems (NEMS) are nano-to-micrometer scale mechanical resonators coupled to electronic devices of similar dimensions. NEMS show promise for fast, ultrasensitive force microscopy and for deepening our understanding of how classical dynamics arises by approximation to quantum dynamics. This article begins with a survey of NEMS and then describes certain aspects of their classical dynamics. In particular, we show that for weak coupling the action of the electronic device on the mechanical resonator can be effectively that of a thermal bath, this despite the device being a driven, far-from-equilibrium system.