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May
21
comment Explicit form of this series expansion?
Well, the first term will tend to dominate: $f(k) \approx e^{-k}$ as $k$ gets large. For $k = 6$, $f(k) = e^{-k}$ up to several digits.
May
21
comment Explicit form of this series expansion?
The upper bound is pretty easy: $f(k) \leq \int_0^\infty e^{-kx^2} \, dx = \frac{1}{2} \sqrt{\pi/k}$ since $f(k)$ is concave up. I doubt there's a simple closed form expression in elementary functions, but there could be using special functions.
May
21
comment Conjecture of the general form of a power series
It doesn't look like it appears in Sloane's. Unless you give us some idea of where this comes it's unlikely we'll be of any help.
May
20
comment What do we call well-founded posets whose elements have a unique height?
Actually, I think I found the right definition in another paper. See my answer below.
May
20
answered What do we call well-founded posets whose elements have a unique height?
May
20
comment What do we call well-founded posets whose elements have a unique height?
A "locally graded poset" is given here as one so that each interval $[x,y]$ is graded, which is pretty close but perhaps not equivalent to what you want.
May
20
awarded  Citizen Patrol
May
19
comment Geometrical Interpertation of Cauchy's Mean Value Theorem
Nice question, BTW!
May
19
comment Geometrical Interpertation of Cauchy's Mean Value Theorem
Once you understand why the theorem is equivalent to the picture, you can also get an idea of why the theorem ought to be true: Take the red line and slide it backwards or forwards without changing the slope. At first it will intersect the curve in at least two points. At the last second before the line no longer intersects the curve, the two points will become one and the line will be tangent to the curve.
May
18
comment “Vectors aren't really numbers” - how sound is that statement?
@columbus8myhw Sorry, didn't mean to derail this into a grammar argument, haha. Mostly I just like Language Log. For the record, I probably would have said "there is less use for...".
May
18
comment “Vectors aren't really numbers” - how sound is that statement?
@columbus8myhw:"...the now-standard pedantry about less/fewer is in fact one of the many false "rules" that have recently precipitated out of the over-saturated solution of linguistic ignorance where most usage advice is brewed." - Mark Liberman, U Penn linguist.
May
17
comment What is the meaning of expressions of the type $f(\cdot)$ (function (dot))?
@toni Try \cdot
May
15
comment Generalized superfactorial notation.
I don't think I've seen this before. I have seen the notation $(x)_n^{(h)}$ or $(x)\downarrow^{n,h}$ for $x(x-h)\ldots (x - (n-1)h)$. See falling factorial
May
15
comment Generating function for set of binary strings of equal block length
If you find the generating function $f(x)$ for binary strings with blocks of length $1$, think about what $f(x^k)$ might mean.
May
15
comment Generating function for set of binary strings of equal block length
But you haven't defined the "block length" of a string. I think perhaps what you mean is "the set of binary strings whose blocks are all of equal length"?
May
15
comment Similarity of orthogonal matrices
The eigenvalues of $M$ are $\pm 1$. Since $\det M = 1$, at least one of the three eigenvalues must be $1$. This means $M$ has a fixed point... can you finish from here?
May
15
comment Generating function for set of binary strings of equal block length
This is rather unclear. Can you say more precisely which binary strings you are talking about?
May
15
comment Why, intuitively, is the order reversed when taking the transpose of the product?
I don't think we really need to attach Feynman's name here, but in general this combinatorial view of matrix multiplication as a sum over walks is very helpful. The time-reversal interpretation.is a pretty useful way of looking at the transpose.
May
14
comment Series expansion for $x_n=x_{n-1}+\log \left(x_{n-1}\right)$
@martin Hmm, don't know. I suppose it's possible there's some brilliant Ramanujan-esque series solution for the sequence :)
May
14
comment Series expansion for $x_n=x_{n-1}+\log \left(x_{n-1}\right)$
I doubt it. Any random recurrence you write is unlikely to have a closed-form solution. Only those with a very special form will have a simple formula.