Topic
Ordered pair
About: Ordered pair is a research topic. Over the lifetime, 435 publications have been published within this topic receiving 6256 citations. The topic is also known as: (a, b) & couple.
Papers published on a yearly basis
1 / 2
Papers
21 Sep 1992
TL;DR: This model describes a partial order over topological relationships and provides a measure to assess how far two relationships are apart from each other and provides answers to three kinds of qualitative space-time inferences.
Abstract: Geographic objects and phenomena may gradually change their location, orientation, shape, and size over time. A qualitative change occurs if the deformation of an object affects its topological relationship with respect to another object. The observation of such changes is particularly interesting, because qualitative changes frequently require different decisions or trigger new actions. Investigations of a closed set of mutually exclusive binary topological relationships led to a formal model to determine for each topological relationship the relationships closest to it. Applied to the entire set of binary topological relationships between spatial regions, this model describes a partial order over topological relationships and provides a measure to assess how far two relationships are apart from each other. The changes to the binary topological relationship caused by such deformations as translation, rotation, reduction, and expansion of an object are mapped onto this graph. The graphs show characteristic traverses for each kind of deformation. Using these characteristic traverses as knowledge about deformations, one can infer from multiple observations the kind of deformation that caused the change and predict the next topological relationship. Particularly, it provides answers to three kinds of qualitative space-time inferences: (1) Given a process and a state, what is the next most likely state? (2) Given an ordered pair of states, what process may have occurred? (3) Given an ordered pair of states and a process, in what states must the two objects have been in between?
389 citations
TL;DR: In this article, the main critical problem that naturally arises in processing Z-number-based information is computation with Z-numbers, which is a more adequate concept for description of real-world information.
301 citations
1 Jan 1990
TL;DR: In this paper, the authors introduce functors yielding to a binary operation whose composition with an arbitrary functions on its left side, its right side or both is shown to be commutative or associative.
Abstract: Summary. In the article we introduce functors yielding to a binary operation its composition with an arbitrary functions on its left side, its right side or both. We prove theorems describing the basic properties of these functors. We introduce also constant functions and converse of a function. The recent concept is defined for an arbitrary function, however is meaningful in the case of functions which range is a subset of a Cartesian product of two sets. Then the converse of a function has the same domain as the function itself and assigns to an element of the domain the mirror image of the ordered pair assigned by the function. In the case of functions defined on a non-empty set we redefine the above mentioned functors and prove simplified versions of theorems proved in the general case. We prove also theorems stating relationships between introduced concepts and such properties of binary operations as commutativity or associativity.
232 citations
Posted Content•
TL;DR: In this article, an ordered pair of linear transformations (i.e., a Leonard pair on a field and a vector space over a field with finite positive dimension) is considered.
Abstract: Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension We consider an ordered pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfy conditions (i), (ii) below
(i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal
(ii) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^*$ is irreducible tridiagonal
We call such a pair a Leonard pair on $V$ We give an overview of the theory of Leonard pairs
212 citations
TL;DR: This work developed basic arithmetic operations such as addition, subtraction, multiplication and division, and some algebraic operations as maximum, minimum, square and square root of continuous Z-numbers.
178 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 25 |
| 2020 | 13 |
| 2019 | 13 |
| 2018 | 12 |
| 2017 | 13 |
| 2016 | 22 |