Velvet Star Monitor

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How does polynomial long division work?

I get normal long division but this doesn't make sense. How can doing it by only dividing by the leading term work? The problem is $$\dfrac{3x^3 - 2x...

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Is the Wronskian determinant positive or negative?

The Wronskian of the general solution $$y(x)=ay_1(x)+by_2(x)+cy_3(x)~,$$ to a third order differential equation, is given by $$ W = \begin{vmatrix} y_1 &a...

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Transpose of product of matrices [duplicate]

How do you prove the following fact about the transpose of a product of matrices? Also can you give some intuition as to why it is so. $(AB)^T = B^TA^T$

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Maximal ideal in a local artinian ring.

I know that an artinian ring $A$ is the union of its units and its zero-divisors. So every non-zero-divisor is an unit. I also know that in a local ring e...

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Power Series Representation of $x^3/(2-x)^3$

I don't need an answer, as this was a question I got wrong on a problem set, but could someone explain this? So, we have to represent f(x)= $x^3$/$(2...

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Expectation for Trinomial distribution

I am trying to understand the proof of $\mathrm{E}[xy]=n(n-1)p_1p_2$ where x,y have a trinomial distribution with pmf: $p(x,y) = \frac{n!}{x!y!(n−x−y)!}p_...

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Why do the interesting antihomomorphisms tend to be involutions?

Given a semigroup $S$, define that an antihomomorphism on $S$ is a function $$* :S \rightarrow S$$ satisfying $(xy)^* = y^*x^*.$ Examples abound. Consider...

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proof for construction of 60 degree angle

I have a construction from the game Euclidea, puzzle 4.2: The puzzle is given point $A$ and line $\overleftrightarrow{BC}$ (just the line -- neither point...

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How to plot a phase portrait for this system of differential equations?

I beg your help.. I'd like the phase portrait for this system: \begin{aligned} \frac{dx}{dt} &= x (7-x-2y) \\ \frac{dy}{dt} &= y (5-y-x) \end...

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A long nasty limit problem: $\lim_{x \to \infty}\left[8e\,\sqrt[\Large x]{x^{x+1}(x-1)!}- 8x^2-4x \ln x - \ln^2 x - (4x + 2 \ln x) \ln 2\pi\right]$

Does the following limit admit a closed-form? $$\lim_{x \to \infty}\left[8e\,\sqrt[\Large x]{x^{x+1}(x-1)!}- 8x^2-4x \ln x - \ln^2 x - (4x + 2 \ln x) \ln ...

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