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Yes, you need to use the inclusion exclusion principle.

Oops. Nah I think that was just a typo as I typed it on the forums

I feel bad lol. I had the same idea as InteGrand, I just mucked up my matlab input when I went to check my answer _______________ \text{Is this identity true?}\\ \text{proj}_W\textbf{v} = \textbf{v} - \text{proj}_{W^\perp}\textbf{v}

A=\begin{pmatrix}3 &-1&2\\ -1&3&2\\ 2&2&0\end{pmatrix} \text{Found in part iv): }A=QDQ^T\text{ where }\\ Q=\begin{pmatrix}\frac{1}{\sqrt3} & \frac{1}{\sqrt2} & -\frac{1}{\sqrt5}\\ \frac{1}{\sqrt3} & -\frac{1}{\sqrt2} & -\frac{1}{\sqrt 6}\\ \frac{1}{\sqrt 3}& 0 & \frac{2}{\sqrt6}\end{pmatrix}\\...

Oh, now that I think about it could it potentially be x(x^2-pi^2)?

f:[-\pi, \pi]\to \mathbb{R}\text{ satisfied }f(-x)=-f(x), \, f(\pi)=0, \, f^{\prime\prime\prime}(x) = -6 \text{Proven in b): The relevant Fourier series is}\\ Sf(x) = \sum_{k=1}^\infty \frac{12(-1)^k}{k^3}\sin(kx) c) Give a simple formula for f(x) \text{Proven in a): }...

I suppose what's really bugging me is the fact that d) is actually asking to explain why b) is true. Looks like more of just a grind rather than testing some nice stuff but I suppose I'll just have to deal with it.

Goes back up to the above query I suppose. Is the only way of really proving it formally just to brute those second order derivatives?

Oh, that was done in a previous part. I understand your point there... ...albeit this was the question. It's the last part that's of interest.

Maybe it was just my unwillingness to compute any second order partial derivatives because the first order partial derivatives were thoroughly untidy. Is there any easy way of contradicting the criteria of Clairaut's theorem then?

\text{Trying to prove that the mixed partial theorem doesn't hold for this function}\\ f(x,y)=\begin{cases}0&\text{ if }(x,y)=(0,0)\\ \frac{xy(x^2-y^2)}{x^2+y^2}&\text{otherwise}\end{cases} \text{But I thought that }f, f_x\text{ and }f_y\text{ are continuous here?}

Whoops. That went over my head.

f,g\text{ are increasing functions and }X,Y\text{ are i.i.d. r.v.s} \text{RTP: }(f(X) - f(Y))(g(X) - g(Y)) \ge 0

Where does the inspiration come from that it just happens to be the image that satisfy this criteria o.O

V\text{ is a finite dimensional V.S. over }\mathbb{C}\text{ and }T:V\to V\text{ is linear} \\\text{Suppose }T^2 = T = T^*\text{ (i.e. idempotent and self-adjoint)}\\ \text{Prove that there exists a subspace of }W\text{ such that }\\ T(\textbf{v}) = \text{proj}_{W}\textbf{v} My approach thus...

Re: Discrete Maths Sem 2 2016 It depends on what the graph looks like. The way you described it is quite arbitrary, because all 3 edges could just be joining two vertices. Or even, all three edges are loops on one single vertex.

Re: Statistics Does sure convergence necessarily imply every other form of convergence?

Re: Several Variable Calculus Can someone compute the directional derivative at (0,0) of this function? f(x,y) = \begin{cases}\frac{xy^2}{x^2+y^4} & \text{if }x\neq 0\\ 0&\text{otherwise}\end{cases} \text{In some cases I end up with }\frac{\partial f}{\partial \textbf{u}} = 0\text{ but...

Re: Statistics \text{I have a joint distribution}\\ f_{(X_1,X_2)}(x_1,x_2) = \frac{1}{2\sqrt{3}\pi} e^{ -\frac12 \left[ \frac{(x_1-4)^2}{3} +(x_2-2)^2 \right] } \text{ for } x\in \mathbb{R},y \in \mathbb{R} Can I just assume that I have a bivariate Gaussian distribution with X1 and X2 independent?

Re: Statistics If I smack an indicator function onto the f(theta;x_1,...,x_n) it would be indicating on [0≤x≤theta] right?