{"title":"A note on the 1-D minimization problem related to solenoidal improvement of the uncertainty principle inequality","authors":"Naoki Hamamoto","doi":"10.1007/s00013-024-02042-5","DOIUrl":"https://doi.org/10.1007/s00013-024-02042-5","url":null,"abstract":"<p>This paper gives a second way to solve the one-dimensional minimization problem of the form : </p><span>$$begin{aligned} min _{fnot equiv 0}frac{displaystyle int limits _0^infty left( f''right) ^2x^{mu +1}dxint limits _0^infty left( {x}^2left( f'right) ^2 -varepsilon f^2right) {{x}}^{mu -1}d{x}}{displaystyle left( int limits _0^infty left( f'right) ^2 {{x}}^{mu }d{x}right) ^2} end{aligned}$$</span><p>for scalar-valued functions <i>f</i> on the half line, where <span>(f')</span> and <span>(f'')</span> are its derivatives and <span>(varepsilon )</span> and <span>(mu )</span> are positive parameters with <span>(varepsilon < frac{mu ^2}{4}.)</span> This problem plays an essential part of the calculation of the best constant in Heisenberg’s uncertainty principle inequality for solenoidal vector fields. The above problem was originally solved by using (generalized) Laguerre polynomial expansion; however, the calculation was complicated and long. In the present paper, we give a simpler method to obtain the same solution, the essential part of which was communicated in Theorem 5.1 of the preprint (Hamamoto, arXiv:2104.02351v4, 2021).</p>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206094","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}
{"title":"The first two k-invariants of (textrm{Top}/textrm{O})","authors":"Alexander Kupers","doi":"10.1007/s00013-024-02036-3","DOIUrl":"10.1007/s00013-024-02036-3","url":null,"abstract":"<div><p>We show that the first two <i>k</i>-invariants of <span>(textrm{Top}/textrm{O})</span> vanish and give some applications.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206093","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}
{"title":"The Chermak–Delgado measure as a map on posets","authors":"William Cocke, Ryan McCulloch","doi":"10.1007/s00013-024-02015-8","DOIUrl":"10.1007/s00013-024-02015-8","url":null,"abstract":"<div><p>The Chermak–Delgado measure of a finite group is a function which assigns to each subgroup a positive integer. In this paper, we give necessary and sufficient conditions for when the Chermak–Delgado measure of a group is actually a map of posets, i.e., a monotone function from the subgroup lattice to the positive integers. We also investigate when the Chermak–Delgado measure, restricted to the centralizers, is increasing.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-02015-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142205983","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Rigidity of solutions to elliptic equations with one uniform limit","authors":"Phuong Le","doi":"10.1007/s00013-024-02040-7","DOIUrl":"10.1007/s00013-024-02040-7","url":null,"abstract":"<div><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>t} u(x) > -1)</span>, and <i>u</i> is bounded in each half-space <span>({x_N<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></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.5,"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}
{"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}
{"title":"One-dimensional quasi-uniform Kronecker sequences","authors":"Takashi Goda","doi":"10.1007/s00013-024-02039-0","DOIUrl":"10.1007/s00013-024-02039-0","url":null,"abstract":"<div><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></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-02039-0.pdf","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":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Common substring with shifts in b-ary expansions","authors":"Xin Liao, Dingding Yu","doi":"10.1007/s00013-024-02038-1","DOIUrl":"10.1007/s00013-024-02038-1","url":null,"abstract":"<div><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></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.5,"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}
{"title":"A note on parallel mean curvature surfaces and Codazzi operators","authors":"Felippe Guimarães","doi":"10.1007/s00013-024-02043-4","DOIUrl":"10.1007/s00013-024-02043-4","url":null,"abstract":"<div><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></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.5,"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}
{"title":"Rotationally symmetric gradient Yamabe solitons","authors":"Antonio W. Cunha, Rong Mi","doi":"10.1007/s00013-024-02032-7","DOIUrl":"10.1007/s00013-024-02032-7","url":null,"abstract":"<div><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></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.5,"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}
{"title":"An improvement of the sharp Li–Yau bound on closed manifolds","authors":"Jia-Yong Wu","doi":"10.1007/s00013-024-02027-4","DOIUrl":"10.1007/s00013-024-02027-4","url":null,"abstract":"<div><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></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.5,"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}