MixedMath - Recent Comments

davidlowryduda on An intuitive introduction to calculus

Mon, 06 Jun 2022 03:14:15 +0000

Thank you. They should now be fixed.

MF on An intuitive introduction to calculus

Mon, 06 Jun 2022 03:14:15 +0000

The graphics aren't loading!

davidlowryduda on Visualizations for Quanta's 'What is the Langlands Program?'

Fri, 03 Jun 2022 03:14:15 +0000

Quanta sent me palettes, and I designed colormaps around these colors. They adjusted some of the colors afterwards too.

CA on Visualizations for Quanta's 'What is the Langlands Program?'

Fri, 03 Jun 2022 03:14:15 +0000

What colormap did you use for the video?

davidlowryduda on colormapplot - like phasematplot, but with colormaps

Wed, 20 Apr 2022 03:14:15 +0000

I like that you made a custom colormap and went with it. I think this sort of experimentation will lead to powerful, informative visualizations.

Congratulations on making a website!

Andrew on colormapplot - like phasematplot, but with colormaps

Sun, 27 Mar 2022 03:14:15 +0000

Some plots in SageMath, what do you think: https://sheerluck.github.io My first web site ever in my life :)

Markus Nascimento on An intuitive overview of Taylor series

Thu, 17 Mar 2022 03:14:15 +0000

Very nice note. It really helped me to understand a bit more about Taylor polynomials’ intuition. Congratulations!

Vaskor Basak on Prime rich and prime poor

Sat, 25 Dec 2021 03:14:15 +0000

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?

davidlowrydud on An intuitive overview of Taylor series

Tue, 19 Oct 2021 03:14:15 +0000

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.

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.

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.

Bob on An intuitive overview of Taylor series

Mon, 18 Oct 2021 03:14:15 +0000

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.

Yavor Kirov on Long live opalstack!

Wed, 14 Apr 2021 03:14:15 +0000

Thank you! I was looking for an opinion of a long-term user of Opalstack so your answer was exactly what I needed.

davidlowrydud on Long live opalstack!

Tue, 13 Apr 2021 03:14:15 +0000

Hi Yavor,

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.

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/.

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).

If you have any particular questions, I can try to answer.

Yavor Kirov on Long live opalstack!

Tue, 13 Apr 2021 03:14:15 +0000

Hello David,

Would you please be so kind to tell us (me) of your (further) experience with Opalstack so far?

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.

Best regards :)

Tomek on On least squares - a question from reddit

Thu, 18 Mar 2021 03:14:15 +0000

God bless you David. Great explanation!

davidlworydud on phase_mag_plot: a sage package for plotting complex functions

Thu, 25 Feb 2021 03:14:15 +0000

I'm glad to hear it! I would also read issues or pull requests on github.

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.

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.

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.

Jan van Delden on phase_mag_plot: a sage package for plotting complex functions

Thu, 25 Feb 2021 03:14:15 +0000

I took the liberty to download your SageMath module and changed a few options.

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.

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...

elio on Trigonometric and related substitutions in integrals

Wed, 23 Dec 2020 03:14:15 +0000

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?

Thanks for this post!

davidlowryduda, in a followup post on Paper: When are there continuous choices for the Mean Value Abscissa?

Wed, 19 Aug 2020 03:14:15 +0000

[…] 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 […]

davidlowrydud on Notes from a talk at the Maine-Quebec Number Theory Conference

Thu, 17 Oct 2019 03:14:15 +0000

Thanks Peter! You're right — and it's a nice reference. I had no idea.

Peter Humphries on Notes from a talk at the Maine-Quebec Number Theory Conference

Mon, 07 Oct 2019 03:14:15 +0000

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.

Nevin Manimalas Blog on Choosing functions and generating figures for "When are there continuous choices for the mean value abscissa?"

Sun, 16 Jun 2019 03:14:15 +0000

[...] as in David Lowry-Duda [...]

davidlowrydud on Choosing functions and generating figures for "When are there continuous choices for the mean value abscissa?"

Sun, 16 Jun 2019 03:14:15 +0000

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.

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 almost true).

PC on Choosing functions and generating figures for "When are there continuous choices for the mean value abscissa?"

Sun, 16 Jun 2019 03:14:15 +0000

Why did you choose 2 for the pt in the L2 part? What if you chose 1.5 or something?

Eli B on Hosting a Flask App on WebFaction on a Non-root Domain

Thu, 02 May 2019 03:14:15 +0000

Thank god, finally one that works.

davidlowrydud on An intuitive introduction to calculus

Mon, 24 Sep 2018 03:14:15 +0000

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 $k$).