it sends a highly focused beam through a medium and to not damage the detector, the center of the beam is stopped by a physical barrier before it hits the detector
@xxxx036 A quadratic is factorable over the reals iff b^2 - 4ac >= 0 as pp said. You want to know when this remains true even if you negate a,b, or c. All that can do is change the inequality to b^2 + 4ac >= 0. Moving to the other side you want to know that b^2 is >= both 4ac and -4ac, aka, b^2 >= 4|ac|
how am I even supposed to get intuition for this physics course lmao the instructor told me I only need probability theory and all this is only applied probability
and then he defines this without batting an eye https://en.wikipedia.org/wiki/Dynamic_structure_factor
I mean, mathematically no problem, but how am I supposed to know what information this gives me when the experiments that give us the functions are called like this
"Experimentally, it can be accessed most directly by inelastic neutron scattering or X-ray Raman scattering."
"the intermediate scattering function and can be measured by neutron spin echo spectroscopy" ah yes thank you
Hello guys. This looks rather basic but anyway... Suppose $f$ is a $C_0^\infty(\mathbb{R}^n\times [0,\infty))$ function with compactly supported in $\mathbb{R}^n$ (i.e. for each fixed $t$, $f_t:\mathbb{R}^n\to \mathbb{R}$ has compact support.
May I write $$\lim_{t\to \infty}\int_{\mathbb{R}^n}f(x,t)\,dx = \int_{\mathbb{R}^n}\lim_{t\to \infty} f(x,t)\,dx?$$
Hi @TedShifrin I wanted to know if you could verify my thinking. It involved two questions from Sec 2.1: 2b and 3:
2b) Assuming I got the original parametrization correct (it feels like I did) the second part asks to produce infinitely many solutions. So to do that I replaced $1$ with $z$: $(x,y) = (\frac{z-t^{2}}{z+t^{2}}, \frac{t}{z+t^{2}})$ where $t \in \mathbb{Z}$ and $z \in \mathbb{R}$.
3) For this one I used the idea of arc length and adding vectors. So based on the diagram in you text it appears that the arc length of the circle up to the point $a$ is equivalent to the straigh…
@dc3rd: If you replace $1$ with $z$, you're no longer on the circle. The issue is to think about putting in rational values of $t$ !! For #3, of course you want to add vectors. The $a(\cos\theta,\sin\theta)$ is clear, yes. But the point is that the vector from the edge of the reel to $P$ is orthogonal to the first vector (that's what pulling taut does). So you have length, but the direction you used was totally random.
@TedShifrin for 3) I just scribbled something down form your suggestion of orthogonality. So using that idea what came to mind was taking the dot product of the radius with the direction vector going from $a$ to $P$. In other words:
Letting $P = (x,y)$
$(a \cos(\theta), a \sin(\theta)) \cdot (x - a \cos(\theta), y - a \sin(\theta)) = 0$. But of course solving for $x$ and $y$ becomes hard because I only have one expression.
Hmm...interesting. I see it, now it is a matter of "mathefying" it. So in english, I would be rotating the segment $\overrightarrow{aP}$ that attaches to the radius $a$ from the origin. Just going to go work on the correct expression.
$(x,y) = (a \cos(\theta) + a \theta, a \sin(\theta) + a \theta)$ would give me the full vector from the origin to $\overrightarrow{OP}$ in a straight line, but I need to rotate the fragment $\overrightarrow{aP}$. This fragment is: $\overrightarrow{aP} = ( a \cos(\theta) + a \theta - a , a \sin(\theta) + a \theta - a)$. But in this form it is pointing in the same direction as the full vector $\overrightarrow{OP}$, so to rotate it in the correct direction the coordinates would be $(y, -(-x))$, since the original vector $OP$ points on the negative…
Forget your wrong stuff and start over. The first vector we know. The second vector is orthogonal to it and has length $a\theta$. What then is the second vector?
Dr. Shifrin! For part (c) of the limits exercise, where you wrote that b) and c) are related, this is how I solved it earlier: I let $P$ be the statement that the limit for $f$ exists, $R$ that the limit of the sum $f + g$ exists, and $Q$ be the statement that the limit for $g$ exists. Then from b) we know that $P \land R \implies Q$. We can write this as $\neg P \lor \neg R \lor Q$.
Then notice that $P \land \neg Q \implies \neg R$ can be written as $\neg P \lor Q \lor \neg R$, and so by commutativity of the logical or, we know both statements are true.
The paper's link is https://arxiv.org/abs/1704.05747.
In this paper, we give a proof to Riemann Hypothesis, please feel free to indicate any omissions. It's my pleasure to communicate with anyboby who has any questions.
my prof has got to be kidding me. If I understand this correctly he first uses $\vec R_k$ as time-dependent processes governed by differential equations given by the poisson-bracket... and then proceeds to use brackets to signify that some function has inputs $(\vec R_1, \vec P_1,..., \vec R_n, \vec P_n)$? Literally one line afterwards?
Also, $R$ and $P$ are treated as functions and as variables in which we differentiate? And $\vec R_i(t)$ is differentiated with respect to $\vec R_k, k = 1,...,N$? I just don't get this... Is this only a complicated phrasing of $\frac{d}{dt}\vec R_i(t) = (\partial_{r^k_1}H,\partial_{r^k_2}H,\partial_{r^k_3}H)^T$ where $(r^k_1,...,r^k_3) = \vec R_k$ is the $k$-th variable of the Hamiltonian
Ah yes that seems to be it
goddammit
Oh, must be $\frac{d}{dt}\vec R_i(t) = (\partial_{p^k_1}H,\partial_{p^k_2}H,\partial_{p^k_3}H)^T$
lmao someone says that literally every time I post an excerpt
en.wikipedia.org/wiki/Liouville%27s_theorem_(Hamiltonian) what is $\frac{d \rho}{d t}=\frac{\partial \rho}{\partial t}+\sum_{i=1}^{n}\left(\frac{\partial \rho}{\partial q_{i}} \dot{q}_{i}+\frac{\partial \rho}{\partial p_{i}} \dot{p}_{i}\right)=0$ even supposed to mean
why is a regular derivative with respect to $t$ different than a partial derivative
physics.stackexchange.com/questions/28630/… "The best way to define this free of mathematical annoyances"... [goes on to define some lattice limit and then integrates over non-measurable, extremely incontinuous function]
To quote a set of lecture notes "Mathematically, the process $\xi_t$ suggested in this paragraph is a mess. One could, at least in principle, construct such a process using a technique known as Kolmogorov's extension theorem. However, it turns out that there is no way to do this in such a way that the sample paths $t \mapsto \xi_t$ are even remotely wellbehaved: such paths can never be measurable [Kal80, Example 1.2.5]. In particular, this implies that there is, even in principle, no way in which we could possibly make mathematical sense of the time average of this process.
loool accidentally stumbled across a ron maimon post
Tbf he's smart. And does nothing which isn't completely physics standard in that post, I think
@Thorgott yeah physics SE is weirdly combative
@Thorgott also, physics theorem: all measurable functions are differentiable
A is only supposed to be measurable (observables are complex-valued elements of $L^2$)
@user2103480 when I did my set theory seminar on the consistent generalization of Fubini the professor (Luecke) commented that in some set theoretic universes the physicists are right
Ok so say that we have $ (C_f)_M := (\overline{k}[x,y]/(f(x,y)))_M $ where $f$ is irreducible, and $M$ is generated by $x$ and $y$. Assuming the maximal ideal of $(C_f)_M$ is generated by one element, $z$, we can eventually write $rx - sy = 0$ in $ (C_f)_M$ because we can write $x = zs, y =zr$.
Since we can assume wlog $s$ a unit in $(C_f)_M$ we can pick polynomials $\tilde{s}, \tilde{r}, g \in \overline{k}[x,y]$ such that:
$f g = \tilde{r} x - \tilde{s} y$
Ultimately I want to show that $f_y (0,0) \neq 0$, but I get to a step where $f_y (0,0) = -\tilde{s}(0,0)/g(0,0)$ and I am not so s…
my favorite moment in my physics lecture this semester was when the professor, after months of never worrying about whether we actually divided by 0 at any point, stopped during a calculation to worry momentarily about whether we are dividing by 0.. and it was a calculation in which we divided by an exponential
This is a follow-up on this post, which is based upon this paper. First, let me set up some definitions, etc.
A Semigroup $S$ is said to be an inverse semigroup provided that for every $x \in X$, there exists a unique element $x^{-1}$ such that $x^{-1}xx^{-1} = x^{-1}$ and $xx^{-1}x=x$.
The se...
Hi @TedShifrin. I believe I got the answer to the question we were discussing yesterday, but I wanted to go through my thought process to make sure it is the right way to approach these things. I also drew a crude diagram to illustrate the main part of my idea.
So to start off, the second vector is: $(a \theta\ \sin(\theta), - a \theta\ \cos(\theta))$
How I got this was the following:
Knowing that the arc length of the second vector is $a \theta$, I could think of this as another vector emanating from the origin with coordinates given by $(a \theta\ \sin(\theta), a \theta\ \cos(\theta))…
No. First, don't call length of a vector arclength. It's just length. :) You corrected yourself at 5 AM, so don't fall back in the same mistake. What are the two unit vectors orthogonal to $(\cos\theta,\sin\theta)$?
Yeah, @dc3rd, my emphasis on geometric thinking may not be good for you :) That said, statistics is really full of projections and geometry, even though statisticians don't teach it that way.
Well, technically it's the same course. They split calculus for business majors over two semesters. First semester we cover derivatives and second we cover antiderivatives
The only issue is that I have some people who haven't taken the first half for several semesters, so I can't really rely on their calculus knowledge being fresh in their minds. (Also, the lack of prep, as you say)
@TedShifrin more the reason for me to integrate this form of thinking. Give myself the opportunity to broaden my thought processes. That's one of my underlying motivations in pursuing math......I also have noticed all of the geometrical objects that appear in statistics hence why I actually do want to understand a geometric way of thinking. Because sure we could only "picture" a distribution in 3 dimensions, but having the understanding of how these objects behave is invaluable.
Also Halmos's essay on math as art comes to mind in my efforts
Yeah. We found at UGA that people that waited more than one (or at worst, two) semesters to go on to the next course were almost always in trouble. I think we even got a flag on the class roll so that teachers could alert the students, but we couldn't force them to retake the course.
Well, @dc3rd, in fact so many concepts in stat are about projections and dot products. Even regression is solving the normal equations (you'll get to that in chapter 5) and finding the projection onto the column space of an appropriate matrix.
I co-directed a masters thesis on the geometric linear algebra in standard statistics stuff. I can send it to you if you are interested (eventually?).
Yes. I'm also doing a linear regression course at the moment and have seen the basic use of matrix for simple linear regressions. So I know the use of linear algebra is only going to get larger as I look at multivariable systems.
Yes I would appreciate looking at the thesis. I always welcome sources of knowledge @TedShifrin :)
Well I'm off to go do some more studying. I'll probably be back in the room in a few hrs asking questions or just absorbing what you masters of the art form discuss
I want the least natural number for which $(n!)^2$ is greater than $n^n$ how can I solve it? The qn I got was if this number is more or less than 1000.
does the following necessarily hold if $(x_n)_{n \in \mathbb{N}}$ is a real valued markov process? $\mathbb{E}[\mathbb{E}[x_{n+1} | x_{n-1}] | x_n ] = \mathbb{E}[x_{n+1} | x_{n-1}]$. It feels like the answer should be no, but I can't find an easy counterexample
I have only encountered stirling only once in a beginner statistical thermodynamics book, and I am presently in high school so I'm afraid I don't get what you meant by that
since its a markov process, we do have $\mathbb{E}[x_{n+1} | x_{n-1}] = \mathbb{E} [\mathbb{E}[x_{n+1} | \sigma(x_0,...,x_n)] | x_{n-1} ] = \mathbb{E}[ \mathbb{E}[x_{n+1} | x_n ] | x_{n-1} ] $, but that doesn't give us what we want
we want to show $\mathbb{E}[x_{n+1} | x_{n-1}] = \mathbb{E} [ \mathbb{E}[x_{n+1} | x_{n-1} ] | x_n]$
which is the equality on the first line, with $x_n$ and $x_{n-1}$ swapped
Given a variety $X$, is an algebraic subset then a subvariety? And would a subvariety be something such that it's a variety on its own with the inclusion map being a morphism?
The paper's link is https://arxiv.org/abs/1704.05747.
In this paper, we give a proof to Riemann Hypothesis, please feel free to indicate any omissions. It's my pleasure to communicate with anyboby who has any questions.
This is a follow-up on this post, which is based upon this paper. First, let me set up some definitions, etc.
A Semigroup $S$ is said to be an inverse semigroup provided that for every $x \in X$, there exists a unique element $x^{-1}$ such that $x^{-1}xx^{-1} = x^{-1}$ and $xx^{-1}x=x$.
The se...
