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Feb
13
comment Why is Euler's Identity stated the way it is?
@Yuval: "...there's not much merit..."? Really?
Feb
12
comment Calculate distance between two points N,W,E,S
Manahattan refers to the idea that in Manhattan which has mostly just east/west and north/south streets in a perfect grid, you can only follow rectilinear streets to get from any point to any other (you can't cut corners or go in diagonals).
Feb
12
revised How can I systematically find a solution to this problem?
added 1 characters in body
Feb
12
comment Solving a set of recurrence relations
From that last relation for $F_n$ you can 'solve' for $E$ (in terms of $F$ and $C$, then substitute into the first relation. Then you'll have a relation on just $F$ and $C$. Since you already have a complete non-recursive solution for $C$, you'll be able to solve for $F$.
Feb
12
comment Calculate distance between two points N,W,E,S
I think for your distances that 'wrap around' you want to add 1 to each. What's the distance between two squares that are adjacent? 1, right? then if there's a square between thew two, the distance should be 2. You're counting the steps going from the center of one square to another, not from the nearest edges of the two.
Feb
12
comment Calculate distance between two points N,W,E,S
@FeRtoll: I see what you're doing with your examples, but it seems a little 'mixed' to me. That is, if the two points appear closer without having to go over the edges, you seem to want to use the Euclidean distance (the Pythagorean formula). But if it seems closer over the edge, you take the Manahattan distance. To get this, in my definition for $d_t$, you'd change $d(p_1,p_2)$ to be the Euclidean distance instead of Manhattan, and keep the rest Manhattan. I don't think you really -want- to do this, I'm just saying that that's how you would do it if you really wanted to.
Feb
12
answered How can I systematically find a solution to this problem?
Feb
12
answered A complete guide to solving questions of the form “how many integer solutions does this sum have”?
Feb
12
answered Calculate distance between two points N,W,E,S
Feb
11
answered Solving a set of recurrence relations
Feb
11
comment shortcut for finding a inverse of matrix
@Pete: yes, I'm there with you. But in some sense, isn't all mathematics about shortcuts? Thinking real hard (finding a proof), so that you don't have to think (apply the theorem, without worrying about the details)?
Feb
11
comment shortcut for finding a inverse of matrix
@Pete: I totally agree, therefore my (weak) warnings. But I think it is a useful thing to lay out explicitly anyway. 'Lightning' calculations on multiplication (In decimal, what's n5 times n5? It's n*(n+1) followed by 25. 65*65 = 4225). It's not deep math, if you mess up 1 operation the whole thing could fall apart, and there's no meaning in the trick (figuring out the trick has some meaning), but it -is- useful.
Feb
11
comment shortcut for finding a inverse of matrix
@abcdefghijklmnopqrstuvwxyz: Well, sorta. if it's nonsingular, the determinant is 0, and so the method will work in that it will also fail when the inverse of a matrix will fail (when it is non-singular).
Feb
11
revised shortcut for finding a inverse of matrix
added 3x3 explanation
Feb
11
comment shortcut for finding a inverse of matrix
Yes...but...it involves the determinant of the 3x3 and all the 2x2 submatrices. I thought that that isn't much of a trick or shortcut; it seems about the same complexity as just plodding through row/column operations to convert the 3x3 into an identity matrix and applying those operations to an identity matrix at the same time. Of course, if there's an expectation that the determinant is 1, then maybe it's appropriate. Also, be warned that the row/column operations are 'meaningful' (you see that they are computing the inverse) but the 'trick' is just blind application of a formula.
Feb
11
answered shortcut for finding a inverse of matrix
Feb
10
comment “Closed” form for $\sum \frac{1}{n^n}$
Right. It's just a clever saying, especially considering that the freshman's dream happens all the time.
Feb
10
comment Upper bound/exact length of decimal expansion of simple fraction
What's the length of the repeating part in the second case?
Feb
10
comment Calculating the formula for a graph
are you trying to justify a formula for that page that is given somewhere else? Or are you trying to come up with a formula out of the blue? Are you trying to fit a set of points on that page loosely (where you formula just has to get close to your points, using regression) or exactly (where the function goes exactly through each given point)?
Feb
10
answered What should be in every grad student's library?