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Determinant and Inverse of Matrices of Real Elements
, 2007
"... In this paper the classic theory of matrices of real elements (see e.g. [12], [13]) is developed. We prove selected equations that have been proved previously for matrices of field elements. Similarly, we introduce in this special context the determinant of a matrix, the identity and zero matrices, ..."
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In this paper the classic theory of matrices of real elements (see e.g. [12], [13]) is developed. We prove selected equations that have been proved previously for matrices of field elements. Similarly, we introduce in this special context the determinant of a matrix, the identity and zero matrices, and the inverse matrix. The new concept discussed in the case of matrices of real numbers is the property of matrices as operators acting on finite sequences of real numbers from both sides. The relations of invertibility of matrices and the “onto” property of matrices as operators are discussed.
Towards the Construction of a Model of Mizar Concepts
, 2008
"... The aim of this paper is to develop a formal theory of Mizar linguistic concepts following the ideas from [14] and [13]. The theory here presented is an abstract of the existing implementation of the Mizar system and is devoted to the formalization of Mizar expressions. The base idea behind the form ..."
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The aim of this paper is to develop a formal theory of Mizar linguistic concepts following the ideas from [14] and [13]. The theory here presented is an abstract of the existing implementation of the Mizar system and is devoted to the formalization of Mizar expressions. The base idea behind the formalization is dependence on variables which is determined by variabledependence (variables may depend on other variables). The dependence constitutes a Galois connection between opposite poset of dependenceclosed set of variables and the supsemilattice of widening of Mizar types (smooth type widening). In the paper the concepts strictly connected with Mizar expressions are formalized. Among them are quasiloci, quasiterms, quasiadjectives, and quasitypes. The structural induction and operation of substitution are also introduced. The prefix quasi is used to indicate that some rules of construction of Mizar expressions may not be fulfilled. For example, variables, quasiloci, and quasiterms have no assigned types and, in result, there is no possibility to conduct typechecking of arguments. The other gaps concern inconsistent and outofcontext clusters of adjectives in types. Those rules are required in the Mizar identification process. However, the expression appearing in later processes of Mizar checker may not satisfy the rules. So, introduced apparatus is enough and adequate to describe data structures and algorithms from the Mizar checker (like equational classes).
Laplace Expansion
"... Summary. In the article the formula for Laplace expansion is proved. ..."
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Helly Property for Subtrees
, 2008
"... We prove, following [5, p. 92], that any family of subtrees of a finite tree satisfies the Helly property. ..."
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We prove, following [5, p. 92], that any family of subtrees of a finite tree satisfies the Helly property.
DOI: 10.2478/v1003701200042 versita.com/fm/ The Rotation Group
"... Summary. We introduce lengthpreserving linear transformations of Euclidean topological spaces. We also introduce rotation which preserves orientation (proper rotation) and reverses orientation (improper rotation). We show that every rotation that preserves orientation can be represented as a compo ..."
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Summary. We introduce lengthpreserving linear transformations of Euclidean topological spaces. We also introduce rotation which preserves orientation (proper rotation) and reverses orientation (improper rotation). We show that every rotation that preserves orientation can be represented as a composition of base proper rotations. And finally, we show that every rotation that reverses orientation can be represented as a composition of proper rotations and one improper rotation.