$f\in L^1(0,\infty)$ monotone, show $\lim_{x\rightarrow \infty} xf(x) = 0$
If monotone decreasing and $\int_0^\infty f(x)dx <\infty$ then $\lim\limits_{x\to\infty} xf(x)=0.$
If you know what a number mod two different primes is (3 and 2) in this case, how can you tell what the mod is of the two products? no match for ‘operator<<’ in ‘std::operator How can I split multiple joined words? I have an array of 1000 or so entries, with examples below:  wickedweather liquidweather driveourtrucks gocompact slimprojector   I would like to be able to split these into their respective words, as:  wicked weather liquid weather drive our trucks go compact slim projector   I was hoping a regular expression my do the trick.  But, since there is no boundary to stop on, nor is there any sort of capitalization that I could possibly key on, I am thinking, that some sort of reference to a dictionary might be necessary?    I suppose it could be done by hand, but why - when it can be done with code! =)  But this has stumped me.  Any ideas? How do I properly clean up Excel interop objects? I'm using the Excel interop in C# (ApplicationClass) and have placed the following code in my finally clause:  while (System.Runtime.InteropServices.Marshal.ReleaseComObject(excelSheet) != 0) { } excelSheet = null; GC.Collect(); GC.WaitForPendingFinalizers();   Although this kind of works, the Excel.exe process is still in the background even after I close Excel. It is only released once my application is manually closed.  What am I doing wrong, or is there an alternative to ensure interop objects are properly disposed of? I am very new to LaTeX and I am writing my thesis. In a chapter, I want to show my codes, which are in Mathematica.   I use the listings package as follows:  \documentclass{article}   \usepackage{listings}   \begin{document}   \lstset{language=Mathematica}    \begin{lstlisting}   Do[i^j,{i,1,10},{j,1,10}]   \end{lstlisting}    \end{document}   However, the output is much different to what the code looks like in the Mathematica IDE (even if I use other languages in \lstset{language=&lt;name&gt;}):    I read some questions on the site but, in some of them, the font or color of the codes were not the same. Could you please help me to display something like the picture below in LaTeX?  If it is important, I am using the BiDi package. Wrap Caption Around TikZ Figure Graph with Cycle and Two-Colorable i think if the graph G has an odd cycle, it's not two-colorable, otherwise it can be two colorable.   i read in one notes that the following is True: we couldent two-colorable any graph G that has cycle.    anyone could clarify me ? SQL function as default parameter value? I tried changing a default parameter value with this:  ALTER PROCEDURE [dbo].[my_sp] @currentDate datetime = GETDATE()   and all the SQL pre-compiler gave me was this error:     Msg 102, Level 15, State 1, Procedure my_sp, Line 8 Incorrect syntax near '('.   I have already created the procedure. (I'm not sure if that's relevant.) I was using a null default value and checking for that later, but that doesn't seem proper. Can I do this in one line?   Update: I was going off of :      [ = default ] Is a default value for the parameter. If a default value is defined, the function can be executed without specifying a value for that parameter.            Note:     Default parameter values can be specified for CLR functions except for the varchar(max) and varbinary(max) data types.         When a parameter of the function has a default value, the keyword DEFAULT must be specified when the function is called to retrieve the default value. This behavior is different from using parameters with default values in stored procedures in which omitting the parameter also implies the default value.   Am I reading this wrong?  Many thanks. Derivative of $x^{x^{\cdot^{\cdot}}}$? Is tipping mandatory at Restaurants and Bars in Germany? In Bavaria, I paid for my drink, I asked her for "Ruckgeld", but only got the integer part of the money back, the rest of it, she kept it for tip! I looked her in the eye and she understood that I wasn't happy with that "initiative", but wouldn't blink an eye!  My company advised me to let it go, mentioning that if I was going to ask for the change, she would swear.  Notice how she didn't return me the correct amount back and then stare at me, demanding for tip, but she would decide for me!  Since, I will be around for a week or so, I was thinking of paying the exact amount of money next time. Will I get in an uncomfortable situation of having the waiter stare at me, demanding a tip? In other words, is tipping mandatory in Germany?    PS: Having experienced the tipping-cruelty in USA, and having tipped generously waiters across the Mediterranean for their supreme behavior, this Saturday night came as a shock, since I had the impression that this awful habit was only a USA-thing (you know, just like the fake audience laughing they have in USA TV series :) ). Does xbox live gold membership stack when I purchase a new one? I currently have an Xbox Live Gold membership that will expire in March 2011. However the price of membership is increasing to $59.99 starting November 1, 2010.   If I buy a 12 month membership now at the current price of $49.99, will the additional 12 months be added on to my existing membership or will I start a new membership which will expire 12 months after I buy it? I have been hit with a homework problem that I just have no idea how to approach. Any help from you all is very much appreciated. Here is the problem  Prove the equation: $a^{\log_b c} = c^{\log_b a}$  Any ideas? Assume you have already completed your undergrad and have been working for a number of years. Does one need to be part of a university or a learning institution to publish papers? 80/90s UK children's drama series about an "alien" boy with radio weapon I've been trying to track down a live-action children's TV series (or short story perhaps) from the 1980s or 1990s, probably on CBBC or CITV.  A few people I've spoken to IRL have suggested The Boy from Space or Sky, but I'm pretty sure it's neither as a lot of the details don't seem to match.  The story was set on Earth (possibly England or Wales) and was about an "alien" boy who was being pursued by a group of people, though I don't remember who they were.  Somehow he encountered siblings, a girl and a boy, and made friends with them. I think they lived on a farm with their parents, as I remember quite a large barn or other building, and they hid(?) the boy in a car which was inside or next to this building. He also slept in this car with a dog (which might have been a German shepherd).  The "alien" boy had a portable "radio" (all silver, very 1980s) that repelled his pursuers by emitting a high-pitched sound when he turned a dial on it, and he used it when out and about with the other children whenever they encountered a pursuer. I seem to remember lots of hills and valleys, lots of green.  In the end it turned out he wasn’t really an alien, but a human from the far future. He was pretending to be an alien as he was from 20000 years into the future and humans' appearance had changed by then, but he had the ability to make himself look like current-day humans. When he was in his car/bed he reverted to his normal appearance, so he would hide under the blankets until he was able to assume current-day human appearance.  In the finale, it was revealed that he was never in any danger – the pursuers were testing him as part of a kind of coming-of-age thing. I remember a scene on a spaceship or something where his elders/parents where shown telling him that he was safe all along and that he "passed".  I also seem to remember them referring to "ancestors" and making sure they didn't find out, so the either the girl or boy could well have been a direct ancestor. Area 51 reputation changes do not show up in the new Top Bar I can not see how mathematically one can divide the numerator and denominator by theta. Can some one explain this in a different way. When you divide two fractions you multiply the first fraction by the reciprocal of the second. This is not the case here.     Divide numerator and denominator by $\theta$. ($\sin\theta$ also works.) $$\eqalign{\lim_{\theta\to0}\dfrac{\sin\theta}{8\theta+\tan\theta}&amp;= \lim_{\theta\to0}\dfrac{\dfrac{\sin\theta}\theta}{8+\dfrac{\sin\theta}\theta\cdot\dfrac1{\cos\theta}}\\&amp;=\dfrac{\lim\limits_{\theta\to0}\dfrac{\sin\theta}\theta}{8+\lim\limits_{\theta\to0}\dfrac{\sin\theta}\theta\lim\limits_{\theta\to0}\dfrac1{\cos\theta}}\\&amp;=\dfrac1{8+1\cdot1}\\&amp;=\dfrac19.}$$ Express $\sqrt{3}\sin\theta - \cos\theta$ as: $a\cos (\theta + \alpha) $ Express $\sqrt{3}\sin\theta - \cos\theta$ as: $a\cos (\theta + \alpha) $  Can someone please explain to me how to go about doing this? How do I prove that 15462227 and 15462229 relatively prime? The characteristic and minimal polynomial of a companion matrix How can I return to the start of a line in a console? How can I return to the start of a line and overwrite what has already been output on the console? The following does not appear to work:  System.out.print(mystuff+'\r');