We consider a class of one dimensional vector Non-linear Schr$\ddot{o}$dinger Equation(NLSE) in an external complex potential with Balanced Loss-Gain(BLG) and Linear Coupling(LC) among the components field. The solvability generic system is investigated for various combinations time modulated LC BLG terms, space-time dependent strength nonlinear interaction potential. use non-unitary transforma...

The notion of linear-time computability is very sensitive to machine model. In this connection, we introduce a class NLT of functions computable in nearly linear time n(log n)O(1) on random access computers. NLT is very robust and does not depend on the particular choice of random access computers. Kolmogorov machines, Schonhage machines, random access Turing machines, etc. also compute exactl...

The notion of properness, well-known for constant linear systems, is generalized to the time-varying case. In a module theoretic framework intrinsic characterizations of input-output system properness are derived. For this, ltrations and the associated Hilbert polyno-mials are used. Classical criteria for properness of polynomial and state representations for constant linear systems are general...

By now it is moderately well understood how to take as input a set of points that lie on an unknown manifold, and produce as output a piecewise linear approximation to the manifold. The question I explore here is: how fast can we do it? Previous solutions achieve a sequential runtime of O(n log n) in low ambient dimensions. I show that if the points are specified in floating point coordinates, ...