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2h
answered Does $G\times H\cong G'\times H'$ imply $G\cong G'$ and $H\cong H'$?
3h
comment Pathological Question involving $C^1$ Criterion for Differentiability
how about $f\colon \mathbb R^2 \to \mathbb R$ given by $f(x,y)=xy$ if $x,y\in \mathbb Q$ and $f(x,y)=-xy$ otherwise ?
23h
comment Continuity Must Hold in an Entire Open Set?
yes Amitai, something like that :)
23h
answered Continuity Must Hold in an Entire Open Set?
Jul
27
comment “continuously differentiable $\subseteq$ Lipshitz continuous” with $f(x) = x^2$
"over a closed and bounded set"!!!
Jul
26
answered Why do the integers, rationals and any countable set have zero measure?
Jul
26
comment Intuition for the Cauchy-Schwarz inequality
@Mehrdad that student can be shown this fact, or requested to clarify the intuition in an exercise.
Jul
26
comment Intuition for the Cauchy-Schwarz inequality
beware a circular argument: it is the CS inequality that allows one to define $\theta$ in the first place.
Jul
26
comment Intuition for the Cauchy-Schwarz inequality
@Mehrdad the relation to the cosine (law) may be not the best way to see that. I like to think of it in terms of orthonormal basis. Given an orthonormal basis, any vector $u$ is simply $\sum_i (u\cdot v_i) \cdot v_i$ by a trivial computation which has nothing to do with CW or the law of cosines. This trivially shows that the dot product gives the components of a vector. When teaching these things I like to use the law of cosines to motivate the definition of the standard inner product. Then CS is shown to relate to a law of cosines (kind of) for any inner product.
Jul
26
answered Intuition for the Cauchy-Schwarz inequality
Jul
24
comment How can I make math joyful to learn?
The analogy with chess is not a very strong one, I find. Chess is a game. Mathematics is a scientific practice. Yes, both require careful analysis, and yes both often exhibit high degrees of elegance. But their aims are completely different. Chess only applies to real life in that it teaches one how to analyse things carefully. Maths applies to real life both in the techniques it employs and, far more importantly, in its direct applicability to model real life situations.
Jul
24
comment How can I make math joyful to learn?
As you say yourself, it's a problem of how math is typically taught, not with maths itself. If I were taught that mathematics is just the boring manipulation of formulas and numbers, I probably would not find it joyful at all. Luckily, I learned maths from good books and a few good professors. In any case, it is not so clear what you are asking as you seem to say "maths is not the way it's commonly taught, so how do you, who obviously don't represent the common maths student, find it joyful".
Jul
23
revised Why the surface of the sphere is not a Euclidean space?
edited body
Jul
22
comment Finite free objects
what do you mean by trivial here? As the question stands it highly unclear what you are looking for.
Jul
22
comment How can tossing a coin be a certain 50/50 chance? Isn't that short-sighted?
yes, the common vernacular is circularity: it's random cause we assume it's random. We just try to wrap it using longer words.
Jul
22
comment How can tossing a coin be a certain 50/50 chance? Isn't that short-sighted?
"... is effectively random" is precisely the problem. It seems the only way to say "the coin flips are truly random" is to assume that the process produces true randomness. I can only strongly suggest the first few chapters of Jaynes' book where these deep issues are discussed in great detail.
Jul
22
comment How can tossing a coin be a certain 50/50 chance? Isn't that short-sighted?
The physical attributes of the coin are only one side of the coin (ahm, pardon the pun). Even if the coin is physically perfectly symmetric etc., how do you separate the act of flipping the coin from from questions of fairness? With a bit of practice, one can learn to flip a coin in such a way so as to change the outcome. One can thus adjust for physical imperfections and make the coin appear fair. Or one can take a perfectly fair coin and make it appear biased. Jaynes addresses this beautifully as well.
Jul
22
comment How can tossing a coin be a certain 50/50 chance? Isn't that short-sighted?
I would conjecture that to be the leading reason.
Jul
22
comment How can tossing a coin be a certain 50/50 chance? Isn't that short-sighted?
I wonder what the down votes are based on.
Jul
22
answered How can tossing a coin be a certain 50/50 chance? Isn't that short-sighted?