Hull Class Research Articles

Hull classes in compact regular frames

Hull classes in compact regular frames

Bounded Equivalence of Hull Classes in Archimedean Lattice-Ordered Groups with Unit

A hull class in a category is an object class H for which each object has a unique minimal essential extension in H. This paper addresses the enormity of the collection of hull classes in the category W of Archimedean l-groups with distinguished weak order unit through consideration of the action on the hull classes of the bounded coreflection \(\textbf {W} \overset {B}\rightarrow \textbf {W}^{\ast }\) onto the subcategory where the units are strong. It is shown that hull classes go forth under B and back under B−1, that the B-equivalence class of a hull class in W always has a top, and that these B-equivalence classes are frequently not sets. The property “top” is related to various other properties that hull classes might have. This paper is the third by us on the complex taxonomy of hull classes in W, and more are planned.

B-saturated Hull Classes in ℓ-groups and Covering Classes of Spaces

W denotes the category of archimedean l-groups with designated weak unit and l-homomorphisms that preserve the weak unit, and B is the bounded coreflection in W. Comp denotes the category of compact Hausdorff spaces with continuous maps, and Y : W → Comp is the familiar Yosida functor. The enormous collection hcW of hull classes in W and the somewhat less enormous collection ccComp of covering classes in Comp are clearly related “via” Y, but rather unclearly in the details. In an earlier paper we investigated the relationship between hcW and ccComp and continue to do so here, now focusing on the role of B. Among other things, (i) we define B-saturated hull classes and the sub-species Y-saturated and type μ, (ii) show that for a hull class H of the latter two types, but not always the first, Y[H] is a covering class, and (iii) describe the various ways the three types relate. This paper is the second installment in our ongoing investigation of the complex taxonomy of hull classes.

Various disconnectivities of spaces and projectabilities of ℓ-groups

Arch denotes the category of archimedean l-groups and l-homomorphisms. Tych denotes the category of Tychonoff spaces with continuous maps, and α denotes an infinite cardinal or ∞. This work introduces the concept of an αcc-disconnected space and demonstrates that the class of αcc-disconnected spaces forms a covering class in Tych. On the algebraic side, we introduce the concept of an αcc-projectable l-group and demonstrate that the class of αcc-projectable l-groups forms a hull class in Arch. In addition, we characterize the αcc-projectable objects in W—the category of Arch-objects with designated weak unit and l-homomorphisms that preserve the weak unit—and construct the αcc-hull for G in W. Lastly, we apply our results to negatively answer the question of whether every hull class (resp., covering class) is epireflective (resp., monocoreflective) in the category of W-objects with complete l-homomorphisms (resp., the category of compact Hausdorff spaces with skeletal maps).

On hull classes of ℓ-groups and covering classes of spaces

Abstract W denotes the category of archimedean ℓ-groups with designated weak unit and ℓ-homomorphisms that preserve the weak unit. Comp denotes the category of compact Hausdorff spaces with continuous maps. The Yosida functor is used to investigate the relationship between hull classes in W and covering classes in Comp. The central idea is that of a hull class whose hull operator preserves boundedness. We demonstrate how the Yosida functor may be used to identify hull classes in W and covering classes in Comp. In addition, we exhibit an array of order preserving bijections between certain families of hull classes and all covering classes, one of which was recently produced by Martínez. Lastly, we apply our results to answer a question of Knox and McGovern about the class of all feebly projectable ℓ-groups.

Functorial polar functions

Abstract W∞ denotes the category of archimedean ℓ-groups with designated weak unit and complete ℓ-homomorphisms that preserve the weak unit. CmpT2,∞ denotes the category of compact Hausdorff spaces with continuous skeletal maps. This work introduces the concept of a functorial polar function on W∞ and its dual a functorial covering function on CmpT2,∞. We demonstrate that functorial polar functions give rise to reflective hull classes in W ∞ and that functorial covering functions give rise to coreflective covering classes in CmpT 2,∞. We generate a variety of reflective and coreflecitve subcategories and prove that for any regular uncountable cardinal α, the class of α-projectable ℓ-groups is reflective in W ∞, and the class of α-disconnected compact Hausdorff spaces is coreflective in CmpT 2,∞. Lastly, the notion of a functorial polar function (resp. functorial covering function) is generalized to sublattices of polars (resp. sublattices of regular closed sets).

Prediction of recursive convex hull class assignments for protein residues

We introduce a new method for designating the location of residues in folded protein structures based on the recursive convex hull (RCH) of a point set of atomic coordinates. The RCH can be calculated with an efficient and parameterless algorithm. We show that residue RCH class contains information complementary to widely studied measures such as solvent accessibility (SA), residue depth (RD) and to the distance of residues from the centroid of the chain, the residues' exposure (Exp). RCH is more conserved for related structures across folds and correlates better with changes in thermal stability of mutants than the other measures. Further, we assess the predictability of these measures using three types of machine-learning technique: decision trees (C4.5), Naive Bayes and Learning Classifier Systems (LCS) showing that RCH is more easily predicted than the other measures. As an exemplar application of predicted RCH class (in combination with other measures), we show that RCH is potentially helpful in improving prediction of residue contact numbers (CN).

A Statistical Examination of Sea Valve Failure Rates

AbstractThis paper presents the results of a study that targeted identification of statistical methods to characterize the failure rates and related probability distributions of sea valves across a broad range of hull classes. Beginning with maintenance records from sea valve assessment, the paper demonstrates the application of various statistical tools and the related strengths and weaknesses of each. These tools involve maximum likelihood estimators, empirical methods, and hypothesized distributions. Using these methods, failure rates and confidence intervals are estimated and compared. Finally this information is combined to formulate a cohesive view that allows a high level of confidence in predicting sea valve failure. This paper demonstrates use of statistical models as a tool to validate and improve maintenance systems, costs, and strategic maintenance management programs.

Polar functions—II: completion classes of archimedean f-algebras vs. covers of compact spaces

There is an inclusion preserving bijection between the class of completion classes of uniformly complete real f-algebras with identity and the partially ordered class of covering classes of compact Hausdorff spaces. In this setting a completion class A is a hull class of uniformly complete f-algebras, with the additional feature that G∈ A if and only if G ∗∈ A . Using an idempotent invariant polar function X and the covering function K derived from it, the main theorem of this article states that the covering class associated with the uniformly complete f-algebras having no proper X -splitting extensions is the class of compact spaces X which equal their K -cover.

Weakly least integer closed groups

An archimedean lattice-ordered groupA with distinguished weak unit has the canonical Yosida representation as an l-group of extended real-valued functions on a certain compact Hausdorff spaceY A. Such an l-groupA is calledleast integer closed, orLIC (resp.,weakly least integer closed, orwLIC) if, in the representation,a ∈A implies [a] ∈A (resp., there isa′ ∈A witha′=[a] on a dense set inY A), where [r] ≡ the least integer greater than or equal tor. Earlier, we have studiedLIC groups, with an emphasis on their a-extensions. Here, we turn towLIC groups: we give an intrinsic (though awk-ward) characterization in terms of existence of certain countable suprema. This results also in an intrinsic characterization ofLIC, previously lacking. Also,wLIC is a hull class (whichLIC is not), and the hullwlA is “somewhere near” the projectable hullpA. The best comparison comes from a (somewhat novel) factoringpA=loc(wpA), wherewpA is the “weakly projectable” hull (defined here), andlocB is the “local monoreflection”; then,wpA≤wlA≤loc(wpA), andpA≤loc(wlA), while with a strong unit, all these coincide. Numerous examples and special cases are examined.

A family of humped volatility models

Recent empirical studies on interest rate derivatives have shown that the volatility structure of interest rates is frequently humped. Several researchers have modelled interest rate dynamics in such a way that humped volatility structures are possible and yet analytical formulas for European options on discount bonds are derived. However, these models are Gaussian, and hence interest rates may become negative. Here, a family of interest rate models is proposed where (i) humped volatility structures are possible; (ii) the interest rate volatility may depend on the level of the interest rates themselves; and (iii) the valuation of interest rate derivative securities can be accomplished through recombining lattices. The second item implies that a number of probability distributions are possible for the yield curve dynamics, and some of them ensure that interest rates remain positive. Proportional models of the Ritchken and Sankarasubramanian type and the Black and Karasinski model are proposed. To ensure computational tractability the embedding of all models in this paper in either the Ritchken and Sankarasubramanian or the Hull and White class of models is demonstrated.