What is the geometrical interpretation of Lipschitz condition? A function $f$ is said to satisfy Lipscitz condition on $D$ if there exists $M&gt;0$ such that for every $x_1,x_2$ belonging to $D$ we have $$|f(x_1)-f(x_2)|\leq M|x_1-x_2|.$$ Now, what is the geometric interpretation of this thing? Does this mean that slope of chord joining any $2$ points on the curve $y=f(x)$ is bounded. Does this imply that derived function $f'$ of $f$ is bounded on $D$ (provided derivative exists on $D$)? Can it be understood with the help of diagram geometrically? I had heard some concept of double cone for this thing. Can someone explain that, too?
Intuitive idea of the Lipschitz function I'm trying to understand intuitively the notion of Lipschitz function.   I can't understand why bounded function doesn't imply Lipschitz function.   I need a counterexample or an intuitive idea to clarify my notion of Lipschitz function.  I need help  Thanks a lot
Does the Dueling Fighting Style prevent using a shield? The Dueling fighting style doesn't seem to state anything about shields, stating that the character can only wield one weapon while fighting to gain the benefit. Does a shield count as a weapon or otherwise stop the damage bonus from occurring? Find the summation of given expression: $\sum_{k=1}^n k^3\binom nk$ I am trying to solve the following question which is Ex3 from Arthur Engel, Problem Solving strategies. Here is the question: $\sum_{k=1}^n k^3 {n \choose k}$ and asks to find the sum. I am sincerely having a hard time trying to figure out how to solve it and get $n^2 (n+3)(2)^{n-3}$. I would really appreciate if someone could show the step-by-step process. Thank you! Let  $f(xy) =f(x)f(y)$ for all $x,y\geq 0$. Show that $f(x) = x^p$ for some $p$.    I am not very experienced with proof. If we let $g(x)=\log (f(x))$ then this is the same as  $g(xy) = g(x) + g(y)$   I looked up the hint and it says let $g(x) = \log f(a^x) $   The wikipedia page for functional equations only states the form of the solutions without proof.  Attempt  Using the hint (which was like pulling a rabbit out of the hat)  Restricting the codomain $f:(0,+\infty)\rightarrow (0,+\infty)$ so that we can define the real function $g(x) =  \log f(a^x)$ and we have $$g(x+y) = g(x)+ g(y)$$   i.e $g(x) = xg(1)$ as $g(x)$ is continuous (assuming $f$ is).  Letting $\log_a f(a) = p$ we get $f(a^x) =a^p $. I do not have a rigorous argument but I think I can conclude that $f(x) = x^p$ (please fill any holes or unspecified assumptions) Different solutions are invited "need not to be neither atomic nor ordered" Is this double/triple negative grammatically sound?  Ordered operations require that space on disk is allocated sequentially, so spatial ordering corresponds to chronology of operations, but actual disk writes need not to be neither atomic nor ordered.   Explanation:   Words neither and nor emphasize on the fact that both conditions are (independently) allowed to remain unsatisfied, that both are optional. To be more formal, "need not be atomic and need not be ordered". But then again, it becomes a concoction of two negations joined by a conjunction. Using Inclusion/Exclusion to solve $x_1+x_2+x_3=15$ with $x_1,x_2\leq 5$ and $x_3\leq 7$ for non negative integers $x_1,x_2,x_3$ I want to solve for:     Number of integers solutions to the equation $x_1+x_2+x_3=15$ with $x_1,x_2\leq 5$ and $x_3\leq 7$ for non negative integers $x_1,x_2,x_3$   How can this be done using the principle of inclusion/exclusion? Are PDO prepared statements sufficient to prevent SQL injection? Let's say I have code like this:  $dbh = new PDO("blahblah");  $stmt = $dbh-&gt;prepare('SELECT * FROM users where username = :username'); $stmt-&gt;execute( array(':username' =&gt; $_REQUEST['username']) );   The PDO documentation says:     The parameters to prepared statements don't need to be quoted; the driver handles it for you.   Is that truly all I need to do to avoid SQL injections?  Is it really that easy?  You can assume MySQL if it makes a difference.  Also, I'm really only curious about the use of prepared statements against SQL injection.  In this context, I don't care about XSS or other possible vulnerabilities. Should I use 'was' or 'were' in this example?     I was always delighted when my brother or one of my sisters was/were asked to do them. How do you do a deep copy of an object in .NET? I want a true deep copy. In Java, this was easy, but how do you do it in C#? Explain why $E(X) = \int_0^\infty (1-F_X (t)) \, dt$ for every nonnegative random variable $X$ Let $X$ be a non-negative random variable and $F_{X}$ the corresponding CDF. Show,   $$E(X) = \int_0^\infty (1-F_X (t)) \, dt$$   when $X$ has : a) a discrete distribution, b) a continuous distribution.   I assumed that for the case of a continuous distribution, since $F_X (t) = \mathbb{P}(X\leq t)$, then $1-F_X (t) = 1- \mathbb{P}(X\leq t) = \mathbb{P}(X&gt;  t)$. Although how useful integrating that is, I really have no idea. GNU find and masking the {} for some shells - which? The man page for GNU find states:    -exec command ;     [...] The  string `{}'  is  replaced  by the current      file name being processed everywhere it occurs in the      arguments to the command, not just in arguments where      it is alone, as in some  versions  of  find.     Both  of  these  constructions might need to be escaped      (with a `\') or quoted to protect them from expansion      by the shell.     That's from the man to find (GNU findutils) 4.4.2.   Now I tested this with bash and dash, and both don't need to have the {} being masked. Here is a simple test:  find /etc -name "hosts" -exec md5sum {} \;    Is there a shell, for which I really need to mask the braces? Note, that it doesn't depend upon whether the file found contains a blank (invoked from bash):   find ~ -maxdepth 1 -type d -name "U*" -exec ls -d {} \;  /home/stefan/Ubuntu One   This changes if the found file is passed to a subshell:  find ~ -maxdepth 3 -type d -name "U*" -exec bash -c 'ls -d {}' \;  ls: cannot access /home/stefan/Ubuntu: No such file or directory ls: cannot access One: No such file or directory   which can be solved by:  find ~ -maxdepth 3 -type d -name "U*" -exec bash -c 'ls -d "$0"' {} \;   in contrast to:   find ~ -maxdepth 3 -type d -name "U*" -exec bash -c 'ls -d "{}"' \;  /home/stefan/Ubuntu One   but that's not what the man page is talking about, is it? So which shell treats {} in a different way? How do I pre-populate a jQuery Datepicker textbox with today's date? Learn Bayesian inference applied to astronomy / astrophysics? I am using Log4Net with the AdoNetAppender to log messages from a simple systray application into a SQL Server 2005 database.  I want to log the machine name along with the log message because this application will be running on multiple machines and I need to know on which one the message originated.  But, I cannot find a way to expose this information via the log4net.Layout.PatternLayout that I am using with the appender.  Is there a way to log the machine name via log4net in this manner? XeTeX: What happened to \mathcal and \mathscr? Recently I switched from pdflatex to xelatex to be able to use OpenType fonts. I like to use variables highlighted using \mathcal for specific variables, while I used to use \mathscr-typeset variables for matrices. But when I compile with XeLaTeX they look exaclty the same.  Old code  \documentclass{standalone} \usepackage{mathrsfs} \begin{document} $\mathcal{C} = \mathscr{C}$ \end{document}   Output        New code  \documentclass{standalone} \usepackage{unicode-math} \begin{document} $\mathcal{C} = \mathscr{C}$ \end{document}   Output        Broken code  I tried to get back the glyphs from mathrsfs, but it failed this way.  \documentclass{standalone} \usepackage{unicode-math} \usepackage{mathrsfs} \begin{document} $\mathcal{C} = \mathscr{C}$ \end{document}   Output        Question: How can I get back the behaviour? In principle I'd like to have the old \mathcal letters from the pdflatex-version of Computer/Latin Modern.    Alternate solutions (Addendum 2016/05/11) Blocking web sites with Windows Firewall Is it possible to use Windows Firewall to block specific web sites for all browsers? Which hydroxyl from either the phosphate or the glycerol is taken during synthesis of a phospholipid head? Show the post's comments on the "Suggested Edits" review interface This is purely a curiosity, but I'm fascinated by mid-word pluralization, even if the word in question is a compound word.  For example, passersby or standersby.  No others have occurred to me. Can you provide other examples, or a link to a resource that enumerates them?  I'm particularly interested in compounds that do not include spaces or hyphens. I just saw Get Out, and there's one thing I don't understand.  Georgina turns out to be controlled by Rose's grandmother; its grandmothers brain in her body. Why then, is Georgina constantly cleaning and being treated like an actual maid.  During the diner scene, we get a glimpse into the kitchen, and we see Georgina holding up a plate in way that suggested that she was made to work and cook. The pose and the fact that she was already standing there holding the plate before Missy enters the kitchen was also quite strange and suggestive.  Also, at the "party" theres a moment that Chris sees Georgina and Missy. For a moment he thinks that Georgina is being involved in the party, but than it turns out the Georgina is just handed empty glasses to take care of.  So this I don't get. If its really grandma inside Georgina, then why is she treated in this way? Could I use an IPS based TV to edit photos in a color-calibrated environment? Over the past weeks, I've been trying to get all of my color issues worked out. Getting closer, but I'm not there. I've calibrated my monitor with a friend's calibrator and that helped. I'm using the profiles provided by the local processor.  It's close. How close depends on the viewing angle.  So I'm thinking IPS monitor might be a good next step.  Googling around, I see that I could get an   Given that IPS based 22/23" monitors run $250-$300, this seems like a lot more monitor for not much more money.  What are the faults in this logic?