20 research outputs found
System of Biblical Characters in the Prose of Grygoriy Skovoroda
No full textThe article focuses on the issue of biblical characters in the prose of Grygoriy Skovoroda. The key characters in the God’s book (Jesus Christ, Moses, Paul and Lot) and their interpretation by Ukrainian writer are analyzed. A series of female biblical characters that help the author to show different interpretations of perception of given characters is represented. The article affirms that the key feature of Grygoriy Skovoroda’s works is reinterpreting the characters of the God’s book and giving them another meaning not just to show his own ideas but to show ordinary life truths
System of Biblical Characters in the Prose of Grygoriy Skovoroda
No full textThe article focuses on the issue of biblical characters in the prose of Grygoriy Skovoroda. The key characters in the God’s book (Jesus Christ, Moses, Paul and Lot) and their interpretation by Ukrainian writer are analyzed. A series of female biblical characters that help the author to show different interpretations of perception of given characters is represented. The article affirms that the key feature of Grygoriy Skovoroda’s works is reinterpreting the characters of the God’s book and giving them another meaning not just to show his own ideas but to show ordinary life truths
System of Biblical Characters in the Prose of Grygoriy Skovoroda
No full textThe article focuses on the issue of biblical characters in the prose of Grygoriy Skovoroda. The key characters in the God’s book (Jesus Christ, Moses, Paul and Lot) and their interpretation by Ukrainian writer are analyzed. A series of female biblical characters that help the author to show different interpretations of perception of given characters is represented. The article affirms that the key feature of Grygoriy Skovoroda’s works is reinterpreting the characters of the God’s book and giving them another meaning not just to show his own ideas but to show ordinary life truths
Topological and fractal properties of real numbers which are not normal
No full textAbstractThe set L of essentially non-normal numbers of the unit interval (i.e., the set of real numbers having no asymptotic frequencies of all digits in their nonterminating s-adic expansion) is studied in details. It is proven that the set L is generic in the topological sense (it is of the second Baire category) as well as in the sense of fractal geometry (L is a superfractal set, i.e., the Hausdorff–Besicovitch dimension of the set L is equal 1). These results are substantial generalizations of the previous results of the two latter authors [M. Pratsiovytyi, G. Torbin, Ukrainian Math. J. 47 (7) (1995) 971–975].The Q∗-representation of real numbers (which is a generalization of the s-adic expansion) is also studied. This representation is determined by the stochastic matrix Q∗. We prove the existence of such a Q∗-representation that almost all (in the sense of Lebesgue measure) real numbers have no asymptotic frequency of all digits. In the case where the matrix Q∗ has additional asymptotic properties, the Hausdorff–Besicovitch dimension of the set of numbers with prescribed asymptotic properties of their digits is determined (this is a generalization of the Eggleston–Besicovitch theorem). The connections between the notions of “normality of numbers” respectively of “asymptotic frequencies” of their digits is also studied
On fractal properties of non-normal numbers with respect to Renyi f-expansions generated by piecewise linear functions
No full textAlbeverio S, Kondratiev Y, Nikiforov R, Torbin G. On fractal properties of non-normal numbers with respect to Renyi f-expansions generated by piecewise linear functions. Bulletin des Sciences Mathématiques. 2014;138(3):440-455.The paper is devoted to the study of fractal properties of subsets of the set of non-normal numbers with respect to Renyi f -expansions generated by continuous increasing piecewise linear functions defined on [0, +infinity). All such expansions are expansions for real numbers generated by infinite linear IFS f = {f(0),f(1),..fn,...} with the following list of ratios Q infinity = (q(0),q(1)...qn,....). We prove the superfractality of the set of Q infinity-essentially non-normal numbers, i.e. real numbers having no asymptotic frequencies of any digits from the alphabet A = {0, 1,, n,...}, for any infinite stochastic vector Q infinity, independently of the finiteness resp. infiniteness of its entropy and independently of the faithfulness resp. non-faithfulness of the family of cylinders generated by these expansions. (C) 2013 Elsevier Masson SAS. All rights reserved
On new fractal phenomena connected with infinite linear IFS
No full textAlbeverio S, Kondratiev Y, Nikiforov R, Torbin G. On new fractal phenomena connected with infinite linear IFS. MATHEMATISCHE NACHRICHTEN. 2017;290(8-9):1163-1176.We establish several new fractal and number theoretical phenomena connected with expansions which are generated by infinite linear iterated function systems. We show that the systems of cylinders of generalized Luroth expansions are, generally speaking, not faithful for the Hausdorff dimension calculation. Using Yuval Peres' approach, we prove sufficient conditions for the non-faithfulness of such families of cylinders. On the other hand, rather general sufficient conditions for the faithfulness of such covering systems are also found. As a corollary, we obtain the non-faithfullness of the family of cylinders generated by the classical Luroth expansion. We also develop new approach to the study of subsets of Q8-essentially non-normal numbers and prove that this set has full Hausdorff dimension. This result answers the open problem mentioned in [2] and completes the metric, dimensional and topological classification of real numbers via the asymptotic behaviour of frequencies their digits in the generalized Luroth expansion. (C) 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinhei
On singularity of distribution of random variables with independent symbols of Oppenheim expansions
No full textFractal properties of singular probability distributions with independent Q*-digits
No full textAbstractThe problem of the determination of the Hausdorff dimension of sets via the special class of coverings generated by special partitions, called Q*-partitions, of the unit interval is discussed. Fractal properties of singular continuous probability measures with independent “Q*-symbols” are studied in details. Explicit formulae for the determination of the Hausdorff dimension of such measures as well as for the determination of the Hausdorff dimension of the topological supports of the corresponding measures are given
On fine fractal properties of generalized infinite Bernoulli convolutions
No full textAbstractThe paper is devoted to the investigation of generalized infinite Bernoulli convolutions, i.e., the distributions μξ of the following random variables:ξ=∑k=1∞ξkak, where ak are terms of a given positive convergent series; ξk are independent random variables taking values 0 and 1 with probabilities p0k and p1k correspondingly.We give (without any restriction on {an}) necessary and sufficient conditions for the topological support of ξ to be a nowhere dense set. Fractal properties of the topological support of ξ and fine fractal properties of the corresponding probability measure μξ itself are studied in details for the case where ak⩾rk:=ak+1+ak+2+⋯ (i.e., rk−1⩾2rk) for all sufficiently large k. The family of minimal dimensional (in the sense of the Hausdorff–Besicovitch dimension) supports of μξ for the above mentioned case is also studied in details. We describe a series of sets (with additional structural properties) which play the role of minimal dimensional supports of generalized Bernoulli convolutions. We also show how a generalization of M. Cooper's dimensional results on symmetric Bernoulli convolutions can easily be derived from our results
Spectral Properties of Image Measures Under the Infinite Conflict Interaction
No full textWe introduce the conflict interaction with two positions between a couple of image probability measures and consider the associated dynamical system. We prove the existence of invariant limiting measures and find the criteria for these measures to be a pure point, absolutely continuous, or singular cotinuous as well as to have any topological type and arbitary Hausdorff dimensio
