Pearson correlation p-value correction
Look and you shall find (a correlation)
Rationalize   $\dfrac{1}{\sqrt[3]{p^2}+\sqrt[3]{pq}+\sqrt[3]{q^2}}.$  How would I go about doing this without wading through lots of algebra?  Is there a trick similar to how you can multiply by $\dfrac{\sqrt a-\sqrt b}{\sqrt a-\sqrt b}$ with square roots?  Thanks in advance! What does the 'cosmological constant' represent? Newtons theory of gravity involves a gravitational constant $G$, however one does not refer to it directly, we speak instead of gravity or the force of gravity.   Now Einsteins introduced a  in his equation for GR, and ever since it's been referred to as such; however, this usage seems strange to me, given the above parallel; it seems to me we ought to be referring to something physical, which can then be quantified and whose quantification involves the cosmological constant.   Does the cosmological constant then refer to a cosmological force? A force that is, contra gravity, repulsive? And which is even weaker than gravity, kicking in on the scale of galaxies rather than solar systems? Spaces in which "$A \cap K$ is closed for all compact $K$" implies "$A$ is closed." Let $X$ denote a topological space. For any $A \subseteq X$, consider two possible conditions on $A$.   $A$ is closed $A \cap K$ is closed, for all compact $K \subseteq X$.   If $X$ is Hausdorff, then clearly (1) implies (2). If $X$ is metrizable, then the converse holds; see .     Questions.         What are some Hausdorff topological spaces in which (2) does not   imply (1)?   For Hausdorff topological spaces: what weaker conditions than metrizability are known to imply that (2) implies (1)? Are there any interesting characterizations of those Hausdorff spaces in which (2) implies (1)? If $f:D\to D’$ is analytic and $u: D'\to R$ is harmonic then the composition of $u$ and $f$ is harmonic in $D$ If $f:D\to D’$ is analytic and $u: D'\to R$ is harmonic then the composition of $u$ and $f$ is harmonic in $D$.  How can I show that the above statement is true/false? Can anyone help me? How was this answer posted after this question was closed? Determine the number of lines within a text file Is there an easy way to programmatically determine the number of lines within a text file? Word for someone who acts like an expert but who has very little knowledge? What is a word to describe an individual who acts like an expert in a subject area, constantly stating facts and correcting people, but who actually has very little knowledge on the subject? Is uncertainty principle a technical difficulty in measurement? Is the uncertainty principle a technical difficulty in measurement or is it an intrinsic concept in quantum mechanics irrelevant of any measurement?  Everyone knows the thought experiment of measuring the position of an electron. One can detect electron's position by hitting it with a photon, due to Compton scattering the collision of the photon with electron will change electron's momentum. This experiment is used to explain uncertainty principle to layman, but it is over simplified, isn't it? It also gives an impression that if there was a better suited experimental method the uncertainty principle becomes irrelevant.   I personally think it is intrinsic as it arises from the non-zero commutator of position and momentum operators irrespective of the measurement process. Am I right?  EDIT: My question is similar to certain extent to  and . The answers there are nice but they focus on explaining basics of quantum mechanics more than they comment on the technical difficulty part. In the answers of question 2 there are statements like "So, it's not a knowledge limit" and "you're sort of correct when you say it's an observational limit" without further comments  To summarize, assume hypothetically we managed to find a way in the future where we can have a look at an electron without disturbing it by measurement or causing its wave function to collapse, would the uncertainty principle still hold in such a case?? Why/Why not? The bijection between homogeneous prime ideals of $S_f$ and prime ideals of $(S_f)_0$ It is well-known that if $S$ is a graded ring, and $f$ is a homogeneous element of positive degree, then there is a bijection between the homogeneous prime ideals of the localization $S_f$ and the prime ideals of $S_{(f)}$, the subring of $S_f$ comprising the homogeneous elements of degree $0$.    This is proposition II.2.5b in Hartshorne, exercise 5.5B in Ravi Vakil's notes $[1]$ (p. $130$ of the February $24$, $2012$ version), and proposition 8.1.21 of Akhil Mathew's notes $[2]$ (p. $136$). Unfortunately I cannot follow any of those proofs to my own satisfaction, perhaps because I'm not well-versed in commutative algebra.  The crux of the proof appears to be to show that, given a homogeneous prime ideal $\mathfrak{p}$ of $S$ not containing $f$, the construction used to obtain a prime ideal $\Psi (\mathfrak{q})$ of $S$ from a prime ideal $\mathfrak{q}$ of $S_{(f)}$ will recover $\mathfrak{p}$ when $\mathfrak{q} = S_f  \mathfrak{p} \cap S_{(f)}$. To be precise, let $\Psi (\mathfrak{q})$ be the homogeneous ideal of $S$ generated by $$\bigcup_{d \in \mathbb{N}} \{ s \in S_d : s / f^d \in \mathfrak{q} \}$$ and let $\Phi (\mathfrak{p}) = S_f \mathfrak{p} \cap S_{(f)}$. It's easy to see that $\Phi \circ \Psi$ acts as the identity on $\operatorname{Spec} A$ (or, for that matter, the set of all ideals of $A$), but I cannot see any obvious reason why $\Psi \circ \Phi$ should act as the identity on the set of prime ideals of $S$ not containing $f$. A detailed proof of this point would be much appreciated.  References  $[1]$ .  $[2]$ Algebraic geometry notes (covers material at the level of the first and second volume of EGA): , and . Remove space around tikz How can I remove ALL space around TiKZ. So given MWE below, I want no space between fbox and the node box (so essentially I want to see 1 line). How to realize this?   Is this space in tikzpicture or in \fbox?   How to remove space between \fbox and node    MWE  \documentclass{article} \usepackage{tikz}  \begin{document} \fbox{%   \begin{tikzpicture}[node distance = 0.8cm and 0.2cm]%     \node[draw] (center) {CENTER}; %   \end{tikzpicture}% } \end{document} I'm stuck underground.   I can't mine any further to reach lava. There is no water nearby, and zombies can't reach me.  I can't see to find any blocks to suffocate my character.  NBTEdit and similar programs don't work on Mac, so I can't edit my save file to escape. I have hardly any blocks so cannot jump and lay my way up. I can't see, and do not have any torches, so can't make a staircase back up. I don't want to make new save as I have an awesome house filled with stuff that took me ages to find.    How can I get back to the surface? Why do I have the "required reviews completed" status on my submission in Elsevier? I had submitted my paper in an Elsevier journal one year back. Now, status update on the submission tracking says "Required reviews completed." However, for the last two weeks still there is no change in the status.  Should I ask the editor what is going on with my submission? I'm trying to connect to another computer in my house via ssh. Both computers have ubuntu and openssh installed. When I run the command "ssh username@ip" on terminal I'm asked for the user's password I'm trying to connect to. I type it but I get this error "Permission denied, please try again." I'm sure the password is correct, then why can't I access? I ran into the "mysterious"  \pdfendlink ended up in different nesting level than \pdfstartlink   error.  According to , "this happens when hyperref is used under pdftex and a citation splits across a page boundary".  The given solution is to manual fix the citation to not split across the page boundary.  However, this workaround isn't feasible for me, as the documents are automatically generated on a server with no user interaction.  Is there any solution to the problem known, perhaps an experimental hyperref action or a patch to pdftex?  (Probably, one could use luatex to solve the problem, but for other reasons, luatex is not an option in this project.)  MWE:  \documentclass[%draft               ]{article} \usepackage{hyperref} \setlength{\textwidth}{3cm} \setlength{\textheight}{3cm}  \begin{document} Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna \href{http://www.tug.org/errors}{aliquyam erat, sed diam voluptua.} \end{document} Would it be possible to break an RSA key, in for example 1 week of time, if the cracker have already spent X number of years building an index of primes by performing every permutation of existing prime keys up to $2^{2048}$?  I understand this would take an immense amount of time to do, but it would only be done once. Obviously, computing such an index would take a considerable amount of time and would have to be done with someone having the right amount of resource. Given the public key, you would be able to look up what the two primes are, and hence retrieve the private key instantly.  Breaking an 256 AES would then be easier, as AES keys are often generated using the RSA private key.  Would it be possible to build such an index? How long would that take in permutations and perhaps with the fastest computer in the world? Suppose a gambler starts with one dollar and plays a game in which he or she wins one dollar with probability $p$ and loses one dollar with probability $1 - p$. Let $f_{n}$ be the probability that he or she first becomes broke at time $n$ for $n = 0, 1, 2\ldots$. Find a generating function for these probabilities.  Is it binomial distribution $?$ and then use the $\operatorname{mx}\left(s\right) = rx\left(\mathrm{e}^{s}\right)$ of the binomial distribution $?$. Help$\ldots!$. I am correcting my quiz. Why is Python running my module when I import it, and how do I stop it? Following minimal document:  \documentclass{report}  \begin{document} \renewcommand*\thechapter{\Roman{chapter}} \tableofcontents \setcounter{chapter}{6} % Problem starts at chapter 7 \chapter{Seven} \chapter{Eight} \section{Eight-one} \section{Eight-two} \end{document}   yields this output (cut to size for your convenience):    Clearly, the chapter (and section) headings hang into the chapter (section) numbers. For the previous chapter numbers, this isn’t a problem since all chapter titles are aligned to the same column and the numbers I–VI are sufficiently narrow.  (The same happens regardless of document type, I’m actually using scrreprt with the same result.)  How can I prevent this? What happens if a Rainbow Servant takes levels in Mystic Theurge? Which is more idiomatic in a bash script: `|| true` or `|| :`? I don't do terribly much shell scripting, so I was a little surprised when I was reading  and I saw the syntax they used in this documentation:     A non-zero return from the command in any submodule causes the processing to terminate. This can be overridden by adding || : to the end of the command.   I had to look up that || : was a shorthand for forcing a command to exit successfully.  Anytime I've had to make a command exit successfully, I used || true.  Is || : considered to be more idiomatic?