MixedMath - Recent Commentshttps://davidlowryduda.comDavid's personal blog.en-usCopyright David Lowry-Duda (2022) - All Rights Reserved.admin@davidlowryduda.comadmin@davidlowryduda.comSat, 03 Sep 2022 19:10:46 +0000Sat, 03 Sep 2022 19:10:46 +0000mixedmathapp/generate_rss.py v0.1https://cyber.harvard.edu/rss/rss.htmlhttps://davidlowryduda.com/static/images/favicon-32x32.pngMixedMathhttps://davidlowryduda.comdavidlowryduda 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 +0000davidlowrydud 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 +0000davidlowrydud 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 +0000davidlworydud 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 +0000elio 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>[…] the fact by Miles — who researches fluid dynamics, is a friend from grad school, and was my coauthor on a paper about the mean value theorem. I do not typically think about fluid dynamics (and did not write the paper), and it’s a bit […]</p>830b520b9426335cee1c37346ada45d8Wed, 19 Aug 2020 03:14:15 +0000davidlowrydud on Notes from a talk at the Maine-Quebec Number Theory Conferencehttps://davidlowryduda.com/notes-from-a-talk-at-the-maine-quebec-number-theory-conference<p>Thanks Peter! You're right — and it's a nice reference. I had no idea.</p>738cfb0f79ca7228c95e7c8bb00ae47cThu, 17 Oct 2019 03:14:15 +0000Peter Humphries on Notes from a talk at the Maine-Quebec Number Theory Conferencehttps://davidlowryduda.com/notes-from-a-talk-at-the-maine-quebec-number-theory-conference<p>Omega results of this form using this method are presented quite nicely in
Chapter 15 of Montgomery and Vaughan; they give applications towards sign
changes of the Chebyshev psi function and of partial sums of the Mobius
function.</p>20666456b5db15ad4f972caa02257f33Mon, 07 Oct 2019 03:14:15 +0000Nevin Manimalas Blog on Choosing functions and generating figures for "When are there continuous choices for the mean value abscissa?"https://davidlowryduda.com/choosing-functions-for-mvt-abscissa<p>[...] as in David Lowry-Duda [...]</p>8389cacf633c54974c3d67b94c3cb1daSun, 16 Jun 2019 03:14:15 +0000davidlowrydud on Choosing functions and generating figures for "When are there continuous choices for the mean value abscissa?"https://davidlowryduda.com/choosing-functions-for-mvt-abscissa<p>At first, we chose 2 because it was the other "obvious" point that we hadn't
yet specified. We already controlled the values at 0, 1, and 3.</p>
<p>But I later realized it doesn't matter what point we choose. What we're really
doing is adding one extra degree of freedom and using it to minimize the L2
norm of the derivative on [0, 4]. If we instead chose 1.5 instead, the middle
steps might look different, but the desired polynomial that has the specified
behavior at the other points (including values of the derivative) and which has
minimal L2 norm would still be found. (I'm assuming that this polynomial is
unique, but in fact I don't know that this is true. If we're in some
pathological case where it's not unique, then what I claim is <em>almost</em> true).</p>9bb1e500118bd3087a3d408d5ac93de4Sun, 16 Jun 2019 03:14:15 +0000PC on Choosing functions and generating figures for "When are there continuous choices for the mean value abscissa?"https://davidlowryduda.com/choosing-functions-for-mvt-abscissa<p>Why did you choose 2 for the pt in the L2 part? What if you chose 1.5 or
something?</p>318e18561d5bb67d8b2924bdd4a6af55Sun, 16 Jun 2019 03:14:15 +0000Eli B on Hosting a Flask App on WebFaction on a Non-root Domainhttps://davidlowryduda.com/hosting-a-flask-app-on-webfaction-on-a-non-root-domain<p>Thank god, finally one that works.</p>3faef856634e9fe856bd643e2e406b09Thu, 02 May 2019 03:14:15 +0000davidlowrydud on An intuitive introduction to calculushttps://davidlowryduda.com/an-intuitive-introduction-to-calculus<p>Thank you. You could also write $P(t) = ae^{kt + C_1}$. But this may give the
impression that $a$ and $C_1$ carry different data — but they don't. To see
the equivalence, you can write $ae^{kt + C_1} = a e^{kt} e^{C_1} = (a e^{C_1})
e^{kt} = A e^{kt}$, where $A = a e^{C_1}$. There are two fundamental degrees of
freedom expressed as constants: the initial amount (which I wrote as $a$ or
$A$) and the rate of increase (expressed as $k$).</p>62f27f2f989187881c589d6be4129087Mon, 24 Sep 2018 03:14:15 +0000