N.b. the article uses d for the dimension, and n for the moment index. I haven't cranked out the details, but I believe that this is already true just assuming the fourth moments of the distance are integers. That is, for n != 1, 2, or 4, the fourth moment is never an integer. Idea of the "brute force" proof:
Take the formula in the article for the 4th moment of the d-dimensional sphere, which is always a rational number. Basically, for n=2^k, the denominator should be divisible by a larger power of two than the numerator (specifically, if I crunched the numbers right, the gap should be k-2). When n is not a power of two, then for any odd prime p dividing n, I believe the denominator should be divisible by a larger power of p than the numerator. This requires calculating exactly how many powers of p divide various factorial expressions, but you get the idea.
When talking about chess: "But even the most powerful program can be defeated by a skilled human player with access to a computer—even a computer less powerful than the opponent."
Is that true? Can a state-of-the-art chess engine plus a grandmaster really outperform just the state-of-the-art chess engine?
Google doesn't open a 500-engineer research lab in Grand Rapids, MI because there are not 500 Google-caliber engineers already living in Grand Rapids, MI. This is not a statement about the average talent there, but simply that the starting pool is not sufficiently large; hence the need to build offices in high density urban areas.
Google doesn't open an office in Denver so that Bay Area engineers have a slightly cheaper place to work, they do it to tap into the talent of folks who already live there.
Edit: I am basing these claims off of my own experience. I work at a satellite office of one of the big tech co's, and the vast majority of my coworkers are people who are either fresh out of a local university, or already lived here for many years before joining the company.
So Google only interviews candidates already living in the cities where the offices are? Google never relocates a candidate from somewhere else?
This is super false. Google doesn’t open an office in Denver because there is already a sufficient talent market in Denver to staff the whole place. There isn’t.
They open an office in Denver so that when a person passes the interviews and will need relocation they can be relocated more cheaply and paid a relatively lower salary in Denver.
They can only get away with this for certain cities that are granted high-status, like Austin, Seattle, Denver. Companies are trying to create similar status facades for e.g. Pittsburgh and Atlanta too.
It absolutely is wage arbitrage for the company— has nothing to do with the preexisting talent base in the given city, except insofar as that talent base confers some type of mitigating high-status effect.
For Denver it’s access to glamorized nature and skiing. For Pittsburgh it’s centralized around the presence of CMU. And even with these effects, these cities are not looked at as all that desirable for many, many candidates.
If you pass the interview as a generalist, Google absolutely prefers for people to work in their Mountain View office.
1. The cost of relocation is minuscule compared to the total cost of employing someone, so I don't buy that argument.
2. Do you have data to back up that Google engineers in Denver get paid less than their Mountain View counterparts? I believe this is true when comparing US to non-US salaries, but at least where I've worked, engineers get paid the same everywhere within the US (and indeed, you can relocate from the Bay Area to cheaper locales without taking a pay cut).
Based on salarytalk.org and h1bdata.info, median software engineer salary in MV is $127,000 - $129,000. In Pittsburgh: $113,000 - $114,000.
Relocating without a pay cut would be uncomparable, since it would involve pay cut dynamics for an established worker. Google is probably atypically generous in all these areas though, and I don’t think it counters the points about status. They want wage arbitrage to alleviate some headcount, doesn’t mean all headcount, and very likely Google would be among the least worried since revenue per headcount is so ludicrously higher than what they pay in total comp anyway. It still doesn’t suggest they open these office locations for some other reason than urban status or co-location with academic center status. Google US locations look mostly exactly like that’s what they are doing.
This is called Granger-causality (and work on it led to a Nobel prize, so it's important and useful)... it's stronger than just correlation, and way easier to determine than true causation, but it's possible that z causes both x and y, and z's effect on x is just more delayed than its effect on y.
But it at least rules out x causing y, which is something.
> but it's possible that z causes both x and y, and z's effect on x is just more delayed than its effect on y.
This is in fact the case with the barometer falling before a storm. Both the falling barometer and the subsequent rain and wind of a storm are consequences of an uneven distribution of heat and moisture in the atmosphere approaching equilibrium under the constraints of Earth's gravity and Coriolis force.
Still doesn't work. Suppose I flip a coin and write the result in two places. I write it on sheet y then sheet x. We have that X == Y, so p(x|y) = 1, p(x|!y) = 0, and time(y) < time(x), but neither causes the other. I can write more later if you have interest, but I gotta run.
As a former mathematican, I was at first a little offended and dismissive of his claim. But, perhaps what one can say is that mathematicians don't seem to distinguish "causation" with "implication". After all, if the barometer goes down, that does imply a storm is coming (perhaps with some increased probability), but it still doesn't cause the storm to come (even with increased probability).
In a simplified closed system, where all you have are barometers and storms, maybe there is no difference between implication and causation; all you know is these variables are correlated. Perhaps once you take every atom in the universe into account, the two start to look the same.
That can't be right, because you can take the barometers out of the closed system, and it will still storm. Correlation isn't causation, and for good reasons.
> In a simplified closed system, where all you have are barometers and storms, maybe there is no difference between implication and causation; all you know is these variables are correlated. Perhaps once you take every atom in the universe into account, the two start to look the same.
Implication it is a very mathematical thing. It is like you know, that y=f(x), and then you write x=g(x), where g is inverse of f. It works both ways, there are no cause, no effect, just link between two variables. If you use math to reason about causal links in reality, you need to use some implicit knowledge which is not represented in formula. It doesn't means that math is bad. Geometry likes euclidian space while we know from Einstein that our space is not euclidian one -- it doesn't mean that geometry is bad. Euclidian geometry just solves some specific problems and doesn't solve others.
Causal link reflects ability to change dependant variable by changing independant one. It is not something like "fundamental property of the Universe", it is our subjective way to structure information about the reality. It seems to me, that physicists believe the other way, that causation is the inherent property of reality. Maybe they are right in their field, but it doesn't work in everyday life. Causal link is an abstraction that helps us to know what we can do to change outcomes.
In this sense there are no causal link between barometer readings and a storm: if you change barometer readings to reflect a fine weather the storm will come anyway. Maybe there is causal link between atmosphetic pressure and a storm? I do not know it, because I see no way to change atmospheric pressure and I'm not educated well enough to understand scientific weather models. Though it is relatively safe for me to believe that low atmospheric pressure causes storm: causational link or correlational one -- it will not change my behaviour, because I cannot change atmospheric pressure. If I'll find a way, than it would be cruicial to figure out the kind of the link, because I'll be able to break something if I'm wrong. But I said that it is relatively safe, because if I suppose that link is causational, I would use that link differently while reasoning about the weather, it will change my other beliefs and probably it will change my behaviour somehow.
So, the main idea is: causation is just our way to structure reality. We are free to choose which links are causal, it is all up to us. If we think it will help us to reason about reality, then we should speak about casuality. And the most important difference between causation and correlation is the ability to change dependant variable by changing independant. If we can change dependant variable that way, than we should mark link as causational. If we cannot, that we should think about that link as about correlational. Implication just do not draw this difference.
I think the first point is only true for symmetric matrices (which includes those that show up in multivariable calc). In general, the eigenvectors need not be orthogonal.
Yep, you could well be right. The image of an ellipse under a linear transform is definitely an ellipse, but I'm not sure about the eigenvectors in the general case.
The symmetric case is by far the most relevant for probability theory though.
