Existence of bi-measurable function between two measurable sets

Lets say A and B are two lebesgue measurable subsets of the real line, both of positive measure. Is it always true that there is bi-measurable function f mapping A to B? If so, how do we prove that?

By bi-measurable I mean that there are measurable subsets A’ and B’ such that AA’ and BB’ are null sets and f restricted to A’ is a bijective function from A’ to B’ such that the function and its inverse are both measurable.

What about a general measure space? It’s easy to construct a counterexample using a discrete space with counting measure. However is there anything we can say about when a general measure space does have the given property and when not? What if we additionally assume that A and B have the same measure?

If that makes a difference you may additionally assume at any point that A and B have finite measure.

submitted by Nevin Manimala Nevin Manimala /u/whatkindofred

Parametrisation for “Generic potato shaped” simply/multiply connected bounded domain in R2

I am trying to produce a “generic potato shaped” simply/multiply connected bounded domain in R2 (e.g. https://en.wikipedia.org/wiki/Simply_connected_space#/media/File:Runge_theorem.svg) with Tikz and pgfplots. The generic potato shape has, despite its name, a quite specific shape, which I fail to reproduce. Does anyone know a parametrisation of the potato shape (or whichever lump one prefers)?

submitted by Nevin Manimala Nevin Manimala /u/Raibyo

On Algebraic Closure of a finite field

This is a la question I have made myself one year ago. I know that the Algebraic Closure of a finite field is denumerable; but, do we know something more about it?

(My question grew up again like a week ago, I found a Paper about the Algebraic closure of F ((t)), of formal power series field (they construct it from the formal power series ring using indexes and powers of Q).

submitted by Nevin Manimala Nevin Manimala /u/gwindorito

“Illustrating Group Theory” – A (free) coloring book about math

Hi All,

For the past year and a half I’ve been developing this “book” about group theory to benefit math educators and students.

This is a coloring book about math that is both digital and on paper, and publicly available.

On paper: http://coloring-book.co/book.pdf

Online: http://coloring-book.co

The mathematical concepts it presents show themselves in illustrations that can be colored on paper or animated and regenerated by interacting with them on the web version. Throughout the book there are thought challenges and coloring challenges to further engage the reader in puzzling over the content.

The book is about symmetry. It uses group theory as the mathematical foundation to discuss its content while heavily relying on visuals to communicate the concepts.

Group theory and other mathematical studies of symmetry are traditionally covered in college level or higher courses. This is unfortunate because these are the most exciting parts of mathematics and they can be introduced with language that is visual, and with words that avoid jargon. Such an introduction is the intention of this “book”.

I hope this can serve as one more resource in the classroom or at home.

submitted by Nevin Manimala Nevin Manimala /u/alexandraberke

Math problem: How to calculate the theoretical cost of theft prevention?

I’m having a debate with a friend abut the value of theft prevention. The question stands that if you were to catch 4 thieves per month using the below figures what would your expected value of theft prevention be?

Problem details: Once 3 items are stolen we consider this a ‘case’. We compile 4 cases per month. Once a case is created we are allowed to ban that thief from the store for 24 more months. Items sell for \$10 each thus a case totals \$30. Mark up on items is 25%.

What we agree upon:

Each item less the markup comes to a cost of \$7.50 and 3 items to a case values the cost of a case at \$22.50 (\$7.50*3). Stopping a case thief and having them repay the lose for that case means you’ve prevented the loss of \$22.50. Then banning that thief and assuming if they were not caught that their thieving trend stayed consistent we valued the bans value at \$540 (\$22.50 * 24). The total expected value at this point would be \$562.5 (\$22.50 + \$540).

The part we don’t agree on:

The value of recuperating the cost of the stolen items. The proposition is that for every item stolen 3 items are needed to sell in order to recuperate the cost of those stolen items (3 items as an item less markup costs \$7.50 and selling 3 items with 25% markup recovers \$7.50). Any items that are sold that weren’t used to recuperate loss would otherwise be profit. In my opinion if the theft WASN’T prevented then you sold 3 items to recuperate you would break even at \$0 and if you did prevent the theft then you would be up \$7.50 from the sale of the 3 items. In either case you are gaining \$7.50 it just depends on where you start being either down by \$7.50 because an item was stolen or you are at \$0 because you prevented the theft. My oppositions opinion is that this ‘resale’ value can be added to the total expected value of a prevent theft.

If anyone has any insight it would be greatly appreciated. Possibly we are both wrong and both miscalculated 😀 Please prove either of us wrong!!

submitted by Nevin Manimala Nevin Manimala /u/ItsNotAGoodTime

Arnold’s proof of Kepler’s laws

I am currently reading Arnold’s “Huygens and Barrow, Newton and Hooke”. In his appendix 1 he discusses a proof that orbits are elliptic. This is not homework, and I am not trying to learn something, so I hope this is the correct place to ask. I think this is an insightful approach (I hear you say “duh, Arnold chose it”).

https://i.imgur.com/tdxeGsM.png

It seems to be missing a logical step in the proof of lemma 3. He says “when complex numbers are squared, the Joukowsky ellipse goes into a Joukowsky ellipse shifted by 2” and as proof simply expresses the formula for w2 as function of z2. But I think he should have mentioned that z2 + 1/z2 is a Joukowsky ellipse, different from the one he just constructed z + 1/z. I am mainly concerned to present this cleanly, without introducing confusion by insisting on a trivial point.

submitted by Nevin Manimala Nevin Manimala /u/humanino

Given the correlation between shares of all n(n−1)/2 pairs among n different companies. How to achieve maximum diversification?

I have the historical data about the stock prices of the companies and can easily calculate the historical correlation between any two of them.

Given this information, how should I go about finding the position sizes that will provide me the maximum diversification?

Example: Consider three companies, out of which two of them are real estate companies and the third one is an automobile company. The correlation between the two real estate companies is higher[as found from the data]. My intuition tells me that I should give less than (1/3) weight to each of the real estate companies, in order to achieve higher diversification.

submitted by Nevin Manimala Nevin Manimala /u/biodegradable19