The Algebraic Structure of Spaces of Intervals: Contribution of Svetoslav Markov to Interval Analysis and its Applications
In Interval Analysis addition of intervals is the usual Minkowski addition of sets. The fact that the additive inverse generally does not exist has been a major obstacle in applications, e.g. constructing narrow enclosures of a solution,В and possibly one of the most important mathematical challenges associated with the development of the theory of spaces of intervals. The work on this issue during the last 50-60 years lead to new operations for intervals, extended concepts of interval, setting the interval theory within the realm of algebraic structures more general than group and linear space. This theoretical development was paralleled by development of interval computer arithmetic. Svetoslav Markov was strongly involved in this major development in modern mathematics and he in fact introduced many of the main concepts and theories associated with it. These include: extended interval arithmetic, directed interval arithmetic, the theory of quasivector spaces. His work lead to practically important applications to the validated numerical computing as well as in the computations with intervals, convex bodies and stochastic numbers. Such advanced mathematical and computational tools are much useful under the conditions of extreme sensitivity that is often inheritably characteristic for biological processes as well as input biological parameters experimentally known to be in certain ranges .....
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