Communication/review Biomath 2 (2013), 1309257, 1–2

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The Algebraic Structure of Spaces of Intervals
Contribution of Svetoslav Markov to Interval Analysis

and its Applications

Roumen Anguelov
Department of Mathematics and Applied Mathematics

University of Pretoria, Pretoria, South Africa
E-mail: roumen.anguelov@up.ac.za

In Interval Analysis addition of intervals is the usual
Minkowski addition of sets: A + B = {a + b :
a ∈ A, b ∈ B}. 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 develop-
ment 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 [11], [5], [10], di-
rected interval arithmetic [13], the theory of quasivector
spaces [15], [16]. 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 [12], [14], [18]. 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 [17].

One of the most important contributions to knowledge
by Svetoslav Markov is in my view the embedding of
the monoid structure of intervals and convex bodies

into a group structure where the natural definition of
multiplication by scalars is also extended in such a way
that it is monotone with respect to inclusion, that is

A ⊆ B =⇒ γA ⊆ γB, γ ∈ R.

The obtained structure is called a quasivector space
[16]. Since the fundamental idea motivating this field
is computations with sets and computing enclosures, the
stated property cannot be really overemphasized. One
should note that there have been several attempts to
embed the quasi-linear space of convex bodies in a more
computationally convenient algebraic structure. The most
well known such attempt is Radström’s embedding into
a linear space [19]. However, this embedding fails to
preserve precisely the monotonicity property mentioned
above. Hence, while the developed by Radström theory is
mathematically correct and elegant, it is quite irrelevant
regarding the embedded set and in fact the field of
Interval Analysis or more generally the field of Set-
Valued Computing. Markov’s concept of quasivector
space manages to capture and preserve the essential
properties of computations with sets (like the stated
monotonicity) while also providing a relatively simple
structure for computing. Indeed, the quasivector space
is a direct sum of a vector(linear) space and symmetric
quasivector space which makes the computations essen-
tially as easy as computations in a linear space.

A wide spectrum of applications is usually a testimony
for the depth of an idea. The ideas of Markov have
certainly wide and far reaching implications. We focus
on one particular direction of development, namely the

Citation: Roumen Anguelov, The Algebraic Structure of Spaces of Intervals: Contribution of Svetoslav
Markov to Interval Analysis, Biomath 2 (2013), 1309257, http://dx.doi.org/10.11145/j.biomath.2013.09.257

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R Anguelov, The Algebraic Structure of Spaces of Intervals: Contribution of Svetoslav Markov to Interval Analysis

algebraic operations for interval-valued functions. I have
had the privilege to have Svetoslav Markov as a teacher
and as a collaborator. The algebraic structure of spaces
of interval functions is the main topic of our more
recent collaborative work. Jointly with Blagovest Sendov
we studied the operations for Hausdorff continuous (H-
continuous) function. It turned out that the linear space
structure of real functions can be extended to the space
of H-continuous interval functions and it is actually the
largest linear space of interval functions. Hence the space
of H-continuous functions has a very special place in
Interval Analysis. Further, we showed that the practically
relevant set, in terms of providing tight enclosures of
sets of real functions, is the set of Dilworth continuous
(D-continuous) interval valued function. Using an earlier
idea of Svetoslav Markov of abstract construction of
interval space over a vector lattice, one can show that
the set of D-continuous function is a quasi-linear space
of intervals over the space of H-continuous functions.
Moreover, the space of H-continuous functions is pre-
cisely the linear space in the direct Markov’s sum
decomposition of the respective quasivector space.

Let us note that the space of H-continuous functions
has applications in various areas of mathematics, e.g.
Real Analysis [1], [2], Approximation Theory [20], val-
idated computing [3], [6] as well as the general theory of
PDEs [9], [7], [4]. The issue of constructing enclosures
is relevant to all mentioned applications. Hence one can
expect future developments in this research direction
which involve the space of D-continuous functions and
Markov’s approach to computing with them.

REFERENCES
[1] R Anguelov, Dedekind order completion of C(X) by Hausdorff

continuous functions, Quaestiones Mathematicae, Vol. 27, 2004,
pp. 153–170.

[2] R Anguelov, Rational Extensions of C(X) via Hausdorff Con-
tinuous Functions, Thai Journal of Mathematics, Vol. 5 No. 2,
2007, pp. 261–272.

[3] R Anguelov, Algebraic Computations with Hausdorff Contin-
uous Functions, Serdica Journal of Computing, Vol. 1 No. 4,
2007, pp. 443–454.

[4] R Anguelov, D Agbebaku, J H van der Walt, Hausdorff
Continuous Solutions of Conservation Laws, In MD Todorov
(editor),Proceedings of the 4th International Conference on

Application of Mathematics in Technical and Natural Sciences
(AMiTaNS’12), (St Constantine and Helena, Bulgaria), Amer-
ican Institute of Physics - AIP Conference Proceedings 1487,
2012, pp 151–158.

[5] R Anguelov R, S Markov, Extended Segment Analysis,
Freiburger Intervall Berichte 81/10, University of Freiburg,
1981.

[6] R Anguelov, S Markov, Numerical Computations with Haus-
dorff Continuous Functions, In: T. Boyanov et al. (Eds.), Nu-
merical Methods and Applications 2006 (NMA 2006), Lecture
Notes in Computer Science 4310, Springer, 2007, 279–286.

[7] R Anguelov, S Markov, F Minani, Hausdorff Continuous Vis-
cosity Solutions of Hamilton-Jacobi Equations, Proceedings
of the 7th International Conference on Large Scale Scientific
Computations, 3-8 June 2009, Sozopol, Bulgaria, Lecture Notes
in Computer Science, 5910 (2010), pp. 241–248.

[8] R Anguelov, S Markov and Bl Sendov, The Set of Hausdorff
Continuous Functions - the Largest Linear Space of Interval
Functions, Reliable Computing, Vol. 12, 2006, pp. 337–363.

[9] R Anguelov, E E Rosinger, Solving Large Classes of Nonlinear
Systems of PDE’s, Computers and Mathematics with Applica-
tions, Vol. 53, 2007, pp. 491–507.

[10] Dimitrova, N., Markov, S., Popova, E., Extended Interval Arith-
metics: New Results and Applications. In: Computer Arithmetic
and Enclosure Methods (Eds. L. Atanassova, J. Herzberger),
North-Holland, Amsterdam, 1992, 225–234.

[11] S Markov, A Non-standard Subtraction of Intervals, Serdica, 3,
1977, pp.359–370.

[12] S Markov, Calculus for interval functions of a real variable.
Computing 22, 325–337 (1979).

[13] S Markov, On Directed Interval Arithmetic and its Applications,
J. UCS 1 (7), 1995, 514–526.

[14] S Markov, An Iterative Method for Algebraic Solution to
Interval Equations, Applied Numerical Mathematics 30 (2–3),
1999, 225–239.

[15] S Markov, On the Algebraic Properties of Convex Bodies and
Some Applications, J. Convex Analysis 7 (1), 2000, 129–166.

[16] S Markov, On Quasilinear Spaces of Convex Bodies and Inter-
vals, Journal of Computational and Applied Mathematics 162
(1), 93–112, 2004.

[17] S Markov, Biomathematics and Interval Analysis: A Prosperous
Marriage, in Ch Christov and MD Todorov (eds), Proceedings
of the 2d International Conference on Application of Mathemat-
ics in Technical and Natural Sciences (AMiTaNS’10), (21–26
June, Sozopol, Bulgaria), American Institute of Physics - AIP
Conf. Proc. 1301, pp. 26–36.

[18] S Markov, R Alt, Stochastic Arithmetic: Addition and Multi-
plication by Scalars, Applied Numerical Mathematics 50, 475–
488, 2004.

[19] H Radström, An embedding theorem for spaces of convex sets.
Proc. Am.Math. Soc. 3, 165–169. (1952)

[20] Bl Sendov, Hausdorff Approximations, Kluwer Academic,
Boston, 1990.

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