What is the difference between "I expected" and "I had expected"?
What are the difference between the following sentence?    I ate apples. I have eaten apples.    When should we use simple past tense? When should we use perfect past tense?
How many ways can we let people into a movie theater if they only have half-dollars and dollars? My interest in combinatorics was recently sparked when I read about the many things that the Catalan numbers count, as found by Richard Stanley. I picked up a copy of Brualdi's Combinatorics, and while browsing the section on counting sequences I found a nice little puzzle that has definitely puzzled me.     Let $m$ and $n$ be nonnegative integers with $n\geq m$. There are $m+n$ people in line to get into a theater for which admission if $50$ cents. Of the $m+n$ people, $n$ have a $50$-cent piece and $m$ have a $\$ 1$ dollar bill. The box offices opens with an empty cash register. Show that the number of ways the people can line up so that change is available when needed is   $$ \frac{n-m+1}{n+1}\binom{m+n}{m}. $$     I first noted that the first person to enter must be one of the $n$ with a half-dollar. Now the register has a half-dollar change. The second person can be either a person with a half-dollar or a dollar. In the first case, the register will now have two half-dollars, in the second case, the register will now have one dollar bill. So it seems to me that when one of the $n$ people with a half-dollar enters, the number of half-dollars in the register increases by $1$, and when one of the $m$ people with a bill enters, the number of half-dollars decreases by $1$ but the number of bills increases by $1$.   I tried to model this by looking at paths in $\mathbb{Z}^2$. The $x$-axis is like the number of half-dollars, and the $y$-axis is the number of bills. You start at $(0,0)$, and you can take steps forward $(1,0)$ or backwards diagonally $(-1,1)$ corresponding to who enters, but you must always stay in the first quadrant of the plane without crossing over the axes. The goal is to make $m+n$ moves, and I figured maybe the number of such paths is counted by $\frac{n-m+1}{n+1}\binom{m+n}{m}$, but I'm not sure how to show this. I don't know if this observation simplifies the problem at all, as I don't know how to finish up. I'd be happy to see how this problem is done, thank you. Let $A$ be a ring with $0 \neq 1 $, which has $2^n-1$ invertible elements and less non-invertible elements. Prove that $A$ is a field. I have to teach the Bayes formula to some undergraduates, in the form:  $$ P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{P(B|A)P(A)}{P(B)} $$  I was wondering if anyone had any really thoughtful or alternative ways of teaching it instead of the standard approach. Methods to compute $\sum_{k=1}^nk^p$ without Faulhaber's formula Find which date is missing in the results I can't update 12.04 LTS. The Update Manager shows this:  CD/DVD 'Ubuntu 12.04 LTS _Precise Pangolin_ - Release i386 (20120423)' is required Please insert the above CD/DVD into the drive '/media/cdrom/' to install software packages from it.   The image I downloaded and burned to CD is from  the PC (Intel x86) desktop CD version. I burned many CDs but the result is the same. Let's say I have two planets that are one hundred thousand lightyears away from each other. I and my immortal friend on the other planet want to communicate, with a strong laser and a tachyon communication device.  I record a message on the tachyon communication device and release the message at exactly the same time as I activate the laser, both of which are directed to the other planet which is one hundred thousand lightyears away. Say it is the year 0 for both of us at the time I did this.  If tachyons existed, then the message would arrive to my friend before the photons in the laser. It would arrive, say, a thousand years earlier. From my vantage point, that message will arrive to her at year 99,999; the same would be true for my friend's vantage point. However, she will only see the laser at year 100,000.  So since she got the message at year 99,999, she immediately sends me a reply back going through the same procedure as I did. She records a message and releases it at the same time as the laser. The tachyons will arrive 1,000 years earlier than the laser, so for me, I will receive the message at year 199,998. I will receive the laser, however, at year 199,999.  It seems to me that communication this way does not violate causality. I will still have received the message after I had sent it.   If tachyons truly violated causality, though, I realize it should arrive at year -1 for her, and so she can reply to me at year -2, which would mess me up by year 0 as I will ask her how she knew I was planning on sending her a message before I sent it. I could send her a different message, which she would end up receiving at year -1, and will end up confusing her as she would have received one message asking her out, and the other asking her how did she know I was asking her out. She then decides I am crazy and sends me a message at year -2 that she does not want to date me, and so she will have both turned me down and entertained me before I have even asked her out.  On the other hand, let's go back to year 0 and add a third device to our list: an Alcubierre drive. After I send out the message and the laser, I get impatient and do not feel like waiting 99,999 years, so I get on my Alcubierre drive spaceship and arrive on her planet at the same year 0. My friend is not in her office, so I leave a note to her also immortal secretary saying I dropped by and that she should expect a message for her in year 99,999.  I then get back on my Alcubierre drive and land back on my planet, still on year 0. Meanwhile, the tachyons and photons I sent out are still racing to arrive to her. By year 99,999, she receives the message just as I Alcubierre drive back to her, and I pick her up for dinner.  But the point of my question is, it seems to me that just going faster than light, if that alone was what you had, would not violate causality. It must be something else. I understand time dilation and that things with mass cannot travel at the speed of light, but using the Alcubierre drive, hypothetically speaking, I was still able to outpace the photons while also having mass. It still did not produce causality problems. Alcubierre drives are also valid solutions to GR.  It seems circular to me to say that what makes traveling faster than light violate causality is because it violates causality (if faster than light communication was divorced from causality problems, then the causality problem would cause itself -- thereby violating causality and, hence, we would scrap it and conclude that there is no causality problem after all).  What is it that I am missing? If someone could help me out, that would be excellent. I've been itching to ask my friend out for a few millenia now. :) Call to a member function on a non-object I would like to know if there is a word or phrase for a systematic campaign to annihilate every descendant of a particular ancestor.   We have the word 'genocide' for killing off an entire race, and plenty of words for killing people based on how they are related to you (fratricide, filicide, et. al.), but looking over the  and the , I couldn't find a term for systematically killing an entire bloodline.   My question is prompted by a US news story about the horrific murder of 8 members of the same family, but those murders are not an example of what I'm thinking of because the children of the victims were left alive, so the motive for the killings was not to kill off the bloodline. The sense that some group had coordinated an attack against just one family is what made me wonder what the "not an entire race, just one bloodline" version of 'genocide' would be called.   Searching the Internet for "kill off" and bloodline sends me straight into the land of vampires, Illuminati, reptilian overlords, and, naturally, Priory of Sion nonsense so I haven't had much luck finding a thread to pull.   It may be that there is no word for it in the sense of murdering a human bloodline, because we just don't attempt kill off our royals down to the last heir often enough to coin a term for it. Maybe there is a scientific term for culling a line from a test population as part of biological research? This is difficult for me to explain, but how do I put words over a symbol? For instance, I want to add words over an equal sign. Anyone know the command/syntax for that?  Also adding ^ doesn't cut it..., I want the text to appear ON TOP. Comparing nullable column values using "IS NULL" versus "= NULL" I'm dealing with nullable values in a column of a table, i'm wondering why this:  with cte as (     select 'c1'::text as c1,null::text as c2     union     select 'c2'::text as c1,null::text as c2 ) select * from cte where c2 is null   returns a result set with the two "rows" having a null value in c2; comparting to this query:  with cte as (     select 'c1'::text as c1,null::text as c2     union     select 'c2'::text as c1,null::text as c2 ) select * from cte where c2 = null   that returns an empty result set!!  so it looks like that field is null is actually different from field = null? the question is... is there "null-compatible operator" to use? so i can write the comparision like: column_name = $1 and argument $1 can be a not null or null value. IF $1 is a null value then it will semantically equals to write as IS NULL, ELSE (if $1 is not a null value) then it would be semantically equals to = $1.  P.D: This question also applies to the case is NOT NULL versus &lt;&gt; NULL Low-RAM Linux Distro Specifically for serving static sites I'm looking for a near-zero-configuration linux distro that is designed specifically for serving static web sites and nothing else on a low-RAM machine (I'm planning on 128MB, but the option of less would be nice).  Any ideas of a linux distro specifically tuned for serving static sites in a small footprint? Ideal correspondence I'm confusing the ideal correspondence theorem. Is the following  right?  Ideal correspondence: Let $f:A \to B$ be a ring homomorphism. Then there is a one-to-one order-preserving correspondence between ideals of $f(A)$ and ideals of $A$ which contain $\ker(f)$ by $\alpha \mapsto f(\alpha)$ (conversely $\beta \mapsto f^{-1}(\beta)$), and prime ideals correspond to prime ideals, and maximal ideals correspond to maximal ideals. How to dynamically create a local variable? How can the Next Generation's unfamiliarity with the Borg be explained after Archer's encounter? In  Starfleet learns about a culture that we (as audience) know as The Borg. According to TNG, the Federation doesn't seem to know anything about these beings when . Although Archer's crew doesn't learn the name of the Borg, it would be reasonable that the Federation would inherit the information about such a menacing enemy and it would also be reasonable to assume the crew of the Enterprise-D to be able to match their encounter with their databases' information.  Obviously, out-of-universe the explanation is that the scripts weren't written in chronological order, but how can the apparent obliviousness about the Borg be explained in-universe? On the Stack Overflow help center article about review queues, , it says on the bottom:  Note: These two queues do not exist on Meta sites.  But there are Late Answer and First Posts review queues on Meta Stack Overflow, as far as the  says. So shouldn't that last sentence be removed? Is there a way to find comments you made on YouTube? Evaluating a PDF to find an unkown constant Evaluate the PDF to find K  $1 = \int_0^{\infty} \frac{k}{(1+x)^4}$  using u substitution I got:  $u=1+x$ and $du= dx$  $\int_0^{\infty} \frac{k}{u^4} \to \frac{k}{\frac{-1}{3} u^3}$   Since $u^4 du = \frac{u^{-4+1}}{-4+1}=$ $\frac{u^{-3}}{-3}$  which evaluating gives:  $1=\frac{k}{\frac{-1}{3(1+x)^3}} \vert_0^{\infty} \to \frac{-3k}{(1+x)^3} \vert_0^{\infty} = 0 -3k$   and $k= \frac{-1}{3}$  somehow though they came up with $\frac{-k}{3} \frac{1}{(1+x)^3}$ after integrating $\int_0^{\infty} \frac{k}{(1+x)^3}$   Was my approach correct to use substitution? Where does my work go wrong? Incompatible Software for next OSX? I understand the move past Snow Leopard meant losing Rosetta, and compatibility with any PPC apps.   Since there are some apps that can't be replaced (for example, old years of TurboTax that I might need to reference) I have a drive to boot Snow Leopard, and one for Yosemite. I was considering the jump to El Capitan, when I saw the question  which got me wondering the question -     Before installing a next generation OS, is there any way to find, in   advance, which apps I have that would be orphaned by the move?   Edit, the answer below from Jean Valjean was what I needed. One app concerning me appeared as follows:    It might be tedious to search many apps, but at least this gives me a chance to view the top ones that will cause me grief. What are recursive triggers? How can recursive triggers be controlled? What are recursive triggers? How can we control them, or prevent infinite recursion?  Are these any risks inherent in them, and any limits that we should be aware of?  Please explain with an example.