Approximations from Anywhere and General Rough Sets

Not all approximations arise from information systems. The problem of fitting approximations, subjected to some rules (and related data), to information systems in a rough scheme of things is known as the \emph{inverse problem}. The inverse problem is more general than the duality (or abstract representation) problems and was introduced by the present author in her earlier papers. From the practical perspective, a few (as opposed to one) theoretical frameworks may be suitable for formulating the problem itself. \emph{Granular operator spaces} have been recently introduced and investigated by the present author in her recent work in the context of antichain based and dialectical semantics for general rough sets. The nature of the inverse problem is examined from number-theoretic and combinatorial perspectives in a higher order variant of granular operator spaces and some necessary conditions are proved. The results and the novel approach would be useful in a number of unsupervised and semi supervised learning contexts and algorithms.Comment: 20 Pages. Scheduled to appear in IJCRS'2017 LNCS Proceedings, Springe

Rough paths in idealized financial markets

This paper considers possible price paths of a financial security in an idealized market. Its main result is that the variation index of typical price paths is at most 2, in this sense, typical price paths are not rougher than typical paths of Brownian motion. We do not make any stochastic assumptions and only assume that the price path is positive and right-continuous. The qualification "typical" means that there is a trading strategy (constructed explicitly in the proof) that risks only one monetary unit but brings infinite capital when the variation index of the realized price path exceeds 2. The paper also reviews some known results for continuous price paths and lists several open problems.Comment: 21 pages, this version adds (in Appendix C) a reference to new results in the foundations of game-theoretic probability based on Hardin and Taylor's work on hat puzzle

Statistics of the separation between sliding rigid rough surfaces: Simulations and extreme value theory approach

When a rigid rough solid slides on a rigid rough surface, it experiences a random motion in the direction normal to the average contact plane. Here, through simulations of the separation at single-point contact between self-affine topographies, we characterize the statistical and spectral properties of this normal motion. In particular, its rms amplitude is much smaller than that of the equivalent roughness of the two topographies, and depends on the ratio of the slider's lateral size over a characteristic wavelength of the topography. In addition, due to the non-linearity of the sliding contact process, the normal motion's spectrum contains wavelengths smaller than the smallest wavelength present in the underlying topographies. We show that the statistical properties of the normal motion's amplitude are well captured by a simple analytic model based on the extreme value theory framework, extending its applicability to sliding-contact-related topics