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seen Jun 24 '13 at 0:06

No, his mind is not for rent
to any god or government.
Always hopeful, yet discontent.
He knows changes aren't permanent,
but change is.

— Rush, Tom Sawyer


Taking an externally-imposed and much-needed break from SE activities.

E-mail (flipped ROT13): zqd˙ʎʌuzʇ@ʇɐʌǝɥʇʌssqɹǝɥɟuɹʎɔ
Any code I've posted here I place under the WTFPL.


Aug
25
comment On isometric affine transformations
I'm already familiar with manipulations in the complex plane, but never really thought to just start from the definitions of the operations themselves (as I said, I took them for granted). Thanks!
Aug
25
revised On isometric affine transformations
added 118 characters in body
Aug
25
comment Where does the Pythagorean theorem “fit” within modern mathematics?
guest, just to give you an idea on how it's all dependent on defining the concept of distance (norm): consider the problem of going from one place to another in a city where the streets form a regular grid. It's technically correct but not very useful in this situation to say that the distance from one house to another house is given by the Euclidean distance. On the other hand, you can define "distance" for this system as the total of the distances of the streets you will have to walk to go from one place to another. This gives rise to the so-called "Manhattan norm".
Aug
25
asked On isometric affine transformations
Aug
25
comment How do you formally prove that rotation is a linear transformation?
Maybe you don't even have to say there is a $\theta$; just say you have a quantity c and another quantity s that are related by $c^2+s^2=1$. At least one can't get preconceived notions about angles.
Aug
25
comment How do you formally prove that rotation is a linear transformation?
As I said... handwave-y. Looking at it again, circular too. :)
Aug
24
comment How do you formally prove that rotation is a linear transformation?
A sketch: matrices represent linear transformations, and rotations can be represented by matrices. A bit handwave-y though.
Aug
24
comment Minimum of the Gamma Function $\Gamma (x)$ for $x>0$. How to find $x_{\min}$?
I'm actually trying to discourage you from looking at the series expansions (well okay, let's make an exception for Stirling); but if you think you can glean new insights, dig in: dlmf.nist.gov/5.7
Aug
24
comment How do you define functions for non-mathematicians?
If we're going for teaching kids, using cigarettes in your example would not seem to be a good idea. Just sayin'...
Aug
24
comment Minimum of the Gamma Function $\Gamma (x)$ for $x>0$. How to find $x_{\min}$?
Américo: As for question b.), there are series expansions (see DLMF for instance), but none of them seem to be practical so I'd steer away from them. I believe PARI/GP has a digamma/polygamma function somewhere, just check the docs.
Aug
24
comment Minimum of the Gamma Function $\Gamma (x)$ for $x>0$. How to find $x_{\min}$?
A general rule of thumb in numerical computing: it's easier to compute (simple) roots of functions to the full precision of your environment than to compute extrema.
Aug
24
comment Minimum of the Gamma Function $\Gamma (x)$ for $x>0$. How to find $x_{\min}$?
For instance, the computation in Mathematica goes something like x /. FindRoot[PolyGamma[x], {x, 1}, WorkingPrecision -> 20] which yields the result 1.4616321449683623413 .
Aug
24
answered Minimum of the Gamma Function $\Gamma (x)$ for $x>0$. How to find $x_{\min}$?
Aug
24
comment Derivative of a product and derivative of quotient of functions theorem: I don't understand its proof
The "logarithmic derivative" form is also useful when reckoning how uncertainties in data might propagate.
Aug
24
comment Infinite limits
IVlad: Whenever you get a result of ∞ (resp. -∞) from a limit calculation, simply consider it to mean that no matter how close you take your independent variable to the value of interest, you will never manage to find an upper bound (resp. lower bound). But there is still the issue of making sure both the left and right hand limits are consistent: one can probably say that the limit of xֿ² as x→0 is ∞, but the limit of xֿ¹ as x→0 does not exist in any sense of the word.
Aug
24
comment Proof that Pi is constant (the same for all circles), without using limits
Chris: that can be shown to be equivalent to "slicing up" the circle to form a "parallelogram" of appropriate dimensions; unfortunately for you this too involves limits.
Aug
24
comment Proof that Pi is constant (the same for all circles), without using limits
Sounds hard; its being transcendental seems to preclude the existence of a proof that won't appeal to the concept of limits.
Aug
24
comment Determine speed of the object at the current time by the non-uniform time sample
Well, using the monomial basis implies having to manipulate a Vandermonde matrix (no matter what norm you're minimizing in), and this can be ill-conditioned depending on point order and distribution. One could instead choose to use a Bernstein or orthogonal basis, and the matrix from these basis sets stands a better chance of being well-conditioned since the basis functions "don't look very much alike". Of course, if sticking with the rule of thumb I gave in a comment earlier, worries of ill-conditioning are probably moot and academic.
Aug
24
comment What is the best way to solve an equation involving multiple absolute values?
When you put it that way, then I certainly agree.
Aug
24
comment Do complex numbers really exist?
I liked the way an old electrical engineering book put it: "there's nothing imaginary about an electrical shock from j500 V!"