University of KashanMathematics Interdisciplinary Research2538-36396220210601Weakly Compatible Maps and Fixed Points9710511147710.22052/mir.2021.240433.1267ENM.ShahsavariDepartment of Mathematics, Qazvin Branch, Islamic Azad University, Qazvin, IranAbdolrahmanRazaniDepartment of Pure Mathematics,
Imam Khomeini International University, Qazvin, I. R. IranGhasemAbbasiDepartment of Mathematics, Qazvin Branch, Islamic Azad University, Qazvin, IranJournal Article20210115Here, the existence of fixed points for weakly compatible maps is studied. The results are new generalization of the results of [5]. Finally, we study the new common fixed point theorems.University of KashanMathematics Interdisciplinary Research2538-36396220210601On Eccentricity Version of Zagreb Coindices10712011148310.22052/mir.2021.240325.1247ENMahdiehAzariDepartment of Mathematics, Kazerun Branch, Islamic Azad University,
P. O. Box: 73135-168, Kazerun, IranJournal Article20201018The eccentric connectivity coindex has recently been introduced (Hua and Miao, 2019) as the total eccentricity sum of all pairs of non-adjacent vertices in a graph. Considering the total eccentricity product of non-adjacent vertex pairs, we introduce here another invariant of connected graphs called the second Zagreb eccentricity coindex. We study some mathematical properties of the eccentric connectivity coindex and second Zagreb eccentricity coindex. We also determine the extremal values of the second Zagreb eccentricity coindex over some specific families of graphs such as trees, unicyclic graphs, connected graphs, and connected bipartite graphs and describe the extremal graphs. Moreover, we compare the second Zagreb eccentricity coindex with the eccentric connectivity coindex and give directions for further studies.University of KashanMathematics Interdisciplinary Research2538-36396220210601On Finding a Relative Interior Point of a Polyhedral Set12113811147610.22052/mir.2021.240315.1243ENMahmoodMehdilooDepartment of Mathematics and Applications, University of Mohaghegh Ardabili, Ardabil, IranJournal Article20201008This paper proposes a new linear program for finding a relative interior point of a polyhedral set. Based on characterizing the relative interior of a polyhedral set through its polyhedral <em>representing sets</em>, two main contributions are made. First, we complete the existing results in the literature that require the non-negativity of the given polyhedral set. Then, we deal with the general case where this requirement may not be met.University of KashanMathematics Interdisciplinary Research2538-36396220210601Big Finitistic Dimensions for Categories of Quiver Representations13914911148510.22052/mir.2021.240439.1273ENRoghayehBagherianDepartment of Mathematics,
Isfahan University of Technology,
Isfahan, I. R. IranEsmaeilHosseiniDepartment of Pure Mathematics, Faculty of Mathematics and Computer Sciences, Shahid Chamran University, Ahvaz, IranJournal Article20210131<span class="fontstyle0">Assume that </span><span class="fontstyle2">A </span><span class="fontstyle0">is a Grothendieck category and </span><span class="fontstyle2">R </span><span class="fontstyle0">is the category of all </span><span class="fontstyle2">A</span><span class="fontstyle0">-representations of a given quiver </span><span class="fontstyle2">Q</span><span class="fontstyle0">. If </span><span class="fontstyle2">Q </span><span class="fontstyle0">is left rooted and </span><span class="fontstyle2">A </span><span class="fontstyle0">has a projective generator, we prove that the big finitistic flat (resp. projective) dimension </span><span class="fontstyle3">FFD(</span><span class="fontstyle2">A</span><span class="fontstyle3">) </span><span class="fontstyle0">(resp. </span><span class="fontstyle3">FPD(</span><span class="fontstyle2">A</span><span class="fontstyle3">)</span><span class="fontstyle0">) of </span><span class="fontstyle2">A </span><span class="fontstyle0">is finite if and only if the big finitistic flat (resp. projective) dimension of </span><span class="fontstyle2">R </span><span class="fontstyle0">is finite. When </span><span class="fontstyle2">A </span><span class="fontstyle0">is the Grothendieck category of left modules over a unitary ring </span><span class="fontstyle4">R</span><span class="fontstyle0">, we prove that if </span><span class="fontstyle3">FPD(</span><span class="fontstyle2">R</span><span class="fontstyle3">) </span><span class="fontstyle4">< </span><span class="fontstyle3">+∞</span><span class="fontstyle2"> </span><span class="fontstyle0">then any representation of </span><span class="fontstyle2">Q </span><span class="fontstyle0">of finite flat dimension has finite projective dimension. Moreover, if </span><span class="fontstyle4">R </span><span class="fontstyle0">is </span><span class="fontstyle4">n</span><span class="fontstyle0">-perfect then we show that </span><span class="fontstyle3">FFD(</span><span class="fontstyle2">R</span><span class="fontstyle3">) </span><span class="fontstyle4">< </span><span class="fontstyle3">+</span>∞ <span class="fontstyle2"> </span><span class="fontstyle0">if and only if </span><span class="fontstyle3">FPD(</span><span class="fontstyle2">R</span><span class="fontstyle3">) </span><span class="fontstyle4">< </span><span class="fontstyle3">+</span>∞<span class="fontstyle0">.</span><br /> University of KashanMathematics Interdisciplinary Research2538-36396220210601Scaling Symmetry and a New Conservation Law of the Harry Dym Equation15115811150610.22052/mir.2021.240441.1270ENMehdiJafariDepartment of Mathematics, Payame Noor University, P. O. BOX 19395-3697, Tehran, IranAmirhesamZaeimDepartment of Mathematics, Payame Noor University, P. O. BOX 19395-3697, Tehran, IranSomayesadatMahdionDepartment of Mathematics, Payame Noor University, P. O. BOX 19395-3697, Tehran, IranJournal Article20210120In this paper, we obtain a new conservation law for the Harry Dym equation by using the scaling method. This method is algorithmic and based on variational calculus and linear algebra. In this method, the density of the conservation law is constructed by considering the scaling symmetry of the equation and the associated flux is obtained by the homotopy operator. This density-flux pair gives a conservation law for the equation. A conservation law of rank 7 is constructed for the Harry Dym equation.University of KashanMathematics Interdisciplinary Research2538-36396220210601Golden Ratio: The Mathematics of Beauty1591708924510.22052/mir.2019.174577.1123ENHamidGhorbaniFaculty of Mathematical Sciences,
Department of Statistics,
University of Kashan, Kashan, IranJournal Article20190310<span class="fontstyle0">Historically, mathematics and architecture have been associated with one another. Ratios are good example of this interconnection. The origin of ratios can be found in nature, which makes the nature so attractive. As an example, consider the architecture inspired by flowers which seems so harmonic to us. In the same way, the architectural plan of many well-known historical buildings such as mosques and bridges shows a rhythmic balance which according to most experts the reason lies in using the ratios. The golden ratio has been used to analyze the proportions of natural objects as well as building’s harmony. In this paper, after recalling the (mathematical) definition of the golden ratio, its ability to describe the harmony in the nature is discussed. When teaching mathematics in the schools, one may refer to this interconnection to encourage students to feel better with mathematics and deepen their understanding of proportion. At the end, the golden ratio decimals as well as its binary digits has been statistically examined to confirm their behavior as a random number generator.</span>