13 research outputs found
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 the -isomorphism of probability and dimensional theories of representations of real numbers and fractal faithfulness of systems of coverings
No full textAsymptotic behavior of the module of the characteristic Cantor distribution function
No full textThe asymptotic behavior of the modulus of a characteristic function of a random variable, the distribution function of which is the classical singular Cantor function, is investigated. The emphasis is on calculating the upper bound of the modulus of the characteristic Cantor distribution function. The probabilistic measure corresponding to Cantor\u27s distribution belongs to the class of Bernoulli\u27s symmetric convolutions, the interest in which is considerable today. Bernoulli\u27s symmetrical convolutions were actively studied by both domestic mathematicians: M. Pratsovyty, G. Turbin, G. Torbin, J. Honcharenko, O. Baranovsky and others, and foreign ones: Erdos P, Peres Y, Schlag W, Solomyak B, Albeverio, S and other. The value of the upper bound of the modulus of the characteristic function plays an important role in the problem of determining the Lebesgue structure of distributions of sums of probably convergent random series with independent discrete terms (random values of the Jessen-Winter type).
The exact value of the upper bound of the module of the characteristic Cantor distribution function is found in the article.
Pages of the article in the issue: 63 - 68
Language of the article: Ukrainia
On a generalization of the concept of normal numbers
No full textThe paper considers the generalization of the concept of normal numbers in the context of the classical s-th representation of real numbers, in relation to the Q_s-representation, first considered by M. Pratsiovytyi. The result of I. Nivena and H. Zukerman is deepened in relation to the metric theory of normal E. Borel numbers. It is shown that the set of all Q_s-normal numbers has a Lebesgue measure 1. The connection between the property of normality and the uniform distribution of the sequence of numbers generated by the shift operator in relation to the corresponding number is established. It was found that the set of all numbers of the segment [0; 1] for which the corresponding sequence generated by the operator of left-hand shift Q_s-digits is uniformly distributed has a full Lebesgue measure. The corresponding theorems deepen the results of the metric theory Q_s-decompositions of real numbers of the segment [0; 1] obtained by M. Pratsiovytyi and G. Torbin.
Pages of the article in the issue: 58 - 62
Language of the article: Ukrainia
Author Correction: A MHz-repetition-rate hard X-ray free-electron laser driven by a superconducting linear accelerator
No full textAnalysis of the effects of a Constructivist-Based Mathematics Problem Solving Instructional Program on the achievement of Grade Five Students in Belize, Central America.
No full textThis thesis examined whether social constructivist activities can improve the mathematical competency of grade five students in Belize, Central America. The sample included 342 students and eight teachers from two rural and urban schools. A switching replication design was employed enabling students in the experimental groups to be taught using social constructivist activities for 12 weeks and the controls exposed to similar instructional practices from weeks 7 to 12. Students‘ performance was assessed using Pre-test, Post test 1 and 2 with an internal consistency of 0.89, 0.90 and 0.93 respectively. As revealed by the repeated measures ANOVA within subject analysis, there were significant differences among the pre-test and post test 1 and 2 results. That is, students in the control groups, who were instructed using a procedural approach from weeks 1 to 6, demonstrated higher gains than the experimental groups who were immersed in social constructivist activities. Furthermore, when the control groups became immersed in similar activities from weeks 7 to 12, they continued to outperform the experimental groups who were exposed to social constructivist activities alone. Hence, due to this unexpected result, the aim of this thesis became to explain why these results came about and what implications for teaching were highlighted by the consideration.
Besides the quantitative results highlighted above, qualitative data was also obtained as part of the study. For example, students were videoed within constructivist math groups and their performance analyzed using Pirie and Kieren‘s (1994) Model of Growth for Mathematical Understanding. The data from the video recording revealed that use of one step math problems did not enabled students to restructure their thinking to solve innovative problems. Data from semi-structured interviews also revealed that some students lacked basic math skills and were not exposed or guided to solve complex problems. Besides the need for careful examination of social constructivist activities on performance, this thesis underscores the importance of relevant teaching and learning activities, the important role of teachers during social constructivist activities and the need to identify suitable forms of assessment to measure performance
Metagenomic insights into diazotrophic communities across Arctic glacier forefields
No full textMicrobial nitrogen fixation is crucial for building labile nitrogen stocks and facilitating higher plant colonization in oligotrophic glacier forefield soils. Here, the diazotrophic bacterial community structure across four Arctic glacier forefields was investigated using metagenomic analysis. In total, 70 soil metagenomes were used for taxonomic interpretation based on 185 nitrogenase (nif) sequences, extracted from assembled contigs. The low number of recovered genes highlights the need for deeper sequencing in some diverse samples, to uncover the complete microbial populations. A key group of forefield diazotrophs, found throughout the forefields, was identified using a nifH phylogeny, associated with nifH Cluster I and III. Sequences related most closely to groups including Alphaproteobacteria, Betaproteobacteria, Cyanobacteria and Firmicutes. Using multiple nif genes in a Last Common Ancestor analysis revealed a diverse range of diazotrophs across the forefields. Key organisms identified across the forefields included Nostoc, Geobacter, Polaromonas and Frankia. Nitrogen fixers which are symbiotic with plants were also identified, through the presence of root associated diazotrophs, which fix nitrogen in return for reduced carbon. Additional nitrogen fixers identified in forefield soils were metabolically diverse, including fermentative and sulphur cycling bacteria, halophiles and anaerobes
Fractal properties of the distributions of some class of random variables of the Jessen-Wintner type
No full textSIGLEAvailable from TIB Hannover: RS 2745(67) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman
