Публикации, представени за конкурса - абстракти

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1 Публикации, представени за конкурса - абстракти 1.) TRANSITIVE FUNCTIONS OVER THE CIRCLE, Proc. of the VII conf. of the Un. Bulg. Math., P. Stoev, Vl. Todorov Let X be a vector field on the smooth manifold M. It is well known that X may be C M. In this note we prove that X does considered as adifferential operator in 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, )

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.

For Q = [0, 1] let us denote by Q τ the τ-dimensional cube (the τ-power of Q). The well known fact is that the set K τ of vertices of Q τ is homeomorphic to the Cantor set of weight τ.

2 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, 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 Q τ the τ-dimensional cube (the τ-power of Q). The well known fact is that the set K τ of vertices of Q τ is homeomorphic to the Cantor set of weight τ. Actually, one may regard K τ as the 0-dimensional skeleton of Q τ. t It turns out that the finite - dimensional skeletons Q ; n of Q τ possess a similar exotic topological characteristics. One can construct, for example, a piece-wise copy of the classical indecomposable continuum of Knaster - Kuratowski. n

3 7.) МАТЕМАТИКА И МАТЕМАТИЧЕСКО ОБРАЗОВАНИЕ, 2004 MATHEMATICS AND EDUCATION IN MATHEMATICS, 2004 Proceedings of the Thirty Third Spring Conference of 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 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 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.)

4 МАТЕМАТИКА И МАТЕМАТИЧЕСКО ОБРАЗОВАНИЕ, 2006 MATHEMATICS AND EDUCATION IN MATHEMATICS, 2006 Proceedings of the Thirty Fifth Spring Conference of 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 Borovets, April 5_8, 2006 ГЪСТОТА НА РЕДИЦАТА sin(n 2 ) В [0, 1] Велика Драгиева, Симеон Стефанов, Владимир Тодоров Получено е едно геометрично доказателство на факта, че редицата sin(n 2 ) 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

5 Strongn -DimensionaClo nriectednesosf Polyhedrons SimeionS tefanovv, ladimirt odorov Department of Mathematics UACG, 1, Hr. Smirnenski Blvd 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 ( X ), p, at positive definite arguments X = A p 2 H n ( 0 A A K ), where K is the field of real or complex numbers and A H is the complex conjugate transpose of A. The Frech et derivative 2 2 n n f ' ( A) : K K 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. p 15.) МАТЕМАТИКА И МАТЕМАТИЧЕСКО ОБРАЗОВАНИЕ, 2007 MATHEMATICS AND EDUCATION IN MATHEMATICS, 2007 Proceedings of the Thirty Sixth Spring Conference of 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 R n 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 f f k. X C

6 16.) МАТЕМАТИКА И МАТЕМАТИЧЕСКО ОБРАЗОВАНИЕ, 2008 MATHEMATICS AND EDUCATION IN MATHEMATICS, 2008 Proceedings of the Thirty Seventh Spring Conference of Borovetz, April 2-6, 2008 TRANSITIVE OPERATORS ON R R* 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 R R + ( )=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 = R R (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 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, 2C nxn. 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 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.

7 21.) CPl 184, Applications of Mathematics in Engineering and Economics edited by G. Venkov, R. Kovacheva, and V. Pasheva 2009 American Institute of Physics /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, 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 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 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

8 Proceedings of the Thirty Ninth Spring Conference of 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 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 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 d n-1 (X) we have d n-1 (X) > 0 and in the same time d n (X) = 0. It follows now that X contains a minimal by inclusion closed subset Y for which d n-1 (Y ) = d n-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 /10/

9 PATHOLOGICAL'' CANTOR MANIFOLDS Vladimir Todorov, Atanas Hamamdjiev Department of Mathematics, University of Architecture, Civil Engineering and Geodesy, 1046 Sofia, Bulgaria; 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 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 /12/ TRANSITIVE PROPERTIES OF SOME PARTIAL DIFFERENTIAL OPERATORS Vladimir Todorov Department of Mathematics, University of Architecture, Civil Engineering and Geodesy, 1046 Sofia, Bulgaria, tt.vladimir@gmail.com n n Abstract: A partial differential operator L : C ( ) C ( ) is by definition transitive, if one n can find a function f C ( ) for which the forward orbit O ( ) p L f L ( f ) p is a n dense subset of C ( ). Here as usual the forward orbit O ( f ) consists of the iterations of L: p L L L L (p times). It is shown [1] that if L( u) bu a a u 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. a L

10 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 n- dimensional 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 /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 2 k, 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

11 Abstract. One of the open questions in generalized Cantor manifolds is whether every Alexandroffmanifold with respect to the class n 2 D G of all spaces whose cohomological dimension dim G is n-2, where G is an abelian group, is a Mazurkiewicz n 2 D G 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 KG -manifold X can not be separated by a Lindeloff ˆ n 1 H M ; G 0 (in particular, any such X is a normally placed subset M with Mazurkiewicz n 2 D G -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 ˆ n 1 H M ; G 0, where G is a principal ideal is a V n,g-continuum provided dimg X = n 1 and 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- ˆ n 1 H M ; G 0 for some group G and n 1, cannot be separated by a tinuum X, satisfying n 1 compactum K with H K G ˆ ; 0 and dimg K n-1. This provides a partial answer to a question 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: /cmb

Hyperbolic matrix type geometric structures

Hyperbolic matrix type geometric structures Stancho Dimiev and Peter Stoev Abstract. In this paper multidimensional hyperbolic matrix algebras are investigated from geometric point of view. A 2-dimensional