Notion Of Tightness Research Articles

TIGHT EVENTUALLY DIFFERENT FAMILIES

AbstractGeneralizing the notion of a tight almost disjoint family, we introduce the notions of a tight eventually different family of functions in Baire space and a tight eventually different set of permutations of $\omega $ . Such sets strengthen maximality, exist under $\mathsf {MA} (\sigma \mathrm {-centered})$ and come with a properness preservation theorem. The notion of tightness also generalizes earlier work on the forcing indestructibility of maximality of families of functions. As a result we compute the cardinals $\mathfrak {a}_e$ and $\mathfrak {a}_p$ in many known models by giving explicit witnesses and therefore obtain the consistency of several constellations of cardinal characteristics of the continuum including $\mathfrak {a}_e = \mathfrak {a}_p = \mathfrak {d} < \mathfrak {a}_T$ , $\mathfrak {a}_e = \mathfrak {a}_p < \mathfrak {d} = \mathfrak {a}_T$ , $\mathfrak {a}_e = \mathfrak {a}_p =\mathfrak {i} < \mathfrak {u}$ , and $\mathfrak {a}_e=\mathfrak {a}_p = \mathfrak {a} < non(\mathcal N) = cof(\mathcal N)$ . We also show that there are $\Pi ^1_1$ tight eventually different families and tight eventually different sets of permutations in L thus obtaining the above inequalities alongside $\Pi ^1_1$ witnesses for $\mathfrak {a}_e = \mathfrak {a}_p = \aleph _1$ .Moreover, we prove that tight eventually different families are Cohen indestructible and are never analytic.

Extensions of a tight function and their continuity in quantum logic

In the present paper, a new variation of a nonnegative real-valued function $$\rho $$ defined on a subfamily of a quantum logic P is proposed, and the notion of tightness of $$\rho $$ is studied. Various crucial results are proved, and subsequently we have obtained extensions, viz. f-extension, $$\delta $$ -extension and $$\sigma $$ -extension of a tight function $$\rho ;$$ continuity of extensions of $$\rho $$ with respect to an approximating family in P is discussed.

On the representation of tight functionals as integrals

We consider a vector lattice \(\mathcal L \) of bounded real continuous functions on a topological space \(X\) that separates the points of \(X\) and contains the constant functions. A notion of tightness for linear functionals is defined, and by an elementary argument we prove with the aid of the classical Riesz representation theorem that every tight continuous linear functional on \(\mathcal L \) can be represented by integration with respect to a Radon measure. This result leads incidentally to an simple proof of Prokhorov’s existence theorem for the limit of a projective system of Radon measures.

Horo-tightness and total (absolute) curvatures in hyperbolic spaces

We prove Gaus-Bonnet-type and Chern-Lashof-type formulas for immersions in hyperbolic space. Moreover we investigate the notion of tightness with respect to horospheres introduced by T.E. Cecil and P.J. Ryan. We introduce the notions of top-set and drop-set, and we prove fundamental properties of horo-tightness in hyperbolic spaces.

A characterization of answer sets for logic programs

Checking if a program has an answer set, and if so, compute its answer sets are just some of the important problems in answer set logic programming. Solving these problems using Gelfond and Lifschitz’s original definition of answer sets is not an easy task. Alternative characterizations of answer sets for nested logic programs by Erdem and Lifschitz, Lee and Lifschitz, and You et al. are based on the completion semantics and various notions of tightness. However, the notion of tightness is a local notion in the sense that for different answer sets there are, in general, different level mappings capturing their tightness. This makes it hard to be used in the design of algorithms for computing answer sets. This paper proposes a characterization of answer sets based on sets of generating rules. From this characterization new algorithms are derived for computing answer sets and for performing some other reasoning tasks. As an application of the characterization a sufficient and necessary condition for the equivalence between answer set semantics and completion semantics has been proven, and a basic theorem is shown on computing answer sets for nested logic programs based on an extended notion of loop formulas. These results on tightness and loop formulas are more general than that in You and Lin’s work.

Tight and Untight Triangulations of Surfaces by Complete Graphs

Triangular embeddings of complete graphs into surfaces are studied through the notion of tightness which is a natural combinatorial generalization of connectedness for graphs. By means of a construction which "couples" two such surfaces to produce a new one, the existence of untight complete triangular embeddings is proved and the known archive of tight ones is broadened. In particular, K30 admits a tight and an untight triangular embedding into the same surface. Therefore, complete graphs may triangulate the same surface in nonisomorphic ways.