MixedMath - Recent Commentshttps://davidlowryduda.comDavid's personal blog.en-usCopyright David Lowry-Duda (2022) - All Rights Reserved.admin@davidlowryduda.comadmin@davidlowryduda.comWed, 13 Nov 2024 14:44:42 +0000Wed, 13 Nov 2024 14:44:42 +0000mixedmathapp/generate_rss.py v0.1https://cyber.harvard.edu/rss/rss.htmlhttps://davidlowryduda.com/static/images/favicon-32x32.pngMixedMathhttps://davidlowryduda.comDLD on Murmurations in Maass formshttps://davidlowryduda.com/maass-murmurations<p>As part of the <a href="https://cmsa.fas.harvard.edu/event/mml2024/">program in Mathematics and Machine
Learning</a> at Harvard's CMSA, we
looked further at what machine learning (both unsupervised and supervised)
techniques have to say about Maass forms, murmurations style.</p>
<p>Despite the data being transcendental and approximate, the machines can say a
lot! We're going to have a preliminary talk about this during the closing
workshop of the program in the coming week. (These talks are recorded —
you can see a preview of our preprint in Edgar Costa's talk, and a preview of a
different preprint in my talk).</p>f4eb1a4ad81a134f79770b6828531a8cFri, 25 Oct 2024 03:14:15 +0000David Lowry-Duda on Technical report on machine learning experiments for the Möbius functionhttps://davidlowryduda.com/ml-mobius-technical<p>All the code to run the experiments is here or in the Int2Int github repository
I linked to. It's not as hard to run as it might look!</p>
<p>I don't know what you mean about primes. But if you set up an experiment along
these lines, I'd be interested to hear about it.</p>75f5f47d9d2f996a73694a1c25aae1b1Thu, 24 Oct 2024 03:14:15 +0000SH on Technical report on machine learning experiments for the Möbius functionhttps://davidlowryduda.com/ml-mobius-technical<p>Can you share the GPU code? Can you predict primes?</p>a38630d0c7a06f04fb75c92a2fcf7fc4Wed, 23 Oct 2024 03:14:15 +0000DLD on Explicit equations for cubic surfaceshttps://davidlowryduda.com/explicit-cubic-surfaces<p>Excellent observation! At first I thought a lot was lost by choosing the points
to be $6$ points in $\mathbb{P}^2(\mathbb{F}_q)$ instead of consisting of
Galois orbits. But actually passing to (at most) sextic extensions and doing
the same analysis barely loses anything.</p>
<p>The analysis is similar.</p>
<p>This is an interesting heuristic. I wonder: can something like this be made
rigorous?</p>77e5c66bed05c72bd2e556e70928ba7aThu, 26 Sep 2024 03:14:15 +0000anon on Explicit equations for cubic surfaceshttps://davidlowryduda.com/explicit-cubic-surfaces<p>Consider an analogous problem over $\mathbb{F}_q$.
We first count the number of ''cubic surfaces'' here, i.e. homogeneous cubic
$4$-variable polynomials over $\mathbb{F}/\mathbb{F}^\times$.</p>
<p>There are $q^{20} - 1$ homogenous, four-variable cubic polynomials with
coefficients in $\mathbb{F}$. The $20$ is a ''stars and bars game'' analysis.
There are $3$ powers that we insert into $4$ regions, which can be done in ${4 + 3 - 1 \choose 3} = 20$ ways.
Accounting for $\mathbb{F}^\times$ invariance, there are $M := (q^{20} - 1)/(q - 1) > q^{19}$ equivalence classes.</p>
<p>Now suppose we want to count the number of choices of any $6$ <strong>rational</strong> points.
The number of points in $\mathbb{P}^2(\mathbb{F}_q)$ is $n := (q^3 - 1)/(q-1) = q^2 + q + 1$.
There are up to $N := {n \choose 6} \lesssim q^{12}$ many choices of $6$
points to construct blowups.</p>
<p>Thus the probability that a random rational cubic surfaces comes "from blowing
up $6$ points" in $\mathbb{P}^2(\mathbb{F}_q)$ is $< q^{-7}$, quite small.</p>ddf528243979cb3bb3a4bed71fbb4586Thu, 26 Sep 2024 03:14:15 +0000DLD on Murmurations in Maass formshttps://davidlowryduda.com/maass-murmurations<p>We've now proved a version of these murmurations. See <a href="https://arxiv.org/abs/2409.00765">this paper on the
arxiv</a> for more.</p>
<p>There is a miraculus aspect: the underlying murmuration function is the same as
the function for <a href="https://arxiv.org/abs/2310.07746">modular murmurations in the weight
aspect</a>. I have no satisfying explanation for
this.</p>cb97a9186abbbf200534d406ea3d33e5Thu, 05 Sep 2024 03:14:15 +0000DLD on Maass forms in the LMFDBhttps://davidlowryduda.com/maass-forms-now-in-lmfdb<blockquote>
<p>I experiment with mirroring some commentary from Mastodon and here. This is
from <a href="https://mathstodon.xyz/@davidlowryduda">@davidlowryduda@mathstodon</a>.</p>
</blockquote>
<p>Rigorous Maass forms are now on the (beta) of the LMFDB! Go check them out.</p>
<p>https://beta.lmfdb.org/ModularForm/GL2/Q/Maass/</p>4dcdc591fd742dfa47082f34230a8640Wed, 05 Jun 2024 03:14:15 +0000DLD on Another year, another TeXLive reinstallationhttps://davidlowryduda.com/another-year-another-texlive<blockquote>
<p>I experiment with mirroring some commentary from Mastodon and here. This is
from <a href="https://mathstodon.xyz/@davidlowryduda">@davidlowryduda@mathstodon</a>.</p>
</blockquote>
<p>Once a year, sometime shortly after March, I try to update something with <code>tlmgr</code> (the texlive manager) and it flails about. It tells me that I'm silly for not noticing that TeXLive has advanced another year, and that I should get with the times and update too.</p>
<p>Then I spend 30 minutes updating my texlive.</p>
<p>And then <em>afterwards</em> I can install the random latex package I needed.</p>
<p>Today is that day for me. And the package was extdash.</p>
<p>I do essentially the same steps every year. This year I put these steps in a public place.</p>c82a50f5c5ceff302f8ea93e78f79111Mon, 15 Apr 2024 03:14:15 +0000DLD on FLT3 at LftCM2024https://davidlowryduda.com/flt3-at-lftcm2024<p>Thank you to Edward van de Meent and Riccardo Brasca for comments and pointing
out errors in earlier versions of this post.</p>81694b0804ac4a835f7465ef1a180c52Tue, 02 Apr 2024 03:14:15 +0000DLD on FLT3 at LftCM2024https://davidlowryduda.com/flt3-at-lftcm2024<blockquote>
<p>I experiment with mirroring some commentary from Mastodon and here. This is
from <a href="https://mathstodon.xyz/@davidlowryduda">@davidlowryduda@mathstodon</a>.</p>
</blockquote>
<p>I attended Lean for the Curious Mathematician 2024 at CIRM.
Along with Riccardo Brasca, Sanyam Gupta, Omar Haddad, Lorenzo Luccioli, Pietro
Monticone, Alexis Saurin, and Florent Schaffhauser, we proved the $n=3$ case of
Fermat's Last Theorem in a <em>good way</em> in Lean.</p>
<p>I wrote a bit more about this experience.</p>4707fb71daeb6011c3bfb73ed27beebeTue, 02 Apr 2024 03:14:15 +0000DLD on Zeros of Dirichlet Series IVhttps://davidlowryduda.com/zeros-of-dirichlet-series-iv<p>Thank you! I've fixed the typo.</p>
<p>I haven't published these yet. I'm going to try to put this in a publishable
form in the next couple of months.</p>c92ce204cb6bf35b76b1d5a2072cf5b8Wed, 06 Mar 2024 03:14:15 +0000DLD on Quanta on Murmurationshttps://davidlowryduda.com/quanta-on-murmurations<blockquote>
<p>I experiment with mirroring some commentary from Mastodon and here. This is
from <a href="https://mathstodon.xyz/@davidlowryduda">@davidlowryduda@mathstodon</a>.</p>
</blockquote>
<p>Quanta wrote an article on murmurations that mentions the work of my
collaborators Bober, Booker, Lee, and I.</p>
<p>We're not the focus, but we're there - and that's pretty cool. Check it out.</p>
<p>I hope to announce another paper in the same direction in the next couple of
weeks, too.</p>d83c5e80417edd57aa712f488c7fcf27Tue, 05 Mar 2024 03:14:15 +0000Chris on Zeros of Dirichlet Series IVhttps://davidlowryduda.com/zeros-of-dirichlet-series-iv<p>Have you published this yet? There is a typo in the second display equation. It
should have a $\Lambda$ instead of a $\lambda$.</p>92cf0eaf79ad6553f40d4435930dfd71Tue, 05 Mar 2024 03:14:15 +0000PJ on Paper: Congruent number triangles with the same hypotenusehttps://davidlowryduda.com/paper-congruent-triangles-same-hypotenuse<p>@RB can't find more. You already found them all</p>33e5fc6e8f9c1af973871e399c59f88eThu, 15 Feb 2024 03:14:15 +0000DLD on Paper: Towards a Classification of Isolated $j$-invariantshttps://davidlowryduda.com/paper-isolated-j-invariants<p>I'll give a small update as a comment. Our paper has been accepted and will
appear in <em>Mathematics of Computation</em> shortly. (And <em>Math Comp</em> was very fast
and forthright with their review, which is nice).</p>0e93676137e85e8832a8c81d8376ed06Wed, 14 Feb 2024 03:14:15 +0000David Lowry-Duda on Examining Excess in the Schmidt Boundhttps://davidlowryduda.com/schmidt-experiment<p>I will also note that I hadn't expected this to be so cut and dry, and I ran a
separate experiment that sampled randomly from the space of polynomials and
went in that direction.</p>
<p>The setup and conclusion are more complicated. But the conclusion is the same.
So I chose to not document or include it.</p>949e5daee523c8d6ec9629f717261c7bTue, 13 Feb 2024 03:14:15 +0000David Lowry-Duda on Examining Excess in the Schmidt Boundhttps://davidlowryduda.com/schmidt-experiment<blockquote>
<p>I experiment with mirroring some commentary from Mastodon and here. This
is from <a href="https://mathstodon.xyz/deck/@davidlowryduda">@davidlowryduda@mathstodon.xyz</a>.</p>
</blockquote>
<p>I study excess in the Schmidt bound for number fields. Schmidt estimated that
there are at most $X^{\frac{n + 2}{4}}$ number fields of degree $n$ with
discriminant up to $X$. But we expect the real number is really of the form
$cX$ for some constant $c$.</p>
<p>To do this, Schmidt gives a region of polynomials to count, where some
polynomial is guaranteed to generate each number field of discriminant up to
$X$. There are two sorts of error there: counting polynomials that yield number
fields with discriminant larger than $X$, and sets of polynomials that all
count the same number field.</p>
<p>Which error is larger?</p>
<p>For heuristic reasons, we should expect counting
polynomials that are <em>too large</em> to be a major source of overcounting. So
perhaps a meaningful question would be: does counting the same number field
multiple times account for much of the error at all?</p>
<p>In this note I describe an experiment and show that no, repeated counting isn't
a significant factor.</p>0f8f1e84fb2cf4729e26fe916679675eTue, 13 Feb 2024 03:14:15 +0000DLD on Plaintext Emailhttps://davidlowryduda.com/plaintext-email<p>I use K-9 on mobile, and a variety of clients on various machines.</p>
<p>The mobile app for gmail cannot send plaintext. By default, the web interface
doesn't — but it can.</p>
<p>Plaintext is worse for marketing. You can't hide links or track image loading
for additional analytics, so it is maybe no surprise that ad-based companies
default to HTML email.</p>64ac535f751e6f3e65e3da05a5ac2843Wed, 24 Jan 2024 03:14:15 +0000gmailer on Plaintext Emailhttps://davidlowryduda.com/plaintext-email<p>What email do you use? gmail doesn't use plaintext?</p>0a22be4f946a5dbc1fa16140cf478e32Tue, 23 Jan 2024 03:14:15 +0000David Lowry-Duda on Bounds on partial sums from functional equationshttps://davidlowryduda.com/bounds-on-partial-sums-from-fe<blockquote>
<p>I experiment with mirroring some commentary from Mastodon and here. This
is from <a href="https://mathstodon.xyz/deck/@davidlowryduda">@davidlowryduda@mathstodon.xyz</a>.</p>
</blockquote>
<p>In my first public note of the year, I comment on applying Landau's method to
partial sums of full and half integral weight modular forms. This is where one
uses a combinations of smoothed sums $\sum_{n \leq X} a(n) (n - X)^k$ to find
reasonable bounds for the sharp sum $\sum_{n \leq X} a(n)$.</p>
<p>Implicitly, this note compares what one can do from my paper with Thorne and
Taniguchi against my sequence of papers with Hulse, Kuan, and Walker — and
identifies where there is hope to get better results in the future.</p>
<p>For additional context: I've been interested in improving these bounds since 2016.
I've now studied the primary bound of $\sum_{n \leq X} a(n)$ in several different ways,
and they all produce the same bound. This hints at some <em>true barrier</em> to our
understanding, but I don't actually understand what this is.</p>44a445214ad21d2d2289c3b9f9797fd5Thu, 18 Jan 2024 03:14:15 +0000DLD on Comments on this sitehttps://davidlowryduda.com/comments-v4<p>Yes, your message came through just fine. For a typical message, it's easy
to extract plain messages from email. If someone were to try to send formatted
content inside an HTML email, that might be a problem.</p>
<p>I use a couple of different email programs depending on context. See
<a href="https://useplaintext.email/">useplaintext.email</a> for more precise software
recommendations that I also generically support.</p>44d28008f4a5d030b2a8297036c162ccWed, 15 Nov 2023 03:14:15 +0000Sarah on Comments on this sitehttps://davidlowryduda.com/comments-v4<p>What email do you use? Gmail doesn't support plain email? Did this comment
go through?</p>7cc64a4ddfbb2ee336c2a5f6ec36f9dbMon, 13 Nov 2023 03:14:15 +0000DLD on Paper: Congruent number triangles with the same hypotenusehttps://davidlowryduda.com/paper-congruent-triangles-same-hypotenuse<p>Good luck! I'll note that the sparseness means that it is <em>extremely unlikely</em>
that a naive search, even very efficiently coded, would beat the approach
guided by elliptic curves I describe in the paper. But I would be very
interested to know a counterexample (if it exists) or a proof (if it exists) to
my conjecture.</p>f10bd640bbb49277eafebbe041a0ddc2Wed, 01 Nov 2023 03:14:15 +0000RB on Paper: Congruent number triangles with the same hypotenusehttps://davidlowryduda.com/paper-congruent-triangles-same-hypotenuse<p>Ah, I missed "primitive" so I will go back to my GP-Pari routines, since I have
an efficient way of finding square pairs summing to a third square.</p>4321d77bb11625f59f4191b333be9a8cWed, 01 Nov 2023 03:14:15 +0000DLD on Paper: Congruent number triangles with the same hypotenusehttps://davidlowryduda.com/paper-congruent-triangles-same-hypotenuse<p>Thank you for your comment RB. The conjecture refers to <em>primitive</em> triangles
for exactly this reason. The description above now includes your example in the
description.</p>8493da67b7967e22b20c7aafa5584217Wed, 01 Nov 2023 03:14:15 +0000