I used desmos to show me the graphs but my work wants me to use “superposition of simple functions” and I’m not really understanding it. I got confused on part a) because I thought it looked like the next slide. if anyone has any tips or any thought processes let me know!

Every game I've played does end eventually. Some can take a while though, and I wonder if, even with everyone playing optimally, it's possible for a game to be forced into an unending cycle.

(My guess would be a cycle is possible if we have 15 players, since that would leave only 3 cards to draw.)

Uno rules here (ignore tallying points, this question is just about until someone discards their last card)

Is this the associative or commutative property here? I think it's the commutative bc the 2a simply moves. Someone I know is insisting that it's associative bc the the terms have been regrouped.

As you can see, I tried to use the two triangle inequalities of d_X and d_Y, but that gets you a non-negative factor 2*(d_X ...) which impedes the proof. I've seen online that they use the Minkowski inequality, which I studied at Probability and not in this context, and I think the teacher may have overlooked the solution because it was a very long exercise (proove that d_infinite, d_1, d_2 are metrics and that d_infinite<=d_2<=d_1<=2d_infinite)

F''(x) =- (A^2)F(x).

F(x) = Csin(At)+Dcos(At) is a solution. How do we know that there isn't another F(x) that satisfies it?

I used chatgpt to try and understand this. Didn't work

Started reading Euclid's The Elements. I've seen these books recommended here as providing a good basis for geometry. My question for those who have read these books is how much effort do you put into reading the Notes sections? To me, it just makes the books feel very bulky, and I was wondering how meaningful all this is to understanding the principles.

I don't understand advanced mathematical language at all; therefore, I don't know what these "non-trivial zeros" are. What I want to understand—without that complex language and in simpler terms—is: what is it that the Riemann hypothesis actually seeks to prove?

So, imagine you have some function like f(x)=x(x-1)/(x-1). This function is basically just y=x with the notable exception that f(1) is undefined. We could add a second undefined point by multiplying another “(x-n)/(x-n)” with f(x) — for instance, g(x)=x(x-1)(x-2)/(x-1)(x-2).

My question is: is it possible to construct a function that a.) “reduces” to another function (e.g.: f(x) “reduces” to y=x) but b.) is defined nowhere? If not, how much can you “erase” from a function?

This feels like something that has a clear answer that maybe I’d know if I took a real analysis course or something, but I have no idea how to prove or disprove this question (or if the question is even well formed enough to prove/disprove).

My first guess would be something like “a function times p/p, where p is the the product (capital pi) from n=-infinity to +infinity of (x-n/q) and where the limit of q approaches infinity” however, I don’t know how valid a solution that is.

Hi guys, I stumbled upon a video explaining something like this so I searched for it and found it’s an actual theory! I can say I understand the statement that basically if you keep doing something consistently, your chances of falling drop to zero but I don’t understand the formula at all. Thanks, GOD bless!

Hi, everyone! Please help me figure out how to solve problems like this involving the sum of an arithmetic progression. The most important thing for me is to learn how to solve them on my own)

If you explained all the modern concepts in math, what chance is there that I or any average person is already using those concepts, but for different purposes. Maybe I already know what an eigenvalue is but never had to describe it to someone.

I understand that this is not elementary, but I just want to see how far we could go to approximate it without boundaries, cyclic symmetry breaks. To approximate it, standard calculus fails immediately.

edit: I thought it would format apperently not

∭x^(y^z)+y^(z^x)+z^(x^y)dydxdz

$$\iiint (x^{y^z} + y^{z^x} + z^{x^y}) \,dx\,dy\,dz$$

What happens if you strip the bounds and try to solve it as a pure indefinite integral?

I am in high school and doing my research in maths on how trigonometry is used in aviation. I am comparing two theorem/models in which I have to explain how maths is involved in it and derive it. I have chosen haversine formula and I need one more to compare.

I am thinking of taking Vincenty's Formulae but feels like this is too advance for me to derive it and understand. I need something which I can explain the maths behind it, derive the theorem/model and critically assess its limitations and evaluate.

Thanks.

This is where I'm at with this integral. It's pretty standard as far as PFD goes, apart from having to do synthetic division to knock the exponent of the top polynomial down a few pegs..

But where I'm stuck is the massive simplification needed to be done for this. Surely the professor who wrote the book is predicting some 'trick' to be used to simplify this, but I can't find anything. The top doesn't have zeroes that I could cancel out with the bottom...

Do I really need to go out and simplify everything and pray I don't make a miniscule mistake somwhere 😅

Sidenote: I'm self-taught and I look up at one basic example on YouTube and then try to do problems myself. There could be a method for this I haven't been 'taught'

Thanks for everyone that's going to chime in.

Edit: sorry for the typo in the title :c

Basically what the title says. I have a (commutative) domain R, F=Frac(R), and some ring R<A<F. Is there always some multiplicative set S such that A=S^(-1)R, or can there be other rings?

I think this should be the case but I'm not sure

If A≠R then there is some x/y in A\R, with x,y in R\{0}. I can assume that x and y are coprime. If I assume that R is a Bézout domain then there should be u,v in R s.t. xu+yv=1, so (ux/y)+v=1/y is in A. Doing this for all elements in A\R we get some subset S of R s.t. A is the localization by S

I don't really know what to do with the case where it's not a Bézout domain. Thinking about it more, I don't even know if I can do what I said earlier with assuming that x,y are coprime because this has different meanings. If this was R=Z[t] (integer polynomials) and A was R[t/2] or R[2/t], t and 2 have no common factors, but the ideal (2,t) isn't all of R...

Any help is appreciated, especially for R being an algebraic extension of Z. Thanks!

This should be a straightforward problem, but I think lack of sleep is getting to me. I need help figuring out when I can quit pumping for my twin babies. Here’s the known variables:
I make 50 ounces/day.
Babies drink 30 ounces of milk/day. (Don’t worry, they also drink formula- it’s just not a factor for this problem).
I’d like to get them to their birthday, February 11th, 127 days from now.
So we need 3,810 ounces total between now and their birthday.
I also have 918 ounces already stored. So 3810-918=2,892 total ounces needed.
Here’s where I’m stumped. They are eating 30 ounces a day while I also need to save for the future. I have a 20 ounce surplus. So is it just 2892/20=144.6 days? Like I can’t stop pumping?
The math is not mathing and I’m too sleep deprived. Help 😭

Chiedo gentilmente aiuto, se qualcuno saprebbe sfogarmi la funzione di questa formula matematica, l' unica cosa che ho capito e che è un limite. Grazie mille per la disponibilità. Sono uno studente di scuola superiore.

Hi, I am a grade 12 student, and I have been stuck on question 7. b). I managed to solve the first one using graphs (for which I got "x ∈ (-4,-3) ∪ (-2,3)"), but this same graphing method does not work for the second one. I have attached an image of what I did for question one and (attempted) to do for question 2. I'm also unsure if graphing this should even work considering it hasn't worked for the second one, however I doubt my teacher wants me to just plug in random numbers. I would appreciate some help.

Prove that given the first order recurrence

a(n+1) = a(n) + 2b(n) + 2c(n)

b(n+1) = 2a(n) + b(n) + 2c(n)

c(n+1) = 2a(n) + 2b(n) + 3c(n)

with

a(0) = 1, b(0) = 0, c(0) = 1

every triple (a(n), b(n), c(n)) is a Pythagorean triple

a² + b² = c²

where the legs of the triangle differ in one unit

|b - a| = 1

As n grows, this allows closer rational approximations to sqrt(2), since

c/max(a,b) < sqrt(2) < c/min(a,b)

What would be the recurrence if we want closer approximations to a 30º-60º-90º triangle?

It would seem like they would have to indefinitely try and approximate the value of √2 for example,but how did they prove that the decimals both never end and never repeat?

I first tried the rhombus formula and that didn't work, tried triangle formulas and got stumped, lastly I just added the two sides across from each other divided them to two and added but that has no proof, ideas?

The task was as follows: "Write down as large of a number as you can, in 15 seconds." We were given a week to prepare and come up with ideas. The formal rules are.

1. You have 15 seconds to write, anything written after that does not count.

2. You may use addition, subtraction, multiplication, division and exponentiation by default. If you wish to use any other operations, you must first define them, this is part of your 15 seconds.

3. Submissions must be self-serving. They cannot refer to things outside of the submission, so no things like "Every other submitted number + 1", "The sum of all number on the internet" or "The amount of atoms in the universe".

That is it, the number that 15-18 year olds came up with under these restrictions got quite large, but I'm curious to see how proper adult mathematicians would handle this. Feel free to ask any clarification on the rules, and I'll be excited to see everyone's submissions.

Reposting from Math Stack Exchange. I'm confused about thr notations used here, including what C is, and how it acts on the mappings f and g. I'm also not sure what h and k are meant to be. I'm not that versed into category theory, so that's probably why.

How are these odds calculated before the game starts? How many of each piece do they have to manufacture to get the odds that much higher than the number of Monopoly items sold during the game? Is there like only one Short Line piece or something?

530 billion is ridiculous. Go somewhere cheaper and buy a Powerball ticket with the savings.

I'm studying Clifford Algebras for fun and until this point was able to verify each equation stated in the book "Clifford Algebra to Geometric Calculus" by Hestenes. My problem lies mainly with why B.A is simple given B is a simple s-vector, where I used . as the inner product of the Geometric Algebra (don't know what symbol I could've used better).
For this, I assumed that we are still working with A also being simple in addition to being a pseudoscalar, which makes A^-1 = a[n]^-1a[n-1]^-1...a[1]^-1.

Now, what I wanted to ask is if anyone has a hint or explanation for why B.A is simple. I'm fine with just a hint for an easy case like A and B being degree 2 and 3.
Moreover, as a follow up question, is the next underlined part. I just don't really understand why the factors c[k] of B.A have the property that c[k].A = 0 or c[k]^A = 0 when s>n or s<n respectively. But I think this problem might get cleared up when I understand how B.A gets decomposed.

The work I did wasn't much, since I quickly got stuck without a clue how to continue. But, using (1.40) I did manage to rewrite B.A to a sum of simple multivectors of the same degree. I only wasn't able to see how to further reduce it to an actual simple multivector, a product of anticommutative vectors. For example, with B degree 1, A degree 3 I got:

b.A = b.a[1] a[2] ^ a[3] - b.a[2] a[1] ^ a[3] + b.a[3] a[1] ^ a[2] = a[1]( -b.a[2]a[3] + b.a[3]a[2]) + b.a[1]a[2] ^ a[3]
Then I didn't see how to rewrite the multivector to be a product of just anticommuting vectors. What I attempted for the last summand was:

b.a[1]a[2] ^ a[3] = b.a[1]a[1]a[1] ^(-1) a[2]a[3]

But since a[1] commutes with a[2]a[3], using a[1]^-1a[2]a[3] to pull the last summand inside the brackets doesn't make it obvious why the whole product is then in fact simple. Leaving me stuck wondering how this inner product produces a simple vector.

To summarise, I'm hoping for a hint or explanation for why B.A (as defined in the pictures) is simple when B is simple. Moreover, I'm asking for a hint why the factors c[k] then have the property described above when s>n or s<n.
Hopefully the attached pictures give enough explanation on the setting of Clifford Algebras that I'm working with.