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-USCopyright for articles published in this journal is retained by the journal.mike@nisc.co.za (Mike Schramm)dubeta@unisa.ac.za (T. Dube)Tue, 15 Oct 2019 16:04:33 +0200OJS 2.4.3.0http://blogs.law.harvard.edu/tech/rss60Some equivalent conditions for Huneke's two conjectures
https://www.ajol.info/index.php/qm/article/view/190370
<p>In this paper we give some equivalent conditions for the C. Huneke's two conjectures concerning the finiteness properties of the local cohomology module H<sup>i</sup><sub>I</sub> (<em>R</em>), where <em>R</em> is a regular local ring, <em>I</em> is an ideal of <em>R</em> and <em>i</em> <span>≥</span> 0 is an integer.</p><p><em><strong>Mathematics Subject Classification (2010):</strong></em> 13D45, 14B15, 13E05.</p><p><strong>Keywords:</strong> Bass number, Betti number, Goldie dimension, local cohomology, regular ring</p>Fatemeh Sohrabi, Kamal Bahmanpour, Ghader Ghasemihttps://www.ajol.info/index.php/qm/article/view/190370Mon, 14 Oct 2019 00:00:00 +0200The strongly invariant extending property for Abelian Groups
https://www.ajol.info/index.php/qm/article/view/190371
<p>We define three specific properties of abelian groups calling them <em>strongly invariant extending property, intermediate fully invariant extending property</em> and <em>strongly intermediate fully invariant extending property</em>. Some results concerning these concepts are proved as for the third one a complete necessary and sufficient conditions is established. Our achievements somewhat extend results due to Birkenmeier et al. published in Comm. Algebra (2001).</p><p><em><strong>Mathematics Subject Classification (2010):</strong> </em>20K10, 20K20, 20K21.</p><p><strong>Keywords:</strong> fi-extending property, si-extending property, i-extending property</p>Andrey R. Chekhlov, Peter V. Danchevhttps://www.ajol.info/index.php/qm/article/view/190371Tue, 15 Oct 2019 00:00:00 +0200Division closed lattice-ordered rings and commutative <i>L</i>*-rings
https://www.ajol.info/index.php/qm/article/view/190374
<p>The paper continues the study of division closed lattice-ordered rings and commutative <em>L</em>*-rings. More interesting properties of division closed lattice- ordered rings are presented and it is shown that under certain conditions such rings are <em>f</em>-rings. The main result on <em>L</em>*-rings is that for a commutative semilocal ring with the identity, it is <em>L</em>* if and only if it is <em>O</em>*.</p><p><strong><em>Mathematics Subject Classification (2010):</em></strong> Primary 06F25.</p><p><strong>Keywords:</strong> Lattice order, partial order, regular division closed, total order, <em>F</em>*-ring, <em>L</em>*-ring, <em>O</em>*-ring</p>Jingjing Ma, Jessica Smithhttps://www.ajol.info/index.php/qm/article/view/190374Tue, 15 Oct 2019 00:00:00 +0200The (<i>p, q</i>)-mixed geominimal surface areas
https://www.ajol.info/index.php/qm/article/view/190376
<p>The (<em>p, q</em>)-mixed geominimal surface areas are introduced. A special case of the new concept is the <em>L<sub>p</sub></em> geominimal surface area introduced by Lutwak. Related inequalities, such as affine isoperimetric inequality, monotonous inequality, cyclic inequality, and Brunn-Minkowski inequality, are established. These new inequalities strengthen some well-known inequalities related to the <em>L<sub>p</sub></em> geominimal surface area.</p><p><strong><em>Mathematics Subject Classification (2010):</em> </strong>52A20, 52A40.</p><p><strong>Keywords:</strong> (<em>p, q</em>)-mixed geominimal surface areas,<em> L<sub>p</sub></em> geominimal surface area, affine isoperimetric inequality, Brunn-Minkowski inequality</p>Yibin Feng, Binwu Hehttps://www.ajol.info/index.php/qm/article/view/190376Mon, 14 Oct 2019 00:00:00 +0200Some new results on <i>H</i> summability of fourier series
https://www.ajol.info/index.php/qm/article/view/190377
<p>In this paper we shall be concerned with <em>H<sub>α</sub> </em>summability, for 0 < <span>α</span> <span>≤ </span>2 of the Fourier series of arbitrary <em>L</em><sup>1</sup>([-<span>π,</span> <span>π</span>]) functions. The methods employed here are a modification of the real variable ones introduced by J. Marcinkiewicz. The needed modications give direct proofs of maximal theorems with respect to <em>A</em><sub>1</sub> weights. We also give a counter-example of a measure such that there is no convergence a.e. to the density of the measure. Finally, we present a Kakutani type of theorem, proving the <em>w</em>*- density, in the space of of probability measures defined on [-π, π] of Borel measures for which there is no<em> H</em><sub>2</sub> summability a.e.</p><p><em><strong>Mathematics Subject Classification (2010):</strong></em> Primary 42B08; Secondary 26A24.</p><p><strong>Keywords:</strong> Fourier series, strong summability, Marcinkiewicz function, <em>A</em><sub>1</sub>-weights</p>Calixto P. Calderón, A. Susana Coré, Wilfredo O. Urbinahttps://www.ajol.info/index.php/qm/article/view/190377Tue, 15 Oct 2019 00:00:00 +0200A generalized contraction proximal point algorithm with two monotone operators
https://www.ajol.info/index.php/qm/article/view/190378
<p>In this work, we introduce a generalized contraction proximal point algorithm and use it to approximate common zeros of maximal monotone operators <em>A</em> and <em>B</em> in a real Hilbert space setting. The algorithm is a two step procedure that alternates the resolvents of these operators and uses general assumptions on the parameters involved. For particular cases, these relaxed parameters improve the convergence rate of the algorithm. A strong convergence result associated with the algorithm is proved under mild conditions on the parameters. Our main result improves and extends several results in the literature.</p><p><em><strong>Mathematics Subject Classification (2010):</strong></em> 47J25, 47H05, 47H09.</p><p><strong>Keywords:</strong> Maximal monotone operator, contraction proximal point algorithm, alternating method, nonexpansive map, resolvent operator</p>Oganeditse A. Boikanyo, Spencer Makgoenghttps://www.ajol.info/index.php/qm/article/view/190378Tue, 15 Oct 2019 00:00:00 +0200Some new results on functions in <i>C</i>(<i>X</i>) having their support on ideals of closed sets
https://www.ajol.info/index.php/qm/article/view/190382
<p>For any ideal <em>P</em> of closed sets in <em>X</em>, let <em>C<sub>P</sub></em>(<em>X</em>) be the family of those functions in <em>C</em>(<em>X</em>) whose support lie on <em>P.</em> Further let <em>C <sup>P</sup><sub>∞</sub> </em>(<em>X</em>) contain precisely those functions f in <em>C</em>(<em>X</em>) for which for each ϵ > 0, {<em>x</em> <span>∈</span> <em>X</em> : <span>|</span><em>f</em>(<em>x</em>)<span>|</span> <span>≥</span> ϵ} is a member of <em>P</em>. Let <em><sup>υ</sup>C<sub>P</sub>X</em> stand for the set of all those points <em>p</em> in <em>βX</em> at which the stone extension <em>f</em>* for each <em>f</em> in <em>C<sub>P</sub></em>(<em>X</em>) is real valued. We show that each realcompact space lying between <em>X</em> and <em>βX</em> is of the form <em><sup>υ</sup>C<sub>P</sub>X</em> if and only if <em>X</em> is pseudocompact. We find out conditions under which an arbitrary product of spaces of the form locally-<em>P</em> or almost locally-<em>P</em>, becomes a space of the same form. We further show that <em>C<sub>P</sub></em>(<em>X</em>) is a free ideal (essential ideal) of <em>C</em>(<em>X</em>) if and only if <em>C <sup>P</sup><sub>∞</sub></em> (<em>X</em>) is a free ideal (essential ideal) of <em>C*</em> (<em>X</em>) + <em>C <sup>P</sup><sub>∞</sub></em> (<em>X</em>) when and only when <em>X</em> is locally-<em>P</em> (almost locally-<em>P</em>). We address the problem, when does <em>C<sub>P</sub></em>(<em>X</em>) or<em> C <sup>P</sup><sub>∞</sub></em> (<em>X</em>) become identical to the socle of the ring <em>C</em>(<em>X</em>). The results obtained turn out to imply a special version of the fact obtained by Azarpanah corresponding to the choice <em>P</em> <span>≡</span> the ideal of compact sets in <em>X</em>. Finally we observe that the ideals of the form <em>C<sub>P</sub></em>(<em>X</em>) of <em>C</em>(<em>X</em>) are no other than the <em>z<sup>◦</sup></em> -ideals of <em>C</em>(<em>X</em>).</p><p><em><strong>Mathematics Subject Classification (2010):</strong></em> Primary 54C40; Secondary 46E25.</p><p><strong>Keywords:</strong> Compact support, pseudocompact space, intermediate ring, pseudocompact support, essential ideal, <em>z<sup>◦</sup></em>-ideal, socle, <em>C</em>-type ring</p>Sudip Kumar Acharyya, Sagarmoy Bag, Goutam Bhunia, Pritam Roojhttps://www.ajol.info/index.php/qm/article/view/190382Tue, 15 Oct 2019 00:00:00 +0200Remarks on <i>w</i>-domination of discrete subspaces
https://www.ajol.info/index.php/qm/article/view/190384
<p>Given a space <em>X</em>, we will say that <em>a class A of subsets of X is dominated by a class B</em> if for any <em>A</em> <span>∈ </span><em>A</em>, there exists a <em>B</em> <span>∈</span> <em>B</em> such that <em>A</em> <span>⊂</span> <em>B̅</em>. In particular, all (closed) discrete subsets of <em>X</em> are countably dominated (which we frequently abbreviate as <em>w</em>-dominated) if, for any (closed) discrete set <em>D</em> <span>⊂</span> <em>X</em>, there exists a countable set <em>B</em> <span>⊂</span> <em>X</em> such that <em>D</em> <span>⊂</span> <em>B̅</em>. In this paper, we investigate the topological properties of spaces in which (closed) discrete subspaces are dominated either by countable subsets or by Lindelöf subspaces.</p><p><strong>Mathematics Subject Classification (2010):</strong> Primary 54D20, 54A25; Secondary 54A35.</p><p><strong>Keywords:</strong> <em>w</em>-domination of discrete subsets, <em>G</em><span><sub>δ</sub></span>-diagonal, star Lindel<span>ö</span>f, semi-stratifiable space</p>Yan-Kui Song, Wei-Feng Xuanhttps://www.ajol.info/index.php/qm/article/view/190384Tue, 15 Oct 2019 00:00:00 +0200Global Italian domination in graphs
https://www.ajol.info/index.php/qm/article/view/190387
<p>An Italian dominating function (IDF) on a graph <em>G</em> = (<em>V</em>,<em>E</em>) is a function f : V <span>→</span> f{0, 1, 2} satisfying the condition that for every vertex <em>v</em> <span>∈</span> <em>V</em> (<em>G</em>) with <em>f</em>(<em>v</em>) = <em>0</em>, either <em>v</em> is adjacent to a vertex assigned 2 under <em>f</em>, or <em>v</em> is adjacent to at least two vertices assigned 1. The weight of an IDF <em>f</em> is the value <em>Σ<sub>v∈V</sub></em> <sub>(<em>G</em>)</sub> <em>f</em>(<em>v</em>). The Italian domination number of a graph<em> G</em>, denoted by <em><sub>γI</sub></em> (<em>G</em>), is the minimum weight of an IDF on <em>G</em>. An IDF <em>f</em> on <em>G</em> is called a global Italian dominating function (GIDF) on <em>G</em> if f is also an IDF on the complement <em>G</em> of <em>G</em>. The global Italian domination number of <em>G</em>, denoted by <em><sub>γgI</sub></em> (<em>G</em>), is the minimum weight of a GIDF on <em>G</em>. In this paper, we initiate the study of the global Italian domination number and we present some strict bounds for the global Italian domination number. In particular, we prove that for any tree <em>T</em> of order <em>n</em> <span> ≥</span> 4, <sub><em>γg</em>I</sub> (<em>T</em>) <span>≤</span> <em><sub>γI</sub> </em>(<em>T</em>) + 2 and we characterize all trees with <em><sub>γgI</sub></em> (<em>T</em>) = <em><sub>γI</sub></em> (<em>T</em>) + 2 and <em><sub>γgI</sub></em> (<em>T</em>) = <em><sub>γ</sub><sub>I</sub></em> (<em>T</em>) + 1.</p><p><em><strong>Mathematics Subject Classification (2010):</strong></em> 05C69.</p><p><strong>Keywords:</strong> Italian dominating function, Italian domination number, global Italian dominating function, global Italian domination number</p>Guoliang Hao, Kangxiu Hu, Shouliu Wei, Zhijun Xuhttps://www.ajol.info/index.php/qm/article/view/190387Tue, 15 Oct 2019 00:00:00 +0200Separating sets in bitopological spaces by <i>P-LSC</i> and <i>L-USC</i> functions
https://www.ajol.info/index.php/qm/article/view/190389
<p>In this paper we characterize the pairs ⟨A<sub>0</sub>,A<sub>1</sub>⟩ and ⟨A<sup>-</sup> ,A<sup>+</sup>⟩ of dis- joint subsets of bitopological space (<em>X, P, L</em>) which can be separated by the function<em> f</em> : <em>X</em> <span>→</span> <em>R</em> simultaneously lower semicontinuous in the topology<em> P</em> and upper semi-continuous in the topology <em>L</em>.</p><p><em><strong>Mathematics Subject Classification (2010):</strong></em> Primary 26A21, 54E55; Secondary 26A15, 54C08.</p><p><strong>Keywords:</strong> Bitopological spaces, upper semicontinuous functions, lower semicontinuous functions, Urysohn's lemma, separating sets</p>Paulina Szyszkowskahttps://www.ajol.info/index.php/qm/article/view/190389Tue, 15 Oct 2019 00:00:00 +0200Generating dual Baer Modules via fully invariant submodules
https://www.ajol.info/index.php/qm/article/view/190391
<p>In this paper, we introduce a concept of a dual <em>F</em>-Baer module <em>M</em> where <em>F</em> is the fully invariant submodule of <em>M</em>, by this means we deal with generating dual Baer modules. We investigate direct sums of dual <em>F</em>-Baer modules <em>M</em> by exerting the notion of relatively dual <em>F</em>-Baer modules. We also obtain applications of dual <em>F</em>-Baer modules to rings and the preradical <em>Z</em> * (<sup>.</sup>).</p><p><em><strong>Mathematics Subject Classification (2010):</strong></em> 16D10, 16D40, 16D70, 16D80.</p><p><strong>Keywords:</strong> Dual Baer module, dual inverse split module, fully invariant submodule, cosingular submodule</p>Tugce Pekacar Calci, Abdullah Harmanci, Burcu Ungorhttps://www.ajol.info/index.php/qm/article/view/190391Mon, 14 Oct 2019 00:00:00 +0200