# Blagovest Sendov

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- Jan 2019

A set A, in the extended complex plane, is called convex with respect to a pole u, if for any x, y ∈ A the arc on the unique circle through x, y, and u, that connects x and y but does not contain u, is in A. If the pole u is taken at infinity, this notion reduces to the usual convexity. Polar convexity is connected with the classical Gauss-Lucas' and Laguerre's theorems for complex polynomials. If a set is convex with respect to u and contains the zeros of a polynomial, then it contains the zeros of its polar derivative with respect to u. A set may be convex with respect to more than one pole. The main goal of this article is to find the relationships between a set in the extended complex plane and its poles.

- Apr 2018

This work investigates the connections between the notion of a locus of a complex polynomial and the polar derivatives. Polar differentiation extends classical derivatives and provides additional flexibility. The notion of a locus was introduced in [8] and proved useful in providing sharp versions of several classical results in the area known as Geometry of Polynomials. The investigations culminated in the work [11]. A need was revealed for a unified treatment of bounded and unbounded loci of polynomials of degree at most n as well as a unified treatment of polar derivatives and ordinary derivatives. This work aims at providing such a framework.

- Nov 2017

The following Rolle’s Theorem for complex polynomials is proved. If p(z) is a complex polynomial of degree n ≥ 5, satisfying p(−i) = p(i), then there is at least one critical point of p in the union D[−c; r] ∪ D[c; r] of two closed disks with centres −c, c and radius r, where c = cot(2π/n), r = 1/ sin(2π/n). If n = 3, then the closed disk (Formula presented) has this property; and if n = 4, then the union of the closed disks D[−1/3; 2/3] ∪ D[1/3; 2/3] has this property. In the last two cases, the domains are minimal, with respect to inclusion, having this property. This theorem is stronger than any other known Rolle’s Theorem for complex polynomials.

- Apr 2017

In this work, we present a nonconvex analogue of the classical Gauss–Lucas theorem stating that the critical points of a polynomial with complex coefficients are in the convex hull of its zeros. We show that if the polynomial p(z) of degree n has nonnegative coefficients and zeros in the sector \(\{z \in \mathcal C: |\arg (z)| \ge \varphi \}\), for some \(\varphi \in [0,\pi ]\), then the critical points of p(z) are also in that sector. Clearly, when \(\varphi \in [\pi /2,\pi ]\), our result follows from the classical Gauss–Lucas theorem. But when \(\varphi \in [0,\pi /2)\), we obtain a nonconvex analogue.

- Apr 2017
- Progress in Approximation Theory and Applicable Complex Analysis

The spirit of the classical Grace-Walsh-Szegő coincidence theorem states that if there is a solution of a multi-affine symmetric polynomial in a domain with certain properties, then in it there exists another solution with other properties. We present two results in the same spirit, which may be viewed as extensions of the Grace-Walsh-Szegő result.

- Jan 2017

A classical theorem in the geometry of polynomials states the following: Let p(z) be a polynomial of degree n ≥ 2 with p(z1) = p(z2) for z1 ≠ z2. Then, the line l passing through the point (z1 +z2)/2 and orthogonal to the segment defined by the points z1 and z2 separates the zeros of p′ (z), or all the zeros of p′ (z) are on the line l. In this paper an inverse of this theorem is proved.

- May 2016

We consider the classical theorem of Grace, which gives a condition for a geometric relation between two arbitrary algebraic polynomials of the same degree. This theorem is one of the basic instruments in the geometry of polynomials. In some applications of the Grace theorem, one of the two polynomials is fixed. In this case, the condition in the Grace theorem may be changed. We explore this opportunity and introduce a new notion of locus of a polynomial. Using the loci of polynomials, we may improve some theorems in the geometry of polynomials. In general, the loci of a polynomial are not easy to describe. We prove some statements concerning the properties of a point set on the extended complex plane that is a locus of a polynomial.

The present book reflects contributions (articles and presentations) to the UNESCO International Workshop QED: Quality of Education and Challenges in a Digitally Networked World, held in Sofia from October 30-31, 2014.
The QED workshop was organised by the State University of Library Studies and Information Technologies (SULSIT), Sofia, Bulgaria, in cooperation with the Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences, and with financial support of the EU project EE74 NETT: Networked Entrepreneurship Training of Teachers, Law and Internet Foundation, and the assistance of British Counsil.
The workshop was a regional event as a follow-up of:
the EDUSummIT 2013 Research-informed strategies to address educational challenges in a digitally networked world, October 1- 2, Washington, USA and
the International Conference of the UNESCO IITE and UNESCO Chairs UNESCO Chairs Partnership on ICTs Use in Education, June 1-5, St. Petersburg in the frame of XV International Forum: Modern Information society formation – problems, perspectives, innovation approach.
The event was hosted by SULSIT which is a unique (in Bulgarian context) research and educational interdisciplinary center integrating studies in the library science, digital technologies, cultural and historical heritage.
It has been successfully implemented and has achieved multi-direction positive results and impact. Useful ideas and practices have been discussed, which will continue to be built up and used in the future including the following ones:
to establish fora and communities of practice for cross-stakeholder ICT in Education communication
to develop and establish a repository of open ICT in Education best practices
to summarise the general ideas, contributions and outcomes of the QED’15 workshop and to submit an article based on it to EDUsummIT’15
to propose new topics to EDUsummIT’15, e.g. related to the upbringing, and the language barriers
to collect feedback from the participants in the workshop
to advice the school Lyuben Karavelov Secondary School in Koprivshtitsa in its endeavour to become a UNESCO associated school and continue its cooperation with the QED community
to extend the teachers’ sessions in terms of participation
to start teacher education master programs in ICT with IITE
to translate in Bulgarian the framework of UNESCO for IT in education
to provide pieces of advice to policy makers to adapt their strategies for ICT in education according to the best world practices;
to bring it back EDUsummIT’17 to Europe (namely – in Bulgaria) as discussed with its former coordinators Joke Voogt and Gerald Knezek.
Although presented in a condensed form here the contributions reflect the main ideas conveyed by their authors at the QED workshop and they will hopefully serve as an inspirational source for further work towards advancing education into the digital age.
The first feedback:
Mariana Patru
I very much enjoyed attending the workshop, meeting exciting participants, as well as visiting an innovative school. The atmosphere was great and I felt that attendees enjoyed sharing their work with pride.
I believe that the Chair has made very good progress and has acquired an internationally recognized international visibility. I am sure that you will further expand the Chair's international status.
Joke Voogt
The teachers in the poster session of QED’15 showed inspirational use of ICT. They are so creative these math teachers! I really admire them for it. As for the school we visited in Koptivshtica I would have liked to attend a lesson in it. The atmosphere was so warm and good. What a pity that it was a Saturday
Boyka Dulgyarova – Director of Lyben Karavelov Secondary School
We at the school were honored to meet such distinguished guests habing dedicated their professional activities for bringing the education in an international context in harmony with the UNESCO standards. Together with all teachers, we will involve our efforts in meeting the requirements for becoming a associated school of UNESCO.
A student from that school: Communicating with the visiors to our school was the happiest day in my life...

- Jun 2015

For every complex polynomial
p
(
z
), closed point sets are defined, called
loci
of
p
(
z
). A closed set Ω ⊆
${\mathbb C}$
* is a locus of
p
(
z
) if it contains a zero of any of its apolar polynomials and is the smallest such set with respect to inclusion. Using the notion locus, some classical theorems in the geometry of polynomials can be refined. We show that each locus is a Lebesgue measurable set and investigate its intriguing connections with the higher-order polar derivatives of
p
.

- Oct 2014

The classical Grace theorem states that every circular domain in the complex plane ℂ containing the zeros of a polynomial p(z) contains a zero of any of its apolar polynomials. We say that a closed domain Ω⊆ℂ * is a locus of p(z) if it contains a zero of any of its apolar polynomials and is the smallest such domain with respect to inclusion. In this work we establish several general properties of the loci and show, in particular, that the property of a set being a locus of a polynomial is preserved under a Möbius transformation. We pose the problem of finding a locus inside the smallest disk containing the roots of p(z) and solve it for polynomials of degree 3. Numerous examples are given.

Let $S(\phi)= \{z:\;|\arg(z)|\geq \phi\}$ be a sector on the complex plane
$\CC$. If $\phi\geq \pi/2$, then $S(\phi)$ is a convex set and, according to
the Gauss-Lucas theorem, if a polynomial $p(z)$ has all its zeros on $S(\phi)$,
then the same is true for the zeros of all its derivatives. In this paper is
proved that if the polynomial $p(z)$ is with real and non negative
coefficients, then the same is true also for $\phi < \pi/2$, when the sector is
not a convex set on the complex plane.

All linear operators $L:\mathcal{C}[z]\to \mathcal{C}[z]$ which decrease the
diameter of the zero set of any $P\in\mathcal{C}[z]$ are found.

- Jan 2013

The Walsh's coincidence theorem states: Let P(z(1), z(2), ... , z(n)) be a multi-affine symmetric polynomial in n complex variables. Then every circular domain containing the points zeta(1), zeta(2), ... , zeta(n) contains at least one point zeta such that P(zeta(1), zeta(2), ... , zeta(n)) = P(zeta, zeta, ... , zeta). The purpose of the paper is to prove the converse that only the circular domains are the closed point sets with this property.

- Dec 2012

Dear mournful family and all present,
We are here to bow and to send to his eternal rest the President of the Bulgarian Academy of Sciences—Academician Stefan Manev Dodunekov. We have come to share together the enormous loss every one of us has suffered.
The family lost their loving and caring father and husband, the mathematical community lost one of its brightest and most active members, the Bulgarian Academy of Sciences lost one of its hopes for a better and more successful development, Bulgaria lost one quiet, but uncompromising patriot ...

- Dec 2012

Dear mournful family and all present,
We are here to bow and to send to his eternal rest the President of the Bulgarian Academy of Sciences—Academician Stefan Manev Dodunekov. We have come to share together the enormous loss every one of us has suffered.
The family lost their loving and caring father and husband, the mathematical community lost one of its brightest and most active members, the Bulgarian Academy of Sciences lost one of its hopes for a better and more successful development, Bulgaria lost one quiet, but uncompromising patriot ...

- Aug 2012

Cell movement is a complex process. Cells can move in response to a foreign stimulus in search of nutrients, to escape predation, and for other reasons. Mathematical modeling of cell movement is needed to aid in achieving a deeper understanding of vital ...

The present book contains the Proceedings of the UNESCO International Workshop QED: Re-Designing Institutional Policies and Practices, to Enhance the Quality of Education through Innovative Use of Digital Technologies, held in Sofia, Bulgaria on 14-16 June 2010. It is the forerunner of the UNESCO activities at the State University of Library Studies and Information Technologies (SULSIT). The QED is the natural continuation of EDUsummIT 2011: Building a Global Community of Policy-Makers, Educators, and Researchers to Move Education into the Digital Age.
The QED workshop was an action taken by UNESCO’s Teaching Policy and Development Section, co-orgnized with the SULSIT, Sofia, Bulgaria, together with The Bulgarian Academy of Sciences, The Bulgarian Ministry of Education, Youth and Science and the Bulgarian National Commission for UNESCO. It has been successfully implemented and has achieved multi-direction positive results and impact. Useful ideas and practices have been discussed, which will continue to be built up and used in the future including the following ones:
• to re-establish the Children in the Information Age Initiative and Conference with the support of UNESCO, IFIP, and other international organizations
• to establish forums and communities of practice for cross-stakeholder ICT in Education communication
• to develop and establish a repository of open ICT in Education best practices (SULSIT) and to develop and launch an initiative for multi-lingual and multi-cultural communication and exchange of best practices in ICT in Education.
Although presented in a condensed form here the contributions reflect the main ideas conveyed by their authors at the QED workshop and will hopefully serve as an inspirational source for further work towards advancing education into the digital age.

- Jan 2011

Let p(z) be an algebraic polynomial of degree n >= 2 and D(p) be the smallest disk containing all zeroes of p(z). According to the classical Theorem of Grace, if p(z) and q(z) are apolar polynomials, then D(p) contains at least one zero of q(z) and D(q) contains at least one zero of p(z). For every polynomial p(z) the minimal closed and simply connected domain Omega(p) subset of D(p), called apolar locus of p(z), is defined, such that Omega(p) contains at least one zero of q(z) and Omega(q) contains at least one zero of p(z). A polynomial q(z), apolar to p(z), is extreme for Omega(p) if all zeroes of q(z) are on the contour of Omega(p). The main result in this paper is: If q(z) is extreme for Omega(p), then all zeroes of q(z) are on a circle. This is an instrument to find the apolar locus of a given polynomial. Specially, it is possible to find sharp Rolle's theorem for complex polynomials.

- Jan 2011

A connection between polar derivatives and apolarity is established and a sharp version of Laguerre Theorem is proved.

- Jan 2010

Let p(z) = (z - z1)(z - z2) (z - zn) be a polynomial of degree n ≥ 2 with distinct zeroes. In this paper is proved that the distance from every zero z1 of p(z) to the closest critical point of p(z) is not bigger than the distance of z1 to any other zero of p(z) divided by 2 sin π/n.

- Jan 2010

The purpose of the paper is to present some old and new conjectures in the geometry of polynomials.

- Jan 2010

Let D(C(p);R(p)) be the smallest disk containing all zeros of the polynomial p(z) = (z - z1)(z - z2) (z - zn). Half a century ago, we conjectured that for every zero zk of p(z), the disk D(zk;R(p)) contains at least one zero of the derivative p'(z). In this paper a stronger conjecture is announced and proved for polynomials of degree n = 3. A similar, stronger than the Smale's mean value conjecture, is formulated.

- Oct 2007

Mathematical methods for image processing make use of function spaces which are usually Banach spaces with integral norms. The corresponding mathematical models of the images are functions in these spaces. There are discussions here involving the value of for which the distance between two functions is most natural when they represent images, or the metric in which our eyes measure the distance between the images. In this paper we argue that the Hausdorff distance is more natural to measure the distance (difference) between images than any norm.

- Oct 2007

The paper is a survey of the literature on the theory of approximation in the Hausdorff metric, and certain related questions.
In the first chapter the definition of the Hausdorff distance is given, together with some of its properties. The relation between the Hausdorff and uniform distances is also discussed.
The second chapter gives a survey of results relating to the calculation of ?-entropy, ?-capacity, and widths relative to the Hausdorff distance.
A central position is occupied by the third chapter, where a number of estimates are given of the best approximation of functions and curves in the plane relative to the Hausdorff distance. A theorem is proved here on the existence of a universal estimate of the best approximation relative to the Hausdorff distance for all bounded functions. The question of the approximation of convex functions and curves by polygons, relative to the uniform and Hausdorff distances, is treated separately.
Chapter 4 is devoted to linear approximations relative to the Hausdorff distance and the convergence of sequences of positive and convex linear operators.
In a short final chapter a new problem in the theory of approximations is proposed.

Classical Rolle's theorem and its analogues for complex algebraic polynomials are discussed. A complex Rolle's theorem is conjectured. 1. Introduction. The classical theorem of Rolle states that if p(x) is a real polynomial, a; b are two dieren t real numbers, a < b, and p(a) = p(b), then there exists 2 (a; b), such that p0( ) = 0. As linear transformations of the complex plane do not change the geometric relations between the zeros and the critical points of a polynomial, we may consider only the points a = 1; b = 1. There are many statements that are considered renemen ts of the classical Rolle theorem. Every such a renemen t has the following structure: Let Kn be the class of real polynomials p(x) of degree n, n 2, with p( 1) = p(1) and n > 0. Then every p 2 Kn has at least one critical point in the interval ( 1 + n; 1 n).

Using massive computer calculations for finding the maximum of the modulus of a complex function with up to 8 complex variables, we verify the Mean Value Conjecture of S. Smale for n ≤ 10. The main burden was that the modulus of this function has many local extremes.

- Oct 2006

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.

We show that the operations addition and multiplication on the set $C(\Omega)$ of all real continuous functions on $\Omega\subseteq\mathbb{R}^n$ can be extended to the set $\mathbb{H}(\Omega)$ of all Hausdorff continuous interval functions on $\Omega$ in such a way that the algebraic structure of $C(\Omega)$ is preserved, namely, $\mathbb{H}(\Omega)$ is a commutative ring with identity. The operations on $\mathbb{H}(\Omega)$ are defined in three different but equivalent ways. This allow us to look at these operations from different points of view as well as to show that they are naturally associated with the Hausdorff continuous functions.

- Dec 2005

Let be a linear operator on the set of monic algebraic polynomials with . Of interest here is the value
for various linear operators. The motivation is that Smale's mean value conjecture may be formulated as for the linear operator

- Jun 2005
- Large-Scale Scientific Computing, 5th International Conference, LSSC 2005, Sozopol, Bulgaria, June 6-10, 2005, Revised Papers

In the present work we show that the linear operations in the space of Hausdorff continuous functions are generated by an
extension property of these functions. We show that the supremum norm can be defined for Hausdorff continuous functions in
a similar manner as for real functions, and that the space of all bounded Hausdorff continuous functions on an open set is
a normed linear space. Some issues related to approximations in the space of Hausdorff continuous functions by subspaces are
also discussed.

- Aug 2002
- Numerical Methods and Applications, 5th International Conference, NMA 2002, Borovets, Bulgaria, August 20-24, 2002, Revised Papers

The purpose of this paper is to demonstrate the potential of a fruitful collaboration between Numerical Analysis and Geometry
of Polynomials. This is natural as the polynomials are still a very important instrument in Numerical Analysis, regardless
many new instruments as splines, wavelets and others. On the other hand, the Numerical Analysis through computers is a powerful
instrument for experimentation in almost every mathematical discipline.

- Jan 2002

In this paper we survey work on and around the following conjecture, which was first stated about 45 years ago: If all the zeros of an algebraic polynomial p (of degree n ≥ 2) lie in a disk with radius r, then, for each zero z1 of p, the disk with center z1 and radius r contains at least one zero of the derivative p'. Until now, this conjecture has been proved for n ≤ 8 only. We also put the conjecture in a more general frameworkinvolving higher order derivatives and sets defined by the zeros of the polynomials.

- Jul 1999

The classical Walsh functions have different generalizations called generalized Walsh functions, Walsh-like functions, multiplicative systems and others. All these generalizations are functions which correspond to the characters in a given zero-dimensional group. The Walsh functions correspond to the characters of the dyadic topological group. Another direction for generalization of the Walsh function is possible, called Walsh-similar functions, which in general do not correspond to the characters in the dyadic topological group and depend on a sequence of real numbers. The richness of the orthonormal systems of Walsh-similar functions offers the possibility to adapt such an orthonormal system to a particular function, which is useful for compression of signals and images. In this paper the convergence of the Fourier-Walsh-similar (FWS) series is studied as a generalization of the respective theorems from the Fourier-Walsh analysis. It is interesting to mention that for every 1≤κ<2 there exist Walsh-similar functions with fractal dimension κ, thus the FWS analysis is in fact a fractal analysis.

- Jul 1999
- Proceedings of the Fourth International Conference

- Feb 1999

A type of multiresolution analysis on the space of continuous functions defined on the dyadic topological group is proposed, depending on free parameters. The appropriate choice of parameters is used to adapt this analysis to a given function.

- Jan 1999

The classical orthonormal system of Walsh functions is generalized in a new direction, called Walsh-similar functions, different from the already well-known ones [5], [7], [9], [14], [21] and from the Generalized Walsh-like functions [10], [11]. The definition of the Walsh-similar functions involves real parameters and allows adaptation of the orthonormal system to a particular function by an appropriate choice of these parameters. A Walsh-similar function may have any given fractal dimension.

- Jan 1999
- Large Scale Computations in Air Pollution Modelling

A classical linear method for approximation of a function f from a Hilbert space H is the following. Take an orthonormal basis
\(\{ {\phi _i}\} _{i = 0}^\infty \)
in H. Calculate the Fourier coefficients
\({c_i}(f) = \left\langle {f,{\phi _i}} \right\rangle ;i = 0,1,2,...,n.\)
Take for an approximation of f the generalized polynomial
$${P_n}(f;x) = \sum\limits_{i = 0}^n {{c_i}} (f){\phi _i}(x).$$
The quality of this approximation depend on the function f, on n and on the choice of the orthonormal system
\(\{ {\phi _i}\} _{i = 0}^\infty \)
The aim of this paper is to present a method for construction of a orthonormal basis adapted to a given function f, based on a generalization of the Walsh functions, called Walsh-similar functions. The advantage of this adapted orthonormal basis is the improvement of the approximation for discontinuous functions and fractal functions. The motivation for this study is the application in the signal and image compression.

- Sep 1997

Conclusions Educational reform needs the synergetic efforts of UNESCO, the EU, national and local governments, policy makers, educators,
business communities, public interest groups, parents, citizens, and non-governmental organizations, such as IFIP and the
International Association for the Evaluation of Educational Achievements (IEA). It is a matter of crucial importance that
countries in transition participate in the EU’s educational and training initiatives and programmes, such as PHARE (including
TEMPUS), COPERNICUS, SOCRATES and LEONARDO, as well as in all UNESCO initiatives and projects.

- Mar 1996

The problem of finding appropriate mathematical objects to model images is considered. Using the notion of acompleted graph of a bounded function, which is a closed and bounded point set in the three-dimensional Euclidean spaceR
3, and exploring theHausdorff distance between these point sets, a metric spaceIM
D of functions is defined. The main purpose is to show that the functionsf∈IM
D, defined on the squareD=[0,1]2, are appropriate mathematical models of real world images.
The properties of the metric spaceIM
D are studied and methods of approximation for the purpose of image compression are presented.
The metric spaceIM
D contains the so-calledpixel functions which are produced through digitizing images. It is proved that every functionf∈IM
D may be digitized and represented by a pixel functionp
n, withn pixels, in such a way that the distance betweenf andp
n is no greater than 2n
−1/2.
It is advocated that the Hausdorff distance is the most natural one to measure the difference between two pixel representations
of a given image. This gives a natural mathematical measure of the quality of the compression produced through different methods.

- Jul 1994
- Third International Conference on Numerical Methods and Applications

- Jan 1994
- Parcella 1994, VI. International Workshop on Parallel Processing by Cellular Automata and Arrays, Potsdam, September 21-23, 1994. Proceedings

- Jan 1994

Using the notion of complete graph of a bounded function, which is a closed and bounded point set in the three dimensional Euclidean space R 3 , and exploring the Hausdorff distance between these sets, a metric space IM D of functions is defined. It is shown that the functions f∈IM D , defined on the square D=[0,1] 2 , are appropriate mathematical models of real world images. The metric space IM D contains pixel functions which are produced through digitizing images. It is proved that every function f∈IM D may be digitized and represented by a pixel function p n , with n pixels, in such a way, that the distance between f, and p n , is no greater than 2n -1/2 . We claim that the Hausdorff distance is the most natural distance to measure the difference between two pixel representations of a given image. This gives a natural mathematical measure for the quality of the compression produced by different methods. An O(nlogn) algorithm is proposed for calculating the image of the Hausdorff difference between two images represented with n pixels each.

- Jan 1993
- Informatics and Changes in Learning, Proceedings of the IFIP TC3/WG3.1/WG3.5 Open Conference on Informatics and Changes in Learning, Gmunden, Austria, 7-11 June, 1993

The topic `Informatics and Changes in Learning' of this conference is very rich and may be considered from historical, present and futuristic points of view. The history of the informatics influence on learning is rather short, but very important as a basis for predicting the future. The present situation is extremely dynamic and far from settled. This opens the door for many futuristic deliberations. In this opening keynote speech, I will share my experience and my visions in using informatics and information technologies in educational theory and in everyday educational practice. I will also discuss some general problems of education in the information era from the point of view of human development and world stability. One of the theses in this paper is that informatics is shaping a new era in the developed world where changes in education are influenced through information technologies, but in the future the character of the information technologies will depend mostly on the enacted education.

- Jan 1992
- Education and Society - Information Processing '92, Volume 2, Proceedings of the IFIP 12th World Computer Congress, Madrid, Spain, 7-11 September 1992

- Jan 1991
- International Youth Workshop on Monte Carlo Methods and Parallel Algorithms

- Jan 1990

In this chapter we shall consider the problem for the best approximation of segment functions with regard to the Hausdorff distance. We shall confine ourselves to the consideration of functions defined on a finite or infinite interval.

- Jan 1990
- Hausdorff Approximations

Converse theorems in approximation theory have their origins in the classical investigations of Bernstein. These theorems are of two types. Some are concerned with the existence of a function with preassigned approximations. In others, the rate of approximation is prescribed, from which one derives certain properties of the approximated function, e. g., membership to a given class of functions.

- Jan 1990
- Hausdorff Approximations

Practical application of the polynomials of best Hausdorff approximation requires numerical methods for the approximate determination of these polynomials. As was already noted §4.1, the polynomial of best Hausdorff approximation need not be unique. Therefore it is possible to construct numerical methods for the determination of the polynomial of best Hausdorff approximation by analogy to the method of E. Remez [1] for the uniform metric only if the conditions for uniqueness considered in §4.1.1 are fulfilled: the function f has a unique polynomial of best Hausdorff approximation of degree n for sufficiently large n if f is λ-monotonic (see Definition 4.4).

- Jan 1990
- Hausdorff Approximations

The concepts of ε-entropy and ε-capacity of function spaces were introduced by A. Kolmogorov [1] to characterize massiveness of these spaces. Let us provide the relevant terminology.

- Jan 1990
- Hausdorff Approximations

Let (M,ρ) be a metric space, i.e., to every pair a, b ∈ M there corresponds a nonnegative number ,ρ(a, b) such that for all a, b, c ∈ M $$\rho (a,b) = \rho (b,a) \geqslant 0$$ (2.1), $$\rho (a,b) = 0 \Leftrightarrow a = b$$ (2.2), $$\rho (a,b) \leqslant \rho (a,c) + \rho (c,b)$$ (2.3).

- Jan 1990
- Hausdorff Approximations

We shall consider multivalued functions with special attention paid to appropriate notation and operations. Actually, we shall need only a very special type of real multivalued functions with closed and bounded convex images, i.e., each image is a closed interval (segment) in the extended real line. Functions that take segment values will be called segment functions.

- Jan 1990
- Hausdorff Approximations

In this chapter we shall consider the question of the approximation of compact sets in the plane various classes of point sets. The Hausdorff distance is quite natural in this situation. We note that interesting research on the approximation of point sets in the plane with respect to Hausdorff distance has been conducted by P. Davis, R. Vitale, and E. Ben-Sabar [1].

- Jan 1990
- Hausdorff Approximations

An operator L, defined on BΩ ⊂ AΩ, is called linear if L(αf + βg;x) = αL(f; x) + βL(g; x), for all f, g ∈ BΩ and for all constants α, β, and it is called positive if L(f; x) ≥ 0 for all x ∈ Ω whenever f(x) ≥ 0 for all x ∈ Ω.

- Dec 1989

This article, taken from the author's keynote speech at the applications section of the Education and Informatics Conference, presents a broad overview of informatics and of the different ways it might be used in education. The paper draws a parallel between energetics (the basis of the industrial revolution) and informatics (the driving force in the move to the new age of information), and points out that informatics may itself be the subject of instruction, or it may be a supporting element in a broad range of subject areas. The paper considers the possible effects of artificial intelligence, and concludes that real education can never be completely automated because it involves creativity, which is arrived at through actions of the individual human being. Comment is made on the positive outcomes which are possible, and negative side-effects are recognized. The paper ends with a set of recommendations for increasing international cooperation in the area.

- Dec 1988

The plenary papers from this International Conference express fundamental ideas and views, which include:(1) No educational system can overlook the problem of computerization.(2) The computerization of education has both enthusiastic supporters and sceptical critics.(3) Well-designed, carefully conducted and precisely evaluated experiments are needed.(4) No computer can replace a teacher.(5) Although the computer is only a tool, it has created a need for reform of the systems and content of educational programmes.(6) The development of information technology means that we must teach students to teach themselves.(7) The desire and need for continuing study must be encouraged and developed.Lastly, the final goal of computerization should be to make people more human.

- Jan 1987

Whitney's famous theorem shows that the error of approximation to a functionf by algebraic polynomials of degree n can be estimated by thenth order modulus of smoothness off. We show that the constants in this theorem can be taken independent ofn.

- Jan 1987
- Fourteen Papers Translated from the Russian

- Jan 1984
- Anniversary Volume on Approximation Theory and Functional Analysis

Academician Lyubomir Iliev enters his seventies with unfailing creative spirit, with an enormous experience as an organizer of science and education of national and worldwide importance. His road as a scientist and still more as an organizer of science is most closely related to the remarkable advance of Bulgarian mathematics and the birth and development of computer technology in our country during the last three decades. The happy fusion of mathematical talent and organizational and political insight allowed Acad. Iliev to establish himself as the most authoritative and most popular leader of the Bulgarian mathematical community.

- Dec 1980

This chapter presents an overview of interval analysis. Interval analysis was introduced at the end of 50s in connection with the development of numerical methods for computers. In interval analysis, the notion derivative is introduced by using the general theory of multivalued functions. Another type of derivative that is naturally connected with the Hausdorff metric will be needed. This derivative is based on a certain extension of the notion limit so that every sequence has a limit. In principle, the derivative that is introduced is close to the subdifferential, which is widely used in convex analysis The definition of intersection of two segments completely coincides with the set-theoretic definition of intersection of two sets. In interval analysis, only finite intervals are usually used, and the arithmetic operations are not defined for all pairs of intervals. Thus, the division by an interval containing zero is excluded. Infinite intervals have been used in the interval arithmetic in a different way.

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