Acta et Commentationes Universitatis Tartuensis de Mathematica https://ojs.utlib.ee/index.php/ACUTM <p><em>Acta et Commentationes Universitatis Tartuensis de Mathematica&nbsp;</em>(ACUTM) is an international journal of pure and applied mathematics.</p> University of Tartu Press en-US Acta et Commentationes Universitatis Tartuensis de Mathematica 1406-2283 On the degree of approximation of continuous functions by a specific transform of partial sums of their Fourier series https://ojs.utlib.ee/index.php/ACUTM/article/view/ACUTM.2021.25.01 <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> Xhevat Krasniqi Copyright (c) 2021 Acta et Commentationes Universitatis Tartuensis de Mathematica 2021-08-06 2021-08-06 25 1 5 19 10.12697/ACUTM.2021.25.01 Corollaries and multiple extensions of Gessel and Stanton hypergeometric summation formulas https://ojs.utlib.ee/index.php/ACUTM/article/view/ACUTM.2021.25.02 <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> Thomas Ernst Per W. Karlsson Copyright (c) 2021 Acta et Commentationes Universitatis Tartuensis de Mathematica 2021-06-21 2021-06-21 25 1 21 31 10.12697/ACUTM.2021.25.02 On Riemann problem in weighted Smirnov classes with general weight https://ojs.utlib.ee/index.php/ACUTM/article/view/ACUTM.2021.25.03 <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> Bilal Bilalov Aysel Guliyeva Sabina Sadigova Copyright (c) 2021 Acta et Commentationes Universitatis Tartuensis de Mathematica 2021-06-21 2021-06-21 25 1 33 56 10.12697/ACUTM.2021.25.03 Geometry of multilinear forms on l_1 https://ojs.utlib.ee/index.php/ACUTM/article/view/ACUTM.2021.25.04 <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> Sung Guen Kim Copyright (c) 2021 Acta et Commentationes Universitatis Tartuensis de Mathematica 2021-06-21 2021-06-21 25 1 57 66 10.12697/ACUTM.2021.25.04 Asymptotic approximation of misclassification probabilities in linear discriminant analysis with repeated measurements https://ojs.utlib.ee/index.php/ACUTM/article/view/ACUTM.2021.25.05 <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> Edward K. Ngailo Dietrich von Rosen Martin Singull Copyright (c) 2021 Acta et Commentationes Universitatis Tartuensis de Mathematica 2021-06-21 2021-06-21 25 1 67 85 10.12697/ACUTM.2021.25.05 Natural vibrations of nanostrips with cracks https://ojs.utlib.ee/index.php/ACUTM/article/view/ACUTM.2021.25.06 <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> Mainul Hossain Jaan Lellep Copyright (c) 2021 Acta et Commentationes Universitatis Tartuensis de Mathematica 2021-06-21 2021-06-21 25 1 87 105 10.12697/ACUTM.2021.25.06 About the number of τ-numbers relative to polynomials with integer coefficients https://ojs.utlib.ee/index.php/ACUTM/article/view/ACUTM.2021.25.07 <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> Mart Abel Helena Lauer Ellen Redi Copyright (c) 2021 Acta et Commentationes Universitatis Tartuensis de Mathematica 2021-06-21 2021-06-21 25 1 107 117 10.12697/ACUTM.2021.25.07 Module bundles and module amenability https://ojs.utlib.ee/index.php/ACUTM/article/view/ACUTM.2021.25.08 <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> Terje Hill David A. Robbins Copyright (c) 2021 Acta et Commentationes Universitatis Tartuensis de Mathematica 2021-06-21 2021-06-21 25 1 119 141 10.12697/ACUTM.2021.25.08 On generalized fractional integral inequalities of Ostrowski type https://ojs.utlib.ee/index.php/ACUTM/article/view/ACUTM.2021.25.09 <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> Hüseyin Yildirim Seda Kilinc Yildirim Copyright (c) 2021 Acta et Commentationes Universitatis Tartuensis de Mathematica 2021-06-21 2021-06-21 25 1 143 151 10.12697/ACUTM.2021.25.09 Thue's equation as a tool to solve two different problems https://ojs.utlib.ee/index.php/ACUTM/article/view/ACUTM.2021.25.10 <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> Sadek Bouroubi Ali Debbache Copyright (c) 2021 Acta et Commentationes Universitatis Tartuensis de Mathematica 2021-06-21 2021-06-21 25 1 153 156 10.12697/ACUTM.2021.25.10 On the Fibonacci quaternion sequence with quadruple-produce components https://ojs.utlib.ee/index.php/ACUTM/article/view/ACUTM.2021.25.11 <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> Orhan Diskaya Hamza Menken Copyright (c) 2021 Acta et Commentationes Universitatis Tartuensis de Mathematica 2021-06-21 2021-06-21 25 1 157 170 10.12697/ACUTM.2021.25.11