I'm using phpmailer to sending emails to users,but the email account which i send with to users,it wasn't created on the same hosting,it created on Office 365, I used SMTP authentication,when i sent emails to users they considered as spams_for example hotmail say :This sender failed our fraud detection checks and may not be who they appear to be .  when i google the problem i found the following issues:  1) Email failed in SPF and DKIM checks: but on cpanel in email authorization,i make SPF and DKIM enabled, so as my hosting company mentioned "that help reduce the amount of spam." . Do i need to setup SPF and DKIM records manually at zone file?   2) My Hosting company advise to setup reverse DNS:but does this important for my problem?   Sending code :   $mail = new PHPMailer(true);  $mail-&gt;Host = "smtp.office365.com";  $mail-&gt;Port       = 587;  $mail-&gt;SMTPSecure = '';  $mail-&gt;SMTPAuth   = true;  $mail-&gt;Username = "xxxxxx";  $mail-&gt;Password = "xxxx";  $mail-&gt;SetFrom($from, 'Name');   $mail-&gt;addReplyTo($from);  $mail-&gt;SMTPDebug  = 2;  $mail-&gt;SingleTo = true ;  $mail-&gt;clearAllRecipients();   for($i = 0 ; $i &lt; count($to) ; $i++)  {       $mail-&gt;addAddress($to[$i]);   }     if(!is_null($cc))   {       foreach ($cc as $email)       {           $mail-&gt;addCC($email);       }   }    $mail-&gt;Subject = $subject;  $mail-&gt;IsHTML(true);  $mail-&gt;MsgHTML($body);    if(!$mail-&gt;send()) {      //echo 'Message could not be sent.';      echo 'Mailer Error: ' . $mail-&gt;ErrorInfo;      //  return false ;  } else {       return true ;  }   So what do you advise me?
This is a tricky one and I've always relied on techniques, such as permission-based emails (i.e. only sending to people you have permission to send to) and not using blatantly spamish terminology.  Of late, some of the emails I send out programmatically have started being shuffled into people's spam folder automatically and I'm wondering what I can do about it.  This is despite the fact that these particular emails are not ones that humans would mark as spam, specifically, they are emails that contain license keys that people have paid good money for, so I don't think they're going to consider them spam  I figure this is a big topic in which I am essentially an ignorant simpleton.
Infinitely number of primes in the form $4n+1$ proof Question: Are there infinitely many primes of the form $4n+3$ and $4n+1$?  My attempt: Suppose the contrary that there exist finitely many primes of the form $4n+3$, say $k+1$ of them: $3,p_1,p_2,....,p_k$  Consider $N = 4p_1p_2p_3...p_k+3$, $N$ cannot be a prime of this form. So suppose that $N$=$q_1...q_r$, where $q_i∈P$  Claim: At least one of the $q_i$'s is of the form $4n+3$:  Proof for my claim: $N$ is odd $\Rightarrow q_1,...,q_r$ are odd $\Rightarrow q_i \equiv 1\ (\text{mod }4)$ or $q_i ≡ 3\ (\text{mod }4)$  If all $q_1,...q_r$ are of the form $4n+1$, then $(4n+1)(4m+1)=16nm+4n+4m+1 = 4(\cdots) +1$  Therefore, $N=q_1...q_r = 4m+1$. But $N=4p_1..p_k+3$, i.e. $N≡3\ (\text{mod }4)$, $N$ is congruent to $1\ \text{mod }4$ which is a contradiction.  Therefore, at least one of $q_i \equiv 3\ (\text{mod }4)$. Suppose $q_j\equiv 3\ (\text{mod }4)$  $\Rightarrow$ $q_j=p_i$ for some $1\leq i \leq k$ or $q_j =3$  If $q_j=p_i≠3$ then $q_j$ | $N = 4p_1...p_k + 3 \Rightarrow q_j=3$ Contradiction!  If $q_j=3$ ($\neq p_i$, $1\leq i \leq k$) then $q_j | N = 4 p_1...p_k + 3 \Rightarrow q_j=p_t$ for some $1 \leq i \leq k$ Contradiction!  In fact, there must be also infinitely many primes of the form $4n+1$ (according to my search), but the above method does not work for its proof. I could not understand why it does not work. Could you please show me?  Regards Socket Tester all indicators light up I used the above GFCI/RCD (up to 250v) socket tester on the 240v black and red live wire of the US split phase ac power system. When I connect the ground, all indicator lights (3 red) got on indicating "Live/Grd Reverse, Missing GRD" although it is connected properly.  What is inside the socket tester? Does it have a live and neutral leg such that it should only be used for ac system with polarity? How about if you use it on the red and black 240v phase to phase? Why would all the indicators light become on (3 red lights)?  The following is the specs of it:  Specifications: Brand Name: HABOTEST Color: Black Material: Flame Retardant ABS Plug Type: US Voltage: 48V-250V 45-65Hz RCD Test: >30mA (EU/UK Plug) RCD Working Voltage: 220V±20V GFCI Test: >5mA (US Plug) GFCI Working Voltage: 110V±20V Voltage Measurement: 48V~250V / 45~65Hz    Accuracy: ±(2.0%+2) Working Environment: 0°C ~ 40°C, 20% ~75%RH Storage Environment: -10°C ~ 50°C, 20% ~80%RH Safety Rating: EN61010-1,-2-030, EN61326-1,CAT II 300V Item Size: 62 * 65 * 55mm / 2.4 * 2.6 * 2.2in Item Weight: UK 64g/2.3oz, EU 59g/2.1oz, US 53g / 1.9oz Package Size: 7.5 * 7.4 * 7.3cm / 3.0 * 2.9 * 2.9in Package Weight: UK 90g/3.2oz, EU 86g/3.0oz, US 79g / 2.8oz View progress towards Announcer, Booster, and Publicist badges I shared a link to a question and now I am wondering whether I am almost going to get an announcer badge.  Can I somehow see which links I shared and how many clicks were on it for the last days?  Update:  Maybe this would be a nice feature. Next to any question you can see who has promoted it the most (number of fresh hits from a referer link) and also the amount of new ip's through search engine's and other referring sites.  Note the progress feature should include the Announcer, Booster, and Publicist badges. Are all integers fractions? Why should we typedef a struct so often in C? I have seen many programs consisting of structures like the one below  typedef struct  {     int i;     char k; } elem;  elem user;   Why is it needed so often? Any specific reason or applicable area? How do camera body motors compare to in-lens motors for focusing? What is the difference between using a motor on the camera body and the motor built into lenses?   Aside from how both compare in terms of speed, what differences would I notice if I use a body with a motor and motor-less lens, and a lens with motor on a body that lacked one?  A comparison chart outlining the various lens motor types would be a great help as high end USM motors for Sigma lenses are faster then their regular versions. I'm looking for a story that was probably written somewhere from the 1940s to the 1960s. I think I read it in an anthology, possibly in the 1970s.  A group of human explorers landed on a planet with primitive aliens, living in hunter-gatherer villages. They hoped to make a profit from trading our goods for—well, I'm not clear on what they wanted from the aliens.  The native people learned the human language before the explorers could learn theirs. Then the natives suddenly started producing technological innovations far beyond what they originally used. I remember that wooden sailing ships suddenly replaced their canoes. All the natives had to do was hear a story once from the explorers before copying the technology in the story.  Finally, the explorers realized that the natives were much, much smarter than us and that Earth faced a serious problem.  I'm fairly certain that it wasn't a Heinlein story, but I remember that it had some qualities similar to his work. Tentative definitions in C and linking How to prevent negative comments/downvoted being deleted by re-posting answers Are nested functions a bad thing in gcc ? disable comments if array exists This is the code which is registering the function for rewrite_endpoints():   function wpa121567_rewrite_endpoints(){  add_rewrite_endpoint( 'comments', EP_PERMALINK );  add_rewrite_endpoint( 'stats', EP_PERMALINK );  }  add_action( 'init', 'wpa121567_rewrite_endpoints' );   NOTE  I don't know if this matters but I will write it.   I have single-CPT.php which is calling  /inc/post-format/single-CPT.php like this:      &lt;?php while ( have_posts() ) : the_post(); ?&gt;  &lt;?php   get_template_part( 'inc/post-format/single-debate');      ...   This is the code which is checking if array key exists, which is  inside my /inc/post-format/single-cpt.php:   ?php if( array_key_exists( 'comments', $wp_query-&gt;query_vars ) ){  the_field('comments_section');   } elseif( array_key_exists( 'stats', $wp_query-&gt;query_vars ) ) {   the_field('stats_section');   } else { // the request is for the main post  }   ?&gt;   This is my comments.php :       &lt;div id="comments" class="comments-area"&gt; &lt;?php // If CPT and not logged in, display a message: if ( 'debate' == get_post_type() &amp;&amp; ! is_user_logged_in() ) { echo '&lt;p class="must-log-in" style="padding-left:20px; font-size:20px;"&gt;You must be logged in to post a comment.' . '&lt;/p&gt;';  echo do_shortcode("[upme_login]");   } ?&gt;     &lt;?php if ( have_comments() ) : ?&gt;         &lt;h3 class="comments-title"&gt;              &lt;?php                 printf( _n('%d comment', '%d comments', get_comments_number(), 'outbox' ),                     number_format_i18n( get_comments_number() ) );             ?&gt;         &lt;/h3&gt;             &lt;ol class="commentlist"&gt;              &lt;?php                 wp_list_comments( array( 'callback' =&gt; 'outbox_comment' ) );             ?&gt;         &lt;/ol&gt;&lt;!-- .commentlist --&gt;          &lt;?php if ( get_comment_pages_count() &gt; 1 &amp;&amp; get_option( 'page_comments' ) ) : // are there comments to navigate through? If so, show navigation ?&gt;         &lt;nav role="navigation" id="comment-nav-below" class="site-navigation comment-navigation clearfix"&gt;             &lt;div class="nav-previous"&gt;&lt;i class="icon-left-open-1"&gt;&lt;/i&gt;&amp;nbsp;&lt;?php echo get_previous_comments_link( __( 'Older Comments', 'outbox' ) ); ?&gt;&lt;/div&gt;             &lt;div class="nav-next"&gt;&lt;?php echo get_next_comments_link( __( 'Newer Comments', 'outbox' ) ); ?&gt;&amp;nbsp;&lt;i class="icon-right-open-1"&gt;&lt;/i&gt;&lt;/div&gt;         &lt;/nav&gt;&lt;!-- #comment-nav-below .site-navigation .comment-navigation --&gt;         &lt;?php endif; ?&gt;      &lt;?php endif; // have_comments() ?&gt;      &lt;?php         // If comments are closed and there are comments, let's leave a little note, shall we?         if ( ! comments_open() &amp;&amp; '0' != get_comments_number() &amp;&amp; post_type_supports( get_post_type(), 'comments' ) ) :     ?&gt;         &lt;p class="nocomments"&gt;&lt;?php _e( 'Comments are closed.', 'outbox' ); ?&gt;&lt;/p&gt;     &lt;?php endif; ?&gt;     &lt;?php  // Don't output the comment form if CPT and user isn't logged in if ( 'debate' != get_post_type() || is_user_logged_in() ) {  comment_form(); }  ?&gt;  &lt;/div&gt;&lt;!-- #comments .comments-area --&gt; &lt;?php } ?&gt;    This is how the comments template is included  inside my single-CPT.php :  if ( comments_open()  )  comments_template( '', true );   I want to disable comments if if( array_key_exists( 'comments', $wp_query-&gt;query_vars ) ){  How can I do this? Foreign key constraint on array member? If $f \in \mathcal{L}^1(\mathbb{R})$ then $\lim_{h\rightarrow 0}\int_{-\infty}^\infty |f(x+h)-f(x)|dx\rightarrow 0$ I am working on some past prelim problems and one of them is the following: If $f \in \mathcal{L}^1(\mathbb{R})$ then $\lim_{h\rightarrow 0}\int_{-\infty}^\infty |f(x+h)-f(x)|dx\rightarrow 0$   Update: Following Jackie Chong's advice, I am attempting this problem with the dominated convergence theorem. Since $f\in L^1(\mathbb{R})$, $f$ can be approximated from below by a sequence of continuous functions $f_n$. Then, I believe I can say that, for all $n$ sufficiently large, $|f_n(x+\frac{1}{n})-f_n(x)|\le 2|f(x)|$. (I'm not sure of this part thought). Then $2|f(x)|$ is integrable and hence a dominating function for $|f_n(x+\frac{1}{n})-f_n(x)|$. So by the dominated convergence theorem we can take the limit inside $\lim_{n\rightarrow \infty}\int|f_n(x+\frac{1}{n})-f(x)|dx=\int \lim_{n\rightarrow \infty}|f_n(x+\frac{1}{n})-f(x)|dx=0$ Is this on the right track?  Another approach that I thought may work, but which I'm less certain of is the following: If $f$ can be approximated by $C^1$ functions (I have no real justification for this, so please correct me if its wrong), then for each $n$ and each $x$ there is some $y(x,n)\in B_{1/n}(x)$ such that $|f(x+\frac{1}{n}-f(x)|=\frac{1}{n}|f'(y(x,n)|$ but from here I don't see how to proceed, because just because $|f|$ is integrable, I can't justify $|f'|$ being integrable, but if there is some way to fix this approach, hints would be appreciated.   Another thing I was thinking was to first approximate $f$ from below by continuous functions $f_n$ then to define $g_{m,n}=|f_n(x+\frac{1}{m})-f_n(x)| \chi_{[-m,m]}$. Then, for each $n$, the function $g_{m,n}$ is continuous on its support, and since it support is compact, it is uniformly continuous there. Can this be used to justify bringing the limit $\lim_{n\rightarrow \infty}$ inside the integral? Then apply the monotone convergence theorem? I know this idea is probably even more sketchy than my previous one, but if it can be fixed please let me know :)  Thanks! $-1$ is a quadratic residue modulo $p$ if and only if $p\equiv 1\pmod{4}$ .NET offers a generic list container whose performance is almost identical (see Performance of Arrays vs. Lists question). However they are quite different in initialization.   Arrays are very easy to initialize with a default value, and by definition they already have certain size:  string[] Ar = new string[10];   Which allows one to safely assign random items, say:  Ar[5]="hello";   with list things are more tricky. I can see two ways of doing the same initialization, neither of which is what you would call elegant:  List&lt;string&gt; L = new List&lt;string&gt;(10); for (int i=0;i&lt;10;i++) L.Add(null);   or  string[] Ar = new string[10]; List&lt;string&gt; L = new List&lt;string&gt;(Ar);   What would be a cleaner way?  EDIT: The answers so far refer to capacity, which is something else than pre-populating a list. For example, on a list just created with a capacity of 10, one cannot do L[2]="somevalue"  EDIT 2: People wonder why I want to use lists this way, as it is not the way they are intended to be used. I can see two reasons:   One could quite convincingly argue that lists are the "next generation" arrays, adding flexibility with almost no penalty. Therefore one should use them by default. I'm pointing out they might not be as easy to initialize. What I'm currently writing is a base class offering default functionality as part of a bigger framework. In the default functionality I offer, the size of the List is known in advanced and therefore I could have used an array. However, I want to offer any base class the chance to dynamically extend it and therefore I opt for a list. For each of the following, give an example of a sequence of functions $f_n$ that converges to f  A. uniformly but not in the mean square sense.  B. in the mean square sense but pointwise nowhere.  I know that for part A the domain of the function cannot be bounded but other then that I'm not sure how to go about the problem Why is RSA encryption significantly faster than decryption? I am designing an implementation of RSA . I recorded computation times in Java by using System.currentTimeMillis(). It returned an encryption time of 0.05 ms and decryption time of 0.55 ms, or a ratio of 1:11.   This seems like a large difference. Is it supposed to be this way?  //here my key has 256 bits  for (;;) {             long begin = System.currentTimeMillis();              for (int i = 0; i &lt; num; i++) {              decrypt();             }              long end = System.currentTimeMillis();              long time = end - begin;              if (time &gt;= 10000) {                 System.out.printf("Average Encryption takes: %.2f ms\n",                         (double) time / num);                 break;             }              num *= 2;         }  p = BigInteger.probablePrime(128, random); q = BigInteger.probablePrime(128, random); N=(p.subtract(one)).multiply(q.subtract(one)); e=BigInteger.probablePrime(32, random); d=e.modInverse(N); private void encrypt() {  C= M.modPow(e,N);  }  private void decrypt() {     RM = C.modPow(d, N);     } Help with math symbols in BibTeX Use 'import module' or 'from module import'? Unlock by fingerprint only