{"title":"Discrete and continuous dynamics of real 3-dimensional nilpotent polynomial vector fields","authors":"Álvaro Castañeda, Salomón Rebollo-Perdomo","doi":"10.1007/s00013-024-02085-8","DOIUrl":"10.1007/s00013-024-02085-8","url":null,"abstract":"<div><p>The aim of this work is to present general properties of the discrete and continuous dynamical systems induced by a large class of 3-dimensional nilpotent polynomial vector fields of arbitrary degree. In the discrete case, we prove that each dynamical system has a unique fixed point and there are no 2-cycles. Moreover, either the fixed point is a global attractor or there exists a 3-cycle which is not a repeller. In the continuous setting, we prove that each dynamical system is polynomially integrable. Particularly, it is proved that the global dynamics of some low degree vector fields is completely understood and that there are invariant surfaces foliated by periodic orbits. As far as we know, this last property has not been shown before in the nilpotent context. We achieve our results by using the approach of polynomial automorphisms to obtain simplified conjugated dynamical systems, instead of considering only the usual linear transformations. Finally, we point out some similarities shared by the discrete and continuous dynamical systems, and we formulate some open questions motivated by our results, which are related with the Markus–Yamabe conjecture and the problem of planar limit cycles.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 4","pages":"415 - 434"},"PeriodicalIF":0.5,"publicationDate":"2024-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143621948","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":"Linear maps preserving the inclusion of fixed subsets into the local spectrum at some fixed vector","authors":"Constantin Costara","doi":"10.1007/s00013-024-02078-7","DOIUrl":"10.1007/s00013-024-02078-7","url":null,"abstract":"<div><p>For a natural number <span>(n ge 2)</span>, denote by <span>(mathcal {M}_{n})</span> the space of all <span>(ntimes n)</span> matrices over the complex field <span>(mathbb {C})</span>. Let <span>(x_0 in mathbb {C}^{n})</span> be a fixed nonzero vector, and fix also two nonempty subsets <span>(K_1, K_2 subseteq mathbb {C})</span>, each having at most <i>n</i> distinct elements. Under the assumption that <span>(|K_1| le |K_2|)</span>, we characterize linear bijective maps <span>(varphi )</span> on <span>(mathcal {M}_{n})</span> having the property that, for each matrix <i>T</i>, we have that <span>(K_2)</span> is a subset of the local spectrum of <span>(varphi (T))</span> at <span>(x_0 )</span> whenever <span>(K_1 )</span> is a subset of the local spectrum of <i>T</i> at <span>(x_0)</span>. As a corollary, we also characterize linear maps <span>(varphi )</span> on <span>(mathcal {M} _{n})</span> having the property that, for each matrix <i>T</i>, we have that <span>(K_1)</span> is a subset of the local spectrum of <i>T</i> at <span>(x_0)</span> if and only if <span>(K_2)</span> is a subset of the local spectrum of <span>(varphi (T))</span> at <span>(x_0)</span>, without the bijectivity assumption on the map <span>(varphi )</span> and with no assumption made regarding the number of elements of <span>(K_1)</span> and <span>(K_2)</span>.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 2","pages":"165 - 176"},"PeriodicalIF":0.5,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-02078-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143108624","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":"Hardy inequalities and uncertainty principles in the presence of a black hole","authors":"Miltiadis Paschalis","doi":"10.1007/s00013-024-02082-x","DOIUrl":"10.1007/s00013-024-02082-x","url":null,"abstract":"<div><p>In this paper, we establish Hardy and Heisenberg uncertainty-type inequalities for the exterior of a Schwarzschild black hole. The weights that appear in both inequalities are tailored to fit the geometry, and can both be compared to the related Riemannian distance from the event horizon to yield inequalities for that distance. Moreover, in both cases the classic Euclidean inequalities with a point singularity can be recovered in the limit where one stands “far enough” from the black hole, as expected from the asymptotic flatness of the metric.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 2","pages":"205 - 218"},"PeriodicalIF":0.5,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143108976","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":"Alternating state complexity of the set of primes and squarefree integers","authors":"Jan-Christoph Schlage-Puchta","doi":"10.1007/s00013-024-02075-w","DOIUrl":"10.1007/s00013-024-02075-w","url":null,"abstract":"<div><p>We show that the set of prime numbers has exponential alternating complexity, proving a conjecture by Fijalkow. We further show that the set of squarefree integers has essentially maximal possible alternating complexity.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 2","pages":"151 - 156"},"PeriodicalIF":0.5,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-02075-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143108804","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}
Jogli G. Araújo, Eudes L. de Lima, Henrique F. de Lima
{"title":"Sharp bounds for the norm of the traceless second fundamental form of complete linear Weingarten spacelike hypersurfaces","authors":"Jogli G. Araújo, Eudes L. de Lima, Henrique F. de Lima","doi":"10.1007/s00013-024-02073-y","DOIUrl":"10.1007/s00013-024-02073-y","url":null,"abstract":"<div><p>We establish sharp bounds for the norm of the traceless second fundamental form of complete linear Weingarten spacelike hypersurfaces immersed in a Lorentzian space form and satisfying a suitable Okumura type inequality. As a consequence of these estimates, we derive new characterization results concerning totally umbilical spacelike hypersurfaces and hyperbolic cylinders of Lorentzian space forms.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 2","pages":"219 - 231"},"PeriodicalIF":0.5,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143108185","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":"On Dirac equations with Hartree type nonlinearity in modulation spaces","authors":"Seongyeon Kim, Hyeongjin Lee, Ihyeok Seo","doi":"10.1007/s00013-024-02079-6","DOIUrl":"10.1007/s00013-024-02079-6","url":null,"abstract":"<div><p>We obtain the local well-posedness for Dirac equations with a Hartree type nonlinearity derived by decoupling the Dirac–Klein–Gordon system. We extend the function space of initial data, enabling us to handle initial data that were not addressed in previous studies.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 2","pages":"195 - 204"},"PeriodicalIF":0.5,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143110082","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":"Linear topological invariants for kernels of differential operators by shifted fundamental solutions","authors":"Andreas Debrouwere, Thomas Kalmes","doi":"10.1007/s00013-024-02070-1","DOIUrl":"10.1007/s00013-024-02070-1","url":null,"abstract":"<div><p>We characterize the condition <span>((Omega ))</span> for smooth kernels of partial differential operators in terms of the existence of shifted fundamental solutions satisfying certain properties. The conditions <span>((POmega ))</span> and <span>((Poverline{overline{Omega }}))</span> for distributional kernels are characterized in a similar way. By lifting theorems for Fréchet spaces and (PLS)-spaces, this provides characterizations of the problem of parameter dependence for smooth and distributional solutions of differential equations by shifted fundamental solutions. As an application, we give a new proof of the fact that the space <span>({ f in {mathscr {E}}(X) , | , P(D)f = 0})</span> satisfies <span>((Omega ))</span> for any differential operator <i>P</i>(<i>D</i>) and any open convex set <span>(X subseteq {mathbb {R}}^d)</span>.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 2","pages":"177 - 193"},"PeriodicalIF":0.5,"publicationDate":"2024-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-02070-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143109116","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":"Rotators-translators to mean curvature flow in (mathbb {H}^2times mathbb {R})","authors":"R. F. de Lima, A. K. Ramos, J. P. dos Santos","doi":"10.1007/s00013-024-02076-9","DOIUrl":"10.1007/s00013-024-02076-9","url":null,"abstract":"<div><p>We establish the existence of one-parameter families of helicoidal surfaces of <span>(mathbb H^2times mathbb R)</span> which, under mean curvature flow, simultaneously rotate about a vertical axis and translate vertically.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 3","pages":"343 - 353"},"PeriodicalIF":0.5,"publicationDate":"2024-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143184868","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 Brill–Noether theory of curves of fixed gonality","authors":"Gerriet Martens","doi":"10.1007/s00013-024-02059-w","DOIUrl":"10.1007/s00013-024-02059-w","url":null,"abstract":"<div><p>Recently the Brill–Noether theory of curves <i>C</i> of both fixed genus and gonality was established. In particular, in this theory (now called the Hurwitz–Brill–Noether theory), all irreducible components of the variety of complete linear series of a fixed degree and dimension on <i>C</i> are obtained from the closures of certain so-called “Brill–Noether splitting loci” (loci which have a rather succinct description). In this paper, a method previously invented for the construction of some of these irreducible components is applied to get simply designed varieties inside the difference between these splitting loci and their closures, i.e., inside the boundary of the splitting loci.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 1","pages":"49 - 61"},"PeriodicalIF":0.5,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-02059-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142963144","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":"On the transitivity of Lie ideals and a characterization of perfect Lie algebras","authors":"Nikolaos Panagiotis Souris","doi":"10.1007/s00013-024-02063-0","DOIUrl":"10.1007/s00013-024-02063-0","url":null,"abstract":"<div><p>We explore general intrinsic and extrinsic conditions that allow the transitivity of the relation of being an ideal in Lie algebras. We also prove that perfect Lie algebras of arbitrary dimension and over any field are intrinsically characterized by transitivity of this type. In particular, we show that a Lie algebra <span>(mathfrak {h})</span> is perfect (i.e., <span>(mathfrak {h}=[mathfrak {h}, mathfrak {h}])</span>) if and only if for all Lie algebras <span>(mathfrak {k}, mathfrak {g})</span> such that <span>(mathfrak {h})</span> is an ideal of <span>(mathfrak {k})</span> and <span>(mathfrak {k})</span> is an ideal of <span>(mathfrak {g})</span>, it follows that <span>(mathfrak {h})</span> is an ideal of <span>(mathfrak {g})</span>.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 1","pages":"9 - 18"},"PeriodicalIF":0.5,"publicationDate":"2024-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142963143","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}