Mathematische Nachrichten最新文献

{"title":"On totally umbilical and minimal surfaces of the Lorentzian Heisenberg groups","authors":"Giovanni Calvaruso,&nbsp;Marco Castrillón-Lopez,&nbsp;Lorenzo Pellegrino","doi":"10.1002/mana.12020","DOIUrl":"https://doi.org/10.1002/mana.12020","url":null,"abstract":"<p>This paper has manifold purposes. We first introduce a description of the Gauss map for submanifolds (both spacelike and timelike) of a Lorentzian ambient space and relate the conformality of the Gauss map of a surface to total umbilicity and minimality. We then focus on surfaces of the three-dimensional Heisenberg group, equipped with any of its left-invariant Lorentzian metrics. We prove that with the obvious exception of the flat case, no totally umbilical surfaces occur. On the other hand, we determine and explicitly describe several examples of minimal and constant mean curvature (CMC) surfaces.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 6","pages":"1922-1942"},"PeriodicalIF":0.8,"publicationDate":"2025-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.12020","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144281473","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Groups with triangle-free graphs on \u0000 \u0000 p\u0000 $p$\u0000 -regular classes","authors":"M. J. Felipe,&nbsp;M. K. Jean-Philippe,&nbsp;V. Sotomayor","doi":"10.1002/mana.202400554","DOIUrl":"https://doi.org/10.1002/mana.202400554","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <mi>p</mi>\u0000 <annotation>$p$</annotation>\u0000 </semantics></math> be a prime. In this paper, we classify the <span></span><math>\u0000 <semantics>\u0000 <mi>p</mi>\u0000 <annotation>$p$</annotation>\u0000 </semantics></math>-structure of those finite <span></span><math>\u0000 <semantics>\u0000 <mi>p</mi>\u0000 <annotation>$p$</annotation>\u0000 </semantics></math>-separable groups such that, given any three non-central conjugacy classes of <span></span><math>\u0000 <semantics>\u0000 <mi>p</mi>\u0000 <annotation>$p$</annotation>\u0000 </semantics></math>-regular elements, two of them necessarily have coprime lengths.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 6","pages":"1796-1807"},"PeriodicalIF":0.8,"publicationDate":"2025-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.202400554","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144281372","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Semiclassical solutions for critical Schrödinger–Poisson systems involving multiple competing potentials","authors":"Lingzheng Kong,&nbsp;Haibo Chen","doi":"10.1002/mana.12012","DOIUrl":"https://doi.org/10.1002/mana.12012","url":null,"abstract":"<p>In this paper, a class of Schrödinger–Poisson system involving multiple competing potentials and critical Sobolev exponent is considered. Such a problem cannot be studied using the same argument for the nonlinear term with only a positive potential, because the weight potentials set <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 <msub>\u0000 <mi>Q</mi>\u0000 <mi>i</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>x</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>|</mo>\u0000 <mn>1</mn>\u0000 <mo>≤</mo>\u0000 <mi>i</mi>\u0000 <mo>≤</mo>\u0000 <mi>m</mi>\u0000 <mo>}</mo>\u0000 </mrow>\u0000 <annotation>$lbrace Q_i(x)|1le i le mrbrace$</annotation>\u0000 </semantics></math> contains nonpositive, sign changing, and nonnegative elements. By introducing the ground energy function and subtle analysis, we first prove the existence of ground state solution <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>v</mi>\u0000 <mi>ε</mi>\u0000 </msub>\u0000 <annotation>$v_varepsilon$</annotation>\u0000 </semantics></math> in the semiclassical limit via the Nehari manifold and concentration–compactness principle. Then we show that <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>v</mi>\u0000 <mi>ε</mi>\u0000 </msub>\u0000 <annotation>$v_varepsilon$</annotation>\u0000 </semantics></math> converges to the ground state solution of the associated limiting problem and concentrates at a concrete set characterized by the potentials. At the same time, some properties for the ground state solution are also studied. Moreover, a sufficient condition for the nonexistence of the ground state solution is obtained.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 6","pages":"1808-1838"},"PeriodicalIF":0.8,"publicationDate":"2025-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144281373","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}

{"title":"Jacobian elliptic fibrations on K3s with a non-symplectic automorphism of order 3","authors":"Felipe Zingali Meira","doi":"10.1002/mana.12018","DOIUrl":"https://doi.org/10.1002/mana.12018","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$X$</annotation>\u0000 </semantics></math> be a K3 surface admitting a non-symplectic automorphism <span></span><math>\u0000 <semantics>\u0000 <mi>σ</mi>\u0000 <annotation>$sigma$</annotation>\u0000 </semantics></math> of order 3. Building on work by Garbagnati and Salgado, we classify the Jacobian elliptic fibrations on <span></span><math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$X$</annotation>\u0000 </semantics></math> with respect to the action of <span></span><math>\u0000 <semantics>\u0000 <mi>σ</mi>\u0000 <annotation>$sigma$</annotation>\u0000 </semantics></math> on their fibers. If the fiber class of a Jacobian elliptic fibration on <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>NS</mo>\u0000 <mo>(</mo>\u0000 <mi>X</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$operatorname{NS}(X)$</annotation>\u0000 </semantics></math> is fixed by <span></span><math>\u0000 <semantics>\u0000 <mi>σ</mi>\u0000 <annotation>$sigma$</annotation>\u0000 </semantics></math>, we determine the possible configurations of its singular fibers and present the equation for its generic fiber. When the Picard number of <span></span><math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$X$</annotation>\u0000 </semantics></math> is at least 12 and <span></span><math>\u0000 <semantics>\u0000 <mi>σ</mi>\u0000 <annotation>$sigma$</annotation>\u0000 </semantics></math> acts trivially on <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>NS</mo>\u0000 <mo>(</mo>\u0000 <mi>X</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$operatorname{NS}(X)$</annotation>\u0000 </semantics></math>, we apply the Kneser–Nishiyama method to find its Jacobian elliptic fibrations up to <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>J</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <annotation>$mathcal {J}_2$</annotation>\u0000 </semantics></math>-equivalence. We use our method to classify them with respect to any non-symplectic automorphism of order 3 in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>Aut</mo>\u0000 <mo>(</mo>\u0000 <mi>X</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$operatorname{Aut}(X)$</annotation>\u0000 </seman","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 5","pages":"1758-1788"},"PeriodicalIF":0.8,"publicationDate":"2025-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.12018","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143930266","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Regularity and separation for Grušin-type p-Laplace operators","authors":"Daniel Hauer,&nbsp;Adam Sikora","doi":"10.1002/mana.12011","DOIUrl":"https://doi.org/10.1002/mana.12011","url":null,"abstract":"<p>We analyze <i>p</i>-Laplace type operators with degenerate elliptic coefficients. This investigation includes Grušin-type <i>p</i>-Laplace operators. We describe a <i>separation phenomenon</i> in elliptic and parabolic <i>p</i>-Laplace type equations, which provide an illuminating illustration of simple jump discontinuities of the corresponding weak solutions. Interestingly validity of an isoperimetric inequality for the considered setting does not imply the continuity of weak solutions to elliptic equations. On the other hand, we can establish global <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mn>1</mn>\u0000 </msup>\u0000 <annotation>$L^1$</annotation>\u0000 </semantics></math>-<span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mi>∞</mi>\u0000 </msup>\u0000 <annotation>$L^infty$</annotation>\u0000 </semantics></math>-regularization and decay estimates of every mild solution of the parabolic Grušin-type <i>p</i>-Laplace equation.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 5","pages":"1727-1757"},"PeriodicalIF":0.8,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.12011","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143930258","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Local existence and nonexistence of fractional Rayleigh–Stokes equations with a superlinear source term","authors":"Yubin Liu,&nbsp;Li Peng","doi":"10.1002/mana.12010","DOIUrl":"https://doi.org/10.1002/mana.12010","url":null,"abstract":"<p>Fractional Rayleigh–Stokes equations can be described as the viscoelasticity of non-Newtonian fluids behavior for a generalized second grade fluid. In this paper, we present the monotone iteration method to investigate the nonlinear fractional Rayleigh–Stokes equations from the perspective of the supersolution. More precisely, we analyze the local existence, boundedness, and convergence of nonnegative mild solutions under the superlinear growth conditions. Further, the local nonexistence results of mild solutions are also given.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 5","pages":"1700-1712"},"PeriodicalIF":0.8,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143930333","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}

{"title":"Convergence of approximating solutions of the Navier–Stokes equations in higher ordered Sobolev norms","authors":"Yuta Koizumi","doi":"10.1002/mana.12009","DOIUrl":"https://doi.org/10.1002/mana.12009","url":null,"abstract":"&lt;p&gt;We show that the approximating solutions &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;{&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mi&gt;j&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;}&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;j&lt;/mi&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mi&gt;∞&lt;/mi&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;annotation&gt;$lbrace u_jrbrace _{j=0}^{infty }$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of the Navier–Stokes equations constructed by Kato with the initial data &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;mi&gt;σ&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$u(0) in L_{sigma }^{n}(mathbb {R}^{n})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; converge to the local strong solution &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;annotation&gt;$u$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; in the topology of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;W&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$W^{k,q}(mathbb {R}^n)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; for all &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$k in mathbb {N}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; provided the convergence in the scaling invariant norm in &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 ","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 5","pages":"1663-1679"},"PeriodicalIF":0.8,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143930331","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}

{"title":"Existence of solutions for nonlocal Schrödinger–Kirchhoff problems with critical Trudinger–Moser nonlinearity","authors":"Jin Liang,&nbsp;Zhi-Cheng Xiong","doi":"10.1002/mana.12013","DOIUrl":"https://doi.org/10.1002/mana.12013","url":null,"abstract":"<p>In this paper, we are concerned with solutions for nonlocal Schrödinger–Kirchhoff problems with critical Trudinger–Moser nonlinearity on a bounded domain with Lipschitz boundary. With the help of the minimax method, the fractional Trudinger–Moser inequality, Ekeland's variational principle, and other techniques, we establish an existence theorem for a nonnegative solution with positive energy to a class of nonlocal Schrödinger–Kirchhoff problems and an existence theorem for a nontrivial solution with negative energy to another class of nonlocal Schrödinger–Kirchhoff problems.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 5","pages":"1680-1699"},"PeriodicalIF":0.8,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143930332","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}

{"title":"Singular integral operators and commutators on two-weight Morrey spaces","authors":"Meichuan Lv,&nbsp;Wenming Li","doi":"10.1002/mana.12007","DOIUrl":"https://doi.org/10.1002/mana.12007","url":null,"abstract":"<p>In this paper, we obtain the mapping properties for the Hardy–Littlewood maximal operator, Calderón–Zygmund singular integral operators, and the commutators of the singular integral operators with the <span></span><math>\u0000 <semantics>\u0000 <mtext>BMO</mtext>\u0000 <annotation>$text{BMO}$</annotation>\u0000 </semantics></math> functions on the two-weight Morrey spaces. We also get the endpoint estimates for the commutators on the two-weight Morrey spaces.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 5","pages":"1713-1726"},"PeriodicalIF":0.8,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143930329","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}

{"title":"Angular distribution toward the points of the neighbor-flips modular curve seen by a fast moving observer","authors":"Jack Anderson,&nbsp;Florin P. Boca,&nbsp;Cristian Cobeli,&nbsp;Alexandru Zaharescu","doi":"10.1002/mana.12016","DOIUrl":"https://doi.org/10.1002/mana.12016","url":null,"abstract":"&lt;p&gt;Let &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;h&lt;/mi&gt;\u0000 &lt;annotation&gt;$h$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be a fixed non-zero integer. For every &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$tin mathbb {R}_+$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and every prime &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;annotation&gt;$p$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, consider the angles between rays from an observer located at the point &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mi&gt;J&lt;/mi&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$(-tJ_p^2,0)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; on the real axis toward the set of all integral solutions &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;y&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$(x,y)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of the equation &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;y&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;≡&lt;/mo&gt;\u0000 &lt;mi&gt;h&lt;/mi&gt;\u0000 &lt;mfenced&gt;\u0000 &lt;mi&gt;mod&lt;/mi&gt;\u0000 &lt;mspace&gt;&lt;/mspace&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;/mfenced&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$y^{-1}-x^{-1}equiv h left(mathrm{ mod;}pright)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; in the square &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;[&lt;/mo&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 ","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 5","pages":"1617-1632"},"PeriodicalIF":0.8,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.12016","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143930387","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}