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Repdigits in base b as product of two k-generalized Pell numbers 以b为基底的数字是两个k-广义佩尔数的乘积
IF 0.7
Afrika Matematika Pub Date : 2025-10-10 DOI: 10.1007/s13370-025-01384-9
Zafer Şiar
{"title":"Repdigits in base b as product of two k-generalized Pell numbers","authors":"Zafer Şiar","doi":"10.1007/s13370-025-01384-9","DOIUrl":"10.1007/s13370-025-01384-9","url":null,"abstract":"<div><p>Let <span>(kge 2)</span> be an integer. The <i>k</i>-generalized Pell sequence <span>((P_{n}^{(k)})_{nge 2-k})</span> is defined by the initial values <span>(0,0,ldots ,0,1)</span>(<i>k</i> terms) and the recurrence <span>(P_{n}^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+ldots +P_{n-k}^{(k)})</span> for all <span>(nge 2)</span>. In this study, we deal with the Diophantine equation </p><div><div><span>$$P_{n}^{(k)}P_{m}^{(k)}=dleft( frac{b^{l}-1}{b-1}right)$$</span></div></div><p>in positive integers <i>n</i>, <i>m</i>, <i>k</i>, <i>b</i>, <i>d</i>, <i>l</i> with <span>(kge 3,lge 2,~2le mle n,)</span> <span>(2le ble 10,)</span> and <span>(1le dle b-1,)</span> and we show that all solutions of this equation are given by </p><div><div><span>$$begin{aligned} P_{2}^{(k)}P_{2}^{(k)}&amp;=(11)_{3},~P_{3}^{(k)}P_{2}^{(k)}=(22)_{4}=(11)_{9}text {, }P_{4}^{(k)}P_{2}^{(k)}=(222)_{3}text { for }kge 3, P_{5}^{(k)}P_{3}^{(k)}&amp;=(2222)_{4}text { for }kge 4, end{aligned}$$</span></div></div><p>and </p><div><div><span>$$P_{5}^{(3)}P_{2}^{(3)}=left( 66right) _{10}.$$</span></div></div></div>","PeriodicalId":46107,"journal":{"name":"Afrika Matematika","volume":"36 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145256187","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The rainbow connected number of several infinite graph families 若干无限图族的彩虹连通数
IF 0.7
Afrika Matematika Pub Date : 2025-10-10 DOI: 10.1007/s13370-025-01381-y
Liam B. Baker, Jonathan Kariv, Ronald J. Maartens
{"title":"The rainbow connected number of several infinite graph families","authors":"Liam B. Baker,&nbsp;Jonathan Kariv,&nbsp;Ronald J. Maartens","doi":"10.1007/s13370-025-01381-y","DOIUrl":"10.1007/s13370-025-01381-y","url":null,"abstract":"<div><p>Let <i>G</i> be a nontrivial connected graph with an edge coloring, and let <span>(u,v in V(G))</span>. A <span>(u-v)</span> path in <i>G</i> is said to be a <i>rainbow path</i> if no color is repeated on the edges of the path. Similarly, we define a <i>rainbow geodesic</i>. A <i>rainbow connected graph</i> <i>G</i> is a graph with an edge coloring such that every two vertices in <i>G</i> are connected by a rainbow path. Further, a <i>strong rainbow connected graph</i> <i>G</i> is a graph with an edge coloring such that every two vertices in <i>G</i> is connected by a rainbow geodesic. The minimum number of colors needed to make a graph rainbow connected is called the <i>rainbow connection number</i>, denoted <span>({{,textrm{rc},}}(G))</span>, and the minimum number of colors needed to make a graph strong rainbow connected is called the <i>strong rainbow connection number</i>, denoted <span>({{,textrm{src},}}(G))</span>. In this paper we determine <span>({{,textrm{rc},}}(G))</span> and <span>({{,textrm{src},}}(G))</span> when <i>G</i> is a <i>n</i>-dimensional rectangular grid graph, triangular grid graph, hexagonal grid graph, and a (weak) Bruhat graph, respectively. We show for all these families that <span>({{,textrm{src},}}(G)={{,textrm{diam},}}(G))</span>.</p></div>","PeriodicalId":46107,"journal":{"name":"Afrika Matematika","volume":"36 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s13370-025-01381-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145256188","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Small energy and Hosoya index among caterpillars with a given degree sequence 毛虫的能量和细谷指数在一定程度序列下较小
IF 0.7
Afrika Matematika Pub Date : 2025-10-10 DOI: 10.1007/s13370-025-01382-x
Eric O. D. Andriantiana, Xhanti Sinoxolo
{"title":"Small energy and Hosoya index among caterpillars with a given degree sequence","authors":"Eric O. D. Andriantiana,&nbsp;Xhanti Sinoxolo","doi":"10.1007/s13370-025-01382-x","DOIUrl":"10.1007/s13370-025-01382-x","url":null,"abstract":"<div><p>The energy <i>En</i>(<i>G</i>) of a graph <i>G</i> is defined as the sum of the absolute values of its eigenvalues. The Hosoya index <i>Z</i>(<i>G</i>) of a graph <i>G</i> is the number of independent edge subsets of <i>G</i>, including the empty set. For any given degree sequence <i>D</i>, we characterize the caterpillar <span>(mathcal {S}(D))</span> that has the minimum <i>Z</i> and <i>En</i>. We also show that <span>(Z(mathcal {S}(D))&lt;Z(mathcal {S}(Y)))</span> and <span>(En(mathcal {S}(D))&lt;En(mathcal {S}(Y)))</span> for any degree sequences <span>(Y=(y_1,dots ,y_n))</span> and <span>(D=(d_1,dots ,d_n))</span> with </p><div><div><span>$$sum _{i=1}^{n}y_i=sum _{i=1}^{n}d_itext { and }sum _{i=1}^{k}y_ile sum _{i=1}^{k}d_i text { for all }1le k le n.$$</span></div></div></div>","PeriodicalId":46107,"journal":{"name":"Afrika Matematika","volume":"36 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s13370-025-01382-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145256184","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Plithogenic product intuitionistic fuzzy graph 生成物直观模糊图
IF 0.7
Afrika Matematika Pub Date : 2025-10-10 DOI: 10.1007/s13370-025-01378-7
T. Bharathi, Arul Priya Sampath
{"title":"Plithogenic product intuitionistic fuzzy graph","authors":"T. Bharathi,&nbsp;Arul Priya Sampath","doi":"10.1007/s13370-025-01378-7","DOIUrl":"10.1007/s13370-025-01378-7","url":null,"abstract":"<div><p>Plithogenic product intuitionistic fuzzy graph is a novel graphical model for representing complex systems characterised by multi-valued dyadic attributes. It provides four or more dyadic attributes to its elements, consisting of <span>({mu })</span> membership values and <span>({nu })</span> non-membership values. The dyadic attribute values of the edges are calculated using the (*) operator. In this paper, various properties and characterisations of Plithogenic product intuitionistic fuzzy graphs, including order, size, path, and cycle, are analysed to show the utility of the Plithogenic product intuitionistic fuzzy graphs. Additionally, weight, strength, the strength of connectedness, and subgraphs of Plithogenic product intuitionistic fuzzy graphs are newly introduced, accompanied by examples and figures, to examine the connectivity between parts and the significance of each part.</p></div>","PeriodicalId":46107,"journal":{"name":"Afrika Matematika","volume":"36 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145256185","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the semisimplicity of the decorated partial Brauer algebras 关于修饰部分Brauer代数的半简单性
IF 0.7
Afrika Matematika Pub Date : 2025-10-10 DOI: 10.1007/s13370-025-01386-7
Amani M. Alfadhli
{"title":"On the semisimplicity of the decorated partial Brauer algebras","authors":"Amani M. Alfadhli","doi":"10.1007/s13370-025-01386-7","DOIUrl":"10.1007/s13370-025-01386-7","url":null,"abstract":"<div><p>The decorated Partial Brauer algebras are finite dimensional diagram algebras contain Brauer algebras, Partial Brauer algebras and the group algebras <span>(Rwidetilde{S_{n}})</span>, where <span>(widetilde{S_{n}})</span> is the wreath product group <span>(mathbb {Z}_{2}wr S_{n})</span> of <span>(mathbb {Z}_{2})</span> with <span>(S_{n})</span>. In this paper, we study the semisimplicity criterion of the decorated partial Brauer algebras using two functors <i>F</i> and <i>G</i>. In particular, we determine for which value of the parameters this algebra is semisimple. This result can be considered as a generalization of Hanlon–Wales conjecture on Brauer algebra.</p></div>","PeriodicalId":46107,"journal":{"name":"Afrika Matematika","volume":"36 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145256186","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Congruences Modulo 2 for the Eighth-Order Mock Theta Function (V_1(q)) 八阶模拟函数模2的同余式 (V_1(q))
IF 0.7
Afrika Matematika Pub Date : 2025-09-30 DOI: 10.1007/s13370-025-01380-z
Hirakjyoti Das
{"title":"Congruences Modulo 2 for the Eighth-Order Mock Theta Function (V_1(q))","authors":"Hirakjyoti Das","doi":"10.1007/s13370-025-01380-z","DOIUrl":"10.1007/s13370-025-01380-z","url":null,"abstract":"<div><p>Not many of the congruence properties of the eighth-order mock theta function <span>(V_1(q))</span>: </p><div><div><span>$$begin{aligned} V_1(q):=sum _{n=0}^infty dfrac{q^{(n+1)^2}left( -q;q^2right) _n}{left( q;q^2right) _{n+1}}=sum _{n=1}^infty v_1(n)q^n end{aligned}$$</span></div></div><p>have been considered to date. We show that there are self-similarities of the coefficients of <span>(V_1(q))</span>. As consequences, we find congruences like the one below. For all <span>(nge 0)</span> and <span>(kge 1)</span>, we have </p><div><div><span>$$begin{aligned} v_1left( 6times 29^{2 k} n+ 6times 29^{2 k-1} s+dfrac{7times 29^{2 k-1}+1}{4}right) equiv 0 pmod {2} end{aligned}$$</span></div></div><p>for <span>(0le s&lt; 29)</span>, <span>(sne 13)</span>.</p></div>","PeriodicalId":46107,"journal":{"name":"Afrika Matematika","volume":"36 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2025-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145211010","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A new conservative numerical method for strongly coupled nonlinear Schrödinger equations 强耦合非线性Schrödinger方程的一种新的保守数值方法
IF 0.7
Afrika Matematika Pub Date : 2025-09-30 DOI: 10.1007/s13370-025-01379-6
Rıdvan Fatih Örs, Canan Köroğlu, Ayhan Aydın
{"title":"A new conservative numerical method for strongly coupled nonlinear Schrödinger equations","authors":"Rıdvan Fatih Örs,&nbsp;Canan Köroğlu,&nbsp;Ayhan Aydın","doi":"10.1007/s13370-025-01379-6","DOIUrl":"10.1007/s13370-025-01379-6","url":null,"abstract":"<div><p>In this paper, a numerical method based on the conservative finite difference scheme is constructed to numerically solve the strongly coupled nonlinear Schrödinger (SCNLS) equation. Conservative properties such as energy and mass of the SCNLS equation have been proven. In particular a fourth-order central difference scheme is used to discretize the the spatial derivative and a second-order Crank-Nicolson type discretization is used to discretize the temporal derivative. It has been shown that the proposed scheme preserves the discrete mass and energy. The existence of discrete solution is also investigated. Several numerical results are given to demonstrate the preservation properties of the new method. Also, the effect of the linear coupling parameters on the evolution of solitary waves is investigated.</p></div>","PeriodicalId":46107,"journal":{"name":"Afrika Matematika","volume":"36 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2025-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145211014","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
About integral product formula for Jack polynomials of two variables 关于二元杰克多项式的积分积公式
IF 0.7
Afrika Matematika Pub Date : 2025-09-30 DOI: 10.1007/s13370-025-01383-w
Béchir Amri
{"title":"About integral product formula for Jack polynomials of two variables","authors":"Béchir Amri","doi":"10.1007/s13370-025-01383-w","DOIUrl":"10.1007/s13370-025-01383-w","url":null,"abstract":"<div><p>In this work, we provide an elementary proof of an integral product formula for Jack polynomials of two variables, extending the well-known case of zonal polynomials. As an application, we derive an explicit integral representation for the Dunk- Bessel function of type <span>(B_2)</span>.</p></div>","PeriodicalId":46107,"journal":{"name":"Afrika Matematika","volume":"36 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2025-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145211015","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Representation-theoretic framework for the linear canonical transform in quantum harmonic analysis 量子谐波分析中线性正则变换的表示理论框架
IF 0.7
Afrika Matematika Pub Date : 2025-09-16 DOI: 10.1007/s13370-025-01377-8
Ishtaq Ahmad
{"title":"Representation-theoretic framework for the linear canonical transform in quantum harmonic analysis","authors":"Ishtaq Ahmad","doi":"10.1007/s13370-025-01377-8","DOIUrl":"10.1007/s13370-025-01377-8","url":null,"abstract":"<div><p>The Linear Canonical Transform (LCT) serves as a powerful generalization of the Fourier and fractional Fourier transforms, with significant implications in signal processing, optics, and quantummechanics. This paper develops a novel representation-theoretic framework for the LCT by leveraging the unitary dual of the Heisenberg group and the metaplectic representation of the symplectic group. Beyond recovering known uncertainty principles, we present refined inequalities that explicitly depend on the LCT parameter matrix and derive new structural results for the spectral decomposition of LCT operators. In particular, we provide a distributional spectral analysis for degenerate LCT cases (<span>(b = 0)</span>), introduce entropic uncertainty bounds tailored to the LCT domain, and propose a group-theoretic formulation of sparsity constraints. These findings significantly extend classical results and offer a deeper understanding of the LCT in both theoretical and applied contexts. We conclude with suggestions for quantum state manipulation via LCTs and numerical illustrations that bridge abstract theory with practical computation.</p></div>","PeriodicalId":46107,"journal":{"name":"Afrika Matematika","volume":"36 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2025-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145062212","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Unveiling quasi inverses of linear pencils in Banach algebra 揭示Banach代数中线性铅笔的拟逆
IF 0.7
Afrika Matematika Pub Date : 2025-09-16 DOI: 10.1007/s13370-025-01374-x
Hassen Khlif
{"title":"Unveiling quasi inverses of linear pencils in Banach algebra","authors":"Hassen Khlif","doi":"10.1007/s13370-025-01374-x","DOIUrl":"10.1007/s13370-025-01374-x","url":null,"abstract":"<div><p>We investigate the essential spectrum of linear pencils in Banach algebras, particularly their behavior under ideal perturbations. Building upon the foundational work of J. Shapiro and M. Snow in [The Fredholm spectrum of the sum and product of two operators, Transactions of the American Mathematical Society, 191 (1974), 387-393] on the Fredholm spectrum in Banach spaces, this study introduces novel characterizations of quasi-inverses and their role in spectral analysis. By leveraging these characterizations, we derive conditions ensuring that the essential spectrum of a linear pencil is confined within a specific sector of the complex plane. Our findings establish a refined connection between Fredholm theory and the algebraic structure of Banach algebras, offering both theoretical advancements and geometric insights into spectral containment. These results extend existing frameworks and open avenues for exploring spectral properties in more general algebraic settings with applications in operator theory and differential equations.</p></div>","PeriodicalId":46107,"journal":{"name":"Afrika Matematika","volume":"36 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2025-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145062211","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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