I cannot remember what is true about actual Hilbert spaces versus what we sometimes call Hilbert spaces but actually aren't. Do $U$ and $U^\dagger$ always have the same domain in a Hilbert space?
I have posted a question about expected value. I have the calculations but I miss something since y result is negative, which cannot be. Could you take a look at the edit part of my question?
Your payoffs are incorrect. If you get 0 matches, you should lose 1 euro, not have a payoff of 0. The way you have structured the game mathematically, the player wins either 0 euros, 1 euro, 2 euros, or c euros. So if c is also positive, then the player can never lose; at worst they 'push.' That's why you're getting that 'c' has to be negative to have 0 expected value.
So do we have that if none of the dice shows the number that we have chosen then the profit is Payout−stake=0−1=−1 euro ? Then at the expected value is the term $\binom{3}{-1}$ defined? Is it defined with negative numbers?
So is it correct that I get a negative number for $c$ ? What do you mean by "push" ? @KevinDriscoll
Yes if none of the dice match your number, the payoff is -1.
I do not think you understand where these binomial coefficients are coming from / what they represent. The small x represents the number of dice that match your chosen number, not the payoff. So we don't need to worry about what happens if we try to take 3 choose -1.
"Push" means that you get your stake back. Neither win nor lose money. Is the correct answer for part c of your question a negative number? No, I don't think so. Given the way that you set up the equations, did you have to get a negative value for c? Yes.
So at (b) my equation is correct, isn't it? At (c) we have to replace only the coefficient by "c" not all the "3" by "c", right? That is my mistake, right? It should be $$E[X]= 0 \Rightarrow 0\cdot p_X(0)+1\cdot p_X(1)+2\cdot p_X(2)+c\cdot p_X(3)=0$$ right? @KevinDriscoll
Indeed, I have been sometimes, back when I was but a lowly grad student. I expect to be treated much more harshly now that I'm just some rando with a PhD.
Oh yes at (b) we have $$E[X]=(-1)\cdot p_X(0)+1\cdot p_X(1)+2\cdot p_X(2)+3\cdot p_X(3)$$ and at (c) we have $$E[X]= 0 \Rightarrow (-1)\cdot p_X(0)+1\cdot p_X(1)+2\cdot p_X(2)+c\cdot p_X(3)=0$$ right? @KevinDriscoll
@leslietownes For practical purposes here, I am neither a physicist nor a mathematician. Closer to the awful hybrid mathematical physicist. And the cases where the domains fail to coincide actually matters to me sometimes. For example, my entire thesis is on some systems where the Hamiltonian fails to be essentially self-adjoint.
well that's weird. i assumed physicists always wanted their hamiltonians to be self-adjoint. i think you're a mathematician.
unless we're in a multiverse where we're all evolving accoridng to different self-adjoint extensions of the same thing. which is actually my theory of the universe.
I mean if it can be self-adjoint, then by all means let's go with that. Sadly when you take the limit that the "short-range" physics of atoms gets squished into a smaller and smaller region while holding the "long-range" physics fixed, it turns out that the result is a non-self-adjoint Hamiltonian
Which it should be, if you think that the short-range physics is not uniquely determined by the long-range physics. The fact that the resulting model is not essentially self-adjoint corresponds to the fact that there is some range of "short-range" physics that can correspond to the same long-range behavior. And the degrees of freedom that you have to choose are precisely the self-adjoint extensions.
I haven't done the algebra, so I'm not sure if the payoff is negative or not. But it wouldn't surprise me. Real life gambling games are always set up so that in the long run the player loses money to the "bank."
Yeah, a house edge of ~0.7% is pretty good. You'll usually only do better playing Blackjack or Video Poker optimally. If you get enough comps you might be able to turn a profit!
I have been watching talks on symplectic and contact geometry recently trying to get a flavor of what's going on, might it be useful for me to study more deeply, etc. So far I have noticed that questions of which manifolds have symplectic/contact structure, if you have almost-symplectic/contract structure can you extend it to genuine symplectic/contact structure, and if two symplectic/contact structures agree on some nbhd or submanifold, say, do they agree everywhere get mentioned all the time.
Clearly they are foundational to the field in the same way that we spent a lot of time in differential topology asking when a manifold has a smooth structure and how many does it have up to diffeomorphism and so on.
From many of the talks I get the sense that people are trying hard to prove *positive* answers to these questions. In some sense, they want manifolds to have these structures, they want to be able to extend almost-structure to genuine one, and for agreement somewhere to imply agreement everywhere.
If such things were true, it seems like you could do a lot constructively by defining structures locally someplace and then guaranteeing that it extends to the whole space.
On the other hand, when I listen to Yakov Eliashberg it seems pretty clear he wants these things to be false. And I *think* I understand why he wants them to be false.
If you can't homotope almost-structures to genuine ones, presumably there is some characteristic obstruction that prevents it. We already know about some of them: these volume and no-squeezing constraints. And the point seems to be that if there are more, then they are novel in the sense that they lead to invariants that are different from the ones that we know about. That's good because presumably the invariants are topolog…
Can anyone comment on if I am getting the right gist here? Positive results are nice because they allow you to convert local constructions into global ones. Negative results are nice because presumably they lead to invariants that you can use to classify things.
the paths they've chosen are somewhat arbitrary. if you vary them you get different results. they probably chose those C_1 and C_2 because the integrals don't get too ugly.
it looks like maybe in (b) they're setting up an example that shows how the value of the result can depend on the particular way you travel from (0,0,0) to (1,1,1).
but the stuff in the screenshot applies to (a) only.
it drives me crazy that the "(a)" and "(b)" and the figure annotations are in times new roman and yet the question is in a sans serif font (maybe a variant of arial).
Lol, yeah. The formatting for this textbook is nutty. I don't want to buy the physical copy though because even my prof said this isn't the best calc text out there.
Agreed, the inconsistent fonts are a nightmare. Very possible this happens because in some LaTeX package, the test body font is controlled in a different place to the enum fonts and someone didn't know or couldn't be bothered to change them all
i was looking for a good horror movie, but couldn't find anything that appealed to me. a lot of the more recent stuff is too pointlessly violent, and i've seen all of the old stuff already.
I watched a show called See. It shows a world in which everyone has lost the power to see so when they are told that in past people could actually see, they would think of that to be a myth.
and one in horror genre is: Servant, though I don't know if anyone would like but I liked it.
I'm doing a reading about Godel's incompleteness theorem for my math history course and I just got up to the part where it discusses statements that are "formally undecidable". I'm frustrated because I learned about one of these statements in a math class this semester and I can't remember the class or the statement -_-
It's at the tip of my brain, so I'm 99% sure this event actually happened lol
Maybe I'll remember what I'm thinking of once I get some sleep.
When checking if the expansion point x=x0 for ode, to solve using series method, is ordinary or not, we normally just check if B(x) or C(x) are analytic. Where the ode is y''+B(x) y' + C(x) y=0. But what if there was an f(x) on the RHS? do we need to also check if that is analytic or not at expansion point? Here is an example, using first order. Given y' + x*y = 1/x^4 and the expansion is around x=0. Clearly if we just look at y'+x y=0, then x=0 is ordinary point.
... Do I need to worry about 1/x^4 or not? Currently I only consider the homogeneous part of the ODE to determine if the point x0 is ordinary or not.
is this true without more assumptions on $f$? if $f$ is entire, and $f f' f'' $ never vanishes, then $f = \exp(az+b)$ for some constants $a,b$. I know this is true if we also assume $f$ is of finite order (by hadamards factorization theorem), but I don't know if this is true in general
ideally (if it were true) one should be able to show $\frac{f'}{f}$ is forced to omit three points, it already omits $\infty$ and $0$
so one would hope there would be a way to say if $\frac{f'}{f} = C$ for some constant $C \neq 0, \infty$, then we contradict our hypothesis.. but i'm beginning to believe there may be a counterexample to this statement if $f$ does not have finite order
Yeah, I mean I'm trying to implement Gamma, and using the reflection formula and Zeta seems like a better and better idea, but I'd prefer to implement Zeta using Gamma instead, and then I want to see how to implement the incomplete Gamma functions.
@Derivative I've been trying to find a closed form of Gamma through the reflection formula that Euler found. I haven't made significant progress. It just seems like it would be possible to manipulate this into Gamma(x) and then generalize to the complexes. desmos.com/calculator/gm1ttzfqs5
However, I don't know enough about Gamma in principle to efficiently find such a closed form with what is given here. It would seem that we have all the pieces, and I just don't know how to fit them together.
This is a really convenient form, too, since I can easily compute secant; the whole motivation here, of course, is uniform convergence across the entire domain of Gamma(z).
Been messing around some more. It looks like a plausible path forward is through derivatives here and manipulating digamma since the derivative of Gamma here is a sum of digamma functions multiplied by the original gamma product. That makes things a bit easier since the integral of these sums are sums involving log gamma.
Though it does look like digamma can be implemented as a piecewise function for $x<0$ and $x>0$ with the former being what appears to be a form of tangent.
jakobian: do you know anything about duality in L^p or L^p_loc spaces, or is this arising somewhere where maybe nobody bothers with that layer of abstraction
one way of seeing this for harmonic functions is noting that harmonic = continuous + mean value property on balls, with the right hand side not depending on what coordinates you choose
we're talking about the same thing. the laplacian is one thing. youcan express it differently in different coordinate systems, but it's not an artifact of any particular choice.
you originally asked about harmonic functions. those are characterizable by a mean value property which is obviously coordinate independent, although this doesn't address the more general issue of the laplacian, which i noted is \pm div grad, which is hopefully coordinate independent.
then you said it's coordinate independent using nablas.
same thing all the way through.
i forget why we didn't use nablas in my geometry class.
ok. 'making the variables match' is not likely to have intrinsic meaning. when one says the laplacian is independent of coordinates, this doesn't mean that the symbolic expression for it 'matches' after you symbolically swap in the old variables for the new ones. it's a fine equation to consider but maybe we are exiting the world of intrinsic stuff.
e.g. x^2 - y^2 is a harmonic polynomial but if i pretend x is r and y is theta and use the expression for the laplacian in polar coordinates r, theta and 'relabel' i get something that isn't 0.
which in a way is good, because having fixed any one set of coordinates the laplacian ought to be only one thing in those coordinates and not two things.
on another note, as part of continued life procrastination, wonderful little article in the economist about ivermectin. the title is "Ivermectin may help covid-19 patients, but only those with worms". the punch line: “Ivermectin doesn’t treat covid,” Dr Bitterman wrote. “It treats parasites (shocker) that kill people when they get steroids that treat covid.”
nasser, is there a particular post you have in mind? i agree that often it's annoying when people downvote without any comment.
there's also the phenomenon of the downvote pile-on, where one or two downvotes on a post seem to make more downvotes more likely. i haven't seen this much lately but i remember it from my earlier times of being active on mse and math overflow
in the early days of math overflow, one or two downvotes on something would often lead to a pile-on effect even if the question was at least marginally appropriate. or so it seemed to me. people were really nervous, perhaps excessively nervous, about MO turning into a homework site. there may have been some overcorrection there.
of course anybody 'we all knew' could post the vaguest question imaginable, or respond to a question with a non-answer, and get upvotes and responses.
MO seems to have settled down.
copper i think if someone goes around upvoting or downvoting the same user too rapidly over a given time period the algorithm automatically flushes the toilet.
Awesome, will update, it offends my sensibilities too. Surely conjugate linear in the first variable is more consistent with the $\langle f, x \rangle$ notation for funtionals?
I am not particularly attached to either other than muscle memory.
we also discuss which natural supplements sold by our advertising partners are most effective in fighting the coronavirus and undoing the effects of vaccines.
ingrates. wait,... i see the convex question ray in the sky, gotta go
i'm expecting a major censure for my last answer.
echos of lou reed sounding in my mind...
i want to send a note to an old(er) friend who helped my mom a lot over the years. i am marveling that there are still places where a name, town & country are sufficient to get delivered.
the USPS used to be pretty good with that. at least if someone had a distinctive name. my dad would sometimes get mail with just our hometown (pop. 60,000ish at the relevant time) on it, because there was no one else with his name in town.
jam, some typesetting there and maybe other issues. the dependence on n is far from clear. the comma in the second part of the definition suggests u might be a function of more than one thing?
is x = (x_1, ..., x_n)? is the x', -x_n some notation for fiddling with one coordinate only of x?
i worked with an economist who wrote "x/c_n" for "x, with the nth coordinate replaced with c_n". not the best notation, particularly if the thing you're putting in there isn't tagged with an n already.
I have a question about problem 9(b) in Chapter 2 of Evans' PDE book. It says if we have $u$ is harmonic in the open upper half ball $U^+$ and $u\in C^2(U^+)\cap C(\bar{U^+})$, $u=0$ for $x\in \bar{U^+}\cap\{x_n=0\}$ Then by the reflection principle we define$$v(x)=u(x)\phantom{fd} \text{ if }\ph...
The question you posted is another question where $u$ is C at the boundary not C^2
im tryna prove something simpler given $u$ $C^2$ and harmonic at the upper closed half ball and zero at the base ($x_n$=0) prove the extension is also $C^2$ at the whole ball and harmonic
the question you posted lowers the smoothness of u at the shell of the ball
i think its just a multivariable exercise showing a function is differentiable 2 times at xn=0
copper do you have a tea to recommend? i am mostly drinking whole foods store label green up to about 3pm, then i switch to herbal 'tea' often just ginger.
