Bound for the cardinality of a minimal generating set of a finite group Given a finite group $G$ of order $n\geq 2$, it obviously has a minimal generating set (a generating set of minimal cardinality) let's say of cardinalty $m$. I am looking to find out if a bound for $m$ can be established. I asked myself this question after I saw a problem which asked to prove that $$|\text{End}(G)|\leq\sqrt[p]{n^n}$$ Where End$(G)$ is the set of endomorphisms of $G$ and $p$ is the greatest prime divisor of $n$. Given the fact that an endomorphism would be uniquely determined by its values on the elements of a generating set of $G$ that problem would be solved by showing that $$m\leq \frac{n}{p}$$ which is an idea for a possible bound. (I'm not looking to find another solution to the problem above, I only mentioned it because it provided an idea for a bound)
Why is the minimum size of a generating set for a finite group at most $\log_2 n$? It seems to be known that the minimum size of a generating set for a finite group of order $n$ is at most $\log_2 n$. Can someone explain why this is true?  Edit: noted that the logarithm is base 2, and $n$ represents the order of the group.
What techniques can be used to define a class in JavaScript, and what are their trade-offs? I prefer to use OOP in large scale projects like the one I'm working on right now. I need to create several classes in JavaScript but, if I'm not mistaken, there are at least a couple of ways to go about doing that. What would be the syntax and why would it be done in that way?  I would like to avoid using third-party libraries - at least at first. Looking for other answers, I found the article  that discusses object-oriented programming in JavaScript. Is there a better way to do inheritance? Mount error: "unknown filesystem type 'exfat'" When trying to mount an exfat filesystem, I get the following error:  Error mounting /dev/sda6 at /media/gkp/Backup: Command-line  `mount -t "exfat" -o "uhelper=udisks2,nodev,nosuid" "/dev/sda6" "/media/gkp/Backup"'  exited with non-zero exit status 32: mount: unknown filesystem type 'exfat'   Exfat is used on some USB sticks and camera sd cards. What can I do to mount this type of filesystem? Conditionals (Past Tense) In terms of conditional sentences, is it correct to say:   if you knew why I WAS here, then I wouldn't need to be here.   or   If you knew why I WERE here, then I wouldn't need to be here.   Which type of conditional is this? Type 2? I was thought at school that there are three types of conditionals. Are there more of them? How can I get suffix matching in a search in SharePoint 2013? I'm new to SharePoint 2013 and I'm quite puzzled, that it seems to be impossible to search for the middle part of a string.  Simple example:  I've added a lot of documents (PDFs, ...) from our Windows Server to the SharePoint Library and I want to add metadata. I try to filter the view of the document library, to only get those documents which I currently want to edit.  The names of the documents are very systematic, composed of naming conventions with abbreviations like:  Superthing2000-b_Suppl_REF_13580_note_1108_04en which means e. g.: Document about Product "Superthing2000", variant "b", Supplement to the manual, ID "13580", from 08/2011 in english language.  Now, I'd like for example to filter out all documents containing ID 13580 or "en" for english, but no chance, as suffix matching is not possible, even if I explicitly search for e. g. *en*.  Is there a way to define a search with kind of a "contains" operator which also finds files when the search string is only a part in the middle of the file name? Even the Windows Explorer search feature is able to do that!! the least possible value for :$ \lfloor \frac{a+b}{c}\rfloor +\lfloor \frac{b+c}{a} \rfloor+\lfloor \frac{c+a}{b} \rfloor $ barchart with pgfplot - missing bar I am using PGFplot to make a barchart. There are 4 entries in my csv file and unfortunately I have only 3 bars in my figure.   my LaTeX code:  \begin{figure}     \centering     \begin{tikzpicture}          \begin{axis}[             x tick label style={/pgf/number format/1000 sep=},                 xlabel=Sequence,             ylabel=Heigth,                 ymin=0.7,             enlargelimits=0.05,             legend style={at={(0.5,-0.15)},                 anchor=north,legend columns=-1},             ybar interval=0.7,             width=\textwidth,             ]             \addplot table [col sep=comma,trim cells=true,y=height] {numericaltests/data/test.csv};              \addplot[red,sharp plot,update limits=false]                 coordinates {(-1,1.00) (34,1.00)}                 node[above] at (axis cs:15,1.00) {};             \legend{Ca2Fe2As2}         \end{axis}     \end{tikzpicture}     \caption{Sequence}     \label{fig:sequenceHeight} \end{figure}   my test.csv file:  sequence,height 1,2.0 2,1.5 3,1.8 4,2.4   Barchart: hierarchical retreival of records in postgresql (This question is only as reference, I will add the answer myself)  Suddenly the search in the expl3 documentation interface3.pdf no longer works. As an example searching for "divis" gives this as the first hit:    In sumatra there is no problem, other files (e.g. source3.pdf) works fine. What could be the reason?) Why is the amount of photos that fit on one card dependent on ISO? Financial proof to get a Schengen visa to France I am planning to apply for a Schengen visa to visit France. I have an attestation d'acceuil, but the visa requires I submit bank statement for the past 3 months. I can get this statement, but the problem is that my company has been paying my salary in cash for the past months, and I place the money into the account by myself.  Can I submit the following?   my accounting book (current account)  letter from the bank stating that I have money in it payslips for the past 3 months   Will that satisfy the financial requirements for the visa? I can't find a definitive answer for this. As far as I know, you can't have multiple __init__ functions in a Python class. So how do I solve this problem? Suppose I have a class called Cheese with the number_of_holes property. How can I have two ways of creating cheese objects...  One that takes a number of holes like this: parmesan = Cheese(num_holes = 15) And one that takes no arguments and just randomizes the number_of_holes property: gouda = Cheese()  I can think of only one way to do this, but this seems clunky: class Cheese():     def __init__(self, num_holes = 0):         if (num_holes == 0):             # Randomize number_of_holes         else:             number_of_holes = num_holes  What do you say? Is there another way? I am trying to build an executable jar program which depends on external jar downloaded. In my project, I included them in the build path and can be run and debug within eclipse.  When I tried to export it to a jar, I can run the program but I can't when I try to press a button which includes function calls and classes from the external jar. I have edited the environment variables (Windows XP) CLASSPATH to include paths of all the external jar, but it doesn't work.  A point to note is that I got compile warnings while exporting my executable jar, but it doesn't show up any description about the warnings.  Would someone kindly provide a thorough guide on how to include an external jar program using eclipse? I am trying to prove the following statement but have trouble  comprehending/going forward with some parts! Here is the statement:     If $A$ is any set, then  $|A|$ $&lt;$ $|P(A)|$   Here is what I have so far:  We need to show that there is an injection from $A$ to $P(A)$ but not a surjection.   A natural choice for an injection is the function $ f(x)$ $=$ $\{x \}$, which in plain English, takes any element $x$ (that is in $A$) and sends it to the one-element set $\{x \}$. Thus $f(x)$ is injective!  To show that there is no surjection, for the sake of contradiction, assume there is a surjection. Here is where I start to have trouble. Surjectivity means that every element of the co-domain is mapped to an element of the domain, correct? Consequently, in this particular case, we are "matching" sets (from $P(A)$) to elements (from $A$) right?  If the above is correct, my problem arises here. I am not sure how to prove that $f$ is not surjective. Unfortunately, I am easily confused by notation so please explain in English.  Thank you in advance!! :) Recently I found a claim saying that  $$  \int_0^\infty \left( \frac{\sin ax}{x}\right)\left( \frac{\sin bx}{x}\right) \mathrm{d}x= \pi \min(a,b)/2  $$ from what I can see this seems to be true. I already know that $\int_{0}^\infty \operatorname{sinc}xy\,\mathrm{d}y = \pi/2$, and so independant of $y$. My suspicion is that this is closely related to the integral above.   Can someone give me some suggestions for evaluating the integral above? Also are there any generalizations for the integral? Eg $$ \int_{0}^{\infty} \left( \prod_{k=1}^N \frac{\sin (a_k \cdot x)}{x} \right) \,\mathrm{d}x $$ Where $a_k, \cdots, a_N$ are arbitrary positive constants. It seems related to the , but there are some subtle differences. What is the difference between Θ(n) and O(n)? I need a compact representation of an array of booleans, does Python have a builtin bitfield type or will I need to find an alternate solution? How to perform a full system dump/backup with a broken screen? So, yesterday my phone took a serious tumble.  It is now essentially useless, as the screen either fails to turn on, or when it does, there is nothing useful to see (just lines of primary colors, mostly).  I believe (from the fact that it still chirps at me when emails come in, etc), that the phone is still functional, otherwise.  I would like to perform a full system dump, so that I can get off any files from the phone (a TMobile SGS2 (hercules) running slimrom 4.3 build 1).  The sdcard probably isn't important, as I can just pull it, but the phone's own internal storage I should like to get a copy of.  When I plug the phone into a usb port, it 'mounts' the usb via MTP, and I can see all the files.  Unfortunately, when I do a select all | copy | paste operation, the paste op dies before transferring all files.  I've tried a couple other ADB shell operations, in hope of flipping regular USB mounting on, but nothing has worked.  Any ideas how I can get a full dump of my poor phone's contents?  Edit  Currently running a adb pull /sdcard/ ... Its copying files presently, but its too early to tell if its going to get evertying I'm expecting.  Edit #2  looks like its going to work...its taking a rather long time, but it appears to be grabbing everything I was expecting it to. Both options affect colors in a similar way, but I can't figure out the difference. What are these three references from episode 9? Book recommendations