Nicolas Villanueva
  • Member for 10 years, 9 months
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Don't understand why this binomial expansion is not valid for x > 1
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13 votes

Notice that if $x>1$, then the summation $$1+x+x^2+x^3+\ldots$$ diverges to $\infty$. And thus cannot be equal to a finite number.

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Simplifying a Trigonometric Expression
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10 votes

We can multiply the numerator and denominator by $\cos(x)$ and get, $$\frac{\cos^4(x) - 2 \cdot \cos^2(x) + 1}{\cos^2(x) \cdot \sin^2(x)}$$ $$= \frac{\left(\cos^2(x) - 1 \right)^2}{\cos^2(x) \cdot \...

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Permutations Problem
5 votes

Notice that if we have boys occupying every other seat, there are 2 total arrangements for boys and girls. If I take the arrangement where the first spot is a boy, we will have the following: $${\_\_}...

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Proving number of subsets of a set size $n$ via induction
3 votes

You think of each subset as a binary string of 0's and 1's, where the $i^{th}$ character in the string is 0 if the $i^{th}$ element in the set is not in the subset. So for your Inductive Hypothesis, ...

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How to solve $(3\log_y 5)(2\log_y 5) / (6\log_y 5)$?
3 votes

By simple arithmetic, $\frac{3 log_y(5) \cdot 2 log_y(5)}{6 log_y(5)} = \frac{6 log^2_y(5)}{6 log_y(5)} = log_y(5)$

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Amount of moving balls on chessboard
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2 votes

This is an incomplete solution, but it's a starting point and I would be open to any further suggestions to get the final solution I hope my assumptions are correct, but I'm going to assume that you ...

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Linear Algebra: Network Flow problem
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2 votes

Your first balanced flow equation is incorrect, it should be: $$x_1 - x_2 = 400$$

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How do I model poker hands as graphs such that I can evaluate using graph isomorphism?
2 votes

You should get a complete bipartite graph $G$ where $V(G) = V_1 \cup V_2$ where $V_1 = \left\{ 2,3,\ldots,10,J,Q,K,A \right\}$ and $V_2 = \left\{ Club, Heart, Diamond, Spade \right\}$. Thus a hand of ...

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Questions about a birth and death process
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2 votes

If we look at $N(t)$ to be the number of people in the room, then $S = \left\{ 0,1,2 \right\}$ with transitions of $\lambda$ from $0 \rightarrow 1$ and $1 \rightarrow 2$, and transitions of $\mu$ from ...

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Guided Random Walk on scale-free graph to have a uniform distribution over all nodes as end-step
1 votes

One possibility is to just have a $K_{3000}$ with self-loops on each vertex in $V$. Give the transition probability for each edge, $e \in E(G)$, to be $\frac{1}{3000}$.

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General solution to homogeneous difference equation
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1 votes

Even if that first equation is a typo, that is the correct general solution. If a proof is necessary, I suggest using Induction on $n$.

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probability of choosing numbers
1 votes

25 total possibilities of picking 2 numbers from $\left[1,\ldots,5\right]$: Out of those 25 possibilities, which we'll represent as pairs $(x,y)$, we see that there are 9 ways of having either $(1,x)$...

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How to calculate the new intersection on the x-axis after rotation of a rectangle?
1 votes

By the Law of Sines and since $b$ is a right angle, $$len(A) = \frac{len(B)}{sin(\frac{\pi}{2}-a)}$$ where $0 \leq a <\pi$.

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meaning of $[x]_{m(x)}$ in congruence classes modulo a polynomial
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1 votes

I believe that, $\left[ x \right]_{m(x)} = b(x) + \left(m(x)\right)$, where $b(x) \in F[\alpha]$ s.t. $deg(b(x)) < deg(m(x))$ Check here for more notes on Modular Arithmetic and Congruence ...

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What does $\ll$ mean?
1 votes

It means significantly smaller than, if I'm not mistaken.

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What is the maximum length of shortest odd cycle in a non-bipartite graph?
1 votes

This problem might be NP-Hard, since it's similar to finding the longest even Hamiltonian path, $p = (v \rightarrow w)$ , in $G$ s.t. $e = (v,w) \in E(G)$.

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Why is 33 1/3% of 240 = 79.992 wrong?
0 votes

Well, this is just being extremely picky about the solution to this problem (Personally). But the only difference is the truncating error in the calculator. For calculators, since they don't store ...

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Back substitute example
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0 votes

In the Euclidean Algorithm finding the solutions to $$3f+280y=1$$ is equivalent to finding the solutions to $$3a-280b=1$$ And it's not abnormal to have solutions that are negative since solutions to ...

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