@EricSilva I heard that Trudinger used to write papers in such a way that people knew he was on top of the PDE game but couldn't really understand and catch up
Essentially, start asking them questions related to the thing you're arguing about, but using the proper jargon so they know you know what you're talking about, sort of
Some people in my department showed some new exact ancient solutions for MCF and it was so much work to show they actually had a solution. This stuff is crazy
The minimax sphere eversion was related to that, I think. They started with the halfway model (Morin surface), nudged it slightly, and let it "flow" until the curvature was the same everywhere and it was a sphere
Suppose $\omega$ is a, say, $W^{1,\infty}$ closed form on a closed set $C\subset\Bbb R^n$ with smooth boundary. Can $\omega$ be extended closed-ly (word ?) to a nbhd of $C$?
Q. How many integers from $0$ through $60$ must you pick in
order to be sure of getting at least one that is odd? at least
one that is even?
Here is my verbal solution 'without using' pigeonhole principle.
A. The list of integers $0,1,2...,60$ has $31$ even integers $30$ odd integers...
In my second approach I took two different sets of pigeonholes for the cases of getting at least one even integer and at least one odd integer. To be sure that at least one integer is odd, I took $\{\{0,1\},\{2,3\},...,\{58,59\},\{60\}\}$ as a partition and to be sure that at least one integer picked is even, I took $\{\{0,1\},\{2,3\},...,\{56,57\},\{58,59,60\}\}$ as a partition.
**At least one integer is odd case :**Using definition of $f$ and pigeonhole principle I found that at least $32$ integers should be picked. If $31$ of them are even, then $32$nd integer must go to any of the sets $\{0,1\}, \{2,3\},...,\{58,59\}$ but not $\{60\}$ ($\because$ definition of $f$).
At least one integer is even case : Using definition of $f$ and pigeonhole principle I found that at least $31$ integers should be picked. If first $30$ of them are odd, then $31$st integer must go to any one of the sets $\{0,1\},\{2,3\},...,\{56,57\},\{58,59,60\}.$ Since every set had exactly one odd integer, our $31$st number must be even according to definition of $f$.
If $(N_t)_{t\geq0}$ is a Poisson process of rate $\lambda$, what would $P\left\{\cap \ _{r=0}^mN_r=2r\right\}$ look like? Is it just $(P\left\{N_r=2r\right\})^m$ ?
From WIkipedia
A logical system or, for short, logic, is a formal system together
with a form of semantics, usually in the form of model-theoretic
interpretation, which assigns truth values to sentences of the formal
language, that is, formulae that contain no free variables.
Accordin...
Suppose I have a statement like so: P such that Q such that R. I want to negate said statement. Is this correct way to go: -P such that -Q such that -R?
Okay, so let $f(x)$ be a continious function defined in the interval $(a,b)$. My book tells that $f(x)$ is bounded in this interval too, but what's the upper bound if you take $f(x) = \frac{1}{|x|}$ d take the interval to be $(0, 5)$
Even stranger: Theorem 2: If $f(x)$ is continious function through $(a,b)$, and it's lower and upper boudns are $M$ and $m$ respectively, then $f$ attains both values in that interval.
This is definately false.
Take $f(x) = \frac{1}{x}$, and let $(a,b)$ = $(3, 5)$
Let $[]$ denote the least integer function, and let $f$ be any real-valued function. Is it true that $|f(x) - \frac{1}{n} [nf(x)]| \le 1$ for every $n \in \Bbb{N}$? Clearly it holds for $n=1$, since in this case the quantity in the absolute value bars is just the fractional part...Wait! Maybe it's trivial. We can rewrite the inequality as $|nf(x) - [nf(x)]| \le n$, and the quantity in the absolute value is always the fractional part of $nf(x)$ which will always be in $[0,1)$, right?
Chat reference: https://chat.stackexchange.com/transcript/message/40613132#40613132 Denote $(n,m)$ where $n,m \in \Bbb{N}$. That is, the pair denotes one component of the map $f : \Bbb{N}^2 \to \Bbb{N}$
In fact, we should have $|nf(x) - [nf(x)]| < 1 \le n$ is true, from which we get the desired inequality $|f(x) - \frac{1}{n}[nf(x)]| < 1$. How does this sound?
However, consider the following list $b = (-5,1),(-4,2),(-3,3),(-2,4),(-1,5),(0,6),(1,7),...$ which is from func(Z,Bool). Then we can use any increasing map $x \mapsto g(x)$ (obvious examples including $x \mapsto nx+m$ for naturals $n,m$ to shift the members of the list $b$ and the ordering will be preserved)
It does seemed the only way to construct an uncountable well ordering is to already have a sequence a,b,c,d,... of uncountable length in the first place
I also need to read why $D$ is an infinite dedekind finite set, then $D \cap \text{Countable} = \text{Finite}$
Meanwhile reading progress on type theory is sloooooooooow
My most memorable experience of solving TISE is that holiday exercise during my 2nd year on triangle potential $V=mx$ that my lecturer gave me to further my curiosity, and I solve that by hand using series methods, without realising the solutions are called Airy functions. It is perhaps the only hard physics problem that I solved without asking anyone else for help
Hello, I'd appreciate some help with a question on (not locally trivial) bundles, particularly on when a map inducing fiberwise isomorphisms is an isomorphism of bundles.
I found the following algorithm generating magic square for a given odd integer n: https://scipython.com/book/chapter-6-numpy/examples/creating-a-magic-square/ I am currently trying to understand and wonder, how to prove it works (or doesn't) in general.
If $(N_t)_{t\geq0}$ is a Poisson process of rate $\lambda$, what would $P\left\{\cap \ _{r=0}^mN_r=2r\right\}$ look like? Is it just $(P\left\{N_r=2r\right\})^m$ ?
I can't seem to figure out what it would look like. The notation is confusing me. Should it be a multiplication, from r=1 to m, of $(P\left\{N_r=2r\right\})$ ?
@NickStephen You want the probability of your Poisson sample landing in the intersection of the events $N_0=0,$ $N_1=2$, $N_2=4$, etc.
I don't know how to express that as a sum, mind. That'd require me to think more about it than I'm willing to.
Based on the comments you've received to your question, you have enough information to solve it yourself.
There's the Schrodinger equation. That's the entirety of what you need.
Yes, it is.
I could. But I'm not going to. If you're not willing to think about what information you've been given, I'm not going to do it for you.
You're given that wavefunction and that it's a solution to the time-independent Schrodinger equation with potential $V(x)$.
That is the entirety of what you need.
in context, it's a physics problem. However, with how you presented the question on MSE, all that's present is the mathematical problem and thus it's suitable for MSE.
So yes, it's a mathematical question as you've presented it.
properly understood? a minute or two.
also, I think you stated it's a zero energy solution.
yes, because you're waiting for someone to do the work for you.
...
So you know the answer, but you want someone to answer it for you.
what you've written in your question is indistinguishable from someone who doesn't have any idea how to solve the problem and just wants an answer. as such, it has pretty much zero appeal.
2
not long. but I'm not a standard person---I'm TAing quantum mechanics this semester
so the time involved is just however much time it takes for me to take two derivatives
and to the extent that that tends to invoke the spookiness of modern physics in order to say how little we really know man...well, it gets under my skin
@Semiclassical That actually happened to me on campus a couple months back, missionary stopped me to get me to come to something which I had no intention of going to and he ended up talking about String Theory
@EricSilva So I learnt an interesting theorem yesterday
Well, by learn I really mean heard. But what it says is that if $X$ is a Riemannian manifold and $Y$ is a locally symmetric space, both of negative curvature, then the geodesic flows on $T_1X$ and $T_1Y$ are $C^0$-conjugate iff $X$ is isometric to $Y$
Q. How many integers from $0$ through $60$ must you pick in
order to be sure of getting at least one that is odd? at least
one that is even?
Here is my verbal solution 'without using' pigeonhole principle.
A. The list of integers $0,1,2...,60$ has $31$ even integers $30$ odd integers...
From WIkipedia
A logical system or, for short, logic, is a formal system together
with a form of semantics, usually in the form of model-theoretic
interpretation, which assigns truth values to sentences of the formal
language, that is, formulae that contain no free variables.
Accordin...
