Prove that a given expression is always an integer
How prove this $\prod_{1\le i<j\le n}\frac{a_{j}-a_{i}}{j-i}$ is integer
Consider this conversation:      John: I am giving free chess lessons.   Mary: Nice! You’re a true teacher.   John: How so?   Mary: A true teacher imparts knowledge without a price tag.   John: But what if teaching is his only source of income? Would he not be a “true” teacher then?   Mark (in response to the question above): You clearly mentioned “free chess lesson”.   I don’t see how Mark’s comment adds anything to the discussion or makes sense in the current discussion. It is “redundant” — that’s one word for it — but is there a more fitting word? How to get the ASCII value of a character Any tools to generate an XSD schema from an XML instance document? I am looking for a tool which will take an XML instance document and output a corresponding XSD schema.  I certainly recognize that the generated XSD schema will be limited when compared to creating a schema by hand (it probably won't handle optional or repeating elements, or data constraints), but it could at least serve as a quick starting point. Proving that kernels of evaluation maps are generated by the $x_i - a_i$ I want to prove that the kernel of the evaluation map $s_a : \mathbb{C}[x_1,\dots,x_n] \rightarrow \mathbb{C}, x_i \mapsto a_i$ where $a = (a_1,\dots, a_n) \in \mathbb{C}^n$ is the ideal generated by $\{x_1 - a_1, \dots, x_n - a_n\}$.    The proof in the book first shows it easily for the case $a = 0$, and says that by substitutions of variables $x_i' = x_i - a_i$, you can show it for the rest of the cases.  I know that $f \in \ker s_a$ iff $f(a) = 0$.  And $\phi : \mathbb{C}[x_1,\dots, x_n] \rightarrow \mathbb{C}[x_1 - a_1, \dots, x_n-a_n], x_i \mapsto x_i - a_i$ is a surjective ring homomorphism.  Please give a hint and not the full answer.  Thanks. Academic dishonest: Possession of unauthorized materials during an exam that you didn't even know existed So unfortunately during an exam, I had the misfortune of using an eraser (with cover) that was not mine and had a small amount of words inside it unknown to me. I did not know this was the case, as I don't have a habit of unraveling erasers or looking at them closely?  However, now I face the problem in trying to defend myself against the accusation of me cheating.  I'm specifically concerned on my university's blank statement of "possession of unauthorized materials during an exam" is considered academic dishonesty. Seems like student conduct has all the power in trying to prove academic dishonesty, despite none of my test containing any of what to me is  unreadable, small notes. Why is the observable universe so big? The observable universe is approximately 13.7 billion years old. But yet it is 80 billion light years across. Isn't this a contradiction? Finding an angle within an 80-80-20 isosceles triangle The vote fraud script breaks legitimate polling questions - A serial downvoter's lament I've downvoted several of the answers on these Meta discussions:       Both questions are typical self evaluations, Stack Exchange runs them on all young Beta sites and as you may have noticed they are essentially polls, a series of example questions from the site that we are supposed to vote on.  Today I noticed that almost all of my votes have disappeared on both evaluations, only an upvote remains in the later one. My votes triggered the serial voting script, something that :      Heh, it appears I just serially downvoted you (I downvoted a lot of the answers here), hopefully the script is smart enough to not reverse my votes. Will check back in 24h, and if it does, I'll rant about it on MSO ;)    My voting behaviour on both evaluations was very simple:   I had the evaluation open in the left monitor, Opened all the example questions in their own tab, in the right monitor, Scanned each question quickly, and voted on the Meta answer (most of my votes were downvotes), Repeat for the next one, and so on...   I didn't spend more than a few seconds on any question, as I'm fairly regular on the site and I've already read all of them while casually browsing the site. In fact, if my memory isn't failing me, I had voted (up/down) and/or commented, and/or voted to close on each of the 20 example questions. I might have spend a minute or two re-reading a couple, but I mostly scanned through them. In any case I think my voting pattern is quite normal, and the script should be somehow tweaked to take serial posts, posted by the same user in quick succession, into consideration, especially on Metas.   Since I had anticipated that the script would kick in, I could have tried to fool it by pausing a bit between votes or trying to look for other Meta posts to vote on, in between voting on the evaluation answers. However I'm a very active voter on Meta Workplace, there aren't many posts I haven't already voted on, and the few that I haven't is probably because I neither agree or disagree with (my opinion on them is simply "meh"). In any case I think my voting behaviour was natural, and although the serial voting script is awesome, this time it caught and reversed a perfectly legitimate behaviour.  Animuson, in , mentions that the script isn't really optimal on Area51 as well:      Yes, please implement this on Area 51 proposals too! Oftentimes a single user would suggest a plethora of "example" questions to their own proposal which are just terrible, yet all of everyone's votes get reversed and the question just sits there at 0. I've noticed several times that my votes have gotten reversed.    I haven't noticed it myself, but it certainly seems plausible, especially early in a proposal's life.  Mysticial found a related discussion, , but it's focused on voting patterns on main sites, where reputation is also a factor. I can understand jumping through hoops to satisfy the script on main sites where, unfortunately, there are a lot of people trying to get some cheap rep or "punish" others with their downvotes, but not on Metas, and especially not on these evaluation posts.   Voting fraud is a problem, and more often than not the script works wonderfully, but in my case it was extremely counterproductive. Critical evaluations are essential for a young Beta site, and their answers typically only get a handful of votes. I wonder how many of these votes have silently gone away...  Fix this please. Minecraft Laucher has no Versions So this if the first time this has ever happened too me and I was wondering what it is. I just downloaded Minecraft and there is no area where i can change my version too 1.8.9/1.7.10 all i see when I do it is blank  Im just Wondering on how come even tho i hit the Repair button on the minecraft.msi download it dosnt fix it... If there is a way too fix this let me know cause my friend wants too play and I cant cause of this.. In my object record type i have created formula field. Now i want my formula field to work as below.  If Basement__c = NA and Property__r.Basement__c = NA OR  Basement__c = Crawl and Property__r.Basement__c = Crawl OR  Basement__c = Slab and Property__r.Basement__c = Slab Then it will return Type__c(formula field) = "MAtch"  And (Another rule) Basement__c DOES NOT contain any of the following "N/A,Crawl,Slab,None and Property__r.Basement__c Does not contain any of this then formula field will return "HAs basement"  How can i achieve this? Please help. Proper use of the IDisposable interface Unable to install Cisco Packet Tracer 7.3 on Ubuntu 20.04 'A / One / At least one student entered the room.' Are these the same? (truth-conditionally) I just wonder if the two following sentences are truth-conditionally the same. Sentence 1 essentially means there existed a student who entered the room, and this situation includes two, three or more students entering the room. Hence, I believe these two sentences are truth-conditionally the same. Are they?        A student entered the room.   At least one student entered the room.      How about        One student entered the room      I gather this is different, as sentence 3 only means one student, not more and not fewer?  Any and all insights would be welcome. What are the best ways to improve a site's position in the Google search results? Whenever I open a .sh file, it opens it in gedit instead of the terminal. I can't find any option similar to Right Click → Open With → Other Application... → Terminal.  How do I open this file in the terminal? Shrink Tempdb log file when reaching a specific size how I can shrink tempdb log file when reaching a specific size?? I'm scheduling job to shrink tempdb but still I face a problem of disk space due to tempdb is full! Should I use  tag for icons instead of ? Polygon Touch detection Google Map API V2 Prove the following using epsilon-delta definition:  Let $a \in \mathbb{R}$ and $f,g$ be defined on $\mathbb{R}$.  If  $$\lim_{x\to a} f(x) = 647 \tag 1$$  and $$\lim_{x\to a} g(x) = \infty \tag 2$$ then   $$\lim_{x\to a} f(x) + g(x) = \infty.$$  $$(1) \implies\forall \epsilon &gt; 0, \exists \delta &gt; 0, s.t. |x-a|&lt;\delta \implies |f(x)-647| &lt; \epsilon$$  $$(2) \implies \forall M &gt; 0, \exists \delta_2 &gt; 0, s.t. |x-a|&lt;\delta_2 \implies |g(x)| &gt; M$$  Not sure about this... I've set up the definitions. I believe we will have to take a minimum between the deltas for first two limits then use it in the third.  Anyone have any clues? Confused at + and ++ in TikZ I can't seem to figure out the difference between + and ++ relative coordinates in TikZ.   I understand ++. ++ specifies delta coordinates that are added to the last current point, the current point is then set to the new absolute current point (correct?):  \draw (0,0) -- ++(1,0) -- ++(0,1) means:  set Current Point to (0,0) add (1,0) to Current Point = (1,0) set Current Point to (1,0) add Current Point (1,0) to (0,1) = (1,1) set Current Point to (1,1)   I can't seem to figure out that + does, when I try it out it looks like the coordinates in the +(x,y) are absolute coordinates rather than delta coordinates, but the manual says otherwise