Information can never be lost. But what does it mean when it's said that information can't be cloned?
I am trying to get an intuitive idea of how the  affects Quantum computation. My understanding is that given a qubit $Q$ in superposition $Q_0 \left| 0 \right&gt; + Q_1 \left| 1 \right&gt;$, NCT states another Qubit $S$ cannot be designed such that $S$ is equivalent to the state of $Q$.   Now the catch is, what does Equivalent mean? It could mean either that:   $S = S_0 \left| 0 \right&gt; + S_1 \left| 1 \right&gt;$ such that $S_0 = Q_0, S_1 = Q_1$. Or it could mean that $S = Q$, meaning that if $S$ is observed to be some value ( for example 0) then $Q$ MUST be that same value, and vice versa.    So it seems that point 2, occurs anyways in entangled systems (particularly cat-states), so I can eliminate that option and conclude that that No Cloning states, given a qubit $Q$, it's impossible to make another qubit $S$ such that:  $S = S_0 \left| 0 \right&gt; + S_1 \left| 1 \right&gt;$ such that $S_0 = Q_0, S_1 = Q_1$.  Is this correct?
Is there a tool to update my PPA sources.list entries after upgrading? Showing $\gcd(2^m-1,2^n+1)=1$ A student of mine has been self-studying some elementary number theory. She came by my office today and asked if I had any hints on how to prove the statement      If $m$ is odd then $\gcd(2^m-1,2^n+1)=1$.    It's been a while since I took number theory and I'm not sure what to do. She said she is learning about congruences, primitive roots, and power residues.  She has not taken any group theory. Experimental observation of matter/antimatter in the universe Ordinary matter and antimatter have the same physical properties when it comes to, for example, spectroscopy. Hydrogen and antihydrogen atoms produce the same spectroscopy when excited, and adsorb the same frequencies. The charge does not make a difference in the potential (regardless if it's generated by a proton or an antiproton) nor in how the positron behaves in this potential (being its mass equal to the mass of an electron)  How can astronomy evaluate if a far galaxy is made of matter or antimatter, given that from the spectroscopy point of view, they behave in the same way? In other words, how do we know that an asymmetry exists between matter and antimatter in the universe? How to make traces on an universal PCB? I hope this question won't be closed as too subjective. I'd like to know best practice - how to make traces on universal PCB's with individual holes without traces (like the following image). My idea is to bend the ends of discrete components and use them to make traces to other components. Is this approach acceptable, or nasty? Although this is called "prototype" PCB, I'd like to use it for my simple final circuits (low frequency, low current applications), because I hope it will save time. Proving the set of points at which a function diverges to $\infty$ is countable Let $f\colon\mathbb{R}\to\mathbb{R}$. Prove that the set    $$\{x \mid \mbox{if $y$ converges to $x$, then $f(y)$ converges to $\infty$}\}$$ is countable.   My book told me to consider  $g(x)=\arctan(f(x))$, then it said "it is easy to see the set is countable." But I still can't understand what it mean. Php string not working in Wordpress Functions.php (trying to fetch 1st category for each blog that post appears in the sidebar) I am trying to fetch 1st category for each blog post that appears in the sidebar of any blogpost landing page. I tried multiple options to add the 2nd block code to the 1st block at the line &quot;.wp_get_post_categories ($post_id).&quot; but I could not find any solution. Please help me with that and if you need closer to the code just let me know. 1st Block $related_blog .=&quot;&lt;div class='related-blog'&gt;                             &lt;div class='row'&gt;                                 &lt;a href=&quot;.get_the_permalink().&quot;&gt;                                                                  &lt;div class='col-lg-4 col-md-4 col-xs-4 blog-image'&gt;                                         &lt;img src=&quot;.$image_related[0].&quot;&gt;                                     &lt;/div&gt;                                      &lt;div class='col-lg-8 col-md-8 col-xs-8 blog-text'&gt;                                         &lt;strong&gt;&lt;p class='related-blog-title'&gt;&quot;.get_the_title().&quot;&lt;/p&gt;&lt;/strong&gt;                                 &lt;/a&gt;                                 &lt;span class='related-blog-sidebar-content '&gt;&quot;.substr(get_the_content(),0,250).&quot;...&lt;/span&gt;                                                                  &lt;p class='related-blog-sidebar-category'&gt;                                 &quot;.wp_get_post_categories ($post_id).&quot;                                                                &lt;/p&gt;                                                                  &lt;a class='related-blog-sidebar-link' href=&quot;.get_the_permalink().&quot;&gt;Read More                                 &lt;/a&gt;                                      &lt;/div&gt;                             &lt;/div&gt;                         &lt;/div&gt;&quot;;  2nd Block &lt;?php $category = get_the_category(); ?&gt; &lt;a  href=&quot;&lt;?php echo get_category_link($category[0]-&gt;cat_ID); ?&gt; &quot;&gt; &lt;?php echo $category[0]-&gt;cat_name; ?&gt; &lt;/a&gt; Calculate: $$\int_{0}^{2\pi}e^{R{ {\cos t}}}\cos(R\sin t+3t)\mathrm{d}t$$ My try: $\displaystyle\int_{0}^{2\pi}e^{R{ {\cos t}}}\cos(R\sin t+3t)dt\\ \displaystyle \int_{|z|=R}e^{\mathfrak{R\textrm{z}}}\cos(\mathfrak{I\textrm{z}}+3(-i\log z)dz\\ \displaystyle \int_{|z|=R}e^{\mathfrak{R\textrm{z}}}\mathfrak{R\textrm{e}}^{(\mathfrak{I\textrm{z}}+3(-i\log z))i}dz\\ \displaystyle \int_{|z|=R}e^{\mathfrak{R\textrm{z}}}\mathfrak{R\textrm{e}^{\mathfrak{I\textrm{z}}}}z^{3}dz\\ \displaystyle \int_{|z|=R}e^{\mathfrak{R\textrm{z}}}\mathfrak{R\textrm{e}^{\mathfrak{I\textrm{z}}}R}z^{3}dz$ and there is nothing here that is not holomorphic, therefore according to Cauchy theorem it must be exactly $0$. How do I wrap a macro definition in an environment? In 10.04, I had a lot of keyboard shortcuts defined using Super/Mod4 and one single other key,   Super+O ran Opera Super+W opened Nautilus pointing to my Work folder, etc.   In 11.04, these do not seem to work -- only Super+R works to run the terminal, and Super+Shift+W successfully runs Nautilus.  Is there some way I can get these to function again?  Adding them in Keyboard Shortcuts does not work, and neither does adding commands in CompizConfig Settings Manager. Find three Poisson-distributed random variables, pairwise independent but not mutually independent An application of derivatives problem- to show that $f(x+2)-f(x)>2$ for $f(x)=x\cos(1/x)$ I have this function $$f(x) = x\cos(1/x)$$ defined for all $x \geq 1$.      How do I prove that    $$f(x+2)-f(x)&gt;2$$    for all $x\geq 1$?   I found $$f'(x)= \cos(1/x) + \frac{\sin(1/x)}{x}$$ and $$f''(x)=-\frac{\cos(1/x)}{x^3}.$$ How do i use this information? Normality assumption in linear regression Cross-referencing problem if counter of theorem-like environments is shared The problem is when i share numbering the \cref can't show name, even by \crefname and \autoref show all name Equation.  Here is an MWE:  %&amp;latex \documentclass{amsart}  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % If you want to insert other packages. Insert them here \usepackage{amsthm} \usepackage{hyperref} \usepackage[nameinlink]{cleveref} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  \begin{document}  \numberwithin{equation}{section}  \newtheorem{The}[equation]{Theorem} \crefname{The}{Theorem}{Theorems}  \newtheorem{Dfn}[equation]{Definition} \newtheorem{Lem}[equation]{Lemma} \newtheorem{Prs}[equation]{Proposition} \newtheorem{Crl}[equation]{Corollary} \newtheorem{Rmk}[equation]{Remark}  \begin{The}\label{T1}    Assume ... \end{The}  ... \begin{Dfn}\label{D1}    Assume ... \end{Dfn}  ... \begin{Lem}\label{L1}    Assume ... \end{Lem}  ... \begin{Prs}\label{P1}    Assume ... \end{Prs}  ... \begin{Crl}\label{C1}    Assume ... \end{Crl}  ... \begin{Rmk}\label{R1}    Assume ... \end{Rmk}  \begin{equation}\label{E1}    Assume ... \bigskip \end{equation}  \noindent \cref{T1} \\ \cref{D1} \\ \cref{L1} \\ \cref{P1} \\ \cref{C1} \\ \cref{R1} \\ \cref{E1} \\  \noindent \autoref{T1} \\ \autoref{D1} \\ \autoref{L1} \\ \autoref{P1} \\ \autoref{C1} \\ \autoref{R1} \\ \autoref{E1} \\  \end{document}     How can I solve it? Everyone I speak to says that a 50mm lens is the best and beats them all as an all-rounder. I was wondering whether to get myself a decent, fast 50mm as well. However I'm still not 100% sure whether I really need such a lens.  I use a Pentax K10D and mainly shoot people (kids, portraits, some street shots). The lens I love and use most is the DA 70mm f/2.4 Limited. I often also use a DA 40mm f/2.8 Limited. They are great little lenses that produce very high quality images but they could be a bit faster.  Is is worth spending 400 quid on a Sigma 50mm f/1.4 in my situation? It seems like a good compromise between the 40mm and 70mm and much faster (in fact, 2 f-stops faster than the DA 70mm f/2.8!). On the other hand would the lens be sufficiently different from my current lenses? OpenID support for Ruby on Rails application What is current state of the art for enabling OpenID login in Ruby on Rails applications?  This is a community wiki with up-to-date answers to this question. Authlogic The most advanced authentication solution seems to be .  It supports OpenID with .  It supports Rails 4 and 3. Rails 2 is supported in the rails2 branch. You may want to watch  (and the ). There is a sample application called . Devise Devise is flexible authentication framework for Rails.  It supports OpenID with . restful_authentication Another authentication library of choice is  Rails plugin.  Seems like you also need to install  plugin. You may want to watch (old, circa 2007) . Ruby OpenID Raw support for OpenID protocol is handled by  library. Suppose that $f(x)$ is continuous on $(0, \infty)$ such that for all $x > 0$,$f(x^2) = f(x)$. Prove that $f$ is a constant function. Suppose that $f(x)$ is continuous on $(0, +\infty)$ such that for all $x &gt; 0$,$f(x^2) = f(x)$. Prove that $f$ is a constant function. My attempt is to show that for any point $a \neq b$ , we have $f(a)=f(b)$. But I have no idea on how to get this. Anyone can help? Where to start with Oracle? Question involving the Cauchy-Goursat Theorem Show $\int_C \frac{dz}{(z-z_0)^n} = 2\pi i$ when $n = 1$ and 0 when $n \gt 1$   I'm not entirely sure how to approach this one.  Any help would be much appreciated. How to fix "The system is running in low-graphics mode" error? How can I recover photo files from a memory card with a corrupt filesystem? I know this is not specifically a photography issue but since many of us with prosumer/professional-level DLSRs use Compact Flash cards, it is bound to happen to others as well.  I have a Sandisk 16GB Ultra card that occasionally had the camera reporting inability to access the card. I had to reinsert the card to continue using it. I was able to read the card and transfer the photo files to my computer drive. Today, the same thing happened again, just that nearing the capacity of the card, it failed one last time and the camera was subsequently unable to read it anymore. Despite reinsertion.  My other CF cards never once exhibited this problem, so it's pretty much isolated to this card.  The card is similarly unrecognisable on my computer CF card reader. I think the filesystem is toast, but I suspect the photo files are still intact. I would like to find ways where I can inspect the raw binary data in the card and hopefully recover the photos?