We study the shrinking and separation properties (two notions well-known in descriptive set theory) for NP and coNP and show that under reasonable complexity-theoretic assumptions, both properties do not hold for NP and the shrinking property does not hold for coNP. In particular we obtain the following results. 1. NP and coNP do not have the shrinking property, unless PH is finite. In general,...

A kinetic study of cadmium and lead biosorption to a brown alga, Kjellmaniella crassiforia, was carried out. The shrinking core model derived by M. Gopala Rao and A. K. Gupta (Chem. Eng. J.24, 181, (1982)) was modified and adapted for description of the rate process of cadmium and lead biosorption to the alga. The biosorption rate process was well described and average apparent diffusion coeffi...

Shrinking cities are widespread throughout the world despite the rapidly increasing global urban population. These cities are attempting to transition to sustainable trajectories to improve the health and well-being of urban residents, to build their capacity to adapt to changing conditions and to cope with major events. The dynamics of shrinking cities are different than the dynamics of growin...

We prove that a Schauder frame for any separable Banach space is shrinking if and only if it has an associated space with a shrinking basis, and that a Schauder frame for any separable Banach space is shrinking and boundedly complete if and only if it has a reflexive associated space. To obtain these results, we prove that the upper and lower estimate theorems for finite dimensional decompositi...

Analytical calculations of the dynamics of a curved domain wall in a nematic liquid crystal are performed. The core of the wall is assumed to form a cylinder, whose axis coincides with the direction of an external magnetic field. The equation of motion for the nematic director field is solved in a comoving coordinate frame by applying a polynomial expansion of the tilt angle with respect to the...