Multivalued generalizations of fixed point results in fuzzy metric spaces

This paper attempts to prove fixed and coincidence point results in fuzzy metric space using multivalued mappings. Altering distance function and multivalued strong {bn}-fuzzy contraction are used in order to do that. Presented theorems are generalization of some well known single valued results. Two examples are given to support the theoretical results

Nonclassical stochastic flows and continuous products

Contrary to the classical wisdom, processes with independent values (defined properly) are much more diverse than white noise combined with Poisson point processes, and product systems are much more diverse than Fock spaces. This text is a survey of recent progress in constructing and investigating nonclassical stochastic flows and continuous products of probability spaces and Hilbert spaces.Comment: A survey, 126 pages. Version 3 (final): former Question 9d4 is solved; 8a1 reformulated. Ref [41] added. For readability, sections are reordered (123456..->142536..). Cosmetic changes, mostly in 1b, 2a, 3d, (4a7) (v3 numbers) and Introductio

On Sharp Identification Regions for Regression Under Interval Data

The reliable analysis of interval data (coarsened data) is one of the most promising applications of imprecise probabilities in statistics. If one refrains from making untestable, and often materially unjustified, strong assumptions on the coarsening process, then the empirical distribution of the data is imprecise, and statistical models are, in Manski’s terms, partially identified. We first elaborate some subtle differences between two natural ways of handling interval data in the dependent variable of regression models, distinguishing between two different types of identification regions, called Sharp Marrow Region (SMR) and Sharp Collection Region (SCR) here. Focusing on the case of linear regression analysis, we then derive some fundamental geometrical properties of SMR and SCR, allowing a comparison of the regions and providing some guidelines for their canonical construction. Relying on the algebraic framework of adjunctions of two mappings between partially ordered sets, we characterize SMR as a right adjoint and as the monotone kernel of a criterion function based mapping, while SCR is indeed interpretable as the corresponding monotone hull. Finally we sketch some ideas on a compromise between SMR and SCR based on a set-domained loss function. This paper is an extended version of a shorter paper with the same title, that is conditionally accepted for publication in the Proceedings of the Eighth International Symposium on Imprecise Probability: Theories and Applications. In the present paper we added proofs and the seventh chapter with a small Monte-Carlo-Illustration, that would have made the original paper too long

Vector analysis for Dirichlet forms and quasilinear PDE and SPDE on metric measure spaces

Starting with a regular symmetric Dirichlet form on a locally compact separable metric space $X$, our paper studies elements of vector analysis, $L_p$-spaces of vector fields and related Sobolev spaces. These tools are then employed to obtain existence and uniqueness results for some quasilinear elliptic PDE and SPDE in variational form on $X$ by standard methods. For many of our results locality is not assumed, but most interesting applications involve local regular Dirichlet forms on fractal spaces such as nested fractals and Sierpinski carpets