Contributions to fuzzy analysis

Gähler, Werner; Gähler, Siegfried

doi:10.1016/s0165-0114(98)00320-0

Cited by 17 publications

(23 citation statements)

References 7 publications

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“…This result has been extended by Bellissard et al [2] to the general case of a Delone set with finite R d -type. Gähler [3] announced recently a very simple proof of the fact that the hull is a projective limit of branched manifolds in the same context. However, in this proof, the sizes of the faces of the branched manifolds remain constant as we go backward in the projective limit and, thus, the number of faces goes to infinity.…”

Section: Remark 12mentioning

confidence: 99%

On the dynamics of \mathbb{G} -solenoids. Applications to Delone sets

Gambaudo¹,

Benedetti²

2003

Ergod. Th. Dynam. Sys.

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A \mathbb{G}-solenoid is a laminated space whose leaves are copies of a single Lie group \mathbb{G} and whose transversals are totally disconnected sets. It inherits a \mathbb{G}-action and can be considered as a dynamical system. Free \mathbb{Z}^d-actions on the Cantor set as well as a large class of tiling spaces possess such a structure of \mathbb{G}-solenoids. For a large class of Lie groups, we show that a \mathbb{G}-solenoid can be seen as a projective limit of branched manifolds modeled on \mathbb{G}. This allows us to give a topological description of the transverse invariant measures associated with a \mathbb{G}-solenoid in terms of a positive cone in the projective limit of the dim(\mathbb{G})-homology groups of these branched manifolds. In particular, we exhibit a simple criterion implying unique ergodicity. Particular attention is paid to the case when the Lie group \mathbb{G} is the group of affine orientation-preserving isometries of the Euclidean space or its subgroup of translations.

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Section: Remark 12mentioning

confidence: 99%

On the dynamics of \mathbb{G} -solenoids. Applications to Delone sets

Gambaudo¹,

Benedetti²

2003

Ergod. Th. Dynam. Sys.

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“…Roughly speaking, they are 1-dimensional "semi-vector" spaces. This concept is not new, and has been used in contexts which differ a lot from the present one: for instance, see [4] for the analysis of some properties of Z 2 -valued matrices, [12] for problems of fuzzy analysis, [28] for problems of measure theory, [29,30] for topological fixed point problems. Then, we introduce the sesqui-tensor product of a semi-vector space with a vector space and the semi-tensor product between semi-vector spaces.…”

Section: Summary Of the Present Papermentioning

confidence: 99%

“…Despite the fact that some authors (see [12]) require that a semi-vector space has a zero vector, here we do not make such a general assumption. On the other hand, interesting properties arise for semi-vector spaces with a zero vector.…”

Section: Semi-vector Spacesmentioning

confidence: 99%

An Algebraic Approach to Physical Scales

Janyška

Modugno

Vitolo³

2009

Acta Appl Math

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This paper is aimed at introducing an algebraic model for physical scales and units of measurement. This goal is achieved by means of the concept of "positive space" and its rational powers. Positive spaces are "semi-vector spaces" on which the group of positive real numbers acts freely and transitively through the scalar multiplication. Their tensor multiplication with vector spaces yields "scaled spaces" that are suitable to describe spaces with physical dimensions mathematically. We also deal with scales regarded as fields over a given background (e.g., spacetime).

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“…Roughly speaking, semivector spaces are "vector spaces" where the scalars are in a semi-field. Although the concept of semi-vector space was investigated over time, there exist few works available in the literature dealing with such spaces [13,11,12,10,4,5,8]. This fact occurs maybe due to the limitations that such concept brings, i.e., the non-existence of (additive) symmetric for some (for all) semi-vector.…”

Section: Introductionmentioning

confidence: 99%

On semi-vector spaces and semi-algebras

Guardia¹,

Chagas²,

Lenzi³

et al. 2021

Preprint

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It is well-known that the theories of semi-vector spaces and semi-algebras -which were not much studied over time -are utilized/applied in Fuzzy Set Theory in order to obtain extensions of the concept of fuzzy numbers as well as to provide new mathematical tools to investigate properties and new results on fuzzy systems. In this paper we investigate the theory of semi-vector spaces over the semi-field of nonnegative real numbers R + 0 . We prove several results concerning semi-vector spaces and semi-linear transformations. Moreover, we introduce in the literature the concept of eigenvalues and eigenvectors of a semi-linear operator, describing in some cases how to compute them. Topological properties of semi-vector spaces such as completeness and separability are also investigated. New families of semivector spaces derived from semi-metric, semi-norm, semi-inner product, among others are exhibited. Additionally, some results on semi-algebras are presented.

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Contributions to fuzzy analysis

Cited by 17 publications

References 7 publications

On the dynamics of \mathbb{G} -solenoids. Applications to Delone sets

On the dynamics of \mathbb{G} -solenoids. Applications to Delone sets

An Algebraic Approach to Physical Scales

On semi-vector spaces and semi-algebras

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