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12:59 AM
Hi chat! i was just trying to understand a proof from a paper on mathematical physics
The above is a three dimensional system of differential equations
Here they have considered a Lyapunov function $L$
To show that the trajectories are confined to a bounded region, we need t show $\dot{L}
I could understand upto (7) but I am not sure how to determine if the inequality in (8) is true or not?
I don't suppose anyone knows the latex for the symbol ⧦
My standard tools aren't cutting it..
I found $\models$ but
also did not realize there was a SE for this
1 hour later…
2:25 AM
@copper Have you heard that the Irish fishermen will take on the Russian navy? These are scary times.
2:55 AM
While doing this question, I had a suspicion that the other stationary point (x=-1) would be the mirror image of the x=1 stationary point. But I didn't know how to mentally confirm this to myself without solving the DE. How could I have confirmed this would be the case without solving?
I notice that dy/dx is odd function. Is it true in general that y will be even?
3:11 AM
@TedShifrin heh heh, good for them! i am surprised ireland is ok with russian exercises under irish airspace.
3:47 AM
@copper I don't think they have much say about it.
@LearningCHelpMeV2 Well, how could you? What have you tried to do?
@TedShifrin It seems to be the case for even odd/polynomials. But not sure about other functions. Do you ever sleep by the way?
It's 8 PM where I am.
Well, can you show that the derivative of any differentiable even function is odd (and vice versa)? This is elementary.
@leslie did munchkin or COVID kidnap you?
4:16 AM
@TedShifrin Yup figured it out thanks
Yo, I got some work finally :>
It's limited to 100 hrs / month, but that's a lot after having not worked for a long time
Coding something in Python (most likely Python)
But I can't disclose any more info on that
4:30 AM
Congrats, @PenAndPaper.
@TedShifrin thx
4:59 AM
@PenAndPaperMathematics Well done. It can be tough. Been there.
Keep looking.
@copper.hat thx
Might get to use the Rasp Pi on this one
I own two and never used them for anything
Rasp Pi 2 model B's
Pretty cool! Which pi? 4b?
5:20 AM
convex psqs have dried up.
5:40 AM
Go for concave.
@copper.hat convex psq?
@soupless its a long running joke. psq = problem statement question (i think) which means a question with no effort shown
i like convex problems
i sometimes look for convex problems to get my maths jollies out
I am familiar as to what psq means. I don't quite understand what 'convex psq' means. Is convex an adjective or psq's about convex things?
with convex in the tags
5:53 AM
Oh, okay. I didn't quite notice that. Maybe I spent way too much time outside the site
you can safely ignore much of what i say
@copper.hat thank you so much and sorry for being late in my response.
just busy in the school.
@CroCo np at all
reading more with this book A Modern Introduction to Linear Algebra
By Henry Ricardo
6:12 AM
wow, modern textbooks are enormous
Still no leslie.
what's with the control sporadically?
@copper.hat it started here.
@robjohn Thanks!
6:26 AM
That doesn't give any hint about why, however.
6:54 AM
gn... 20 minutes too late
1 hour later…
8:21 AM
If $E_1/F$ and $E_2/F$ are Galois, then $Gal(E_1E_2/F)$ can be embedded into $Gal(E_1/F)\times Gal(E_2/F)$ by restriction maps. Does that imply there is an exact sequence? : $1\to Gal(E_1E_2/F)\to Gal(E_1/F)\times Gal(E_2/F)\to Gal(E_1\cap E_2/F)\to 1$. I don't understand the last map. I wonder if it can be defined.
@robjohn still no @leslie.
8:54 AM
@Koro Maybe in the Cafe?
9:07 AM
@soupless not even in Café.
I tried in whatever, quid: meta.se but no @leslietownes.
9:23 AM
Who wants to join in the search of a prime factor of $2022!-1$ ?
@soupless Since leslie has starred comments, it is pretty easy to find their chat profile (just click on their name after the comment). Then look under "recent".
@Peter I wouldn't start under 2023 ;-)
@Peter I tried using yafu, and it crashed instantly
@Peter Mathematica says it is (probably) not a prime
Mathematica is looking... FactorInteger[2022!-1,2]
9:43 AM
turned out, 2022 has both ternary and decimal expansions only consists of 0 and 2.
and there are only finitely many numbers with that property.
2022!-1 is surely not a prime (the result "composite" is always true in the usually used primality tests). The smallest prime factor probably has more than 20 digits.
@love_sodam Nice observation !
@Peter Actually, that observation is not mine. math.stackexchange.com/a/4348504/668308
@love_sodam How can we prove that only finite many numbers satisfy this property ?
@Peter Ramsey theoretic proof argument(?) shows that. See the link.
10:24 AM
@mohan10216 This also happens to the activity page of a user.
10:37 AM
@mohan10216 There are posts on meta.SE and math.meta.SE regarding this.
@mohan10216 which post is that?
10:52 AM
Q: Prove that $f(A \cap f^{-1}(B)) = f(A) \cap f(f^{-1}(B)) = f(A) \cap B$

nicksabanLet $f: X \to Y$, and let $A \subseteq X$ And $B \subseteq Y$. Prove the following equivalence $$f(A \cap f^{-1}(B)) = f(A) \cap f(f^{-1}(B)) = f(A) \cap B.$$

This new format is really bad...
@mohan10216 that looks okay. Where were you seeing the folded MathJax?
I found it on the main questions page, and it looks okay to me there, also
11:21 AM
Is it a problem with my side then?
11:36 AM
No, there are the reports I linked
12:34 PM
It feels like calculus is the godfather of real-analysis.
I was very excited when I wanted to learn calculus in my 7th grade, but now it has turned into a mindless computation.Maybe calculus is just algebra in disguise.
1:02 PM
it kind of is
Can anyone help me to interpret this statement: " let the symmetric group $S_5$ act on the set $X$, then the orbit of an $x \in X$ cannot have $3$ or $4$ elements."
Well, an orbit is just all the elements you can get from $x$
and this says it cannot be a 3/4 element set
So is an orbit the cardinality of the set $X$, or am I misinterpreting what you are saying?
what do you mean by it cannot be a 3/4 element set, I don't quite understand.
I'm quite new to Galois theory, so it's a little hard for me.
The orbit doesn't have to be all of $X$
@mohan10216 That the number of its elements cannot be 3 nor 4
@mohan10216 Which book is that statement from?
1:10 PM
it's from this set of notes i'm beginning to work through: degraaf.maths.unitn.it/algnotes/galois.pdf
So just to clarify @Jakobian: the number of elements in set $X$ cannot be 3 or 4.
when did I say that
Sorry if this is trivial, i'm trying to make sure I don't understand something incorrectly.
Oh it's wrong...
I inferred wrong from your comment then.
I clearly said something else
@mohan10216 Page number?
6, right at the top.
1:17 PM
$S_5x = \{\sigma x : \sigma\in S_5\}$
this is the orbit of $x$ under the action by $S_5$
what they mean is that this set cannot have 3 or 4 elements
by set you mean the orbit?
not $X$
@mohan10216 Yes. They are not saying the set being acted on (X) has 3 or 4 elements, but rather the orbit, which is a subset of X defined according to what Jakobian said.
Oh that makes sense, thanks.
@Jakobian: Sorry about that, I didn't know what an orbit was.
@mohan10216 Its ok =)
1:29 PM
Orbits are nice after you get to know them a bit better. Their name is also intuitive (imagine a cyclic group acting on a set X, draw the elements of X as dots on your paper, and draw the generator acting on X via connecting the dots with arrows corresponding to repeated action of the generator).
ummm...Am I supposed to know what those words mean? e.g. generator, cyclic group etc... Should I be learning group theory before Galois theory then?
So there are a couple of group theoretic terminology: symmetric group, act , orbit.
@mohan10216 Probably yes.
I mean I know a little bit about group theory, i've solved some group theory questions but never have I ever seen those words before...
@mohan10216 From a textbook?
No, my math tutor gave me some problems
I think he may have gotten them from a textbook but i'm not sure.
1:32 PM
@mohan10216 What kind of problems? Do you remember some of it?
It was to determine if an equation or smth like that was abelian, which was easy and he gave me some questions on functions bijectivity, thats all I remember tho, this was like 2 years ago. I probably did hear about cyclic groups coz I remember a little about cayley diagrams but that's about it.
@mohan10216 Absolutely, yes, you should study group theory first.
Any good resource? @anakhro
@mohan10216 Depends on a few things. What is your background in math so far? And what sort of style of notes have you liked or not liked in the past?
@mohan10216 Oh ok. Thanks for sharing.
1:40 PM
I have experience with calculus, functions, trig, abstract algebra(homomorphisms, unique factorisations, etc), algebra-precalc, geometry, to name a few. I prefer notes which are not too long (> 500 pages) and explain the content in a very basic manner. I also prefer if they don't start off assuming you know the notations but build up the usage of the notations.
Okay, that's not too restrictive. Milne's notes will probably suffice: jmilne.org/math/CourseNotes/GT.pdf
Being comfortable with the first 5 chapters is probably enough to get you started with Galois theory. However, for Galois theory I would recommend Ian Stewart's textbook which is a lot more user-friendly and I think suits your preferences you mentioned (it goes through all the requisite ring and field theory to begin with).
They're the same notes my tutor recommended to me.
BTW: do you know what a primitive root is?
can you give me an example.
@anakhro Chapter 6 contains Solvable groups.
thanks @anakhro: Stewarts book looks really good from just the first few pages, looking forward too reading =)
@mohan10216 I think the wikipedia article gives more than enough examples to get started. If you want a specific example explained for you, let us know what is troubling you.
1:53 PM
ok thanks, it's a habit of mine too ask things first without reading, and boy have I gotten in trouble XD
@mohan10216 I also have that habit.
@love_sodam This is a matter of preference, but I view it as more pedagogically sound to learn about solvable groups in the context of Galois theory, rather than before touching any Galois theory.
For reference, Stewart introduces simple and solvable groups halfway through his book on Galois theory. I don't think it's required to start getting into Galois theory at all.
@anakhro Is Rotman's An introduction to the theory of groups good? [I am asking this because I also want to learn group theory]
@Prithubiswas I would not recommend Rotman if you are learning group theory for the first time.
Yeah, just had a look and it said graduate text.
Seems like you need to have done an undergrad course first...
Wanna try learning together ? @Prithubiswas
1:59 PM
@anakhro But it seems simpler than Dummit and foote.
group theory
@mohan10216 Probably not because college will murder me.
It's not simpler than Dummit and Foote. Dummit and Foote is a standard textbook for all the common abstract algebra requirements for grad school.
lol, ok.
@anakhro Then why Dummit and foote seems impenetrable to me? Maybe I am missing something?
2:01 PM
@Prithubiswas What is your background in mathematics? What course have you taken in pure math, and what part of Dummit and Foote do you get to before you start to struggle?
@anakhro High-school.
That's probably enough to explain why you are facing issues.
I am guessing you don't have extensive experience with proofs?
Are you doing the IB?
@anakhro Well I can't live without proofs.
You might find it a lot easier to enter the world of abstract algebra by reading a book on linear algebra first (one that focuses on the theory of vector spaces). The proofs there would be much more accessible and would help familiarize yourself with some more formal concepts in mathematics before you move on to Dummit & Foote.
Not to mention that I think linear algebra is a requirement for D&F, I believe.
2:05 PM
@anakhro are you a teacher or smth, you know alot...
if not you must be a grad student or phd
@anakhro I haven't ever read linear algebra.
@Prithubiswas that's a good place to start. I think you will appreciate abstract algebra a bit more after doing a more pure intro to linear algebra.
Take a look here math.stackexchange.com/questions/2377980/… @Prithubiswas
you can find the one right for you there probably
personally I use Nicholson Linear Algebra with applications. very basic and goes into a lot of detail, but slowly.
Also, there are notes like these which are pretty decent if you just want a cursory overview rather than having to dive into a textbook: alistairsavage.ca/mat2141/notes/MAT2141-Linear_Algebra_I.pdf
If I find smth basic you have too find it really easy coz i'm dyslexic and fun fact I couldn't read till I was ten.
i'm 15 now
2:09 PM
The nice thing about these notes in particular are that you have a brief intro to some concepts from abstract algebra at the end.
Ok I have decided to study rotman. It looks easy.
It's not easy.
@anakhro Why it is not easy?
It's not aimed at undergrads, let alone high school students with no experience of pure mathematics. It's aimed at graduate students who have already seen abstract algebra.
From the intro.
In other words, it assumes intimate familiarity with linear algebra, and mild familiarity with groups, rings, and fields.
2:26 PM
@anakhro Still easy.
It's amusing when the author themselves can't convince someone that the book is not for them.
@anakhro IMO everything is easy once one understands the precise logical reasoning behind the proofs and the syntax of the mathematical statements. The hard part is coming up with an idea to construct the proofs.
A pretty common but infantile approach to mathematics.
It's understandable you'd have such a misconception at your age and skill level.
Conveniently lines up with your profile picture. :P
@anakhro I guess then clear up my misconception :P
There are lots of books written on the philosophy of mathematics and what it means to write proofs, what a proof is, how mathematicians in practice find proofs, etc.
2:36 PM
@anakhro And?
Prithu: you want to study abstract algebra?
Did you try Herstein's topics in algebra or Gallian's abstract algebra?
@Prithubiswas The point is that there are many people with far more expertise in mathematics, research, and pedagogy than anyone in this chat who all basically disagree with your statement.
Herstein is a common suggestion. Was he the one who did function composition "in the right order", though?
@anakhro Oh ok. I guess I am wrong.
i don't think so anakhro, I could be wrong though. I think he did consider the composition in the familiar order.
@Koro I tried only Dummit. Didn't seem to work for me though .
2:44 PM
@Koro I commend you if this is familiar to you. :P
Ah, yes @anakhro. I confuse Herstein's topics in algebra with Herstein's Abstract algebra.
Does he not use the same notation in the other book?
In the former, $y=xf$ whereas in the latter $y=f(x)$.
Interesting that he changed it up.
If I could go back in time and change a convention, it would probably be that.
elements should be on the left
I have seen some people say how the former notation is better but I don't really understand how.
anakhro, can you please explain that to me?
2:51 PM
In a lot of cases it makes more intuitive sense due to the diagrammatic idea of a function.
A little hard without a chalkboard, but if you imagine the so-called bubble diagrams for functions, an element x goes in on the left and pops out on the right
Similarly, viewing f as a "machine" that takes an input and produces an output, you'd expect the element to be placed in the machine, not the machine placed around the element.
Function composition also looks better since it follows the diagrammatic rule of left to right.
Which is probably the main reason, I'd say.
i'll stick with the latter notation :). I am used to that :).
Indeed, being used to one notation is more than enough reason to stick with it.
As long as we don't use that as reason to say it's the "best" notation, I think we are good.
1 hour later…
3:55 PM
Hmm... I am pretty bad at proofs.
So maybe rotman isn't for me.
I seem to only know basic algebra and high-school geometry.
Try linear algebra out.
It's a natural first step in pure mathematics .
4:25 PM
@anakhro I have Axler. Is it fine?
Yes, that's adequate.
Some might disagree with his determinant-free approach, but you just want to learn some math and have fun, right?
@anakhro Yup
Then it will work perfectly fine.
Hey, guys. I'm going to need some feedback. I'm going to start writing a paper on the infinite sum I found. So far, I've got two lemmas with proofs. Can I get some feedback as I work on this?
You can post it, sure. Whether someone will take a look depends on how long it is.
4:37 PM
Well it'll grow in length over time. It'll be easier to analyze piece by piece as I progress forwards.
Out of curiosity, what is the final result?
4:51 PM
What do you mean?
It's for the quotient algorithm I found.
Is that what you're asking?
I was just wondering what the major result of the paper is.
You just mentioned "infinite sum" above, I am not around enough to know if you mentioned it in the past.
Yes, the infinite sum computes quotients for at least rationals. Not sure how to prove for irrationals and transcendentals quite yet, though.
how mysterious
5:18 PM
Ok, here's what I have so far: incongruous-yew-7c4.notion.site/…
Might want to reword lemma 1 so it is easy to read
Alright, how might I word it better?
Like "Let $n$ be a power of 2, then $$\frac{2^{\left\lfloor\log_2(n)\right\rfloor}}{n} = 2.$$"
Or "If n is a power of 2, then..."
Oh I forgot a +1, but you get my point
Lemma 1 is also so simple that you might as well combine Lemma 1 with Lemma 2 in the form of a \\begin{cases}...\\end{cases}.
@anakhro I'm not sure how to do that.
Can you show me?
5:29 PM
Right, it defines cases, but how best do I rewrite the two as a single lemma using this notation?
Lemma 1 is case 1 and lemma 2 is case 2?
Yes, but what do I put in place of $f(n)$? Just say $$\text{Lemma 1 }:= \begin{cases}...\end{cases}$$?
The expression is the same in lemma 1 and 2, yes?
just for different x or n
Ok, I see what you're saying now. I can define cases based on the domain of $x$.
Yes, they're the same expression. What about the proof, though?
I'll keep this in the back of my mind. In the mean time, I'm going to focus on more drafting and explaining how to obtain the sum.
6:03 PM
You know when your mind goes blank while staring at the screen and you're thinking about what to write and the best you can get is this?
Formal proof of my infinite sum. Done. QED, nerds. Get owned.
@AMDG Yes. As a dumb idiot I haven't experienced anything different from that.
I probably just need more coffee and stuff or something idk
6:22 PM
Is integral the commonly used adjective for "having the character of an integer"?
I've heard people say "integral portion" and whatnot when referring to the integer part of certain things.
While I'm at it, any comments on the new paragraphs thus far?
I don't see any comments on the draft.
6:43 PM
@TedShifrin RE: humor has no place in formal papers: See https://incongruous-yew-7c4.notion.site/An-Infinite-Sum-Suitable-for-the-Efficient-Computation-of-Quotients-95ce4cebc9c54ab2a7e8fc59306ce464

I'm leaving this in and there's nothing anyone can do about it.
(par. 2)
6:57 PM
Well, actually, I distinguish between high-caliber research papers and undergraduate textbooks. I have injected humor in the latter numerous times, but probably never in the former.
I guess there's a first for everything
Though I'd say this paper would fit in well with second year calc students I bet considering its domain of interest.
It's just division and they'll have been introduced to summation notation by then.
Well, first you have to get it accepted for publication :)
Of course, of course... how do I do that? lol
I'm only drafting right now of course
If you're aiming at middle-level undergraduates, then something like the MAA Mathematics Magazine or College Mathematics Journal might be appropriate, but you'll see.
Lots of things get submitted and don't get accepted.
Well it's intended to be for efficient computation of quotients by binary computers. Which journal would be most relevant in this case?
7:01 PM
I don't know the comparable journals for computer science/programming. But there must be a few.
I'm sure I'll find something. I just want fast divides for everyone. I don't think anyone is going to complain about that; I think getting my paper accepted, for that reason, is practically guaranteed (provided I can explain it well enough).
I'm not convinced.
Why, because you aren't convinced that it's competent for this purpose?
That's why I'm drafting this now.
I would want an actual computer scientist who knows stuff to give an opinion. I don't think you're going to change the way calculators and computers compute arithmetic.
7:18 PM
I am an actual computer scientist.
I'm just self-taught.
Well, when it comes to publication, typically they want "expert opinions" and one or two known professionals write reviews/referee reports.
I am a professional, but I'm not recognized as such because no one knows me or what I'm capable of, simply put, so I understand what you mean. This, in part, is to give me some sort of credibility since I don't have credentials from a university.
As I said, I cannot begin to judge and I don't want to. This is too far removed from any mathematics that I actually care about.
It's actually quite the opposite if you take a moment to glance at what I have drafted so far. It's more math than anything, and it likely won't have more than a single block of pseudocode in the paper.
Read carefully what I said.
7:24 PM
Oh I see now.
Fair enough.
An old student of mine, who has a masters in math, is a community college math professor, and is completing his PhD in CS, might be interested in looking at it and offering an opinion when you are done.
I would be very grateful if you would ask him to take a look at my draft when I have completed it.
7:55 PM
@AMDG He said he would. So work on!!
Sweet! Thank you for asking him!
I'm actually making a good amount of progress, thankfully. Much faster than I expected. I've made my way to the principal identity for deriving the recursive formula, and then the infinite sum as a limit by converting the recursive sum to an explicit formula.
(again, the notion link is to a live draft. Updates as I work on it in real time.)
1 hour later…
8:59 PM
I am a bit confused over how to apply Duhamel's Principle (en.wikipedia.org/wiki/Duhamel%27s_principle) when the forcing term depends on $t$, if it is in fact possible. In the first equations above the Wiki "Contents," it looks like the forcing term loses its dependence on $t$ when it gets moved to the initial condition, but in the "General considerations" section, it looks like it becomes a function of both $x$ and $s$.
9:11 PM
I'm not looking at the link, but why are there so many variables? Where did $s$ come from?
I think $s$ is a new notion of time introduced when swapping from the inhomogeneous PDE to the homogeneous PDE. It gets integrated away in the end and everything goes back to $t$ time.
I've usually seen $s$ to mean arclength or... some sort of length if not that specifically. Can't quite remember.
Not a relevant remark.
If you need more irrelevant remarks, I'm always available.
So $s$ is just a fixed value of $t$.
9:19 PM
Anyways, I wasn't finished, though I'm not sure how relevant it is now. Time and arclength tend to be virtually equivalent in particular contexts, especially when dealing with arclength parametrizations of a function in the form $(x(t), y(t))$.
Yes, initial condition is originally for $t=0$, but then they're talking about $P^s f$, which is a solution where the time domain is restricted to $[s,\infty)$ instead. So you're going to have "initial conditions" for every $t=s$, which looks like having the forcing term $f(x,t)$ to start with.
@AMDG I am an expert on differential geometry. This is totally NOT related to anything @user10478 is asking about.
Free association is not always helpful in chat.
I apologize.
They mention $P^sf$ being a retarded solution in the link, what does that mean?
Time $t=0$ is being delayed to time $t=s$.
so delayed, weird word to use
9:23 PM
Retard = slow down.
Delayed isn't right either. It's a different approach to the problem.
It must in some sense be related to giving a solution of the inhomogeneous problem as a convolution of the forcing term with the fundamental solution (the solution when the forcing term is a delta function).
@TedShifrin So until the end of the solution process when the integration with respect to $s$ is performed, $s$ is always treated as a constant, thus the notation-indifference between saying the new initial condition is a function of just $x$, or a function of $x$ and $s$?
Well, $s$ is of course $t$. It's talking about specifying boundary conditions when $s=t$ for every $s$ as being morally equivalent to having a forcing term $f(x,s)$ or $f(x,t)$ for every $t$.
You are specifying different initial conditions at different times $s$. So it really is a function of both $x$ and $s$ (or $t$).
Is there anyone here who's good with grammar? I'm trying to send my professor an email to ask for a recommendation letter, and I don't know if this sentence has too many commas: Recognizing that this is fairly short notice, if the time frame does not feel feasible, I completely understand.
I would put the I completely understand second. Recognizing is a participle modifying "I."
The commas are not the issue.
Oooh, so like: Recognizing that this is fairly short notice, I completely understand if the time frame does not feel feasible.
9:34 PM
"does not feel" should just be "is not" :)
Yes, much better.
@UnderMathUate short notice, okay if cannot
@TedShifrin Lol, ok yeah, I see why you say that.
@LeakyNun You mean send it like this? With no subject line or context?
@UnderMathUate it's a joke
@LeakyNun I know
@UnderMathUate yeah, he'll understand
why use many words if few words do trick
9:42 PM
Am man few word. Why more word if few word do trick?
return to monke
i needed a laugh
Tough crowd I see. youtu.be/rKC38du7h8k
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