Публикации, представени за конкурса - абстракти
1.) TRANSITIVE FUNCTIONS OVER THE CIRCLE, Proc. of the VII conf. of the
Un. Bulg. Math., 343 - 346.
P. Stoev, Vl. Todorov
Let X be a vector field on the smooth manifold M. It is well known that X may be
considered as adifferential operator in C   M  . In this note we prove that X does
not admit a transitive element whenever it has a closed integral curve.
2.)
Доклади на Българската академия на науките
Comptes rendus de l’Academie bulgare des Sciences
Tome 47, No 10, 1994
PHYSIQUE
Physiqae des solides
FIRST ORDERP HASET RANSITION: INTERPEASE SURFACE MORPHOIOGY AND EVOTUTION
' S. Bushev , V. Todorov*
(Submitted by Academician M. Bofissov on July 27, 1994)
Introduction. The interphases urfacea t first order phaset ransition is disturbed
at microscopicle vel[ 1-4] beciuse of two main teasons: 1 ) oversaturation; 2 ) anisotropy
of the crystal surface energy. The chief macro and microscopice vents normally determining
tle crystallization are presented in [5]. In that paper we suggest a generalized
mathematica method for studying the morfhology and evolution of the interphase su rface at
that transition.
3.)
CONCEPT FOR TOPOLOGICAL MODEL OF NEW PHASE
FORMATION (with S. Bushev and N. Miloshev), Compt. Rend. of Bulg. Acad. Sci., vol. 53,
No 9, 39 - 42, 2000.
4.)
5.)
6.)
Applications of Mathematics in Engineering and Economics
30-тh Jubilee International Conference, June 7 - 11, 2001, Sozopol,
Technical Uniuersi,ty of Sofia (Bulgaria), 2005, 183-185.
Finite - dimensional skeletons of cardinal cubes
C.I. Zarеva and V.T. Todorov
Abstract: Let t  w0 be some cardinal number (we denote by w0  0 the countable cardinality).
For Q = [0, 1] let us denote by Qτ the τ-dimensional cube (the τ-power of Q). The well
known fact is that the set Kτ of vertices of Qτ is homeomorphic to the Cantor set of weight τ.
Actually, one may regard Kτ as the 0-dimensional skeleton of Qτ .
t
It turns out that the finite - dimensional skeletons Qn ; n   of Qτ possess a similar exotic
topological characteristics. One can construct, for example, a piece-wise copy of the classical
indecomposable continuum of Knaster - Kuratowski.
7.)
МАТЕМАТИКА И МАТЕМАТИЧЕСКО ОБРАЗОВАНИЕ, 2004
MATHEMATICS AND EDUCATION IN MATHEMATICS, 2004
Proceedings of the Thirty Third Spring Conference of
the Union of Bulgarian Mathematicians
Borovets, April 1–4, 2004
EUCLIDEAN MEMBRANES
Atanas Hamamdjiev, Simeon Stefanov, Vladimir Todorov
It is a classical result from dimension theory that each n-dimensional compact space
contains an n-dimensional Cantor manifold. An important example, after Urysohn,
is a minimal compact subset containing an essential n-system. In this note it is shown
that each PL pseudomanifold with boundary is minimal (in the sense of Zorn) with
respect to some essential system whose frame coincides with its boundary.
8.)
МАТЕМАТИКА И МАТЕМАТИЧЕСКО ОБРАЗОВАНИЕ, 2005
MATHEMATICS AND EDUCATION IN MATHEMATICS, 2005
Proceedings of the Thirty Fourth Spring Conference of
the Union of Bulgarian Mathematicians
Borovets, April 6–9, 2005
A LOWER BOUND FOR THE DIMENSION DIAMETERS OF
CERTAIN SETS WITH RESPECT TO ESSENTIAL SYSTEMS
Vladimir T. Todorov
Let E = {(Ai,Bi)}, i = 1, . . . , n be an essential system in the normal space X. We
prove in this paper, that if U is a finite open covering of X and ord U≥ n, then some
element of U intersects two opposite faces of E. Various consequences of this result
are discussed.
9.)
МАТЕМАТИКА И МАТЕМАТИЧЕСКО ОБРАЗОВАНИЕ, 2005
MATHEMATICS AND EDUCATION IN MATHEMATICS, 2005
Proceedings of the Thirty Fourth Spring Conference of
the Union of Bulgarian Mathematicians
Borovets, April 6–9, 2005
USING CALCULUS FOR SOLVING CUBIC EQUATIONS
Vladimir T. Todorov
The problem of solving cubic equations is not among the favorite subjects in teaching
at the standard high schools in Bulgaria. Probably the reason is that in the “algebraic” method of view the procedure of solving the cubic equations requires some
knowledge about what is a complex number.
In this note we make an attempt to offer a “non algebraic” point of view for solving
cubic equations using just the typical humble abilities of our middle school educational
system at present.
10.)
МАТЕМАТИКА И МАТЕМАТИЧЕСКО ОБРАЗОВАНИЕ, 2006
MATHEMATICS AND EDUCATION IN MATHEMATICS, 2006
Proceedings of the Thirty Fifth Spring Conference of
the Union of Bulgarian Mathematicians
Borovets, April 5-8, 2006
ВРЪЗКИ МЕЖДУ ОТНОШЕНИЯ НА ДЪГИ И ОТСЕЧКИ
Владимир Т. Тодоров, Михаил М. Константинов, Петър В. Стоев
В тази бележка е предложен метод за пресмятане на по дадени пропорции в дъги и (или) отсечки. Ще отбележим, че
такива задачи (според нас) не се разглеждат достатъчно добре в нашите средни училища. Като слефствия се
получават разнообразни (побякога известни) резултати. Като пример е получен аналог на теоремата на Чева за
описана окръжност.
11.)
12.)
МАТЕМАТИКА И МАТЕМАТИЧЕСКО ОБРАЗОВАНИЕ, 2006
MATHEMATICS AND EDUCATION IN MATHEMATICS, 2006
Proceedings of the Thirty Fifth Spring Conference of
the Union of Bulgarian Mathematicians
Borovets, April 5_8, 2006
ГЪСТОТА НА РЕДИЦАТА sin(n2) В [0, 1]
Велика Драгиева, Симеон Стефанов, Владимир Тодоров
Получено е едно геометрично доказателство на факта, че редицата sin(n2) е гъста в интервала [-1,1].
13.)
Applications of Mathematicsin Engineering& Economics,32
eds. M. S. Marinov and M. D. Todorov
Softtrade Ltd., Sofia. 2007
Strongn -DimensionaClo nriectednesosf
Polyhedrons
SimeionS tefanovV, ladimirT odorov
Department of Mathematics UACG, 1, Hr. Smirnenski Blvd.
1421 Sofia, Bulsaria
Abstract. Let ε be a positive number and n be an integer. By definition, the
metric space (X,p) is (n,ε)- connected between its subsets P and Q if the
n-dimensionar diameter of every partition between P and Q is greater than ε.
In this note we study minimal (n,ε) – connected polyhedra.
14.)
Applications of Mathematics in Engineering & Economics’32
eds. M. S. Marinov and M. D. Todorov
Softtrade Ltd., Sofia, 2007
Theory of Differentiation of Matrix Power Functions
M. M. Konstantinov1, J. K. Boneva1, V. T. Todorov1,
and P.H. Petkov2
1Department of Mathematics,
University of Architecture, Civil Engineering and Geodesy
1046 Sofia, Bulgaria
2Department of Automatics, Technical University of Sofia
1756 Sofia, Bulgaria
Abstract. In this paper we derive a theory for differentiation of matrix power
functions X  f p ( X ), p   , at positive definite arguments X = A
(0 A A  K
H
n2
), where K is the field of real or complex numbers and
'
AH is the complex conjugate transpose of A. The Frech´et derivative f p ( A) : K
n2
 Kn
2
of f at A is an invertible Lyapunov operator which is studied in detail. The results obtained are
applicable to the perturbation analysis of certain classes of non-linear matrix equations as well
as to the affine approximation of matrix power function.
15.)
МАТЕМАТИКА И МАТЕМАТИЧЕСКО ОБРАЗОВАНИЕ, 2007
MATHEMATICS AND EDUCATION IN MATHEMATICS, 2007
Proceedings of the Thirty Sixth Spring Conference of
the Union of Bulgarian Mathematicians
St. Konstantin & Elena resort, Varna, April 2–6, 2007
DIMENSION-RAISING FUNCTIONS AND MULTIPLE
INTEGRALS
Vladimir T. Todorov, Petar V. Stoev,
Mihail M. Konstantinov, Simeon T. Stefanov
Let G  Rn be a connected domain in the n¡dimensional space. It is well known
that in fact G is a space-filling curve. At this juncture it is not astonishing that one
can represent the multiple integral over G as a single one. Moreover, every metrizable
compactum X is the image of the classical Cantor set C under a continuous map
·κ :C  X. Then we can put

X
f   f k.
C
16.)
МАТЕМАТИКА И МАТЕМАТИЧЕСКО ОБРАЗОВАНИЕ, 2008
MATHEMATICS AND EDUCATION IN MATHEMATICS, 2008
Proceedings of the Thirty Seventh Spring Conference of
the Union of Bulgarian Mathematicians
Borovetz, April 2-6, 2008
TRANSITIVE OPERATORS ON RR
*
Simeon T. Stefanov, Vladimir T. Todorov
Let be a topological space and   . The map :!is said to be transitive
if there exists an element 2for which the forward orbit RR +()=f()j2g is a dense subset of . In this paper we
consider an example of a transitive map in the non countable product = RR (recall that has an uncountable weight).
17.)
МАТЕМАТИКА И МАТЕМАТИЧЕСКО ОБРАЗОВАНИЕ, 2008
MATHEMATICS AND EDUCATION IN MATHEMATICS, 2008
Proceedings of the Thirty Seventh Spring Conference of
the Union of Bulgarian Mathematicians
Borovetz, April 2-6, 2008
HIGHER ORDER FRECHET DERIVATIVES OF MATRIX
POWER FUNCTIONS*
Mihail Mihaylov Konstantinov, Juliana Kostadinova Boneva,
Petko Hristov Petkov, Vladimir Todorov Todorov
We study higher order Frechet derivatives of matrix power functions 7! p, 2Q,
2Cnxn. The results obtained may be applied to the accuracy estimation of Taylor
approximations of matrix power functions as well as to the perturbation analysis of
non linear matrix equations.
18.)
МАТЕМАТИКА И МАТЕМАТИЧЕСКО ОБРАЗОВАНИЕ, 2008
MATHEMATICS AND EDUCATION IN MATHEMATICS, 2008
Proceedings of the Thirty Eighth Spring Conference of
the Union of Bulgarian Mathematicians
Borovetz, April 1-5, 2009
A COMPATIBLE METRIC FOR COMPUTING THE
DIMENSION DIAMETERS OF SUBSETS OF ESSENTIAL
SYSTEMS
Vladimir Todorov, Atanas Hamamdjiev, Simeon Stefanov
Let X be a normal space. In this note we propose a construction of a pseudometric r
in X which allows an easy calculation of the n-dimensional diameters of some subsets
of X. Some consequences of this result are discussed.
19.) Дискусия.
20.) Дискусия.
21.)
CPl 184, Applications of Mathematics in Engineering and Economics
edited by G. Venkov, R. Kovacheva, and V. Pasheva
© 2009 American Institute of Physics 978-0-7354-0750-3/09/
The coquatemion algebra and complex partial
differential equations
Stancho Dimiev*, Mihail Konstantinov† and Vladimir Todorov**
* Institute of Mathematics and Informatics,
of the Bulgarian Academy of Sciences,
Sofia 1113, Bulgaria,
E-mail: sdimiev@math.bas.bg,
Univ. of Architecture, Civil Engineering and Geodesy,
Sofia, bul. Hristo Smirnenski N1, Bulgaria
^Univ. of Architecture, Civil Engineering and Geodesy,
Sofia, bul. Hristo Smirnenski N1, Bulgaria
"Univ. of Architecture, Civil Engineering and Geodesy,
Sofia, bul Hristo Smirnenski N1, Bulgaria
E-mail: tt.vladimir®gmail.com
Abstract. In this paper we consider the problem of differentiation of coquaternionic functions. Let
us recall that coquatemions are elements of an associative non-commutative real algebra with zero
divisor, introduced by James Cockle (1849) under the name of split-quaternions or coquaternions.
Developing two type complex representations for Cockle algebra (complex and paracomplex ones)
we present the problem in a non-commutative form of the  -type holomorphy. We prove that
corresponding differentiable coquaternionic functions, smooth and analytic, satisfy PDE of complex,
and respectively of real variables. Applications for coquaternionic polynomials are sketched.
Keywords: Coquaternions, non-commutative differential, complex and paracomplex representations,
complex PDE, hyperbolic PDE.
22.)
МАТЕМАТИКА И МАТЕМАТИЧЕСКО ОБРАЗОВАНИЕ, 2010
MATHEMATICS AND EDUCATION IN MATHEMATICS, 2010
Proceedings of the Thirty Ninth Spring Conference of
the Union of Bulgarian Mathematicians
Albena, April 6–10, 2010
ЗА ИЗВЪНКЛАСНИТЕ ФОРМИ НА ОБУЧЕНИЕ ПО
МАТЕМАТИКА
Стоянка Вълчева, Владимир Тодоров
Средствата и “енергията”, които се отделят през последните години за преподаване на
“нехуманитарни науки” за съжаление са нерастяща функция на времето и това изглежа е
тендеция не само в България. Ето защо според нас е важно да се усилва и узаконява така
наречената “извънкласна” форма на преподаването и обучението по математика в средните
училища. Това ще допринесе за запазването на интереса към математиката у тези ученици,
които имат желание и (или) по-изявени способности за изучаване на естествените науки.
23.)
МАТЕМАТИКА И МАТЕМАТИЧЕСКО ОБРАЗОВАНИЕ, 2010
MATHEMATICS AND EDUCATION IN MATHEMATICS, 2010
Proceedings of the Thirty Ninth Spring Conference of
the Union of Bulgarian Mathematicians
Albena, April 6–10, 2010
ИЗПОЛЗВАНЕ НА СИСТЕМАТА MATLAB В
ТЕХНИЧЕСКИТЕ УНИВЕРСИТЕТИ*
Михаил Константинов, Владимир Тодоров, Галина Пелова,
Юлиана Бонева
Разгледани са някои аспекти на обучението по математика с помощта на програмната система
MATLAB1 в техническите университети. Специално внимание е отделено на разделите
aналитична геометрия и диференциални уравнения.
24.)
МАТЕМАТИКА И МАТЕМАТИЧЕСКО ОБРАЗОВАНИЕ, 2010
MATHEMATICS AND EDUCATION IN MATHEMATICS, 2010
Proceedings of the Thirty Ninth Spring Conference of
the Union of Bulgarian Mathematicians
Albena, April 6–10, 2010
EVERY n-DIMENSIONAL SEPARABLE METRIC SPACE
CONTAINS A TOTALLY DISCONNECTED (n − 1)-DIMENSIONAL
SUBSET WITH NO TRUE QUASI-COMPONENTS
Vladimir Todorov, Petar Stoev
The quasi-component Q(x) of a point x of a topological space X is by definition the intersection of all open and
closed subsets of X, every one of which contains x. If a quasi-component consists of more than one point, it is
called a true quasi-component. In this note we give a simple construction of (at least) (n − 1)-dimensional totally
disconnected subspace Y of a given n-dimensional separable metric space X such that every quasi-component
in Y is a single point.
25.)
МАТЕМАТИКА И МАТЕМАТИЧЕСКО ОБРАЗОВАНИЕ, 2011
MATHEMATICS AND EDUCATION IN MATHEMATICS, 2011
Proceedings of the Fortieth Jubilee Spring Conference
of the Union of Bulgarian Mathematicians
Borovetz, April 5–9, 2011
MINIMAL SUBSPACES WITH MAXIMAL DIMENSIOANAL
DIAMETERS
Vladimir Todorov
Suppose that X is a compact metric space with dimX = n. Then for the n − 1
dimensional diameter dn-1(X) we have dn-1(X) > 0 and in the same time dn(X) = 0.
It follows now that X contains a minimal by inclusion closed subset Y for which
dn-1(Y ) = dn-1(X). Under these conditions Y is a Cantor manifold [7]. In this note
we prove that every such subspace Y is even a continuum Vn. Various consequences
are discussed.
26.)
CPl 184, Applications of Mathematics in Engineering and Economics
edited by G. Venkov, R. Kovacheva, and V. Pasheva
© 2010 American Institute of Physics 978-0-7354-0750-3/10/
“PATHOLOGICAL'' CANTOR MANIFOLDS
Vladimir Todorov, Atanas Hamamdjiev
Department of Mathematics, University of Architecture, Civil Engineering
and Geodesy, 1046 Sofia, Bulgaria; vttp@yahoo.com
The n-dimensional compact topological spaceis is called to be a CantorManifotd(CM), if it is not a sum
of two proper closed subsets with (n-2)- dimensional intersection. It is by definition a Strongly Cantor
Manifold (SCN), if it is not a sum of countable sum of proper closed subsets with (n-2)- dimensional
intersections. We shall call our space a Pathological Cantor Manifold (PCM), if it is CM, but not SCM.
In this note we give some examples to investigate “how bad” can be the structure of PCM’s.
27.)
Questions and answers in General topology
30 (2012), pp. 93-102
GENERALIZED CANTOR MANIFOLDS AND
INDECOMPOSABLE CONTINUA
V. TODOROV AND V. VALOV
(Communicated by Yasunao Hattori)
Abstract. We review results concerning homogeneous compacta and discuss some open questions. It
is established that indecomposable continua are Alexandroff (resp., Mazurkiewicz, or strong
Cantor) manifolds with respect to the class of all continua. We also provide some new proofs of Bing’s
theorems about separating metric compacta by hereditarily indecomposable compacta.
28.)
CPl 184, Applications of Mathematics in Engineering and Economics
edited by G. Venkov, R. Kovacheva, and V. Pasheva
© 2012 American Institute of Physics 978-0-7354-0750-4/12/
TRANSITIVE PROPERTIES OF SOME
PARTIAL DIFFERENTIAL OPERATORS
Vladimir Todorov
Department of Mathematics, University of Architecture, Civil Engineering
and Geodesy, 1046 Sofia, Bulgaria, e-mail: tt.vladimir@gmail.com
Abstract: A partial differential operator L : C  (  n )  C  (  n ) is by definition transitive, if one
can find a function f  C  (  n ) for which the forward orbit OL ( f )   Lp ( f ) | p   is a
dense subset of C  (  n ) . Here as usual the forward orbit OL ( f ) consists of the iterations of
L: Lp  L  L  L (p times). It is shown [1] that if
L(u )  bu   aaau
0|a | p
is a partial differential operator with constant coefficients then L is transitive if it does not contains a
term of the type bu; in other words if b=0. In this note we show that it is possible to omit the condition
b=0 for operators of first order as well for some operators of a Dirac type.
29.)
Houston Journal of Mathematics
© 2012 University of Houston
Volume 38, No. 2, 2012
GENERALIZED CANTOR MANIFOLDS AND HOMOGENEITY
A. KARASSEV, P. KRUPSKI, V. TODOROV, AND V. VALOV
Communicated by Charles Hagopian
Abstract. A classical theorem of Alexandroff states that every n-dimensional compactum X contains an ndimensional Cantor manifold. This theorem has a number of generalizations obtained by various authors. We
consider extension-dimensional and infinite dimensional analogs of strong Cantor manifolds, Mazurkiewicz
manifolds, and Vn - continua, and prove corresponding versions of the above theorem. We apply our results to
show that each homogeneous metrizable continuum which is not in a given class C is a strong Cantor manifold
(or at least a Cantor manifold) with respect to C. Here, the class C is one of four classes that are de_ned in terms
of dimension-like invariants. A class of spaces having bases of neighborhoods satisfying certain special onditions
is also considered.
30.)
Applications of Mathematics in Engineering and Economics (AMEE 2012)
edited by G. Venkov, R. Kovacheva, and V. Pasheva
© 2012 American Institute of Physics 978-0-7354-1111-1/12/
Cyclic hypercomplex number systems
Stancho Dimiev*, Peter S toev *' and Vladimir Todorov*'*
Department of Mathematics, University of Architecture, Civil Engineering
and Geodesy, 1046 Sofia, Bulgaria;
* sdimiev@math.bas.bg
*’ peteruasg@abv.bg
*’* tt.vladimir@gmail.com
Abstract. We consider formal hypercomplex number systems of special type, namely
introduced here polynomial type number systems. The corresponding matrix algebras of this
type are always associative of real dimension 2k, k is an integer. Our problem is related with
the extension of function theory over the mentioned algebras, like the classical case in the
dimension 2, i.e. k=1. The difference is in the fact the classical case we have division and in
the considered here algebras we have non-divisions ones.
31.)
ALEXANDROFF TYPE MANIFOLDS AND
HOMOLOGY MANIFOLDS
V. TODOROV AND V. VALOV
Abstract. One of the open questions in generalized Cantor manifolds is whether every
n 2
Alexandroffmanifold with respect to the class DG
of all spaces whose cohomological
n 2
dimension dimG is ≤n-2, where G is an abelian group, is a Mazurkiewicz DG manifold.
In our search for the answer of this question we introduce and investigate different
types of connectedness between disjoint subsets of compacta, and prove that any two
n
disjoint open subset of a strong K G -manifold X can not be separated by a Lindeloff
n 1
normally placed subset M with Hˆ
M ; G  0 (in particular, any such X is a
n 2
Mazurkiewicz DG
-manifold). We also establish a result implying that if X is arcwise
connected complete metric space which is either a homology n-manifold over a group G
or a product of n metric spaces, then X is a Mazurkiewicz arc n-manifold.
Remark: the paper is proposed to Transaction of AMS.
32.)
ALEXANDROFF MANIFOLDS AND HOMOGENEOUS
CONTINUA
A. KARASSEV, V. TODOROV, AND V. VALOV
Abstract. We prove the following result announced in [18]: Any homogeneous, metric ANR-continuum
is a V n,G-continuum provided dimG X = n ≥ 1 and Hˆ
n 1
M ; G  0 , where G is a principal ideal
domain. This implies that any homogeneous n-dimensional metric ANR-continuum is a V n-continuum
in the sense of Alexandroff [1]. We also prove that any finite-dimensional homogeneous metric con-
M ; G  0 for some group G and n ≥ 1, cannot be separated by a
Hˆ n1  K ; G   0 and dimG K ≤ n-1. This provides a partial answer to a question
tinuum X, satisfying Hˆ
compactum K with
n 1
of Kallipoliti-Papasoglu [11] whether any two-dimensional homogeneous Peano continuum cannot be
separated by arcs.
Author’s draft: to be published in the Canadian Mathematical Bulletin dol:10.4153/CMB-2013-010-8