Classical structure of rough set theory was first formulated by Z. Pawlak in [6]. The foundation of its object classification is an equivalence binary relation and equivalence classes. The upper and lower approximation operations are two core notions in rough set theory. They can also be seenas a closure operator and an interior operator of the topology induced by an equivalence relation on a u...

THEOREM 4. Let a be an arbitrarily often differentiable function on the real line having countably many isolated zeros s1, S2, . . . offinite orders ml, M2, ..., respectively. Denote by A the multiplication operator mapping x into Ax = ax and let E be any linear space satisfying X = E (3 R*. Then there is a unique continuous linear op rator A-l :X -X whose range is orthogonal to E and is such t...

Both electroencephalography (EEG) and magnetoencephalography (MEG) are currently used to localize brain activity. The accuracy of source localization depends on numerous factors, including the specific inverse approach and source model, fundamental differences in EEG and MEG data, and the accuracy of the volume conductor model of the head (i.e., the forward model). Using Monte Carlo simulations...

Dynamics with noncommutative coordinates invariant under three dimensional rotations or, if time is included, under Lorentz transformations is developed. These coordinates turn out to be the boost operators in SO(1, 3) or in SO(2, 3) respectively. The noncommutativity is governed by a mass parameter M . The principal results are: (i) a modification of the Heisenberg algebra for distances smalle...

We reconsider Chern-Simons gauge theory on a Seifert manifold M (the total space of a nontrivial circle bundle over a Riemann surface Σ). When M is a Seifert manifold, Lawrence and Rozansky have shown from the exact solution of Chern-Simons theory that the partition function has a remarkably simple structure and can be rewritten entirely as a sum of local contributions from the flat connections...

we introduce a new concept of general $g$-$eta$-monotone operator generalizing the general $(h,eta)$-monotone operator cite{arvar2, arvar1}, general $h-$ monotone operator cite{xiahuang} in banach spaces, and also generalizing $g$-$eta$-monotone operator cite{zhang}, $(a, eta)$-monotone operator cite{verma2}, $a$-monotone operator cite{verma0}, $(h, eta)$-monotone operator cite{fanghuang}, $h$-...