possessives and number (sing./pl) I'm revising a text containing these two sentences:      a) Students’ sex was coded by using their first names.        b) Students’   birthday was self-reported in the survey.   I'm sure the correct version should have "students' birthdays" because the rule in English is to use the plural for the possessed objects, even if the possessors only have one each.  But if I apply this logic to sentence a) about sex, I get  "students' sexes" ...  which seems ridiculous. Why? They are both potentially plural, yet each student has only one sex, and one birthday.   I wonder if the parameter "sex" has only a superficial resemblance to the parameter "birthday". Have you any ideas? Many thanks.
Do you pluralize the singular possessions of / items or people associated with individual members of a plural group? In a situation where say a group (or at least a plurality) of men is being addressed — for example on a sign passed by many married men — which is correct?  &quot;Remind your wife.&quot;  or  &quot;Remind your wives.&quot;
Is there an official list of Disney Princesses? In $Sets$, the initial object is $0 = \emptyset$. We have $C \times \emptyset = \emptyset$ for any set $C$. As there are no maps to the empty set, we don't get much of the universal property of the product.  I am now wondering what is special about arbitrary categories $\mathscr{C}$ with $C \times 0 \cong 0$ for all objects $C$ of $\mathscr{C}$, where $0$ is the initial object of $\mathscr{C}$.   From Sheaves in Geometry and Logic by MacLane (p.194), in any topos $\mathscr{C}$, any arrow $k:C \to 0$ is an isomorphism. The universal property of the product would then also not yield very much.   Are there any categories of higher interest? If A and B are symmetric is AB=BA true? Question  if we have matrices $A$ and $B$ which are symmetric. Both of these satisfy: $$  A=A^T$$ Is $AB=BA$ true ?  Attempt to solve  Now if $A$ and $B$ are symmetric then $AB$ most be symmetric too.  $$ AB=(AB)^T=B^TA^T=BA $$  problem is how do you proof that if $A$ and $B$ are symmetric also $AB$ is symmetric ?  If somone could provide some insight on this that would ge highly appreciated.  Thanks,  Tuki I have an array of Foo objects. How do I remove the second element of the array?  I need something similar to RemoveAt() but for a regular array. xticks steps using Pgfplots / dateplot Using right-hand Riemann sum to evaluate the limit of $ \frac{n}{n^2+1}+ \cdots+\frac{n}{n^2+n^2}$ I'm asked to prove that $$\lim_{n \to \infty}\left(\frac{n}{n^2+1}+\frac{n}{n^2+4}+\frac{n}{n^2+9}+\cdots+\frac{n}{n^2+n^2}\right)=\frac{\pi}{4}$$ This looks like it can be solved with Riemann sums, so I proceed:  \begin{align*} \lim_{n \to \infty}\left(\frac{n}{n^2+1}+\frac{n}{n^2+4}+\frac{n}{n^2+9}+\cdots+\frac{n}{n^2+n^2}\right)&amp;=\lim_{n \to \infty} \sum_{k=1}^{n}\frac{n}{n^2+k^2}\\ &amp;=\lim_{n \to \infty} \sum_{k=1}^{n}(\frac{1}{n})(\frac{n^2}{n^2+k^2})\\ &amp;=\lim_{n \to \infty} \sum_{k=1}^{n}(\frac{1}{n})(\frac{1}{1+(k/n)^2})\\ &amp;=\lim_{n \to \infty} \sum_{k=1}^{n}f(\frac{k}{n})(\frac{k-(k-1)}{n})\\ &amp;=\int_{0}^{1}\frac{1}{1+x^2}dx=\frac{\pi}{4} \end{align*}  where $f(x)=\frac{1}{1+x^2}$. Is this correct, are there any steps where I am not clear? For the recently reported production (January 2017) of metallic hydrogen in the laboratory - what is the evidence exactly? My google news page has burst forth the news that metallic hydrogen has been produced in a laboratory using a diamond anvil cell. Only  mentions (at this particular moment) that it has been called into question at - at least - premature. That can be read about in Nature:   Here I only want to ask about the evidence. Is it only the sudden change in visual appearance change as recorded in a camera on top of the microscope, at a specific temperature and pressure? That it "looks shiny?" Or is there actually more direct data than that?  note: I am not asking if the data is sufficient or not, nor if it is convincing, or metallic or not. Just a narrowly defined question about the extent of the observations that have lead to the proposal that a metallic state was achieved.     above: potentially three states of hydrogen as viewed in a diamond anvil cell, from . Why is my "Committed" memory so much higher than my actual RAM space? I have run two virus scans and neither of them have found anything so far (Malware Bytes and Avira Rescue). During gaming I will get Windows popups saying that my system is low of memory and recommends closing the game. This can also happen while browsing. My computer seems to become sluggish a lot and will hang up often, such as when clicking a link from a messenger program to open in my browser.   I am unsure what is causing this. My memory usage on the processes tab doesn't add up to such a high percentage of my actual memory.    As a side note, is my Disk Usage supposed to spike up to 100% sometimes? Research seems to say so. Non-numerical citation mark like [Str91] [Samar] [Tro77] Find: $$\lim\limits_{x \to 1} \left( {\frac{x}{{x - 1}} - \frac{1}{{\ln x}}} \right) $$  Without using L'Hospital or Taylor approximations  Thanks in advance Find when three numbers have the same remainder when divided by the same number If three numbers 112, 232, and 400 are each divided by the number D, each of their quotients will have the same remainder R. Find R where R>1  How should I approach this? Adding execute permission in opt folder I am trying to install oracle database in Ubuntu 16.04 LTS. I had unzipped the oracle database file in opt folder,but now when I am trying to give execute right to the sqldeveloper file using chmod +x sqldeveloper.sh, it's throwing this error:  chmod: changing permissions of 'sqldeveloper.sh': Operation not permitted Most of the time one or the other feels better, but every so often, "which" vs. "what" trips me up.   So, what's the exact difference and when should you use one or the other? I have a script that updates my desktop background every day with the NASA APOD. Since upgrading to 11.10, that script stopped working, presumably because I'm using gconftool-2 to actually update the desktop background. Is there a new way to do this under Unity? Traditional statistical tests, like the two sample t-test, focus on trying to eliminate the hypothesis that there is no difference between a function of two independent samples. Then, we choose a confidence level and say that if the difference of means is beyond the 95% level, we can reject the null hypothesis. If not, we "can't reject the null hypothesis". This seems to imply that we can't accept it either. Does it mean we're not sure if the null hypothesis is true?   Now, I want to design a test where my hypothesis is that a function of two samples is the same (which is the opposite of traditional statistics tests where the hypothesis is that the two samples are different). So, my null hypothesis becomes that the two samples are different. How should I design such a test? Will it be as simple as saying that if the p-value is lesser than 5% we can accept the hypothesis that there is no significant difference? This might seem a stupid question but why do Ethernet cables have 8 wires? Cat5 cables were just using 4 of the 8 wires, so only 4 are actualy 'needed'. Why not 12 or 16 wires? This is a question in general, not specific to any method or data set. How do we deal with a class imbalance problem in Supervised Machine learning where the number of 0 is around 90% and number of 1 is around 10% in your dataset.How do we optimally train the classifier.  One of the ways which I follow is sampling to make the dataset balanced and then train the classifier and repeat this for multiple samples.   I feel this is random, Is there any framework to approach these kind of problems. Optimizing a screenshot for adding in a document and printing I keep reading in economics journals about a particular result used in random utility models. One version of the result is: if $\epsilon_i \sim_{iid}, $ Gumbel($\mu, 1), \forall i$, then:   $$E[\max_i(\delta_i + \epsilon_i)] = \mu + \gamma + \ln\left(\sum_i \exp\left\{\delta_i \right\} \right),  $$  where $\gamma \approx 0.52277$ is the Euler-Mascheroni constant.  I've checked that this makes sense using R, and it does.  The CDF for the Gumbel$(\mu, 1)$ distribution is:  $$G(\epsilon_i) = \exp(-\exp(-(\epsilon_i - \mu)))$$  I'm trying to find a proof of this and I've had no success.  I've tried to prove it myself but I can't get past a particular step.   Can anyone point me to a proof of this?  If not, maybe I can post my attempted proof up to where I get stuck. Vitali-type set with given outer measure