I came across this sentence in an elementary grammar book:     Another class of animals is fish. Scientists believe there are more than 24,000 kinds of fish. Fish live in the water and come in many shapes and sizes. They have scales, fins, and gills. Most are cold-blooded.   Shouldn’t this be written "Another class of animals are fish" ?  Using “is” doesn’t sound right to me.
Essentially, I wonder that if X is a plural noun, should we use a plural or singular verb in a sentence such as "X are a subject that [verb] ..."? I believe that we should always use the singular verb form there (for example, "Eatable flowers are a subject that comes up ..."). However, I was surprised to find this sentence on the  page:     Eatable flowers are a subject that come up in almost all cooking classes eventually and I am a big fan of them.   as well as this on Google Books:     Tools are a subject that come up at our house from time to time.   I think they are typos, but I would like a double check to be sure.
Where can I find every gnome? SHOWPLAN permission denied when including execution plan in Data Explorer If I select the "Include execution plan" checkbox in the Data Explorer, the query fails:     SHOWPLAN permission denied in database 'StackOverflow'.     I see that  following maintenance. I don't know whether there was any recent maintenance. It's been several days since the data was updated and longer since an update was deployed, if that's related. Understanding p-value I know that there are lots of materials explaining p-value. However the concept is not easy to grasp firmly without further clarification.  Here is the definition of p-value from Wikipedia:     The p-value is the probability of obtaining a test statistic at least   as extreme as the one that was actually observed, assuming that the   null hypothesis is true. ()   My first question pertains to the expression "at least as extreme as the one that was actually observed." My understanding of the logic underlying the use of p-value is the following: If the p-value is small, it's unlikely that the observation occurred assuming the null hypothesis and we may need an alternative hypothesis to explain the observation. If the p-value is not so small, it is likely that the observation occurred only assuming the null hypothesis and the alternative hypothesis is not necessary to explain the observation. So if someone wants to insist on a hypothesis he/she has to show that the p-value of the null hypothesis is very small. With this view in mind, my understanding of the ambiguous expression is that p-value is $\min[P(X&lt;x),P(x&lt;X)]$, if the PDF of the statistic is unimodal, where $X$ is the test statistic and $x$ is its value obtained from the observation. Is this right? If it is right, is it still applicable to use the bimodal PDF of the statistic? If two peaks of the PDF are separated well and the observed value is somewhere in the low probability density region between the two peaks, which interval does the p-value give the probability of?  The second question is about another definition of p-value from Wolfram MathWorld:     The probability that a variate would assume a value greater than or   equal to the observed value strictly by chance.   ()   I understood that the phrase "strictly by chance" should be interpreted as "assuming a null hypothesis". Is that right?  The third question regards the use of "null hypothesis". Let's assume that someone wants to insist that a coin is fair. He expresses the hypothesis as that relative frequency of heads is 0.5. Then the null hypothesis is "relative frequency of heads is not 0.5." In this case, whereas calculating the p-value of the null hypothesis is difficult, the calculation is easy for the alternative hypothesis. Of course the problem can be resolved by interchanging the role of the two hypotheses. My question is that rejection or acceptance based directly on the p-value of the original alternative hypothesis (without introducing the null hypothesis) is whether it is OK or not. If it is not OK, what is usual workaround for such difficulties when calculating the p-value of a null hypothesis?     I posted a new  that is more clarified based on the discussion in this thread. The question is:     Are there square matrices $A,B$ over $\mathbb{C}$ s.t. $A^2+B^2=AB$ and $BA-AB$ is non-singular?   From $A^2+B^2=AB$ one could obtain $A^3+B^3=0$. Can we get something from this?  Edit:  $$A^2+B^2=AB\implies\\ A(A^2+B^2)=A^2B \implies\\ A^3+AB^2=A^2B \implies \\ A^3+(A^2+B^2)B=A^2B \implies \\  A^3+A^2B+B^3=A^2B \implies \\ A^3+B^3=0 $$ Book about classical mechanics Can a website know about my browsing history? When I visit a website it sometimes puts cookies on my browser. Usually cookies keep track of browsing info from that particular site. If I don't delete those cookies and browse other websites, will the cookies be able to know which site I am browsing? Also, can any website read browsing history which is saved by browsers? Can any website know about my browsing history by help of cookies, browsing histories saved by browsers, or by any scripting language e.g. javascript? How to use find command to search for multiple extensions I can get all jpg images by using:    find . -name "*.jpg"     But how can I add png files to the results as well? The question is to calculate the radius of convergence of $\sum\limits_{n=0}^\infty n!x^{n^2}$.  I dont know how to calculate R.O.C for such type of questions involving $n^2$ as power of x. How do we do this? Any help appreciated. Generating random numbers in C What are post-installation triggers?  These are executed when upgrading software. Show that unit circle is not homeomorphic to the real line If $a^{n}-1$ is prime then $a=2$ and $n$ is prime? I was doing some basic Number Theory problems and came across this problem :     Show that if $a$ and $n$ are positive integers with $n\gt 1$ and $a^{n} - 1$ is prime, then $a = 2$ and $n$ is prime   My Solution : (Sloppy)        $a^{n}-1$ = $(a-1)$ . $(a^{n-1} + a^{n-2} + ... + a + 1)$   This means that $(a-1)$ | $(a^{n}-1)$    But $(a^{n}-1)$ is prime   So , $(a-1)$ = 1 $\Rightarrow$ $a = 2$   Now , let $n$ be composite   $n = kl$ , where $1 \lt k \lt n$ and $1 \lt l \lt n$   Now , $a^{kl}-1$ = $(a^{k}-1)$ . $(a^{k.{(l-1)}} + a^{k.{(l-2)}} + ... + a^k + 1)$   This means that $2^{n} -1$ is composite    Hence , we have achieved a contradiction      My Question : Am I correct ? In the sentence      It's one of the best films that has/have ever been made   What should the verb agreement be? Prove that any conjugate of $a$ has the same order as $a$. I am trying to prove that any conjugate of $a$ has the same order as $a$.  Let $G$ be a group and let $a \in G$.  An element $b \in G$ is called a conjugate of $a$ if $b=xax^{-1}$.  My professor gave us a hint.  Prove $a^k=e$ iff $b^k=e$ for all $k \in \mathbb{Z}$ implies $o(a) = o(b)$.  My first move is to $b^k = (xax^{-1})^k = (x^{-1})^ka^kx^k = (x^{-1})^kex^k = (x^{-1})^kx^k=x^{-k+k}=x^0$  I am not sure that I am doing this correctly?  I am trying to get $b^k=e$.  However when I raise $(xax^{-1})^k$, I am not sure if I what I have done is correct?  I am kind of confused here. $m$ doesn't come in the sequence $a_n=[n+\sqrt{n}+\frac{1}{2}]$ iff m is a perfect square Let $(a_n)$ be a sequence of positive integers defined by  $a_n=[n+\sqrt{n}+\frac{1}{2}]$  Show that a natural number $m$ doesn't occur in the sequence iff it is a square. Does "until [date]" mean "before that date"? What does until mean in the following?     You need to deliver this product within 2 days (until August 18, 2011)   to meet your deadline and get paid.   Does this mean that I have to deliver the product by 23:59 on 17 August 2011 or 23:59 on 18 August 2011? The order time is 21:04 on August 16 2011. How to pass an ArrayList to a varargs method parameter? Representing every positive rational number in the form of $(a^n+b^n)/(c^n+d^n)$ About a month ago, I got the following :      For every positive rational number $r$, there exists a set of four positive integers $(a,b,c,d)$ such that    $$r=\frac{a^\color{red}{3}+b^\color{red}{3}}{c^\color{red}{3}+d^\color{red}{3}}.$$      For $r=p/q$ where $p,q$ are positive integers, we can take    $$(a,b,c,d)=(3ps^3t+9qt^4,\ 3ps^3t-9qt^4,\ 9qst^3+ps^4,\ 9qst^3-ps^4)$$   where $s,t$ are positive integers such that $3\lt r\cdot(s/t)^3\lt 9$.      For $r=2014/89$, for example, since we have $(2014/89)\cdot(2/3)^3\approx 6.7$, taking $(p,q,s,t)=(2014,89,2,3)$ gives us $$\frac{2014}{89}=\frac{209889^3+80127^3}{75478^3+11030^3}.$$   Then, I began to try to find every positive integer $n$ such that the following proposition is true :   Proposition : For every positive rational number $r$, there exists a set of four positive integers $(a,b,c,d)$ such that $$r=\frac{a^\color{red}{n}+b^\color{red}{n}}{c^\color{red}{n}+d^\color{red}{n}}.$$  The followings are what I've got. Let $r=p/q$ where $p,q$ are positive integers.   For $n=1$, the proposition is true. We can take $(a,b,c,d)=(p,p,q,q)$. For $n=2$, the proposition is false. For example, no such sets exist for $r=7/3$. For even $n$, the proposition is false because the proposition is false for $n=2$.   However, I've been facing difficulty in the case of odd $n\ge 5$. I've tried to get a similar set of four positive integers $(a,b,c,d)$ as the set for $n=3$, but I have not been able to get any such set. So, here is my question.     Question : How can we find every odd number $n\color{red}{\ge 5}$ such that the following proposition is true?      Proposition : For every positive rational number $r$, there exists a set of four positive integers $(a,b,c,d)$ such that $$r=\frac{a^n+b^n}{c^n+d^n}.$$   Update : I posted this question on .  Added :  asks the case $n=3$. How to safely remove proprietary Intel driver I recently installed the proprietary Intel drivers for my Toshiba Satellite A105 s4284 running Ubuntu 14.10. Nouveau was working just fine, I was just attempting to make it so that I was able to run CS:S (no need for explanation on that.)  Long story short, I went and downloaded the Intel Graphic Drivers for Linux* from . After downloading and installing the .deb, I ran intel-linux-graphics-installer which opened up a window that downloaded &amp; installed the open Intel drivers. I then rebooted, and it's slow. Very slow, especially on 3D effects and when I hit Alt+Tab.  Now, I don't know if the previously installed Nouveau drivers are conflicting with these new Intel drivers (if that's the issue, what should I do about fixing it?) What I do know is that my system isn't behaving right and I want it to (obviously.) If removing the Nouveau drivers isn't the solution, how can I safely remove the just-installed Intel drivers?  Does anyone have any suggestions? What is a plain English explanation of "Big O" notation? I'd prefer as little formal definition as possible and simple mathematics.