Recent content by ArcanaNoir

Wowowowow! Thanks so much :)

The number of terms from 500 to 1000 including both 500 and 1000 is actually 501. To make it easier to see, subtract 500 so we are counting the number of terms between 0 and 500 including both 0 and 500. Since you are starting at 0, and not 1, you actually have one extra term. In your second...

It means to argue by counting the same thing two different ways. For example choosing a committee and then a subcommittee versus choosing a subcommittee and then the remaining regular committee members would represent the equality \binom{n}{k} \binom{k}{r}=\binom{n}{r} \binom{n-r}{k-r} .

Homework Statement Give a combinatorial proof that (n-r)\binom{n+r-1}{r} \binom{n}{r}=n\binom{n+r-1}{2r} \binom{2r}{r} Homework EquationsThe Attempt at a Solution I interpreted the right side of the equation as: There are n grad students and r undergrads. First, from the n grad students...

Here is a picture with the triangle that I think might be relevant. The arc A is the arc of the inner circle that is inside the left hand circle. My professor told me that for values near 0, sin(x) and arcsin(x) are approxiately x, and that is why we can estimate arcsin(3(1-r)) as O(1-r)...

)pdate: one of my professors has basically dismissed the formula given in the chapter, and instead is focusing on the needed estimate, that the length of A is O(1-r). I don't really see why. He said (something like) \sin(\theta)<3(1-r) implies this. I may have misinterpreted his statements as he...

Homework Statement My class is working through chapter 2 of Newman's Analytic Number Theory text (on partitions). We have come to a part where he states that "elementary geometry gives the formula" (for the length of arc A) 4r\text{arcsin}\frac{\sqrt(2)(1-r)}{\sqrt(r)} We are attempting to...

How can you express the measure of angle 3? Can you express the measure of angle 5 in the same way? [edit] I think I was transposing the 4 and the 6. >_<

Let {Q} be a collection of cubes covering a set E in R^n. Prove that there is a countable sub collection {Q}' of these cubes which covers E and \cup {\frac{1}{2}Q} \subseteq \cup {Q}' , and the number of cubes in the subcollection containing any given point of E is less than something depending...

Homework Statement Let {Q} be a collection of cubes covering a set E in R^n. Prove that there is a countable sub collection {Q}' of these cubes which covers E and \cup (\frac{1}{2}){Q} \subseteq \cup {Q}' , and the number of cubes in the subcollection containing any given point of E is less...

You're so inspirational ILS, you're at the level where you are feeding me insight telepathically!

Okay I made "progress". Didn't solve it but I have more effort to offer. I remembered the Euclidean algorithm can be used to find the gcd of two numbers. it goes like this: a=q_0b+r \\ b=q_1r_0+r_1 \\ r_0 = q_2r_1+r_2 and so on until the last non-zero remainder, which will be the gcd. So I...

The problem asks to find a generator of the principal ideal <6+7i, 5+3i> in Z[i]. It is my understanding that such a generator must be a greatest common divisor of 6+7i and 5+3i. So, let's call this guy d, we should have d(a+bi)=6+7i and d(c+di)=5+3i. I'm not really sure how to find d. If I...

*first snow first snow first snow first snow*! :biggrin: As a bonus, it's self shoveling!

Thank you so much ILS. This analysis makes me want to cry!