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MIT PRIMES/Art of Problem Solving

CROWDMATH 2018: Neural Codes

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Topic
First Poster
Last Poster
i Welcome to CrowdMath
CrowdMath   6
N Dec 29, 2018 by JGeneson
CrowdMath is an open project that gives all high school students the opportunity to collaborate on a large research project with top-tier research mentors and an exceptional peer group. MIT PRIMES and Art of Problem Solving are working together to create a place for students to experience research mathematics and discover ideas that did not exist before.

Below we've tried to unpack and explain the things you will find on the CrowdMath page:

Resources and Problems

The problems in the CrowdMath project are open, unsolved problems in mathematics. We will be discovering new truths that were unknown before. The problems will become available in early 2018. Until then, we expect participants to be studying the readings in our Resources section of the CrowdMath page.

The Resources for this project are background ideas that you will need to understand and make progress on the CrowdMath problems. We've chosen resources that are directly relevant to the project: the problems are defined explicitly in terms of ideas that you will find in our resources. All of the problems we will pose are thematically linked and all of the resources we post will be as well. We will be releasing the resources roughly in the order that you should be reading them.

Each resource also has exercises to help clarify the key ideas and give practice with them. You can discuss the exercises by clicking either the "View Discussions" or "Start New Topic" button.

Eligibility

CrowdMath is designed to give very well-prepared high school and college students (as of 1/1/18) experience with math research. Very advanced middle school students are also welcome to participate. We know that the problems will be interesting to a broader range of people, but we want to create a specific opportunity for the upcoming generation of math and science researchers.

Be polite and constructive

This rule is simple, but important. The goal here is to learn to collaborate. Be nice!

Make your comments as easy to understand as possible

Polymath is a conversation. Assume that many people will be reading anything you write. Take a little time to make sure you write as clearly as possible and all of your collaborators will appreciate it.


Mentors

We have plenty of people watching and ready to help out when needed. However, we also know that there are many mathematicians out there who will find the CrowdMath project interesting and will want to help out. If you'd like to take part, send us a note at crowdmath@aops.com.

Dissemination of results and intellectual property

Polymath projects are inherently massively collaborative. Done correctly, it should be impossible to determine the lines between one person's work and the rest of the group. As such, we agree that the results created must be attributed to all CrowdMath contributors.
6 replies
CrowdMath
Dec 28, 2017
JGeneson
Dec 29, 2018
polynomial
pritam01   0
Jun 2, 2020
What I've tried so far:

<Describe what you have tried so far here. That way, we can do a better job helping you!>

Where I'm stuck:

<Describe what's confusing you, or what your question is here!>
0 replies
pritam01
Jun 2, 2020
0 replies
AP order series
Smoothe   1
N Nov 23, 2019 by lilcritters
Well I just found out an amazing result but cannot prove it.
Claim-The series $1^{n}$, $2^{n}$, $3^{n}$,......., $n^{n}$ is an $n-th$ order $AP$.
Please try to prove this.
Any help will be appreciated.

Smoothe

1 reply
Smoothe
Nov 19, 2019
lilcritters
Nov 23, 2019
crowdmath 2019 project on formalization
JGeneson   4
N Jul 18, 2019 by isar
Hello CrowdMath participants,

I would like your opinion about one of the projects for CrowdMath 2019.

Anthony Bordg has proposed a CrowdMath project on formalization using the Isabelle proof assistant. The purpose of the project will be to learn how to use Isabelle and to convert some math proofs so that Isabelle can verify them.

We would like to know what math proofs you are interested to formalize with Isabelle. One possibility is the first CrowdMath paper (which was published this year in EJOC: http://www.combinatorics.org/ojs/index.php/eljc/article/view/v25i1p5).

Are there other papers or proofs that you would be interested to formalize with Isabelle?
Please post any ideas here. For more information, see Anthony's original post below:

[quote=isar]Hi,

my name is Anthony Bordg, I'm a postdoc researcher at the University of Cambridge (UK). I would like to suggest an idea for a new CrowdMath project in 2019. This project will use the Isabelle proof assistant developed at Cambridge to formalize some fun mathematics. You can have a look at the Isabelle https://isabelle.in.tum.de/ webpage for further information.
What I have in mind is something similar to a previous collaborative project involving undergraduate students in Germany mentored by a few researchers. They formalized the so-called DPRM theorem using Isabelle, and their work resulted in the following publication https://easychair.org/publications/preprint/GhvC.
The formalization of mathematics has made nice progress over the last few decades, and now a substantial chunk of the undergraduate math curriculum has been formalized (see https://isabelle.in.tum.de/dist/library/HOL/index.html and https://www.isa-afp.org/). We might soon have a complete formalization of the Mathematical Tripos.
Be aware that formalized mathematical proofs should not be mistaken for automated proofs. The Isabelle proof assistant only checks that man-made proofs written with it are correct. Hence, writing mathematics with a proof assistant remains a highly creative endeavor.
I will try to remove as many barriers to participation as possible for those of you eager to take part in that project. We have tutorials and manuals for Isabelle, and there is a mailing list where anyone can ask for help. Moreover, I will be available on this forum to answer questions, and I will provide exercises to get started. Isabelle is very easy to install, easy to use, and it comes with a very nice level of automation. We should use GitHub https://github.com/ to share code.
At this stage I'm open to ideas of formalizations, so please let me know your favorite theorem/theory you would like to formalize. Jesse Geneson suggested that we could formalize the CrowMath paper published this year called Bounds on Parameters of Minimally Nonlinear Patterns http://www.combinatorics.org/ojs/index.php/eljc/article/view/v25i1p5 . This is certainly a nice possibility. The selected project will depend on its feasibility, the background available in the core Isabelle/HOL library and in the Archive of Formal Proofs, and its interest and originality.[/quote]
4 replies
JGeneson
Nov 25, 2018
isar
Jul 18, 2019
What is this? -A Fourth Grader
eagleup   4
N Apr 25, 2019 by AnimalLover111
I am a fourth grader so I have no idea how to do this thing.
4 replies
eagleup
Mar 2, 2019
AnimalLover111
Apr 25, 2019
More Cushing-Pascoe problems
DCushing   24
N Jan 4, 2019 by SW1
It’s been great to see all this work so I thought I’d post a few extra problems for people to try:

Let P_k be the set Of k-powerful numbers.

1) Enunerate P_3 as a_1<a_2<a_3 ...

Show that lim inf |a_{n+1}-a_{n}| = infinity

2) Let P be a polynomial with integer coefficients and at least 3 simple roots. Is it true that P(n) is
powerful only finitely often?

3) Pell’s Equation can be used to generate infinitely many pairs of powerful numbers. Can this or something similar be used to generate infinitely many pairs of powerful numbers inside a coprime arithmetic progression

I have a few more problems that I can post in due course









24 replies
DCushing
Jan 12, 2018
SW1
Jan 4, 2019
Major Breakthrough (from #of intervals to #of codewords - one dimensional)
lilcritters   12
N Nov 23, 2018 by haha0201
I hope this has not been posted by anyone else...because then I'd be violating rules of CrowdMath...


First, define $P\prime$ to be the number of codewords possible with $P$ endpoints. For example, $2\prime=2 $ as in the diagram (assuming the segment is open, not closed):

IMAGE

giving the neural code [0,1].
However, $2\prime$ could also equal 3, as in this diagram (still assuming the endpoints are open, not closed):

IMAGE

giving the neural code [10, 00, 01]. Thus, $2\prime$=2 and 3. I found out that
$$  P\prime = \begin{cases} [\frac{P+1}{2}+1,P+2]\in \mathbb{N} & \text{if } P \equiv 1 (mod 2) \\ [\frac{P}{2},P+2]\in \mathbb{N} & \text{if } P \equiv 0 (mod 2) \end{cases}  $$for all $P$>0$\in\mathbb{N}$.

*$\mathbb{N}=1,2,3, \cdots$
12 replies
lilcritters
Jun 15, 2018
haha0201
Nov 23, 2018
2019 PRIMES
Mathlete2017   6
N Nov 23, 2018 by haha0201
This might be a bit early, but does anyone have an idea as to the topic of 2019 PRIMES (I want to make sure I actually understand the topic before it starts).
6 replies
Mathlete2017
Aug 19, 2018
haha0201
Nov 23, 2018
A corrolary to the abc-conjecture
SCP   1
N Nov 7, 2018 by JGeneson
This is just a fun fact taken from my master thesis, related to Problem $4$ which is formulated very rough (and so this is not giving "the" solution).

Analogously to the original congruent numbers (see e.g. https://en.wikipedia.org/wiki/Congruent_number), one can define a 3D version, being the cubefree part of the volume of a right triangular pyramid with (6) integer side lengths.
In the 2D case, one can get an infinite family (even all) integer solutions by Pythagorean triples.
Similarly, in the 3D case, one has the infinite family generated by Sounderson's formula (see generating formula at https://en.wikipedia.org/wiki/Euler_brick )

In the normal setting, any squarefree number which is congruent, is the area of infinitely many primitive right triangles with integer side lengths.
As a corrolary of the abc-conjecture, this is not the case for 3D congruent numbers when restricting to the solutions generated by Sounderson's formula.



1 reply
SCP
Sep 12, 2018
JGeneson
Nov 7, 2018
PQ tree algorithm
A1234   1
N Nov 7, 2018 by JGeneson
This paper references the "PQ tree algorithm." I looked this up and have found information about what PQ trees are. However, I haven't been able to find a clear explanation of what the algorithm does and how it solves Theorem 1.6 in this paper. Can someone please explain the PQ tree algorithm or link to a paper or article that explains it?
1 reply
A1234
Jul 9, 2018
JGeneson
Nov 7, 2018
Rules for 1d codes
A1234   15
N Jul 12, 2018 by lilcritters
I don't fully understand the rules for neural codes. Do 00, 000, 0000 etc. count as codewords? Is the space a line segment, or all of $\mathbb{R}$? If so, can the place fields go to $\pm\infty$ (for example, is a place field at $[0, \infty)$ allowed)? Can place fields be single points?

Thanks in advance,
A1234
15 replies
A1234
Jan 5, 2018
lilcritters
Jul 12, 2018
More Cushing-Pascoe problems
DCushing   24
N Jan 4, 2019 by SW1
It’s been great to see all this work so I thought I’d post a few extra problems for people to try:

Let P_k be the set Of k-powerful numbers.

1) Enunerate P_3 as a_1<a_2<a_3 ...

Show that lim inf |a_{n+1}-a_{n}| = infinity

2) Let P be a polynomial with integer coefficients and at least 3 simple roots. Is it true that P(n) is
powerful only finitely often?

3) Pell’s Equation can be used to generate infinitely many pairs of powerful numbers. Can this or something similar be used to generate infinitely many pairs of powerful numbers inside a coprime arithmetic progression

I have a few more problems that I can post in due course









24 replies
DCushing
Jan 12, 2018
SW1
Jan 4, 2019
a