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 Aug14 revised Rotate a point in circle about an angle corrected error in RHS Aug14 comment Rotate a point in circle about an angle Bleh, I took time typing out the rotation matrix! :) Aug14 answered Rotate a point in circle about an angle Aug14 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 $\exp(-1)$. Aug14 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! :) Aug14 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! Aug14 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. Aug14 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". Aug14 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). Aug13 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! Aug13 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. Aug13 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. Aug13 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. Aug13 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 AT, 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). Aug12 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. Aug12 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! Aug12 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? Aug12 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? Aug12 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. Aug12 comment Are there smooth analogs to polynomial splines As mentioned, the best solution is dependent on the configuration of your given points. Obviously if your point-set seems to be coming from an oscillatory function, a monotonic method would be unsuitable. The usual constraint I keep seeing in practice is that the local extrema in your interpolant should either be in the same place as the extrema of your data, or that the new extremum (e.g. for an interpolant between two points with near-identical ordinates) be of the same order of magnitude as the points in the neighborhood, viz. "locally monotonic".