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Rigidity of solutions to elliptic equations with one uniform limit 具有一个均匀极限的椭圆方程解的刚性
IF 0.6 4区 数学
Archiv der Mathematik Pub Date : 2024-08-20 DOI: 10.1007/s00013-024-02040-7
Phuong Le
{"title":"Rigidity of solutions to elliptic equations with one uniform limit","authors":"Phuong Le","doi":"10.1007/s00013-024-02040-7","DOIUrl":"https://doi.org/10.1007/s00013-024-02040-7","url":null,"abstract":"<p>Let <span>(uge -1)</span> be a solution to the semilinear elliptic equation <span>(-Delta u = f(u))</span> in <span>(mathbb {R}^N)</span> such that <span>(lim _{x_Nrightarrow -infty } u(x',x_N) = -1)</span> uniformly in <span>(x'in mathbb {R}^{N-1})</span>, <span>(lim _{trightarrow +infty } inf _{x_N&gt;t} u(x) &gt; -1)</span>, and <i>u</i> is bounded in each half-space <span>({x_N&lt;lambda })</span>, <span>(lambda in mathbb {R})</span>. Here <span>(f:[-1,+infty )rightarrow mathbb {R})</span> is a locally Lipschitz continuous function which satisfies some mild assumptions. We show that <i>u</i> is strictly monotonically increasing in the <span>(x_N)</span>-direction. Under some further assumptions on <i>f</i>, we deduce that <i>u</i> depends only on <span>(x_N)</span> and it is unique up to a translation. In particular, such a solution <i>u</i> to the problem <span>(Delta u = u + 1)</span> in <span>(mathbb {R}^N)</span> must have the form <span>(u(x)equiv e^{x_N+alpha }-1)</span> for some <span>(alpha in mathbb {R})</span>.</p>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206095","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 remark on the norm of the parallel sum 关于平行和的规范的评论
IF 0.6 4区 数学
Archiv der Mathematik Pub Date : 2024-08-16 DOI: 10.1007/s00013-024-02048-z
Ali Zamani
{"title":"A remark on the norm of the parallel sum","authors":"Ali Zamani","doi":"10.1007/s00013-024-02048-z","DOIUrl":"https://doi.org/10.1007/s00013-024-02048-z","url":null,"abstract":"<p>It is shown that if <span>(a!:!b)</span> is the parallel sum of the two positive definite elements <i>a</i> and <i>b</i> of a <span>(C^*)</span>-algebra, then for any <span>(s, tin [0, 1])</span>, </p><span>$$begin{aligned} big Vert a!:!bbig Vert le frac{1}{2}left( Vert aVert !:!Vert bVert + frac{Vert aVert :Vert bVert }{Vert aVert +Vert bVert }sqrt{left( Vert aVert -Vert bVert right) ^2 +4left| a^{1-s}b^{t}right| left| a^{s}b^{1-t}right| },right) . end{aligned}$$</span><p>This inequality, which is sharper than the inequality <span>(big Vert a!:!bbig Vert le Vert aVert !:!Vert bVert )</span>, generalizes an earlier related inequality.</p>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206096","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
One-dimensional quasi-uniform Kronecker sequences 一维准均匀克罗内克序列
IF 0.6 4区 数学
Archiv der Mathematik Pub Date : 2024-08-14 DOI: 10.1007/s00013-024-02039-0
Takashi Goda
{"title":"One-dimensional quasi-uniform Kronecker sequences","authors":"Takashi Goda","doi":"10.1007/s00013-024-02039-0","DOIUrl":"https://doi.org/10.1007/s00013-024-02039-0","url":null,"abstract":"<p>In this short note, we prove that the one-dimensional Kronecker sequence <span>(ialpha bmod 1, i=0,1,2,ldots ,)</span> is quasi-uniform over the unit interval [0, 1] if and only if <span>(alpha )</span> is a badly approximable number. Our elementary proof relies on a result on the three-gap theorem for Kronecker sequences due to Halton (Proc Camb Philos Soc, 61:665–670, 1965).</p>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206097","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Common substring with shifts in b-ary expansions b-ary 扩展中带有移位的共同子串
IF 0.6 4区 数学
Archiv der Mathematik Pub Date : 2024-08-08 DOI: 10.1007/s00013-024-02038-1
Xin Liao, Dingding Yu
{"title":"Common substring with shifts in b-ary expansions","authors":"Xin Liao, Dingding Yu","doi":"10.1007/s00013-024-02038-1","DOIUrl":"https://doi.org/10.1007/s00013-024-02038-1","url":null,"abstract":"<p>Denote by <span>(S_n(x,y))</span> the length of the longest common substring of <i>x</i> and <i>y</i> with shifts in their first <i>n</i> digits of the <i>b</i>-ary expansions. We show that the sets of pairs (<i>x</i>, <i>y</i>), for which the growth rate of <span>(S_n(x,y))</span> is <span>(alpha log n)</span> with <span>(0le alpha le infty )</span>, have full Hausdorff dimension. Our method relies upon some estimation of the spectral radius of matrices.</p>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935933","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 parallel mean curvature surfaces and Codazzi operators 关于平行平均曲率曲面和科达齐算子的说明
IF 0.6 4区 数学
Archiv der Mathematik Pub Date : 2024-08-08 DOI: 10.1007/s00013-024-02043-4
Felippe Guimarães
{"title":"A note on parallel mean curvature surfaces and Codazzi operators","authors":"Felippe Guimarães","doi":"10.1007/s00013-024-02043-4","DOIUrl":"https://doi.org/10.1007/s00013-024-02043-4","url":null,"abstract":"<p>We use an intrinsic Klotz–Osserman type result for surfaces in terms of Codazzi operators to study surfaces with parallel mean curvature and non-positive Gaussian curvature in product spaces.</p>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935929","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
An improvement of the sharp Li–Yau bound on closed manifolds 封闭流形上尖锐李-尤约束的改进
IF 0.6 4区 数学
Archiv der Mathematik Pub Date : 2024-08-06 DOI: 10.1007/s00013-024-02027-4
Jia-Yong Wu
{"title":"An improvement of the sharp Li–Yau bound on closed manifolds","authors":"Jia-Yong Wu","doi":"10.1007/s00013-024-02027-4","DOIUrl":"https://doi.org/10.1007/s00013-024-02027-4","url":null,"abstract":"<p>In this paper, we give a generalization of Zhang’s recent work about a sharp Li–Yau gradient bound on compact manifolds by extending Hamilton’s gradient estimates. In particular, we take a special auxiliary function to indicate that our estimate is a slight improvement of Zhang’s result.</p>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Rotationally symmetric gradient Yamabe solitons 旋转对称梯度山叶孤子
IF 0.6 4区 数学
Archiv der Mathematik Pub Date : 2024-08-06 DOI: 10.1007/s00013-024-02032-7
Antonio W. Cunha, Rong Mi
{"title":"Rotationally symmetric gradient Yamabe solitons","authors":"Antonio W. Cunha, Rong Mi","doi":"10.1007/s00013-024-02032-7","DOIUrl":"https://doi.org/10.1007/s00013-024-02032-7","url":null,"abstract":"<p>This short note deals with compact and complete and non-compact gradient Yamabe solitons (<i>M</i>, <i>g</i>, <i>f</i>) such that it has metric of constant scalar curvature. Firstly, we give a new proof of triviality for gradient compact Yamabe solitons. Also, under some integral conditions, we are able to improve a result due to Ma and Miquel (Ann Global Anal Geom 42:195–205, 2012). Finally, we obtain that the Yamabe metric becomes rotationally symmetric. Results for <i>k</i>-Yamabe solitons are also obtained here.</p>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935930","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Partial Dedekind zeta values for ideal classes of the real quadratic field $${mathbb {Q}}$$ ( $$sqrt{9m^{2}+2m})$$ 实二次域 $${mathbb {Q}}$( $$sqrt{9m^{2}+2m})$$ 的理想类的部分戴德金兹塔值
IF 0.6 4区 数学
Archiv der Mathematik Pub Date : 2024-08-06 DOI: 10.1007/s00013-024-02037-2
Ahmad Issa, Boushra Darrag
{"title":"Partial Dedekind zeta values for ideal classes of the real quadratic field $${mathbb {Q}}$$ ( $$sqrt{9m^{2}+2m})$$","authors":"Ahmad Issa, Boushra Darrag","doi":"10.1007/s00013-024-02037-2","DOIUrl":"https://doi.org/10.1007/s00013-024-02037-2","url":null,"abstract":"<p>In this paper, we obtain some of the partial Dedekind zeta values for ideal classes of the real quadratic field <span>({mathbb {Q}})</span>(<span>(sqrt{D}))</span>, where <span>(D = 9m^{2}+2m)</span> is a square-free positive integer and <span>(m equiv 2)</span> (mod 3) is an odd positive integer.</p>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935935","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Combinatorial constructions of repairable threshold schemes 可修复阈值方案的组合构造
IF 0.6 4区 数学
Archiv der Mathematik Pub Date : 2024-08-06 DOI: 10.1007/s00013-024-02034-5
Jinghui Zhao, Xiuling Shan, Zihong Tian, Guohui Hao
{"title":"Combinatorial constructions of repairable threshold schemes","authors":"Jinghui Zhao, Xiuling Shan, Zihong Tian, Guohui Hao","doi":"10.1007/s00013-024-02034-5","DOIUrl":"https://doi.org/10.1007/s00013-024-02034-5","url":null,"abstract":"<p>A repairable threshold scheme is a threshold scheme in which a player can securely reconstruct a lost share with the help from a subset of players. In 2018, Stinson and Wei proposed distribution designs to construct repairable threshold schemes. In this paper, we use packing quadruples as distribution designs to distribute subshares of a ramp scheme. The resulting repairable threshold scheme has lower computational complexity and repairing degree.</p>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935934","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 hypersurfaces of section for geodesic flows on convex hypersurfaces 凸超曲面上大地流的全局超曲面剖面
IF 0.6 4区 数学
Archiv der Mathematik Pub Date : 2024-07-31 DOI: 10.1007/s00013-024-02018-5
Sunghae Cho, Dongho Lee
{"title":"Global hypersurfaces of section for geodesic flows on convex hypersurfaces","authors":"Sunghae Cho, Dongho Lee","doi":"10.1007/s00013-024-02018-5","DOIUrl":"https://doi.org/10.1007/s00013-024-02018-5","url":null,"abstract":"<p>We construct a global hypersurface of section for the geodesic flow of a convex hypersurface in Euclidean space admitting an isometric involution. This generalizes the Birkhoff annulus to higher dimensions.</p>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141867933","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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