info@viam.science.tsu.ge (+995 32) 2 30 30 40 (+995 32) 2 18 66 45

Alexander Kharazishvili

  • Discret Mathematics Department
    Head / Chief Researcher
  • Office: room 310, University str.11, 0186 Tbilisi, Georgia
  • Phone: +995 322 370442
  • E-mail: kharaz2@yahoo.com
  • Research fields: Discrete Mathematics, Mathematical Logic, Set Theory, Real Analysis, Measure Theory

Summary

Scientific Title:
1984 - Doctor of Mathematical and Physical's Sciences
2009 - Associate member of Georgian National Scientific Academy
2014 - Academician of Georgian National Scientific Academy

Work Experiences:
1. I. Vekua Institute of Applied Mathematics at Tbilisi State University, Young Researcher (1973-1976).
2. I. Vekua Institute of Applied Mathematics at Tbilisi State University, Senior Researcher (1976-1983).
3. I. Vekua Institute of Applied Mathematics at Tbilisi State University, Head of Department of Discrete Mathematics, Chief Researcher(1983- until today).
4. University of Lodz, Invited Professor , Poland (1994-1998).
5. TSU, A. Razmadze Mathematical Institute, Chief Researcher (2006-Until today).

Projects

Talks

  • On the generalized nonmeasurability of Vitali sets and Bernstein sets, by Alexander Kharazishvili (Speaker) at Conference of A. Razmadze Mathematical Institute, Tbilisi, Georgia, 16-18 February, 2021.
  • Some applications of the almost invariant sets , by Alexander Kharazishvili (Speaker) at Scientific Session Dedicated to the memory of professor Shalva Pkhakadze, Tbilisi, Georgia, 03 May, 2019.
  • Some remarks on the Steinhaus property for invariant extensions of the Lebesgue measure, by Alexander Kharazishvili (Speaker) at 46th Winter school in abstract analysis, Klenci, Czech Republic, 13-20 January, 2018.
  • On a multi-dimensional version of Sylvester's problem, by Alexander Kharazishvili (Speaker) at Workshop in Discrete Mathematics II, Tbilisi, Georgia, 27-28 November, 2017.
  • On generalized Delaunay systems, by Alexander Kharazishvili (Speaker) at Workshop in Discrete Mathematics, Tbilisi, Georgia, from 31 October to 01 November, 2016.
  • On the cardinal number of the family of all invariant extensions of a nonzero $\sigma$-finite invariant measure, by Alexander Kharazishvili (Speaker) at A. Razmadze Mathematical Institute Conference, Tbilisi, Georgia, 14-18 December, 2015.
  • ON A THEOREM OF LUZIN AND SIERPINSKI, by Alexander Kharazishvili (Speaker) at A. Razmadze Mathematical Institute Conference, Tbilisi, Georgia, 02-06 December, 2013.
  • On sums of absolutely non-measurable functions , by Alexander Kharazishvili (Speaker) at International Conference dedicated to the 120th anniversary of S. Banach, Lviv, Ukraine, 17-21 September, 2012.
  • Measurability properties of real-valued functions, by Alexander Kharazishvili (Speaker) at 40th Winter School in Abstract Analysis, Klenci, Czech Republic, 14-21 January, 2012.
  • On Some Topologic-Geometrical Properties of External Bisectors of a Triangle, by Alexander Kharazishvili (Speaker) at 10th International Conference on Geometry and Applications, Varna, Bulgaria, 03-09 September, 2011.
  • On Non-Elementary Methods in Elementary Geometry, by Alexander Kharazishvili (Speaker) at 9th International Conference on Geometry and Applications, Varna, Bulgaria, from 05 September to 09 October, 2009.
  • Some applications of almost disjoint families of sets, by Alexander Kharazishvili (Speaker) at International Conference on Modern Problems in Applied Mathematics, Tbilisi, Georgia, 07-09 October, 2008.

Publications

  • Alexander Kharazishvili, ON π-WEIGHTS AND EXTENSIONS OF INVARIANT MEASURES, Transactions of A. Razmadze Mathematical Institute, Volume 177, Issue 2, Pages 315 - 317, I. Javakhishvili Tbilisi State University, 2023.
  • Alexander Kharazishvili, On Some Version of Random Variables, Transactions of A. Razmadze Mathematical Institute, Volume 177, Issue 1, Pages 143 - 146, I. Javakhishvili Tbilisi State University, 2023.
  • Alexander Kharazishvili, Quasi-invariant measures on topological groups and ω-powers, Georgian Mathematical Journal, Walter de Gruyter, 2023.
  • Alexander Kharazishvili, Nonmeasurable products of absolutely negligible sets in uncountable solvable groups, Georgian Mathematical Journal, Volume 30, Issue 3, Pages 397 - 402, Walter de Gruyter, 2023.
  • Alexander Kharazishvili, On $T_2$-negligible $S_2$-absolutely nonmeasurable sets in the Euclidean plane, Bull. TICMI,  2021 /25, no. 1, 77-81, VIAM, 2021.
  • Alexander Kharazishvili, On nonmeasurable uniform subsets of the Euclidean plane, Trans. A. Razmadze Math. Inst., 2021 / 175, no. 2, 285-286, TSU, 2021.
  • Alexander Kharazishvili, On the generalized nonmeasurability of some classical point sets, Trans. A. Razmadze Math. Inst., 2021 / 175, no. 1, 151-153, TSU, 2021.
  • Alexander Kharazishvili, On oscillations of real-valued functions, Trans. A. Razmadze Math. Inst., 2021 /  175, no. 1, 63-67, TSU, 2021.
  • Alexander Kharazishvili, On finite sums of periodic functions, Georgian Math. J., 2020 / 27, no. 2, 265-269, De Gruyter, 2020.
  • Alexander Kharazishvili, Elements of Combinatorial Geometry, Part II, 258 pp., Georgian National Academy of Sciences., 2020.
  • Alexander Kharazishvili, Some remarks on Sierpiński-Zygmund functions in the strong sense, Georgian Math. J., 2020 / 27, no. 4, 569-575, De Gruyter, 2020.
  • Alexander Kharazishvili, A measure zero set in the plane with absolutely nonmeasurable linear sections, Trans. A. Razmadze Math. Inst., 2020 / 174, no. 3, 359-362, TSU, 2020.
  • Alexander Kharazishvili, On the generalized nonmeasurability of Vitali sets and Bernstein sets, Georgian Math. J., 2021 / 28, no. 4, 575–579, De Gruyter, 2020.
  • Alexander Kharazishvili, On orbits without the Baire property, Georgian Math. J., 2019 / 26, no. 4, 625-628, De Gruyter, 2019.
  • Alexander Kharazishvili, On the Steinhaus property and ergodicity via the measure-theoretic density of sets, Real Anal. Exchange, 2019 / 44, no. 1, 217-228, Michigan State University Press, 2019.
  • Alexander Kharazishvili, The Beginnings of Set Theory, Part 3, 250 pp., Georgian National Academy of Sciences., 2019.
  • Alexander Kharazishvili, Some remarks on the Steinhaus property for invariant extensions of the Lebesgue measure, Eur. J. Math., 2019 / 5, no. 1, 81-90, Springer, 2018.
  • Alexander Kharazishvili, A note on the Borel types of some small sets, Georgian Math. J., 2018 / 25, no. 3, 419-425, De Gruyter, 2018.
  • Alexander Kharazishvili, A note on the uniqueness property for Borel $G$-measures, Real Anal. Exchange, 2018 / 43, no. 1, 223-233, Michigan State University Press, 2018.
  • Alexander Kharazishvili, A characterization of sets containing absolutely nonmeasurable subsets, Georgian Math. J., 2017 / 24, no. 2, 211-215, De Gruyter, 2017.
  • Alexander Kharazishvili, On some classes of negligible subsets of the Euclidean plane, Georgian Math. J., 2018 / 25, no. 1, 41-46, De Gruyter, 2017.
  • Alexander Kharazishvili, Strange Functions in Real Analysis, 3rd expanded edition, 440 pp., Chapman and Hall/CRC, 2017.
  • Alexander Kharazishvili, Acute triangles in the context of the illumination problem, Ann. Sofia Univ., Fac. Math and Inf., 103, 2016, 39–44., FMI, 2016.
  • Alexander Kharazishvili, On bijective continuous images of absolute null sets, Ukrainian Math. J., 2016 / 67, no. 8, 1277-1282, Springer, 2016.
  • Alexander Kharazishvili, On negligible and absolutely nonmeasurable subsets of uncountable solvable groups, Trans. A. Razmadze Math. Inst., 2016 / 170, no. 1, 69-74, Elsevier , 2016.
  • Alexander Kharazishvili, On the cardinal number of the family of all invariant extensions of a nonzero $\sigma$-finite invariant measure, Trans. A. Razmadze Math. Inst., 2016 / 170, no. 2, 200-204, Elsevier , 2016.
  • Alexander Kharazishvili, On the difference between a Vitali-Bernstein selector and a partial Vitali-Bernstein selector, Georgian Math. J., 2016 / 23, no. 3, 387-391, De Gruyter, 2016.
  • Alexander Kharazishvili, Elements of combinatorial geometry I, 282 p, Georgian National Academy of Sciences., 2016.
  • Alexander Kharazishvili, Absolute null subsets of the plane with bad orthogonal projections, Real Anal. Exchange, 2016 / 41, no. 1, 233-243, Michigan State University Press, 2016.
  • Alexander Kharazishvili, On inscribed and circumscribed convex polyhedra, Proc. A. Razmadze Math. Inst., 2015 / 167, 123-129, TSU, 2015.
  • Alexander Kharazishvili, On the cardinalities of $at$-sets in a real Hilbert space, Georgian Math. J., 2015 / 22, no. 2, 259-263, De Gruyter, 2015.
  • Alexander Kharazishvili, A partition of an uncountable solvable group into three negligible subsets, Bull. TICMI, 2015 / 19, no. 1, 37-44, VIAM, 2015.
  • Alexander Kharazishvili, Three-colorings of the Euclidean plane and associated triangles of a prescribed type, Georgian Math. J., 2015 / 22, no. 3, 393-396, De Gruyter, 2015.
  • Alexander Kharazishvili, Set theoretical aspects of real analysis, Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2015. xxii+433 pp. ,  CRC Press, 2015.
  • Alexander Kharazishvili, A characterization of uncountable sets in terms of their self-mappings and large invariant subsets, Georgian Math. J., 2014 / 21, no. 3, 297-302, De Gruyter, 2014.
  • Alexander Kharazishvili, To the existence of projective absolutely nonmeasurable functions, Proceedings of A. Razmadze Mathematical Institute Vol. 166 (2014), 95–102, TSU, 2014.
  • Alexander Kharazishvili, On a theorem of Luzin and Sierpinski, Proc. A. Razmadze Math. Inst. 164(2014), 109–115, TSU, 2014.
  • Alexander Kharazishvili, On measurability properties of Bernstein sets, Proceedings of A. Razmadze Mathematical Institute Vol. 164 (2014), 63–70, TSU, 2014.
  • Alexander Kharazishvili, On partitions of the real line into continuum many thick subsets, Real Anal. Exchange, 2013-14 / 39, no. 2, 459-468, Michigan State University Press, 2014.
  • Alexander Kharazishvili, On some real-valued step-functions with strange measurability properties, Georgian Math. J., 2014 / 21, no. 1, 83-87, De Gruyter, 2014.
  • Alexander Kharazishvili, On countable almost invariant partitions of $G$-spaces, Ukr. Math. J. 66, 572–579, Springer, 2014.
  • Alexander Kharazishvili, On some pathological homomorphisms of uncountable commutative groups, Proc. A. Razmadze Math. Inst., 2013 / 162, 136-140, TSU, 2013.
  • Alexander Kharazishvili, Sums of absolutely nonmeasurable functions, Georgian Math. J., 2013 / 20, no. 2, 271-282, De Gruyter, 2013.
  • Alexander Kharazishvili, On the measurability properties of well-orderings, Georgian Math. J., 2013 / 20, no. 3, 533-545, De Gruyter, 2013.
  • Alexander Kharazishvili, Additive properties of certain classes of pathological functions, Real Anal. Exchange, 2012-13 / 38, no. 2, 475-486, Michigan State University Press, 2013.
  • Alexander Kharazishvili, Some unsolved problems in measure theory, Proc. A. Razmadze Math. Inst., 2013 / 162, 59-77, TSU, 2013.
  • Alexander Kharazishvili, On absolutely nonmeasurable homomorphisms of commutative groups, Studia Sci. Math. Hungar., 2013 / 50, no. 3, 287-295, Akademiai Kiado , 2013.
  • Alexander Kharazishvili, The Beginnings of Set Theory, Part 2, 238 p., Ilia State University Press, 2012.
  • Alexander Kharazishvili, A large group of absolutely nonmeasurable additive functions, Real Anal. Exchange, 2011-12 / 37, no. 2, 467-476, Michigan State University Press, 2012.
  • Alexander Kharazishvili, Some properties of $at$-sets and $ot$-sets in a Hilbert space, Proc. A. Razmadze Math. Inst., 2012 / 159, 147-151, TSU, 2012.
  • Alexander Kharazishvili, Some topologic-geometrical properties of external bisectors of a triangle, Georgian Math. J. , 2012 / 19, no. 4, 697-704, De Gruyter, 2012.
  • Alexander Kharazishvili, Ergodic measures and the definability of subgroups via normal extensions of such measures, Theory Stoch. Process., 2012 / 18, no. 1, 58-64, Institute of Mathematics, Ukrainian National Academy of Sciences , 2012.
  • Alexander Kharazishvili, Some geometric consequences of Ramsey's combinatorial theorem, Bull. TICMI, 2012 / 16, no. 1, 34-42, VIAM, 2012.
  • Alexander Kharazishvili, Some discrete geometric structures and associated algorithms, TSU / Proc. A. Razmadze Math. Inst., 2011 / 155, 138-144, TSU, 2011.
  • Alexander Kharazishvili, On a relationship between absolutely nonmeasurable functions and Sierpiński-Zygmund functions, De Gruyter / Georgian Math. J., 2011 / 18, no. 2, 249-258, De Gruyter, 2011.
  • Alexander Kharazishvili, On ordinary differential equations with bad right-hand sides, TSU / Proc. A. Razmadze Math. Inst., 2011 / 155, 135-137, TSU, 2011.
  • Alexander Kharazishvili, Measurability properties of Vitali sets, Amer. Math. Monthly, 2011 / 118, no. 8, 693-703, Mathematical Association of America, 2011.
  • Alexander Kharazishvili, A combinatorial problem on translation-invariant extensions of the Lebesgue measure, Expo. Math., 2011 / 29, no. 1, 150-158, Elsevier , 2011.
  • Alexander Kharazishvili, Some combinatorial problems on the measurability of functions with respect to invariant extensions of the Lebesgue measure, Acta Universitatis Carolinae. Mathematica et Physica, Vol. 51 (2010), No. Suppl, 57--65, Charles University in Prague, 2010.
  • Alexander Kharazishvili, Mathematical Sketches, part 2, 204 p, Ilia State University Press, 2010.
  • Alexander Kharazishvili, Aleks Petre Kirtadze, On Nonmeasurable Subgroups of Uncountable Solvable Groups, Georgian Mathematical Journal, Volume 14, Issue 1, 435-444, De Gruyter, 2010.
  • Alexander Kharazishvili, On the Measurability of Additive Functionals, Georgian Mathematical Journal, Volume 14, Issue 1, 135-143, De Gruyter, 2010.
  • Alexander Kharazishvili, On Absolutely Nonmeasurable Sets and Functions, Georgian Mathematical Journal, Volume 15, Issue 2, 317-325, De Gruyter, 2010.
  • Alexander Kharazishvili, Tengiz Tetunashvili, On Some Coverings of the Euclidian Plane with Pairwise Congruent Circles, Amer. Math. Monthly, 117, no 5, 414-423, Mathematical Association of America, 2010.
  • Alexander Kharazishvili, On a relationship between the measurability and continuity of real-valued functions, Georgian Mathematical Journal, Volume 1, Issue 4, 649-661, De Gruyter, 2010.
  • Alexander Kharazishvili, ON NONMEASURABLE UNIONS OF MEASURE ZERO SECTIONS OF PLANE SETS, Proc. A. Razmadze Math. Inst. 154(2010), 137–143, Ltd. “Polygraph Tbilisi”, 2010.
  • Alexander Kharazishvili, On almost measurable real-valued functions, Studia Scientiarum Mathematicarum Hungarica, Volume 47: Issue 2, 257–266, Akademiai Kiado, 2010.
  • Alexander Kharazishvili, PIECEWISE AFFINE APPROXIMATIONS OF CONTINUOUS FUNCTIONS OF SEVERAL VARIABLES AND GALE POLYHEDRA, Proc. A. Razmadze Math. Inst. 152(2010), 133–140, Ltd. “Polygraph Tbilisi”, 2010.
  • Alexander Kharazishvili, Some combinatorial problems on the measurability of functions with respect to invariant extensions of the Lebesgue measure, Acta Universitatis Carolinae. Mathematica et Physica, Vol. 51 (2010), No. Suppl, 57--65, Charles University in Prague, 2010.
  • Alexander Kharazishvili, FINITE FAMILIES OF NEGLIGIBLE SETS AND INVARIANT EXTENSIONS OF THE LEBESGUE MEASURE, Proc. A. Razmadze Math. Inst. 151(2009), 119–123, Ltd. “Polygraph Tbilisi”, 2009.
  • Alexander Kharazishvili, METRICAL TRANSITIVITY AND NONSEPARABLE EXTENSIONS OF INVARIANT MEASURES, Taiwanese Journal of Mathematics Vol. 13, No. 3 pp. 943-949 , Mathematical Society of the Republic of China, 2009.
  • Alexander Kharazishvili, On sums of real-valued functions with extremely thick graphs, Expositiones Mathematicae Volume 27, Issue 2, Pages 161-169, Elsevier , 2009.
  • Alexander Kharazishvili, On Nonmeasurable Functions of Two Variables and Iterated Integrals, Georgian Mathematical Journal, Volume 16, Issue 4, 705-710, De Gruyter, 2009.
  • Alexander Kharazishvili, On thick subgroups of uncountable σ-compact locally compact commutative groups, Topology and its Applications Volume 156, Issue 14, Pages 2364-2369, Elsevier , 2009.
  • Alexander Kharazishvili, ALMOST MEASURABLE REAL-VALUED FUNCTIONS AND EXTENSIONS OF THE LEBESGUE MEASURE, Proc. A. Razmadze Math. Inst. 150(2009), 135–138, Ltd. “Polygraph Tbilisi”, 2009.
  • Alexander Kharazishvili, Tengiz Tetunashvili, On some combinatorial problems concerning geometrical realizations of finite and infinite families of sets, Georgian Math. J., 15, no 4, 665-675 , De Gruyter, 2008.
  • Alexander Kharazishvili, Tengiz Tetunashvili, Combinatorial properties of families of sets and Euler-Venn diagrams, Trans. A.Razmadze Math. Inst., 146, 115-119, GCI , 2008.
  • Alexander Kharazishvili, The Beginnings of Set Theory, Part 1, 227 p, I. Chavchavadze State University Press, 2008.
  • Alexander Kharazishvili, ON A BAD DESCRIPTIVE STRUCTURE OF MINKOWSKI’S SUM OF CERTAIN SMALL SETS IN A TOPOLOGICAL VECTOR SPACE, Theory of Stochastic Processes Vol. 14 (30), no. 2, 2008, pp. 35–41, Institute of Mathematics, Ukrainian National Academy of Sciences, 2008.
  • Alexander Kharazishvili, A Nonseparable Extension of the Lebesgue Measure Without New Nullsets, Real Anal. Exchange 33(1): 263-274 , Michigan State University Press, 2008.
  • Alexander Kharazishvili, Aleks Petre Kirtadze, On measurability of algebraic sums of small sets, Studia Scientiarum Mathematicarum Hungarica, Volume 45: Issue 3, 433–442, Akademiai Kiado, 2008.
  • Alexander Kharazishvili, Aleks Petre Kirtadze, On Weakly Metrically Transitive Measures and Nonmeasurable Sets, Real Anal. Exchange 32(2): 553-562, Elsevier , 2007.
  • Alexander Kharazishvili, Aleks Petre Kirtadze, On extensions of partial functions, Expositiones Mathematicae, Volume 25, Issue 4, Pages 345-353, Elsevier , 2007.
  • Alexander Kharazishvili, Some Combinatorial Properties of Finite Line-Systems in the Euclidean Plane, Georgian Mathematical Journal, Volume 14, Issue 4, 681-686, De Gruyter, 2007.
  • Alexander Kharazishvili, Mathematical Sketches, Part 1, 198 p, Ilia State University Press, 2007.
  • Alexander Kharazishvili, ERGODIC COMPONENTS AND NONSEPARABLE EXTENSIONS OF INVARIANT MEASURES, Proc. A. Razmadze Math. Inst. 145(2007), 95–99, GCI, 2007.
  • Alexander Kharazishvili, SOME ASPECTS OF THE MEASURE EXTENSION PROBLEM, Proceedings of A. Razmadze Mathematical Institute Vol. 143 (2007), 73–78, GCI, 2007.
  • Alexander Kharazishvili, On Some Mutual Positions of Hyperplanes in a Finite-Dimensional Affine Space, Georgian Mathematical Journal, Volume 13 Issue 1, 101-108, De Gruyter, 2006.
  • Alexander Kharazishvili, On additive absolutely nonmeasurable Sierpiński-Zygmund functions., Real Anal. Exchange 31 (2) 553 - 560, Michigan State University Press, 2006.
  • Alexander Kharazishvili, On Measurable Sierpiński-Zygmund Functions, Journal of Applied Analysis Volume 12 Issue 2, 283-292, De Gruyter, 2006.
  • Alexander Kharazishvili, On Decompositions of a Cube into Cubes and Simplexes, Georgian Mathematical Journal, Volume 13 Issue 2, 285-290, De Gruyter, 2006.
  • Alexander Kharazishvili, Strange functions in real analysis. 2nd ed, Pure and Applied Mathematics (Boca Raton) 272(275), 415p., Chapman & Hall/CRC, 2006.
  • Alexander Kharazishvili, Nonmeasurable sets and functions, North-Holland Mathematics Studies,  Elsevier Science B.V., Amsterdam,  xii+337 pp. , Elsevier , 2004.