Proceedings of the American Mathematical Society, Series B最新文献

{"title":"Radial symmetry for an elliptic PDE with a free boundary","authors":"L. Hajj, H. Shahgholian","doi":"10.1090/bproc/88","DOIUrl":"https://doi.org/10.1090/bproc/88","url":null,"abstract":"<p>In this paper we prove symmetry for solutions to the semi-linear elliptic equation <disp-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Delta u equals f left-parenthesis u right-parenthesis in upper B 1 comma 0 less-than-or-equal-to u greater-than upper M comma in upper B 1 comma u equals upper M comma on partial-differential upper B 1 comma\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi mathvariant=\"normal\">Δ<!-- Δ --></mml:mi>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mspace width=\"1em\" />\u0000 <mml:mtext> in </mml:mtext>\u0000 <mml:msub>\u0000 <mml:mi>B</mml:mi>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mspace width=\"2em\" />\u0000 <mml:mn>0</mml:mn>\u0000 <mml:mo>≤<!-- ≤ --></mml:mo>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mo>></mml:mo>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mspace width=\"1em\" />\u0000 <mml:mtext> in </mml:mtext>\u0000 <mml:msub>\u0000 <mml:mi>B</mml:mi>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mspace width=\"2em\" />\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mspace width=\"1em\" />\u0000 <mml:mtext> on </mml:mtext>\u0000 <mml:mi mathvariant=\"normal\">∂<!-- ∂ --></mml:mi>\u0000 <mml:msub>\u0000 <mml:mi>B</mml:mi>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo>,</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">begin{equation*} Delta u = f(u) quad text { in } B_1, qquad 0 leq u > M, quad text { in } B_1, qquad u = M, quad text { on } partial B_1, end{equation*}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</disp-formula>\u0000 where <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M greater-than 0\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:mo>></mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">M>0</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is a constant, and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B 1\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>B</mml:mi>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">B_1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is the unit ball. Under certain assumptions on the r.h.s. <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f left-parenthesis u right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mo stretchy=\"f","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"105 2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122631208","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Slopes of modular forms and reducible Galois representations, an oversight in the ghost conjecture","authors":"John Bergdall, R. Pollack","doi":"10.1090/bproc/136","DOIUrl":"https://doi.org/10.1090/bproc/136","url":null,"abstract":"The ghost conjecture, formulated by this article’s authors, predicts the list of \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000-adic valuations of the non-zero \u0000\u0000 \u0000 \u0000 a\u0000 p\u0000 \u0000 a_p\u0000 \u0000\u0000-eigenvalues (“slopes”) for overconvergent \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000-adic modular eigenforms in terms of the Newton polygon of an easy-to-describe power series (the “ghost series”). The prediction is restricted to eigenforms whose Galois representation modulo \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000 is reducible on a decomposition group at \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000. It has been discovered, however, that the conjecture is not formulated correctly. Here we explain the issue and propose a salvage.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131903362","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A construction principle for proper scoring rules","authors":"Jonas R. Brehmer","doi":"10.1090/bproc/98","DOIUrl":"https://doi.org/10.1090/bproc/98","url":null,"abstract":"Proper scoring rules enable decision-theoretically principled comparisons of probabilistic forecasts. New scoring rules can be constructed by identifying the predictive distribution with an element of a parametric family and then applying a known scoring rule. We introduce a condition which ensures propriety in this construction and thereby obtain novel proper scoring rules.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129960586","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A note on the asymptotic behavior of radial solutions to quasilinear elliptic equations with a Hardy potential","authors":"K. Itakura, Satoshi Tanaka","doi":"10.1090/bproc/100","DOIUrl":"https://doi.org/10.1090/bproc/100","url":null,"abstract":"<p>The quasilinear elliptic equation with a Hardy potential <disp-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal d normal i normal v left-parenthesis StartAbsoluteValue x EndAbsoluteValue Superscript alpha Baseline StartAbsoluteValue nabla u EndAbsoluteValue Superscript p minus 2 Baseline nabla u right-parenthesis plus StartFraction mu Over StartAbsoluteValue x EndAbsoluteValue Superscript p minus alpha Baseline EndFraction StartAbsoluteValue u EndAbsoluteValue Superscript p minus 2 Baseline u equals 0 in bold upper R Superscript upper N Baseline minus StartSet 0 EndSet\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"normal\">d</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">i</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">v</mml:mi>\u0000 </mml:mrow>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo stretchy=\"false\">|</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo stretchy=\"false\">|</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mi>α<!-- α --></mml:mi>\u0000 </mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo stretchy=\"false\">|</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mi mathvariant=\"normal\">∇<!-- ∇ --></mml:mi>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo stretchy=\"false\">|</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>p</mml:mi>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:mi mathvariant=\"normal\">∇<!-- ∇ --></mml:mi>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mfrac>\u0000 <mml:mi>μ<!-- μ --></mml:mi>\u0000 <mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo stretchy=\"false\">|</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo stretchy=\"false\">|</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>p</mml:mi>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mi>α<!-- α --></mml:mi>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 </mml:mrow>\u0000 </mml:mfrac>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo stretchy=\"false\">|</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo stretchy=\"false\">|</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>p</mml:mi>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:mi>u</mml:mi>","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130948742","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Basic properties of 𝑋 for which the space 𝐶_{𝑝}(𝑋) is distinguished","authors":"J. Ka̧kol, A. Leiderman","doi":"10.1090/bproc/95","DOIUrl":"https://doi.org/10.1090/bproc/95","url":null,"abstract":"<p>In our paper [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99] we showed that a Tychonoff space <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\u0000 <mml:semantics>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is a <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Delta\">\u0000 <mml:semantics>\u0000 <mml:mi mathvariant=\"normal\">Δ<!-- Δ --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">Delta</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-space (in the sense of R. W. Knight [Trans. Amer. Math. Soc. 339 (1993), pp. 45–60], G. M. Reed [Fund. Math. 110 (1980), pp. 145–152]) if and only if the locally convex space <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Subscript p Baseline left-parenthesis upper X right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>p</mml:mi>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">C_{p}(X)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is distinguished. Continuing this research, we investigate whether the class <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Delta\">\u0000 <mml:semantics>\u0000 <mml:mi mathvariant=\"normal\">Δ<!-- Δ --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">Delta</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Delta\">\u0000 <mml:semantics>\u0000 <mml:mi mathvariant=\"normal\">Δ<!-- Δ --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">Delta</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-spaces is invariant under the basic topological operations.</p>\u0000\u0000<p>We prove that if <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X element-of normal upper Delta\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo>∈<!-- ∈ --></mml:mo>\u0000 <mml:mi mathvariant=\"normal\">Δ<!-- Δ --></mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">X in Delta</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"phi colon upper X right-arrow upper Y\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>φ<!-- φ --></mml:mi>\u0000 <mml:mo>:</mml:","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132604226","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Scaled homology and topological entropy","authors":"B. Hou, Kiyoshi Igusa, Zihao Liu","doi":"10.1090/bproc/182","DOIUrl":"https://doi.org/10.1090/bproc/182","url":null,"abstract":"In this paper, we build up a scaled homology theory, \u0000\u0000 \u0000 \u0000 l\u0000 c\u0000 \u0000 lc\u0000 \u0000\u0000-homology, for metric spaces such that every metric space can be visually regarded as “locally contractible” with this newly-built homology. We check that \u0000\u0000 \u0000 \u0000 l\u0000 c\u0000 \u0000 lc\u0000 \u0000\u0000-homology satisfies all Eilenberg-Steenrod axioms except the exactness axiom whereas its corresponding \u0000\u0000 \u0000 \u0000 l\u0000 c\u0000 \u0000 lc\u0000 \u0000\u0000-cohomology satisfies exactness axiom for cohomology. This homology can relax the smooth manifold restrictions on the compact metric space such that the entropy conjecture will hold for the first \u0000\u0000 \u0000 \u0000 l\u0000 c\u0000 \u0000 lc\u0000 \u0000\u0000-homology group.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127325334","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Randomly perturbed ergodic averages","authors":"Jaeyong Choi, Karin Reinhold-Larsson","doi":"10.1090/bproc/61","DOIUrl":"https://doi.org/10.1090/bproc/61","url":null,"abstract":". We consider a class of random ergodic averages, containing averages along random non–integer sequences. For such averages, Cohen & Cuny obtained uniform universal pointwise convergence for functions in L 2 with (cid:2) max(1 , log(1+ | t | )) dμ f < ∞ via a uniform estimation of trigonometric polynomials. We extend this result to L 2 functions satisfying the weaker condition (cid:2) max(1 , log log(1+ | t | )) dμ f < ∞ . We also prove that uniform universal pointwise convergence in L 2 holds for the corresponding smoothed random averages or for random averages whose kernels exhibit sufficient decay at infinity.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123779001","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Analytic continuation of generalized trigonometric functions","authors":"P. Ding","doi":"10.1090/bproc/119","DOIUrl":"https://doi.org/10.1090/bproc/119","url":null,"abstract":"Via a unified geometric approach, certain generalized trigonometric functions with two parameters are analytically extended to maximal domains on which they are univalent. Some consequences are deduced concerning radius of convergence for the Maclaurin series, commutation with rotation, continuation beyond the domain of univalence, and periodicity.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128249757","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The inductive McKay–Navarro conditions for the prime 2 and some groups of Lie type","authors":"L. Ruhstorfer, A. S. Fry","doi":"10.1090/bproc/123","DOIUrl":"https://doi.org/10.1090/bproc/123","url":null,"abstract":"For a prime \u0000\u0000 \u0000 ℓ\u0000 ell\u0000 \u0000\u0000, the McKay conjecture suggests a bijection between the set of irreducible characters of a finite group with \u0000\u0000 \u0000 \u0000 ℓ\u0000 ′\u0000 \u0000 ell ’\u0000 \u0000\u0000-degree and the corresponding set for the normalizer of a Sylow \u0000\u0000 \u0000 ℓ\u0000 ell\u0000 \u0000\u0000-subgroup. Navarro’s refinement suggests that the values of the characters on either side of this bijection should also be related, proposing that the bijection commutes with certain Galois automorphisms. Recently, Navarro–Späth–Vallejo have reduced the McKay–Navarro conjecture to certain “inductive” conditions on finite simple groups. We prove that these inductive McKay–Navarro (also called the inductive Galois–McKay) conditions hold for the prime \u0000\u0000 \u0000 \u0000 ℓ\u0000 =\u0000 2\u0000 \u0000 ell =2\u0000 \u0000\u0000 for several groups of Lie type, namely the untwisted groups without non-trivial graph automorphisms.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"67 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124342554","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Sumsets and fixed points of substitutions","authors":"F. Dekking","doi":"10.1090/bproc/132","DOIUrl":"https://doi.org/10.1090/bproc/132","url":null,"abstract":"<p>In this paper we introduce a technique to determine the sumset <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A plus upper A\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mi>A</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">A+A</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, where <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\u0000 <mml:semantics>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is the indicator function of the 0’s occurring in a fixed point <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"x\">\u0000 <mml:semantics>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">x</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of a substitution on the alphabet <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartSet 0 comma 1 EndSet\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">{0,1}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>.</p>","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116287318","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}