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> Taylor & Francis en-US Quaestiones Mathematicae 1607-3606 Copyright for articles published in this journal is retained by the journal. Practical central binomial coefficients https://www.ajol.info/index.php/qm/article/view/217083 <div class="page" title="Page 1"> <div class="layoutArea"> <div class="column"> <p>A practical number is a positive integer n such that all positive integers less than <em>n</em> can be written as a sum of distinct divisors of <em>n</em>. Leonetti and Sanna proved that, as <em>x → +∞</em>, the central binomial coefficient (<em>2n/n</em>) is a practical number <em>n </em>for all positive integers <em>n ≤ x</em> but at most <em>O</em>(<em>x<sup>0.88097</sup></em>) exceptions. We improve this result by reducing the number of exceptions to exp(<em>C</em>(log <em>x</em>)<sup>4/5</sup> log log <em>x</em>), where C &gt; 0 is a constant.</p> </div> </div> </div> Carlo Sanna Copyright (c) 2021-11-08 2021-11-08 44 9 1141 1144 Weak compactness of almost L-weakly and almost M-weakly compact operators https://www.ajol.info/index.php/qm/article/view/217086 <div class="page" title="Page 1"> <div class="layoutArea"> <div class="column"> <p>In this paper, we investigate conditions on a pair of Banach lattices <em>E</em> and <em>F</em> that tells us when every positive almost L-weakly compact (resp. almost M-weakly compact) operator <em>T : E −→ F</em> is weakly compact. Also, we present some necessary conditions that tells us when every weakly compact operator <em>T : E −→ F</em> is almost M-weakly compact (resp. almost L-weakly compact). In particular, we will prove that if every weakly compact operator from a Banach lattice E into a Banach space <em>X</em> is almost L-weakly compact, then <em>E</em> is a KB-space or <em>X</em> has the Dunford-Pettis property and the norm of <em>E</em> is order continuous.</p> </div> </div> </div> Farid Afkir Khalid Bouras Aziz Elbour Safae El Filali Copyright (c) 2021-11-08 2021-11-08 44 9 1145 1154 Weak exceptional sequences https://www.ajol.info/index.php/qm/article/view/217087 <div class="page" title="Page 1"> <div class="layoutArea"> <div class="column"> <p>We introduce weak exceptional sequence of modules which can be viewed as another modification of the standard case, different from the works of Igusa- Todorov [IT17] and Buan-Marsh [BM18]. For hereditary algebras it is equivalent to standard exceptional sequences. One important new feature is: if the global dimension of an algebra is greater than one, then the size of a full sequence can exceed the rank of the algebra. We use both cyclic and linear Nakayama algebras to test combinatorial aspects of this new sequence. For some particular classes, we give closed form formulas which returns the number of a full weak exceptional sequences, and compare them with the number of exceptional sequences of types A and linear radical square zero Nakayama algebras [Sen19-2].</p> </div> </div> </div> Emre Sen Copyright (c) 2021-11-08 2021-11-08 44 9 1155 1171 Computations and global properties for traces of Bessel’s Dirichlet form https://www.ajol.info/index.php/qm/article/view/217088 <div class="page" title="Page 1"> <div class="layoutArea"> <div class="column"> <p>By an approximation method we compute explicitly traces of the Dirichlet form related to the Bessel process with respect to discrete measures as well as measures of mixed type. Then some global properties of the obtained Dirichlet forms, such as conservativeness, irreducibility and compact embedding for their domains are discussed.</p> </div> </div> </div> Ali BenAmor Rafed Moussa Copyright (c) 2021-11-08 2021-11-08 44 9 1173 1196 Infinitely many solutions for a class of sublinear fractional Schrödinger-Poisson systems https://www.ajol.info/index.php/qm/article/view/217089 <div class="page" title="Page 1"> <div class="layoutArea"> <div class="column"> <p>In this paper, we consider the following nonlinear fractional Schr<span style="font-weight: 400;">ö</span>dinger-Poisson system <span style="font-weight: 400;">{(<em>−∆)</em></span><em><sup><span style="font-weight: 400;">s</span></sup><span style="font-weight: 400;">u + V (x)u + K(x)φu = a(x)|u|</span><sup><span style="font-weight: 400;">q−1</span></sup><span style="font-weight: 400;">u, x ∈ R</span><sup><span style="font-weight: 400;">3</span></sup><span style="font-weight: 400;">, (−∆)</span><sup><span style="font-weight: 400;">t</span></sup><span style="font-weight: 400;">φ = K(x)u</span><sup><span style="font-weight: 400;">2</span></sup><span style="font-weight: 400;">, x ∈ R</span><sup><span style="font-weight: 400;">3</span></sup></em><span style="font-weight: 400;">} </span>where <em>s,t</em> ∈ (0,1) and <em>4s + 2t</em> ≥ 3,0 &lt; q &lt; 1, and a,K,V ∈ L∞(R<sup>3</sup>). When <em>a, V</em> both change sign in R<sup>3</sup>, by applying the symmetric mountain pass theorem, we prove that the problem has infinitely many solutions under appropriate assumptions on <em>a, K, V</em>.</p> </div> </div> </div> Wen Guan Lu-Ping Ma Da-Bin Wang Jin-Long Zhang Copyright (c) 2021-11-08 2021-11-08 44 9 1197 1207 A note on Tingley’s problem and Wigner’s theorem in the unit sphere of L<sup>∞</sup>(Г)-type spaces https://www.ajol.info/index.php/qm/article/view/217091 <div class="page" title="Page 1"> <div class="layoutArea"> <div class="column"> <p>Suppose that <em>f : S<sub>X</sub> → S<sub>Y</sub></em> is a surjective map between the unit spheres of two real L<sup>∞</sup>(Γ)-type spaces <em>X</em> and <em>Y</em> satisfying the following equation<em> {∥f(x) + f(y)∥,∥f(x) − f(y)∥} = {∥x + y∥,∥x − y∥} (x,y ∈ SX).&nbsp;</em>We show that such a mapping <em>f</em> is phase equivalent to an isometry, i.e., there exists a function <em>ε : SX</em> → {−1,1} such that <em>εf</em> is an isometry. We further show that this isometry is the restriction of a linear isometry between the whole spaces. These results can be seen as a combination of Tingley’s problem and Wigner’s theorem for L∞(Γ)-type spaces.</p> </div> </div> </div> Dongni Tan Xiaolei Xiong Copyright (c) 2021-11-08 2021-11-08 44 9 1209 1217 Shephard type problem for <i>L<sub>p</sub></i>-mixed Blaschke-Minkowski homomorphisms https://www.ajol.info/index.php/qm/article/view/217153 <div class="page" title="Page 1"> <div class="layoutArea"> <div class="column"> <p>In 2006, Schuster introduced the concept of Blaschke-Minkowski homomorphism of convex bodies. In this paper, we introduce the <em>L<sub>p-</sub></em>mixed Blaschke-Minkowski homomorphism in <em>L<sub>p</sub></em>-Brunn-Minkowski theory. We then further study the Shephard type problem involving an affirmative answer and two negative answers for <em>L<sub>p</sub></em>-mixed Blaschke-Minkowski homomorphism.</p> </div> </div> </div> Bin Chen Weidong Wang Peibiao Zhao Copyright (c) 2021-11-09 2021-11-09 44 9 1219 1232 On best proximity points of interpolative proximal contractions https://www.ajol.info/index.php/qm/article/view/217156 <div class="page" title="Page 1"> <div class="layoutArea"> <div class="column"> <div class="page" title="Page 1"> <div class="layoutArea"> <div class="column"> <p>In this paper, we introduce some kinds of interpolative proximal contractions named as Reich-Rus-<span style="font-weight: 400;">Ć</span>iri<span style="font-weight: 400;">ć</span> and Kannan types. Then taking into account the aforementioned mappings, we give a few best proximity point results. As special cases we obtain some fixed point results for interpolative contractions. To support the theory, some examples are provided.</p> </div> </div> </div> </div> </div> </div> Ishak Altun Ayşenur Taşdemir Copyright (c) 2021-11-09 2021-11-09 44 9 1233 1241 On property (<i>A</i>) of the amalgamated duplication of a ring along an ideal https://www.ajol.info/index.php/qm/article/view/217157 <div class="page" title="Page 1"> <div class="layoutArea"> <div class="column"> <p>The main purpose of this paper is to totally characterize when the amalgamated duplication <em>R</em> <span style="font-weight: 400;">⋈</span> <em>I</em> of a ring <em>R</em> along an ideal <em>I</em> is an <em>A</em>-ring as well as an <em>SA</em>-ring. In this regard, we prove that <em>R</em> ◃▹ <em>I</em> is an <em>SA</em>-ring if and only if <em>R</em> is an <em>SA</em>-ring and <em>I</em> is contained in the set of zero divisors Z(<em>R</em>) of <em>R</em>. As to the Property (<em>A</em>) of <em>R</em> <span style="font-weight: 400;">⋈</span> <em>I</em>, it turns out that its characterization involves a new concept that we introduce in [6] and that we term the Property (<em>A</em>) of a module <em>M</em> along an ideal <em>I</em>. In fact, we prove that <em>R </em><span style="font-weight: 400;">⋈ </span><em>I</em> is an <em>A</em>-ring if and only if <em>R</em> is an <em>A</em>-ring, <em>I</em> is an <em>A</em>-module along itself and if <em>p</em> is a prime ideal of <em>R</em> such that <em>p</em> ⊆ Z<em><sub>R</sub></em>(<em>I</em>)∪Z<em><sup>I</sup></em>(<em>R</em>), then either <em>p</em> ⊆ Z<em><sub>R</sub></em>(<em>I</em>) or <em>p</em> ⊆ Z<em><sup>I</sup></em> (<em>R</em>), where Z<em><sup>I</sup></em> (<em>R</em>) := {a ∈ <em>R</em> : a + <em>I</em> ⊆ Z(<em>R</em>)}.</p> </div> </div> </div> Youssef Arssi Samir Bouchiba Copyright (c) 2021-11-09 2021-11-09 44 9 1243 1259 Global behavior and oscillation of a third order difference equation https://www.ajol.info/index.php/qm/article/view/217158 <div class="page" title="Page 1"> <div class="layoutArea"> <div class="column"> <p>In this paper, we solve and study the global behavior of the admissible solutions of the difference equation <em><span style="font-weight: 400;">x</span></em><sub><span style="font-weight: 400;">n+1</span></sub><span style="font-weight: 400;"> = </span><em><span style="font-weight: 400;">x</span></em><sub><span style="font-weight: 400;">n</span></sub><em><span style="font-weight: 400;">x</span></em><span style="font-weight: 400;"><sub>n-2 / </sub></span><em><span style="font-weight: 400;">ax</span></em><sub><span style="font-weight: 400;">n-1</span></sub><span style="font-weight: 400;"> + </span><em><span style="font-weight: 400;">bx</span></em><sub><span style="font-weight: 400;">n-2&nbsp;</span></sub><span style="font-weight: 400;"><em>n</em> = 0,1,...., </span>where <em>a,b</em> &gt; 0 and the initial values <em>x</em><sub>−2</sub>, <em>x</em><sub>−1</sub>, <em>x</em><sub>0</sub> are real numbers. We study also the oscillation of the admissible solutions of the aforementioned difference equation and give some illustrative examples.</p> </div> </div> </div> R. Abo-Zeid Copyright (c) 2021-11-09 2021-11-09 44 9 1261 1280 Addendum to: “Variations of classical selection principles: An overview” https://www.ajol.info/index.php/qm/article/view/217159 <div class="page" title="Page 1"> <div class="layoutArea"> <div class="column"> <p>In this addendum we give a few remarks about Subsection 6.2 of the paper “Variations of classical selection principles: An overview,” Quaestiones Mathematicae, https://doi.org/10.2989/16073606.2019.1601646.</p> </div> </div> </div> Ljubiŝa D.R. Kočinac Copyright (c) 2021-11-09 2021-11-09 44 9 1281 1282