The purpose of this note is to compare the rings of continuous functions, integer-valued or real-valued, in pointfree topology with those in classical topology. To this end, it first characterizes the Boolean frames (= complete Boolean algebras) whose function rings are isomorphic to a classical one and then employs this to exhibit a large class of frames for which the functions rings are not o...

A benchmarking procedure ranks real-valued acts by the probability that they outperform a benchmark β; that is, an act f is evaluated by means of the functional V (f) = P (f ≥ β). Expected utility is a special case of benchmarking procedure, where the acts and the benchmark are stochastically independent. This paper provides axiomatic characterizations of preference relations that are represent...

in this paper, we first present a new important property for bouligand tangent cone (contingent cone) of a star-shaped set. we then establish optimality conditions for pareto minima and proper ideal efficiencies in nonsmooth vector optimization problems by means of bouligand tangent cone of image set, where the objective is generalized cone convex set-valued map, in general real normed spaces.

In 1934, H. Whitney asked how one can determine whether a real-valued function on a closed subset of Rn is the restriction of a Cm-function on Rn. A complete answer to this question was found much later by C. Fefferman in the early 2000s. Here, we work in an o-minimal expansion of a real closed field and solve the C1-case of Whitney’s Extension Problem in this context. Our main tool is a defina...

The negative database (NDB) is the negative representation of original data. Existing work has demonstrated that NDB can be used to preserve privacy and hide information. However, most work about NDB is based on binary representation. In some applications which are naturally descripted in real-valued space, the binary negative database is hard to be applied appropriately. Therefore, the real-va...

Braman [1] described a construction where third-order tensors are exactly the set of linear transformations acting on the set of matrices with vectors as scalars. This extends the familiar notion that matrices form the set of all linear transformations over vectors with real-valued scalars. This result is based upon a circulant-based tensor multiplication due to Kilmer et al. [4]. In this work,...

Let f be a real-valued real analytic function in several variables. Then we associate with each algebraic local cohomology class u with support in f = 0 a distribution (generalized function) ρ(u) in terms of the residue of fλ + with respect to λ at a negative integer. Then ρ constitutes a homomorphism of modules over the sheaf of analtyic functions but not over the sheaf of differential operato...

In this note we present a new treatment of the pointfree version of the semicontinuous quasi-uniformity based on the new tool of the ring of arbitrary (not necessarily continuous) real-valued functions made available recently by J. Gutiérrez Garćıa, T. Kubiak and J. Picado [Localic real functions: a general setting, Journal of Pure and Applied Algebra 213 (2009) 1064-1074]. The purpose is to sh...

Cylindrical Wiener processes in real separable Banach spaces are defined, and an approximation theorem involving scalar Wiener processes is given for such processes. A weak stochastic integral for Banach spaces involving a cylindrical Wiener process as integrator and an operator-valued stochastic process as integrand is defined. Basic properties of this integral are stated and proved. A class o...