Browsing by Author "Mashurov F."

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    LAGRANGE’S THEOREM AND 2- CONTINUED FRACTION EXPANSION
    (СДУ хабаршысы - 2019, 2019) Kadyrov Sh. ; Mashurov F.
    Abstract. The simple continued fraction theory is a sub-branch of number theory that is well developed. One of the classical results is due to Lagrange which states that the simple continued fraction expansion of a real number has eventually periodic expansion if and only if it is quadratic irrational. Similar results are not available when one considers N-continued fraction expansion which is not so well developed theory. In this article, authors aim to provide computational evidence when a quadratic irrational may not necessarily have eventually periodic 2-continued fraction expansion. Moreover, a proof is provided for a special type of real numbers for which Lagrange’s theorem does hold.
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    Semi-Regular Continued Fractions with Fast-Growing Partial Quotients
    (MDPI Journal, 2024) Mashurov F.; Kadyrov Sh.; Kazin A.
    In number theory, continued fractions are essential tools because they provide distinct representations of real numbers and provide information about their characteristics. Regular continued fractions have been examined in great detail, but less research has been carried out on their semi-regular counterparts, which are produced from the sequences of alternating plus and minus ones. In this study, we investigate the structure and features of semi-regular continuous fractions through the lens of dimension theory. We prove a primary result about the Hausdorff dimension of number sets whose partial quotients increase more quickly than a given pace. Furthermore, we conduct numerical analyses to illustrate the differences between regular and semi-regular continued fractions, shedding light on potential future directions in this field.
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    Unified computational approach to nilpotent algebra classification problems
    (Sciendo Journal, 2021) Kadyrov Sh.; Mashurov F.
    In this article, we provide an algorithm with Wolfram Mathematica code that gives a unified computational power in classification of finite dimensional nilpotent algebras using Skjelbred-Sund method. To illustrate the code, we obtain new finite dimensional Moufang algebras.