Zdenka Kolar-Begović

Full Professor
Department of Mathematics
Josip Juraj Strossmayer University of Osijek
Trg Ljudevita Gaja 6
Osijek, HR-31000, Croatia¸
phone: +385-31-224-811
fax: +385-31-224-801
email:  zkolar @ mathos.hr
office:  2 (ground floor)

 


Research Interests

 nonassociative algebraic structures 

 geometry

Degrees

PhD in Mathematics, University of Zagreb, Croatia, 2003 

Msc in Mathematics, University of Zagreb, Croatia, 1999

Bsc in Mathematics and Physics,  University of Osijek, Croatia, 1993


Publications

Journal Publications

  1. V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, Cubic structure, Glasnik Matematički 52/2 (2017), 247-256
    In this paper we examine the relationships between cubic structures, totally symmetric medial quasigroups, and commutative groups. We prove that the existence of a cubic structure on the given set is equivalent to the existence of a totally symmetric medial quasigroup on this set, and it is equivalent to the existence of a commutative group on this set. We give also some interesting geometric examples of cubic structures. By means of these examples, each theorem that can be proved for an abstract cubic structure has a number of geometric consequences. In the final part of the paper, we prove also some simple properties of abstract cubic structures.
  2. R. Kolar-Šuper, Z. Kolar-Begović, V. Volenec, Steiner point of a triangle in an isotropic plane, Rad HAZU, Matematičke znanosti. 20/528 (2016), 83-95
    The concept of the Steiner point of a triangle in an isotropic plane is defined in this paper. Some different concepts connected with the introduced concepts such as the harmonic polar line, Ceva’s triangle, the complementary point of the Steiner point of an allowable triangle are studied. Some other statements about the Steiner point and the connection with the concept of the complementary triangle, the anticomplementary triangle, the tangential triangle of an allowable triangle as well as the Brocard diameter and the Euler circle are also proved.
  3. V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, Kiepert hyperbola in an isotropic plane, Rad HAZU, Matematičke znanosti. (2016), prihvaćen za objavljivanje
    The concept of the Kiepert hyperbola of an allowable triangle in an isotropic plane is introduced in this paper. Important properties of the Kiepert hyperbola will be investigated in the case of the standard triangle. The relationships between the introduced concepts and some well known elements of a triangle will also be studied.
  4. Z. Kolar-Begović, R. Kolar-Šuper, V. Volenec, Equicevian points and equiangular lines of a triangle in an isotropic plane, Sarajevo Journal of Mathematics 11/23 (2015), 101-107
    The concepts of equicevian points and equiangular lines of a triangle in an isotropic plane are studied in this paper. A number of significant properties of the introduced concepts are considered.
  5. V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, Affine Fullerene C_60 in a GS-Quasigroup, Journal of Applied Mathematics 2014 (2014), 1-8
    It will be shown that the affine fullerene C60, which is defined as an affine image of buckminsterfullerene C60, can be obtained only by means of the golden section. The concept of the affine fullerene C60 will be constructed in a general GS-quasigroup using the statements about the relationships between affine regular pentagons and affine regular hexagons. The geometrical interpretation of all discovered relations in a general GS-quasigroup will be given in the GS- quasigroup $C((1/2)(1+sqrt 5))$.
  6. V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, Reciprocity in an isotropic plane, Rad HAZU, Matematičke znanosti. 519/18 (2014), 171-181
    The concept of reciprocity with respect to a triangle is introduced in an isotropic plane. A number of statements about the properties of this mapping is proved. The images of some well known elements of a triangle with respect to this mapping will be studied.
  7. J. Beban-Brkić, V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, Cosymmedian triangles in an isotropic plane, Rad HAZU, Matematičke znanosti. 515/2013 (2013), 33-42
    In this paper the concept of cosymmedian triangles in an isotropic plane is defined. A number of statements about some important properties of these triangles will be proved. Some analogies with the Euclidean case will also be considered.
  8. V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, Crelle-Brocard points of the triangle in an isotropic plane, Mathematica Pannonica 24/2 (2013), 167-181
    In this paper the concept of Crelle-Brocard points of the triangle in an isotropic plane is defined. A number of statements about the relationship between Crelle-Brocard points and some other significant elements of a triangle in an isotropic plane are also proved. Some analogies with the Euclidean case are considered as well.
  9. J. Beban-Brkić, V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, On Gergonne point of the triangle in isotropic plane, Rad HAZU, Matematičke znanosti. 515/2013 (2013), 95-106
    Using the standard position of the allowable triangle in the isotropic plane relationships between this triangle and its contact and tangential triangle are studied. Thereby different properties of the symmedian center, the Gergonne point, the Lemoine line and the de Longchamps line of these triangles are obtained.
  10. V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, Affine regular icosahedron circumscribed around the affine regular octahedron in GS--quasigroup, Commentationes Mathematicae Universitatis Carolinae 53/3 (2012), 501-507
    The concept of the affine regular icosahedron and affine regular octahedron in a general GS- quasigroup will be introduced in this paper. The theorem of the unique determination of the affine regular icosahedron by means of its four vertices which satisfy certain conditions will be proved. The connection between affine regular icosahedron and affine regular octahedron in a general GS- quasigroup will be researched. The geometrical representation of the introduced concepts and relations between them will be given in the GS- quasigroup $mathbb{ C} ((frac{1}{2}(1+sqrt 5))$.
  11. Z. Kolar-Begović, A short direct characterization of GS-quasigroups, Czechoslovak Mathematical Journal 61/136 (2011), 3-6
    The theorem about the characterization of a GS- quasigroup by means of a commutative group in which there is an automorphism which satisfies certain conditions, is proved directly.
  12. V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, Affine-regular hexagons in the parallelogram space, Quasigroups and Related Systems 19 (2011), 353-358
    The concept of the affine-regular hexagon, by means of six parallelograms, is defined and investigated in any parallelogram space and geometrical interpretation in the affine plane is also given.
  13. V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, ARH-quasigroups, Mathematical Communications 16 (2011), 539-550
    In this paper, the concept of an ARH-quasigroup is introduced and identities valid in that quasigroup are studied. The geometrical concept of an affine-regular heptagon is defined in a general ARH-quasigroup and geometrical representation in the quasigroup $C(2 cos pi/7)$ is given. Some statements about new points obtained from the vertices of an affine-regular heptagon are also studied.
  14. V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, Kiepert triangles in an isotropic plane, Sarajevo Journal of Mathematics 7/19 (2011), 81-90
    In this paper the concept of the Kiepert triangle of an allowable triangle in an isotropic plane is introduced. The relationships between the areas and the Brocard angles of the standard triangle and its Kiepert triangle are studied. It is also proved that an allowable triangle and any of its Kiepert triangles are homologic. In the case of a standard triangle the expressions for the center and the axis of this homology are given.
  15. V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, ARO-quasigroups, Quasigroups and Related Systems 18 (2010), 213-228
    In this paper the concept of ARO-quasigroup is introduced and some identities which are valid in a general ARO-quasigroup are proved. The "geometric" concepts of midpoint, parallelogram and affine-regular octagon is introduced in a general ARO-quasigroup. The geometric interpretation of some proved identities and introduced concepts is given in the quasigroup $C(1+sqrt2/2)$.
  16. R. Kolar-Šuper, Z. Kolar-Begović, V. Volenec, Dual Feuerbach theorem in an isotropic plane, Sarajevo Journal of Mathematics 18 (2010), 109-115
    The dual Feuerbach theorem for an allowable triangle in an isotropic plane is proved analytically by means of the so-called standard triangle. A number of statements about relationships between some concepts of the triangle and their dual concepts are also proved.
  17. V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, Steiner's ellipses of the triangle in an isotropic plane, Mathematica Pannonica 21/2 (2010), 229-238
    The concept of the Steiner's ellipse of the triangle in an isotropic plane is introduced. The connections of the introduced concept with some other elements of the triangle in an isotropic plane are also studied.
  18. R. Kolar-Šuper, Z. Kolar-Begović, V. Volenec, Thebault circles of the triangle in an isotropic plane, Mathematical Communications 15 (2010), 437-442
    In this paper the existence of three circles, which touch the circumscribed circle and Euler circle of an allowable triangle in an isotropic plane, is proved. Some relations between these three circles and elements of a triangle are investigated. Formulae for their radii are also given.
  19. V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, Thebault's pencil of circles in an isotropic plane, Sarajevo Journal of Mathematics 18 (2010), 237-239
    In the Euclidean plane Griffiths's and Thebault's pencil of the circles are generally different. In this paper it is shown that in an isotropic plane the pencils of circles, corresponding to the Griffiths's and Thebault's pencil of circles in the Euclidean plane, coincide.
  20. Z. Kolar-Begović, R. Kolar-Šuper, V. Volenec, Brocard angle of the standard triangle in an isotropic plane, Rad HAZU, Matematičke znanosti. 503 (2009), 55-60
  21. V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, Heptagonal triangle as the extreme triangle of Dixmier-Kahane-Nicolas inequality, Mathematical Inequalities and Applications 12/4 (2009), 773-779
  22. Z. Kolar-Begović, V. Volenec, LGS-quasigroups, Quasigroups and Related Systems 17 (2009), 77-90
  23. V. Volenec, J. Beban-Brkić, R. Kolar-Šuper, Z. Kolar-Begović, Orthic axis, Lemoine line and Longchamp's line of the triangle in I_2., Rad HAZU, Matematičke znanosti. 503 (2009), 13-19
  24. Z. Kolar-Begović, R. Kolar-Šuper, V. Volenec, The second Lemoine circle of the triangle in an isotropic plane, Mathematica Pannonica 20/1 (2009), 79-86
  25. V. Volenec, Z. Kolar-Begović, Affine regular decagons in GS-quasigroups, Commentationes Mathematicae Universitatis Carolinae 49/3 (2008), 383-395
  26. Z. Kolar-Begović, R. Kolar-Šuper, V. Volenec, Angle bisectors of a triangle in I_2, Mathematical Communications 13/1 (2008), 97-105
  27. R. Kolar-Šuper, Z. Kolar-Begović, V. Volenec, Apollonius circles of the triangle in an isotropic plane, Taiwanese journal of mathematics 12/5 (2008), 1239-1249
    The concept of Apollonius circle and Apollonius axes of an allowable triangle in an isotropic plane will be introduced. Some statements about relationships between introduced concepts and some other previously studied geometric concepts about triangle will be investigated in an isotropic plane and some analogies with the Euclidean case will be also considered.
  28. R. Kolar-Šuper, Z. Kolar-Begović, V. Volenec, J. Beban-Brkić, Isogonality and inversion in an isotropic plane, International Journal of Pure and Applied Mathematics 44/3 (2008), 339-346
  29. V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper, Two characterizations of the triangle with the angles $ frac{pi}{7}, frac{2 pi}{7}, frac{4 pi}{7}$, International Journal of Pure and Applied Mathematics 44/3 (2008), 335-338
  30. Z. Kolar-Begović, R. Kolar-Šuper, Six concyclic points, Mathematical Communications 12 (2007), 255-256
  31. R. Kolar-Šuper, Z. Kolar-Begović, V. Volenec, The first Lemoine circle of the triangle in an isotropic plane, Mathematica Pannonica 18/2 (2007), 253-263
  32. Z. Kolar-Begović, V. Volenec, The meaning of computer search in the study of some classes of IM-quasigroups, Croatian Journal of Education 53 (2007), 293-297
  33. J. Beban-Brkić, R. Kolar-Šuper, Z. Kolar-Begović, V. Volenec, On Feuerbach's theorem and a pencil of circles in I_2, Journal for Geometry and Graphics 10/2 (2006), 125-132


Books

  1. Z. Kolar-Begović, R. Kolar-Šuper, Lj. Jukić Matić, Mathematics Education as a Science and a Profession, Odjel za matematiku i Fakultet za odgojne i obrazovne znanosti, Zagreb, 2017.
  2. Z. Kolar-Begović, R. Kolar-Šuper, I. Đurđević, Higher Goals in Mathematics Education , Odjel za matematiku, Fakultet za odgojne i obrazovne znanosti , Osijek, 2015.
  3. M. Pavleković, Z. Kolar-Begović, R. Kolar-Šuper, Mathematics teaching for the future, Odjel za matematiku, Fakultet za odgojne i obrazovne znanosti, Osijek, 2013.



Projects

Participation (as researcher) in work of the following projects:

  • Non associative algebraic structures and their application (Neasocijativne algebarske strukture i njihove primjene), Ministry of Science, Education and Sports of the Republic Croatia, Department of Mathematics, University Of Zagreb, Principal investigator: Vladimir Volenec

  • Geometric and algebraic geometric structures (Geometrije i algebarsko geometrijske strukture), Ministry of Science, Education and Sports of the Republic Croatia, Department of Mathematics, University Of Zagreb, Principal investigator: Vladimir Volenec

Professional Activities

Editorial Boards
Editor in Chief of the Journal Osječki matematički list (since 2012)

 

Committee Memberships
Member of the Scientific and Organizing Committee of the International Colloquium Mathematics and Children  (Osijek 2007, Osijek 2009, Osijek 2011, Osijek 2013, Osijek 2015)
Head of the Organizing Committee of the 5th International Scientific Colloquium Mathematics and Children (Osijek 2015)
 
Refereeing/Reviewing
Osječki matematički list
Mathematical Reviews 

Teaching

Konzultacije (Office Hours): Četvrtak (16:00 h).  Konzultacije su moguće i po dogovoru.

Elementarna Geometrija

Analitička geometrija

Konstruktivna Geometrija


Prijedlog tema diplomskih radova

1. Geometrija zlatnog reza

2. Inverzija u ravnini i primjene

3. Značajni pravci u geometriji trokuta

  Prijedlog tema završnih radova

1. Algebarska metoda rješavanja konstruktivnih zadataka

2. Euleorova kružnica

Personal

  • Birthdate: March 24, 1969
  • Birthplace: Sremska Mitrovica
  • Citizenship: Croatian
  • Family: Married with two children (Dolores, Alojzije)