When I was writing a literature survey on Moore-Penrose pseudoinverse (literatures like this one, and this one), I encountered with the following equality which was named as reverse order law: (AB)+=B∗(A∗ABB∗)+A∗. They claimed this equality, by noticing AB=((A∗)+A∗A)(BB∗(B∗)+)=(A∗)+(A∗ABB∗)(B∗)+, and then applying a 'general reverse order law' (PNQ)+=Q+N+P+ here. In these papers, this law (PNQ)+=Q+N+P+ was widely used to derive equalities on (AB)+. My question here is that (PNQ)+=Q+N+P+ cannot be true for arbitrary matrix P,N,Q. Are there any extra requirements for it to hold, how is it proved, and how does it fit in the above case? Are there any references? References on the first discovery of equality (AB)+=B∗(A∗ABB∗)+A∗ are also appreciated.

In a random survey 250 people participated. Out of 250 people who took part in the survey, 40 people listen to Pink Floyd. 30 people listen to Metallica and 20 people listen to John Denver. If 10 people listen to all three then find the no. of people who listen only Pink Floyd.

The volume of a sphere is increasing at a rate of 3 cubic centimetres per second. How fast is the surface area increasing when the radius is 2 centimetres ?
A)$3c{m}^2/s$
B)$6c{m}^2/s$
C)$12c{m}^2/s$
D)$1c{m}^2/s$

Determine the average value of function $y=Asi{n}^{2}x$ in the range x=0 to $x=\mathrm{\pi <!--\; \pi \; -->}$.

Find the average angular speed of the minute hand of a normal clock in 30 minutes.

Find the average value $f_{ave}$ of the function f on the given interval.
$f(x)=3x^2+8x,[-1,2]$

A negative potential energy is possible. Explain

Find the average value fave of the function f on the given interval.
$f(t)=e\sin <!--\; \; -->(t),\cos <!--\; \; -->(t),[0,\frac{\mathrm{\pi <!--\; \pi \; -->}}{2}]$

What is key in user research?

On the generalized Sierpinski space
In Sierpiński topology the open sets are linearly ordered by set inclusion, i.e. If $S=\{0,1\}$, then the Sierpiński topology on S is the collection $\{\varphi ,\{1\},\{0,1\}\}$ such that $\mathrm{\varphi <!--\; \varphi \; -->}\subset <!--\; \subset \; -->\{0\}\subset <!--\; \subset \; -->\{0,1\}$ we can generalize it by defining a topology analogous to Sierpiński topology with nested open sets on any arbitrary non-empty set as follows: Let X be a non-empty set and I a collection of some nested subsets of X indexed by a linearly ordered set $(\mathrm{\Lambda <!--\; \Lambda \; -->},\le <!--\; \le \; -->)$ such that I always contains the void set $\varphi$ and the whole set X, i.e.
$I=\{\mathrm{\varnothing <!--\; \varnothing \; -->},A_{\mathrm{\lambda <!--\; \lambda \; -->}},X:A_{\mathrm{\lambda <!--\; \lambda \; -->}}\subset <!--\; \subset \; -->X,\mathrm{\lambda <!--\; \lambda \; -->}\in <!--\; \in \; -->\mathrm{\Lambda <!--\; \Lambda \; -->}\}$
such that $A_{\mathrm{\mu <!--\; \mu \; -->}}\subset A_{\mathrm{\nu <!--\; \nu \; -->}}$ whenever $\mathrm{\mu <!--\; \mu \; -->}\le <!--\; \le \; -->\mathrm{\nu <!--\; \nu \; -->}$.
Then it is easy to show that I qualifies as a topology on X.
My questions are:
(1) Is there a name for such a topology in general topology literature?
(2) Is there any research paper studying such type of compact, non-Hausdorff and connected chain topologies?

Is the invariant subspace problem open for invertible maps?
Let $T:H\to <!--\; \to \; -->H$ be a bounded linear operator with bounded inverse on the separable complex Hilbert space. Does T preserve a closed proper non-trival invariant subspace?
I'm aware the question is (famously) open for bounded linear maps, and of partial results, but no survey (or Tao's blog, etc) seem to address the invertible case.
If it is open, does a positive or negative answer imply the answer in the non-invertible case?

Finite Math (Probability/Venn Diagram)
13) A survey revealed that 25% of people are entertained by reading books, 39% are entertained by watching TV, and 36% are entertained by both books and TV. What is the probability that a person will be entertained by either books or TV? Express the answer as a percentage.
Is this problem stated correctly? How can 36% of the people be entertained by both books and TV, when only 25% of the people are entertained by reading books?
EDIT
Here are two other questions from the exam, that the instructor said followed the same logic as the question above.
14) Of the coffee makers sold in an appliance store, 5.0% have either a faulty switch or a defective cord, 1.6% have a faulty switch, and 0.2% have both defects. What is the probability that a coffee maker will have a defective cord? Express the answer as a percentage.
15) A survey of senior citizens at a doctor's office shows that 42% take blood pressure-lowering medication, 45% take cholesterol-lowering medication, and 13% take both medications. What is the probability that senior citizen takes either blood pressure-lowering or cholesterol-lowering medication? Express the answer as a percentage.

Homotopy equivalence between two mapping tori of compositions
For any maps $s:X\to <!--\; \to \; -->K$ there is defined a homotopy equivalence
$T(d\circ <!--\; \circ \; -->s:X\to <!--\; \to \; -->X)\to <!--\; \to \; -->T(s\circ <!--\; \circ \; -->d:K\to <!--\; \to \; -->K);(x,t)\mapsto <!--\; \mapsto \; -->(s(x),t).$
Here, T(f) denotes the mapping torus of a self-map $f:Z\to <!--\; \to \; -->Z$ (not necessarily a homeomorphism). It is very surprising to me that this holds with no extra conditions on d and s. I'm guessing that the homotopy inverse is the map:
$T(s\circ <!--\; \circ \; -->d)\to <!--\; \to \; -->T(d\circ <!--\; \circ \; -->s),(k,t)\mapsto <!--\; \mapsto \; -->(d(k),t).$
If the above is a genuine homotopy inverse, then the map:
$(x,t)\mapsto <!--\; \mapsto \; -->(d(s(x)),t)$
would have to be homotopic to the identity somehow. However, after banging my head against the wall on this for a while I can't come up with a valid homotopy. So my questions are:
Is the map $T(s\circ <!--\; \circ \; -->d)\to <!--\; \to \; -->T(d\circ <!--\; \circ \; -->s)$ I've defined above actually a homotopy inverse? If so, what is the homotopy from the composition I wrote down above to the identity map?
Is there a better one that makes the homotopy obvious?

A survey is mailed to a random sample of residents in a city asking whether or not they think the current mayor is doing an acceptable job. What type of bias do you think would most likely be introduced in this type of situation? You can argue for more than one answer but please select the answer that would be the main source of bias. (This question is from your textbook) Group of answer choices
A. Response bias.
The wording of the survey may be confusing or provoke a certain response.
B. Undercoverage.
The entire population is not reached,
C. Nonresponse bias.
People who feel strongly about the mayor are more likely to respond.

Solve clustering problem. After step $1$ i.e putting each data-point to a cluster now calculate the mean or average of all of the data-points in a specific cluster. let suppose We have some data-points like $A(2,3),B(4,5),C(6,7),D(8,9)$ in a cluster. How can we calculate their average?

Calculate an average or typical time. If I would just use the median (or other "normal" types of calculating an average), for example $23:59$ and $00:01$ would yield $12:00$ when it should $00:00$. Is there a better method?