How to draw Square Diagonal?
How to solve 5x5 grid with 16 diagonals?
How can I take a screenshot with Selenium WebDriver? Is multiplication not just repeated addition? If multiplication is not repeated addition, what is it? How do you define multiplication? All of us had to memorize the multiplication table in elementary school, but how did they come up with it if so many people claim that it is wrong to think of it as repeated addition? Voodoo magic? What can I do to manage my phone's internal storage? Relation bewteen Hessian Matrix and Curvature According to ,  It describes the local curvature of a function.   AFAIK, for one-variable function $f(x)$, its local curvature is  $$\kappa = \frac{|f''|}{(1 + f'^2)^{3/2}},$$ and its Hessian matrix is  $$\mathcal{Hess}(f) = [f''],$$ right? And here is my problem, I think the local curvature is not just described by its Hessian matrix, because $f'$ also has its role in it, doesn't it?  And furthermore, for 2-variable function $f(x,y)$, its Hessian matrix is  $$\mathcal{Hess}(f) = \left[ \begin{array}{cc} f_{xx}'' &amp; f_{xy}'' \\ f_{xy}'' &amp; f_{yy}'' \end{array} \right].$$ How does it relate to the local curvature of $f(x,y)$? I have several entries in my list of references that include of the order of 40 authors. Is there a way to make bibtex automatically reduce these lists to e.g. the first name + "et al."?  I don't want to edit the BibTeX entries manually and I cannot use biblatex. If at all possible I would also make as little changes as possible to the bibliography style.  There are similar questions, but none of those had a concise answer:       natbib e.g. seems to change only the reference within the text, but not the entry in the bibliography.  EDIT: I am using \bibliographystyle{unsrt} as style definition. Unable to install LanguageTool 2.5 extension for LibreOffice 4.2 on Ubuntu 14.04 I am not able to install LanguageTool extension for LibreOffice. Getting exception when I try to install it.   p.s. Somebody says that the solution is installing openoffice.org-java-common package but Ubuntu 14.04 does not have this package in its repository.  System details:  OS: Ubuntu 14.04 LTS  LibreOffice: 4.2 (4.2.3.3)  LanguageTool: 2.5  Java: Oracle JDK 1.8.0_05     Exception message: Why is cpu_idle called as the last function of the start_kernel routine? I'm reading the  function in /init/main.c.  The last function called in start_kernel is rest_init, and at the end of rest_init, cpu_idle is called.   Why is it called as the last function of the start_kernel routine, and what does it do? What is the purpose of the word 'self'? Deleted Ubuntu 20 alongside windows 10 and unable to install again Why WordPress choose data serialization over json_encode? In my little age with WordPress, I've seen WordPress itself and its friendly plugins are using PHP serialize() in storing data into db in many cases. But in a recent search I found a serious community support for the json_encode() over the serialize().     And I personally tested an associative array with both of 'em, that shows:  serialize() stores 342 chars json_encode() stores 285 chars  Why I'm asking this? I'm on a project while I'm going to store repeating meta fields to a post. Where:  Data would be basically in English, but sometimes can be Bengali Data would be associative array, 3 level deep (I hope I understood levels correctly):   array(     1 =&gt; array(         'key'=&gt;'value',         'key2'=&gt;'value'     ),     2 =&gt; array(         'key'=&gt;'value',         'key2'=&gt;'value'     ) )  I've checked the postmeta table's meta_value field it's a longtext,  a length of 4,294,967,295 chars (4GB). So I need a robust solution into storing things. Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$. What is the best way to get a sine wave from a square wave? Why does .NET use banker's rounding as default? According to the documentation, the  method uses a round-to-even algorithm which is not common for most applications. So I always end up writing a custom function to do the more natural round-half-up algorithm:  public static decimal RoundHalfUp(this decimal d, int decimals) {     if (decimals &lt; 0)     {         throw new ArgumentException("The decimals must be non-negative",              "decimals");     }      decimal multiplier = (decimal)Math.Pow(10, decimals);     decimal number = d * multiplier;      if (decimal.Truncate(number) &lt; number)     {         number += 0.5m;     }     return decimal.Round(number) / multiplier; }   Does anybody know the reason behind this framework design decision?  Is there any built-in implementation of the round-half-up algorithm into the framework? Or maybe some unmanaged Windows API?  It could be misleading for beginners that simply write decimal.Round(2.5m, 0) expecting 3 as a result but getting 2 instead. Value of $\frac{x^n -a^n}{ x-a}$? We have a view for a type of node that includes a Private File.  We cannot see a way to add an exposed filter, such as "Contains any word", and have it filter based on the file's name.  Are we missing something obvious? Suppress black box at end of line Under TeX, when I write a line too long, it prints a fat black box at the end of the line.  That is probably a feature, but I dislike it.  How to turn that OFF? Is there a command-line option or something else?   I usually use LaTeX, but I have a complicated file which happens to be plain TeX. How do Horcruxes work? A few questions regarding the Horcruxes.   Had Voldemort actually succeeded in using a Horcrux, would he come back the same age (biologically and/or aesthetically) he was when he made the Horcrux, or the age he would biologically be if he had never died (half died)? If Harry was a Horcrux, then in the graveyard, when Wormtail added Harry's blood to the potion to bring Voldemort back, didn't that technically use up the Horcrux inside of Harry? So in Deathly Hallows when Voldemort kills Harry, shouldn't Harry have actually died rather than just have the Horcrux inside him destroyed, seeing as it was technically used in the Goblet of Fire? Number of non-isomorphic groups of order 21 Why are American-style options worth more than European-style options? Why are American-style options worth more than European-style options?   I realize I can exercise American-style options anytime before the  expire, but I can only exercise European-style options during their  "exercise period" (usually right when they expire, but no earlier).   So it makes sense that an American option is worth at least as much as  a European option.   But why is it worth more? If I exercise my American option early, I  may make more money than waiting until expiration, but I also may make  less. Mathematically, there's no advantage, since I'm equally likely  to make as much money by waiting for expiration.   Mathematically speaking, is there ever a good reason to exercise an  American option early?   I realize people don't always behave logically, but even the formulas  that valuate options show that American options are worth more.   EDIT (answering comments):    Many of the comments people made would apply to lookback options  (), but not to American  options.  American options can do everything European options can and more. I  understand that, and that means American options can't be worth any  LESS than European options, but I'm still not clear on how you would  mathematically calculate this extra value.  @Aaronaught: "The difference between an American and European  option is the difference between getting N chances to get it right (N  being the number of days 'til expiration) and getting just one  chance. It should be easy to see why you're more likely to profit  with the former, even if you can't accurately predict price  movement."    REPLY: I don't think you really get N chances. Once you exercise the  option, that's it, no more chances. And, if you decide not to exercise  the option today, and the underlying's price falls, you can't go back  in time and exercise it yesterday.    @jdsweet: Think of it this way, if you traveled back through time  one month - with perfect knowledge of AAPL's stock price over that  period - which happens to peak viciously then return to its old price  at the end of the period - wouldn't you pay more for an American  option?    REPLY: Well, no. I'd simply buy a cheaper European option that expires  right at the spike. Or, I'd buy a European option that expires later,  and sell it when the spike occurs. American options don't give you  precognition, so you're still guessing as to when to sell/exercise.    @jdsweet: "one of the more common reasons people exercise (as  opposed to sell) an American option before expiration is if an  unexpected dividend (larger than remaining time value of the option)  was just announced that's going to be paid before the option contract  expires"    REPLY: This makes sense. I'm actually trading FOREX options, so there  is no dividend. In that scenario, are European and American options  worth the same? How can I set Excel to always import all columns of CSV files as Text? Though I try to avoid it, I occasionally have to open a CSV file in Excel. When I do, it formats columns containing numbers, which makes them useless for my purposes. As far as I can tell, the only way to prevent this from happening on import is to rename the file so the extension isn't .csv and use the import wizard to specify the format of each column individually. For files with 50-60 columns, this is impractical.  Since every answer for this oft-asked question on the internet suggests either some means of converting the formatted numbers back once the file is open (which won't work for me - I want to solve the general problem, not a few specific cases) or manually selecting the format type of each column (which I don't want to do), I'm looking for a way to set a global preference or style such that all columns of all CSV files opened are always formatted as text.  I know about "armoring" the numbers with quotes, too, but the files I get don't come like that and I was hoping to avoid having to pre-process the files so Excel doesn't screw them up.  Is there a way to do specifically this: Always format all columns in opened CSV files as text, without manually selecting each column every time during import?  I'm using Excel 2003, but I'll take answers for 2007 if that's what you know.