When I write a long limit there is a big gap between the limit and the function. How can I fix this? Furthermore is there a way to make the limits smaller so as not to be longer than the limit symbol? Also why is that gap there and that size difference between the length of the symbol and the limit under it? Shouldn't all be under it?  \documentclass[12pt]{article} \usepackage{amsmath} \usepackage{unicode-math} \begin{document}  $\displaystyle \lim_{n\to +\infty}x(n)$  \end{document}
I want to get the following:    I tried the following  \[    a(i,k) \leftarrow     \min\Big\{0, r(k,k) + \sum_{i^{'} | i^{'}\notin \{i,k\}}\max\{0, r(i{'},k)\}\Big\} \]   and got    How can I make the part under the summation extend to left and right without making a space between plus and max as in the first figure?
I've seen many boads with weird zig-zag shaped tracks just like the one below. Why are they designed like that? Is it possible record videos of my screen? Suppose that I would like to record some video sessions of Angry Birds on my phone.   Are there any apps that I can use? I have root permissions. How are dashes implemented in TeX? How does LaTeX implement the differences between the dashes?  -, --, and --- all print different dashes, but - is not an active character: it has catcode 12, the same as digits like 2. Create default home directory for existing user in terminal How do I do this? I created a user without a home directory and now I want to create one. Not just a folder called new_user, I want the default home directory with all it's folders and hidden files, etc. Kinetic energy is simple, right?! from the equation it seems as though if you know the speed and mass you'll know the work. Let's try to prove it. let's assume we know the acceleration and distance for now. $$w=mda$$ So the distance is $$d= 1/2 *at^2$$ Then  $$t = \sqrt{2d/a}$$ So $a*t$, which is the speed, is  $$a*\sqrt{2d/a} = a* \sqrt{2d}/\sqrt{a} = \sqrt{2da} = \sqrt{2w/m}$$ So regardless of the distance or acceleration, if you know the work and the mass, you can calculate the speed which is  $$v=\sqrt{2w/m}$$  What if you know the speed and the mass? You can easily calculate the work. $$v = \sqrt{2w/m}$$ Then  $$v^2 = 2w/m$$  That would mean  $$m*1/2*v^2=W $$ Rearrange it and  $$W= 1/2 *m *v^2$$ would be the equation.  What if you have an initial speed and a new speed and want calculate the amount of work done after an initial speed to get the new speed? You'll just subtract the total work from the initial speeds work.  $$1/2mv^2 - 1/2ms^2$$  where s is  the initial speed and v is the new speed. However if you do it the other way which is  $$d= 1/2at^2 + vt$$ and solve it the answer would be $$1/2ms^2  + 1/2msv$$ Why aren't they equal? Facebook OAuth Data Capturing You have already activated X, but your Gemfile requires Y How to detect when a variable changes value How can I easily detect when a variable changes value? I would like the execution of the program to break on the debugger whenever a specified variable changes value. Right now I'm using Eclipse's debugger. Counterexample for $A,B\triangleleft G,G/A\cong B \Rightarrow G/B\cong A$ Converting WKT to SHP? Navigate folder structure in Open/Save File Dialog with keyboard? I've seen Breaking Bad complete series 2-3 times, but I never found out how Walter White was able to poison Brock.  I mean did he himself directly feed the Valley of the Lilly berries to Brock or did he let Saul do the work? I can't figure it out. Why is the gap between the translation and the last line of the gloss so wide in my interlinear gloss? I notice if I take away some letters, the translation line goes to where it should be, but this big block of white space is pretty untenable. Ideas?    \documentclass[11pt,a4paper]{book}  \setcounter{secnumdepth}{4} \usepackage{gb4e} \usepackage{tipa} \usepackage{setspace} \usepackage[T1]{fontenc} \begin{document} \begin{exe} \ex \glll Jichax    Qamaqix lij parki,  aynacha,    amsta,  parki, q'awanaks,   khuyt'asis  sarawjchix. \\ \{jicha-x(a)    Qamaqi-x(a) lij(u)  parki   aynacha amsta   parki q'awa-nak(a)-_{c}$\oslash$-s(a)   khuy(u)-t'a-s(i)-s(a)   sara-wj(a)-ch(i)-i-x(a)\}    \\ now-\textsc{top}    Fox-\textsc{top}    all acclivity   declivity   ascent  acclivity ravine-\textsc{pl-acc-ad} whistle-\textsc{m}-\textsc{refl}-\textsc{subr} go-\textsc{bfr-cnj-3sim-top}\\ \glt `And so, Fox must've gone whistling on all the ascents, slopes, ascents, mountainsides, and even in the ravines.' \end{exe} \end{document} Auto-incrementing field in feature class using ArcGIS Desktop? So I'm relatively new to ArcGIS and ArcCatalog.  I have created a field name of PROJECTID in a feature class in a Personal Geodatabase.  The data type is TEXT.  This will be a ten digit ID.  The first five characters will be a acronym.  The sixth character will be a digit that represents the type of feature.  It will be a number.  The last four characters will be numbers that should be incremented starting at 0001.    Example:     WATER-1-0001  Screen shot of the field:   So my question is how can I do this in ArcCatalog.  I've seen some Python code that will autoincrement like the following:  rec=[OBJECTID] def autoIncrement(a):   global rec   return "water" + str(a) + "-" +  format(rec, '04d')   Again I'm a newbie when it comes to ArcGIS . Now I am getting an error NameError name 'OBJECTID' is not defined. Questions on "All Horse are the Same Color" Proof by Complete Induction I'm bugged by the following that's summarized on p. 109 of .     False theorem: All horses are the same color.      Proof by induction:      $\fbox{$P(n)$ is the statement: In every set of horses of size $n$, all $n$ horses are the same color.}$   $\fbox{Base Case or $P(1)$:}$ One horse is the same color as itself. This is true by inspection.   $\fbox{Induction Step:}$ Assume $P(k)$ for some $k \geq 1$.   $\fbox{Proof of $P(k + 1) :$}$   Since $\{H_1, H_2, ... , H_n\}$ is a set of $n$ horses, the induction hypothesis applies to this set. Thus, all the horses in this set are the same color.   Since $\{H_2, H_3, ... , H_{n+1}\}$ is also a set of $n$ horses, the induction step likewise holds for this set. Thus, all the horses in this set are the same color too.         Therefore, all $n +1$ horses in $\{H_1, H_2, H_3, ... , H_n , H_{n+1}\}$ are the same color. QED.   The issue the instructor was trying to point out is clearly valid. For the case $n = 1$, there is only ${H_1}$. So this case says nothing about possible overlapping elements of each set of $(n + 1)$, for instance $H_2$ in the above proof.    But it was proposed in the class discussion that this was the only problem. Had you been able to prove $P(2)$ true, then a proof of the above format would have been fine.    My interpretation is that yes, you could prove all horses are the same color, if you can prove that any set of two horses will be the same color. But this format would not work.  Why not? The problem I see is that the above proof is for the existence of at least one particular pair of sets of horses of sizes $n$ and $n + 1$, such that in each set, all horses are the same color.  Particularly when the set of size $n$ is a subset of the set of size $n + 1$.  In order to prove the induction step, don't you need to prove that sets of sizes $n$ and $n + 1$, do not necessarily contain the same, overlapping elements?  You could prove that any horse can be added to a set of 2 horses.  Take the last two, and they must be the same color, and so on.  Wwhatever color the first two happen to be, all other horses must thus be the same color.    Am I misinterpreting the example, or am I making a logical error? Thanks in advance. Can you take Gorrilla Glue in Hold Luggage on a plane? Show that $K[x,xy,xy^2,\dots]$ is not Noetherian Here is the problem I am stuck on: Fix a field $K$ and consider the subring $A \leq K[x,y]$ generated by $K \cup \{x,xy,\dots,\}$. Show that $A$ is not Noetherian.  I figure that taking ideals $I_n = (x,xy,\dots,xy^n)$ should give an infinite strictly ascending chain, which would establish that $A$ is not Noetherian, but I cannot figure out how to show that each $I_n$ is a proper subset of $I_{n+1}$. Any help here? What is the safest way to clean up /boot partition? I have 200 MB assigned for the /boot partition.  Whenever I try to update the kernel, I receive an error message that basically states /boot is full.  What can I do to cleanup /boot and remove/backup the older kernels? Using adjustbox along with table footnote Intuition for uniform continuity of a function on $\mathbb{R}$