Inverse function of ArcTan1(y/x) = θ (Domain: (−∞,+∞), Range: (−π/2,π/2))
is Tan1(θ) = y/x (Domain: (−π/2,π/2), Range: (−∞,+∞)).

Inverse function of ArcTan2(x,y) = θ (Domain: (−∞,+∞), Range: (−π,π))
is Tan2(θ) = x,y (Domain: (−π,π), Range: (−∞,+∞)).

Please note, Tan1(θ) and Tan2(θ) are the same function but with different Ranges and Domains so I consider them different functions and name them differently.

My question are:

• Did I define Tan2(θ) = x,y correctly?
• Is it also a multivariable function like its inverse (i.e. ArcTan2(x,y))?
• Does it input one variable (i.e. θ) and output 2 variables (x,y)? If it does, what kind of function is that and how do you graph that?

Thank you!

There are 2 teams of 12 players playing 12 1-on-1 matches, with the captains of each team assigning their players to each match independently (without knowing the other captain's selections).

A) what are the odds of a particular player X from team 1 playing particular player Y from team 2?

and

B) what are the odds of X playing Y in the 6th match of the day?

A couple posts I saw recently here on reddit got me thinking of an idea I was messing around with in Python a while back. As I understand it, zʷ ∈ ℂ can have more than one answer (as in complex roots). It seems like there would be some area of math where these types of functions would be defined using sets containing complex numbers so that they could be analyzed as injective functions. I couldn't find anything on google, and my math skills aren't good enough to figure out the intricacies of it myself beyond the simple class I created in Python.

For more context: Iirc from what I was able to code at the time, a singleton set containing the complex number z could be raised to the power 1/n to obtain a set with n elements, but subsequent operations would make the set explode into larger nested sets. A lot of other operations would have been impossible since the outputs could end up with infinite elements (I'm guessing). On top of that, floating-point arithmetic made it kind of janky overall. Maybe a symbolic representation would have worked better, but I wouldn't dare try something that ambitious for the sake of mild curiosity.

As stated, the only motivation I could think of is turning complex powers/roots/logarithms into injective functions. Whether or not any legitimate mathematicians have done something similar, could there be other applications?

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https://preview.redd.it/3otyse1ifgrb1.png?width=927&format=png&auto=webp&s=c3045eb99efe3570278cf2513fb8739ecd14cde9

I was self-studying Real Analysis and came across a very interesting and difficult problem that I believe I solved but am not so sure. The question was to find a sequence {x_n} such that for any y that is a real number there exists a subsequence {x_ni} converging to y. My first instinct was to divide the naturals into 3 categories: If n mod(3) is congruent with 0 we create a mapping of increasing positive rationals, if n mod(3) is congruent with 1 we do the same with decreasing positive rationals, finally if n mod(3) is congruent with 2 we map n to 0. This way the subsequence consisting of only n such that n mod(3) is congruent with 2 converges to 0, if y is positive we use the density of rationals in ℝ to always find an n mod(3) congruent to 1 that maps to a rational between y and our previous term and if you choose your sequence properly such that the supremum of your sequence is y then we can obviously observe that our sequence converges to y, we apply then the same argument if y is negative. The obvious problem I came across is that by the same density of rationals in the reals our "mapping of naturals to increasing rationals" cannot exist as we write our function as F and choose F(1)=0 and seek our next F(2) but by the density of rationals in the reals we find that regardless of our choice of F(2) there exists a rational between 0 & F(2) which must be included in our mapping as the point is for it to be surjective and this then implies their exists some n>2 such that 0<F(n)<F(2) violating the claim that F is increasing throwing a wrench into our prior idea as now our prior idea of choosing some F(m)<F(n)<y where m<n to continue the sequence may not exist as m<n no longer promises F(m)<F(n). My second idea was to create a surjective but not injective mapping where given any rational q there exist infinite n satisfying F(n)=q, therefore we apply the same idea as before but this time we need not worry about there not being an m<n satisfying F(m)<F(n)<y as there are infinite F(n) equal to our desired rational and therefore as long as we order our mapping properly we can simply choose the first n satisftying m<n. My question is then can such a mapping exist? At first glance I thought not but then I realized it should be like mapping ℕ to ℕ^3 as we choose the first two components to be the numerator and denominator of the rational of our choice and the third component be what instance of our rational we're on, that is the first 1/2 is (1,2,1) then the second (1,2,2) etc. If I'm not mistaken ℕ has the same cardinality as ℕ^3 but I could be wrong and even then there may be some implicit contradiction with my application, for example I'm not entirely certain of the idea of a mapping where given some m and a desired q we may always find f(n)=q where m<n.

To restate my claim as this post is quite long. We design a mapping F of ℕ to ℕ^3 satisfying the criteria that for any combination of a, b, & m there exists n satisfying F(n)=(a,b,c) where c is any arbitrary natural number and m<n, we then write a second G mapping N^3 to Q where G(a,b,c)=a/b. Our sequence is then the composition of G(F(n)) we then select our y and create a monotone subsequence which has a supremum of y and this is possible as we then observe that as ℚ is dense in ℝ we can select any 𝜖 of the form1/(2^n) there will always be a rational q we can add to our sequence such that y-𝜖<q<y allowing y to be the supremum of our sequence (or infimum if you simply rearrange a few things) and by the way we defined F there is always an n>m such that G(F(n))=q allowing us to finally create a subsequence of our original sequence that converges to y. Conveniently unlike my original solution we need not select some odd modulus based formulation of our sequence as we've already accounted for 0 & negative numbers as our mapping of G(F(n)) includes mapping ℕ to negatives and 0.

I'm simply curious if this properly answers the question and also if there may be another (preferably more simple) solution to the question as my solution is quite ugly even in the event that it is correct. Also apologies for the notation being quite ugly, if there's any confusion just ask me and I'll try to explain.

I have watched multiple videos on int. by parts, int. by parts the second time, and I've gone through my textbook, and haven't had time to ask a tutor yet, so here I am... but I 100% cannot find an answer.

The "go to" example everyone seems to show is this:

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https://preview.redd.it/8acx5bj30grb1.jpg?width=2342&format=pjpg&auto=webp&s=e054fe297c1f13672ddb23cb724385862281e60e

Everyone showing this example always chooses u = e^(2x). (Green text in the image.) Fine.

But in the second round (red text), he chooses u = e^(2x) again.

I have found that sometimes, especially in more complicated examples, I will — at this point — switch my u and dv choices... and everything goes to hell.

Is there an unspoken rule that if you pick a u-sub in the first round, that you MUST pick the same u-sub in the second (or more) rounds?

I literally cannot find an answer to this and I'm not smart enough to understand what's going on. I think that if I switch the u- and dv- choices in the second round, I'm reversing the process and effectively undoing the work I've done, which defeats the whole purpose of the integration?

Thank you :)

I’m currently taking Calc 5 (partial differential equations) and learning about the Fourier Series. One thing that’s not becoming clear to me is the vector properties that are assigned to functions.

I understand how linear algebra works for vectors, like n-dimensional vector spaces representing vectors with n components or how orthogonality means that the relative angle between two vectors is 90 degrees or how the norm of a vector is its length. However, I can’t seem to find an intuition for how these properties connect to functions.

What do vector spaces represent for functions? What is a dot product of two functions and what does it mean for two functions to be “orthogonal”? What is the actual meaning behind the “norm” of a function?

Basically, why are these linear algebra / vector analogues being applied to functions?

Let P={p_1,...,p_k} be finitely many distinct points of R^n. I'm asked to prove that R^n - P is simply connected. Now I've already done this. But I really don't like my solution. Basically what I did was drew an arbitrary loop and show that it contracts to a point. If it happens that it would intersect one of the missing points, I "pivot" the loop around the missing point and carry on. Then since the points are finite this process eventually terminates.

I really don't like my solution and it feels really clunky. Instead I had the idea of covering my space with open sets such that every open set has at most 1 of the missing points in it. Then I want to deformation retract each open set and "stitch" it all together. The thought being that all the open sets without any missing points will retract to a point, and the open sets with a point missing deformation retract to S^{n-1}

Basically I'm wondering if this is a fruitful approach? I'm pretty new to algebraic topology so I'm not so sure what I can get away with. I'm not even really sure how to express this properly. Any advice or resources would be much appreciated.

This is something I think intuatively, but I don't know if there's any reason to believe it.

A bus roote runs in a loop between A and B.

I'm waiting at a bus stop, somewhere on the loop, for a bus going from A to B.

Each bus stop on the loop has the same average bus frequency (I assume this would have to be the case anyway).

I see a B to A bus drive past on the opposite side of the road. Inutatively to me I think this means my wait is probably half over or more. For example if I started a stop watch when I got to the bus stop and it read X minutes when I saw the bus opposite, then I can expect my bus will be less than X minutes away.

Is there any validity to my intuition?

The diameter of a nonempty subset E of a metric space (X,d) is defined to be

diam(E) = sup{d(x,y): x,y ∊ E}.

Show that {E_k}_{k=1}^∞ if is a decreasing sequence of closed nonempty subsets of a complete metric space whose diameters tend to zero, then ∩_{k=1}^∞ E_k consists of precisely one point.

Wanna know if it makes sense what I wrote:

By assumption we know E_1 ⊇ E_2 ⊇ E_3 ⊇... E_n ⊇ ... , where E_k is nonempty and closed for all n ∊ ℕ. Now for each k ∈ ℕ we choose x_k ∊ E_k. Then we get a sequence (x_n)_{n ≧ k} in E_k. Now by definition of sup we know d(x_n,x_k) ≤ diam(E_k). Let ℇ > 0, by assumption there is an N ∈ ℕ such that diam(E_k) < ℇ for all k ≧ N. Hence we have for all n ≧ k ≧ N that d(x_n,x_k) < ɛ, proving that (x_n) is Cauchy in (X,d). Since X is complete (x_n) converges to an element x ∊ X. In particular each subsequence (x_n)_{n ≧ k} converges to x, for all k ∊ ℕ. Since each E_k is closed we have x ∈ E_k for all k ∈ ℕ.

Now we show there is only a single element in the intersection: Let x,y be elements in ∩_{k=1}^∞ E_k. Then x,y ∈ E_k for all k ∈ ℕ. But then d(x,y) ≤ diam(E_k). Taking the limit k --> ∞ we get d(x,y)=0 and so x=y.

I'm curious to see what approach people take to show that the attached limit equals the attached value.

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Summation Limit

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Value

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Approach

Hi, I am studying auto correlation in time series data, from the basic usage, I understand that I can use acf to extract how much correlation and confident interval for each lag. For example, giving the data like this, I can use plot_acf to show correlation with lag

from statsmodels.graphics.tsaplots import acf, plot_acf
import matplotlib.pyplot as plt
import numpy as np

# Fake data
x = np.arange(500) / 20
data = np.cos(x)

# Create figure and axis objects
fig, axs = plt.subplots(2, 1, figsize=(10, 10))

# Plot data and autocorrelation
plot_acf(data[:51], lags=50, ax=axs[0])
axs[0].set_title('Autocorrelation')
axs[0].grid()
axs[0].legend()
axs[1].plot(data)
axs[1].set_title('Data')
axs[1].grid()
plt.show()

and this is the plot

https://preview.redd.it/6e8wzhwujfrb1.png?width=861&format=png&auto=webp&s=5c1341def605bae48ba21a3e2283a718352ff998

From the plot, I understand I can use up to maximum 4 lags with high confident (because it's out of shaded area). However, when I extract acf like this, I see that the interval plot with confident like this

result = acf(data[:51], nlags=50, alpha=0.05)
plt.plot(result[0][:], label='Cofficient')
plt.plot(result[1][:, 0], label='Upper confident interval')
plt.plot(result[1][:, 1], label='Lower confident interval')
plt.xlabel('Lags')
plt.legend()
plt.grid()
plt.show()

https://preview.redd.it/vjpzhl7qkfrb1.png?width=893&format=png&auto=webp&s=bc157fdaca9ac280fc0733089b58aebe8d0ad845

To correctly show the shaded area like plot_acf, I need to subtract interval from coefficient (which also applied in plot_acf) like below

result = acf(data[:51], nlags=50, alpha=0.05)
plt.plot(result[0][:], label='Cofficient')
plt.plot(result[1][:, 0] - result[0][:], label='Upper confident interval')
plt.plot(result[1][:, 1] - result[0][:], label='Lower confident interval')
plt.xlabel('Lags')
plt.legend()
plt.grid()
plt.show()

https://preview.redd.it/zxb9nwmllfrb1.png?width=846&format=png&auto=webp&s=9481e145fe40cfe25a19f816be31302c9b5c6814

Why is it? I think the implementation from this page https://www.itl.nist.gov/div898/handbook/eda/section3/autocopl.htm but can not found the clear explanation for this.

I plan to start studying mathematics from the school curriculum in order to get a good base and then start studying discrete mathematics and linear algebra. Which book or books will help me get this database? I know that there is a khan academy, but unfortunately English is not my native language and I still can't understand it well by ear, so I'm looking for a book.

i would like to know the probability of winning if i have say 1000 entries out of 2 million entries everyday over a 40 day period? How is this calculated?

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thanks

I am trying to figure out how to express 27^6 as a power with a base of 9.

This is a review, so I can see that the final answer is 9^9, but I can't for the life of me figure out how to get there. So far I can get to...

-27^6 =(3^3)^6

But I can't figure out where to go next. Any help would be greatly appreciated.

Edit: thanks for the help everyone! I understand what to do. Much appreciated

The answer i got was: x = 3/2 + sqrt(5)/2. x = 3/2 - sqrt(5)/2. But the 3th answer apparently is also x = -3/2 could someone exlain how. Thanks in advance.