As the title says, I'm trying to find the formula that calculates the average cost for a game of chance.
Here are the rules of the game -
There is a 25% chance to win a game
The player wants to win exactly 3 games in one day. After that he stops playing.
There is a maximum of 10 games per day
Th...
I am contemplating asking a question on MSE, but i'm not sure if it will be closed. As part of a research project, I am working on evaluating the integral of $\sin(t^7) \,dt$. Is there anywhere in math where this integral arises(over any bound) that is needed to solve some real world problem? I was thinking there may be something in Fourier analysis but I still haven't found anything. Would this be an appropriate question to ask on MSE or MO?
i'm aware that when I am looking for applications in engineering, physics etc to ask on other sites, but since it is about math application I'm not sure if it should be MSE or MO.
So to summarise: Can i post this without it being closed? If so, which site should I post it on?
MSE seems like an OK place for it. MO does not seem like an OK place for it. you might mention where you have looked, or tried to look. and (if there is a useful story) where the focus on that integral came from. people on MSE like to see surrounding context and a search for answers, even if the search didn't lead anywhere.
as an attempt at answering the question, it would surprise me if that integral has ever come up anywhere in a physical model (but, i have been surprised before). i would expect that people would ask in the comments, where did this come up for you?
why 7 and not 12 or 6.5? exponents in physical problems tend to come from somewhere.
oh. often for something like that, the only significance of the integrand is that it's something where you can't write down a nice formula for the antiderivative and thus evaluate the definite integral 'explicitly' using the FTC.
i agree that any given example of such a thing would be somewhat contrived, although the general phenomenon of 'function that doesn't have a nice antiderivative' is not contrived.
the integral of sin(t^2) dt somehow comes up in optics. the 'fresnel integral.' you might peek around the physical origins of the 2 exponent in that and see if there's some way of varying the hypotheses (perhaps with a modified physical law on a different exponent) that would incorporate sin(t^p) for other p.
none? i don't know? the broad message is just, you can numerically compute definite integrals without antiderivatives. that's a useful message.
this particular example, or any particular example, who cares.
if anything, it tells us that 'having a nice formula for an antiderivative' is not that significant of a property of a function, although a lot of the toy examples in calculus textbooks enjoy that property.
you might think about int 0..1 cos(x) dx for example. it's not "impossible" in the sense that there's a "nice" formula for it (namely sin(1)). but what is sin(1)? how would you compute it to three digits?
that's a question that comes up even if you do have a formula for the antiderivative.
Because $[a,b]=\cap (a-1/n,b+1/n)\in \mathfrak B$. For any rational r in the interval, $r=\cap (r-1/n,r+1/n)$. So the set of rationals in the interval(being a countable union) is Borel set.
Let the set of rationals in the interval be E. So [a,b] intersection E^c is also a Borel set.
@leslietownes :-)
It’s so sad how some big companies make fool of customers. Like a purchase is made by a customer and the product being covered under warranty is not repaired due to arbitrary reasons by the company. In such cases, many customers just decide to never ever purchase anything from that company. They don’t file lawsuit which I’m sure they’ll win if they do, because filing lawsuits will take time and they don’t have time.
time and money, at least in jurisdictions where by default you pay your own legal fees even if you win.
in such places, you can basically harm a customer as much as you want, as long as you don't want to harm them more than the cost of hours of attorneys' time
it sometimes works. it deters a lot of the worst abuses.
it can succeed in punishing the wrongdoer even if it doesn't succeed in compensating people who were affected. also, attorneys get paid, so there's that.
Hii, I'm stuck on a question today. Question is to find $\dfrac{d^2x}{dy^2}+ 20$ when $y(x) = (x^x)^x$, $x>0$.
I started it by differentiating $y$ wrt $x$ and got $\dfrac{dy}{dx} = xy(2\log(x) + 1)$. And so, $\dfrac{dx}{dy} = \frac{1}{xy(2\log(x) + 1)}$.
I know I can now differentiate $\dfrac{dx}{dy}$ wrt $y$ so as to obtain $\dfrac{d^2x}{dy^2}$ but I think there might be a clever technique to solve it.
Suppose we're given a function $f(x)=e^{-x^2}$ and we want to make a piecewise linear interpolation on the interval $[0,10]$ under the following constraints:
We divide the interval $[0,10]$ into two intervals $[0,a]$ and $[a,10]$ and make equidistant partitions into $n_1$ and $n_2$ intervals re...
@HelpMeToUnderstandContours here’s one approach that you might like: Suppose that you want to find dy/dx and you know that y= g(x). Write f=y-g(x) to have f=0. Use chain rule to get: $f_x+ f_y dy/dx=0$.
It’s no different at all from usual implicit differentiation; it just sorts your algebra slightly. Of course, the implicit function theorem gives hypotheses and proves that it is valid.
@TedShifrin How? At first, I thought I've to do more work to simplify $\frac{1}{\sqrt{1-\sin x}}\cdot \frac{\cos x}{2\sqrt{\sin x}}$. But that doesn't seem so.
@Yai0Phah Agreed. He was just saying that to get the RS as an algebraic variety took some analysis too. Maybe we can get to RR and avoid Serre duality :)
bart: interesting observation. i'd guess that a lot of MSE stuff is driven by coursework in schools, and that as the pandemic persisted, more people found avenues outside of historical ones for asking/resolving questions.
In mathematics, more specifically ring theory, the Jacobson radical of a ring
R
{\displaystyle R}
is the ideal consisting of those elements in
R
{\displaystyle R}
that annihilate all simple right
R
{\displaystyle R}
-modules. It happens that substituting "left" in place of "right" in the definition yields the same ideal, and so the notion is left-right symmetric. The Jacobson radical of a ring is frequently denoted by
J
(
R...
I'm thinking about maximal ideals of Z[$\sqrt {-5}]$.
I think that if I consider the natural map $\phi: Z[x]\to \frac{Z[x]}{(x^2+5)}$, then ideals of the Gaussian integer ring are same as the ideals of Z[x] which contain the principal ideal (x^2+5).
This is by correspondence theorem (one of the isomorphism theorems).
But I don't know how to show that every maximal ideal of the Gaussian ring contains a prime no.
correspondence theorem talks about ideals to ideals but doesn't say that maximal ideals also "correspond".
or Z[x] is a commutative ring with 1. (x^2+5) is irreducible so (x^2+5) is a maximal ideal of Z[x] and therefore, the quotient ring by (x^2+5) is a field. What is a maximal ideal of a field? There is no maximal ideal of a field. So vacuously the statement is true. But this looks wrong.
(x^2+5) is not a maximal ideal of Z[X], so that's incorrect
it's also incorrect that there is no maximal ideal in a field. every non-trivial ring contains a maximal ideal. in any field, the zero ideal is the unique maximal ideal.
@Koro think about how the correspondence works and whether it plays nicely with inclusions
@Thorgott I think yes. The inclusions are respected in the following sense: If I, J are two ideals containing kernel and satisfying I$\subset J$. Then, $I \mod K\subset J\mod K$, where K:= kernel.
@copper.hat Yes, if a comment thread gets long, autoflags are raised. I tend to be pretty brutal with moving those threads to chat, particularly if it is a back-and-forth between only two or three people.
Sometimes I've done my best teaching on this site in such comment streams. And I don't think it's bad. Usually, I ask the OP to write an answer based on our discussion once he/she's understood. Sometimes that happens.
@TedShifrin I don't think that it is bad, but the model is that SE is a Q&A site, and any content which is meant to be preserved should be put into either a question or answer, and not comments. Comments are supposed to be ephemeral.
so now we have correspondence between two partially ordered sets (ideals of $R/I$ and ideals of $R$ containing $I$, each partially ordered by inclusion) and you have observed this correspondence preserves the partial order
@Xander That's why it's good for the OP to write up the proof once he understands it. I used to get very upset when someone who wanted rep would post the solution while the OP was clearly interacting with me and figuring it out somewhat for himself.
Anyhow, I think I'm mostly phasing myself out of this place.
I'm also annoyed that someone who posts zillions of questions about Spivak's text and problems has not accepted the answer I gave him. It was a rare situation where Spivak just messed up the exposition in the text, I believe.
No, quite long after that, Koro. It was Pedro Tamaroff who dragged me into chat and that's how I got to know him. I don't remember how I landed at MSE in the first place.