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  • 1
    Electronic Resource
    Electronic Resource
    Lattice operators underlying dynamic systems (1996)
    Springer
    Set-valued analysis 4 (1996), S. 119-134 
    Details
    ISSN: 1572-932X
    Keywords: 06A23 ; 34A60 ; 68U10 ; 93C15 ; complete lattice ; algebraic dilation and erosion ; algebraic opening and closing ; semicontinuity ; differential inclusion ; contingent cone ; reachable set ; exit tube ; viability kernel ; invariance kernel
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper investigates algebraic and continuity properties of increasing set operators underlying dynamic systems. We recall algebraic properties of increasing operators on complete lattices and some topologies used for the study of continuity properties of lattice operators. We apply these notions to several operators induced by a differential equation or differential inclusion. We especially focus on the operators associating with any closed subset its reachable set, its exit tube, its viability kernel or its invariance kernel. Finally, we show that morphological operators used in image processing are particular cases of operators induced by constant differential inclusion.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00425961
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Minkowski operations and vector spaces (1995)
    Springer
    Set-valued analysis 3 (1995), S. 33-50 
    Details
    ISSN: 1572-932X
    Keywords: 47H04 ; 68U10 ; mathematical morphology ; Minkowski sum ; Minkowski subtraction ; set-convolution ; internal set-convolution ; Steiner selection
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Mathematical morphology started as a set of tools for analysing images by the use of transformations based on set-theoretical operations which are the Minkowski sum and subtraction. It was first developed for the analysis of binary images. Its extension to grey-level images was a later development with the extension of the Minkowski operations to real-valued functions in terms of sup-convolution and inf-convolution. The purpose of this paper is to define a type of convolution between set-valued maps, to study its properties, and to establish some associated differential relations. This set-convolution map allows us to extend the Minkowski sum and substraction to multivalued functions and to functions with vectorial values.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01033640
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    Paper (German National Licenses)
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