One problem is to prove that there is no smooth retract of the closed unit ball onto its boundary whose restriction to the boundary is the identity
One problem has 2 parts, the first of which is one we did in analysis about proving $SO(n)$ is a manifold, finding the dimension, and tangent space at the identity, but part two is proving that $SO(3)$ is homeomorphic to $\mathbb{RP}^3$, which seems fun
2 problems are from GP, one of which is to show that a manifold is locally a graph, the other is to show that if the entries in a matrix are non-negative, then it has a non-negative real eigenvalue
The graph one I assume is a graph over one of the standard coordinate planes. That's super important. E.g., how do you prove that $y^2=x^3$ isn't a 1-manifold?
The Frobenius theorem about positive eigenvalues is used in econ a ton.
I have not encountered manifolds but still I think that if I read then it makes me feel that I can understand or at least have an intuition about many complex surfaces ,shapes ?..
The final problem is to find a map $F:\{(x,y):y\ge 0\}$ such that for any $p\in F^{-1}(0)$, $dF_p$ is surjective, $F^{-1}(0)$ intersects the real axis, but $\partial F^{-1}(0) \ne F^{-1}(0) \cap \{(x,0) \}$
Let D be a Euclidean domain with euclidean valuation ν. Then if a and b are associates in D, then ν(a) = ν(b). ? a = b . u for some unit u in D applying euclidean valuation on both sides ?
By basically taking a point and looking at the rotation which fixes its axis, by angle equal to $\pi$ times the magnitude, with some orientation chosen to decide which side you're going
The nice thing is that $f(x) = f(-x)$
But this is kind of different than what I was looking for
Like, I can't nicely identify things otherwise, because if two points are antipodal, they share an axis of rotation, but the only way the direction of rotation is not arbitrary is if you rotate by $\pi$
This function extends it and makes sure that it's not adding new equivalencies
Let D be a Euclidean domain with euclidean valuation ν. Then if a and b are associates in D, then ν(a) = ν(b). ? a = b . u for some unit u in D applying euclidean valuation on both sides ?
For what it's worth, anyone who's interested in increasing the visibility of female mathematicians might want to consider adding a name to this page, or creating an article for a name already on this page: en.wikipedia.org/wiki/Wikipedia:WikiProject_Women_in_Red/… it could use some crowd-sourced effort
@Fawad no what i do is i made a system of equations: I have practice in making partial fractions so I know how the splitting occurs. your problem is quadratic upon cubic. 3-2 =1 hence the numerator will have integers and denominator will have increasing degree (x-3) terms.
By Zhang's result on Twin Primes. For any sequence (including finite) of distinct integers $\geq 246, \ a = (a_1, a_2, \dots)$ there exists an infinite sequence $p = (p_1, p_2, \dots )$ of distinct primes, such that $2a_i + p_j $ is prime for all $i, j= 1, 2, \dots$
In other words certain subse...
By Zhang's result on Twin Primes. For any sequence (including finite) of distinct integers $\geq 246, \ a = (a_1, a_2, \dots)$ there exists an infinite sequence $p = (p_1, p_2, \dots )$ of distinct primes, such that $2a_i + p_j $ is prime for all $i, j= 1, 2, \dots$
In other words certain subse...
More sanity checks, is there any difference between choosing points uniformly in $[0,1]^2$ and choosing the $2$ coordinates each uniformly in $[0,1]$ independently?
@Astyx well first of all, the book that was recommended to me, contained not a single proof :P
so I switched to something more rigourous
and it's going ok, I have one small question!
They say that $\lim_{n\to\infty}x_n=\lim_{n\to\infty}ax^{k-1}=\infty$, but I don’t see why that is true for $x<1$ (it’s obvious for $x>1$). I understand that the series diverges, because $\lim_{k\to\infty}ax^{k-1}\neq0$, but I don’t see why it equals infinity.
(also, isn't it a typo? shouldn't it be $k\to\infty$ instead of $n\to\infty$)
I think I basically have to show that $\lim_{n\to\infty}(-1)^na^k=\infty$ for $a>1$
which can't be true. I'll conclude for now that they wrote a misfortunate "typo" (infinity sign while meaning divergence), that didn't really matter, because all we needed was that that limit didn't equal zero
Pick a topological space $X$ of arbitrary cardinality, consider the set $Y=X\cup\{x\}$ which is the same space with a single point added and define a topology on $Y$ by taking the open sets in $X$ and adding $x$ to all of them. Now the singleton $\{x\}$ is dense in $Y$
@DHMO I need help. If $y=a\sin (kx-\theta +\phi)$ and $y=A\sin(kx-t\theta ) + B\sin(kx-t\theta$ then $a=\sqrt{A^2+B^2}$ and $\phi$ = \arctan \dfrac{B}{A}$
@DHMO I need help. If $y=a\sin (kx-\theta +\phi)$ and $y=A\sin(kx-t\theta ) + B\sin(kx-t\theta$ then $a=\sqrt{A^2+B^2}$ and $\phi$ = \arctan \dfrac{B}{A}$
By Zhang's result on Twin Primes. For any sequence (including finite) of distinct integers $\geq 246, \ a = (a_1, a_2, \dots)$ there exists an infinite sequence $p = (p_1, p_2, \dots )$ of distinct primes, such that $2a_i + p_j $ is prime for all $i, j= 1, 2, \dots$
In other words certain subse...
