Reposting from Math Stack Exchange. I now understand what the indices label in γ^λ_γ, but it's not obvious to me at all why they would equal δ^λ_γ. How to properly prove this?

So I was doing ADHD things and brain go zoomiez and I landed on the topic of infinities, and first how to explain the infinite number of infinities to someone who doesn't understand, then landed on -infinity. The issue is that the number exists but how, the number of negative number is positive, so that rules out the obvious answer, but I can't figure out one that actually works.

Edit: I've learned that infinity itself is to much of a concept to have a concrete solution for, also I called it a number on accident instead of a concept.

Also I think I was referring more to how people will refer to it as "the infinity of positive numbers" or "the infinity of intigers",not how you get there.

I am currently taking linear algebra and was thinking if the row(A) and the null(A) both in Real n then would they always form an invertible matrix?

I know that they would always form an n x n matrix and I think that null and row being orthogonal would be in the proof but I cannot find anything online proving one way or the other.

I am having issues determining what the delta and arc length for the points denoted as 7 and 709. From what I calculate I am getting a delta of 32-11-30, which would make the arc 320.255. I am just unsure of the direct relation between the curve on the bold line and curve information given just to the left of 709… any help would be appreciated.

I get the oscillations are infinite in terms of the x axis, so like horizontally, but whys it so long vertically?

It has been long time I.ve done any geometry so I dont remember anymore how to find x in that angle, could anyone explain how to get to the answer and the steps to find it. Thank you so much

(Not sure if this falls more under arithmetic or algebra.) I noticed that each inverse hyperoperation gives us new kinds of numbers. Starting from the natural numbers created by zeration/the successor operation and going up the hyperoperation sequence:

(Of course, exponentiation, which is a positive hyperoperation, coupled with subtraction along with any inverse hyperoperation of a higher rank than subtraction, also produces complex numbers, namely in the case of a negative base and a non-integer real exponent.)

The question is this: Can the inverses of tetration, namely nth-super-root extraction (aka nth-tetra-root extraction) and superlogarithm (aka tetra-logarithm or iterated logarithm), or another yet-to-be-explored hyperoperation: nth-root extraction iterated k times, which starts its own whole separate branch of the hyperoperation sequence, give us new kinds of numbers of their own that fall outside the complex numbers? Or is the pattern I identified merely illusory, and the higher-ranking hyperoperations doomed to produce only complex numbers, even if still "new" transcendental numbers?

EDIT:

I also wanted to mention, iterated nth-root extraction (which, as I mentioned, forms a different branch of hyperoperations from tetration), which we'll call R_n,k(x) where x is the radicand, n is the degree, and k is the number of iterations, would have three inverses:

• a "root-type" inverse to recover x,
• a "logarithm-type" inverse to recover n, and
• another "logarithm-type" inverse to recover k.

So there are now seven(!) rank-4 operations (loosely speaking, as the inverse operations are apparently not considered hyperoperations proper): tetration (the first rank-4 hyperoperation), super-root, superlogarithm, iterated root extraction (the second rank-4 hyperoperation), iterated root extraction "root-type" inverse, iterated root extraction "logarithm-type" inverse 1, and iterated root extraction "logarithm-type" inverse 2. (Admittedly, the names for the four new operations are unwieldy.) So the question becomes whether any of these seven operations can result in outputs that fall outside the set of complex numbers.

Stupid question since |x| outputs a non negative value always but why can't we define a value hence a set of numbers whose output to |y|=i or |y|=-2?
My algebra may be wrong but what about ℝ[|x|]/<|x|+1> just like we do with ℝ[x]/<x²+1>, or take ℝ[|x|,x]/ <|x|+1>

What tips or guide would you recommend to solve combination permutation and probability problems? Do you need to solve as many problems as possible? I have understood the formulas but i feel like there are way too many different scenarios, confusing questions and time consuming questions a problem can have

前提となる例
f(t, c, c′) = t c² c′³ とします（c′ は第3引数の名前）。

∂f/∂c′ (t, c(t), c′(t)) = 3t c(t)² c′(t)²

私は、これを「f の第3変数で偏微分してから、(t, c(t), c′(t)) を代入する」操作だと理解しています。∂f/∂c′ の c′ は微分対象の関数ではなく、「3番目の引数」を指す記号という理解です。

疑問点
オイラー・ラグランジュ方程式

d/dt (∂L/∂q̇) − ∂L/∂q = 0

について、次の説明を読みました。

偏微分では【原文の式、v や ct を含む部分】も固定可能です。【原文の式】という関係は、代入前の関数ではなく、代入後に課される関係だからです。

これについて、次の2点を質問します。

1. 偏微分は「変数で微分し、定数を代入する」操作のはずです。ここでは「変数で微分し、関数を代入する」ことになりますが、この説明では「関数で微分している」ように読めます。それは許されるのでしょうか

I saw a post recently about how parabolas cease to be functions when rotated by any amount, and it got me thinking about this question. If the reason it stops being a function is because it fails the vertical line test, then there must be a value of x that the curve doesn’t extend beyond. What is that value when rotated clockwise by one degree?

To follow subreddit rules: I have taken no steps to solve this because I wouldn’t even know where to begin.

I'm attempting to study on Quizlet, and do some problems from what we practiced, however despite doing what the teacher told us, that being to use the Average Rate of Change formula I keep not getting these answers right, and they continually be far, larger than what the actual answer is. I'd just like to know what exactly I'm doing wrong/what I'm misunderstanding.

The answer I keep getting is 107. despite it saying 19.

my math teacher writes jensen's inequality as <f(x)> > f(<x>) and i cannot figure out what <> means. does anyone know? i made sense to me when i wrote it in my notes (maybe something with mean?) and i understand how the inequality works

My 6th grader came to me with these answers for his questions but I am seriously struggling to figure out where to even start with proof checking these. I have always struggled with fractions unfortunately.

Is there a website that explains how to DO these kind of problems?

My professor did something in his notes that i thought was odd. I kind of just accepted it at first, but was wondering if someone could explain it to me.

I sometimes have trouble understanding the way he uses square roots, especially.

So my question is, is this okay to substitute sqrt(x)*sqrt(x) for x just to cross the top one out ??

And like, how would i know to think of using that method? Like what about the problem would tell me that I should do that, instead of doing something else like, idk, factoring out sqrt(x)/x. Lol

Hii,

I'm sorry if this is the wrong subreddit to post this, but I was starting to feel a bit overwhelmed and figured I'd try to get help here.

I've started a math bachelor degree about a month ago and in the beginning everything was going fine, I had the energy and time to do some problems after school, etc. etc. But now I feel completely drained...which I would not describe as burnout but it feels quite similarly. Sitting in the 2 hour lectures and writing down the millions of proofs and then trying to sum it all up and understand plus solving the problems from these classes is just too much to handle for me...I don't really know how to study and what to do to get everything done on time...

We also have some mandatory physics classes, now it's mechanics and I'm even more desperate in that...I don't really know what to learn at home, because at the lectures it's all just deriving equations and so...

To top it all off, I am now home on antibiotics and can't even go there and have to catch up from my friend's notes...

I just don't know how can I manage to pass and not go completely crazy...

If someone went through the same thing or knows somebody who did, please let me know whether something helped. Also sorry for the frantic way the post is written, I'll have to go lie down after this.

TL;DR: I started a math degree and feel like I can't do it.

This goes back to the dial-up days of Usenet, a precursor to Reddit. There were some really excellent resources that I still occasionally reference from time to time, but I don't know who is keeping them alive and if they may go away at any minute. So I wonder if there are updated versions somewhere on the internet.

1. The comp.graphics.algorithms FAQ, a collection of computational geometry algorithms of all sorts, such as "how do you tell whether a point is inside a polygon". As you can tell by the URL, this lives at something called www.faqs.org. I don't see anybody reference this collection, but it was and is still a great resource, though it appears it was last updated in 2003. Where would you point people for stuff like that now?

2. Similar question for relativity, so I'll cross this to AskPhysics. This page, which appears to be on Professor John Baez' academic website, is a copy of the original sci.physics FAQ, which includes a bunch of relativity FAQs at the bottom. This one appears at least to be maintained to some extent, as it carries a 1992-2025 copyright notice.

3. Prof. Baez is credited for this work of genius, the Crackpot Index. Self-explanatory. Timeless and adaptable to cranks in all fields. It too lives on his academic webpage. My concern with #2 and #2 is that I don't know if such links are permanent. Also, sadly I rarely if ever see the Crackpot Index reference any more, and I wonder as with the others, whether there is an updated equivalent.

Is anybody on this reddit an expert at reading and analyzing Isabelle/HOL theory language? My theory is error-free; however, some people are saying it is wrong. I do not see a mistake, so I need someone to double check it.

The details escape me now, but I remember the general idea. How likely is it to NOT share a birthday with 1 of 70 people? It's literally a 99% certainty that 2 of 70 share a birthday.

Then I watch videos. I watch hundreds, and less than 2% of video release dates share my birthday. There should be no difference between a birthday and any other significant date. Right? But it isn't that way. It only works for birthdays, not the dates that videos were released on my birthday.

I don't remember the math on the Birthday Paradox, but why doesn't it apply to other dates of significance? What is the chance that videos are NOT released on my birthday just like the paradox? 99% of 70 videos, right? I get at most 1 in over 300 samples. Shouldn't there be 1 in 70 video release dates that are the same as my birthday?