Blagovest Sendov

Bulgarian Academy of Sciences | BAS · Institute of Mathematics and Informatics

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' a...

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 G...

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...

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...

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.

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 pap...

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 G...

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, i...

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 ref...

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 thi...

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 w...

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.

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 t...

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...

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...

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 Studie...

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 conn...

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

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.

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

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 simil...

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 th...

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...

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 transf...

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.

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. O...

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...

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

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...

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 throu...

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...

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. Anot...

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.

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 orth...

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) =...

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 mat...

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 pur...

Today, the tendency in developing numerical methods is oriented exclusively to the use of computers. The character of these methods is naturally influenced to be easily performed by computers. Many of the classical numerical methods were discovered long before the existence of the computers, but still represent a practical interest. One very basic...

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...

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....

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.

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...

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 b...

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.

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).

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...

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...

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.

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 t...

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 need...

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.

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 deve...

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...