Let E be an ideal of L0 over a σ-finite measure space (Ω,Σ, μ), and let E∼ stand for the order dual of E. For a real Banach space (X, ‖ · ‖X ) let E(X) be a subspace of the space L0(X) of μ-equivalence classes of strongly Σ-measurable functions f : Ω −→ X and consisting of all those f ∈ L0(X) for which the scalar function ‖f(·)‖X belongs to E. For a real Banach space (Y, ‖ · ‖Y ) a linear opera...

In this paper, the notions of anti-reflexive kernel, symmetric kernel, reflexive closure, and symmetric closure of a soft set relation are first introduced, respectively. Then, their accurate calculation formulae and some properties are obtained. Finally, soft set relation mappings and inverse soft set relationmappings are proposed, and some related properties are discussed. © 2010 Elsevier Ltd...

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...

Given a subspace S of operators on a Hilbert space, and given two operators X and Y (not necessarily in S), when can we be certain that there is an operator A in S such that AX = Y ? If there is one, can we find some bound for its norm? These questions are the subject of a number of papers, some by the present authors, and mostly restricted to the case where S is a reflexive algebra. In this pa...

We provide and analyze a based average for two convex functions, based on a kernel function. It covers several known averages such as the arithmetic average, epigraphical average, and the proximal average. When applied to the Fitzpatrick function and the conjugate of Fitzpatrick function associated with a monotone operator, our average produces an autoconjugate (also known as selfdual Lagrangia...

We exhibit an aggressive optimizing compiler for a functionalprogramming language which includes a first-class forward automatic differentiation (AD) operator. The compiler’s performance is competitive with FORTRAN-based systems on our numerical examples, despite the potential inefficiencies entailed by support of a functional-programming language and a first-class AD operator. These results ar...

Let R be a commutative Noetherian ring. The k-torsionless modules are defined in [7] as a generalization of torsionless and reflexive modules, i.e., torsionless modules are 1-torsionless and reflexive modules are 2-torsionless. Some properties of torsionless, reflexive, and k-torsionless modules are investigated in this paper. It is proved that if M is an R-module such that G-dimR(M)

Every nilpotent operator on a complex Hilbert space is shown to be quasi-similar to a canonical Jordan model. Further, the para-reflexive operators are characterized generalizing a result of Deddens and Fillmore. A familiar result states that each nilpotent operator on a finite dimensional complex Hubert space is similar to its adjoint. One proof proceeds by showing that both a nilpotent operat...