Reciprocity theorem Puns

Been trying to find one simple solution for this but I haven't get lucky yet.

http://en.wikipedia.org/wiki/Quadratic_reciprocity

I am a bit confused. I have an example where taking reciprocal of squeeze functions flips the larger or equal than symbols. I am confused how that works, and I don't know the name for finding this information either since when I search reciprocal its just that of a single number, not a function with large or equal then symbols etc.

1 <= x/sinx <= 1/cosx

Reciprocal is: 1 >= sinx/x >= cosx

I understand how the fractions switch when reciprocal is taken but I dont understand how the symbols and why the symbols change. Thanks.

I was reading about the reciprocal pythagorean theorem on wikipedia, and I came across this "transformation" of the theorem: https://imgur.com/a/NxmXucn.

My question is, why can 1/a^2 + 1/b^2 = 1/d^2 be transformed into 1/(xz)^2 + 1/(yz)^2 = 1/(xy)^2, given that x y and z are a pythagorean triple? I get why 1/(xz)^2 + 1/(yz)^2 = 1/(xy)^2 is true, but I do not understand why every a, b and d that is a solution to the reciprocal pythagorean theorem can also be written as 1/(xz)^2 + 1/(yz)^2 = 1/(xy)^2.

Thanks in advance!

Naïve undergrad here!

I'm currently taking a functional analysis course, and we recently covered the standard open mapping theorem, closed graph theorem, and uniform boundedness principle in what I figured was full generality. We spent several lectures going over detailed proofs of these results.

However, I was recently googling around when I came across this theorem, that gives all three results as 'quick' corollaries (albeit I hardly understand a lot of the terminology being used in the statement).

Are there any other similar high level results in mathematics that yield big theorems as simple corollaries? Like for a hypothetical example, a theorem that -as a corollary- proves stokes theorem via a simple communitive diagram. I'm vaguely aware of a few results like this from category theory, but I don't know any specific names.

I recently animated four proofs of the fact that the sum of a positive real number and its reciprocal is greater than or equal to 2. The four visual proofs come from Roger Nelsen's static diagram that can be found here: https://www.maa.org/sites/default/files/Nelsen99959975.pdf

I ran a poll to figure out which proof people like best, and I was kind of surprised that the answer was the "Calculus proof" because I find that one to require the most "background knowledge." No one justified why they prefer that one over the more geometric ones, so I thought I would bring it here to see which proof you all like best and to get feedback about why you prefer one over the other. You can either check the static diagram above, or you can check the short animations of the proofs below:

Tatami diagram: https://youtu.be/IghOHBl0Do8

Pythagorean theorem: https://youtu.be/zBUK8C6wSqs

Triangle areas : https://youtu.be/BkQvnriVblY

Calculus : https://youtu.be/qoVNsahM9vI

This is partially some self promotion of the animations I am doing (as I think visual proofs can help people think about mathematics in different ways), but I truly am interested in why the Calculus proof has been the clear winner on my other poll when it is my least favorite proof and I almost chose to not animate that one at all.

PS. Some people think this fact is trivial or not worth proof, but I will note that it is equivalent to the two-variable Arithmetic Mean-Geometric Mean inequality.

Sometimes, even good news isn't quite good enough.

I've posted several times here about some of the peculiar difficulties I've dealt with in my journey toward a PhD in mathematics, either in threads, or as submitted posts like this one. The central problem I've been dealing with is that my research topic—the Collatz Conjecture—and the tools I've been using to study it (harmonic analysis, analytic number theory, and—most recently—non-archimedean (functional) analysis) are all completely outside the purview of the expertise of my university's mathematical faculty.

For most of my time in graduate school, the most troubling manifestation of this problem was that I wasn't certain I would be able to get a PhD, seeing as there was no one in my orbit capable of rendering judgment on the merit of my work. The agreement with I'd reached with my department was that if I could get something of mine published in a reputable journal, that would suffice as the "expert approval" needed to justify conferring upon me the doctoral degree that I've been working toward all this time. Unfortunately, my attempts to get myself published have not met with success—and, certainly, the backlog of excess submissions caused by the pandemic has only made matters worse.

In mid November 2021, however, I received some truly wonderful news: my department decided that they would not require me to get something published. They will accept whatever original work I have done.

Although I have no evidence for this, given the way the head of graduate studies phrased the message, I have a strong suspicion that when I informed my advisor I had independently rediscovered a good deal of the contents of W. M. Schikhof's PhD dissertation (Non-Archimedean Harmonic Analysis, 1967), that was what convinced them that I was worthy of their gracious leap of faith.

While this news has definitely taken a great deal of stress off my shoulders, me being me—that is, obsessive—I've found a new, daunting psychological difficulty to nail to onto my skull:

I'm worried that I don't deserve it, because I haven't done enough.

The positives:

1. I know for a fact that my work is cutting-edge, insofar as novelty goes. The only major antecedent work in a comparable vein that I can point to is Tao's 2019 paper on the Collatz Conjecture—but, even then, the similarity is only in the fact that our approaches share essentially the same central object of study; otherwise, they couldn't be more different. His take is

I took Real Analysis this semester so I had series and sequences on my mind. I had the random idea to look at the sum of the reciprocals for the Hailstone sequence of some numbers. So, taking 5 and applying the Collatz function iteratively you would get,

[5, 16, 8, 4, 2, 1]

Wherein I would stop. Then I would look at,

1/5 + 1/16 + 1/8 + 1/4 + 1/2 + 1/1 = 2.1375

I plotted this number with respect to its initial value (I started calling these seeds just to help me out).

Here are some of my findings.

Fig 1. Hailstone Sequences, Length of Sequence, Sum of Reciprocals

This first image was to get my head around everything. The first graph is of the hailstone sequences for the first 1000 integers. You can see how all the lines eventually fall down to 1.

The second graph plots the length of the sequences, from the starting seed to 1, against the seed's value. It seems to grow sort of logarithmically which is neat.

The third plot is my work. The sum of reciprocals for the sequences. The first thing to notice is that strange band at the bottom, followed by a gap, followed by a cluster of points. Very interesting. I'll comment more on the bottom band later. Second thing of note is how slowly it increases. This makes sense because the reciprocal of large numbers are small, so we have to go out very far to see any noticeable change. We know the harmonic series diverges and I am just taking different subsequences of the harmonic sequence. If there was some upper bound on these sums of sequences that could say something interesting. But I don't know.

Fig 2. Sum of Reciprocals up to 10,000

Here is a graph of just the sums of reciprocals but up to 10,000 integers. The interesting thing here is that you can see more pronounced bands forming. The lowest band around 2.0, a thin band about 2.3-2.4, a dense band around around 2.5, a thick band from 2.1 to 3.0, and then an upper band about 3.2. The bands themselves are interesting but what is even more interesting to me is the gaps between the bands.

Fig 3. Sum of Reciprocals up to 10E5

This figure is just to

I don't want to step on anybody's toes here, but the amount of non-dad jokes here in this subreddit really annoys me. First of all, dad jokes CAN be NSFW, it clearly says so in the sub rules. Secondly, it doesn't automatically make it a dad joke if it's from a conversation between you and your child. Most importantly, the jokes that your CHILDREN tell YOU are not dad jokes. The point of a dad joke is that it's so cheesy only a dad who's trying to be funny would make such a joke. That's it. They are stupid plays on words, lame puns and so on. There has to be a clever pun or wordplay for it to be considered a dad joke.

Again, to all the fellow dads, I apologise if I'm sounding too harsh. But I just needed to get it off my chest.

Do your worst!

I'm surprised it hasn't decade.

I was recently watching a college lecture where the professor prefaced a discussion of game theory with "and the concept of a prisoner's dilemma is one of those things I am confident saying you are not an educated person in any meaningful sense if you have never taken the trouble to understand or learn it. The idea that we can map out the conditions under which cooperators will defect given individual incentives, even despite the fact that the collective incentive can be to cooperate for a higher payoff, is so fundamental to understanding the problems of the 21st century (like Climate Change) that I think it's only fair that we set our bar/expectations for the educated person high enough that they would know this enough to be able to act on it."

This got me thinking: what is your list (or solitary individual entry for what could become a larger list) of things every 21st Century person who likes to think of themselves as having achieved a serious education should know, but probably doesn't? I say "probably doesn't" because a list of what an educated person should know in general would be too long and (for the most part) too obvious for my purposes here (i.e. please don't say something like "the earth orbits the sun"). I also want people to emphasize knowing things that are considered groundbreaking in their respective fields and that may even have a practical or important connection to a larger issue, particularly the larger issues that we will be counting on the "educated (but common) person" to address in this century.

Pick any discipline you want, but try to meet my criteria. Here's mine! (A list like this is bound to sound opinionated and self-congratulatory because it's an attempt to list the things you think you already know but that many others don't, but for the same reason that the "rationalist" community has chosen a vaguely positive adjective for itself, and only aspirationally rather than narcissistically, I want you to put aside the self-conscious worry that you sound self-indulgent and just do your best to outline the greatest ideas an education can impart for someone aspiring to a true education)

These are not in any particular order from most to least important, but more "what occurred to me first," and it is bound to be horribly incomplete or include things it shouldn't--that's where you come in!

Here goes nothing:

1. Graham Oppy's concept of rational belief-formation as a process of "worldview comparison" where people follow the

For context I'm a Refuse Driver (Garbage man) & today I was on food waste. After I'd tipped I was checking the wagon for any defects when I spotted a lone pea balanced on the lifts.

I said "hey look, an escaPEA"

No one near me but it didn't half make me laugh for a good hour or so!

Edit: I can't believe how much this has blown up. Thank you everyone I've had a blast reading through the replies 😂

It really does, I swear!

Because she wanted to see the task manager.

Ever wonder why the bitch/jerk (i.e., asshole) is constantly fawned over or why people go out of their way to be friends with them? The asshole will never reciprocate the adoration, affection, or friendship.

Why is the "mean" girl popular?

Why is the pompous jerk promoted at work?

The only thing I could conjure was -- psychologically -- if the asshole was being mean to others and not to me, then I must be doing something right. Leading to an "appeal to authority" like train of thought.

Corollary to this theorem are the assholes who can be downright mean, but state they are "just being honest".

They’re on standbi

Pilot on me!!

Nothing, he was gladiator.

Dad jokes are supposed to be jokes you can tell a kid and they will understand it and find it funny.

This sub is mostly just NSFW puns now.

If it needs a NSFW tag it's not a dad joke. There should just be a NSFW puns subreddit for that.

Edit* I'm not replying any longer and turning off notifications but to all those that say "no one cares", there sure are a lot of you arguing about it. Maybe I'm wrong but you people don't need to be rude about it. If you really don't care, don't comment.

BamBOO!

When I got home, they were still there.

I won't be doing that today!

[Removed]

Where ever you left it 🤷‍♀️🤭

This morning, my 4 year old daughter.

Daughter: I'm hungry

Me: nerves building, smile widening

Me: Hi hungry, I'm dad.

She had no idea what was going on but I finally did it.

Thank you all for listening.

There hasn't been a post all year!

You take away their little brooms

It was about a weak back.

What did 0 say to 8 ?

" Nice Belt "

So What did 3 say to 8 ?

" Hey, you two stop making out "