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, 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

Some properties of the Blumberg's hyper-log-logistic curve

The paper considers the sigmoid function definedthrough the hyper-log-logistic model introduced by Blumberg. We study the Hausdorff distance of this sigmoid to the Heaviside function, which characterises the shape of switching from 0 to 1. Estimates of the Hausdorff distance in terms of the intrinsic growth rate are derived. We construct a family of recurrence generated sigmoidal functions based on the hyper-log-logistic function. Numerical illustrations are provided

Imprecise Probability and Chance

Understanding probabilities as something other than point values (e.g., as intervals) has often been motivated by the need to find more realistic models for degree of belief, and in particular the idea that degree of belief should have an objective basis in “statistical knowledge of the world.” I offer here another motivation growing out of efforts to understand how chance evolves as a function of time. If the world is “chancy” in that there are non-trivial, objective, physical probabilities at the macro-level, then the chance of an event e that happens at a given time is e goes to one continuously or not is left open. Discontinuities in such chance trajectories can have surprising and troubling consequences for probabilistic analyses of causation and accounts of how events occur in time. This, coupled with the compelling evidence for quantum discontinuities in chance’s evolution, gives rise to a “(dis)continuity bind” with respect to chance probability trajectories. I argue that a viable option for circumventing the (dis)continuity bind is to understand the probabilities “imprecisely,” that is, as intervals rather than point values. I then develop and motivate an alternative kind of continuity appropriate for interval-valued chance probability trajectories

The Linear Space of Hausdorff Continuous Interval Functions

In this paper we discuss the algebraic structure of the space H(X) of finite Hausdorff continuous interval functions defined on an arbitrary topological space X. In particular, we show that H(X) is a linear space over R containing C(X), the space of continuous real functions on X, as a linear subspace. In addition, we prove that the order on H(X) is compatible with the linear structure introduced here so that H(X) is an Archimedean vector lattice

Convergence of sequences of semi-continuous functions

In this paper we investigate how three well-known modes of convergence for (real-valued) functions are related to one another. In particular, we consider order convergence, pointwise convergence and continuous convergence of sequences of nearly finite normal lower semi-continuous functions. There is a natural comparison to be made between the results we obtain for convergence of sequences of semi-continuous functions, and classic results on the convergence of sequences of measurable functions.http://www.elsevier.com/locate/jmaahb2016Mathematics and Applied Mathematic

The set of Hausdorff continuous functions — the largest linear space of interval functions. Reliable Computing 12, 337 – 363

Abstract. Hausdorff continuous (H-continuous) functions are special interval-valued functions which are commonly used in practice, e.g. histograms are such functions. However, in order to avoid arithmetic operations with intervals, such functions are traditionally treated by means of corresponding semi-continuous functions, which are real-valued functions. One difficulty in using H-continuous functions is that, if we add two H-continuous functions that have interval values at same argument using point-wise interval arithmetic, then we may obtain as a result an interval function which is not H-continuous. In this work we define addition so that the set of H-continuous functions is closed under this operation. Moreover, the set of H-continuous functions is turned into a linear space. It has been also proved that this space is the largest linear space of interval functions. These results make H-continuous functions an attractive tool in real analysis and provides a bridge between real and interval analysis