@TedShifrin By assuming that $f$ has at least 4 roots, say $x_1 < x_2 < x_3 < x_4$. Then since $f(x_i)=0=f(x_{i+1})$, there exists $c_i \in (x_i,x_{i+1})$ such that $f'(c_i) = \frac{f(x_{i+1})-f(x_i)}{x_{i+1}-x_i}=0$.
Well, one good thing came from my idiocy with non-orientable Gauss Bonnet. I'm now going to go up to Irvine and see Dick Palais and Chuu-Lian Terng for Chinese dinner :P
And tbh I think I'm probably making a bit more progress on math than when I used it to procrastinate tbh. Going through some AT now (gotta at least learn fundamental group and covering spaces properly, having done problems, before class this fall)
Maybe something like... if $d$ is negative, then for $n > 0$, $\log(cn + d) < \log(cn)$, and if $d$ is positive, then for $n > 1$, $\log(cn + d) < \log(cn + d + d(n - 1)) = \log n + \log(c + d)$.
Let $A \subseteq \Bbb{R}$, let $a$ be a limit point of $A$, and $f,g : A \to \Bbb{R}$ functions. If $\lim_{x \to a} f(x) = \infty$ and $\lim_{x \to a} g(x) = L \in (0,\infty)$, then $\lim_{x \to a} f(x)g(x) = \infty$.
I first tried to solve this problem "from the definitions", but I couldn't...
Well, by definition we have that for every $\epsilon > 0$, there exist $\delta>0$ such that $|x-a| < \delta$ implies $f(x) > \epsilon$ and $K-\epsilon < g(x) < K + \epsilon$. Hence $f(x)g(x) > \epsilon(K-\epsilon)$. My idea was to choose $\epsilon > 0$ so that $\epsilon (K-\epsilon) > \epsilon$...
But that's true if and only $K-\epsilon > 1$, which isn't necessarily the case.
(continuing what i was saying) You can imagine that when $x$ is close to $a$, 1. $f(x)$ can be as big as you want 2. $g(x)$ is roughly $L$
which means you want to be in the situation where e.g. 2. $g(x) > L/2$ (which should be possible as long as $x$ is close enough to $a$) 1. and then $f(x) > M/(L/2)$
Saying $g(x) > ?$ is harder. But you know that $g(x) \to L$, so what that means is - for anything smaller than $L$, e.g. $L/2$, or $L/3$, or $0.99999L$ - once you are close enough to $a$, then $g(x)$ would be bigger than that thing ($L/2$ or $L/3$ or $0.99999L$)
$L/2$ is just a convenient choice unless I want to troll and do $0.314159265L$ instead
and meh I'm too lazy to write a full fledged answer based on this. If you are working on an assignment or something like that though, it maybe a good exercise to write that out once yourself
Write (1) and (2) as a system of HOMOGENEOUS equations ($0$ on the rhs). It has a nontrivial solution. Therefore the matrix of coefficients has determinant 0.
The tables below show the values of predicates P(x, y), Q(x, y), and S(x, y) for every possible combination of values of the variables x and y. The row number indicates the value for x and the column number indicates the value for y. The domain for x and y is {1, 2, 3}.
https://i.imgur.com/tJeR8WA.jpg
Indicate whether the quantified statements is true or false ∃x ∀y P(y, x)
I cant quite figure out how to deal with the swapped x,y --> y,x
I understand perfectly when the order of x,y is flipped like this ∀x∃y y2 = x, but they've described a table and how to read x,y on it, so I dont know whether I invert the manner in which i would be reading it or what
"Let $f(x)=x/3$ and $g(x)=3x$, $f,g:\mathbb R \to \mathbb R$. Determine whether they are topologically conjugate".
My thinking: h=1/x clearly satisfies $h\circ f = g \circ h$ but is not a homeomorphism on the real line since it is not continuous at x=0. But this does not show that $h$ is the only function that can be chosen. How do I determine this more generally?
I have to evaluate $\int_0 ^{1/2} dx/((1+x^2)\sqrt(1-x^2)) ... I substitute x=1/t to and then it simplified into a simple integral. Is there any objection or incorrect mathematics in this substitution?
so they are wrong about that line. However they actually never used it because they didn't go as far as they should have and stopped at det(conjugate(transpose(A))) * det(A) = I
from there the correct deduction is conjugate(det(transpose(A))) * det(A) = 1
@ForeverInactive I received your chat messages. If you want you can visit my new youtube channel (in my profile) and there you will see my new email address. If you want you can contact me and we can talk privately there. Take care!
@Abcd I am not sure which high school textbooks are good actually. But for some high school mathematics, you may want to see Basic Mathematics by Serge Lang.
@Abcd If you want you can buy the books on cambridge.org that are endorsed by Cambridge for the GCE O level and A level exams. Those are usually more well written than other books I have come across.
I do not care about popularity or rankings. Most rankings are superficial to me. I just judge a book, course, syllabus, or teaching method based on its own merit.
Most math teachers I have come across know very little about math, and teach their students the wrong things.
The same might be said of those who write books for school students.
hello, i need help with the integral e^(x^0.5). Or, i see the solution in front of me, but i don't understand the substitution made there (u := x^0.5). It seems like i dont understand the concept of integration by substitution right... i thought the derivative of that what i substitute has to appear as a factor in my integrand, but this is not the case here?..
But can someone please explain this thing: Sum of the products of the elements of any row or column with the cofactors of the corresponding elements of any other row or column is 0?
I have to evaluate $\int_0 ^{1/2} dx/((1+x^2)\sqrt(1-x^2)) ... I substitute x=1/t to and then it simplified into a simple integral. Is there any objection or incorrect mathematics in this substitution?
@tatan mathematica gives the result as simply $\frac{1}{\sqrt{2}}\tan^{-1}\sqrt{\frac{2}{3}}$, so the suggestion that said substitution simplifies matters seems reasonable
if A*adj(A)=adj(A)*A=det(A)I, then upon transposing both sides you get $A^\top (\text{adj } A)^\top = (\det A)I_n = (\det A^\top)I_n$
and since the adjugate of A^top should satisfy $A^\top (\text{adj }A^\top)=\det (A^\top)I_n$, we see that they'd better agree
(not sure that's a sufficient proof tho. all it shows is that the two matrices satisfy the same equation; one has to further argue that only one such matrix exists.)
Still having a little bit of trouble seeing this. How to prove the following Let ord(a) = n in group G, n in Naturals. prove that $ord(a^k)|n$ for any k in integers
So the subgroup generated by $a$ will be $e,a,a^2,...,a^{n−1}$
@Abcd start from $A(\text{adj }A)=\det(A)I_n$. Since adj(A) is proportional to the inverse of A, the two commute and you can equally well say that $(\text{adj }A)A=\det(A)I_n$.
(look at the action of $\Bbb Z$ on $G$ by sending $(n,b)$ to $b^n$, and deduce the first theorem I quoted, using the fact that every subgroup of $\Bbb Z$ is cyclic)
@LeakyNun Doesn't that theorem come from the fact order is defined as the least positive integer $m$ to satisfy $b^m = e$. And thus no matter what ord(b) | m ?
i wanted ur opinion ive asked this before. I failed my classic differential geometry class the previous semester and now i have 2 option. Take the course again or take measure theory class. What would be more beneficial. (my interests are more towards algebra). Taking a new course is always more fun but .. i dont know..
If I understood correctly, that union and intersection is basically saying that the Liuoville numbers are those that are arbitrarily close to the center of the band of irrationals that surrounds each rational
Let $A \subseteq \Bbb{R}$, let $a$ be a limit point of $A$, and $f,g : A \to \Bbb{R}$ functions. If $\lim_{x \to a} f(x) = \infty$ and $\lim_{x \to a} g(x) = L \in (0,\infty)$, then $\lim_{x \to a} f(x)g(x) = \infty$.
I first tried to solve this problem "from the definitions", but I couldn't...
