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Let V be an infinite-dimensional vector space over a field of characteristic not equal to 2. We classify ideals of the Lie algebra gl(V) of all linear transformations of the space V., 9 pages

The paper is dedicated to the memory of an outstanding ukrainian mathematician and teacher, one of the founders of Kyiv school of representation theory and ring theory, professor of Taras Shevchenko National University of Kyiv Volodymyr Vasylovych Kyrychenko. Volodymyr Vasylovych was born in 1942 in Penza. He graduated Kyiv University with honors in 1964. He defended his PhD thesis “Representations of hereditary, completely decomposable and Bass orders” under the supervision of D.K. Faddeev. He defended his habilitation “Modules and structural theory of rings” in 1986. He started to work at Kyiv University as an assistant of the Algebra and Mathematical Logic Department in 1967. Since then he served as an Associate Professor and Professor of the Algebra and Mathematical Logic Department. He was the head of the Geometry Department from 1988 till 2016. His main scientific results lay in the field of structural ring theory. In particular, he introduced the notion of the quiver for some special casses of rings and developed the structural theory of hereditary semiserial semidistributive rings. He is the author of about 250 scientific papers and 5 fundamental monographies. In his monographies the theory of semiperfect rings is presented by means of correspond- ing quivers. The monography “Finite dimensional algebras” written jointly with professor Y. Drozd is regarded one of the classical works in algebra and translated from Russian to Chineese, English and Spanish. He was an outstanding teacher and 36 PhD thesis were defended under his supervision. His lectures on analytical geometry and ring theory were always top ranked among students. Professor Kyrychenko was one of the founders of the journal “Algebra and Discrete Mathematics”. He was an editor of a couple of mathematical journals. In 1997 he was one of the founders of the series of international algebraic conferences in Ukraine that were conducted every two years. Since that he was one of the main organizers of each of them., Статтю присвячено пам’ятi видатного українського математика i педагога, одного iз засновникiв київської школи теорiї зображень i теорiї кiлець, професора Київського нацiонального унiверситету iменi Тараса Шевченка Володимира Васильовича Кириченка.

Biotope spaces were introduced by Marchevsky-Steinhaus in for the needs of mathematical biology, namely the study of ecosystems. Biotope distance is defined on the set of all subsets of some finite set X. The distance between any subsets A1 and A2 of X is calculated by the rule: d(A1, A2) = (0, if A1 = A2 = ∅; |A1⊕A2| |A1∪A2| , if A1, A2 ∈ B(X)).We introduce a new generalization of a biotope metric to the infinite case using supernatural or Steinitz numbers. A supernatural number (or Steinitz number) is an infinite formal product of the form Y p∈P p kp where P is the set of all primes and kp ∈ N ∪ {0, ∞}. On the set of all periodic {0, 1}-sequences with the period that is a divisor of some supernatural u; we define the metric dB for any infinite periodic sequences x¯ and y¯ by the rule: dB(¯x, y¯) = dBn (¯xn, y¯n) where n is a common period of periodic sequences x¯ and y¯, and the formula dB(¯xn, y¯n) denotes the biotope distance between the first n coordinates of sequences x¯ and y¯ in the finite biotope metric space Bn. We denote the periodic biotope space that is defined by some Steinitz number u as B(u). If u is a finite Steinitz number, i.e. u is a positive integer, then B(u) is isometric finite biotope space Bu. We also prove that the introduced metric between such two periodic sequences does not depend on a choice of a common period.A family of such introduced periodic biotope spaces is naturally parametrized by supernatural numbers. More precisely, the family of these spaces forms a lattice that is isomorphic to the lattice of supernatural numbers. Moreover, each of these spaces B(u) is invariant with respect to the shift.We prove that the diametr of any periodic biotope space equals 1. We also show that any finite subset of a countable biotope space introduced in is isometric embedding in the periodic biotope space B(u) for any u., У статтi введено узагальнення бiотопної метрики на нескiнченний випадок. Побудовано родину перiодичних бiотопних просторiв, елементами яких є перiодичнi {0, 1}-послiдовностi, перiоди яких є дiльниками супернатуральних чисел. Причому введена метрика мiж двома такими перiодичними послiдовностями не залежить вiд вибору спiльного перiоду. Доведено, що перiодичнi бiотопнi простори природним чином параметризуються супернатуральними числами. Точнiше, родина цих просторiв утворює вiдносно операцiї включення решiтку, iзоморфну решiтцi супернатуральних чисел. Кожен iз введених таким чином просторiв є iнварiантним вiдносно зсуву, тобто зсув є iзометрiєю для довiльного з таких просторiв.

The article is devoted to the 150th anniversary of the birth of outstanding Ukrainian mathematician Georgy Feodosiyovych Voronoi who has woldwide recognition. Georgy Voronoi was born on April 28, 1868 in the small town of Zhuravka near Chernihiv. Georgy Voronoi first followed school at Berdyansk and later attended the Pryluky gymnasium from which he graduated in 1885. His first scientific paper was entitled "Decomposition of the polynomials on factors based on the properties of the roots of quadratic equation" published in Elementary Math. J. in Kyiv in 1885. Latter he studied in the University of St Petersburg on the Faculty of Physics and Mathematics. He was awarded a Master’s Degree in 1894 for the dissertation on the algebraic integers associated with the roots of an irreducible cubic equation. Andrey Markov was his supervisor. He worked as math professor at the University of Warsaw and published a number of scientific papers with significant results. He wrote his doctoral thesis on algorithms for continued fractions. The main scientific results of Georgy Voronoi were published in his papers "Sur quelques propriet´ es des formes quadratiques positives´ parfaites" and "Recherches sur les paralleloedres primitifs". In the last of them Georgy Voronoi introduced mathematical objects that are nowadays called Voronoi diagrams or Voronoi tessellations. Voronoi diagram is a partitioning of a plane into cells based on the distance to points in a specific subset of the plane. Nowadays Voronoi diagrams are one of the main objects of investigation in computational geometry. They are used in robotics, navigation systems, ecology, economic models, architecture, and clothing design, etc. There are many generalizations of Voronoi diagrams. We describe his life path and significant dates and events in his life. We consider also Georgy Voronoi’s main mathematical results and publications., Статтю присвячено 150-й рiчницi вiд дня народження видатного українського математика Георгiя Феодосiйовича Вороного. Описано його життєвий шлях, основнi математичнi результати i публiкацiї.