How to show C being a club if C is defined as a collection of limit ordinals $< \kappa$?? I know that definition of club means closed and unbounded so I have to use proof by cases to show that C is closed and C is unbounded. What I got for unbounded is that since $\kappa \in C$ so that implies sup($C \cap \kappa) = \kappa$ but that's just a one liner.
Hi there, I'm reading a book in control systems and the author states "The Routh–Hurwitz criterion is a necessary and sufficient criterion for the stability of linear systems. The method was originally developed in terms of determinants". Could you please state any reference that discusses this issue with determinants. Thanks
I have a more complex compound interest question involving partial sums. Suppose one opened a savings account with principal P at rate i compounded monthly, then also contributed an amount c every month. What would be the closed form to find the total value of the savings account after k terms?
In other words, (((P + Pi) + c) + [((P + Pi) + c)]*i) + c) + [(((P + Pi) + c) + [((P + Pi) + c)]*i) + c)]*i...
So does anyone have a good example of an existence proof for a bijection between two sets? I was just reading that the multiplicative property of the totient function can be proven by such a proof, but it only outlines the proof, as does not include the parts I cant work out myself
@AkivaWeinberger but I think that would be a good thing, at least some aspect of our species would survive, as the artificial intelligence that we made, but that's from the perspective of someone that believes the extinction of the human race is inevitable
@AlessandroCodenotti I don't know a lot of number theory at all, but you could read a copy of David Burton's introduction to elementary number theory. Contains this proof in depth.
@TedShifrin hi thanks you're answer regarding my question about the semidefinite cone. You wrote: "Assuming you mean symmetric when you say semidefinite, the vector space of symmetric $n\times n$ matrices has dimensions $n(n+1)/2$. The positive semidefinite cone is the closure of an open subset of this." Do mean, that semidefinite cone is a closed set if you write closure? Or does is mean something more specific?
Hi, I have a question: I have a paper that says the following: Let T be a stochastic matrix. It says that "Note also that stochastic matrices preserve the $L_1 norm$, i.e. for every vector u, $L_1(u.T)=L_1(u)$". What does it mean by saying "L_1 norm"?
I don't understand! Is it a function? since he states L_1(u.T) [which vector u and matrix T]
Note that I understand Linear Algebra, basics of abstract algebra, etc.
Hey there, everyone! Does anyone know about Walton-Meek Gregory patch? I am having trouble to compute its normal and to solve its zero singularity. Does anyone know to correctly calculate and compute these? Thanks!
Reading more about modular forms, finally getting to some number theory :) Unfortunately I'm currently hanging on a lovely "obviously this is equivalent to" statement
then there is a Haar measure on $H$, called $\mathrm d\mu$, such that $\text{Lebesgue measure on $\Bbb R^{n+1}$} = d\mu \times \text{Lebesgue measure on $\Bbb R$}$
$H$ was defined to be $\{ (x_0, \cdots, x_n) \mid \sum x_i = 0 \}$
btw the inverse map $H \times \Bbb R \to \Bbb R^{n+1}$ sends $(x,t)$ to $(x_i + \frac1{n+1} t)_i$
Well, with all the typos you've made, it can't tell if you're doing it correctly. You need to take a unit vector in the normal direction, or else it won't work.
Ah I remember the good ol' days during my bachelor, when asking a question on main would immediately get someone subtly calling you a moron and answering your question
Right, @JackM. You can think of one in terms of multiplying by an elementary matrix on the left and the other in terms of multiplying by an elementary matrix on the right.
Yikes, it's crazy how people can post totally wrong answers and think they're correct.
A youngster who wrote a detailed answer explaining how $f'(x^2)$ was the derivative of $f$ with respect to $x^2$. He removed it when I said it was totally wrong.
But makes me wonder how many just-plain-wrong answers are out there.
Hello, I have a pde u_x(x,y)=0 with initial condition u(x,x^3)=h(x). I found the characteristics (t+C,D) with C,D constants. How do I continue from here to find u? Thanks
And then a French mathematician comes along and writes $(f(x^2))'$, which I totally discourage students from doing, because it leads to all sorts of crap. Differentiate functions, not formulas.
Hi @TedShifrin, I am a newbie here, its my first time. I am a computer science undergrad and now pursuing an open university BS Mathematics. I am taking Vector Calculus this year, and I found your YouTube videos on multivariable calculus to be pretty terrific resource. Thanks for giving back to the community.
Oh, @Quasar, I'm glad they help. It was my students' idea, actually. I tried to get them to record my differential geometry lectures my last semester, too, but they were too tired.
@Eran: So the function depends only on $y$, right? The way they gave you the initial values is confusing, since they use $x$. Can you rewrite that in terms of $y$?
This amounts to what I was saying, I guess. I said $u_x=0$ means $u(x,y)=g(y)$. What I said was, knowing $u(x,x^3)=h(x)$, can you figure out what $g(y)$ is?
So you know the function on the graph of $y=x^3$. That determines it everywhere, because you start at any point $(x,y)$ and go along the characteristic until it hits the curve $y=x^3$, and there it has the value that was given to you.
@TedShifrin So I go along until I hit the curve y=x^3, and there it has the value. But we said the value is constant for all y-axis, so this is the value I'm looking for in the solution of u?
right. Just last question about the method of characteristics, so you wouldn't solve it like that: dx/dt=1--> x=t+C, dy/dt=0-->y=D? You would draw/think about it geometrically?
If you wanted to be more algebraic about it, I'd probably proceed as such: characteristics tell you that $x=x_0+t$, $y=y_0$. That intersects the curve $y=x^3$ when $y_0 = (x_0+t)^3\implies t=y_0^{1/3}-x_0$, i.e. the intersection is at $(x,y)=(x_0+(y_0^{1/3}-x_0),y_0) = (y_0^{1/3},y_0)$
At which point the question is once again: $u(y_0^{1/3},y_0)=?$
If $A^m\to A^n$ is surjective while $m<n$, take another homomorphism $A^n\to A^m$ by mapping the first $m$ coordinates identically to $A^m$. The composition of the two arrows gives a surjective endomorphism $A^m\to A^m$, which must be an isomorphism.
But clearly this composition cannot be an isomorphism or $A$ would be trivial.
Well, $(y_0^{1/3},y_0)$ is the intersection between two curves: the characteristic $y=y_0$ and the curve $y=x^3$. You know that $u$ is constant on the former curve. What do you know about $u$ on the latter curve?
Then apply nakayama’s lemma(a special case of Cayley-Hamilton) which states that $IM=M$ for some ideal $I\subset A[x]$, then for some i\in I, $i\cdot m=m$ for all $m\in M$.
Another is characteristics, which gives $x=x_0+t,y=y_0+t$. That means that the function is unchanged along the line $x-y=x_0-y_0$
which really means that $u$ must be a function of $x-y$ alone, i.e. $u(x,y)=f(x-y)$. in other words, characteristics gives you a way of deducing the smart change of variables
If you would like I have another question about pde that I thought I solved good but when I check it in the end it doesn't work (but also straight lines lol)
u_x+u_y=u^2. Inital condition: u(x,0)=h(x). I defined g(t)=u(x_0+t,t) and then differentiated and found g'(t)=g^2(t). I solved the ODE and received g(t)=-1/(t+C), and then returned to u with u(x,y)=-1/(y+C). I used the inital condition and got C=-1/h(x) --> u(x,y)=h(x)/(1-yh(x)). When I check this solution with differentiating and calculating u_x+u_y=u^2, I don't get the right answer.
If something is not explained good I'll explain again :)
Is it true that $$\operatorname*{argmin}_{\lambda \in \mathbb{R}^n : p_j = \sum_{i \in [n]} \lambda_i p_i} \|\lambda\|_1 = \mathrm{e}_j$$ for any $j \in [n]$?
That is, if we express a vector as an affine combination of itself and a bunch of other vectors, does the trivial combination (assigning 1 to that vector and 0 to the other vectors) minimize the size of the coefficients (in the $\ell_1$ sense)?
This is not true for the $\ell_2$ norm.
Forgot to add the constraint $\lambda \cdot \vec{1} = 1$.
