prime zeta function when $0<s<1$
How does $ \sum_{p<x} p^{-s} $ grow asymptotically for $ \text{Re}(s) < 1 $?
Real-time updates for new review tasks Proper notation for distinct sets Consider I have two sets that neither one is a subset of the other (for example, the set of prime numbers and the set of odd integers). Is there a specific notation that combines the meanings of both ⊄ and ⊅ ?  Also, is there a notation to mean that two sets are completely distinct and share no elements in common (for example, the sets of odd and even integers)?  ≠ seems very inappropriate, and I'd like to avoid it.  Context: My area is actually Computer Science. I'm using this refine my vocabulary in order to teach my students about  (which is basically about sets and subsets of IP addresses), and to write some exercises for them. I'll probably use the words incomparable and disjoint as suggested. Financial proof to get a Schengen visa to France I am planning to apply for a Schengen visa to visit France. I have an attestation d'acceuil, but the visa requires I submit bank statement for the past 3 months. I can get this statement, but the problem is that my company has been paying my salary in cash for the past months, and I place the money into the account by myself.  Can I submit the following?   my accounting book (current account)  letter from the bank stating that I have money in it payslips for the past 3 months   Will that satisfy the financial requirements for the visa? Uniform convergence for sequences of functions In classic OLS regression it is well-known that $(\mathbf{I}-\mathbf{H})\mathbf{y}=\mathbf{r}$, where $\mathbf{I}$ is the identity matrix, $\mathbf{H}$ is the , $\mathbf{y}$ is the vector of response (dependent) values, and $\mathbf{r}$ is the vector of residuals.  My question is whether the scalar $\mathbf{y}^T(\mathbf{I}-\mathbf{H})\mathbf{y}=\mathbf{y}^T\mathbf{r}=\mathbf{y}^T\mathbf{y}-\mathbf{y}^T\mathbf{\hat{y}}$ has some meaningful interpretation. In bash, calling foo would display any output from that command on the stdout.  Calling foo &gt; output would redirect any output from that command to the file specified (in this case 'output').  Is there a way to redirect output to a file and have it display on stdout? Including null values in unpivoted column Looking for sci-fi short story anthology with a particular story of an Earth/Mars war children inheriting the surface I read this in the early to mid 80's.  It was a textbook sized hardcover anthology with color illustrations throughout.  Each story had a few images, some were full page illustrations.  The particular story I remember the most involved Earth survivors living underground.  Robots controlled the war from the surface.  A teenage girl and boy decided to sneak away to the surface as no one had been up there in a long time.  The kids discover greenery and a healthy environment, with rusted robot guardians.  The robots had been keeping the humans fooled that the war was still going on, and had kept them under the surface.  Mars had been wiped out from nearly the first volleys of missiles.    The robots judge the teenagers as worthy and let them remain on the surface.  The kids had silver jumpsuits that dissolved when they got wet.  The robot illustrations were blocky, almost weathered stone-like statues.  They had nearly spherical feet and square simple heads.  The faces had eyes and simple mouths.  -edit- The robots were similar in shape to the one in this picture, but smoother, and no arms.  The illustrations with them had the robots being several stories tall.  I kind of remember that the kids emerged from the belly of one or something like that. What primitives are needed to generically implement public-key cryptography? $t^3$ is never equal to Let $t$ be a real number such that $t^2=at+b$ for some positive integers $a$ and $b$. Then for any choice of positive integers $a$ and $b$, $t^3$ is never equal to: (A). $4t+3$ (B). $8t+5$ (C). $10t+3$ (D). $6t+5$  We have $t^3 = at^2+bt$ and so for (A) we have $t^3 = 4t+3 = 4(at^2+bt)+3$. I don't see how to continue since this is a cubic equation. Is there an easier way of finding a contradiction? Recall the definition of the Weyl Chambers: A Weyl Chamber is a region of $V \setminus \bigcup_{\alpha \in \Phi} H_{\alpha}$, where $V$ is underlying Euclidean space, and $H_\alpha$ the hyperplane that is perpendicular to $\alpha$ the elements of  $\Phi$ in $V$.   I want to show that:   The Weyl chambers are open, convex and connected.   My attempts:   A single hyperplane $H_\alpha$ separates the space $V$ into two half-spaces: positive half-space $H_\alpha^+=\{x \in V \mid (\alpha,x)&gt;0\}$ and negative half-space $H_\alpha^-=\{x \in V \mid (\alpha,x)&lt;0\}$. It is clear that $V \setminus H_\alpha = H_\alpha^+\cup H^-_\alpha.$ As the result, the Weyl chambers can be regarded as components of  $$V \setminus \bigcup_{\alpha \in \Phi} H_{\alpha}= \bigcap_{\alpha \in \Phi} \left( V \setminus H_{\alpha} \right)= \bigcap_{\alpha \in \Phi} \left( H_\alpha^+\cup H_\alpha^- \right).$$ By the discussion above, a Weyl chamber is simply a $\textbf{intersection}$ of finite half-spaces. It is the case since one may write $$\left( H_\alpha^+\cup H_\alpha^- \right) \cap \left( H_\beta^+\cup H_\beta^- \right) =\left( H_\alpha^+\cap H_\beta^+ \right)  \cup\left( H_\alpha^+\cap H_\beta^- \right)  \cup \left( H_\alpha^-\cap H_\beta^+ \right)  \cup\left( H_\alpha^-\cap H_\beta^- \right).$$ Therefore openness and convexity of the Weyl chambers are straightforward since half-spaces are open and convex, so are their intersections. Moreover, we would replace $V$ by $\Bbb R^{\dim V}$ since they are isomorphic. Now we can prove that the Weyl chamber is connected by showing that every convex set in $\Bbb R^{\dim V}$ is connected. And I'm sure that proof of this statement existed. So am I done? A word or short phrase for 'better in every way'? When we're comparing two or more things, there are times when we have to weigh the advantages or disadvantages of each one.  There are other times when one of the things is better for every way in which it's being judged, and the final decision is not based upon trade-offs.  I know that I've heard a phrase for this (I want to say it was two words), but I've had no luck in remembering it.    I don't think 'clearly superior' or 'clear winner' quite fit the bill, as that might still suggest that the disadvantages are significantly outweighed by the advantages.  Is there such a phrase, or even a single word, that would convey this meaning without using idioms? Content is not allowed in Prolog SAXParserException I need your help with references. I have 1 main reference section, and I want to include a second list of references as an appendix. For the main reference I use:  \newpage  \bibliographystyle{plainnat}  \bibliography{library}   The problem is that I don't know how to include the second list using a different bib file, and then calling the function \nocite{*} to show every sources inside the file.   I have the appendix part workout. I need your help to design the function to call the second bibliography, and that does't interfere with the main reference:   \newpage  \appendix  \section{Other Sources} \label{app:ReferencesB}   In addition, I will prefer that the word References is not shown after Other Sources.  Thank you so much!! If I push or hit an object in space will it rotate or move along a straight line? When my receiver that I have hooked up to my pc turns on it causes my monitors to turn black is there a way to stop this? Consider the following piece of Markdown code:  This is some regular text.      &gt;&gt;&gt; def factorial(n):     ...     return 1 if n &lt; 2 else n * factorial(n - 1)     ...   * This is a list item.      &gt;&gt;&gt; def factorial(n):     ...     return 1 if n &lt; 2 else n * factorial(n - 1)     ...    Notice that the second code block is preceded by a list item.   Why is the factorial code not properly formatted under these circumstances? To demonstrate the problem, I'll insert the Markdown code show above:    This is some regular text.  &gt;&gt;&gt; def factorial(n): ...     return 1 if n &lt; 2 else n * factorial(n - 1) ...     This is a list item                 def factorial(n):       ...     return 1 if n &lt; 2 else n * factorial(n - 1)       ...               This applies to numbered as well as unnumbered lists. How to install mods for the new Minecraft launcher Understanding linear regression In class we've seen that $a$ (the weights) must satisfy $$X^T (y-Xa) =0$$  Here $X$ is a $(n\times d)$ matrix (so we have $n$ samples in $\mathbb R^d$)  let's denote the residuals $r = y-Xa$. In our class notes, it is written that     The normal equations require the residuals to be orthogonal to each of the columns of $X$.   Why?     Therefore, the solution of the linear regression is a projection of $y$ onto the subspace spanned by $v_1 , \ldots , v_d$ (the columns of $X$)   Can you please explain this? I'm new at Linux and I just download Ubuntu 12.04 LTS.  I found a spot that showed me how join my Windows Active Directory.  It has me add couple lines into the 'hosts' but I'm not able to edit the file because I do not have the permission.  I'm not the owner.  I use chmod with it rwx and still it will not let edit the file.  How do you change the permission?  Thanks