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Pustam | पुस्तम | পুস্তম🇳🇵One of the best things I saw this week: a paper uncovering alien signals in the Riemann Zeta function. April Fools always brings peak creativity. 😅<a href="https://mathstodon.xyz/tags/Riemann" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Riemann</a> <a href="https://mathstodon.xyz/tags/Zeta" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Zeta</a> <a href="https://mathstodon.xyz/tags/ZetaFunction" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#ZetaFunction</a> <a href="https://mathstodon.xyz/tags/RiemanZetaFunction" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#RiemanZetaFunction</a> <a href="https://mathstodon.xyz/tags/AprilFool" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#AprilFool</a> <a href="https://mathstodon.xyz/tags/AprilFools" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#AprilFools</a> <a href="https://mathstodon.xyz/tags/AprilFoolsDay" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#AprilFoolsDay</a> <a href="https://mathstodon.xyz/tags/Creativity" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Creativity</a> <a href="https://mathstodon.xyz/tags/Math" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Math</a> <a href="https://mathstodon.xyz/tags/Maths" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Maths</a> <a href="https://mathstodon.xyz/tags/NumberTheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#NumberTheory</a> <a href="https://mathstodon.xyz/tags/PeakCreativity" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#PeakCreativity</a> <a href="https://mathstodon.xyz/tags/Nerd" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Nerd</a> <a href="https://mathstodon.xyz/tags/Nerds" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Nerds</a> <a href="https://mathstodon.xyz/tags/Humor" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Humor</a> <a href="https://mathstodon.xyz/tags/Humour" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Humour</a> <a href="https://mathstodon.xyz/tags/Alien" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Alien</a> <a href="https://mathstodon.xyz/tags/AlienSignals" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#AlienSignals</a>

Jochen FrommWill the toughest problem in <a href="https://fediscience.org/tags/math" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#math</a> ever be solved? David Whitehouse about the Riemann hypothesis <a href="https://www.spectator.co.uk/article/will-the-toughest-problem-in-maths-ever-be-solved/" rel="nofollow noopener noreferrer" translate="no" target="_blank">https://www.spectator.co.uk/article/will-the-toughest-problem-in-maths-ever-be-solved/</a>Recently there has been some progress - a proof creates stricter limits on potential exceptions to the famous <a href="https://fediscience.org/tags/Riemann" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Riemann</a> hypothesis <a href="https://www.quantamagazine.org/sensational-proof-delivers-new-insights-into-prime-numbers-20240715/" rel="nofollow noopener noreferrer" translate="no" target="_blank">https://www.quantamagazine.org/sensational-proof-delivers-new-insights-into-prime-numbers-20240715/</a>

ƧƿѦςɛ♏ѦਹѤʞ<a href="https://social.zvavybir.eu/@zvavybir" class="u-url mention" rel="nofollow noopener noreferrer" target="_blank">@zvavybir</a> It does diverge. It has no sum. However, the uniquely valued <a href="https://mastodon.social/tags/Riemann" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Riemann</a> <a href="https://mastodon.social/tags/ZetaFunction" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#ZetaFunction</a> can be analytically continued into the left half-plane where we find zeta(-1)=-1/12 (which 'looks like' 1+2+...). <a href="https://mastodon.social/tags/Ces%C3%A0ro" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Cesàro</a> <a href="https://mastodon.social/tags/summation" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#summation</a> will get you part of the way there also, and, as you say, yields the same result; presumably due to some ultimate cosmic logical rightness :-) I very strongly recommend BP's superb exposition of this issue <a href="https://www.youtube.com/watch?v=YuIIjLr6vUA" rel="nofollow noopener noreferrer" translate="no" target="_blank">https://www.youtube.com/watch?v=YuIIjLr6vUA</a> <a href="https://mastodon.social/tags/maths" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#maths</a> <a href="https://mastodon.social/tags/AnalyticContinuation" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#AnalyticContinuation</a> <a href="https://mastodon.social/tags/Ramanujan" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Ramanujan</a>

Pustam | पुस्तम | পুস্তম🇳🇵DOMINATED CONVERGENCE THEOREM Lebesgue's dominated convergence theorem provides sufficient conditions under which pointwise convergence of a sequence of functions implies convergence of the integrals. It's one of the reasons that makes <a href="https://mathstodon.xyz/tags/Lebesgue" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Lebesgue</a> integration more powerful than <a href="https://mathstodon.xyz/tags/Riemann" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Riemann</a> integration. The theorem an be stated as follows:Let \((f_n)\) be a sequence of measurable functions on a measure space \((\mathcal{S},\Sigma,\mu)\). Suppose that \((f_n)\) converges pointwise to a function \(f\) and is dominated by some Lebesgue integrable function \(g\), i.e. \(|f_n(x)|\leq g(x)\ \forall n\) and \(\forall x\in\mathcal{S}\). Then, \(f\) is Lebesgue integrable, and\[\displaystyle\lim_{n\to\infty}\int_\mathcal{S}f_n\ \mathrm{d}\mu=\int_\mathcal{S}f\ \mathrm{d}\mu\] <a href="https://mathstodon.xyz/tags/ConvergenceTheorem" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#ConvergenceTheorem</a> <a href="https://mathstodon.xyz/tags/Convergence" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Convergence</a> <a href="https://mathstodon.xyz/tags/DominatedConvergenceTheorem" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#DominatedConvergenceTheorem</a> <a href="https://mathstodon.xyz/tags/Lebesgue" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Lebesgue</a> <a href="https://mathstodon.xyz/tags/MeasurableFunction" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#MeasurableFunction</a> <a href="https://mathstodon.xyz/tags/LebesgueFunction" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#LebesgueFunction</a> <a href="https://mathstodon.xyz/tags/LebesgueIntegration" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#LebesgueIntegration</a> <a href="https://mathstodon.xyz/tags/RiemannIntegration" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#RiemannIntegration</a> <a href="https://mathstodon.xyz/tags/MeasureSpace" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#MeasureSpace</a>

Jon Awbrey<a href="https://mathstodon.xyz/tags/SignRelationalManifolds" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#SignRelationalManifolds</a> • 4 • <a href="http://inquiryintoinquiry.com/2022/11/05/sign-relational-manifolds-4-2/" rel="nofollow noopener noreferrer" target="_blank">http://inquiryintoinquiry.com/2022/11/05/sign-relational-manifolds-4-2/</a>Another set of notes I found on this theme strikes me as getting to the point more quickly and though they read a little rough in places I think it may be worth the effort to fill out their general line of approach.<a href="https://mathstodon.xyz/tags/RepresentationInvariantOntology" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#RepresentationInvariantOntology</a>• <a href="https://web.archive.org/web/20150302021003/http://stderr.org/pipermail/inquiry/2003-April/thread.html#439" rel="nofollow noopener noreferrer" target="_blank">https://web.archive.org/web/20150302021003/http://stderr.org/pipermail/inquiry/2003-April/thread.html#439</a>• <a href="https://web.archive.org/web/20141220180218/http://stderr.org/pipermail/inquiry/2003-April/000439.html" rel="nofollow noopener noreferrer" target="_blank">https://web.archive.org/web/20141220180218/http://stderr.org/pipermail/inquiry/2003-April/000439.html</a>• <a href="https://web.archive.org/web/20141220180220/http://stderr.org/pipermail/inquiry/2003-April/000440.html" rel="nofollow noopener noreferrer" target="_blank">https://web.archive.org/web/20141220180220/http://stderr.org/pipermail/inquiry/2003-April/000440.html</a><a href="https://mathstodon.xyz/tags/Peirce" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Peirce</a> <a href="https://mathstodon.xyz/tags/Semiotics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Semiotics</a> <a href="https://mathstodon.xyz/tags/SignRelations" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#SignRelations</a> <a href="https://mathstodon.xyz/tags/Riemann" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Riemann</a> <a href="https://mathstodon.xyz/tags/SergeLang" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#SergeLang</a> <a href="https://mathstodon.xyz/tags/Manifolds" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Manifolds</a> <a href="https://mathstodon.xyz/tags/DifferentialGeometry" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#DifferentialGeometry</a> <a href="https://mathstodon.xyz/tags/DifferentialLogic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#DifferentialLogic</a>

Jon Awbrey<a href="https://mathstodon.xyz/tags/Peirce" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Peirce</a> <a href="https://mathstodon.xyz/tags/Kant" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Kant</a> <a href="https://mathstodon.xyz/tags/Riemann" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Riemann</a> <a href="https://mathstodon.xyz/tags/SignRelationalManifolds" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#SignRelationalManifolds</a> <a href="https://inquiryintoinquiry.com/2022/11/01/sign-relational-manifolds-2-2/" rel="nofollow noopener noreferrer" target="_blank">https://inquiryintoinquiry.com/2022/11/01/sign-relational-manifolds-2-2/</a>Applications of <a href="https://mathstodon.xyz/tags/Manifolds" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Manifolds</a> are illustrated by the following excerpts from Doolin & Martin's Introduction to <a href="https://mathstodon.xyz/tags/DifferentialGeometry" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#DifferentialGeometry</a> for Engineers.<a href="https://web.archive.org/web/20110612002240/http://suo.ieee.org/ontology/thrd28.html#04056" rel="nofollow noopener noreferrer" target="_blank">https://web.archive.org/web/20110612002240/http://suo.ieee.org/ontology/thrd28.html#04056</a>Manifolds came up in connection with <a href="https://mathstodon.xyz/tags/Ontology" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Ontology</a> and <a href="https://mathstodon.xyz/tags/Semiotics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Semiotics</a> due to the issues of <a href="https://mathstodon.xyz/tags/Perspectivity" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Perspectivity</a>, <a href="https://mathstodon.xyz/tags/OntologicalRelativity" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#OntologicalRelativity</a>, and <a href="https://mathstodon.xyz/tags/Interoperability" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Interoperability</a> among multiple ontologies. As I see it, those are the very sorts of problems manifolds were invented to handle.

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