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A Diophantine Equation With Powers of Three Consecutive $$k-$$ Fibonacci Numbers 包含三个连续 $$k-$$ 斐波那契数幂的二阶方程
IF 2.2 3区 数学
Results in Mathematics Pub Date : 2024-04-19 DOI: 10.1007/s00025-024-02156-w
Carlos A. Gómez, Jhonny C. Gómez, Florian Luca
{"title":"A Diophantine Equation With Powers of Three Consecutive $$k-$$ Fibonacci Numbers","authors":"Carlos A. Gómez, Jhonny C. Gómez, Florian Luca","doi":"10.1007/s00025-024-02156-w","DOIUrl":"https://doi.org/10.1007/s00025-024-02156-w","url":null,"abstract":"<p>The <i>k</i>–generalized Fibonacci sequence <span>({F_n^{(k)}}_{nge 2-k})</span> is the linear recurrent sequence of order <i>k</i> whose first <i>k</i> terms are <span>(0, ldots , 0, 1)</span> and each term afterwards is the sum of the preceding <i>k</i> terms. The case <span>(k=2)</span> corresponds to the well known Fibonacci sequence <span>({F_n}_{nge 0})</span>. In this paper we extend the study of the exponential Diophantine equation <span>(left( F_{n+1}right) ^x+left( F_{n}right) ^x-left( F_{n-1}right) ^x=F_{m})</span> with terms <span>(F_r^{(k)})</span> instead of <span>(F_r)</span>, where <span>(rin {n+1,n,n-1,m})</span>.</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":"1 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140628402","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Note on the Matrix Moment Problem and its Relation with Matrix Biorthogonal Polynomials 关于矩阵矩问题及其与矩阵双多项式关系的说明
IF 2.2 3区 数学
Results in Mathematics Pub Date : 2024-04-19 DOI: 10.1007/s00025-024-02161-z
Juan Carlos García-Ardila
{"title":"A Note on the Matrix Moment Problem and its Relation with Matrix Biorthogonal Polynomials","authors":"Juan Carlos García-Ardila","doi":"10.1007/s00025-024-02161-z","DOIUrl":"https://doi.org/10.1007/s00025-024-02161-z","url":null,"abstract":"<p>In this manuscript, we study the operator moment problem for a block tridiagonal non-symmetric matrix. To study this problem, we used the relation between the resolvent function and a sequence of left-matrix orthogonal polynomials with respect to the sesquilinear form defined from the operator.</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":"112 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140628390","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Markov and Division Inequalities on Algebraic Sets 代数集上的马尔可夫不等式和除法不等式
IF 2.2 3区 数学
Results in Mathematics Pub Date : 2024-04-17 DOI: 10.1007/s00025-024-02153-z
Leokadia Bialas-Ciez, Jean-Paul Calvi, Agnieszka Kowalska
{"title":"Markov and Division Inequalities on Algebraic Sets","authors":"Leokadia Bialas-Ciez, Jean-Paul Calvi, Agnieszka Kowalska","doi":"10.1007/s00025-024-02153-z","DOIUrl":"https://doi.org/10.1007/s00025-024-02153-z","url":null,"abstract":"<p>A compact set <span>(Esubset mathbb {C}^N)</span> satisfies the Markov inequality if the supremum norm on <i>E</i> of the gradient of a polynomial <i>p</i> can be estimated from above by the norm of <i>p</i> multiplied by a constant polynomially depending on the degree of <i>p</i>. This inequality is strictly related to the Bernstein approximation theorem, Schur-type estimates and the extension property of smooth functions. Additionally, the Markov inequality can be applied to the construction of polynomial grids (norming sets or admissible meshes) useful in numerical analysis. We expect such an inequality with similar consequences not only on polynomially determining compacts but also on some nowhere dense sets. The primary goal of the paper is to extend the above definition of Markov inequality to the case of compact subsets of algebraic varieties in <span>(mathbb {C}^N)</span>. Moreover, we characterize compact sets satisfying such a Markov inequality on algebraic hypersurfaces as well as on certain varieties defined by several algebraic equations. We also prove a division inequality (a Schur-type inequality) on these sets. This opens up the possibility of establishing polynomial grids on algebraic sets. We also provide examples that complete and ilustrate the results.</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":"222 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140612671","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Self-dual Polytope and Self-dual Smooth Wulff Shape 自偶多面体和自偶光滑乌尔夫形
IF 2.2 3区 数学
Results in Mathematics Pub Date : 2024-04-16 DOI: 10.1007/s00025-024-02159-7
Huhe Han
{"title":"Self-dual Polytope and Self-dual Smooth Wulff Shape","authors":"Huhe Han","doi":"10.1007/s00025-024-02159-7","DOIUrl":"https://doi.org/10.1007/s00025-024-02159-7","url":null,"abstract":"<p>For any Wulff shape <i>W</i>, its dual Wulff shape and spherical Wulff shape <span>(widetilde{W})</span> can be defined naturally. A self-dual Wulff shape is a Wulff shape equaling its dual Wulff shape exactly. In this paper, we prove that a polytope is self-dual if and only if its spherical Wulff shape is a spherical convex body of constant width. We also prove that a smooth Wulff shape is self-dual if and only if for any interior points <i>P</i> of <span>(widetilde{W})</span> and for any point <i>Q</i> of the intersection of the boundary of <span>(widetilde{W})</span> and the graph of its spherical support function (with respect to <i>P</i>), the image of <i>Q</i> under the spherical blow-up (with respect to <i>P</i>) is always a point of <span>(widetilde{W})</span>. Moreover, we give an affirmative answer to the problem posed by M. Lassak which says that “Do there exist reduced spherical <i>n</i>-dimensional polytopes (possibly some simplices?) on <span>(mathbb {S}^n)</span>, where <span>(nge 3)</span>, different from the <span>(1/2^n)</span> part of <span>(mathbb {S}^n?)</span>”.</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":"81 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577940","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Logarithmic Vector Bbundles on the Blown-Up Variety 吹胀变体上的对数向量束
IF 2.2 3区 数学
Results in Mathematics Pub Date : 2024-04-13 DOI: 10.1007/s00025-024-02157-9
Sukmoon Huh, Min-Gyo Jeong
{"title":"Logarithmic Vector Bbundles on the Blown-Up Variety","authors":"Sukmoon Huh, Min-Gyo Jeong","doi":"10.1007/s00025-024-02157-9","DOIUrl":"https://doi.org/10.1007/s00025-024-02157-9","url":null,"abstract":"<p>We study the logarithmic vector bundles associated to arrangements of smooth irreducible curves with small degree on the blow-up of the projective plane at one point. We then investigate whether they are Torelli arrangements, that is, they can be recovered from the attached logarithmic vector bundles.</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":"15 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140578069","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Global Nilpotent Reversible Centers with Cubic Nonlinearities Symmetric with Respect to the x-Axis 具有与 x 轴对称的立方非线性的全局无势可逆中心
IF 2.2 3区 数学
Results in Mathematics Pub Date : 2024-04-13 DOI: 10.1007/s00025-024-02163-x
Montserrat Corbera, Claudia Valls
{"title":"Global Nilpotent Reversible Centers with Cubic Nonlinearities Symmetric with Respect to the x-Axis","authors":"Montserrat Corbera, Claudia Valls","doi":"10.1007/s00025-024-02163-x","DOIUrl":"https://doi.org/10.1007/s00025-024-02163-x","url":null,"abstract":"<p>Let <span>(P_3(x,y))</span> and <span>(Q_3(x,y))</span> be polynomials of degree three without constant or linear terms. We characterize the global centers of all polynomial differential systems of the form <span>(dot{x} = y+ P_3(x,y))</span>, <span>(dot{y} =Q_3(x,y))</span> that are reversible and invariant with respect to the <i>x</i>-axis.</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":"28 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140578071","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Ground States for the Nonlinear Schrödinger Equation with Critical Growth and Potential 具有临界增长和潜能的非线性薛定谔方程的基态
IF 2.2 3区 数学
Results in Mathematics Pub Date : 2024-04-13 DOI: 10.1007/s00025-024-02166-8
Jin-Cai Kang, Chun-Lei Tang
{"title":"Ground States for the Nonlinear Schrödinger Equation with Critical Growth and Potential","authors":"Jin-Cai Kang, Chun-Lei Tang","doi":"10.1007/s00025-024-02166-8","DOIUrl":"https://doi.org/10.1007/s00025-024-02166-8","url":null,"abstract":"<p>We investigate a class of the nonlinear Schrödinger equation in <span>( mathbb {R}^N)</span></p><span>$$begin{aligned} -Delta u +V(x)u=|u|^{2^*-2}u+lambda |u|^{p-2}u, end{aligned}$$</span><p>where <span>(Nge 3)</span>, <span>(lambda &gt;0)</span> and <span>(pin (2,2^*))</span> with <span>( 2^*=frac{2 N}{N-2})</span>. Here, <span>(V(x)=V_1(x))</span> for <span>(x_1&gt;0)</span> and <span>(V(x)=V_2(x))</span> for <span>(x_1&lt;0)</span>, where <span>(V_1,V_2 )</span> are periodic in each coordinate direction. By providing a splitting Lemma corresponding to non-periodic external potential, we obtain the existence of ground state solution for the above problem. It is worth to mention that the arguments used in this paper are also valid for the Sobolev subcritical problem studied by Dohnal et al. (Commun Math Phys 308:511–542, 2011).</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":"32 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577990","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Achievement Set of Generalized Multigeometric Sequences 广义多重计量序列的成就集
IF 2.2 3区 数学
Results in Mathematics Pub Date : 2024-04-13 DOI: 10.1007/s00025-024-02158-8
Dmytro Karvatskyi, Aniceto Murillo, Antonio Viruel
{"title":"The Achievement Set of Generalized Multigeometric Sequences","authors":"Dmytro Karvatskyi, Aniceto Murillo, Antonio Viruel","doi":"10.1007/s00025-024-02158-8","DOIUrl":"https://doi.org/10.1007/s00025-024-02158-8","url":null,"abstract":"<p>We study the topology of all possible subsums of the <i>generalized multigeometric series</i> <span>(k_1f(x)+k_2f(x)+dots +k_mf(x)+dots + k_1f(x^n)+dots +k_mf(x^n)+dots ,)</span> where <span>(k_1, k_2, dots , k_m)</span> are fixed positive real numbers and <i>f</i> runs along a certain class of non-negative functions on the unit interval. We detect particular regions of this interval for which this achievement set is, respectively, a compact interval, a Cantor set and a Cantorval.</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":"130 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577992","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Reformulations and Generalizations of Hoffman’s and Genčev’s Combinatorial Identities 霍夫曼和根切夫组合特性的重述和概括
IF 2.2 3区 数学
Results in Mathematics Pub Date : 2024-04-13 DOI: 10.1007/s00025-024-02160-0
Chun-Ying He, Feng Qi
{"title":"Reformulations and Generalizations of Hoffman’s and Genčev’s Combinatorial Identities","authors":"Chun-Ying He, Feng Qi","doi":"10.1007/s00025-024-02160-0","DOIUrl":"https://doi.org/10.1007/s00025-024-02160-0","url":null,"abstract":"<p>In the paper, the authors intrinsically observe that Hoffman’s combinatorial identity discovered in 1992 and Genčev’s combinatorial identities extended in 2024 can be reformulated in terms of complete Bell polynomials. They provide alternative proofs and offer some elementary generalizations of these combinatorial identities.</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":"3 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577959","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Note on BV Continuity of Discrete Maximal Operators 关于离散最大算子 BV 连续性的说明
IF 2.2 3区 数学
Results in Mathematics Pub Date : 2024-04-13 DOI: 10.1007/s00025-024-02164-w
Feng Liu, Xiao Zhang
{"title":"A Note on BV Continuity of Discrete Maximal Operators","authors":"Feng Liu, Xiao Zhang","doi":"10.1007/s00025-024-02164-w","DOIUrl":"https://doi.org/10.1007/s00025-024-02164-w","url":null,"abstract":"<p>We prove <span>(textrm{BV})</span> continuity of a class of discrete maximal operators associated to a general function <span>(Phi )</span>, which cover the classical one-dimensional discrete uncentered Hardy–Littlewood maximal operator. The main result not only improves and generalizes some known ones, but also answers a question originally posed by Liu and Wu in 2019.</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":"16 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140578065","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","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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