Minimal LateX installation needed for a simple document
What is the minimum distribution for LaTeX
Very specific paragraph shape If $[G' : G'']\leq p^2$, then $G'$ is abelian. Problem : Let $G$ be a p-group and $G'$ denote the commutator subgroup of $G$. If $[G' : G'']\leq p^2$, then $G'$ is abelian.  It is easy to prove it for the case of $[G' : G'']=1$ since G is solvable. But it is too hard to prove the other cases. Please give me any advice or solutions. Thanks in advance. Is it possible to somehow mark a System.Array as immutable.  When put behind a public-get/private-set they can't be added to, since it requires re-allocation and re-assignment, but a consumer can still set any subscript they wish:  public class Immy {     public string[] { get; private set; } }   I thought the readonly keyword might do the trick, but no such luck. Is it possible to instantiate an object at runtime if I only have the DLL name and the class name, without adding a reference to the assembly in the project? The class implements a interface, so once I instantiate the class, I will then cast it to the interface.  Assembly name:     library.dll    Type name:      Company.Project.Classname     EDIT: I dont have the absolute path of the DLL, so Assembly.LoadFile won't work. The DLL might be in the application root, system32, or even loaded in the GAC. When speaking precisely or technically, one would say that "Homo erectus and homo sapiens are two species of hominid" rather than "Homo erectus and homo sapiens are two species of hominids." The hominid here should be singular because we are speaking about instances of a single class ("class" being used here in its broader sense, not in the sense of taxonomic grouping).  Now let's consider more common parallel constructions replacing the word species with other words.      Tide and Wisk are two brands of detergent.       Tide and Wisk are two brands of detergents.      Cats and dogs are two types of pet you can buy at Pets-R-Us.       Cats and dogs are two types of pets you can buy at Pets-R-Us.   I feel that using the singular noun to specify the class is more grammatically correct, but sounds stilted in conversation. Almost everyone I hear making these constructions uses the plural form. What are your thoughts about singular vs. plural here?  Edit  Here's a more extreme example that may help clarify the issue. Consider the following sentence:     It was my first attempt at calming an angry crowd, and I just        stood there while people were hurling all kinds of insults at me.   Here I think the singular insult sounds strange, if not downright wrong.  Further Edit  I just noticed in A Treasury for Word Lovers (Morton S. Freeman, 1983) a section entitled "Types of Errors." This is a book about (American) English usage by an English professor and editor, which purports to be a "practical guide for serious writers and readers." I wish I could say this discovery satisfies the question, but in fact I now feel somewhat farther from the truth, if there is any single "truth" in this matter. But at least I don't have to worry about being wrong when using the plural form of the class in these constructions. limit of Holder norms: $\sup\limits_{x\in [a,b]} f(x) = \lim\limits_{n\rightarrow\infty} \left(\int_a^b (f(x))^n \;dx\right)^{\frac{1}{n}}$ I am having a problem computing the pearson correlation coefficient of data sets with possibly zero standard deviation (i.e. all data has the same value).  Suppose that I have the following two data sets:  float x[] = {2, 2, 2, 3, 2}; float y[] = {2, 2, 2, 2, 2};   The correlation coefficient "r", would be computed using the following equation:  float r = covariance(x, y) / (std_dev(x) * std_dev(y));   However, because all data in data set "y" has the same value, the standard deviation std_dev(y) would be zero and "r" would be undefined.  Is there any solution for this problem? Or should I use other methods to measure data relationship in this case? Describing groups with given presentation? $\langle x,y\ |\ xy=yx,x^5=y^3\rangle$ and $\langle x,y\ |\ xy=yx,x^4=y^2\rangle$. I'm trying to describe the groups with presentations $\langle x,y\ |\ xy=yx,x^5=y^3\rangle$ and $\langle x,y\ |\ xy=yx,x^4=y^2\rangle$. I have some problems getting a good picture of what they look like...   For the first one, $xy=yx$ allows us to say that any element in the group can be written as $x^iy^j$ for some $i,j$. More precisely, using $x^5=y^3$, we can always reduce $j$ to $0,1$ or $2$, whereas $i$ could have any value. So I was thinking about $\mathbb{Z} \times C_3$. But the other way around (reducing $i$ this time) it's also $C_5 \times \mathbb{Z}$. But then I should be able to prove that both descriptions are equivalent, which I can't do...  For the second, I proceed similarly and get to the same kind of problem...  Could you tell me where I'm going wrong?  Thank you very much in advance! What is the correct way to write 1.5 hours? I'm a bit confused in describing 1.5hrs in words.   Is writing one and a half an hour correct or should it be one and half hour?  E.g:     I'll see you there in one and a half an hour.     OR       I'll see you there in one and half hour.   Or is there any other correct way of writing this? Proof of $a^n+b^n$ divisible by $a+b$ when $n$ is odd I read somewhere that     $(a^n - b^n)$         It is always divisible by $a-b$.   When $n$ is even it is also divisible by $a+b$.   When $n$ is odd it is not divisible by $a+b$.      and     $(a^n + b^n)$         It is never divisible by $a-b$.   When $n$ is odd it is divisible by $a+b$.   When $n$ is even it is not divisible by $a+b$.      I wonder what's the proof for this.  First postulate is clear. $(a-b)$ would or would not be a factor. Any light on others? How do I cd into a directory in the home folder? I understand that the alphabet for the English language is not strictly English as languages such as French, Dutch and many more use the same alphabet, with few additions in other languages. Is Roman alphabet the common term to refer to this set of symbols for writing texts? Orthogonal projections with $\sum P_i =I$, proving that $i\ne j \Rightarrow  P_{j}P_{i}=0$ I am teaching how to convert from one base to another base, however, I can't find a way to do something like this:    I know little about Tikz, the only thing I have been able to produce is a simple square with a couple of lines over it, so I don't even know where to start. Any suggestions? I don't mind typing all the numbers, what I want to do is to automatize the process of creating the lines. Let $\kappa, \lambda,\theta$ be infinite cardinals of $M$ where $M$ is a transitive model of ZFC. Let $\mathcal{P}=Fn(\kappa\times \omega,2)$ (finite functions from $\kappa\times \omega$ to 2) and G is a (M,P)-generic set. I want to show $(\lambda^\theta)^{M[G]}=(max(\kappa,\lambda)^\theta)^M$. One direction is easy $(\lambda^\theta)^M \leq (\lambda^\theta)^{M[G]}$ and $(\kappa^\theta)^M\leq (\kappa^\theta)^{M[G]}\leq ((2^\omega)^\theta)^{M[G]}=(2^\theta)^{M[G]}\leq (\lambda^\theta)^{M[G]}$ where $(\kappa \leq 2^\omega)^{M[G]}$ is by the forcing construction. But now I am not quite sure how to get the other direction. Any thoughts? I have Ubuntu 14.04 installed on my laptop. Can I clone the entire system with apps, settings, etc on a USB 3 pendrive, and then boot from that on other computers as a portable Ubuntu installation to take around with me?  I am trying a function that Timeshift has that's called "Clone the current system on another device". I've selected the pendrive as the target device. Timeshift was talking about this being bootable. But is it going to bootable on only this laptop, or on any system? For example, on an Intel Mac, or a Windows computer without the grub bootloader? Is that Timeshift function what I am looking for, or do I have to do something else?  Edit: The Timeshift clone failed to boot on the first system I tested it on. (A Dell Hackintosh that usually boots from USB just fine.) Creating compressed GeoTIFF with QGIS Clipper tool? I have a geo-tiff created by Pix4D which is about 375 Mb in file size.   I have clipped it to a smaller area using Qgis 2.8(using Raster ‣ Extraction ‣ Clipper), and the output file is about 900 MB in size.  Why did the clipping process generate such a large file, when the source file was smaller?   How do I run the clipper tool so that it produces a smaller size? GNU screen: Running commands in different window when screen is started Finding what methods a Python object has Given a Python object of any kind, is there an easy way to get the list of all methods that this object has?  Or,  if this is not possible, is there at least an easy way to check if it has a particular method other than simply checking if an error occurs when the method is called? Efficiently storing query-affecting settings in a database I want to know how to efficiently store query-affecting settings in a large analytical database (MS SQL Server). By efficiency I mean no hardcode (1) and fast/optimal query processing (2).  My example  I've got a massive table of sales orders. Let's simplify it to two columns:  SalesOrder  OrderID      int      Primary Key  OrderDate    date   I need to retrieve the list of orders for the last X days and this X should be configurable. That's why I introduce a table of settings where I define it:  Admin_Configuration  SettingName   (PK)|      SettingValue  ------------------|------------------  DaysBack          |          30      It allows me not to hardcode this "30" in query definitions. Then I would run queries like:  SELECT OrderId   FROM SalesOrder as S   JOIN Admin_Configuration as A     ON A.SettingName = 'DaysBack'   WHERE [OrderDate] &gt; DATEADD(DAY, -A.SettingValue, GETDATE());   Performance  It avoids hardcoding, but leads to a performance problem. This query runs much slower than it would've, had I simply hard-coded 30 in the WHERE clause.  My understanding is that optimizer doesn't know how large my SettingValue is and assumes the worst - that all rows would be retrieved from the SalesOrders table, thus leading to a suboptimal query plan.  Trying a scalar function yielded even worse results. I'm desperate enough to have a separate view which stores these hardcoded settings but I've been refraining from it with hopes for a better solution.  Here is the original query plan (with a join): 4 seconds elapsed   Here is the plan with subquery, as :  SELECT orderid FROM [SalesOrder] as s WHERE s.OrderDate &gt; dateadd(DAY,    (SELECT -NumValue    FROM Admin_Configuration where SettingName = 'DaysBack'   )     , getdate())   7 seconds elapsed, slightly slower   As for the ideal performance: I just hardcoded 20, the query took 2 seconds. It needs mentioning that the Admin_Configuration table contains only 18 rows, hence it shouldn't affect performance.  SELECT OrderId   FROM SalesOrder as S   JOIN Admin_Configuration as A     ON A.SettingName = 'DaysBack'   WHERE [OrderDate] &gt; DATEADD(DAY, -20, GETDATE());   Using parameters would mean that I'd have to replace views (that I use now) with stored procedures/table-valued functions, which is unacceptable in my case