Is it possible to add new functions in the class if the implementation of this class is hidden to user. But still if its required to add a new function to this class.
Backgrounder:  The  (Pointer to IMPLementation) is a technique for implementation hiding in which a public class wraps a structure or class that cannot be seen outside the library the public class is part of.  This hides internal implementation details and data from the user of the library.  When implementing this idiom why would you place the public methods on the pimpl class and not the public class since the public classes method implementations would be compiled into the library and the user only has the header file?  To illustrate, this code puts the Purr() implementation on the impl class and wraps it as well.  Why not implement Purr directly on the public class?  // header file: class Cat {     private:         class CatImpl;  // Not defined here         CatImpl *cat_;  // Handle      public:         Cat();            // Constructor         ~Cat();           // Destructor         // Other operations...         Purr(); };   // CPP file: #include "cat.h"  class Cat::CatImpl {     Purr(); ...     // The actual implementation can be anything };  Cat::Cat() {     cat_ = new CatImpl; }  Cat::~Cat() {     delete cat_; }  Cat::Purr(){ cat_-&gt;Purr(); } CatImpl::Purr(){    printf("purrrrrr"); }
Could gravity accelerate light? Gravity causes anything with energy to accelerate toward the source. Black holes, for example, have such strong gravity that they pull in light and don't let any escape. But can acceleration still apply to light? The speed of light is constant, of course, but why are photons affected by gravity yet aren't accelerated by it?  Edit: My main question is why photons aren't affected in the same way as most other particles. I'm perfectly aware that it cannot surpass lightspeed, but I want to know what makes it unaffected by acceleration while other particles are affected. Let $A$ be a diagonalizable matrix and $\lambda$ be an eigenvalue of A. Prove that rank($\lambda I - A$) = rank(($\lambda I - A)^2$). How can super massive black holes have a lower density than water? I heard on a podcast recently that the  at the centre of some galaxies could have densities less than water, so in theory, they could float on the substance they were gobbling up... can someone explain how something with such mass could float?  Please see the below link for the podcast in question: I've got a custom loop that I'm using to display some Real Estate listings that will be available within 60 days. I'm calling it with the following function:   &lt;?php  $sixtydays = date('Y/m/d', strtotime('+60 days')); $paged = (get_query_var('paged')) ? get_query_var('paged') : 1; $query = new PostsOrderedByMetaQuery(array(   'post_type' =&gt; array('post', 'real-estate'),   'meta_key' =&gt; 'Time Available',   'meta_compare' =&gt; '&lt;=',   'meta_value' =&gt; $sixtydays,   'paged' =&gt; $paged,   'orderby_meta_key' =&gt; 'Price',   'orderby_order'    =&gt; 'ASC' )); ?&gt; &lt;?php while ($query-&gt;have_posts()) : $query-&gt;the_post(); ?&gt;   While the loop works great, I can't get it to paginate. It shows the first 10 (my default) posts but doesn't show the pagination. The only way to display all posts is to show them on one page by adding 'posts_per_page' =&gt; -1, I have similar loops on other pages that have no problem paginating. The only difference with this one is that there are two meta keys that are filtering the posts.   I'm using WP Page Navi for this and the rest of my pages. I'm closing the loop and adding the pagination using the following code:   &lt;?php endwhile; // End the loop. Whew. ?&gt; &lt;?php wp_pagenavi(); ?&gt; &lt;?php wp_reset_query(); ?&gt;   How can I go about fixing this? How can I make a .jar file executable? Suppose $a_1, a_2,\dots , a_n$ are $n$ positive real numbers with $a_1a_2 \dots a_n = 1$.   Then what is the minimum value of $(1 + a_1)(1 + a_2). . .(1 + a_n)$ ?    I think $(1 + a_1)(1 + a_2). . .(1 + a_n)$ takes its minimum value when $a_1=a_2=\dots=a_n=1$ and thus the minimum value is $2^n$.  I don't know how to prove it. Please help. I am in the process of completing some homework and have come across a question that says: "What is the steady-state current?" It is asking in reference to a circuit that includes two capacitors in parrallel and a DC source of 24V and an AC source given by $\epsilon = 20 \cos(120 \pi t)\ \text{V}$, where $t$ is in seconds. I assume that it is asking when the current is constant. Because naturally the current is not constant due to the AC source, would I just consider the DC source for this? I have a third party php library which I would like to use in a module. How do I do that in Drupal 8? In drupal 7, I have done things like the following code, but libraries_get_path() doesn't exist in Drupal 8, and I'm not exactly sure how things change, not that we use namespaces (of which I am a beginner).  require(libraries_get_path('twitter-api-php') . '/TwitterAPIExchange.php');   And, where should I place my library? Should I create a new libraries directory and put it in the root, just like modules and themes are now in the root? How can I use function's parameter in it? Python 3 with QGIS on Mac? Why does correlation matrix need to be positive semi-definite and what does it mean to be or not to be positive semi-definite? I have been researching the meaning of positive semi-definite property of correlation or covariance matrices.   I am looking for any information on   Definition of positive semi-definiteness; Its important properties, practical implications; The consequence of having negative determinant, impact on multivariate analysis or simulation results etc. Is there a way to preserve a weapon in Magicka within a saved multiplayer game Limits of Poisson's ratio in isotropic solid Why does Bash's source not need the execution bit? Determinant of circulant matrix Find the determinant of the following matrix in the terms of $a_1,a_2,\cdots,a_n$ explicitly, $$ \begin{bmatrix} a_1 &amp; a_2 &amp; a_3 &amp; \cdots &amp; a_n\\ a_2 &amp; a_3 &amp; a_4 &amp; \cdots &amp; a_1\\ a_3 &amp; a_4 &amp; a_5 &amp; \cdots &amp; a_2\\ \vdots &amp; \vdots &amp; \vdots &amp; \ddots &amp; \vdots\\ a_n &amp; a_1 &amp; a_2 &amp; \cdots &amp; a_{n-1}\\ \end{bmatrix} $$ When the determinant is zero? Conditional Expectation of Bernoulli R.V. Let $X_1, X_2,\ldots, X_n$ be iid bernoulli r.v. with parameter $p$. Let $S=X_1+\cdots+X_n$ and $Y=X_1X_2$. Compute $\mathbb{E}(Y\mid S)$.  I know that $\mathbb{E}(X_1\mid S) = S/n$. So If I could split $\mathbb{E}(Y\mid S)$ into two then I would be done. But I don't think that is allowed. How to proceed? Is using only HTTPS bad in a SEO perspective? Show $100+t$ is reducible in $\mathbb{Z}[[t]]$ $\mathbb{Z}[[t]]$ is the ring of formal power series with integer coefficients, and the problem asks to show that $100+t$ is reducible. This means I need to find two power series, $\sum_{i=0}^\infty a_i t^i$ and $\sum_{j=0}^\infty b_j t^j$ with $a_i,b_j\in\mathbb{Z}$, that when multiplied give $100+t$. Since the integers are an integral domain, I know that neither of these power series may be finite, but I'm having a really hard time figuring out how to come up with two series that will telescope properly. Any suggestions? Show a connected metric space (X,d) is $\epsilon$-chainable for $\epsilon &gt;0$       $\epsilon$-chainable Definition      (X,d) is $\epsilon$-chainable if given any two points $a,b \in X$, $\exists$ a finite set $\{ x_0,x_1,...,x_n\}$ of points in X with $x_0 = a$, $x_n=b$ and $d(x_{n-1},x_n)&lt;\epsilon$, $\forall$ n      Clopen Definition      A set is clopen $iff$ its boundary is empty      Connected Metric Definition      Let $(X,d)$ be a metric space. It is connected $iff$ it has no proper subsets which are clopen      The following are equivalent:         $X$ is connected   The only subsets of $X$ that are clopen are $\emptyset$ and $X$   $\nexists$ non-empty, disjoint, open sets $A$ and $B$ such that $X=A\bigcup B$   $\nexists$ non-empty, disjoint, closed sets $A$ and $B$ such that $X=A\bigcup B$         Theorem      Let $(X,d)$ be a metric space, and let $S$ be any subset of $X$. Then $S$ is closed $iff$ $S^c$ is open.     So how would someone show this exactly?  Let the set $A$ be the set of all elements in $(X,d)$ that can be reached from $a\in X$ in a finite number of steps with length less than $\epsilon$.  Does this mean that there should be no isolated points? (an isolated point would be at distance infinity from any other point, or distance zero from itself, since $\epsilon &gt;0$ we don't want this?)  Knowing that $X$ is connected $\implies$the only subsets of X that are clopen are $\emptyset$ and $X$  Also, by construction of A;   A is the set of all $\epsilon$-chains between any element and $a$  And, so $A^c = \emptyset$ if $A$ was $\epsilon$-chainable (because A has to have a path from all elements to a, for this to be true, the complement wouldn't have anything in it)  Leaving only $A$ as the remaining clopen set  Does this make any sense? Or should an alternative be used, and how to formalize it.  Thanks. Let $F$ be a field, let $\omega$ be a primitive $n$th root of unity in an algebraic closure of $F$. If $a$ in $F$ is not an $m$th power in $F(\omega)$ for any $m\gt 1$ that divides $n$, how to show that $x^n -a$ is irreducible over $F$?