We discuss the question of which properties of smallness in the sense of measure and category (e.g. being a universally null, perfectly meager or strongly null set) imply the properties of smallness related to some tree forcing notions (e.g. the properties of being Laver-null or Miller-null). 1. Notation and terminology Most of the notation used in this paper is standard. The symbol ω denotes t...

It is shown to be consistent with set theory that every set of reals of size א1 is null yet there are א1 planes in Euclidean 3-space whose union is not null. Similar results are obtained for circles in the plane as well as other geometric objects. The proof relies on results from harmonic analysis about the boundedness of certain maximal operators and a measure theoretic pigeonhole principle.

This paper is mainly concerned with scrambled sets for Luroth map. It shown that all have null Lebesgue measure and there exists a set full Hausdorff dimension.

We present some properties regarding diffusion and semi-convexity for fuzzy set multifunctions defined on a ring of subsets of an abstract nonvoid space and taking values in the family of all nonvoid closed subsets of a real normed space. Key–Words: diffused, semi-convex, fuzzy set multifunction, non-atomicity, Darboux property, regular, multimeasure, null-additive.

Let (X,Y ) be a random variable consisting of an observed feature vector X ∈ X and an unobserved class label Y ∈ {1, 2, . . . , L} with unknown joint distribution. In addition, let D be a training data set consisting of n completely observed independent copies of (X,Y ). Usual classification procedures provide point predictors (classifiers) Ŷ (X,D) of Y or estimate the conditional distribution ...

The generalized fuzzy valued θ-Choquet integrals will be established for the given μ-integrable fuzzy valued functions on a general fuzzy measure space, and the convergence theorems of this kind of fuzzy valued integral are being discussed. Furthermore, the whole of integrals is regarded as a fuzzy valued set function on measurable space, the double-null asymptotic additivity and pseudo-double-...

We consider the control problem for generalized heat equation a Schroedinger operator on domain with reflection symmetry respect to hyperplane. show that if this system is null-controllable, then so its respective parts and corresponding cost does not exceed one whole domain. As an application, we obtain null-controllability results half-spaces, orthants, sectors of angle $\pi/2$. byproduct, al...

For a strongly inacessible cardinal $\kappa$, we investigate the relationships between following ideals: - ideal of meager sets in ${<}\kappa$-box product topology "null" sense [Sh:1004] (arXiv:1202.5799) nowhere stationary subsets (naturally defined) set $S_{\rm pr}^\kappa \subseteq \kappa$. In particular, analyse provable inequalities characteristics for these ideals, and give consistency res...

Functional dependencies are an integral part of database design. However, they are only defined when we exclude null markers. Yet we commonly use null markers in practice. To bridge this gap between theory and practice, researchers have proposed definitions of functional dependencies over relations with null markers. Though sound, these definitions lack some qualities that we find desirable. Fo...