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Mar
18
comment How to study math to really understand it and have a healthy lifestyle with free time?
@user21820 But if that's the standard we want to use, then the same is true for mathematics or statistics as well. Most of that work takes place in companies, defense labs, and R&D consulting firms who are notorious for only going after low-risk demo-ware. As a machine learning professional, I've seen this both with software and with formal math time and again, especially in mathematical finance. Why try fancier learning algorithms if simple OLS will get something (crappy) out the door right now? So if that is the criticism, then it applies just as well to math too.
Feb
23
comment Examples of non-Riemann integrable functions that appear “in nature”?
@JohnDonn Yeah, that's too bad. Once you do have enough points you should click on my user name to see other questions that I have answered and down vote those too, much as you seem to have followed me from the question about whether software abstraction is "mathematical" to this. If you follow me and downvote more, I may have to report it to a moderator. On this question though, I welcome a downvote. After reading your comment, which is correct, I still don't believe it applies at all to my answer or the original question. When you can, you should signal your disagreement with a down vote.
Feb
23
comment Examples of non-Riemann integrable functions that appear “in nature”?
Well then it depends on what you mean. If you mean that because Brownian motion is continuous w.p. 1, and can derive a technical definition of the random-variable-valued "thing" that "is equal" to its "integral" ... then sure, that pedantic definition satisfies. That's clearly a very different thing than what the question is asking about, which is for examples where an attempt to do a naive Riemann integration leads to an unusual outcome that doesn't have the same properties of a conventional integral. Brownian motion certainly counts for that: what's the derivative of the integral of B_t?
Feb
23
comment Examples of non-Riemann integrable functions that appear “in nature”?
Which type of integrability are you talking about? Riemann? Lebesgue? Lebesgue-Stieltjes? Ito? Stratanovich? Henstock-Kurzweil? ... be specific.
Feb
21
comment How to study math to really understand it and have a healthy lifestyle with free time?
@JohnDonn You've clearly never done software engineering. One of the core concepts in all of software engineering is to create abstractions that explicitly prevent manual implementation. Even apart from that, you've got things like functional programming with e.g. Haskell, which is essentially a manifestation of category theory and how to solve practical problems with it.
Dec
24
comment Easy example why complex numbers are cool
What definition of "cool" are we using here?
Jan
14
comment Splitting a sandwich and not feeling deceived
Thus, anyone cutting pieces has an incentive to make sure the worst off piece is at least good enough that whoever randomly draws it will decide to keep it, so that everyone gets to keep theirs too. My guess is this leads to some Nash equilibrium kind of thing where making equal cuts leads to the highest expected value.
Jan
14
comment Splitting a sandwich and not feeling deceived
I wonder if anyone has ever looked into a Dictator Game-based stochastic variation. In experiments with things like Dictator Game, people tend to reject unfair offers, even if it makes them uniformly worse off. That is, they are usually willing to pay some "fairness premium" and simply deny any form of splitting the resources unless they are equal to or better than all of the other players. If we assume humans will use this heuristic, then we could take any N pieces of a divided sandwich and randomly assign them to people in a Dictator Game, where the "worst-off" person is the dictator.
Jan
9
comment Pedagogy: How to cure students of the “law of universal linearity”?
Another point is that Greek symbols still "feel" "mathy" and are seen as "expected math symbols" even by young students. So using Greek symbols might actually be counter-productive: to the extent that a student has anxiety about what a symbol is allowed to mean and whether it has some special meaning, using Greek letters over more "regular" things like x and y might only make the anxiety worse. But using a symbol like a tree, fire hydrant, tulip, pineapple, or anything that's totally not at all possibly confusable with "official mathy type stuff" could have a better chance of working.
Nov
21
comment How to study math to really understand it and have a healthy lifestyle with free time?
Can you elaborate?
Nov
5
comment Is it worth pursuing a statistics minor? I want to go to pure math grad school.
Also, all else equal, extra time to absolutely destroy the Math GRE subject test is worth much more than the minor. So if classes for the minor would prevent you from utterly crushing the Math GRE, don't downweight that -- it could be and important consideration. (This is conditional on the desire to go to math grad school. The stats minor is probably more valuable unconditionally). Check this link for some other considerations.
Nov
5
comment Is it worth pursuing a statistics minor? I want to go to pure math grad school.
Another question: do you attend an undergraduate school where you feel the school's reputation for math majors is good enough to propel you into the pure math grad schools of your choice? If your undergrad school is extremely strong, then it's probably fine to spend extra classes on more well-roundedness (such as in stats). If the best PhD programs will cast any amount of skepticism on your undergrad math program's rigor, you'll be better served just cramming in as many advanced math classes as you can to prove you're good enough at it to be admitted.
Oct
9
comment Best applications-oriented introductory calculus textbooks?
I am more saying that there is some base rate of curiosity which is a required pre-req for understanding how a physical problem motivates a mathematical model. And most students who find they want or need to take calculus do not have that base rate of curiosity.
Oct
9
comment Best applications-oriented introductory calculus textbooks?
I understand. I guess I said it poorly. What I more mean is: a student who passes through calculus once is not equipped to understand the motivations for using calculus. Sure, they can recite platitudes about how it's used in some field, but that's about all they are in any position to appreciate. Which is probably why most "applications" sections (as you point out) just contain platitudes like "This can be used in fluid dynamics..." I'm not saying my point here is right. Just that my prior is that you can't motivate calculus from scratch with applications; it doesn't work well.
Oct
9
comment Best applications-oriented introductory calculus textbooks?
Maybe the book you're looking for is hard to find because it's just not very useful. In my own lectures, which admittedly focus more on probability, I've never seen students take to problems because of physical motivations. But after mastering a technique, like e.g. moment generating functions, then they are eager to see how it applied. I might just have outlier experiences, but both in my undergrad and grad school, and as a lecturer, it just never worked this way even for bright students. Is there evidence that it worked this way for bright students in the past?
Oct
9
comment Best applications-oriented introductory calculus textbooks?
The scope of the question is also hard to understand. For instance, in my own education I had to plod through two courses in advanced calculus, just doing lots of problems, mastering limits, beating all algebra mistakes out of my hand over and over. After that I read "Div Grad Curl and All That" and it was wonderful. If I had read such a book before doing all that gross rote algebra, it wouldn't have meant anything. Similarly, I had to slog through differential equations before coming across some useful boundary values problems books that motivated things with physics.
Oct
9
comment Best applications-oriented introductory calculus textbooks?
"justifying its inclusion in a liberal education for purposes other than contemptible ones like using it as a weeder for medical school or business school applicants." This kind of normative statement seems unnecessary in the question. I, for one, am glad these things help weed out such applicants, even though there are many better reasons to study the calculus.
Oct
9
comment What is the estimator of $P(X=0)$ for $X\sim\operatorname{Poisson}$
That's why I said, "as an aside." And by 'dimensionality' in my comment, I did not mean cardinality of the sample set. I meant the dimension of the space to which each observation belongs.
Oct
9
comment What is the estimator of $P(X=0)$ for $X\sim\operatorname{Poisson}$
As an aside, you may also consider the properties of the James-Stein Estimator if you ever encounter an example of this with dimensionality of three or more.
Aug
7
comment Set of nodes to remove to leave zero edges with maximal summed node values.
Thank you! I suspected it had to be hard. The ad hoc algorithm was exponential time, and usually even for bad first attempts it's hard to make something exponentially slow unless it's just a hard problem.