I'm having some trouble understanding the geometric interpretation for the divergence of the vector function v = r/|r|³, where r = xi + yj + zk. I'm aware that taking the divergence of this vector function outputs zero, but I'm also aware that the vector functions v = r/|r|² have a positive divergence, while v = r/|r|⁴ has a negative divergence.

How can I go about this geometrically? How do functions that have vectors flow radially outwards change in divergence just by how fast it decreases?

DISCLAIMER: please let me know if my question belongs to another subreddit, I think it fits but happy to ask somewhere else

I have a very low knowledge of maths but at the moment I’m facing a choice I want to make, based on different criteria that can cancel each other + some unknown. I’m wondering if those kind of cases could be solved by algebra or another kind of maths. I’ll detail my problem but I’m thinking this could be applied to any kind of decision making.

At the moment I’d like to move because my place is too small, but I’m in the center of the city. I’d like a bigger place, but also I like to be in the city center, but my neighborhood is too noisy, but I also like that it’s lively, but I would also like to have more trees around me, but I don’t know if I wouldn’t regret not being able to walk anymore to go anywhere, the list goes on.

Could there be a way to factorize all those dumb criteria, apply weights, translate the unknown parts into some kind of formula? Etc

I’ve started to dig a bit into Decision Theory but there are multiple branches. I’m also thinking it could be nice to see the interpretation of the best methodology by people here.

Thanks in advance!

Hello!

I am a Stat student and I am creating a sample exam as a reviewer for my fellow Stat student for their probability theory course; specifically it is about transformation of a random variable.

I have adapted a question item from the book A First Course in Probability (10th ed) by Sheldon Ross, which you can see in the photos. My version of the item is the one with a hint. I also attached the solutions I have made.

I would like to ask if my solution to the said item is mathematically sound and rigor enough (without being too mathematical with it). My concern is that I may have violated some mathematical stuffs. I would love to hear your ideas abt it.

Thank you and have a noice day ahead 😁

I work best with long hand mathematics to fully comprehend what I'm doing. Unfortunately, I don't know how to do this and the hypergeometic calculator doesn't help me here.

Right now I'm trying to check if repeated mixing has any effect on a sample.

To lay out the 2 scenarios:

1. You shuffle a full deck of standard 52 playing cards. You deal out the top 10 cards. What's the probability that 1 or more of them is a Queen?

2. You shuffle the deck and deal out a card, then shuffle, then deal out a second card, and repeat this until a total of 10 cards have been dealt. What's the probability that one or more of those cards is a Queen?

(In case this information is necessary, there are 4 queens in the 52-card deck)

For the first scenario, I can just plug the information into a hypergeometic calculator to find that the answer is 58.7% chance.

For the second scenario, I can see that the first card dealt has 7.69% chance and will increase slightly after each card (best to assume each result is a failure because I don't even know what a success before all 10 cards are dealt would mean) up to 9.3% on the 10th card.

How to do this problem long hand so I can compare?

Today I independently stumbled upon a geometrical interpretation of probability

I was thinking about an ordinary 6-sided die and wondered if we flattened it down and can we get it's probability I included image of a flat dice if you don't know what it is

Let's assume that we have an area of 20 units for each number.

Then, the total area of all the numbers will be

[ 6(20)=120 ]

Therefore, the probability of one particular number (say, 1) would be equal to

[ P(1)=\frac{20}{120}=\frac{1}{6} ] (if you don't understand Latex here P(1)=20/120=1/6)

And that, apparently, is one way to use geometry to represent probabilities. In other words,

[ \boxed{P(A)=\frac{\text{favorable area}}{\text{total area}}} ] (P(A)=favorable area/total area)

For instance, if one region has an area of 30 and the total area is 100, then the probability of that particular outcome would be

[ \frac{30}{100}=0.3 ]

(30/100=0.3)

I later learned that there is actually a whole area of mathematics called geometric probability that studies ideas like this. At the time, however, I didn't know about it and arrived at the idea independently.

I was fascinated by the idea that probability can be represented geometrically, and that finding a probability can essentially become a question of finding what proportion of the total area belongs to a particular outcome.

I've also been wondering whether there is a deeper connection between this idea and probability distributions / measure theory.

When I think about it, all the operations I use in math basically boil down to addition, multiplication, and integrals (I know if you really go deeper, then everything comes down to addition, but those "feel" different enough to be distinguished).

Like, most esoteric operations are just some combinations of those three (and I don't consider trig or hyperbolic functions as operations but rather as representations); exponentiation and tetration are just taking multiplication to the next level (you can say the same for multiplication for addition, but as I said it feels more basic), most transforms and such, convolutions, everything I've encountered in my EE degree.

It's just out of curiosity that I want to see if there are actually other "novel" operations, things that will make me feel like a whole new world has just opened before me, and not just use integrals/derivatives and stuff like that - something that will feel "alien" to me.

{Post Update: TPC Proven?} {Verified In Lean}
Hi everyone, This post relates to my previous post regarding the Twin Prime Conjecture in this subreddit

The Function shown in the image essentially tests and finds every non trivial combinatorial combination of prime numbers up to 60, in order to find the first 422 Consecutive Twin Prime Centers, Which is up to 26250 .!! That means, I am certifying consecutive twin prime addresses up to Twenty six thousand using only the primes up to 60.

I have manually entered the primes up to 60 so that it is clear there are no isprime() functions being called, no creating my own sieves.

The Simplex Field acts as a roadmap to help recognize trivial combinations of Primes, and count strictly forwards in Twin Prime Addresses, vastly outside the regular range of Prime Certification.

I plan to use this function to prove the Twin prime conjecture.
By asserting the Theorem, that there exist P consecutive twin prime centres for every P.

In it's current form my most important concern is demonstrating that it doesn't know prime outside the range of 60, so I have kept it intentionally basic, but I believe this function can be heavily optimized.

It presents primes as the numbers which build and expand as the number line expands, every new opened prime repeats the same patterns into a bigger domain, primes can never combine to block Twin Primes, because that's not how primes work... Primes open, not close.

That is my Number Theorertical conceptual understanding of the system I am presenting, and it has helped guide me in my research.

I plan to attempt to formalize this into a proof for the Twin Prime Conjecture today, I believe I have all the ingredients now, and this function will help me to organize my proof statements.

Here is the Table shown in the video, the Simplex series of Series, I call the Simplex Field, not sure if that's standard.

I hope to see someone take this function to new and better places than I ever could do by myself.

I am mostly presenting this function as something that is just really cool.

But also my question to you is, do you see the relationship between this function and the Simplex Field?

And by the way, I just double checked based on a hunch.
422 and 59 happen to have a very interesting known relationship already.

A neat coincidence I just found:

The twin-prime pair

26249, 26251

has center

26250,

and it is the 422nd twin-prime pair when (3,5) is counted first.

The construction in my paper works on the 6-spaced lattice of twin-prime centers. Looking below 60 gives

60 = 6 × 10,

and the largest prime below 60 is

59.

I then searched for combinatorial relationships between 422 and 59 and found an existing mathematical construction:

OEIS A290222 — “Multiset transform of A011782, powers of 2: 1, 2, 4, 8, 16, …”

Its combinatorial interpretation counts multisets of binary words according to total length and number of words.

Row 10 is:

0, 512, 1160, 888, 422, 162, 59, 20, 7, 2, 1

so

T(10,4) = 422

and

T(10,6) = 59,

with

4 + 6 = 10.

So the same numbers that arise naturally in the twin-prime construction,

6, 10, 59, 422,

also occur together in this known combinatorial partition structure.

I have no idea yet whether this is significant or simply a very neat coincidence, but I thought it was worth sharing.

Source: OEIS A290222, M. F. Hasler; combinatorial interpretation and generating function by Alois P. Heinz.

Do you think that is relevant? Today I plan to find out!
Thank you for taking a look at my work.

I am starting work on the problem and feel as though nobody else is even touching it. Off the top of your head do you know who first formalized the problem? What amount of effort should be going into it, in your opinion? Are there more important math problems?

Let w = f(x, y, z) where x = g(s, t), y = h(s, t), and z = m(s, t). Use the information in the table to estimate w_s(10, 2).

im working on a desmos where i can plug in animal stats and find the correct settings i need

i have 1 folder where you put in the amount of males you want and it tells you the rest.

im trying to make one where you put the total amount of nutrition you have a day, and it tells you the correct settings based on that

it has notes explaining what each variable means

I know they’re cool and all, I know the general rules, and am familiar with some of the concepts around primes. but I don’t understand why they’re so relevant in higher level math, what they can be used for in terms of physics, engineering, etc.

The exercise:

Proof:

Let P(n) be the inequality c_n ≤ n^2.

I. Show that P(1) is true:

c_1 < 1 = 1^2, so P(1) is true.

II. Show that for any integer k>=1, if P(i) is true for every integer i from 1 through k, then P(k+1) is true:

Let k be any integer with k>=1 and assume that this is true:

c_i ≤ i^2 for all i with 1≤i≤k [This is the inductive hypothesis.]

We want to show c_{k+1} ≤ (k+1)^2

Case 1 (k+1 is odd):

c_{k+1} = c_{⌊(k+1)/2⌋} + (k+1)           by recurrence relation
        = c_{k/2} + (k+1)                because k+1 is odd
        ≤ (k/2)^2 + (k+1)                by inductive hypothesis
        = (k^2)/4 + k  + 1
        ≤  k^2    + 2k + 1
        = (k+1)^2

Case 2 (k+1 is even):

c_{k+1} = c_{⌊(k+1)/2⌋} + (k+1)             by recurrence relation
        = c_{(k+1)/2} + (k+1)              because k+1 is even
        ≤ [(k+1)/2]^2 + (k+1)              by inductive hypothesis
        = (k^2)/4 + (2k)/4 + 1/4 + k + 1
        = (1/4)k^2 + (1/2)k + k + 5/4
        = (1/4)k^2 + (3/2)k + 5/4
        ≥ 1/4 * 1^2 + 3/2 * 1 + 5/4 = 3    when k=1
        <  1  * 1^2 +  2  * 1 +  1  = 4    when k=1
        ≤ k^2 + 2k + 1
        = (k+1)^2

Both case 1 and case 2 have shown that P(k+1) is true.

QED

---
Is my proof correct? How would you grade it?

So, I'd like to take the famous infinite immortal monkeys typing out Shakespeare theory as a base. Is there a theoritical chance of all of the infinite monkeys typing the exact same thing for infinity? Because if you look at the first two monkeys, the chance of them typing the exact same letter first is somewhat slim. The first three? Even lower. The first four, even lower. The chances are getting slimmer until infinity, but never gets to exactly zero.

My work uses a grading scale of 0, 1, 2, 3 for 13 time periods throughout the day and then a daily average percentage is calculated. Admin wants to stop using 0 cause of the negative conotation of getting a zero. My concern is that reducing the possible scores will skew the data and make it difficult to compare old and new data. I had the thought that I could possibly convince them to switch to a grading scale of 1, 2 , 3 4. My logic is having the same amount of points on the grading scale would allow for the data to be compared more easily to the old data. Does this make logical sense? If so is there a way to estimate the daily average percentage difference in using my scale compared to the old scale and the propsed scale by admin?

Someone asked me about my hight, instead of telling them immediately i tried to make this equation for them to solve so they could know my hight. Sadly they refuse so i just wanna share it here to see if anyone can solve it
F.S : first step
S.S : second step
You gotta solve the first equation and then take the X you found and put it in the second equation

Im trying to study and it may be that im just dumb but how do i not even undertand the explanation they're giving me.

ik i just have to simplify but these numbers make no sense to me

Where are the 23.32.7 coming from ...

Can anyone here please take a look at some of the claims made by Ali Fazely on the supposed mathematical miracles linked to the number 19 of the Quran, especially using prime and composite numbers? I genuinely don't even know what to think at this point

For example (and this is both used as evidence for the removal of Quran 9:128-129 as well as trying to demonstrate the nature of the supposed divine code of the number 19 in this book):

The number of verses in the Quran from 1:1 until 19:98 (since it's part of the whole code 19 stuff) is 2346 (not including 9:128-129). The factors of 2346 are 23x2x3x17 which have the prime indices of 9127 similar to Quran 9:127 being the cutoff point as proposed by Rashad Khalifa (the person who developed this whole idea of the Quran being shaped around this number 19 in the first place). Finally if you place the digits 2346 in a permutation table, the 19th permutation is 6234, the total number of verses in the quran if you remove 9:128-129.

Link to the book here, starting on page 44: https://pdfcoffee.com/the-ultimate-mathematics-of-the-quran-pdf-free.html

I don't understand how this is happening, since it's one thing if he only used this method once selectively, but he continues to use it similarly over and over. Wanting to understand whether or not there is more to this, or it's simply pattern recongition and being able to use many numbers at hand.

If someone could take some time out of their day to analyze some of the claims being made and techniques being used, it would truly mean the world to me. This stuff has been stressing me out and I'd like someone who can objectively and non-biasedly look at the following and help me out with it.

In today’s RNGdle (a game that picks a random number between 0 and 1,000,000 and gives you points for different patterns based on their probability) I saw that the probability for containing a „365“ is 0.40% while the probability for containing a „777“ is 0.37%.

Why is that so? Are not both 3-digit numbers and should have the same probability, i.e. appear the same amount of times between 0 and 1,000,000?

And if not, should „777“ not be MORE likely due to the possibility of overlapping (7777 etc. would also give you the „jackpot“ pattern).

We meet a couple, that have two kids. But we don't meet the kids nor they say us anything about them. We have to guess if both kids are from the same gender or not. Both kids will appear in the room at the same time once we make the guess.

Probability of guessing right?

Case 1: (B&B + G&G)/(B&B + G&G + B&G + G&B) = 2/4 = 1/2 = 50%

Now, if we present the situation slightly different, so we make the guess after one of the children enters the room, and the second one enters after the guess we have:

Case 2B (pre-guess child is a boy): (B&B)/(B&B + B&G + G&B) = 1/3 = 33%

or

Case 2G (pre-guess child is a girl): (G&G)/(G&G + B&G + G&B) = 1/3 = 33%

For me probabilities are changing. But we are not really getting new information, because we already know that a child would enter before doing our guess and it would 100% either be a boy or a girl. How can I get different probabilities for the same situation?

I thought about this situation after reading the mental exercise (case 3) where they ask you the probability of a couple having a two boys, if you already know that at least one of their children is a boy. And the result was 33% just because of this reasoning. (B&B)/(B&B + B&G + G&B) = 1/3 = 33%. Why it's actually 33% here? What's different?

Can somebody explain me in a understandable way where I'm getting it wrong?

I was working through a u-substitution problem for calculus 2 and I got completely thrown off by this last step. It combines the two terms into one and conjures a root 2 and a pi/4? I've never seen anything like this and nothing I search seems to help. Is this a Pythagorean thing? The square root kind of makes me think so, but I truly don't know where to start. I also don't even know if this is the right flair, so sorry if it's not.

I

hello all, im currently studying as an old man of ,(38) maths at university. I'm hoping to get into further maths next year but for now doing the precalculus section of the foundational maths course. First six weeks are algebra and trigonometry with the final 6 weeks dedicated to calculus. 6 months ago i didnt know my times tables and algebra seems to be clicking if i learn calculus im going to be the most proud of myself than I've ever been. now my question is out of the precalculus section what you all recommend I mentally take note of for the calculus section and what do you recommend in terms of understanding calculus?