Recent content by madilyn

I think Sterling's formula works and is cleaner than AM-GM on odd factors. I should've done that. I got the production function in the geometric part wrong, otherwise I realized that approach works too after a few simplifying assumptions (as you've pointed out, this bound is really weak)...

I see - thanks! Yeah. I've been staring at my approach and it seems to get me to \mathcal{O}\left(\dfrac{\ln n}{\sqrt[n]{n}}\right), which is still too loose. I really admire the people who led all the way to the PNT.

I will think about your approach. It seems to me that my approach would suffice if I can show that Sorry I accidentally posted before answering your other questions. I'm still thinking about your approach. Our professor did emphasize that we didn't need to and shouldn't rely on the density of...

It means {\displaystyle \limsup_{x\rightarrow\infty}\left[p\left(n\right)/\left(\dfrac{\log n}{\log\log n}\right)\right]<\infty}. Alternatively, \exists c,n_{0} c\geq0 and for all n\geq n_{0}, such that \left|p\left(n\right)\right|\leq c\dfrac{\log n}{\log\log n}. I believe that means...

Haha, yes it should read "odd". The left hand side of the inequality is the arithmetic mean while the right hand side is the geometric mean: {\displaystyle \dfrac{\sum_{i=1}^{k}\ln\left(2i-1\right)}{k}\geq\sqrt[k]{\prod_{i=1}^{k}\left(2i-1\right)}} I don't think this cuts through the problem...

Homework Statement Let f(n) denote the number of unique prime factors of some positive integer n > 1. Prove that f(n) \in \mathcal{O}\left(\dfrac{\log n}{\log \log n}\right) Homework EquationsThe Attempt at a Solution Since every prime number except 2 is prime, an upper bound on the number...

The chart strangely illustrates that dropping out of college puts you practically ahead of those with a Master's/PhD for 25 years (18 to 42 years old). Factoring in discounted cash flow, dropping out of college seems to be the dominant strategy according to this data.

I'm looking for a book or two that details affine spaces and transformations, then differential geometry of surfaces in affine spaces, starting at a level suitable for a year 1-2 undergraduate. In particular, I'd like to understand a few properties (e.g. what's the gradient and curvature at a...

Thanks! :) I'll get a colleague to reach out to you.

Oh no, that's unfortunate. Is it in bad taste if my employer listed an opportunity in the careers section?

Is there a job listing section on PF? The company that I'm working for is interested to hire more physicists but we're finding it hard to reach out to the physics community specifically. It can be made free to use, or perhaps you can charge a nominal fee to the companies per listing to...

Let's say I'm applying electrical currents to a certain part of a human test subject and measuring certain deflections in his heart readings during this period. Before I increase the electrical currents, which could be dangerous, I'm interested to see if the changes in electrical currents are...

Stephen, thanks for your prompt answers as always! 1. Unfortunately no, I don't have a very sophisticated understanding of the meaning of a confidence interval (I wouldn't be able to write a philosophical debate about it). But I do have a basic grasp of the pitfalls. What's one school of...

I've been figuring out the use of the nonparametric bootstrap and if I understand correctly, this is the procedure: 1. Take an original sample, a vector x = (x1, ..., xn) 2. Generate k vectors, each called a 'bootstrap sample', of the same length as x by random sampling (with replacement)...

Oh sorry, I came up with the questions myself so it isn't a homework problem. I see. What would you intuitively interpret the 89.2% figure as if not the fraction of days I expect to have positive weight gain? Ah, this made a lot of sense! Thanks! Yes, I figured this is a difficult problem...