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On continuous convergence of nets of multifunctions

Przemski M.

Demonstratio Mathematica

2011

Vol. 44, nr 1

181--200

This paper expands the classical concept of the continuous convergence of nets of multifunctions introduced by Cao, Reilly and Vamanamurthy in [7]. We introduce some new types of properties of convergence of such nets which guarantee the upper or lower semicontinuity of the limit multifunction. Furthermore, we obtain some analogous results concerning generalized continuity properties of multifunctions.

Modelowanie zależności topologicznych za pośrednictwem międzyobiektowych związków w programowym środowisku GEOBA

Kłodziński E., Betliński G.

Roczniki Geomatyki

2004

T. 2, z. 2

121-128

This paper describes a general concept of a two models: a network model and a multilevel terrain division model, which are implemented in the Program Environment GEOBA that supports production of object-oriented geographical information systems. The network model bases on a well-known vector data model, so-called arc-node, that topologically links arcs to nodes and polygons. Arc-node topology is showed on the figure 1 and it includes two main aspects of vector data: 1. connectivity - it's an identification of connected arcs by recording start and end node for each arc, 2. contiguity - it's an identification of adjacent polygon by recording the "left" and "right" polygons of each arc. There are two abstract classes in the network model showed on the figure 2: the first of them Łuk representing an arc and the second Węzeł corresponding with a node. These classes are associated by a "z węzła" (from node) and "do węzła " (to node) relations. In the case of Łuk class, a children are representing in reality such the objects as gas pipelines, streets, and so on, however a children of Węzeł class are corresponding with such the objects as a gas station or a street intersection. The n-level division P of the terrain area O is such a sequence { Ojm }, j ∈ N, 1<=m ≤ n, of the areas belonging to O, that (p - a point, B - an area border): 1. m = 1 ⇒ j = 1 ∧ O11 = O ; // there is only one area at the first level of the terrain division and this area is equal whole input area O, 2. ∀ i,j ∈ N, 1< m ≤ n, p ∈ ( Oim ∩ Ojm ) ⇔ p ∈ B(Oim) ∧ p ∈ B(Ojm) // the areas at the same level have to be disjointed (only points belonging to the area borders can be common) 3. ∀ 1 < m < k ≤ n, p ∈ ( Oik ∩ Ojm) ∧ p ∉ B(Oik) ⇒ Oik ⊆ Oj // an area at the level m (excluding level m=1) have to be included in an area at the level m-1. An example of such a 3-level terrain division can be a division of a country territory into a states which are next divided into a counties. The multilevel terrain division model is showed on the figure 3 and it basis on a network model which was extended with an additional classes Łuk Obszaru, Węzeł Obszaru and Obszar. The class łuk Obszaru is a child class of a class Łuk and the class Węzeł Obszaru is the specialization of a class Węzeł Each of these two classes is associated by zawiera sie w. (contains in) relation with an abstract class Obszar which is a base class for a classrepresenting all elements (e.g. a state or a county) of a multilevel terrain division.The most important features of these described in short models are following: Instead of recording a special datum especially designated for topological relationships, a connectivity,a contiguity and an area including are recording like the all normal object associations. The set of objects being within the operation reach is calculated basis on the object associations. The data base contains not overlapped arcs only. The system model and his documentation are made in the same manner for all, including topological, object relationships.