Acta et Commentationes Universitatis Tartuensis de Mathematica <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 Spectrum and genus of commuting graphs of some classes of finite rings <p>We consider commuting graphs of some classes of finite rings and compute their spectrum and genus. We show that the commuting graph of a finite CC-ring is integral. We also characterize some finite rings whose commuting graphs are planar.</p> Jutirekha Dutta Walaa Nabil Taha Fasfous Rajat Kanti Nath ##submission.copyrightStatement## 2019-08-09 2019-08-09 23 1 5 12 10.12697/ACUTM.2019.23.01 Pseudo-symmetric structures on almost Kenmotsu manifolds with nullity distributions <p>The object of the present paper is to characterize Ricci pseudosymmetric and Ricci semisymmetric almost Kenmotsu manifolds with (<em>k</em>; <em>μ</em>)-, (<em>k</em>; <em>μ</em>)′-, and generalized (<em>k</em>; <em>μ</em>)-nullity distributions. We also characterize (<em>k</em>; <em>μ</em>)-almost Kenmotsu manifolds satisfying the condition <em>R</em> ⋅ <em>S</em> = <em>LꜱQ</em>(<em>g</em>; <em>S</em><sup>2</sup>).</p> U. C. De Dibakar Dey ##submission.copyrightStatement## 2019-08-09 2019-08-09 23 1 13 24 10.12697/ACUTM.2019.23.02 Some Hermite–Hadamard and Ostrowski type inequalities for fractional integral operators with exponential kernel <p>We rstly establish Hermite–Hadamard type integral inequalities for fractional integral operators. Secondly, we give new generalizations of fractional Ostrowski type inequalities through convex functions via Hölder and power means inequalities. In accordance with this purpose, we use fractional integral operators with exponential kernel.</p> Hüseyin Budak Mehmet Zeki Sarikaya Fuat Usta Hüseyin Yildirim ##submission.copyrightStatement## 2019-08-09 2019-08-09 23 1 25 36 10.12697/ACUTM.2019.23.03 Some (p, q)-analogues of Apostol type numbers and polynomials <p>We consider a new class of generating functions of the generalizations of Bernoulli and Euler polynomials in terms of (<em>p</em>, <em>q</em>)-integers. By making use of these generating functions, we derive (<em>p</em>, <em>q</em>)-generalizations of several old and new identities concerning Apostol–Bernoulli and Apostol–Euler polynomials. Finally, we define the (<em>p</em>, <em>q</em>)-generalization of Stirling polynomials of the second kind of order <em>v</em>, and provide a link between the (<em>p</em>, <em>q</em>)-generalization of Bernoulli polynomials of order <em>v</em> and the (<em>p</em>, <em>q</em>)-generalization of Stirling polynomials of the second kind of order <em>v</em>.</p> Mehmet Acikgoz Serkan Araci Ugur Duran ##submission.copyrightStatement## 2019-08-09 2019-08-09 23 1 37 50 10.12697/ACUTM.2019.23.04 Fejér type integral inequalities related with geometrically-arithmetically convex functions with applications <p>A new identity involving a geometrically symmetric function and a differentiable function is established. Some new Fejér type integral inequalities, connected with the left part of Hermite–Hadamard type inequalities for geometrically-arithmetically convex functions, are presented by using the Hölder integral inequality and the notion of geometrically-arithmetically convexity. Applications of our results to special means of positive real numbers are given.</p> S. S. Dragomir M. A. Latif E. Momoniat ##submission.copyrightStatement## 2019-08-09 2019-08-09 23 1 51 64 10.12697/ACUTM.2019.23.05 Lucas numbers of the form (2t/k) <p>Let <em>L<sub>m</sub></em> denote the <em>m<sup>th</sup></em> Lucas number. We show that the solutions to the diophantine equation (2<em><sup>t</sup></em>/<em>k</em>) = <em>L<sub>m</sub></em>, in non-negative integers <em>t</em>, <em>k</em> ≤ 2<sup><em>t</em>−1</sup>, and <em>m</em>, are (<em>t</em>, <em>k</em>, <em>m</em>) = (1, 1, 0), (2, 1, 3), and (<em>a</em>, 0, 1) with non-negative integers <em>a</em>.</p> Nurettin Irmak László Szalay ##submission.copyrightStatement## 2019-08-09 2019-08-09 23 1 65 70 10.12697/ACUTM.2019.23.06 Estimations of Riemann–Liouville k-fractional integrals via convex functions <p>The <em>k</em>-fractional integrals introduced by S. Mubeen and G. M. Habibullah in 2012 are a generalization of Riemann–Liouville fractional integrals. Some estimations of these fractional integrals via convexity have been established.</p> Ghulam Farid ##submission.copyrightStatement## 2019-08-09 2019-08-09 23 1 71 78 10.12697/ACUTM.2019.23.07 Notes on certain analytic functions concerning some subordinations <p>By making use of the principle of subordination, we investigate a certain subclass of analytic functions. Such results as subordination and superordination are given. The related sandwich-type results are also presented.</p> Şahsene Altınkaya Shigeyoshi Owa Sibel Yalçin ##submission.copyrightStatement## 2019-08-09 2019-08-09 23 1 79 85 10.12697/ACUTM.2019.23.08 A new approach in topology via elements of an ideal <p>New approaches in topology or related branches of mathematics have contributed in a valuable way to the science, and have yielded various new topics for investigation. The main goal of this paper is to examine a new approach and so a new form of open sets via elements of an ideal. The concept of α<sup>*</sup><sub>ɪ</sub>-open sets is introduced and discussed.</p> Erdal Ekici ##submission.copyrightStatement## 2019-08-09 2019-08-09 23 1 87 94 10.12697/ACUTM.2019.23.09 About the convergence type of improper integrals defining fractional derivatives <p>This article continues the analysis of the class of fractionally differentiable functions. We complete the main result of [4] that characterises the class of fractionally differentiable functions in terms of the pointwise convergence of certain improper integrals containing these functions. Our aim is to present an example, which shows that in order to obtain all fractionally differentiable functions, one may not replace the conditional convergence of those integrals by their absolute convergence.</p> B. Kalam G. Vainikko ##submission.copyrightStatement## 2019-08-09 2019-08-09 23 1 95 102 10.12697/ACUTM.2019.23.10 An Eneström–Kakeya theorem for new classes of polynomials <p>See PDF</p> William "Ty" Frazier Robert Gardner ##submission.copyrightStatement## 2019-08-09 2019-08-09 23 1 103 115 10.12697/ACUTM.2019.23.11 A new characterization of symplectic groups C2(3n) <p>We prove that symplectic groups <em>C</em><sub>2</sub>(3<em><sup>n</sup></em>), where <em>n</em> = 2<em><sup>k</sup></em> (<em>k</em> ≥ 0) and (3<sup>2<em>n</em></sup> + 1)=2 is a prime number, can be uniquely determined by the order of the group and the number of elements with the same order.</p> Behnam Ebrahimzadeh Reza Mohammadyari ##submission.copyrightStatement## 2019-08-09 2019-08-09 23 1 117 124 10.12697/ACUTM.2019.23.12 Comparison of machine learning methods for crack localization <p>In this paper, the Haar wavelet discrete transform, the artificial neural networks (ANNs), and the random forests (RFs) are applied to predict the location and severity of a crack in an Euler–Bernoulli cantilever subjected to the transverse free vibration. An extensive investigation into two data collection sets and machine learning methods showed that the depth of a crack is more difficult to predict than its location. The data set of eight natural frequency parameters produces more accurate predictions on the crack depth; meanwhile, the data set of eight Haar wavelet coefficients produces more precise predictions on the crack location. Furthermore, the analysis of the results showed that the ensemble of 50 ANN trained by Bayesian regularization and Levenberg–Marquardt algorithms slightly outperforms RF.</p> Helle Hein Ljubov Jaanuska ##submission.copyrightStatement## 2019-08-09 2019-08-09 23 1 125 142 10.12697/ACUTM.2019.23.13 Natural vibrations of stepped nanobeams with defects <p>Exact solutions for the transverse vibration of nanobeams based on the nonlocal theory of elasticity are presented. The nanobeams under consideration have piecewise constant dimensions of cross sections and are weakened with crack-like defects. It is assumed that the stationary cracks occur at the re-entrant corners of steps and that the mechanical behaviour of the nanomaterial can be modelled with the Eringen's nonlocal theory. The influence of cracks on the natural vibration is prescribed with the aid of additional local compliance at the weakened cross section. The local compliance is coupled with the stress intensity factor at the crack tip. A general algorithm for determination of eigenfrequencies is developed. It can be used in the case of an arbitrary finite number of steps and cracks.</p> Jaan Lellep Artur Lenbaum ##submission.copyrightStatement## 2019-08-09 2019-08-09 23 1 143 158 10.12697/ACUTM.2019.23.14