So, I have a function $\pi(n)$ and I want it to satisfy these constraints: $\pi(\pi(n))=n$ $n\in\mathbb{N}\iff\pi(n)\in\mathbb{N}$ For a set $p\in\mathbb{N}$, then $\text{gcd}(\pi(n),p)=1$ if $n\neq p$. Is $\pi(n)=p-n$ the only function that satisfies these constraints?
Good morning all, a question from a "physical" paper but maybe some could know:
I don't understand equation (3). What does "reduced matrix element" mean. By what is it reduced?
These are scalar products in some Hilbert space, and the $\Gamma$ stand for irreducible representations, according to which the Hilber space "vectors" $\Psi$ are labelled, each symmetry species contains some $\gamma$ type vectors. Somehow it seems the construction "factors out" the symmetry part. But how this is done in detail I don't understand.
"each symmetry species contains some γ type vectors." -> "each symmetry species contains some $\mu$, $\nu$ or $\gamma$ vectors.
n f(n) f'(n) Estimate
2 0 +2 0
3 2 +2 1.5
4 4 +3 4
5 7 +4 7.5
6 11 +5 12
7 16 +6 17.5
8 22 +7 24
9 29 +8 31.5
10 37 +9 40
f(n) are from observations and I need to find f. After I observed the delta between the values I came to the conclusion that f'(n) = n - 1 after which I then did $\int_{}^{} n -1 dn \rightarrow \frac{(x-2)x}{2}$ as shown in the estimate column. The values are bit off and I can't figure out why. Can someone help? :D
How one can prove that if zeta function has pole at 1 implies there are infinitely many prime; My guess is to show first that summation 1/p diverges which implies that no. of p is infinite; but how to reach 1/p divergent series?
@IanRiley: You can look at the $n$th order difference series, thats stuff like $\Delta^n x_i = \Delta^{n-1}x_{i+1}-\Delta^{n-1}x_i$ with $\Delta^0 = 1$. But idk if there is something like a statistical theory on that.
Volatility is the most sensical to me, but it seems like that word has been captured by finance/stock people, and used to mean either standard deviation, or a High - Low
Using the triangle inequality, $|a+b+c|\ge |a|-|b|-|c|$ etc. This works the same way as the diverse root bounds for polynomials. Show that for $Re(s)>c$ and greater than the given fraction, the function value is decidedly non-zero. — LutzL19 mins ago
Using the triangle inequality, $|a+b+c|\ge |a|-|b|-|c|$ etc. This works the same way as the diverse root bounds for polynomials. Show that for $Re(s)>c$ and greater than the given fraction, the function value is decidedly non-zero. — LutzL19 mins ago
The parametrization of the curve. For instance, given $\gamma = \lambda\beta$ where $\gamma$ and $\beta$ are curves and $\lambda$ is a scalar, I'd assume the curvature of $\gamma$ is $\lambda$ times the curvature of $\beta$.
@gian good to always be aware of if you are parameterized by arclength. Just to be able to check the right formula. However, from what I remember, you are using the correct one (technically for both cases).
@anakhro "with definition ": I dropped the word "my" ... so my definition would be the second derivative, for $\Bbb \to \Bbb$ functions. For other cases the reciprocal radius of the kissing circle is better. But also in that case for curves in 2D I think curvature scales linearly. For higher dimensional surfaces its a different question.
How scales the radius of a kissing ball of a 2D surface in 3D if you scale the lengths? I dunno probably differently than linear?
Hi guys, running this by you to make sure I'm not crazy. I am asked to prove -- $A$ is $n \times n$. Prove $\mathbb{C}^n = R(A) \oplus N(A)$. But in general this is not true. Consider $A(x,y) = (y, 0)$. Then $R(A) = \{(x, 0)| x \in \mathbb{C}\} = N(A);$ where The sum is not even direct, and $R(A) + N(A) = \{(x, 0)| x \in \mathbb{C}\} \ne \mathbb{C}^2$
being careful, $A$ is linear -- $A(x, y) + A(a, b) = (y,0)+(b,0) = (y+b, 0) = A(x+a, y+b)$; and $c \cdot A(x,y) = c \cdot (y,0) = (cy, 0) = A(cx, cy)$
A necessary condition as far as I remember is $R(A^k) = R(A^{k+1})$ for the least such $k$, in which case $\mathbb{C}^n = R(A^k) \oplus N(A^k)$
@JoeShmo: No, it's always true. And, in fact, they're orthogonal complements (either with respect to real inner product or with respect to Hermitian inner product).
You're messing yourself up by writing vectors as rows instead of columns.
Bad, bad JoeShmo.
So your matrix is $\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$. Now check.
@anakhro sorry to disturb you, I read that you were doing contact geometry. I am currently having this course on thermodynamics and I can't help but notice the things like Legendre transformations that appear in classical mechanics as well as in stat mech. Thinking about how a Legendre transformation links a tangent bundle to the cotangent bundle and the symplectic structure of classical mechanics, I was also trying to find something similar in thermodynamics.
My instructor tells me that it has something to do with contact geometry. Do you have any notion of such a similarity?
@Rick Nothing to do with UCSD. AoPS is their own program for kids from 2nd grade through high school. I am teaching a calculus class, which has precalculus knowledge as a prerequisite; all of the students in the class are taking AP calculus in high school simultaneously, so I'm trying to make it a bit deeper.
So, I have a group generated by $a,b$ where $a^4=b^4=1$ and the subgroup $\langle a^2,b\rangle$ is the dihedral group on the square. As a guess, is this, by chance, the group of rotations on the octahedron?
I'm specifically referring to rotations, not general symmetries. Though, I imagine that there could be obvious things I'm missing and that the "group of rotations on an octahedron" isn't applicable here. (The reason I mentioned it is that when I noticed I had the dihedral group as a subgroup, I immediately interpreted $a$ as a "half reflection")
You should talk to Mathein and Tobias and others ... probably representation theory is going to be the way to go.
@Leaky: To do it brute force is very tricky (it took my roommate and me about a week when we were taking graduate differential geometry). However, it's easy to prove.
Yes, and I'm trying to predict what the behavior of the subgroup will be as I go, and checking to see if it lines up with what the subgroup actually ends up being.
@Rithaniel, $S_8$ is extremely complicated. Even doing this for $A_5$ and $S_5$ would be challenging, except that there there's some geometry going on.
But you should chat with the algebra guys, not me.
I remember that Spivak (who taught the course) made a big deal about equality of mixed partials and called it "IT" (because it was the key to all of diff geo, hence "That's IT"). My recollection is that we had something we called Super-IT that we needed for this proof.
Well, @Balarka, when McCrory and I wrote our beautiful little paper on cusps of Gauss maps on surfaces, there was an intricate Chern class computation that we each did something like 12 times, and we had 23 different answers. I would have given a kingdom for Mathematica or something back then ... It would have been so much easier than messing up high school algebra with zillions of variables.
The computation I was messing up was when I was trying to recollect the proof that if $f_t : M \to N$ are homotopic then $f_0^* - f_1^* = dP + Pd$ because my batchmate, who's reading forms, asked me what the intuition for $P$ is. There is a slick proof using Cartan's magic formula, by looking at the 1-parameter family of forms $\omega_t = f_t^* \eta$, which is obtained from taking Lie derivative along $\partial/\partial t$ in $M \times I$ of $F^* \omega$ (where $F$ is the homotopy map).
one of the problems in our mod rep pset involves computing the number and dimension of the simple modules of $K[G]$ where $K = \overline{\Bbb F_2}$ and $G = S_5$
@Balarka: Do you believe this? Because the torus in $\Bbb R^3$ is very non-flat, does it feel to you like he's got the right notion of constant torsion? I need to work that out.
Maybe you can write down coordinates tangential to the meridian and the parallel curves, and use Euler's formula since it makes constant angle with the meridians
Well, I guess since the two sets of circles on the torus are orthogonal, that's not surprising: constant angle with one set means constant angle with the other set.
I'm calculating, though, how quickly you have to turn horizontally if you move at constant speed around the meridian circles.
Anyone have idea how to do this?
