Displaying similar documents to “Morphology for Image Processing. Part I”

N-Dimensional Binary Vector Spaces

Kenichi Arai, Hiroyuki Okazaki (2013)

Formalized Mathematics

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The binary set {0, 1} together with modulo-2 addition and multiplication is called a binary field, which is denoted by F2. The binary field F2 is defined in [1]. A vector space over F2 is called a binary vector space. The set of all binary vectors of length n forms an n-dimensional vector space Vn over F2. Binary fields and n-dimensional binary vector spaces play an important role in practical computer science, for example, coding theory [15] and cryptology. In cryptology, binary fields...

Semantics of MML Query - Ordering

Grzegorz Bancerek (2013)

Formalized Mathematics

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Semantics of order directives of MML Query is presented. The formalization is done according to [1]

Introduction to Formal Preference Spaces

Eliza Niewiadomska, Adam Grabowski (2013)

Formalized Mathematics

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In the article the formal characterization of preference spaces [1] is given. As the preference relation is one of the very basic notions of mathematical economics [9], it prepares some ground for a more thorough formalization of consumer theory (although some work has already been done - see [17]). There was an attempt to formalize similar results in Mizar, but this work seems still unfinished [18]. There are many approaches to preferences in literature. We modelled them in a rather...

Relational Formal Characterization of Rough Sets

Adam Grabowski (2013)

Formalized Mathematics

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The notion of a rough set, developed by Pawlak [10], is an important tool to describe situation of incomplete or partially unknown information. In this article, which is essentially the continuation of [6], we try to give the characterization of approximation operators in terms of ordinary properties of underlying relations (some of them, as serial and mediate relations, were not available in the Mizar Mathematical Library). Here we drop the classical equivalence- and tolerance-based...

Free Interpretation, Quotient Interpretation and Substitution of a Letter with a Term for First Order Languages

Marco Caminati (2011)

Formalized Mathematics

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Fourth of a series of articles laying down the bases for classical first order model theory. This paper supplies a toolkit of constructions to work with languages and interpretations, and results relating them. The free interpretation of a language, having as a universe the set of terms of the language itself, is defined.The quotient of an interpreteation with respect to an equivalence relation is built, and shown to remain an interpretation when the relation respects it. Both the concepts...

Representation Theorem for Stacks

Grzegorz Bancerek (2011)

Formalized Mathematics

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In the paper the concept of stacks is formalized. As the main result the Theorem of Representation for Stacks is given. Formalization is done according to [13].

Inner Products, Group, Ring of Quaternion Numbers

Fuguo Ge (2008)

Formalized Mathematics

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In this article, we define the division of the quaternion numbers, we also give the definition of inner products, group, ring of the quaternion numbers, and we prove some of their properties.MML identifier: QUATERN2, version: 7.8.10 4.100.1011