{"title":"On fractional parabolic equations with Hardy-type potentials","authors":"Veronica Felli, Ana Primo, Giovanni Siclari","doi":"10.1142/s0219199723500621","DOIUrl":"https://doi.org/10.1142/s0219199723500621","url":null,"abstract":"<p>A classification of local asymptotic profiles and strong unique continuation properties are established for a class of fractional heat equations with a Hardy-type potential, via an Almgren–Poon monotonicity formula combined with a blow-up analysis.</p>","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"109 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140072623","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the cohomology of NC(−2) in positive characteristic","authors":"Eric Larson","doi":"10.1142/s0219199723500670","DOIUrl":"https://doi.org/10.1142/s0219199723500670","url":null,"abstract":"<p>Let <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mrow><mi>C</mi><mo>⊂</mo><msup><mrow><mi>ℙ</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></math></span><span></span> be a general Brill–Noether curve. A classical problem is to determine when <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mrow><msup><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msup><mo stretchy=\"false\">(</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>C</mi></mrow></msub><mo stretchy=\"false\">(</mo><mo stretchy=\"false\">−</mo><mn>2</mn><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo><mo>=</mo><mn>0</mn></mrow></math></span><span></span>, which controls the quadric section of <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mrow><mi>C</mi></mrow></math></span><span></span>.</p><p>So far this problem has only been solved in characteristic zero, in which case <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mrow><msup><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msup><mo stretchy=\"false\">(</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>C</mi></mrow></msub><mo stretchy=\"false\">(</mo><mo stretchy=\"false\">−</mo><mn>2</mn><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo><mo>=</mo><mn>0</mn></mrow></math></span><span></span> with finitely many exceptions. In this paper, we extend these results to positive characteristic, uncovering a wealth of new exceptions in characteristic <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mrow><mn>2</mn></mrow></math></span><span></span>.</p>","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"278 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140072617","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Exceptionally simple super-PDE for F(4)","authors":"Andrea Santi, Dennis The","doi":"10.1142/s0219199723500530","DOIUrl":"https://doi.org/10.1142/s0219199723500530","url":null,"abstract":"<p>For the largest exceptional simple Lie superalgebra <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>F</mi><mo stretchy=\"false\">(</mo><mn>4</mn><mo stretchy=\"false\">)</mo></math></span><span></span>, having dimension <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">(</mo><mn>2</mn><mn>4</mn><mo>|</mo><mn>1</mn><mn>6</mn><mo stretchy=\"false\">)</mo></math></span><span></span>, we provide two explicit geometric realizations as supersymmetries, namely as the symmetry superalgebra of super-PDE systems of second- and third-order, respectively.</p>","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"106 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140072585","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Algebraic versions of Hartogs’ theorem","authors":"Marcin Bilski, Jacek Bochnak, Wojciech Kucharz","doi":"10.1142/s0219199723500669","DOIUrl":"https://doi.org/10.1142/s0219199723500669","url":null,"abstract":"<p>Let <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>𝕂</mi></math></span><span></span> be an uncountable field of characteristic <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mn>0</mn></math></span><span></span>. For a given function <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>f</mi><mo>:</mo><msup><mrow><mi>𝕂</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>→</mo><mi>𝕂</mi></math></span><span></span>, with <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo>≥</mo><mn>2</mn></math></span><span></span>, we prove that <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>f</mi></math></span><span></span> is regular if and only if the restriction <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>f</mi><msub><mrow><mo>|</mo></mrow><mrow><mi>C</mi></mrow></msub></math></span><span></span> is a regular function for every algebraic curve <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>C</mi></math></span><span></span> in <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>𝕂</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span><span></span> which is either an affine line or is isomorphic to a plane curve in <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>𝕂</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span><span></span> defined by the equation <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>X</mi></mrow><mrow><mi>p</mi></mrow></msup><mo stretchy=\"false\">−</mo><msup><mrow><mi>Y</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>=</mo><mn>0</mn></math></span><span></span>, where <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi><mo><</mo><mi>q</mi></math></span><span></span> are prime numbers. We also show that regularity of <span><math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"><mi>f</mi></math></span><span></span> can be verified on other algebraic curves in <span><math altimg=\"eq-00013.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>𝕂</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span><span></span> with desired geometric properties. Furthermore, if the field <span><math altimg=\"eq-00014.gif\" display=\"inline\" overflow=\"scroll\"><mi>𝕂</mi></math></span><span></span> is not algebraically closed, we construct a <span><math altimg=\"eq-00015.gif\" display=\"inline\" overflow=\"scroll\"><mi>𝕂</mi></math></span><span></span>-valued function on <span><math altimg=\"eq-00016.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>𝕂</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span><span></span> that is not regular, but all its restrictions to nonsingular algebraic curves in <span><math altimg=\"eq-00017.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>𝕂</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span><span></s","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"278 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140072482","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Curvature of new Kähler metrics on the total space of Griffiths negative vector bundle and quasi-Fuchsian space","authors":"Inkang Kim, Xueyuan Wan, Genkai Zhang","doi":"10.1142/s0219199723500591","DOIUrl":"https://doi.org/10.1142/s0219199723500591","url":null,"abstract":"<p>We study Kähler metrics on the total space of Griffiths negative holomorphic vector bundles over Kähler manifolds. As an application, we construct mapping class group invariant Kähler metrics on <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi mathvariant=\"cal\">ℬ</mi><mo stretchy=\"false\">(</mo><mi mathvariant=\"cal\">𝒮</mi><mo stretchy=\"false\">)</mo></math></span><span></span>, the holomorphic tangent bundle of Teichmüller space of a closed surface <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>S</mi></math></span><span></span>. Consequently,we obtain a new mapping class group invariant Kähler metric on the quasi-Fuchsian space <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">QF</mtext></mstyle><mo stretchy=\"false\">(</mo><mi>S</mi><mo stretchy=\"false\">)</mo></math></span><span></span>, which extends the Weil–Petersson metric on the Teichmüller space <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi mathvariant=\"cal\">𝒯</mi><mo stretchy=\"false\">(</mo><mi>S</mi><mo stretchy=\"false\">)</mo><mo>⊂</mo><mstyle><mtext mathvariant=\"normal\">QF</mtext></mstyle><mo stretchy=\"false\">(</mo><mi>S</mi><mo stretchy=\"false\">)</mo></math></span><span></span>. We also calculate its curvature and prove non-positivity for the curvature along the tautological directions.</p>","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"122 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140072620","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Einstein Lie groups, geodesic orbit manifolds and regular Lie subgroups","authors":"Nikolaos Panagiotis Souris","doi":"10.1142/s0219199723500682","DOIUrl":"https://doi.org/10.1142/s0219199723500682","url":null,"abstract":"<p>We study the relation between two special classes of Riemannian Lie groups <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>G</mi></math></span><span></span> with a left-invariant metric <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>g</mi></math></span><span></span>: The Einstein Lie groups, defined by the condition <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mstyle><mtext mathvariant=\"normal\">Ric</mtext></mstyle></mrow><mrow><mi>g</mi></mrow></msub><mo>=</mo><mi>c</mi><mi>g</mi></math></span><span></span>, and the geodesic orbit Lie groups, defined by the property that any geodesic is the integral curve of a Killing vector field. The main results imply that extensive classes of compact simple Einstein Lie groups <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">(</mo><mi>G</mi><mo>,</mo><mi>g</mi><mo stretchy=\"false\">)</mo></math></span><span></span> are not geodesic orbit manifolds, thus providing large-scale answers to a relevant question of Nikonorov. Our approach involves studying and characterizing the <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>G</mi><mo stretchy=\"false\">×</mo><mi>K</mi></math></span><span></span>-invariant geodesic orbit metrics on Lie groups <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>G</mi></math></span><span></span> for a wide class of subgroups <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>K</mi></math></span><span></span> that we call (weakly) regular. By-products of our work are structural and characterization results that are of independent interest for the classification problem of geodesic orbit manifolds.</p>","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"133 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140072621","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Luigi Montoro, Berardino Sciunzi, Alessandro Trombetta
{"title":"A comparison principle for a doubly singular quasilinear anisotropic problem","authors":"Luigi Montoro, Berardino Sciunzi, Alessandro Trombetta","doi":"10.1142/s0219199723500608","DOIUrl":"https://doi.org/10.1142/s0219199723500608","url":null,"abstract":"<p>In this paper, we prove a comparison principle for sub-supersolutions to a singular quasilinear problem that involves the anisotropic Finsler operator <disp-formula-group><span><math altimg=\"eq-00001.gif\" display=\"block\" overflow=\"scroll\"><mrow><mo stretchy=\"false\">−</mo><msubsup><mrow><mi mathvariant=\"normal\">Δ</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>H</mi></mrow></msubsup><mi>u</mi><mo>:</mo><mo>=</mo><mo stretchy=\"false\">−</mo><mspace width=\"-.17em\"></mspace><mo>div</mo><mo stretchy=\"false\">(</mo><msup><mrow><mi>H</mi></mrow><mrow><mi>p</mi><mo stretchy=\"false\">−</mo><mn>1</mn></mrow></msup><mo stretchy=\"false\">(</mo><mo>∇</mo><mi>u</mi><mo stretchy=\"false\">)</mo><mo>∇</mo><mi>H</mi><mo stretchy=\"false\">(</mo><mo>∇</mo><mi>u</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo><mo>.</mo></mrow></math></span><span></span></disp-formula-group> As a main consequence, we obtain a uniqueness result for weak solutions to the problem (℘). The proof is carried out also proving a sharp regularity result of the solutions up to the boundary. Our results are new even in the euclidean case.</p>","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"31 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140072485","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ulrich Bauer, Anibal M. Medina-Mardones, Maximilian Schmahl
{"title":"Persistent homology for functionals","authors":"Ulrich Bauer, Anibal M. Medina-Mardones, Maximilian Schmahl","doi":"10.1142/s0219199723500554","DOIUrl":"https://doi.org/10.1142/s0219199723500554","url":null,"abstract":"<p>We introduce topological conditions on a broad class of functionals that ensure that the persistent homology modules of their associated sublevel set filtration admit persistence diagrams, which, in particular, implies that they satisfy generalized Morse inequalities. We illustrate the applicability of these results by recasting the original proof of the Unstable Minimal Surface Theorem given by Morse and Tompkins in a modern and rigorous framework.</p>","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"67 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2023-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140072618","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Fractional anisotropic Calderón problem on complete Riemannian manifolds","authors":"Mourad Choulli, El Maati Ouhabaz","doi":"10.1142/s0219199723500578","DOIUrl":"https://doi.org/10.1142/s0219199723500578","url":null,"abstract":"<p>We prove that the metric tensor <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>g</mi></math></span><span></span> of a complete Riemannian manifold is uniquely determined, up to isometry, from the knowledge of a local source-to-solution operator associated with a fractional power of the Laplace–Beltrami operator <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi mathvariant=\"normal\">Δ</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span><span></span>. Our result holds under the condition that the metric tensor <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>g</mi></math></span><span></span> is known in an arbitrary small subdomain. We also consider the case of closed manifolds and provide an improvement of the main result in [A. Feizmohammadi, T. Ghosh, K. Krupchyk and G. Uhlmann, Fractional anisotropic Calderón problem on closed Riemannian manifolds, preprint (2021); arXiv:2112.03480].</p>","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"109 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2023-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140072490","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Harmonic maps on locally conformal almost cosymplectic manifolds","authors":"C. Gherghe, G. Vîlcu","doi":"10.1142/s0219199723500529","DOIUrl":"https://doi.org/10.1142/s0219199723500529","url":null,"abstract":"","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"15 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2023-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139263090","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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