Quaestiones Mathematicae https://www.ajol.info/index.php/qm <p><em>Quaestiones Mathematicae</em> is devoted to research articles from a wide range of mathematical areas. Longer expository papers of exceptional quality are also considered. Published in English, the journal receives contributions from authors around the globe and serves as an important reference source for anyone interested in mathematics.</p><p>Read more about the journal <a href="http://www.nisc.co.za/products/12/journals/quaestiones-mathematicae" target="_blank">here</a>. </p> en-US Copyright for articles published in this journal is retained by the journal. publishing@nisc.co.za (Publishing Manager) journals@nisc.co.za (Editorial Office) Thu, 21 Apr 2022 17:28:02 +0000 OJS http://blogs.law.harvard.edu/tech/rss 60 A Zariski topology on integrally closed maximal subrings of a commutative ring https://www.ajol.info/index.php/qm/article/view/224108 <p>Let <em>R</em> be a commutative ring and <em>R<sup>i.c</sup></em><span style="font-size: 11.6667px;">&nbsp;</span>(R) denotes the set of all integrally closed maximal subrings of <em>R</em>. It is shown that if <em>R</em> is a non-field <em>G</em>-domain, then there exists <em>S</em> ∈ <em>X</em><sup>i.c</sup>(R) with (<em>S</em> : |<em>R</em>) = 0. If <em>K</em> is an algebraically closed field which is not absolutely algebraic, then we prove that the polynomial ring K [X] has an integrally closed maximal subring with zero conductor too; a characterization of integrally closed maximal subrings of <em>K</em> [<em>X</em>] with (non-)zero conductor is given. It is observed that, an integrally closed maximal subrings <em>S</em> of <em>K</em> [<em>X</em>] is a principal ideal domain (PID) if and only if <em>M</em> = <em>Sq</em> for some q<sup>-1 </sup>∈&nbsp;<em>K</em> \ <em>S</em>, where <em>M</em> is the crucial maximal idea of the extension S ⊆ K [X]. We show that if <em>f</em> (X,Y ) is an irreducible polynomial in <em>K</em> [<em>X</em>,<em>Y</em> ], then there exists an integrally closed maximal subring <em>S</em> of <em>K</em> [X, Y ] with (<em>S</em> : <em>K</em> [X, Y ]) = <em>f</em>(<em>X,Y</em> )<em>K</em> [<em>X,Y</em> ]. It is proved that, if <em>R</em> is a ring and <em>S</em> (<em>I</em>) = {<em>T</em> ∈ X<em><sup>i:c</sup></em>(<em>R</em>)&nbsp; |&nbsp;<em>I</em> <em>⊆</em> <em>T</em>}, where <em>I</em> is an ideal of <em>R</em>, then S :=&nbsp; {<em>S</em>(<em>I</em>) | <em>I</em> is an ideal of <em>R</em>} is a topology for closed sets on <em>X</em><sup><em>i:c</em></sup>(<em>R</em>). We show&nbsp; that this space has similar properties such as those one in the Zariski spaces on <em>Spec</em>(<em>R</em>) or <em>K<sup>n</sup></em> (the affine space). In particular, if <em>K</em> is a field which is not algebraic over its prime subring, then <em>X<sup>i.c </sup></em>(<em>K</em> [<em>X<sub>1</sub></em>...,<em>X<sub>n</sub></em>]) is irreducible and if in addition <em>K</em> is algebraically closed, then we prove a similar full form of the Hilbert Nullstellensatz for <em>K</em>[<em>X</em><sub>1</sub>,...,<em>X<sub>n</sub></em>]. Moreover, if <em>R</em> is a non-field <em>G</em>-domain or <em>R</em> = <em>K</em> [<em>X</em>], where <em>K</em> is an algebraically closed field which is not algebraic over its prime subring, then ∅ ≠ <em>gen</em> (<em>X<sup>i.c</sup></em>(<em>R</em>)) = {<em>S</em> ∈&nbsp;<em>X<sup>i.c</sup></em>(<em>R</em>) | (<em>S : R</em>) = <em>0</em>}. We determine exactly when the space <em>X<sup>i.c</sup></em>(<em>R</em>) is a <em>T<sub>i </sub></em>- space for<em> i</em> = 0, 1,2. In particular, we show that if <em>X<sup>i:c</sup></em> is T<sub><em>1</em></sub>-space then <em>R</em> is a Hilbert ring and |X<em><sup>i:c</sup></em>(R)| ≤ 2<em><sup>|Max</sup><sup>(R)|</sup></em>. Finally, we determine when the space <em>X<sup>i.c</sup> </em>is connected.</p> Alborz Azarang Copyright (c) https://www.ajol.info/index.php/qm/article/view/224108 Tue, 19 Apr 2022 00:00:00 +0000 On a type of superlinear growth variational problems https://www.ajol.info/index.php/qm/article/view/224141 <p>In this note, we propose an elementary method to study the existence and uniqueness of solutions to a type of variational problems which arise naturally in the theory of large deviations. This type of problems involves a movable boundary and may not have the coercivity condition in general. Our method is elementarily based on direct analysis over the space of absolutely continuous functions and specific properties of the underlying functional.</p> Zhi Wang, Xiangfeng Yang Copyright (c) https://www.ajol.info/index.php/qm/article/view/224141 Tue, 19 Apr 2022 00:00:00 +0000 A two-variable Dirichlet series and its applications https://www.ajol.info/index.php/qm/article/view/224142 <p>We define a two-variable Dirichlet series associated with two arithmetic functions, which is related to the Riemann zeta function, the Dirichlet <em>L</em>-function, the Dirichlet series associated to the harmonic numbers, and truncated multiple zeta functions. Using the periodic Euler-Maclaurin summation formula, we obtain a representation in terms of an ordinary Dirichlet series, which leads to the explicit evaluation of its values at nonpositive integers. We also find a reciprocity formula, which provides some symmetric formulas involving Bernoulli and associated numbers.</p> Mehmet Cenkci, Abdurrahman Ünal Copyright (c) https://www.ajol.info/index.php/qm/article/view/224142 Tue, 19 Apr 2022 00:00:00 +0000 <i>k</i> - Fibonacci numbers close to a power of 2 https://www.ajol.info/index.php/qm/article/view/224143 <p>A generalization of the well-known Fibonacci sequence is the <em>k</em>-generalized Fibonacci sequence <em>F</em><sup><em>(ƙ)</em></sup> := (<em>F<sub>n</sub></em><sup>(<em>ƙ</em>)</sup> )<sub><em>n</em> ≥ 2 - ƙ</sub> whose first <em>ƙ</em> terms are 0,..., 0,1 and each term afterwards is the sum of the preceding <em>ƙ</em> terms. In this paper, by using a lower bound to linear forms in logarithms of algebraic numbers due to Matveev and some argument of the theory of continued fractions, we find all the members of <em>F</em><sup>(ƙ) </sup>which are close to a power of 2. This paper continues and extends the previous work of Chern and Cui which investigated the Fibonacci numbers close to a power of 2.</p> Jhon J. Bravo, Carlos A. Gómez, Jose L. Herrera Copyright (c) https://www.ajol.info/index.php/qm/article/view/224143 Tue, 19 Apr 2022 00:00:00 +0000 Multivalued generalized graphic θ-contraction on directed graphs and application to mixed Volterra-Fredholm integral inclusion equations https://www.ajol.info/index.php/qm/article/view/224145 <p>The purpose of the present work is to introduce a generalized graphic θ-contraction conditions on a family of mappings defined on subsets of a metric space endowed with a set-transitive directed graph, and discuss common fixed point results without considering any kind of commutativity and continuity of the family of mappings. Useful examples illustrate the applicability and effectiveness of the given notions and results. We apply our result to the problem of existence of solutions of a pair of mixed Volterra-Fredholm integral inclusion equations followed by a numerical example.</p> Lakshmi Kanta Dey, Hiranmoy Garai, Hemant Kumar Nashine, Can Huu Nguyen Copyright (c) https://www.ajol.info/index.php/qm/article/view/224145 Tue, 19 Apr 2022 00:00:00 +0000 On Rall's 1/2-conjecture on the domination game https://www.ajol.info/index.php/qm/article/view/224153 <p>The 1/2-conjecture on the domination game asserts that if G is a traceable graph, then the game domination number<sub><em> γg</em></sub>(<em>G</em>) of <em>G</em> is at most . A traceable graph is a 1/2-graph if <em><sub>γg</sub></em>(<em>G</em>) = <em>n(G)/2&nbsp;</em>holds. It is proved that the so-called hatted cycles are 1/2-graphs and that unicyclic graphs fulfill the 1/2-conjecture. Several additional families of graphs that support the conjecture are determined and&nbsp;computer experiments related to the conjecture described.</p> Csilla Bujtás, Vesna Iršič, Sandi Klavžar, Kexiang Xu Copyright (c) https://www.ajol.info/index.php/qm/article/view/224153 Wed, 20 Apr 2022 00:00:00 +0000 Joint discrete universality for periodic Zeta-functions. III https://www.ajol.info/index.php/qm/article/view/224174 <p>In the paper, a joint theorem on the approximation of collections of analytic functions by generalized discrete shifts of zeta-functions with periodic coefficients is obtained. The latter result extend theorems of [<em>9</em>].</p> Antanas Laurinčika Copyright (c) https://www.ajol.info/index.php/qm/article/view/224174 Thu, 21 Apr 2022 00:00:00 +0000 The powers of two as sums over partitions https://www.ajol.info/index.php/qm/article/view/224175 <p>In this paper, we investigate two methods to express the natural powers of 2 as sums over integer partitions. First we consider a formula by N. J. Fine that allows us to express a binomial coefficient in terms of multinomial coefficients as a sum over partitions. The second method invokes the central binomial coefficients and the logarithmic differentiation of their generating function. Some experimental results suggest the existence of other methods of decomposing the power of 2 as sums over partitions.</p> Mircea Merca Copyright (c) https://www.ajol.info/index.php/qm/article/view/224175 Thu, 21 Apr 2022 00:00:00 +0000 Topological connectednesses and congruences https://www.ajol.info/index.php/qm/article/view/224176 <p>It is known that the connectednesses of topological spaces in the sense of Preuβ is the topological analogue of the Kurosh-Amistsur radicals of algebraic structures in a categorical sense. Here this connection is further explored. As in universal algebra, a congruence on a topological space has been defined. It is shown that a connectedness can be characterized in terms of conditions on congruences which are the precise topological analogues of those conditions that characterize the radical classes of rings in terms of ideals.</p> Stefan Veldsman Copyright (c) https://www.ajol.info/index.php/qm/article/view/224176 Thu, 21 Apr 2022 00:00:00 +0000 Notes on star covering properties https://www.ajol.info/index.php/qm/article/view/224177 <p>In this paper, we show the following statements:<br>(1) There exists a pseudocompact star Lindelöf Tychonoff space which is not star σ-compact.<br>(2) There exists a Tychonoff pseudocompact star countable (hence, star Lindelöf) space having a pseudocompact, G<sub>δ</sub> regular closed subspace which is not star Lindelöf.<br>(3) Assuming 2<sup>ℵ<sub>0</sub></sup> = 2<sup>ℵ<sub>1</sub></sup>, there exists a normal star countable (hence, star Lindelöf) space having a G<sub>δ</sub> regular closed subspace which is not star Lindelöf.<br>(4) Let X be a space, then A(X) is star Lindelöf if and only if <em>e</em>(X) ≤ ω.<br>The statement (1) gives a negative answer to Song [13, Remark 2.3].</p> Yan-Kui Song, Wei-Feng Xuan Copyright (c) https://www.ajol.info/index.php/qm/article/view/224177 Thu, 21 Apr 2022 00:00:00 +0000 Frame congruences via subkernels https://www.ajol.info/index.php/qm/article/view/224189 <p>To study the quotient of algebras, like frames, whose algebraic structures are determined by a partial order, it is often more common to think about sub-kernels of homomorphisms between such algebras. So, in this paper, we first introduce the concept of a pre-congruence on a frame and then characterize them as the sub-kernels of the frame homomorphisms. Second, we characterize the frame congruences as the intersection of a pre-congruence and its inverse. Finally, we prove the decomposition and isomorphism theorems for frame homomorphisms.</p> Halimeh Moghbeli Copyright (c) https://www.ajol.info/index.php/qm/article/view/224189 Thu, 21 Apr 2022 00:00:00 +0000 Quadratic model updating for damped gyroscopic systems https://www.ajol.info/index.php/qm/article/view/224191 <p>This paper is concerned with the problem of the optimal approximation for a given matrix pencil (M<sub>a,</sub>D<sub>a,</sub>G<sub>,</sub>K<sub>a</sub>,N<sub>a</sub>) under the spectral constraint and the symmetric constraint. Such a problem arises in finite element model updating for<br>damped gyroscopic systems. By using constrained optimization theory and matrix derivatives, an explicit formulation for the solution of the problem is established. The efficiency and accuracy of the proposed method is numerically verified by a simple five-degree-of-freedom system.</p> Yongxin Yuan Copyright (c) https://www.ajol.info/index.php/qm/article/view/224191 Thu, 21 Apr 2022 00:00:00 +0000 Matrix characterization of asymptotically deferred equivalent sequences https://www.ajol.info/index.php/qm/article/view/224194 <p>In 1932 Agnew [1] introduced the concept of deferred Cesáro means. Taking inspiration from this new approach, in this paper we introduce deferred asymptotically equivalent sequences using statistical convergence and prove some important results. This will be accomplished through a series of regularity type theorems.</p> Rabia Savaş Copyright (c) https://www.ajol.info/index.php/qm/article/view/224194 Thu, 21 Apr 2022 00:00:00 +0000 Durrmeyer variant of Apostol-Genocchi-Baskakov operators https://www.ajol.info/index.php/qm/article/view/224197 <p>We study the approximation behavior of the Durrmeyer form of Apostol-Genocchi polynomials with Baskakov type operators including K-functional and second-order modulus of smoothness, Lipschitz space and find the rate of convergence for continuous functions whose derivative satisfies the condition of bounded variation. In the last section, we estimate weighted approximation behavior for these operators.</p> Naokant Deo, Sandeep Kumar Copyright (c) https://www.ajol.info/index.php/qm/article/view/224197 Thu, 21 Apr 2022 00:00:00 +0000 Double outer-independent domination number of graphs https://www.ajol.info/index.php/qm/article/view/224198 <p>Let <em>G</em> be a graph with no isolated vertex. A set <em>D</em> ⊆ <em>V</em> (<em>G</em>) is a double outer-independent dominating set of G if V (G)\D is an independent set and |<em>N</em>[<em>v</em>]∩<em>D</em>| ≥ 2 for every <em>v</em> ∈<em> V</em> (<em>G</em>), where <em>N</em>[<em>v</em>] denotes the closed neighbourhood of <em>v. </em>The minimum cardinality among all double outer-independent dominating sets of <em>G</em> is the double outer-independent domination number of <em>G</em>. In this paper, we continue with the study of this parameter. For instance, we give some relationships that exist between this parameter and other domination invariants in graph. Also, we present tight bounds and show some classes of graphs for which the bounds are achieved. Finally, we provide closed formulas for the double outer-independent domination number of rooted product graphs, and characterize the graphs reaching these expressions.</p> Abel Cabrera Martínez Copyright (c) https://www.ajol.info/index.php/qm/article/view/224198 Thu, 21 Apr 2022 00:00:00 +0000