MixedMath - Recent Comments

davidlowryduda on Initial thoughts on visualizing number fields

Thu, 02 Mar 2023 03:14:15 +0000

At CIRM in Luminy, a great alternative idea was proposed. The idea is to replicate parts of Mumford's famous picture on $\mathrm{Spec}(\mathbb{Z}[x])$, but presumably focus on only a single horizontal line (which corresponds to a fixed number field). This would display splitting behavior of various primes with some sort of line plot.

I think this is worth checking out!

davidlowryduda on Slides from a talk at Concordia University

Wed, 01 Mar 2023 03:14:15 +0000

I mean compute an arbitrarily close approximation here. And I should note that there are a fixed set of Maass forms with eigenvalues in some range $\frac{1}{4} < \lambda < \lambda_2$, so there are a precise list of forms that I am trying to find close approximations to.

AA on Slides from a talk at Concordia University

Wed, 01 Mar 2023 03:14:15 +0000

What does it mean to compute something that is uncomputable?

DLD on On Mathstodon

Mon, 02 Jan 2023 03:14:15 +0000

@davidlowryduda@mathstodon.xyz.

PD on Making Plots of Modular Forms

Mon, 02 Jan 2023 03:14:15 +0000

Thanks for the quick reply and your code! I was working already with few libs.

DLD on Making Plots of Modular Forms

Mon, 02 Jan 2023 03:14:15 +0000

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mrita on On Mathstodon

Sun, 01 Jan 2023 03:14:15 +0000

What is your name on mastodon?

PD on Making Plots of Modular Forms

Sun, 01 Jan 2023 03:14:15 +0000

I see your plots here. Do you have an idea how to plot precisely this curve?

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).

No hurry in answering. This is a project I am doing in parallel to my life.

davidlowryduda on The gamma function, beta function, and duplication formula

Sun, 16 Oct 2022 03:14:15 +0000

Dear Nishant,

I believe this is called the duplication formula because it's almost a formula of the form $\Gamma(z)\Gamma(z)$, i.e. this is almost a formula fro two copies of $\Gamma(z)$. Maybe calling it the "near duplication formula" wasn't catchy enough.

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 The Gamma Function, which contains far more if you're interested.

Nishant on The gamma function, beta function, and duplication formula

Wed, 12 Oct 2022 03:14:15 +0000

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.

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?

davidlowryduda 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.

davidlowryduda 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!

davidlworyduda 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.