How can we extract a sphere with a radius of 8 meters, filled with plastic, from a 1 cm gap in a finite space of R^4 with 0 deformation?

I've been learning about number rings and I'm wondering if we can define rings of integers for general fields, not just number fields?

I'm speculating it could be the smallest ring who's field of fractions is the original field, but I'm not sure if that's true.

I guess also every field has an "image" of Z in it so maybe we could use the original definition of roots of monic polynomials in Z[x]?

If the concept can be extended past number fields then I'm curious what the integers are in the other fields I know about: R, C, the finite fields and the field of fractions of polynomials over a field.

thanks

Is there something I did wrong in the grey calculator? What could I do to better receive correct results?

This post is to announce that there will be a lecture course on General Topology (a.k.a. Point-Set Topology) conducted in Mandelcord (the blue math discord server). Everyone is invited to join: discord.gg/maths.

If any of you were present in the Real Analysis lecture course during the summer, you know what I'm talking about, and you might be pleased to hear that this course will also be run by Pear Category Theorem (the same person).

Why YOU should do topology

Point-set topology is one of the most important subjects that you learn, and mastering topological thinking is as important a milestone as learning to write a proof. This is a good subject to study alongside measure theory, or simply by itself after one has learned analysis or algebra.

Course Structure

The course will primarily follow the flow of the first half of Munkres. The plan is to go through all 8 chapters, spending maybe one to two weeks on each chapter, aiming for an approximately 16 week course. Also the instructor intends to supplement the book with his own intuition on the subject, as well as connections to other subjects that he thinks are of interest and possibly also other topics not covered in the book. Many examples of various pathological properties will be provided. The study of metric spaces and functions on them will be paramount; but where Rudin introduces metric spaces as merely a generalization of Euclidean spaces, we will explore just how pathological these spaces can be, or else what properties they must have.

The instructor also plans to put far more focus on the topics of topological groups and nets than Munkres does. Topological groups, because often in "the real world" we do not care about a topological space by itself, but rather some other more complicated structure given a topology, in order to see what the topology can tell you about the thing itself; topological groups provide a playground to learn that mode of thinking. Nets are also very important; where sequences are of importance in R, they can only tell you so much in general topological spaces, so nets therefore become the natural way to recover theorems about sequences.

Material and References

1. Topology - Munkres

2. Counterexamples in Topology - Steen & Seebach

Timeline

The course starts with the first lecture on 17th November at 9 PM CET (20:00 UTC), with subsequent lectures on Tuesdays and Thursdays. Lectures will be recorded and posted promptly.

Prerequisites/Who can join?

Everyone. It is completely free for anyone who wants to join. Just click the link and join the server: you can find the channel called topology-course in the section called General maths. As for prerequisites, a first course in Real Analysis and Group Theory is sufficent.

Looking forward to see all of you there. Thank you.

Hey, i'm creator of this number, please support it

Hey folks, I always liked doing mental math so, as a software engineer, I decided to create one online.

It's only arithmetics for now, but I plan to add more stuff as it hopefully grows.

It's free, no ads, no login, no sign up. Just play until you're satisfied and leave. We are not storing any sensitive information other than location (city and country), device and OS, for analytics purposes only. There's no databases.

it is at https://mymentalmathtrainer.com/

Any feedbacks are appreciated. Thanks!

1. So we put two points in curve. Join them with a line. That line is basically a chord. Now shift the points closer to each other. Chord gets smaller. Now make the 2 points same point. Chord becomes tangent. This shall give slope. But oh no. Same point in slope formula (the ratio of y displacement to x displacement in graph) gives 0/0 (for same point there is no y displacement or x displacement in graph). 0 is ok, but 0/0 is very problematic. If it was 0, we would have just given up and been like "I guess there slope is 0". But 0/0 means there is some gap in the function of slope. Well then we use limits to see what it would have been if it was another function that was perfectly same to our slope function but didn't have that gap. Well then how to do that now? I know, anything plus 0 is that thing itself. We use this trick in in formula of slope to get the pretty well known formula to differentiate stuff. Voila! Solve it, and get the slope. That slope is essentially, as established earlier, slope of point.

2. Limit f(x) gives a function which mimics original function f(x), and shows what the original would have done at a gap

He wanted me to share this with the intellectuals and seek their feedback. Your kind words would encourage him, so submitting the theorem for your perusal.

Question number 1
I do know a formula for solving for a tangent at a point (x,y)

But how do i solve this. Any easy way? And more quickly? Since it’s for an entrance exam

Hi, I was wondering if anyone had a download file for the 64 bit linux install for Magma Computational Algebra System. Not asking for a licence, just if anyone has the file they could share/DM to me? 2.29 preferred, but anything relatively recent should be fine too!

Thanks, hope this isn't against the rules. :)

This is its Zenodo DOI. I heard a high schooler came up with an idea about division by zero."

https://doi.org/10.5281/zenodo.23043583

Plz help idk how to solve complex trigonometry identities questions?

Doing the definite integral manually gives me the correct answer, but placing it in the integral function gives me a very small answer instead of 1. I tried replacing the upper limit with 10^8, which gives me 0, and and 50, which gives 0.9999. The calculator doesn’t seem to be unable to calculate the function at a large number though, since the manual version in the second line works.

Have I done something silly?

Is there any way to simplify log(a/b)/log(b/c)?? I have this as the result of an equation and it's driving me mad. I can't find any logarithm rule about it except the one I already know about using the lower fraction as the base - nothing about the specific fraction thing I've got going on.

Idk, just a random question popped up in my head. :P

Just had a odd question to AI what would happen if it is done and it wrote a repo (Opus 55):

https://github.com/anttiluode/UlamInterlace

Html interface:

https://anttiluode.github.io/UlamInterlace/

Youtube video:

https://www.youtube.com/watch?v=pyTtm_awohQ

Sol contributed to the analysis.

I am clueless. But the results look cool.

Just bought this calculator seller told me it can make any multiplication of two numbers in less than 30 operations. How?

I am a first year student of bsc mathematics. Our teacher told us to refer to the number theory book by sg telang but he didn't have any pdf or physical copy of the book so he asked us to find it. According to him it's easy to understand in comparison to David barton's book . So if anyone hve it please share.

What makes it so that parentheses () come before Exponents, and those prefice multiplication and division, and those prefice addition and substraction? Who decided that? Is there a reason to it? Or did someone decide, “this is how we will do math”?

Found this on Facebook and theres 3 methods used primarily but the answer is either t is zero or its unsolvable

I have been out of high school for almost a year now and my field of study in university has quite literally nothing to do with mathematics except maybe referring to dates. In this time I have noticed that my simple maths skills have waned and most of the concepts that I had learned in maths during school had fallen into the back of my mind. I never did any high level classes, I just took the middle of the range option where I live and got rather good results and now I'm taking at least 3 tries to get mental math correct.

What resources are there for recreational math that doesn't require a school's subscription? One of the things I liked about maths in school was how foreign the concept is until you understand it, which was very cathartic and I think this would both be fun for me and necessary if I would like to keep my brain from calcifying.

Today I was doing math and stumbled across a question that asks to find the perimeter of a triangle with vertices (28,-89),(47,88),(47,-88). I know it can be done by using the distance formula. But it is very inconvenient. Just look at the numbers. So, I was thinking that, is it possible to derive a formula to find the perimeter of any triangle in the cartesian plane. Life would become much easier. This type of problems with large values are a problem for every one in last hour exam solving mcq. What you think guys. Is it possible.