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Since we know that we amplified the 16S regions which we know the rough size of (there might be a bit of variability across species, but not that much), we were expecting to see a big band at that size.

And that is exactly what we saw!

First we had to prepare the gel, put the gel comb, and pipette the samples into wells present in the gel:

To see the DNA, we added ethidium bromide to the samples, which is a substance that that both binds to DNA and is fluorescent.

Because it interacts heavily with DNA, ethidium bromide is a mutagen, and the biology people sure did treat the dedicated electrophoresis bench area with respect! Figure 4. "Gel electrophoresis dedicated bench area to prevent ethidium bromide contamination.".

The UV transilluminator we used to shoot UV light into the gel was the Fisher Scientific UVP LM-26E Benchtop 2UV Transilluminator. The fluorescent substance then emitted a light we can see.

As barely seen at Figure 8. "Fischer Scientific UVP LM-26E Benchtop 2UV Transilluminator illuminated gel." due to bad photo quality due to lack of light, there is one strong green line, which compared to the ladder matches our expected 16S length. What we saw it with the naked eyes was very clear however.

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Young's interference experiment by

Ciro Santilli 40 Updated 2025-07-16

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Not every

x

belongs to the elliptic curve over a non quadratically closed field by

Ciro Santilli 40 Updated 2025-07-16

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One major difference between the elliptic curve over a finite field or the elliptic curve over the rational numbers the elliptic curve over the real numbers is that not every possible

x

generates a member of the curve.

This is because on the Equation "Definition of the elliptic curves" we see that given an

x

, we calculate

x^{3} + a x + b

, which always produces an element

y^{2}

But then we are not necessarily able to find an

y

for the

y^{2}

, because not all fields are not quadratically closed fields.

For example: with

a = 1

and

b = 1

, taking

x = 1

gives:

y^{2} = 1^{3} + 1 \times 1 + 1 = 3

(1)

and therefore there is no

y \in Q

that satisfies the equation. So

x = 1

is not on the curve if we consider this elliptic curve over the rational numbers.

That

x

would also not belong to Elliptic curve over the finite field

F_{4}

, because doing everything

mod 4

we have:

0 * 0 1 * 1 2 * 2 3 * 3 = 0 = 1 = 4 = 9 = 0 = 1 mod 4 mod 4 mod 4 mod 4

(2)

Therefore, there is no element

y \in F_{4}

such that

y \times y = 2

y \times y = 3

, i.e.

2

and

3

don't have a multiplicative inverse.

For the real numbers, it would work however, because the real numbers are a quadratically closed field, and

3 \in R

For this reason, it is not necessarily trivial to determine the number of elements of an elliptic curve.

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Lamp by

Ciro Santilli 40 Updated 2025-07-16

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Elliptic curve over the rational numbers by

Ciro Santilli 40 Updated 2025-07-16

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Edward Teller by

Ciro Santilli 40 Updated 2025-07-16

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www.youtube.com/playlist?list=PLVV0r6CmEsFw1phnddYWXtVkRW8eUVlqx Edward Teller interview by Web of Stories (1996) Date shown at: www.webofstories.com/play/edward.teller/1. Listener: John H. Nuckolls

Video 1.

Witnessing the test explosion Edward Teller interview by Web of Stories (1996)

Source.

Video 2.

Edward Teller, An Early Time

. Source. Comissioned by the Los Alamos National Laboratory in 1979. Producer: Mario Balibreraa.

youtu.be/Goim-4MF_uE?t=338; holy fuck he almost cut his foot off on a stupid tram accident!
youtu.be/Goim-4MF_uE?t=457: he plays the piano
youtu.be/Goim-4MF_uE?t=965: he drove Szilard to propose to Einstein the Einstein-Szilard letter

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Largest known ranks of an elliptic curve over the rational numbers by

Ciro Santilli 40 Updated 2025-07-16

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web.math.pmf.unizg.hr/~duje/tors/rankhist.html gives a list with Elkies (2006) on top with:

y^{2} + x y + y = x^{3} - x^{2} - 20067762415575526585033208209338542750930230312178956502 x + 34481611795030556467032985690390720374855944359319180361266008296291939448732243429

(1)

TODO why this non standard formulation?

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Reduction of an elliptic curve over the rational numbers to an elliptic curve over a finite field mod p by

Ciro Santilli 40 Updated 2025-07-16

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This construction takes as input:

and it produces an elliptic curve over a finite field of order

p

as output.

The constructions is used in the Birch and Swinnerton-Dyer conjecture.

To do it, we just convert the coefficients

a

and

b

from the Equation "Definition of the elliptic curves" from rational numbers to elements of the finite field.

For example, suppose we have

a = 3/4

and we are using

p = 11

For the denominator

4

, we just use the multiplicative inverse, e.g. supposing we have

\frac{3}{4} \to 3 \times 4^{- 1} mod 11 = 3 \times 3 mod 11 = 9 mod 11

(1)

where

4^{- 1} = 3 mod 11

because

4 \times 3 = 1 mod 11

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 Pinned article: Introduction to the OurBigBook Project

Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.

Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!

Video 1.

Intro to OurBigBook

. Source.

We have two killer features:

topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculus
Articles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1.
Screenshot of the "Derivative" topic page
. View it live at: ourbigbook.com/go/topic/derivative
Video 2.
OurBigBook Web topics demo
. Source.
local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
Figure 2.
You can publish local OurBigBook lightweight markup files to either https://OurBigBook.com or as a static website
.
Figure 3.
Visual Studio Code extension installation
.
Figure 4.
Visual Studio Code extension tree navigation
.
Figure 5.
Web editor
. You can also edit articles on the Web editor without installing anything locally.
Video 3.
Edit locally and publish demo
. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.
Video 4.
OurBigBook Visual Studio Code extension editing and navigation demo
. Source.
Internal cross file references done right:
Infinitely deep tables of contents:
Figure 6.
Dynamic article tree with infinitely deep table of contents
.
Live URL: ourbigbook.com/cirosantilli/chordate
Descendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade

All our software is open source and hosted at: github.com/ourbigbook/ourbigbook

Further documentation can be found at: docs.ourbigbook.com

Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact

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