I want to know that can we have the following inequality for $x&gt;0$, $y&gt;0$?  $$\log(1+x/y) \leq x/y$$
I want to prove that: $\ln(x+1)&lt; x$.  My idea is to define:  $f(x) = \ln(x+1) - x$, so:  $f'(x) = \dfrac1{1+x} - 1  = \dfrac{-x}{1+x} &lt; 0, \text{ for }x &gt;0$.  Which leads to $f(x)&lt;f(0)$, so $\ln (x+1)-x&lt;0$.  Is that a valid proof? Any other ideas?  Thanks.
Exponent and abelian groups Why can't humans throw objects at bullet speed when the force involved seems so small? The force that propels a bullet should at least be similar in intensity to the force experienced as recoil and the force exerted on the target. But these forces seem really low (the equivalent of a shove).  If all it takes to make a small object achieve bullet speed is the equivalent of a shove, why can't humans throw objects at these speeds? Figure reference \ref{fig:result} renders as ?? in PDF Evaluate $\lim\limits_{n\to\infty}\frac{\sum_{k=1}^n k^m}{n^{m+1}}$ By considering:  $$\lim_{n\to\infty}\frac{\sum_{k=1}^n k^1}{n^{2}} = \frac 1 2$$ $$\lim_{n\to\infty}\frac{\sum_{k=1}^n k^2}{n^{3}} = \frac 1 3$$ $$\lim_{n\to\infty}\frac{\sum_{k=1}^n k^3}{n^{4}} = \frac 1 4$$  Determine if this is true: $$\lim_{n\to\infty}\frac{\sum_{k=1}^n k^m}{n^{m+1}} = \frac 1 {{m+1}}$$  If it is, prove it.  If it is not, evaluate $\lim\limits_{n\to\infty}\frac{\sum_{k=1}^n k^m}{{m+1}}$. Wrong hyperref with repeated counter I'm having an interesting problem with hyperref.  Apparently, when a counter is reset by \setcounter{mycounter}{0} then hyperref always links to the first possible instance of the counter and not the one where the respective label sits.   Check this MWE out:   % !TEX TS-program = xelatexmk \documentclass{article} \usepackage[paperwidth=6in,paperheight=3in]{geometry} \usepackage{hyperref} \hypersetup{colorlinks=true, allcolors=blue, } \newcounter{mycounter}\setcounter{mycounter}{0} \newcommand\countme{\refstepcounter{mycounter}\themycounter}  \begin{document}  First line \countme{} has some interesting text.  The second line \countme \label{onelabel} has \textbf{more} interesting text.  As we saw at line \ref{onelabel} ...  \newpage  A new page brings new numbers \setcounter{mycounter}{0}   A very different first line \countme{} has other interesting text.  The second line \countme \label{otherlabel} has \textbf{different} interesting text.  As we saw at line \ref{otherlabel} ... \end{document}   As you can see in the screenshot below, the pop up preview of the \ref on the second page doesn't refer to the second line with \label{otherlabel} but to the one on the first page (also has the tooltip "Go to page 1" -- in TeXShop).   Am I not understanding something fundamental about hyperref?      The context for all this is an environment that resets my counter to 0 at begin.  And all subsequent \refs to labels in a second or third iteration of the environment point to the very first instance of that environment, not the correct one where the label sits. How assign shortcut to custom function? During my upcoming holiday to the USA I'm thinking about buying a new guitar as the exchange rate is extremely favorable. Excuse my naivety but I have never flown with a guitar before. What is a recommended way to go about this? Could I take it as carry on luggage or would it have to be checked? If checked luggage what kind of precautions should I take to keep it safe? Are there any problems with flying US budget carriers? If I do buy the guitar I'm expecting to have to make two flights with it, one domestic flight with Virgin America and an international flight with Air NZ. Inheritance of static members in PHP In PHP, if a static attribute is defined in the parent class, it cannot be overridden in a child class. But I'm wondering if there's any way around this.   I'm trying to write a wrapper for someone else's (somewhat clunky) function. The function in question can be applied to lots of different data types but requires different flags and options for each. But 99% of the time, a default for each type would suffice.  It would be nice if this could be done with inheritance, without having to write new functions each time. For example:  class Foo {     public static $default = 'DEFAULT';      public static function doSomething ($param = FALSE ) {         $param = ($param === FALSE) ? self::$default : $param;         return $param;     } }  class Bar extends Foo {     public static $default = 'NEW DEFAULT FOR CHILD CLASS'; }  echo Foo::doSomething() . "\n";  // echoes 'DEFAULT'  echo Bar::doSomething() . "\n";  // echoes 'DEFAULT' not 'NEW DEFAULT FOR CHILD CLASS'  // because it references $default in the parent class :( This thing came up in a combinatorics course I am taking.  Choose a fixed set of primes $p_1,p_2,\dots,p_k$ and let $A_n$ be number of integers in $\{1,2,\dots,n\}$ which are not divisible by any of the $p_i$'s. $A_n$ is given by $ n - \sum_{1\leq i_1 \leq k} \lfloor \frac{n}{p_{i_1}} \rfloor + \sum_{1\leq i_1 &lt; i_2 &lt; k} \lfloor \frac{n}{p_{i_1}p_{i_2}}\rfloor - \dots $. Now, if $n$ is divisible by each of the $p_i$'s then we have the simpler expression : $A_n = n \prod_{i=1}^{k}(1-\frac{1}{p_i})$(added later : note I am not assuming $n = \prod_{i=1}^{k}p_i$ here, this holds true whenever $n = c \prod_{i=1}^{k} p_i $ for some integer c, since each $\lfloor \frac{n}{p_{i_1} \dots p_{i_j}} \rfloor = \frac{n}{p_{i_1} \dots p_{i_j}}\ )$.  Another student pointed out that even if $n$ does not have $\prod_{i=1}^{k} p_i$ as a factor, the approximation $B_n = n \prod_{i=1}^{k}(1-\frac{1}{p_i})$ to $A_n$ is quite close in some specific cases. It is easy to see that $\lim_{n\to\infty}\frac{A_n - B_n}{n}=0$ as $\lim_{n\to\infty}\frac{1}{n}\lfloor \frac{n}{p_{i_1}p_{i_2}\dots p_{i_j}} \rfloor \to \frac{1}{p_{i_1}p_{i_2} \dots p_{i_j}},$ however that is not strong enough to imply that $A_n - B_n$ will be always close. Is there any way to analyze how well this approximation will perform in general? I am interested in the worst case for small to moderately sized $n$.  Just to give a feel of how close $A_n$ and $B_n$ can get, assuming the set of primes is $\{2,3,7\}$ we have (assuming my program is correct):  n     A(n)     B(n) 17    5    4.86   27    8    7.71   37    11      10.57   107   31      30.57   1111  318     317.43   3001  858     857.43   4007  1145    1144.86   5000  1429    1428.57 . Just wanted some feedback to ensure I did not make any mistakes with this proof. Thanks!  Since $G$ is abelian, every subgroup is normal. Since $G$ is simple, the only subgroups of $G$ are $1$ and $G$, and $|G| &gt; 1$, so for some $x\in G$ we have $x\neq 1$ and $\langle x\rangle\leq G$, so $\langle x\rangle = G$. Suppose $x$ has either infinite order or even order. Then $\langle x^2\rangle \leq G$ but $\langle x^2\rangle \neq \langle x \rangle$, a contradiction. So $x$, and therefore $G$, has odd order, and by Feit-Thompson, $G\cong Z_p$ for some prime $p$.  Edit: Thanks, I see that Feit-Thompson is too much.  Since $G$ is abelian, every subgroup is normal. Since $G$ is simple, the only subgroups of $G$ are $1$ and $G$, and $|G| &gt; 1$, so for some $x\in G$ we have $x\neq 1$ and $\langle x\rangle\leq G$, so $\langle x\rangle = G$. Suppose $x$ has infinite order. Then $\langle x^2\rangle \leq G$ but $\langle x^2\rangle \neq \langle x \rangle$, a contradiction. So $x$, and therefore $G$, has finite order. Suppose $x$ has composite order $n$ so for some $p &gt; 1$ that divides $n$, $\langle x^p \rangle$ is a proper non-trivial subgroup of $G$, so $G$ is not simple. So $G$ is a cyclic group of prime order. Some things are easier to implement just by hand (code), but some are easier through WF. It looks like WF can be used to create (almost) any kind of algorithm. So (theoretically) I can do all my logic in WF, but it's probably a bad idea to do it for all projects.  In what situations is it a good idea to use WF and when will it make things harder then they have to be? What are pros and cons/cost of WF vs. coding by hand? What is the difference between a HMAC and a hash of data? I have a Linux CentOS server, the OS+packages used around 5GB. Then, I transferred 97GB data from a Windows server to two folders on this Linux server, after calculated the disk usage, I see the total size of the two folders is larger than the disk used size.  Run du -sh on each folder, one use 50GB, the other one use 47GB  But run df -h, the used space is 96GB. (50GB + 47GB + 5GB) > 96GB  Is there any problem? Those two folders contain lots of files (1 million+). Thanks. Why is foo.hasOwnProperty('__proto__') equal to false? With "magic quotes" disabled, why does PHP/WordPress continue to auto-escape my POST data? Why can current only flow in loops? In the following image:   Why can't current flow across the following wire? It's a simple question, but I've kind off always wondered.  Thanks! Calculate the Factor Group $(\mathbb{Z}_4 \times \mathbb{Z}_6)/\langle(0,2)\rangle$ I had an article, but I lost it.  It showed and described a couple of C/C++ tricks that people should be careful. One of them interested me but now that I am trying to replicate it I'm not being able to put it to compile.  The concept was that it is possible to change by accident the value of a const in C/C++  It was something like this:  const int a = 3;          // I promise I won't change a const int *ptr_to_a = &amp;a; // I still promise I won't change a int *ptr; ptr = ptr_to_a;  (*ptr) = 5;               // I'm a liar; a is now 5   I wanted to show this to a friend but now I'm missing a step.  Does anyone know what's missing for it to start compiling and working?  ATM I'm getting invalid conversion from 'const int*' to 'int*' but when I read the article I tried and it worked great. How to allow non-superusers to mount any filesystem? Is it possible to allow some particular users (e.g. members of a group) to mount any filesystem without superuser privileges on Linux?  Another question might have been "in what ways a user can harm a system by mounting filesystems?" How to draw simple table