Let $P$ denotes the property that if there exists a surjection from set $A$ to set $B$, then there exists an injection from $B$ to $A$. It's apparent that $P$ can be proved in ZFC. My question is that if $P$ and AC are equivalent in the ZF theory?
For sets $A,B$, let $|A|\leq^*|B|$ say that there exists an onto map $f:B\rightarrow A$ or $A=\emptyset$.   My question is, is $$\forall A,B(|A|\leq^*|B|\longrightarrow |A|\leq|B|)$$ equivalent to the axiom of choice?  Thanks
"invalid use of incomplete type" error with partial template specialization The following code:  template &lt;typename S, typename T&gt; struct foo {    void bar(); };  template &lt;typename T&gt; void foo &lt;int, T&gt;::bar() { }   gives me the error  invalid use of incomplete type 'struct foo&lt;int, T&gt;' declaration of 'struct foo&lt;int, T&gt;'   (I'm using gcc.) Is my syntax for partial specialization wrong? Note that if I remove the second argument:  template &lt;typename S&gt; struct foo {    void bar(); };  template &lt;&gt; void foo &lt;int&gt;::bar() { }   then it compiles correctly. Why isn't Derivative equal to NeumannValue? Script for display wrapper class Account list of records? Hi i am trying to write script for display list of records from wrapper class.   @remoteaction     public static wrappercls getaccounts(){         string account = 'select id,name,accountnumber from account';         listacc = database.query(account);         count = listacc.size();         system.debug('#####'+count);         wrappercls wra = new wrappercls(count,listacc);         return wra;     }      public class wrappercls{         public integer count11;         public list&lt;account&gt; account;          public wrappercls(integer count11,list&lt;account&gt; account){             this.count11 = count11;             this.account = account;         }     }    This class returns list of account records and record count parameter.Need to display these records in page.Is it possible if possible Please help me on this..  Script  &lt;script&gt;   function getallaccounts(){     exremote.getaccounts(         function(result,event){             if(event.status){              }             else {              }        },{escape:true}     ); } &lt;/script&gt;   Thanks in Advance... I tried to reproduce the results from the  unfortunately the video link is broken. And I might have overlooked some related settings.  The spot light is barley visible on the ground.    Increasing the Density of the Volume Scatter Node darkens the sky texture but leads to some light cone from the car's headlight. (only 10 samples in preview).    Which settings do I need to change or should I try a different approach?  Integrator settings are default and spotlight settings were copied from the example found in the docs.  Related:    (pre 2.7)  (BI only) When I make an animation, it always starts out slow, then speeds up, then slows down at the end.  I cannot figure out how to fix this. Let $\int_{- \infty}^{\infty} f(x) dx =1$. Then show that $ \int_{- \infty}^{\infty} \frac{1}{1+ f(x)} dx = \infty.$ Maximum number of files in one ext3 directory while still getting acceptable performance? I have an application writing to an ext3 directory which over time has grown to roughly three million files. Needless to say, reading the file listing of this directory is unbearably slow.  I don't blame ext3. The proper solution would have been to let the application code write to sub-directories such as ./a/b/c/abc.ext rather than using only ./abc.ext.  I'm changing to such a sub-directory structure and my question is simply: roughly how many files should I expect to store in one ext3 directory while still getting acceptable performance? What's your experience?  Or in other words; assuming that I need to store three million files in the structure, how many levels deep should the ./a/b/c/abc.ext structure be?  Obviously this is a question that cannot be answered exactly, but I'm looking for a ball park estimate. I have been reading Stack Exchange for years - mostly not logged in, and have never posted (this is my first). I'm simply not very talkative, and very good at googling.  Recently a few occasions have come up when I've wanted to comment on a question.  I wanted either to ask for, or provide, clarification which did not warrant a new answer.  How do I gain enough reputation to comment without:   asking questions which have already been answered or that I can figure out on my own, or posting my comment like as an answer (i.e. being  rude)? Can a dual national visit USA under VWP after visa refusal for other nationality? Recently I was refused an E-2 visa under Section 214b. To be honest I don't really know why because I have strong ties with my home country (house, bank account, family) so my lawyer sent an appeal to the legalnet and I'm  waiting for an answer.  I have a second nationality that I used to travel in USA within the last 15 months under ESTA. A few days after my E-2 refusal I checked my ESTA status  and it is still authorized.  Can I travel to the States right now or not?  I have a purchased business over there and have to take care of my assets. tikz relative coordinates Suppose one defines coordinates a and b. One may draw a line connecting these two coordinates (\draw (a) -- (b)). Suppose one desires to draw a line that is shifted upward 1 unit relative to a and b. A first guess is the command \draw (a){}+(0,1) -- (b){}+(0,1). This command, however, connects the point a+(0,1) to the point b, presumably because addition is subsequent to the draw command. How does one obtain the desired result (drawing from a+(0,1) to b+(0,1)?  MWE:  \documentclass{article} \usepackage{tikz} \usetikzlibrary{calc} \begin{document} \begin{tikzpicture}  \draw[fill] (0,0) circle (2pt) coordinate (a);  \draw[fill] (5,0) circle (2pt) coordinate (b);  \draw (a) -- (b);  \draw (a){}+(0,1) -- (b){}+(0,1); \end{tikzpicture} \end{document}     Edit: How does one draw coordinates at a point shifted relative to the arithmetic average of the coordinates of two points (something akin to \node at ($(((s1)+(s2))/2)+(0,1)$)). When are attributive nouns plural? Sorry for the title, it is not very evident and intuitive but I really do not how to tell it better...  Well, you know, several times, or better, many times, we use this form:  If I want to say:      "development of special weapons was   the first point in Hitler's   program..."   I will say this (a better form):      "special weapon development was..."   OK...  is it      "special weapon development"   or      "special weapons development" (note   the plural...)   what's the grammar rule in order to understand how to use this very used form? I'm not that into dynamic programming languages but I've written my fair share of JavaScript code. I never really got my head around this prototype-based programming, does any one know how this works? var obj = new Object(); obj.prototype.test = function() { alert('Hello?'); }; var obj2 = new obj(); obj2.test();  I remember a lot discussion I had with people a while back (I'm not exactly sure what I'm doing) but as I understand it, there's no concept of a class. It's just an object, and instances of those objects are clones of the original, right? But what is the exact purpose of this &quot;.prototype&quot; property in JavaScript? How does it relate to instantiating objects? Update: correct way var obj = new Object(); // not a functional object obj.prototype.test = function() { alert('Hello?'); }; // this is wrong!  function MyObject() {} // a first class functional object MyObject.prototype.test = function() { alert('OK'); } // OK  Also these  really helped a lot. Are there uncountably many surjections from $\mathbb N$ to $\mathbb N$? Statement: There are uncountably many surjections from $\mathbb N$ to $\mathbb N$.  Is the following proof valid:  The real numbers are uncountable.   There exists a bijection between the set of real numbers and the set of all functions from $\mathbb N$ to $\{0,1\}$.   Given a function, $f$, from $\mathbb N$ to $\{0,1\}$, construct a surjection from $\mathbb N$ to $\mathbb N$, denoted $g$, as follows:  We map the $n$th composite number to $n$, and the nth prime, to $f(n)$.   Clearly this is a surjection since the composite numbers are countable. Clearly different functions $f$ lead to different surjections $g$. Since there are uncountably many $f$s, there are uncountably many surjections $g$. I had some NEF files in a folder that I didn't do anything with. When importing them in Lightroom, no previews and a "file unsupported or damaged message" So I tried with Nikons ViewNX, nothing. Even not in DxO or PhotoMechanic.  I had a look with EXIFTool and the corrupt file prints me only this:  ExifTool Version Number         : 8.39 File Name                       : _DSC5559.NEF Directory                       : /Users/tunafish/Desktop File Size                       : 12 MB File Modification Date/Time     : 2010:11:29 12:46:56+01:00 File Permissions                : rwxrwxrwx Error                           : File format error   I saw another question on photo.stack  and tried to copy over the makernotes from a good file (_DSC5556.NEF) with  exiftool -tagsfromfile _DSC5556.NEF -makernotes _DSC5559.NEF   but EXIFTool gives me this error  Error: Not a valid TIFF - _DSC5559.NEF 0 image files updated 1 files weren't updated due to errors   I have uploaded 2 NEF's, with _DSC5559.NEF being the corrupt one here: How to remove duplicate values from a multi-dimensional array in PHP How can I remove duplicate values from a multi-dimensional array in PHP?  Example array:  Array (     [0] =&gt; Array     (         [0] =&gt; abc         [1] =&gt; def     )      [1] =&gt; Array     (         [0] =&gt; ghi         [1] =&gt; jkl     )      [2] =&gt; Array     (         [0] =&gt; mno         [1] =&gt; pql     )      [3] =&gt; Array     (         [0] =&gt; abc         [1] =&gt; def     )      [4] =&gt; Array     (         [0] =&gt; ghi         [1] =&gt; jkl     )      [5] =&gt; Array     (         [0] =&gt; mno         [1] =&gt; pql     )  ) gdal_translate- why jpg compressed tif is 2 times greater than jpg file? Are research survey questions expected to be handled with close votes? Given a matrix $A \in R^{n \times n}$ which is normal ($AA^H=A^HA$ where $A^H$ is hermitian of $A$) and nilpotent ($A^k=0$ for some $k$). Now we need to show that $A=0$. (This is essentially exercise 5(b) in sec. 80 on p.162 of Paul R. Halmos' Finite-Dimensional Vector Spaces.) I tried to show in the following way, we know that, $AA^H=A^HA$ pre-multiply by $A^{k-1}\implies A^kA^H=A^{k-1}A^HA$ Now, we have $0 = A^{k-1}A^HA$, since $A$ nilpotent. I am not sure how to proceed from here to show $A=0$. Can someone help me in this problem? How do I alter a language item link? I am looking to modify the default output of Drupal 8 of links in a certain situation, the language switcher. Ideally I want to be able to do that in a twig template and not have to produce a  that overwwrites that whole link generation system and add a BIG if for the cases I need to overwrite.  I think the output layout should be controllable in the twig template similar to  (this answer outputs the same url for every languages)  I am using bootstrap as a base theme so i'd like to use the bootstrap markup for my links (btn btn-primary, dropdown)  Here is my code I am trying to make links--language-block.html.twig  {%- if links|length == 2 -%}     {# show only alternate language button #}     {%- for key, item in links -%}         {%- if not item.attributes['#options']['set_active_class'] -%} {# &lt;--- this is always true!? #}             &lt;a href="{{ item.link['#url'] }}" class="btn btn-primary"&gt;{{ item.link['#title'] }}&lt;/a&gt; {# #url is always the same what ever the language it is pointing to! #}         {%- endif -%}     {%- endfor -%} {%- elseif links|length &gt; 2  -%}     {# show selected language in button and other languages in drop down #}     &lt;button class="btn btn-primary dropdown-toggle" type="button" data-toggle="dropdown"&gt;{{ selectedLanguage }}&lt;span class="caret"&gt;&lt;/span&gt;&lt;/button&gt;     &lt;ul class="dropdown-menu"&gt;         {% for key, item in links %}             {%- if not item.attributes['#options']['set_active_class'] -%} {# &lt;--- this is always true!? #}             &lt;li&gt;&lt;a href="{{ item.link['#url'] }}"&gt;{{ item.link['#title'] }}&lt;/a&gt;&lt;/li&gt;             {% endif %}         {% endfor %}     &lt;/ul&gt; {%- endif -%}   Anyone has an idea on how to do this? 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