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Aug
14
comment Rotate a point in circle about an angle
That's funny (but not surprising due to equivalence), I on the other hand started from the Argand plane to get to my answer; essentially expanding a point z=x+iy into polar form: $r\exp(i\theta)$, and considered the real and imaginary parts of when I subtract an angle $\theta'$ from the point (corresponding to clockwise rotation).
Aug
14
revised Rotate a point in circle about an angle
corrected error in RHS
Aug
14
comment Rotate a point in circle about an angle
Bleh, I took time typing out the rotation matrix! :)
Aug
14
answered Rotate a point in circle about an angle
Aug
14
comment Finding $\lim_{x \to \infty} \left[ {x^{x+1} \over (x+1)^x} - { (x-1)^x\over x^{x-1}}\right]$
You will have to take logarithms at some point and come up with an expression whose limit is -1. But since this is the limit for the logarithm, the limit for the original expression should be $\exp(-1)$.
Aug
14
comment Asimov quote about “eight million trillion” arrangements of amino acids
Indeed the world will be a much messier place if permutation of protein sequences were in fact the case! :)
Aug
14
comment Finding $\lim_{x \to \infty} \left[ {x^{x+1} \over (x+1)^x} - { (x-1)^x\over x^{x-1}}\right]$
Mathematica confirms the limit is 1/e; as for figuring out how this result was arrived at, just to give you a hint, this is an ∞-∞ type indeterminate form. Manipulate it into something where L'Hôpital can apply (you may also have to invoke logarithmic differentiation at some point). Good luck!
Aug
14
comment Family of functions with two horizontal asymptotes
Without looking at the wikipedia page ;) , the error function, the hyperbolic tangent, and the arctangent would have sigmoidal behavior. As a matter of fact, the integral of any "bell-shaped" curve will have sigmoidal behavior.
Aug
14
comment Asimov quote about “eight million trillion” arrangements of amino acids
There is "the source" for such things: the *.pdb files in the Protein Data Bank (rcsb.org/pdb/home/home.do ) will carry the sequence of amino acids in "one-letter" format which chemical drawing software can subsequently use for drawing. And correct, the insulin in different mammals has "conserved sequences".
Aug
14
comment Asimov quote about “eight million trillion” arrangements of amino acids
(OT) Kaestur: 22 for archaean life-forms actually; pyrrolysine is another amino acid their RNA has a codon for (which is usually a stop codon in members of the other domains).
Aug
13
comment find minimum of a function defined by integration in Mathematica
Bah, I don't really hang around in StackOverflow... XD too bad my comment was late!
Aug
13
comment find minimum of a function defined by integration in Mathematica
Also, it should be f[t_]:=... ; you need delayed evaluation instead of immediate evaluation of the RHS. Also changing the LHS of the definition to f[t_?NumericQ] might help. This is because FindMinimum[], unlike Plot[], does not have the HoldAll attribute.
Aug
13
comment find minimum of a function defined by integration in Mathematica
What's g[t,x]? If you have an explicit expression, we might manage to be more helpful.
Aug
13
comment A nicer proof of Lagrange's 'best approximations' law?
This is also in Lorentzen and Waadeland's "Continued Fractions with Applications" ; a nice book, if you can get to read it.
Aug
13
comment Is the rank of a matrix the same of its transpose? If yes, how can I prove it?
Probably the "sledgehammer" approach to a "walnut" problem, but I'd just have done a singular value decomposition of A and A<sup>T</sup>, note that one decomposition is expressible in terms of the other, and then show that the two diagonal matrices resulting from the two decompositions have the same rank (and nullity too).
Aug
12
comment Irrationality of powers of $\pi$
The sketch of it at least: if $\pi$ were algebraic, Lindemann-Weierstrass would imply $\exp(2\pi i)$ is transcendental (proving e is transcendental is another story)... and you can fill in the rest.
Aug
12
comment On multiplying quaternion matrices
I'm slowly becoming more enlightened, but I suppose I have to brush up a bit more on the theory (sigh). Thank you kindly, Matt!
Aug
12
comment On multiplying quaternion matrices
Matt, please pardon me for asking, but could you explain to this non-expert how the isomorphism (as I understand it, you're saying that the same set of "rules" apply to the system in Qiaochu's first comment and his answer here) is shown/applied?
Aug
12
comment On multiplying quaternion matrices
Sound reasoning, but could you maybe expand on "multiplying in the opposite order gives you essentially the same multiplication" please?
Aug
12
comment Are there smooth analogs to polynomial splines
An approximant based on sines and cosines (i.e. the Fourier basis) would in principle be infinitely differentiable, but again this is not always appropriate. For most physical applications anyway one is usually content with C^2 (to use a physical example, the usual quantities of interest are position, velocity, and acceleration; "jerk" not so much.); people who solve boundary-value problems might sometimes want a C^4 approximant (quintic splines are the usual basis), but SFAIK nobody bothers with higher orders of continuity in practice.