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It can – consider y = x^5 at the origin.

The total of n integers will have a remainder of either 0, 1, 2, 3, …, or (n-1) upon division by n. This implies that the possible fractional parts for the average of n whole numbers are just: 0, 1/n, 2/n, 3/n, …, (n-1)/n. E.g. If 1 subject: only can have decimal of 0. If 2 subjects: only can...

$\noindent Note that if $\lambda$ is an eigenvalue of $Q$ with unit eigenvector $\vec{v}$, then$ $$\begin{align*}\lambda &= \lambda\left\langle \vec{v},\vec{v}\right\rangle \\ &=\langle \lambda\vec{v},\vec{v}\rangle \\ &= \langle Q \vec{v},\vec{v}\rangle \\ &= \langle \vec{v},Q^{*}...

Note f'(x) = 5 for all x, so f'(1) = 5.

Well you wrote one up here before, so maybe you'd find that easiest to "memorise" for yourself: Note that it needs to be adapted slightly to deal with the complex case, but it's not too big a deal. You can also probably find many proofs online. There are twelve proofs here, but they seem to...

Hint: Use the axioms to show that a + a0 = a.

B (as I said in my first line)

Answer: B tells the complete lie. Assume C tells the complete truth (so B and C are the two who did it). Then we can see that: • B must be telling the complete lie (since neither D nor A did it, yet B says these two did it) • A and D both tell half-truths (each identifies precisely one of the...

Let the tensions be T1 and T2, and obtain two simultaneous equations by decomposing forces in the horizontal and vertical direction (using right-angle trigonometry), and using that the acceleration in each of these directions should be 0 (assuming we are in equilibrium).

Also the first term on your RHS has a mistake/typo (shouldn't have y).

No

Also, if V is a subset of U, then f(V) is automatically a subset of f(U) (true for any function and sets (that make sense)). (Or maybe you meant that it doesn't state that f(V) is open.)

Generally this is correct, but consider the implications of f being an invertible (locally) continuous map.

What have you tried so far? Did you try maybe considering the inverse function theorem?

$\noindent Have you tried using the chain rule?$

$\noindent We can show this using a limit comparison test and the $p$-test. $\Big{(}$Note that the integrand is continuous on $[0,\infty)$ and is asymptotically equivalent to $\frac{1}{x^{\frac{\alpha}{2}-2}}$ as $x\to \infty$, for $\alpha > 0$.$\Big{)}$$

We don't know it ("it" refers to the population standard deviation (or really the standard deviation of the assumed underlying distribution)), since that's only the sample standard deviation.

Oh yeah, I forgot to include the n in front of the matrix when getting the 1,1 entry of the inverse. It should be n in the denominator indeed (for final answer).

It depends on the hypothesis test and the situation at hand. Common ones are when you have asymptotic normality under a null hypothesis (which is often the case from the Central Limit Theorem), you can use a Z-test ( https://en.wikipedia.org/wiki/Z-test ), or you could use a Chi-Squared test...

Also I think the final answer's denominator should be n^2 (not n).