MixedMath - Recent Commentshttps://davidlowryduda.comDavid's personal blog.en-usCopyright David Lowry-Duda (2022) - All Rights Reserved.admin@davidlowryduda.comadmin@davidlowryduda.comMon, 30 Jan 2023 16:00:42 +0000Mon, 30 Jan 2023 16:00:42 +0000mixedmathapp/generate_rss.py v0.1https://cyber.harvard.edu/rss/rss.htmlhttps://davidlowryduda.com/static/images/favicon-32x32.pngMixedMathhttps://davidlowryduda.comDLD on On Mathstodonhttps://davidlowryduda.com/on-mathstodon<p><a href="https://mathstodon.xyz/@davidlowryduda">@davidlowryduda@mathstodon.xyz</a>.</p>8a19677b8293b549c1e68b3f6bdd7029Mon, 02 Jan 2023 03:14:15 +0000PD on Making Plots of Modular Formshttps://davidlowryduda.com/making-plots-of-modular-forms<p>Thanks for the quick reply and your code! I was working already with few libs.</p>3652a9161a1da72b7017f21331085fe6Mon, 02 Jan 2023 03:14:15 +0000DLD on Making Plots of Modular Formshttps://davidlowryduda.com/making-plots-of-modular-formsThis comment is larger than 2500 bytes, which is above the limit for this RSS feed. Please view it directly at the url.76b1770669dd48d2cee6e4661c490f7aMon, 02 Jan 2023 03:14:15 +0000mrita on On Mathstodonhttps://davidlowryduda.com/on-mathstodon<p>What is your name on mastodon?</p>038b45b7433595c2ae0b18e4454e210aSun, 01 Jan 2023 03:14:15 +0000PD on Making Plots of Modular Formshttps://davidlowryduda.com/making-plots-of-modular-forms<p>I see your plots here. Do you have an idea how to plot precisely <a href="https://en.wikipedia.org/wiki/Modular_lambda_function#/media/File:Lambda_function.svg">this curve?</a></p> <p>I am simulating few mathematical things (I am not a mathematician, only engineer) and would like to identify the coordinates of the points of the curve for making a comparison with tetraetion curves. I know how to start python scripts and was using sometime matplotlib (No clue of Sage).</p> <p>No hurry in answering. This is a project I am doing in parallel to my life.</p>4cc69f0a266c153c2ef30d014082ae25Sun, 01 Jan 2023 03:14:15 +0000davidlowryduda on The gamma function, beta function, and duplication formulahttps://davidlowryduda.com/the-gamma-function-beta-function-and-duplication-formula<p>Dear Nishant,</p> <p>I believe this is called the duplication formula because it's <em>almost</em> a formula of the form $\Gamma(z)\Gamma(z)$, i.e. this is <em>almost</em> a formula fro two copies of $\Gamma(z)$. Maybe calling it the "near duplication formula" wasn't catchy enough.</p> <p>It was first proved by Adrien-Marie Legendre, hence the name. Legendre also gave the notation $\Gamma$ for the gamma function. I read this in Artin's book <em>The Gamma Function</em>, which contains far more if you're interested.</p>49ed0a9f58115f4f875bdf7c674763a9Sun, 16 Oct 2022 03:14:15 +0000Nishant on The gamma function, beta function, and duplication formulahttps://davidlowryduda.com/the-gamma-function-beta-function-and-duplication-formula<p>I just had a query that why is this form called duplication formula, and above all why Legendre duplication formula. Any information shared will find me highly motivated.</p>e79257fc23bc7fda02819ca9bae1a599Wed, 12 Oct 2022 03:14:15 +0000davidlowryduda on An intuitive introduction to calculushttps://davidlowryduda.com/an-intuitive-introduction-to-calculus<p>Thank you. They should now be fixed.</p>efde99d93ddf70913680b375312f4a9cMon, 06 Jun 2022 03:14:15 +0000MF on An intuitive introduction to calculushttps://davidlowryduda.com/an-intuitive-introduction-to-calculus<p>The graphics aren't loading!</p>70eba4e5a21e513016e99975c319742aMon, 06 Jun 2022 03:14:15 +0000davidlowryduda on Visualizations for Quanta's 'What is the Langlands Program?'https://davidlowryduda.com/quanta-langlands-viz<p>Quanta sent me palettes, and I designed colormaps around these colors. They adjusted some of the colors afterwards too.</p>7f5ba165d0d033b0e7c4b0a1350428c5Fri, 03 Jun 2022 03:14:15 +0000CA on Visualizations for Quanta's 'What is the Langlands Program?'https://davidlowryduda.com/quanta-langlands-viz<p>What colormap did you use for the video?</p>0b3d422c2e5b166e6e6e172ec943919eFri, 03 Jun 2022 03:14:15 +0000davidlowryduda on colormapplot - like phasematplot, but with colormapshttps://davidlowryduda.com/colormapplot<p>I like that you made a custom colormap and went with it. I think this sort of experimentation will lead to powerful, informative visualizations.</p> <p>Congratulations on making a website!</p>4fd4b32003ca4ead78f7be544348e232Wed, 20 Apr 2022 03:14:15 +0000Andrew on colormapplot - like phasematplot, but with colormapshttps://davidlowryduda.com/colormapplot<p>Some plots in SageMath, what do you think: <a href="https://sheerluck.github.io">https://sheerluck.github.io</a> My first web site ever in my life :)</p>66b898eac5341e5ff7b7442b209e4fb5Sun, 27 Mar 2022 03:14:15 +0000Markus Nascimento on An intuitive overview of Taylor serieshttps://davidlowryduda.com/an-intuitive-overview-of-taylor-series<p>Very nice note. It really helped me to understand a bit more about Taylor polynomials’ intuition. Congratulations!</p>53ed03f5caa13e5f14dfd77ed31f9cf7Thu, 17 Mar 2022 03:14:15 +0000Vaskor Basak on Prime rich and prime poorhttps://davidlowryduda.com/prime-rich-and-prime-poor<p>What polynomials are allowable for prime-poor polynomials? Could I claim that I have found a better example of a prime-poor polynomial than $x^{12}+488669$ by presenting the example $(x+3)^{12}-488601$, for example?</p>3dc95bd2293930cf465b867debd35ebaSat, 25 Dec 2021 03:14:15 +0000davidlowryduda on An intuitive overview of Taylor serieshttps://davidlowryduda.com/an-intuitive-overview-of-taylor-series<p>Hi Bob! The behavior of $c$ is actually quite subtle. It's not true that $c$ is actually a constant. For "nice" functions, what is true is that the mean value $c$ varies continuously over intervals (except at finitely many points). Combined with a form of Darboux's theorem (stating that every function that is the derivative of another function has the intermediate value property, even if it's not continuous) is enough.</p> <p>I published a paper with Miles Wheeler (preprint available at https://arxiv.org/abs/1906.02026) in the American Mathematical Monthly that showed that the mean values generically can be chosen to vary locally continuously on the right endpoint, the key analytic ingredient.</p> <p>Making this rigorous is substantially more complicated than other proofs. As a heuristic, I like that it suggests the right shape of Taylor's formula (which is often nonobvious to a beginner), but I don't think it's the right way to actually go about proving it.</p>1d16e4fb91641622374a2b5c08866b19Tue, 19 Oct 2021 03:14:15 +0000Bob on An intuitive overview of Taylor serieshttps://davidlowryduda.com/an-intuitive-overview-of-taylor-series<p>It's been a while since there was a comment here, but I'll give it a try in the hope for a response. Your proof gives me a sense of validation in my own work, although I still have questions. I wrote a very similar idea here: https://math.stackexchange.com/q/4277898/225136 . As you can see from a comment that someone left, the proof becomes tricky to show that f^(n+1)(c) in the last term can be treated as a constant within the integral. I contend that it is inherently constant. Just because we are rewriting this term with respect to t when we integrate, it does not mean we are varying the value of f^(n+1)(c). This mysterious value depends only on the right endpoint of the integration, not on t (the variable of integration). My heuristic is that any continuous function f^(n+1)(t) that is integrated over [a,b] has an average value over the integral (which is constant of course). The mean value theorem tells us that this constant value is f^(n+1)(c). So if know all of the f^(k)(a) constants of integration for each derivative up to n+1, we can iteratively integrate to recover the original f(t) function starting with y = f^(n+1)(c) as a proxy for the actual f^(n+1)(t). As you stated in your write up, starting the integration is extremely convenient since it develops the Taylor polynomial as well as the remainder term which elegantly falls out in Lagrange form. However, I still have difficulty explaining how to justify rigorously why f^(n+1)(c) can be treated as a constant in the integral. You made a note 1 that you may return to a rigorous proof of this in a future post but I haven't found it.</p>883520f477fbea2a50aa17d0aaa165dcMon, 18 Oct 2021 03:14:15 +0000Yavor Kirov on Long live opalstack!https://davidlowryduda.com/long-live-opalstack<p>Thank you! I was looking for an opinion of a long-term user of Opalstack so your answer was exactly what I needed.</p>7af7efe9dd6a79909c5d00114c40be92Wed, 14 Apr 2021 03:14:15 +0000davidlowryduda on Long live opalstack!https://davidlowryduda.com/long-live-opalstack<p>Hi Yavor,</p> <p>I'm very happy with Opalstack. After the initial transition, there have been almost no problems. I'll note that for a few months, opalstack's email setup was rocky, but they figured that out. I'd recommend it.</p> <p>They've been pretty responsive to problems, too. You can see what's troubling people now on their community/support forum: https://community.opalstack.com/. </p> <p>I know a somewhat common source of frustration is that opalstack doesn't have GNU Mailman for mailing list hosting and doesn't have plans to add it. But this is the only possible deal-breaker I'm currently aware of (and it's not one that affects me).</p> <p>If you have any particular questions, I can try to answer.</p>c19491d4a49861b98bf55322623b4657Tue, 13 Apr 2021 03:14:15 +0000Yavor Kirov on Long live opalstack!https://davidlowryduda.com/long-live-opalstack<p>Hello David,</p> <p>Would you please be so kind to tell us (me) of your (further) experience with Opalstack so far? </p> <p>I myself had to migrate away from WebFaction and I am wondering if now migrating some of my sites to Opalstack is a good idea.</p> <p>Best regards :)</p>c227fdcb553d1adb6612ad4918adfde0Tue, 13 Apr 2021 03:14:15 +0000Tomek on On least squares - a question from reddithttps://davidlowryduda.com/a-response-to-ftyous-question-on-reddit<p>God bless you David. Great explanation!</p>662c14e45d7d24ea1f3db6b0caeefd94Thu, 18 Mar 2021 03:14:15 +0000davidlworyduda on phase_mag_plot: a sage package for plotting complex functionshttps://davidlowryduda.com/phase_mag_plot-a-sage-package-for-plotting-complex-functions<p>I'm glad to hear it! I would also read issues or pull requests on github.</p> <p>I think it would not be particularly hard to incorporate this into official sage. I have a local build of sage that includes a variant of phase_mag_plot as the default complex_plot. There are a few additional usability things I need to do before submitting this to the official sage.</p> <p>I'd also like to write the general colormap functionality in a good way to be included too, but I haven't gotten around to that yet.</p> <p>To learn more about contributing directly to sage (which is fun and exciting, and there are several great programmers who routinely contribute), check out https://doc.sagemath.org/html/en/developer/index.html.</p>04753f6df3687b8cedd2207430d7445cThu, 25 Feb 2021 03:14:15 +0000Jan van Delden on phase_mag_plot: a sage package for plotting complex functionshttps://davidlowryduda.com/phase_mag_plot-a-sage-package-for-plotting-complex-functions<p>I took the liberty to download your SageMath module and changed a few options. </p> <p>Since I was not completely clear on the interaction between lightness and color (defined by the argument of the function to be displayed) I decided to weigh these differently. Color, from the colorwheel by 0.8 and lightness by 0.2. I computed the color first and weighed later. Changed the modulus operation to argument-floor(argument) to solve a problem for arguments near 0. Added the option to choose the number of subdivisions which are used to stress particular moduli or phases.</p> <p>I read both books, love them, and already implemented the extended phase portrait into Maple, c (including domain coloring in general) and Ultra Fractal. Since it was time to learn sage and had to learn how to program, I used your module as my first project. And as you stated: it is time to learn how to incorporate a module into sage, without having to load it. No idea yet, but onwards we go...</p>75ba1c8645715735ed9754e90b9bf933Thu, 25 Feb 2021 03:14:15 +0000eliot on Trigonometric and related substitutions in integralshttps://davidlowryduda.com/trigonometric-and-related-substitutions-in-integrals<p>For Euler substitutions, do you mind explaining the reasoning for the t term in x*sqrt(a) + t. Is there anywhere else I can read up on this?</p> <p>Thanks for this post!</p>b814cdde0967ee9128ff5aea69d8553bWed, 23 Dec 2020 03:14:15 +0000davidlowryduda, in a followup post on Paper: When are there continuous choices for the Mean Value Abscissa?https://davidlowryduda.com/paper-continuous-choices-mvt<p>This is an update with (unexpected) good news. My collaborator Miles Wheeler and I were given the Halmos-Ford award for our paper...</p>b5c78a9ba29923c6e2391a709c7bd4e0Mon, 19 Oct 2020 03:14:15 +0000