https://ojs.utlib.ee/index.php/ACUTM/issue/feed Acta et Commentationes Universitatis Tartuensis de Mathematica 2021-08-09T01:57:16+03:00 Imbi Traat imbi.traat@ut.ee Open Journal Systems <p><em>Acta et Commentationes Universitatis Tartuensis de Mathematica&nbsp;</em>(ACUTM) is an international journal of pure and applied mathematics.</p> https://ojs.utlib.ee/index.php/ACUTM/article/view/ACUTM.2021.25.01 On the degree of approximation of continuous functions by a specific transform of partial sums of their Fourier series 2021-08-09T01:56:45+03:00 Xhevat Krasniqi xhevat.krasniqi@uni-pr.edu <p><span class="fontstyle0">Using the Mean Rest Bounded Variation Sequences or the Mean Head Bounded Variation Sequences, we have proved four theorems pertaining to the degree of approximation in sup-norm of a continuous function </span><em><span class="fontstyle2">f </span></em><span class="fontstyle0">by general means </span><span class="fontstyle2">τ<sup><span class="fontstyle3">λ</span></sup></span><span class="fontstyle3"><sub>n;A</sub></span><span class="fontstyle0">(</span><em><span class="fontstyle2">f</span></em><span class="fontstyle0">) of partial sums of its Fourier series. The degree of approximation is expressed via an auxiliary function&nbsp;</span><em><span class="fontstyle2">H</span></em><span class="fontstyle0">(</span><em><span class="fontstyle2">t</span></em><span class="fontstyle0">) </span><span class="fontstyle4">≥ </span><span class="fontstyle0">0 and via entries of a matrix whose indices form a strictly increasing sequence of positive integers </span><span class="fontstyle2">λ </span><span class="fontstyle0">:= {</span><span class="fontstyle2">λ</span><span class="fontstyle0">(</span><span class="fontstyle2">n</span><span class="fontstyle0">)}<sup>∞</sup></span><sub><span class="fontstyle3">n</span><span class="fontstyle6">=1</span></sub><span class="fontstyle0">.</span></p> 2021-08-06T16:16:25+03:00 Copyright (c) 2021 Acta et Commentationes Universitatis Tartuensis de Mathematica https://ojs.utlib.ee/index.php/ACUTM/article/view/ACUTM.2021.25.02 Corollaries and multiple extensions of Gessel and Stanton hypergeometric summation formulas 2021-08-09T01:57:16+03:00 Thomas Ernst thomas@math.uu.se Per W. Karlsson tyk@ut.ee <p><span style="font-size: small;"><span class="fontstyle0">We find some new simple hypergeometric formulas in the footsteps of the important article by Gessel and Stanton. These are multiple reduction formulas, multiple summation formulas, as well as multiple transformation formulas for special Kampé de Fériet functions and Appell functions. The hypergeometric summation formulas have special function arguments in </span><span class="fontstyle2">Q </span><span class="fontstyle0">and parameter values in </span><span class="fontstyle2">N </span><span class="fontstyle0">or </span><span class="fontstyle2">C</span></span><span class="fontstyle0"><span style="font-size: small;">. The proofs use Pfaff-Kummer transformation, Euler transformation, or an improved form of Slater reversion.</span></span> </p> 2021-06-21T18:01:22+03:00 Copyright (c) 2021 Acta et Commentationes Universitatis Tartuensis de Mathematica https://ojs.utlib.ee/index.php/ACUTM/article/view/ACUTM.2021.25.03 On Riemann problem in weighted Smirnov classes with general weight 2021-08-09T01:57:13+03:00 Bilal Bilalov b_bilalov@mail.ru Aysel Guliyeva aysel_guliyeva20@mail.ru Sabina Sadigova s_sadigova@mail.ru <p><span class="fontstyle0"><span style="font-size: small;">Weighted Smirnov classes in bounded and unbounded domains are defined in this work. Nonhomogeneous Riemann problems with a measurable coefficient whose argument is a piecewise continuous function are considered in these classes. A Muckenhoupt type condition is imposed on the weight function and the orthogonality condition is found for the solvability of nonhomogeneous problem in weighted Smirnov classes, and the formula for the index of the problem is derived. Some special cases with power type weight function are also considered,<br>and conditions on degeneration order are found.</span></span> </p> 2021-06-21T18:10:32+03:00 Copyright (c) 2021 Acta et Commentationes Universitatis Tartuensis de Mathematica https://ojs.utlib.ee/index.php/ACUTM/article/view/ACUTM.2021.25.04 Geometry of multilinear forms on l_1 2021-08-09T01:57:10+03:00 Sung Guen Kim sgk317@knu.ac.kr <p><span style="font-size: small;"><span class="fontstyle0">We characterize extreme, exposed and smooth points in the Banach space </span><span class="fontstyle2"><em>L</em></span><span class="fontstyle0">(</span></span><em><sup><span class="fontstyle3">n</span></sup><span class="fontstyle4"><span style="font-size: small;">E</span></span></em><span style="font-size: small;"><span class="fontstyle0">) of continuous </span><span class="fontstyle4"><em>n</em></span><span class="fontstyle0">-linear forms on </span><span class="fontstyle4"><em>E</em></span><span class="fontstyle0">, and in its subspace </span><span class="fontstyle2"><em>L</em></span></span><span class="fontstyle3"><em><span style="font-size: xx-small;">s</span></em></span><span class="fontstyle0"><span style="font-size: small;">(</span></span><em><sup><span class="fontstyle3">n</span></sup><span class="fontstyle4"><span style="font-size: small;">E</span></span></em><span style="font-size: small;"><span class="fontstyle0">) of symmetric </span><span class="fontstyle4"><em>n</em></span><span class="fontstyle0">-linear forms on </span><span class="fontstyle4"><em>E </em></span><span class="fontstyle0">when </span><span class="fontstyle4"><em>E </em></span><span class="fontstyle0">= </span><span class="fontstyle4"><em>l</em></span></span><span class="fontstyle5"><span style="font-size: xx-small;">1 </span></span><span style="font-size: small;"><span class="fontstyle0">and </span><span class="fontstyle4"><em>E </em></span><span class="fontstyle0">= </span><span class="fontstyle4"><em>l</em></span></span><span class="fontstyle5"><span style="font-size: xx-small;">1</span></span><sup><span class="fontstyle3"><em>m &nbsp;</em></span></sup><span style="font-size: small;"><span class="fontstyle0">for </span><em><span class="fontstyle4">n,m&nbsp;∈</span> </em><span class="fontstyle6">N </span><span class="fontstyle0">with </span><em><span class="fontstyle4">n,m </span><span class="fontstyle2">≥ </span></em><span class="fontstyle0">2</span></span> .</p> 2021-06-21T18:19:32+03:00 Copyright (c) 2021 Acta et Commentationes Universitatis Tartuensis de Mathematica https://ojs.utlib.ee/index.php/ACUTM/article/view/ACUTM.2021.25.05 Asymptotic approximation of misclassification probabilities in linear discriminant analysis with repeated measurements 2021-08-09T01:57:03+03:00 Edward K. Ngailo edward.ngailo@duce.ac.tz Dietrich von Rosen dietrich.von.rosen@slu.se Martin Singull martin.singull@liu.se <p><span style="font-size: small;"><span class="fontstyle0">We propose asymptotic approximations for the probabilities of misclassification in linear discriminant analysis when the group means follow a growth curve structure. The discriminant function can classify a new observation vector of </span><span class="fontstyle2"><em>p </em></span></span><span class="fontstyle0"><span style="font-size: small;">repeated measurements into one of several multivariate normal populations with equal covariance matrix. We derive certain relations of the statistics under consideration in order to obtain asymptotic approximation of misclassification errors for the two group case. Finally, we perform Monte Carlo simulations to evaluate the reliability of the proposed results.</span></span> </p> 2021-06-21T19:17:43+03:00 Copyright (c) 2021 Acta et Commentationes Universitatis Tartuensis de Mathematica https://ojs.utlib.ee/index.php/ACUTM/article/view/ACUTM.2021.25.06 Natural vibrations of nanostrips with cracks 2021-08-09T01:57:01+03:00 Mainul Hossain mainul.hossain@ut.ee Jaan Lellep jaan.lellep@ut.ee <p><span class="fontstyle0"><span style="font-size: small;">Employing the main equations of the theory of plates accounting for the rotational inertia the transverse vibrations of nanobeams and nanostrips are investigated. The nano strips under consideration have piecewise constant dimensions of cross sections. The nanosheets are weakened by cracks at re-entrant corners of steps. While the material behavior corresponds to the Eringen’s nonlocal theory of elasticity it is assumed that the cracks produce additional local compliance, which can be evaluated with the aid of the stress intensity factor at the crack<br>tip. A numerical algorithm for determination of natural frequencies of nanosheets is developed.</span></span> </p> 2021-06-21T19:28:52+03:00 Copyright (c) 2021 Acta et Commentationes Universitatis Tartuensis de Mathematica https://ojs.utlib.ee/index.php/ACUTM/article/view/ACUTM.2021.25.07 About the number of τ-numbers relative to polynomials with integer coefficients 2021-08-09T01:56:58+03:00 Mart Abel mart.abel@tlu.ee Helena Lauer helena.lauer@mail.com Ellen Redi eredi@tlu.ee <p><span style="font-size: small;"><span class="fontstyle0">We show that for all polynomials </span><span class="fontstyle2"><em>Q</em></span><span class="fontstyle0">(</span><span class="fontstyle2"><em>x</em></span><span class="fontstyle0">) with integer coefficients, that satisfy the extra condition </span><em><span class="fontstyle3">|</span><span class="fontstyle2">Q</span></em><span class="fontstyle0">(0) </span><em><span class="fontstyle3">· </span><span class="fontstyle2">Q</span></em><span class="fontstyle0">(1) </span><span class="fontstyle3"><em>|</em></span></span><span style="font-size: small;"><span class="fontstyle0"> ≠ 1, there are infinitely many positive integers </span><span class="fontstyle2"><em>n </em></span><span class="fontstyle0">such that </span><span class="fontstyle2"><em>n </em></span><span class="fontstyle0">is a </span><span class="fontstyle2"><em>τ</em></span></span><span style="font-size: small;"><span class="fontstyle0">-number relative to the polynomial </span><span class="fontstyle2"><em>Q</em></span><span class="fontstyle0">(</span><span class="fontstyle2"><em>x</em></span><span class="fontstyle0">). We also find some examples of polynomials </span><span class="fontstyle2"><em>Q</em></span><span class="fontstyle0">(</span><span class="fontstyle2"><em>x</em></span></span><span style="font-size: small;"><span class="fontstyle0">) for which 1 is the only </span><span class="fontstyle2"><em>τ</em></span><span class="fontstyle0">-number relative to the polynomial </span><span class="fontstyle2"><em>Q</em></span><span class="fontstyle0">(</span><span class="fontstyle2"><em>x</em></span></span><span style="font-size: small;"><span class="fontstyle0">) and some examples of polynomials </span><span class="fontstyle2"><em>Q</em></span><span class="fontstyle0">(</span><span class="fontstyle2"><em>x</em></span><span class="fontstyle0">) with |</span><em><span class="fontstyle2">Q</span></em><span class="fontstyle0">(0) </span><em><span class="fontstyle3">· </span><span class="fontstyle2">Q</span></em><span class="fontstyle0">(1)</span><span class="fontstyle3"><em>|</em></span></span><span style="font-size: small;"><span class="fontstyle0">= 1, which have infinitely many positive integers </span><span class="fontstyle2"><em>n </em></span><span class="fontstyle0">such that </span><span class="fontstyle2"><em>n </em></span><span class="fontstyle0">is a </span><span class="fontstyle2"><em>τ</em></span></span><span style="font-size: small;"><span class="fontstyle0">-number relative to the polynomial </span><span class="fontstyle2"><em>Q</em></span><span class="fontstyle0">(</span><span class="fontstyle2"><em>x</em></span></span><span style="font-size: small;"><span class="fontstyle0">). In addition, we prove one result about the generators of a </span><span class="fontstyle2"><em>τ</em></span><span class="fontstyle0">-number.</span></span></p> 2021-06-21T19:42:31+03:00 Copyright (c) 2021 Acta et Commentationes Universitatis Tartuensis de Mathematica https://ojs.utlib.ee/index.php/ACUTM/article/view/ACUTM.2021.25.08 Module bundles and module amenability 2021-08-09T01:56:56+03:00 Terje Hill terjehill@fau.edu David A. Robbins david.robbins@trincoll.edu <p><span style="font-size: small;"><span class="fontstyle0">Let </span><span class="fontstyle2"><em>X </em></span><span class="fontstyle0">be a compact Hausdorff space, and let </span><em><span class="fontstyle3">{</span><span class="fontstyle2">A</span></em></span><span class="fontstyle4"><em><span style="font-size: xx-small;">x </span></em></span><span style="font-size: small;"><span class="fontstyle0">: </span><em><span class="fontstyle2">x </span><span class="fontstyle3">∈ </span><span class="fontstyle2">X}</span></em></span><span style="font-size: small;"> <span class="fontstyle0">and </span><em><span class="fontstyle3">{</span><span class="fontstyle2">B</span></em></span><span class="fontstyle4"><em><span style="font-size: xx-small;">x </span></em></span><span style="font-size: small;"><span class="fontstyle0">: </span><em><span class="fontstyle2">x </span><span class="fontstyle3">∈ </span><span class="fontstyle2">X}</span> </em><span class="fontstyle0">be collections of Banach algebras such that each </span><span class="fontstyle2"><em>A</em></span></span><span class="fontstyle4"><em><span style="font-size: xx-small;">x </span></em></span><span style="font-size: small;"><span class="fontstyle0">is a </span><span class="fontstyle2"><em>B</em></span></span><span class="fontstyle4"><em><span style="font-size: xx-small;">x</span></em></span><span style="font-size: small;"><span class="fontstyle0">-bimodule. Using the theory of bundles of Banach spaces as a tool, we investigate the module amenability of certain algebras of </span><span class="fontstyle2"><em>A</em></span></span><span class="fontstyle4"><em><span style="font-size: xx-small;">x</span></em></span><span style="font-size: small;"><span class="fontstyle0">-valued functions on </span><span class="fontstyle2"><em>X </em></span><span class="fontstyle0">over algebras of </span><span class="fontstyle2"><em>B</em></span></span><span class="fontstyle4"><em><span style="font-size: xx-small;">x</span></em></span><span style="font-size: small;"><span class="fontstyle0">-valued functions on </span><span class="fontstyle2"><em>X.</em></span></span></p> 2021-06-21T19:49:43+03:00 Copyright (c) 2021 Acta et Commentationes Universitatis Tartuensis de Mathematica https://ojs.utlib.ee/index.php/ACUTM/article/view/ACUTM.2021.25.09 On generalized fractional integral inequalities of Ostrowski type 2021-08-09T01:56:54+03:00 Hüseyin Yildirim hyildir@ksu.edu.tr Seda Kilinc Yildirim sedakilincmath@gmail.com <p><span class="fontstyle0"><span style="font-size: small;">We obtain new generalizations of Ostrowski inequality by using generalized Riemann{Liouville fractional integrals. Some special cases are also discussed.</span></span> </p> 2021-06-21T19:57:47+03:00 Copyright (c) 2021 Acta et Commentationes Universitatis Tartuensis de Mathematica https://ojs.utlib.ee/index.php/ACUTM/article/view/ACUTM.2021.25.10 Thue's equation as a tool to solve two different problems 2021-08-09T01:56:52+03:00 Sadek Bouroubi sbouroubi@usthb.dz Ali Debbache a_debbache2003@yahoo.fr <p><span style="font-size: small;"><span class="fontstyle0">A Thue equation is a Diophantine equation of the form </span><span class="fontstyle2"><em>f</em></span><span class="fontstyle0">(</span><span class="fontstyle2"><em>x; y</em></span><span class="fontstyle0">) = </span><span class="fontstyle2"><em>r</em></span><span class="fontstyle0">, where </span><span class="fontstyle2"><em>f </em></span></span><span style="font-size: small;"><span class="fontstyle0">is an irreducible binary form of degree at least 3, and </span><span class="fontstyle2"><em>r </em></span><span class="fontstyle0">is a given nonzero rational number. A set </span><span class="fontstyle2"><em>S </em></span></span><span style="font-size: small;"><span class="fontstyle0">of at least three positive integers is called a </span><span class="fontstyle2"><em>D</em></span></span><span class="fontstyle3"><span style="font-size: xx-small;">1<sup>3</sup></span></span><span style="font-size: small;"><span class="fontstyle0">-set if the product of any of its three distinct elements is a perfect cube minus one. We prove that any </span><span class="fontstyle2"><em>D</em></span></span><span class="fontstyle3"><span style="font-size: xx-small;">1<sup>3</sup></span></span><span style="font-size: small;"><span class="fontstyle0">-set is finite and, for any positive integer </span><span class="fontstyle2"><em>a</em></span><span class="fontstyle0">, the two-tuple </span><em><span class="fontstyle4">{</span><span class="fontstyle2">a, </span></em><span class="fontstyle0">2</span><em><span class="fontstyle2">a}</span> </em></span><span style="font-size: small;"><span class="fontstyle0">is extendible to a </span><span class="fontstyle2"><em>D</em></span></span><span class="fontstyle3"><span style="font-size: xx-small;">1<sup>3</sup></span></span><span style="font-size: small;"><span class="fontstyle0">-set 3-tuple, but not to a 4-tuple. Using the well-known Thue equation 2</span><span class="fontstyle2"><em>x<sup>3</sup></em></span></span> <em><span style="font-size: small;"><span class="fontstyle4">- </span><span class="fontstyle2">y</span></span></em><sup><span class="fontstyle3">3 </span></sup><span class="fontstyle0"><span style="font-size: small;">= 1, we show that the only cubic-triangular number is 1.</span></span></p> 2021-06-21T20:06:36+03:00 Copyright (c) 2021 Acta et Commentationes Universitatis Tartuensis de Mathematica https://ojs.utlib.ee/index.php/ACUTM/article/view/ACUTM.2021.25.11 On the Fibonacci quaternion sequence with quadruple-produce components 2021-08-09T01:56:48+03:00 Orhan Diskaya orhandiskaya@mersin.edu.tr Hamza Menken hmenken@mersin.edu.tr <p><span class="fontstyle0"><span style="font-size: small;">This paper examines the Fibonacci quaternion sequence with quadruple-produce components, and demonstrates a golden-like ratio and some identities for this sequence. Its generating and exponential generating functions are given. Along with these, its series and binomial sum formula are established.</span></span> </p> 2021-06-21T20:12:11+03:00 Copyright (c) 2021 Acta et Commentationes Universitatis Tartuensis de Mathematica