I have the following code \sum_{i&lt;j}^{}\sum Cov(X_i, X_j) which renders     How can I move the i&lt;j subscript to the center underneath the two sigmas (which will result in a "shift")? Preference will be given to the simplest solution.
I would like to place a lower limit under two consecutive \sums, somewhere in between. So I have \sum_{j=1}^{\infty}\sum_{k=1}^{\infty} and would like to place j&gt;k in the lower limit between the two sums. I believe the multiple integral operator has this kind of functionality.
an analytic function from unit disk to unit disk with two fixed point prove that if  $f:\mathbb{D}\rightarrow\mathbb{D}$ is analytic with two distinct fixed point then $f$ is identity. I thought if one of the fixed points was zero by schwarz lemma this statement is easily proved. but what can I do if fixed points were nonzero? Is the W statistic output by wilcox.test() in R the same as the U statistic? While using the top bar to navigate across communities, I noticed that the reputation displayed on meta sites on that list does not match the one of their corresponding main sites:    I have noticed on past situations that reputation gained takes a bit to "update" on the display on Meta sites (that is, if you gain rep on SO, for example, it will take some time for you to see your updated rep on MSO).  However, I have never noticed this behavior here, on the Sites List of the top bar. Furthermore, the "delay" here seems to be more severe, as the "outdated" rep showed is of several days ago (for example, I turned 3k on SO almost exactly a week ago, but the rep shown on MSO is still &lt;3k).  Any idea what could be causing this behavior?    Update: 2 months later this persists: my main site rep updates correctly but their corresponding meta rep remains the same as in the image above.     Update: As per Catija's  I removed the site (MSO) and put it back in, but it didn't update the rep.   However, after removing one site (TWP Meta), waiting at least a day, and then adding it back I see that although the reputations still don't match, the number displayed did changed a bit (previously 29,924 but now 33,112): Valid references on origins of Machine Learning, Statistical Learning and Data Mining How do I limit the number of rows returned by an Oracle query after ordering? Is there a way to make an Oracle query behave like it contains a MySQL limit clause? In MySQL, I can do this: select *  from sometable order by name limit 20,10  to get the 21st to the 30th rows (skip the first 20, give the next 10). The rows are selected after the order by, so it really starts on the 20th name alphabetically. In Oracle, the only thing people mention is the rownum pseudo-column, but it is evaluated before order by, which means this: select *  from sometable where rownum &lt;= 10 order by name  will return a random set of ten rows ordered by name, which is not usually what I want. It also doesn't allow for specifying an offset. I have a raster aerial photo.  I have county wide parcel data.  I want to clip/screen shot the raster at each individual parcel.  Think of this application - a post card with an image of your house --here is your house--  I want to have each parcel attribute to contain information + hyperlink of the aerial photo.  Any other solutions welcome, but this is the only method I could think of. Although I am unsure of how to achieve it efficiently.  Using ArcInfo 10. Is it possible to load a TikZ library locally? How to disable the physical sim and only use the eSIM on iPhone XS/iPhone XS Max? I'm wondering if there is a way to turn off the Physical sim and use only the eSim on my iPhone XS Max. I travel a lot and I'm loving the use of eSIM.   I understand that I can turn off the eSIM if its required, but I wanted to know if there was any way to turn the Physical Sim off and leave only the eSim ON.  Note - I have an unlocked iPhone XS Max. How do you add a timer to a C# console application Just this - How do you add a timer to a C# console application? It would be great if you could supply some example coding. Is there a command for large middle delimiters consistent with \bigl and \bigr? After browsing through related threads, I am now of the understanding (please correct me if I am wrong) that the rule of thumb is to use the \bigl,\bigr pair with brackets, parentheses, etc. for operators like sums, products, and integrals, and \left,\right in all other cases.  I only recently found out about \middle , which has solved what had been a vexing problem for me: delimiters for function arguments (slashes, bars) not scaling properly. Leafing through my cheatsheet has not been of much help, so I wish to ask: is there an equivalent of \middle for \bigl and \bigr, or would the use of \middle along with that pair not be gauche? I'm designing an algorithm to do the following: Given array A[1... n], for every i &lt; j, find all inversion pairs such that A[i] &gt; A[j]. I'm using merge sort and copying array A to array B and then comparing the two arrays, but I'm having a difficult time seeing how I can use this to find the number of inversions. Any hints or help would be greatly appreciated. How can I execute a terminal command (like grep) from my Objective-C Cocoa application? This is what I have come up with so far, but I am rather lost at lines 3-4...  $$\begin{align} (a,b)\in A\times (B\setminus C)\iff&amp; a\in A \land b\in (B\setminus C)\\ \iff&amp; a\in A \land (b\in B \land \lnot(b\in C))\\ \iff&amp; (a\in A \land b\in B) \land (a\in A \land b\notin C)\\ \iff&amp; (a\in A \land b\in B)\setminus (a\in A \land b\in C)\\ \iff&amp; (a,b)\in (A\times B)\setminus ((a,b)\in A\times C)\\ \iff&amp; (a,b)\in (A\times B)\setminus (A\times C) \end{align}$$ How can I compute the integral $\int_{0}^{\infty} \frac{dt}{1+t^4}$? I have to compute this integral $$\int_{0}^{\infty} \frac{dt}{1+t^4}$$ to solve a problem in a homework. I have tried in many ways, but I'm stuck. A search in the web reveals me that it can be do it by methods of complex analysis. But I have not taken this course yet. Thanks for any help. Recover EXT4 partition I was trying to resize my EXT4 partition and something went wrong. I lost it. The space in once consumed is now being represented as Unallocated. How do I recover the partition? Finding how multiplication and addition behave on $\mathbb{F}_4$ without any result I'm currently tutor in an undergraduate course and the students are asked to find how addition and multiplication behave on the field with 4 elements: $\{0,1,x,y\}$. In the solution the teacher gave me, he uses the fact that $(\mathbb{F}_4^*, \cdot )$ is cyclic with order 3 to find that $x$ and $y$ are both generators and that $(\mathbb{F}_4,+)$ is a group of order 4 to deduce that $0=1+1+1+1=(1+1)(1+1)$ and hence, $(1+1)=0$ since a field is integral.   However, I'm not satisfied with this reasoning because this question is asked to student in first year and they juste have been introduced to fields (and this was part of an analysis course so they won't see any result soon). So I tried to find how it behaves by myself, using as few results as possible.  Multiplication is the easy part, $0\cdot a$ and $1 \cdot a$ is trivial for all $a\in \mathbb{F}_4$. Then, $x\cdot y\not = 0$ because it is invertible, and it can't be equal to $x$ or $y$ because that would imply the other element to be equal to $1$. Hence, $x\cdot y=1$. We are left to find what is $x^2$ (and similarly, $y^2$). $x^2$ cannot be $0$, nor $x$. If $x^2=1$, then $y=(x^2)y=x(xy)=x\cdot 1=x$. Contradiction. Hence $x^2=y$ and by the same reasoning, we have $y^2=x$.  It was easy with the multiplication, but I can't figure out any non-trivial result concerning addition. My intuition tells me that distributivity should be used here but I'm feeling like it's just converting a problem into another one because we don't know anything about addition.  Does anyone here have an idea of how addition can be "found" without using any advanced result (or as few results as possible)?  Thanks in advance. In how many different ways can you split $10$ people into two groups with the same amount of people?  My attempt: Since the order in which you choose someone doesn't matter, I chose to calculate the combinations: $10 \choose 5$ Evaluating the product $\prod\limits_{k=1}^{n}\cos\left(\frac{k\pi}{n}\right)$ Recently, I ran across a product that seems interesting.  Does anyone know how to get to the closed form:  $$\prod_{k=1}^{n}\cos\left(\frac{k\pi}{n}\right)=-\frac{\sin(\frac{n\pi}{2})}{2^{n-1}}$$  I tried using the identity $\cos(x)=\frac{\sin(2x)}{2\sin(x)}$ in order to make it "telescope" in some fashion, but to no avail. But, then again, I may very well have overlooked something.  This gives the correct solution if $n$ is odd, but of course evaluates to $0$ if $n$ is even.   So, I tried taking that into account, but must have approached it wrong.   How can this be shown? Thanks everyone. How can I use Netflix on Ubuntu? What is the best class or classes in 3.5e to build a character that uses both a sword and either arcane magic or psionics which can be fluffed as arcane magic and focuses on fighting powerful individual creatures?    I have a foundness for those that can sling spell along with blade, like Dak'kon from Planescape: Torment.  So, given that, how do I make the best Gishy character I can?  Specifically, the group is looking for a character focused on delivering high damage or other debilitating effects to an individual target.  There is no need to balance sword and magic, though I would prefer to have the melee abilities remain at least relevant through at least mid-levels.  This question:  is related.  But I am specifically looking to optimize for high targeted damage output and have no need to balance between sword and spell if focusing on one is more effective.  I have considered the duskblade and that may be the best answer, but as I have limited experience with 3.5 I want to find out if I should use a pure duskblade, a duskblade which steps into other classes, or some other class I have not considered.  In this particular case, the characters will start at level 7, and equipment rules are liberal as long as the characters can afford it.  I'm not sure how long this will last so I am focused on the next few levels instead of trying to plan all the way out to 20.  It will be a party of 5.  Not all of the rest of the characters have been fully fleshed out, but we will have a cleric and a wizard.  The other two are more up in the air but they one is looking at various rogue classes and the other is looking at some sort of melee class, with this gishy rounding out the team.  The initial encounter is up in the air, but there is a definite fondness for gryphons.  We do know that we will be working up to dragon in relatively short order, which is a factor in wanting to focus on large damage output to individual creatures as opposed to being able to deal with numerous weaker creatures.  The setting is "modified Forgotten Realms".    I personally own absolutely no 3.5 books, but we have access to a fair bit and just about anything from WoTC is permitted.  I am new to 3.5, but I have a fair bit of roleplaying experience including AD&amp;D.    Thematically, I envision the character as a generally honorable, but working as a monster hunter and mercenary.  I definitely favor swords though specifically which sword is flexible.  Dak'kon from Planescape would be an excellent example of what I would like to build, as would The Witcher.  Based on Kryan's excellent answer, it seems to be leading up to using duskblade and a specialist wizard to lead into Abjurant Champion.