A Mathematical Model for Simplifying Representations of Objects in a Geographic Information System

The study of operations on representations of objects is well documented in the realm of spatial engineering. However, the mathematical structure and formal proof of these operational phenomena are not thoroughly explored. Other works have often focused on query-based models that seek to order classes and instances of objects in the form of semantic hierarchies or graphs. In some models, nodes of graphs represent objects and are connected by edges that represent different types of coarsening operators. This work, however, studies how the coarsening operator "simplification" can manipulate partitions of finite sets, independent from objects and their attributes. Partitions that are "simplified first have a collection of elements filtered (removed), and then the remaining partition is amalgamated (some sub-collections are unified). Simplification has many interesting mathematical properties. A finite composition of simplifications can also be accomplished with some single simplification. Also, if one partition is a simplification of the other, the simplified partition is defined to be less than the other partition according to the simp relation. This relation is shown to be a partial-order relation based on simplification. Collections of partitions can not only be proven to have a partial- order structure, but also have a lattice structure and are complete. In regard to a geographic information system (GIs), partitions related to subsets of attribute domains for objects are called views. Objects belong to different views based whether or not their attribute values lie in the underlying view domain. Given a particular view, objects with their attribute n-tuple codings contained in the view are part of the actualization set on views, and objects are labeled according to the particular subset of the view in which their coding lies. Though the scope of the work does not mainly focus on queries related directly to geographic objects, it provides verification for the existence of particular views in a system with this underlying structure. Given a finite attribute domain, one can say with mathematical certainty that different views of objects are partially ordered by simplification, and every collection of views has a greatest lower bound and least upper bound, which provides the validity for exploring queries in this regard.

Hyalotekite, (Ba,Pb,K)(4)(Ca,Y)(2)Si-8(B,Be)(2)(Si,B)(2)O28F, a Tectosilicate Related to Scapolite: New Structure Refinement, Phase Transitions and a Short-Range Ordered 3B Superstructure

Hyalotekite, a framework silicate of composition (Ba,Pb,K)(4)(Ca,Y)(2)Si-8(B,Be)(2) (Si,B)(2)O28F, is found in relatively high-temperature(greater than or equal to 500 degrees C) Mn skarns at Langban, Sweden, and peralkaline pegmatites at Dara-i-Pioz, Tajikistan. A new paragenesis at Dara-i-Pioz is pegmatite consisting of the Ba borosilicates leucosphenite and tienshanite, as well as caesium kupletskite, aegirine, pyrochlore, microcline and quartz. Hyalotekite has been partially replaced by barylite and danburite. This hyalotekite contains 1.29-1.78 wt.% Y2O3, equivalent to 0.172-0.238 Y pfu or 8-11% Y on the Ca site; its Pb/(Pb+Ba) ratio ranges 0.36-0.44. Electron microprobe F contents of Langban and Dara-i-Pioz hyalotekite range 1.04-1.45 wt.%, consistent with full occupancy of the F site. A new refinement of the structure factor data used in the original structural determination of a Langban hyalotekite resulted in a structural formula, (Pb1.96Ba1.86K0.18)Ca-2(B1.76Be0.24)(Si1.56B0.44)Si8O28F, consistent with chemical data and all cations with positive-definite thermal parameters, although with a slight excess of positive charge (+57.14 as opposed to the ideal +57.00). An unusual feature of the hyalotekite framework is that 4 of 28 oxygens are non-bridging; by merging these 4 oxygens into two, the framework topology of scapolite is obtained. The triclinic symmetry of hyalotekite observed at room temperature is obtained from a hypothetical tetragonal parent structure via a sequence of displacive phase transitions. Some of these transitions are associated with cation ordering, either Pb-Ba ordering in the large cation sites, or B-Be and Si-B ordering on tetrahedral sites. Others are largely displacive but affect the coordination of the large cations (Pb, Ba, K, Ca). High-resolution electron microscopy suggests that the undulatory extinction characteristic of hyalotekite is due to a fine mosaic microstructure. This suggests that at least one of these transitions occurs in nature during cooling, and that it is first order with a large volume change. A diffuse superstructure observed by electron diffraction implies the existence of a further stage of short-range cation ordering which probably involves both (Pb,K)-Ba and (BeSi,BB)-BSi.