Hello I have a silly question: How can I show that $\lim\limits_{x \to \infty} \dfrac{\log(x+1)}{\log{x}}=1$. Thank you.
I have to solve: $\lim\limits_{x\to \infty} \frac{\ln(x+1)}{\ln(x)}$. Can you give me any hints to go? Thanks a lot!
Interpretation of interaction term Suppose $X_1,\ldots,X_n$ are a random sample of $U(-\theta,\theta)$ with $\theta&gt;0$. How can find $$ E\left(|X_1|\ \big|\max |X_i| \right) $$ that $1\leq i\leq n$? Using same modulus for RSA sudo find / -name "*" | xargs grep -sn --color=auto "-j"   Above command returns below:  grep: invalid option -- 'j' Usage: grep [OPTION]... PATTERN [FILE]... Try `grep --help' for more information. ...   How do I search for string -j? Prove the equality: $\det\left[\begin{smallmatrix} -2a &a+b &a+c \\ b+a&-2b &b+c \\ c+a&c+b &-2c \end{smallmatrix}\right] = 4(a+b)(b+c)(c+a)$ i have to prove:   $$\begin{vmatrix} -2a &amp;a+b  &amp;a+c \\   b+a&amp;-2b  &amp;b+c \\   c+a&amp;c+b  &amp;-2c  \end{vmatrix} = 4(a+b)(b+c)(c+a)$$   I have tried many calculations between the rows and columns of the determinant to get to the answer i want to solve the exercise however none of them gave me something right. Can someone help? Namespace constant in C# How to create a fancy table of contents for Beamer How can I create a fancy-looking table of contents for a Beamer presentation?    \documentclass{beamer} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc}  \begin{document}  \frame{ \tableofcontents }  \section{Foo bar} \frame{} \section{Foo bar} \frame{} \section{Foo bar} \frame{} \section{Foo bar} \frame{} \section{Foo bar} \frame{} \section{Foo bar} \frame{} \section{Foo bar} \frame{}  \end{document} Is there a tool that combines zcat and cat transparently? When handling log files, some end up as gzipped files thanks to logrotate and others not. So when you try something like this:  $ zcat *   you end up with a command line like zcat xyz.log xyz.log.1 xyz.log.2.gz xyz.log.3.gz and then with:  gzip: xyz.log: not in gzip format   Is there a tool that will take the magic bytes, similar to how file works, and use zcat or cat depending on the outcome so that I can pipe the output to grep for example?  NB: I know I can script it, but I am asking whether there is a tool out there already. What does the 'B' (or bonus) mean on a basketball scoreboard? $2 \otimes_{R} 2 + x \otimes_{R} x$ is not a simple tensor in $I \otimes_R I$ Hot and Week tabs not shown to non-logged in users (except on SO) postfix - how to prevent users from sending with another users name Even though my postfix server cannot be used without authentification (open relay), sending mail in another users name after logging in is still possible like this:  EHLO domain.org auth plain eW91IGxpdHRsZSBkaXJ0eSBiYXN0YXJkIDstKQ== 235 2.7.0 Authentication successful mail from: otheruser@domain.org 250 2.1.0 Ok rcpt to: recipient@otherdomain.org 250 2.1.5 Ok data 354 End data with &lt;CR&gt;&lt;LF&gt;.&lt;CR&gt;&lt;LF&gt; Please send me your account info. . 250 2.0.0 Ok: queued as D40692A61AA quit   How can this be prevented? Reflection to Identify Extension Methods In C# is there a technique using reflection to determine if a method has been added to a class as an extension method?  Given an extension method such as the one shown below is it possible to determine that Reverse() has been added to the string class?  public static class StringExtensions {     public static string Reverse(this string value)     {         char[] cArray = value.ToCharArray();         Array.Reverse(cArray);         return new string(cArray);     } }   We're looking for a mechanism to determine in unit testing that the extension method was appropriately added by the developer.  One reason to attempt this is that it is possible that a similar method would be added to the actual class by the developer and, if it was, the compiler will pick that method up. Solve the recurrence relation:$ T(n) = \sqrt{n} T \left(\sqrt n \right) + n$ SemanticImport vs Import differ on same CSV file? How do you reset timers? How to "flatten" a multi-dimensional array to simple one in PHP? It's probably beginner question but I'm going through documentation for longer time already and I can't find any solution. I thought I could use implode for each dimension and then put those strings back together with str_split to make new simple array. However I never know if the join pattern isn't also in values and so after doing str_split my original values could break.  Is there something like combine($array1, $array2) for arrays inside of multi-dimensional array? Back-transformation of regression coefficients I'm doing a linear regression with a transformed dependent variable. The following transformation was done so that the assumption of normality of residuals would hold. The untransformed dependant variable was negatively skewed, and the following transform made it close to normal:   $$Y=\sqrt{50-Y_{orig}}$$  where $Y_{orig}$ is the dependent variable on the original scale.   I think it makes sense to use some transformation on the $\beta$ coefficients to work our way back to the original scale. Using the following regression equation,   $$Y=\sqrt{50-Y_{orig}}=\alpha+\beta \cdot X$$  and by fixing $X=0$, we have   $$\alpha=\sqrt{50-Y_{orig}}=\sqrt{50-\alpha_{orig}}$$  And finally,  $$\alpha_{orig}=50-\alpha^2$$  Using the same logic, I found  $$\beta_{orig}=\alpha\space(\alpha-2\beta)+\beta^2+\alpha_{orig}-50$$  Now things work very well for a model with 1 or 2 predictors; the back-transformed coefficients resemble the original ones, only now I can trust the standard errors. The problem comes when including an interaction term, such as  $$Y=\alpha+X_1\beta_{X_1}+X_2\beta_{X_2}+X_1X_2\beta_{X_1X_2}$$  Then the back-transformation for the $\beta$s are not so close to the ones from the original scale, and I'm not sure why that happens. I'm also unsure if the formula found for back-transforming a beta coefficient is usable as is for the 3rd $\beta$ (for the interaction term). Before going into crazy algebra, I thought I'd ask for advice... Recently, I was wondering how division of aleph numbers would work. First, I thought about how finite cardinality division would work. What I came up with was that the result of $A/B$ where $A$ and $B$ are both cardinalities, is the number of times that each element of $B$ had to be mapped to an element of $A$ in order to ensure that all elements of $A$ were mapped to.  Extending this to aleph numbers, specifically, I thought that $\aleph_1/\aleph_0$ would be $\aleph_1$. The reasoning behind this is that there are an infinite ($\aleph_1$) number of real numbers between any two natural numbers. As such, that mapping would need to be applied.  Can anyone validate this idea? Is this how division of aleph numbers actually works, or am I totally off base?  Thanks. Got a bunch of .mp4 videos (info shows H.264 Video + AAC Audio) that are chapters of a larger video.  I'd like to merge them, but not sure which is the best way.  I don't have Quicktime Pro, but I'll get it if that's the best way to do this.  Edit: Just to clariify this is Mac OSX 10.6 I'm on.