So wait, what's even the point of ##\pi## here? You've just come up with a complicated way of writing down the integers.
Here's a fact. If you construct an injection f from Z to R, then define a + which I'll call +' by f(x)+'f(y) = f(x+y), then f(Z) with +' will have a group structure, but it...
But you said 1 is in the group. If 1 is in the group, and it's closed under addition, then 11 must be in the group. Unless you are defining addition weirdly. What is 1+1+1+...+1 supposed to be equal to?
The point of what I wrote is that I assumed we already had the first k elements of the sequence which you dropped from your final list, but your sequence also works. I would consider what you wrote a proof.
It's possible if you're taking a class they will require you to write something more...
That's probably a bit unfair. Only two planes have crashed from implementing a solution to this. Do you think this hasn't prevented two planes crashes in the history of the regulation?
You spend a lot of , big words justifying the existence of ##n_{k'}## that both doesn't seem necessary to me and sounds very confusing.
I would just say something like, given ##n_1,...,n_k## in the subsequence, we know there is some ##n>n_k## such that ##a_n<L##. Let ##n_{k+1}=n##. And do...
Hmmm.
Population growth of humans isn't really purely exponential but not a terrible example if it makes sense.
Moore's law of computing power might be intuitive - 20 years ago computers couldn't even run the first ten frames of a modern video game etc.
What about the second peak in July that was bigger than the April one? And why are the cases per day supposed to be so b temporally correlated with lack of scent complaints, but the actual average review doesn't go up and down?
Manufacturing problems can result in your primary supply chain...
The graph of reviews mentioning lack of smell by month seems pretty unconvincing to me.
I would guess the real problem is either a new customer base trying the candles, or a decrease in manufacturing quality because of supply chain issues.
The graph also suggests that maybe 3% additional...
What is your career goal? In math, the point of a post doc is to spend three years building a cv to try to apply for a tenure track position. The point is not to do a second PhD. Maybe physics is different, but it feels like you're not focusing on the right stuff here.
Even if your goal...
If you think that when you have ten dollars you also have one dollar, then why does the statement that you have one dollar mean you cannot have ten dollars?
Either you mean you have exactly 1 dollar, or you mean you have at least 1 dollar. You seem to be confusing the two statements in various...
Right, so 2018 is in the range, which means f(2018) = 1/2018 as you figured out.
Now what are the possible graphs of this function?
Well we know between 1/2019 and 2019 it's just 1/x. One possible choice is that for x < 1/2019 Ave x > 2019, it wiggles around however you want so that f(x) is...
I think this problem is intentionally subtle. If there was supposed to be a recursive solution like fresh proposed thinking about, they wouldn't have needed f to be continuous.
This is exactly the point. If y is in the range, f(y)=1/y.
So is 2018 in the range of f? This is not obvious! For example, we know 2020 is not in the range of f, since f(2020) is not 1/2020.
I agree the graph of f is interesting. Once you have described why 2018 is in the range of f I think...
Fresh, I don't see how you work backwards from there. I think you need to use this fact:
Let y be a point in the range of f. What is f(y)?
Also, they didn't tell us this function is continuous just for fun.
The claim is that all the people of the world come together to trade the stock given the information that they have, and when the dust settles the stock will be trading at the right price. Once you understand this, I think things become more clear
So there are a couple things here
1.) If people...
There's a huge field of physics that involves taking divergent integrals and getting numbers
https://en.m.wikipedia.org/wiki/Renormalization
there are also other areas where performing algebra on divergent values can be useful. As one example, you can count the number of integer points in a...
I agree that the first programming problems will be very tough for someone who can't program. The first time my wife had to do a similar problem in python, with no instructions on how to install python, went something like
What's python?
Ok, well we can install basic python just to get you...
I think elo rewards the lower rated person more points for winning than it takes away when they lose, so it's a random walk that drifts back towards the starting point.
Every ellipse starts with a circle of radius 1, which just gives the identity matrix. The area of that is ##\pi\det(C)## by trivial math.
You then scale each axis by changing the value The area is proportional to the scaling, and the determinant is also proportional to the scaling. You then...
Exactly. You actually picked these functions (in simplified form)
f(1)=sin(1)
g(1)=1
The thing about s is a distraction. Any function is entirely defined as f(1)=c for some number c.
(1) seems pretty straightforward. Let x be a point in the ellipse. Then it's of the form Cu+d. If it's in the polygon, ##Ax \leq b##. Plugging in Cu+d for x gives the result.
For part 2, do you know why the area of the ellipse is proportional to det(C)?