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?
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.
@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".
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.
Let $f: X \to Y$, and let $A \subseteq X$ And $B \subseteq Y$. Prove the following equivalence
$$f(A \cap f^{-1}(B)) = f(A) \cap f(f^{-1}(B)) = f(A) \cap B.$$
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.
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."
@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.
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).
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.
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.
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).
@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.
@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.
@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?
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.
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.
@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.
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.
@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?
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.
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?
@TedShifrin RE: humor has no place in formal papers: See 1.1.5.2 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.
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.
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.
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 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.
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.
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.
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'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.)
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 $s$.
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.
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.
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.
Let $f: X \to Y$, and let $A \subseteq X$ And $B \subseteq Y$. Prove the following equivalence
$$f(A \cap f^{-1}(B)) = f(A) \cap f(f^{-1}(B)) = f(A) \cap B.$$
