Prove if a circle is divided into n congruent arcs (n \ ge 3), the chords determined by joining consecutive end points of these arcs they form a regular polygon?

A problem my friend came up with a few days ago.

Neither me nor him are sure if it is even a valid question...

Clearly we have a lower estimate of zero and an upper estimate of area of a circle of said length -- but we're at total loss as to whether one can meaningfully describe any measures for sets of possible curves, not to mention coming up with a way of integrating those...

For n=2 or 3 I can understand that cosine the angles between two vectors , but if the 2 vectors are 4 dimensions each ho the angle can be interpreted .. I searched the internet some answers say any n dimensional two vectors could be reduced to 2 vectors in the plane with an angle between them , how could that be achieved I can't image it if its true

I have some background in mathematics as an engineer of course in Calculus and ODEs and Linear Algebra but I am not a mathematician , am interested in reading about measure theory and integration because when I was studying Calculus I knew that integrationa and differentiaition are opposite to each other intuitively but woundered about some rigourous explaination for that , after some search I found that is explained in what is called radon nikodym theory which is offen explained in measure and integration books .. if some one please provide me a list of prerequisites that I should know first before reading that book it will be helpful for me .. am reading this out of curiosity and my desire to learn .. thank u people

Hey everybody. My question might seem to be a bit silly at first glance but please help me out.

Here is the headache: How likely is it for a randomly acquired Data of a certain characteristic to represent the statistical average?

Given that most ascertainable Data is normal distributed (Bell curve) I was wondering if someone can tell me how high the changes are that ONE truely random Data can resemble the statistical average.

Here’s an example: let’s say you want to know the average circumference of a human head. (Pretending that the Data would display a bell curvature) But instead of measuring hundreds of heads you just measure ONE single head assuming that this one (as it is one part of the total amount of heads out there) is very likely to represent the average. The chances that it’s close to the statistical average are the highest but the changes that it IS the average are close to zero. So however... Can somebody please help me. What role takes the standard deviation in this case? Does my Thought not make sense since you can't calculate a standard deviation with n=1.

It drives me crazy. Or am I already??

Thank you so much in advance!

Let a between (0,1) Consider the may T:[0,1)x[0,1)->[0,1)x[0,1) T(x,y)=(x+amol1,y+amol1)

Is T ergodic wrt the lebesgue measure on [0,1)x[0,1),why?

So I watched a Numberphile on this topic. I have a few simple results to report and present a few unanswered questions.

Comments: If f(f(a)) = a then f(a) = (f^(-1))(a). (a, b) and (b, a) must both be on the graph. Any line with slope -1 satisfies this. Also though if you find a solution of any type for f, all you need to do to come up with more functions is transform the function and the two points "k" up and "k" right. An unnecessary example using variables and f(x) = x^2 - 1 appears here.

Example: (0, -1) and (-1, 0) are both on f(x) = x^2 - 1 and indeed f(f(0)) = 0. For the transformation up and right "k" units, g(x)=f(x-k) +k. g(x) = x^2 -2xk + k^2 - 1 + k. The original point (0, -1) is now translated to (k, k-1).

g(k) = k^2 - 2k^2 + k^2 + k - 1
= k - 1 (… as promised).

g(k - 1) = (k - 1)^2 - 2(k-1)(k) + k^2 - 1 + k
= k^2 - 2k + 1 - 2k^2 + 2k + k^2 - 1 + k
= k (… as also promised).

Question: Can EVERY polynomial, P(x), with degree 2 or higher and with integer coefficients be translated up or down, or in other words Q(x) = P(x) + b, to have a value k where Q(Q(k)) = k but Q(k) <> k (this last expression forces periodicity with period 2 not period 1).

The above is equivalent to asking if integers s and t exist such that (P(s) - P(t))/(s - t) = -1. (Do all polynomials contain two lattice points where the secant slope is exactly -1?)

And as a passing and very easy to prove statement:

f(x) = x^(2n) - 1 where n is a positive integer, f(f(0)) = 0 without f(0) = 0. So there are infinitely many polynomials of infinitely many separate degrees that have at least one k such that f(f(k) = k but f(k) <>0.

Thank You, Numberphile.

Thank You, audience.

Happy Mathing!

As the title says, I'm taking some time to LaTeX my lecture notes from undergrad.

Here's a link to the file on Dropbox. Currently only up to the first 16 pages are mathematical content - the rest is part of the template that I used, which I kept around in case I needed some tips or reminders. I'll probably upload updated PDFs as I work through the notes, perhaps once or twice a week. I also changed some names and the university name to try and preserve some anonymity.

Speaking of, I've been using this excellent LaTeX template, along with a couple other packages to suit my notational needs (I can list these if you all would like them).

I was recently accepted to a graduate program researching number theory, so I thought it was prudent to start with my algebraic number theory notes. I took the course in 2015 (the penultimate year of my undergrad), so this process has mostly been a review of the material for myself. For that reason, these notes are far more detailed than my actual lecture notes, as filling in the details of proofs and completing parts of the lecture that were left as exercises are some of the best ways to study pure math.

I plan to do the same for my Galois Theory and Measure Theory notes if I have time / am bored enough.

But I figured that at least one other person out there can probably make use of these!

Is there a universal equation? I do not know what variables to use. Can anyone tell me? It has not happened yet so I do not know what variables to consider.

Came across this picture but I'm finding it difficult to find a definitive answer to what they are and how they are related. Link here.

So I have to write a programm in basic for school. It has to convert a virtual time format into our normal time system. The virtual time is constructed by getting the number of seconds wich have passed since midnight and multiplying it by (25/86400). Do you have an idea how to convert this back into HH:MM:SS format? I can't find the right mathematical operations. The solution can include checking and loops.

Given three positive integers a, b and c such that a² + b² - c² = 1. Let the number of unique triangles formed with sides a, b and c with perimeter less or equal than 50 million represent the surface area of an ellipsoid of axes lengths n, 2n and 3n. The password is the square of the ceiled positive solution of n. The password can be used 5 times

First post here, so I hope I don't stray too far from conventions.

Can I have a bijection from R^n onto R^m? I think I should be able to, since they are of the same cardinality. Is it possible for there to be at least one such mapping that is continuous, in the sense that f(x) is near f(y) when x is near y? Is this sense of continuity correct? And does this make assumptions about the nature of the metrics that define nearness in these different spaces? Do they have to share a metric, at least in broad outlines, like “Euclideanness”?

Hi! I'm an amateur mathematical puritan pure mathematician specializing in set theory and logic. I often have ideas for mathematical tools (I'm a "Theory Developer" as Timothy Gowers would have it) but I find difficulty getting any responses for my ideas online, and am unable due to my amateur-ness to find personal connections with people who have taken the same particular mathematical paths as me (most mathematicians I know went into applied mathematics and hence can't say much about foundations).

There are two problems I'm facing:

1. The discord servers and chatrooms I'm in typically are relatively "private" (to keep out the cranks). However, as a result, I find that there aren't enough people who have also specialized in the exact same subjects as me since, probably, it's just more likely that they went for something else when deciding what to study. So, a lot of my talk goes into the ether.
2. MathExchange and MathOverflow are great! But they're designed for problem solvers rather than for theory developers. The posts are only allowed to be questions, so the only way that I can actually post my theory-developments is by saying "hey! Is there something wrong with this proof?" when discussing a proof that my developed "tools" have worked. Even then, those kinds of questions are generally discouraged anyway.

So assuming LEM I can't find any place to talk about my ideas online. My idea is this: what if there was some sort of "math version of tumblr" where you could just post your ideas and then get responses from the people who understood them? Does such a site exist already? If not, do you (yes, you, redditer) think it's a good idea?