The generalized inequality defined by a proper cone $K$ is that $x \ge_{K} y$ if $x-y \in K$ for $x,y \in K$. Does this means that for any $x \in K$, we have $x \ge_{K} 0$ since $x - 0 = x \in K$ ?  EDIT: My definitions of cones and proper cones follow the ones in "" by S. Boyd, in which 0 is contained in cones. On page 53 of the book, there is a statement about generalized inequalities:  $x \le_{K} y$ if and only if $\lambda^T x \le \lambda^T y$ for all $\lambda \ge_{K^*} 0$  where $K^*$ is the dual cone of the proper cone $K$, so $K^*$ is also a proper cone.  The reason I ask this question is that if for any $\lambda \in K^*$, we have $\lambda \ge_{K^*} 0$, why not just using $\lambda \in K^*$ instead of $\lambda \ge_{K^*} 0$ in the above statement?
I am reading Boyd &amp; Vandenberghe's Convex Optimization. The authors use curved greater than or equal to (\succcurlyeq)  $$f(x^*) \succcurlyeq \alpha$$  and curved less than or equal to (\preccurlyeq)  $$f(x^*) \preccurlyeq \alpha$$  Can someone explain what they mean?
parsing NSJSONReadingAllowFragments The question is to show that for any $x,y \in\Re^n$  $||x+y||\cdot||x-y||\le||x||^2+||y||^2$  with equality if and only if $\langle x,y \rangle= 0.$  I have that $||x+y||^2=||x||^2+2 \langle x,y \rangle+||y||^2$ , which helps me prove equality but I'm lost on how to show this and show that the inequality part. I own a Nikon D40X and just the kit lens and I'm considering buying either a macro lens or some extension tubes.  I don't necessarily want to do true 1:1 macro, just some close ups of bugs, fungi and flowers. Is it possible to activate an iPhone 4 without a SIM? So we bought an iPhone 4 from the Apple store. But we didn't get a micro-SIM as there are no PAYG tariffs on micro-SIM available.  Is there anyway to activate the device without getting hold of a SIM? I don't want to sign up to a contract just to develop with. Most of something is plural or singular? I am writing a literature review. My question is which sentence is correct?   Most of the existing work focus on... Most of the existing work focuses on ... Most of the existing studies focus on ...   Thanks I have a question, to aid my understanding, about the proof that differentiability implies continutity.$\mathstrut$    Differentiability Definition  When we say a function is differentiable at $x_0$, we mean that the limit:  $$‎f^{\prime} ‎(x) = \lim_{x\to x_0} \frac{f(x) - f(x_0)}{x-x_0}$$ exists.  Continuity Definition  When we say a function is continuous at $x_0$, we mean that: $$\lim_{x\to x_0} f(x) - f(x_0) = 0$$  Theorem: Differentiability implies Continuity: If $f$ is a differentiable function at $x_0$, then it is continuous at $x_0$.  Proof:  Let us suppose that $f$ is differentiable at $x_0$. Then $$ \lim_{x\to x_0} \frac{f(x) - f(x_0)}{x-x_0} =  ‎f^{\prime} ‎(x) $$  and hence  $$ \lim_{x\to x_0} f(x) - f(x_0) = \lim_{x\to x_0} \left[ \frac{f(x) - f(x_0)}{x-x_0} \right] \cdot \lim_{x\to x_0} (x-x_0) = 0$$  We have therefore shown that, using the definition of continuous, if the function is differentiable at $x_0$, it must also be continuous.  My Question  The proof seems to execute the following steps:   Assume the function is continuous at $x_0$ Show that, with little algebra, we can change this into an equivalent question about differentiability at $x_0$. With this little bit of algebra, we can show that if a function is differentiable at $x_0$ it is also continuous.   What I am slightly unsure about is the apparent circularity. In my mind it seems to say, if a function is continuous, we can show that if it is also differentiable, then it is continuous. Rather than what I was expecting, namely, if a function is differentiable, we can show it must be continuous.  Hopefully my confusion is clear. Any help will be greatly appreciated. I've tried different ways to install iTunes, but to no avail. I tried with Wine, but it didn't work. Is it possible to have iTunes on Ubuntu? Am I Running as a Service I am currently writing a little bootstrap code for a service that can be run in the console. It essentially boils down to calling the OnStart() method instead of using the ServiceBase to start and stop the service (because it doesn't run the application if it isn't installed as a service and makes debugging a nightmare).  Right now I am using Debugger.IsAttached to determine if I should use ServiceBase.Run or [service].OnStart, but I know that isn't the best idea because some times end users want to run the service in a console (to see the output etc. realtime).  Any ideas on how I could determine if the Windows service controller started 'me', or if the user started 'me' in the console? Apparantly  is not the answer. I thought about using commandline args, but that seems 'dirty'.  I could always see about a try-catch statement around ServiceBase.Run, but that seems dirty. Edit: Try catch doesn't work.  I have a solution: putting it up here for all the other interested stackers:      public void Run()     {         if (Debugger.IsAttached || Environment.GetCommandLineArgs().Contains&lt;string&gt;("-console"))         {             RunAllServices();         }         else         {             try             {                 string temp = Console.Title;                 ServiceBase.Run((ServiceBase[])ComponentsToRun);             }             catch             {                 RunAllServices();             }         }     } // void Run      private void RunAllServices()     {         foreach (ConsoleService component in ComponentsToRun)         {             component.Start();         }         WaitForCTRLC();         foreach (ConsoleService component in ComponentsToRun)         {             component.Stop();         }     }   EDIT: There was another question on StackOverflow where the guy had problems with the Environment.CurrentDirectory being "C:\Windows\System32" looks like that may be the answer. I will test today. Motivated by :   Which finite groups are the group of units of some ring? Which finite groups are the group of units of some finite ring? Which finite abelian groups are the group of units of some commutative ring? Which finite abelian groups are the group of units of some finite commutative ring?   It  that even for fields this is not simple to answer.  I expect the general question (Which groups are the group of units of some ring?, with no finiteness hypotheses) is even harder. I have been struggling to install VirtualBox Guest Additions in a Debian Virtual Machine (Debian 7, Debian 8 and Debian 9). Association Bonus not removed upon account deletion Recently a user asked a question on Meta asking about his deleted account (I won't draw attention to him or his question unless it is really necessary but shouldn't be hard to find - at least until the question gets deleted1 as I don't know if it is relevant to this question).  However, while looking at his account, it appears as if he received the association bonus on all of his accounts, including the new Stack Overflow account he just created, but does not have any account with >200 rep.    The most likely scenario is that the deleted account had more than 200 rep and triggered  the association bonus, but that leads to 2 questions:   Why did his previously associated accounts retain the association bonus? Why did he get an association bonus on his brand new account?   Is this a  or ?  1 - The post in question was deleted by the OP several hours ago but before then the account was confirmed by a mod to be deleted due violations of a sock puppety nature (which I think everyone suspected at the time).  Given that the account in question was a sock puppet, I think it highlights an important reason for removing the association bonus when the account that triggered the bonus is deleted.  Theoretically, as long as he keeps getting the association bonus, he can go to any site and already have more than enough privileges to do damage. How do I find out what fonts are used in a document/picture? Is it possible to identify the font used in a specific document/picture?  Answers to this question should identify:   Possible methods to do this (perhaps one answer per method) and adequately describe how to use it (as opposed to merely stating it); Ways of finding the identified fonts, if possible (free or not); and Any prerequisites associated with the method used, if required (for example, "In order to use method X, your document has to be in format Y").   This question is meant as an FAQ, based on an . Its aim is to facilitate the community with the general procedures involved in font identification. Similar cases are solved on a per-usage basis on 's  tag. One of the joys of high-school mathematics is summing a complicated series to get a “closed-form” expression. And of course many of us have tried summing the harmonic series $H_n =\sum \limits_{k \leq n} \frac{1}{k}$, and failed. But should we necessarily fail?      More precisely, is it known that $H_n$ cannot be written in terms of the elementary functions, say, the rational functions, $\exp(x)$ and $\ln x$? If so, how is such a theorem proved?    Note. When I started writing the question, I was going to ask if it is known that the harmonic function cannot be represented simply as a rational function? But this is easy to see, since $H_n$ grows like $\ln n+O(1)$, whereas no rational function grows logarithmically.  Added note: This  asks a similar question for “elementary integration”. I guess I am asking if there is an analogous theory of “elementary summation”. Can't resolve issue: iptables: No chain/target/match by that name Debian users have a kind of "wayback machine":  , which is really great.  There are regressions sometimes, and if I don't have local deb package with old version, I'm getting sad sometimes. Having a wayback machine like this is very helpful.  I can't find analogue of it for Ubuntu. Have I missed it? Simple example of non-arithmetic ring (non-distributive ideal lattice) Can anyone provide a simple concrete example of a non-arithmetic commutative and unitary ring (i.e., a commutative and unitary ring in which the lattice of ideals is non-distributive)? How do I get a list of all files (and directories) in a given directory in Python? I am trying to add jQuery to my page (it is not being added by default). The file is adding to the page but after drupal.js causing the error:  SCRIPT5009: 'jQuery' is undefined  drupal.js, line 5 character 1    The code that adds drupal.js to the page in common.inc  ... 'misc/drupal.js' =&gt; array(           'data' =&gt; 'misc/drupal.js',           'type' =&gt; 'file',           'scope' =&gt; 'header',           'group' =&gt; JS_LIBRARY,           'every_page' =&gt; TRUE,           'weight' =&gt; -1,           'preprocess' =&gt; TRUE,           'cache' =&gt; TRUE,           'defer' =&gt; FALSE,         ), ...   In my module hook_init I have:    drupal_add_js('misc/jquery.js', array(     'group' =&gt; JS_LIBRARY,     'weight' =&gt; -20,     'cache' =&gt; TRUE,   ));   I would think with a lower weight it would load before drupal.js?  The header of the page contains the following scripts:  &lt;script type="text/javascript" src="http://mysite.com/misc/drupal.js?mqdxrf"&gt;&lt;/script&gt; &lt;script type="text/javascript" src="http://mysite.com/misc/jquery.js?mqdxrf"&gt;&lt;/script&gt; &lt;script type="text/javascript" src="http://mysite.com/misc/jquery.once.js?mqdxrf"&gt;&lt;/script&gt; &lt;script type="text/javascript" src="http://mysite.com/sites/default/files/languages/nl_c9aC92_wtuULa79-hTFcGPM53oqk1yKwciOa_Q3yO28.js?mqdxrf"&gt;&lt;/script&gt; &lt;script type="text/javascript" src="http://mysite.com/sites/all/modules/lang_dropdown/msdropdown/jquery.dd.js?mqdxrf"&gt;&lt;/script&gt; &lt;script type="text/javascript" src="http://mysite.com/sites/all/modules/lang_dropdown/lang_dropdown.js?mqdxrf"&gt;&lt;/script&gt; &lt;script type="text/javascript" src="http://mysite.com/sites/all/modules/devel/devel_krumo_path.js?mqdxrf"&gt;&lt;/script&gt; How to retrieve all keys (or values) from a std::map and put them into a vector? This is one of the possible ways I come out:  struct RetrieveKey {     template &lt;typename T&gt;     typename T::first_type operator()(T keyValuePair) const     {         return keyValuePair.first;     } };  map&lt;int, int&gt; m; vector&lt;int&gt; keys;  // Retrieve all keys transform(m.begin(), m.end(), back_inserter(keys), RetrieveKey());  // Dump all keys copy(keys.begin(), keys.end(), ostream_iterator&lt;int&gt;(cout, "\n"));   Of course, we can also retrieve all values from the map by defining another functor RetrieveValues.  Is there any other way to achieve this easily? (I'm always wondering why std::map does not include a member function for us to do so.) Is there any difference between “which” and “that”?