Improve Your Algebra Skills with Practice Problems

With the help of a suitable transformation and Fubini I want to determine the integral
${\int <!--\; \int \; -->}_V{x}^{3}yd{\mathrm{\lambda <!--\; \lambda \; -->}}_2(x,y),$
where $V$ is the open subset of ${\mathbb{R}}_{+}^2$ bounded by the following curves:
$\begin{array}{rl}& x^2+y^2=4\\ & x^2-<!--\; -\; -->y^2=2\\ & x^2-<!--\; -\; -->y^2=1\end{array}$
I know how to do that. The only problem is finding $V.$ Is it
$V=\{(x,y)\in <!--\; \in \; -->{\mathbb{R}}^2:1<x^2-<!--\; -\; -->y^2<2,0<x^2+y^2<4\}$
Because then I set
$\begin{array}{rl}& x^2+y^2=v\\ & x^2-<!--\; -\; -->y^2=u\end{array}$
and get
$(x,y)=(\sqrt{1/2(v-<!--\; -\; -->u}),\sqrt{1/2(u+v)})$
So either my transformation is wrong or the limits of the intervalls I chose.
Thanks for any kind of help.

Proving functions to be Big Oh
How do I determine if there exists a function $f$, such that
$f(n)=\mathcal{O}(\log <!--\; \; -->n),$
but
$2^{f(n)}\ne \mathcal{O}(n).$
Is true or false?
I tried using the c and No method but cannot come up with a solution

How do you solve ${\displaystyle n+5<11}$ ?

How do you solve ${\displaystyle 18\le x÷6}$ ?

I am self-studying introduction to ergodic theory and found this problem (application of Poincare recurrence theorem) which I couldn't do.
I need to show the set, $E=\{x\in <!--\; \in \; -->[0,1]:\text{decimal representation of }x\text{ has }5\text{ infinitely often}\}$, has measure 1. How can this be extended to show that almost all elements in $[0,1]$ contain the block (this sequence of numbers appear consecutively and in the same order) 123 in decimal representation infinitely many times?
What I did: I know, by Poincare recurrence theorem, for almost all elements in $[0.5,0.6)$ have 5 infinitely often in the decimal representation. I am not sure how the Poincare recurrence theorem can be used to extend this to the whole of $[0,1]$. Any hints would be helpful.

Solving a rational equation with multiple and nested fractions
This is the equation to solve: ${\displaystyle \frac{{\displaystyle \frac{x+{\displaystyle \frac{1}{2}}}{{\displaystyle \frac{1}{2}}+{\displaystyle \frac{x}{3}}}}}{{\displaystyle \frac{1}{4}}+{\displaystyle \frac{x}{5}}}}=3$
What I did:
$x+{\displaystyle \frac{1}{2}}={\displaystyle \frac{2x+1}{2}}$
${\displaystyle \frac{x}{3}}+{\displaystyle \frac{1}{2}}={\displaystyle \frac{2x+3}{6}}$
${\displaystyle \frac{x}{5}}+{\displaystyle \frac{1}{4}}={\displaystyle \frac{4x+5}{20}}$
${\displaystyle \frac{2x+1}{2}}÷<!--\; ÷\; -->{\displaystyle \frac{2x+3}{6}}={\displaystyle \frac{2x+1}{2}}\times <!--\; \times \; -->{\displaystyle \frac{6}{2x+3}}={\displaystyle \frac{6x+3}{2x+3}}$
${\displaystyle \frac{6x+3}{2x+3}}÷<!--\; ÷\; -->{\displaystyle \frac{4x+5}{20}}={\displaystyle \frac{6x+3}{2x+3}}\times <!--\; \times \; -->{\displaystyle \frac{20}{4x+5}}=3$
$⟹<!--\; ⟹\; -->{\displaystyle \frac{120x+60}{(2x+3)(4x+5)}}=3$
I know how to solve the equation. But right now I tried several times and I got wrong answers.
So I appreciate your help. Thank you.

Convergence of a series
Does this series converge?
$\underset{x=2}{\overset{n}{\sum <!--\; \sum \; -->}}{\left(\frac{1}{x}\right)}^{\left(\frac{1}{x}\right)}$
I tried hard but stil had problems...
Could someone help me?

Find the smallest number $\mathrm{\alpha <!--\; \alpha \; -->}$, such that for all $x,y,z$
$\mathrm{\alpha <!--\; \alpha \; -->}(x^2-<!--\; -\; -->x+1)(y^2-<!--\; -\; -->y+1)(z^2-<!--\; -\; -->z+1)\ge <!--\; \ge \; -->(xyz{)}^2-<!--\; -\; -->xyz+1$
My work so far:
Let $f(t)=t^2-<!--\; -\; -->t+1$. Then $f(t)\ge <!--\; \ge \; -->\frac{3}{4}$
If $x=0,y=z=\frac{1}{2}$, then
$\mathrm{\alpha <!--\; \alpha \; -->}\ge <!--\; \ge \; -->\frac{16}{9}$

Let $(X,\mathcal{A},\mathrm{\mu <!--\; \mu \; -->})$ be a measure space let f and $f_1,f_2,\dots <!--\; \dots \; -->$ be nonnegative real-valued $\mathcal{A}$-measurable functions on X that are $\mathrm{\mu <!--\; \mu \; -->}$-integrable.
Suppose that they satisfy
i) $\{f_n\}$ converges to f $\mathrm{\mu <!--\; \mu \; -->}$-a.e., and
ii) $\int <!--\; \int \; -->fd\mathrm{\mu <!--\; \mu \; -->}=\underset{n}{lim}\int <!--\; \int \; -->f_{n}d\mathrm{\mu <!--\; \mu \; -->}$.
Show that $\underset{n}{lim}\int <!--\; \int \; -->|f_n-<!--\; -\; -->f|d\mathrm{\nu <!--\; \nu \; -->}=0$.
I'm thinking I want to bound $|f_n-<!--\; -\; -->f|$ from above $\mathrm{\mu <!--\; \mu \; -->}$-a.e. by something $\mathrm{\mu <!--\; \mu \; -->}$-integrable, independent of n, and use the DCT, but I'm not successful yet. Any hints?

What is 15.36 divided by 4.8?

Probability of 2 students being chosen the both have under 100 books at home
Suppose we select two students at random from the class of fifteen. What is the probability that both students chosen have less then 100 books at home?
Data provided is the amount of books each student has at home. There are 6 out of the 15 students that have less than 100 books:
33, 47, 48, 52, 56, 66, 173, 186, 251, 300, 323, 417, 450, 1280, 2000
I tried doing 6/15 x 5/14 = 30/210 or a 14% chance. Is this correct?
Thank you!

What skills should I learn before going for a PhD in numerical analysis, numerical linear algebra and scientific computing?
I have two years in my hand after which I would be applying for graduate school most probably in the US. I already have a masters degree in pure mathematics and have very limited computer skills. For some reason, eventually I got interested in applied mathematics.
So, now I am going for a second masters (which has not yet started because of the pandemic) in applied mathematics which is heavy on the above-mentioned topics. As of computer skills, I have started learning C++ but I don't know what else should I learn. It would be very helpful if someone can give a roadmap which would prepare me to finally embark for a PhD in one of these areas i.e., in and around numerical analysis.
What math and computer science courses should I take? What software should I learn? Any general advice you would like to give is also welcome. Book recommendations would also be very helpful for me.Thanks!

Assuming there's a variable I want to measure, but I have only very noisy instrument to do so. So I want to take multiple measurements so that I have a better chance to recover the state of the variable. Hopefully, with each measurement, my instrument can report the result as a Gaussian distribution , with the mean to be the most likely state of variable and the standard deviation suggests a rough possible region of the state.
My problem now is that I don't know how to combine these multiple measurements to get a sensible answer. My guess is that it would be nice if I can get a new gaussian from these results, with the mean centered at the expectation value of the state of the variable, and a standard deviation to reflect how confident I am about the result...
I tried to teach myself about gaussians, and probabilities, but I just couldn't get my head around...please can someone help me?

I have a solid laundry detergent with measurement instructions advising a ratio of 4 litres (1 gallon) hot water to 15 ml (1 tbs) of detergent.
If someone wanted a variable measurement, such as how much detergent one would use with 1 litre of water, or 500 ml of water, how would they calculate this ratio?

Let $(\mathrm{\Omega <!--\; \Omega \; -->},\mathcal{A},\mathrm{\mu <!--\; \mu \; -->})$ be a finite measure space, $p>2$ and $f,g:\mathrm{\Omega <!--\; \Omega \; -->}\to <!--\; \to \; -->\mathbb{R}$ with $|f|,|g|\le <!--\; \le \; -->1$.
How can we show that ${\Vert f-<!--\; -\; -->g\Vert }_{L^p(\mathrm{\mu <!--\; \mu \; -->})}^p\le <!--\; \le \; -->2^{p-<!--\; -\; -->2}{\Vert f-<!--\; -\; -->g\Vert }_{L^2(\mathrm{\mu <!--\; \mu \; -->})}^2$?
The trick must obviously be writing ${\Vert f-<!--\; -\; -->g\Vert }_{L^p(\mathrm{\mu <!--\; \mu \; -->})}^p=2^p{\Vert \frac{f-<!--\; -\; -->g}{2}\Vert }_{L^p(\mathrm{\mu <!--\; \mu \; -->})}^p$ and noting that $|f-<!--\; -\; -->g|\le <!--\; \le \; -->2$, but how do we need to proceed?

Data that follows a distribution with non-finite expectation
I find it difficult to get around the idea of some random variables following some distributions (such as the Cauchy Distribution) not to have finite means.
How does one actually interpret data from such random variables? Do we give up and not investigate sample means or ...? Please help clarify this.

I feel that a radian measurement is a real number because we use it widely in calculus and the evaluation of limits of real and complex functions; but I've never heard, for example, someone consider
$\underset{x\to <!--\; \to \; -->0}{lim}\frac{\sin <!--\; \; -->x}{x}=1$
when x is in degrees.
Is an angle in degrees not a mathematical quantity, but a physical quantity?
Is a degree measurement a real number?

How do you solve ${\displaystyle 4x-5>x-11}$ ?

Trending proof for fairly simple fraction
(Please humour the physicist!)
Why does $\left(\frac{1-<!--\; -\; -->a}{1-<!--\; -\; -->a^b}\right)\to <!--\; \to \; -->\frac{1}{b}$ as $a\to <!--\; \to \; -->1$?
This came from a calculation involving flow measurement of gases, and although I can see and compute the answer, I don't understand how this general result occurs.

What is the GCF of 210 and 252?