I'm sure you've all seen the diagrams and/or 3D visualizations of gravity; the ball sitting on a piece of fabric which makes it sink down. They've also started using it in the videos that explain gravitational waves. Two objects will be circling each other on the fabric and will emit waves.  I know this is an oversimplification of what's going on, but it is quite misleading and has left me a little bit puzzled as to why gravitational waves are infact the size of protons, not the size of planets like the videos suggest.   So I guess my question is, why are they in fact so small, and why can't we detect them from astral bodies in our solar system that we can actually detect a physical force from?
Assume two black holes in the most common size range, spiraling into each other until they merge. The event releases significant amounts of energy via gravitational waves, which warp the space-time.  If the distortion is powerful enough, it would get noticed in daily life, or perhaps even detected by the proprioceptors in the human body.  How close, as an order of magnitude estimate, would you have to be to the merger to perceive it like that, without any instrument?
Let H={e,(12)(34)} and K ={e, (12)(34), (13)(24), (14)(23)} be subgroups of $ S_4$ where e is identity element. Then which of following is true   H and K are normal subgroups of $S_4$ H is normal in K and K is normal in$ A_4$ H is normal in $A_4$, but not normal in $ S_4$ K is normal in $S_4$ but H is not   How should this question be approached, checking left and right cosets of these subgroup is very time consuming. Checking gh$ g^{-1}$ $\in $ H for every g in $ S_4$ is also time consuming. Does DTIME(O(n)) = REGULAR? Why does Quantum Field Theory use usually Lagrangians rather than Hamiltonains?  I heard many reasons, but I'm not sure which is true.   Some say it's just a matter of beauty, so Lagrangians are more beautiful because they don't break/separate the space-time variables (so space-time is a single variable, like in the Klein-Gordon Lagrangian and Hamiltonian).  Some say that Hamiltonians are not always Lorentz invariant.  Could someone explain in a little bit more details? How do the likely/unlikely macros in the Linux kernel work and what is their benefit? Story where vampires are created by drop absorbed on neck? I have been trying to identify a book that I read maybe 25-30 years ago (1987-1992), in which there are vampire or vampire-like creatures, but instead of being created in the traditional way (e.g. bite on the neck), something drops a drop of a fluid on the back of their neck, which is then absorbed into the body and transforms them. I have a vague notion that some sort of tree is what creates these drops. The drops may have had a shiny metallic appearance.  I've tried to find this book before and always came up short. I'm starting to think the whole thing was a dream and I never read it at all. But I really feel like it was a book or at least a short story. The use of “So X as to be Y” First of all, note that this is not a duplicate question of another one asking about the usage of so as to, since this one is asking about the structure of so . . . as to. . . .  I understand this might have been a better question at the , since I’m not a native English speaker. I am asking it here since I want to get some professional advice in regard to this question, since I don’t seem to have found anything similar to this on the Internet.  At the Oxford Dictionaries Online (OD) I have found the following definition of ludicrous:     So foolish, unreasonable, or out of place as to be amusing.   I understand the phrase so as to, simply meaning in order to and preceding the purpose for which an action is done.  So by itself is just therefore, hence, thus, or a way of emphasizing an adjective or adverb.  As to in itself means with regard to or according to, as pointed out in .  In the OD definition of ludicrous, the use of so looks fair to me, that is, it clearly emphasizes the high extent to which something is foolish or unreasonable.  The use of as to doesn’t work in any way I’ve mentioned in this case (neither as in with regard to nor according to). I think the dictionary I’ve referred to has missed one, if not more, meaning of the phrase as to.  In this case, it appears to have likely been used as an alternative to that when used in the form So X, that Y, meaning something is or happens X (a combination of adjectives or adverbs) enough to cause Y to happen.  Of course, I don’t mean that in the structure so X as to Y the phrase as to is absolutely equivalent to that, since substituting as to with that changes the structure of Y.  Does all this mean that the phrase as to is a great alternative to that (although changing the structure of Y) whenever the sentence is of the form so X, that Y (where X is a combination of one or more adjectives or adverbs)?  I am new here and so I’m not too familiar with the variety of existing tags here, so if anyone sees an additional one that would fit, please add it to this question. I would also be glad to see some other corrections of my grammar. Recovering a accidently deleted file in Ubuntu I'm using ubuntu 9.10 and accidently deleted a specific file within a directory on my system. I checked in the Trash bin and could not locate the file. I deleted it 10mins ago, are there any tools available that I can recovery this file? It's a really important file.  Any help would be highly appreciated  My partition is ext4 Custom Edit control inside a ExtJS Editor grid Why do we multiply the magnetic flux by the winding number? From Euler we've learned that $z=re^{i\theta}$.  And it's easy to see that $|z|^2=r^2$, since $re^{i\theta}\times re^{-i\theta}=r^2$.  Why must we use e to represent these numbers correctly? It seems that I could arbitrarily choose a different exponent $z=r\pi^{i\theta}$ and get the same size for $z$ as I did before: $|z|^2=r\pi^{i\theta}\times r\pi^{-i\theta}=r^2$  What did I miss? Why is find not accepting '-exec cp {} dir +'? I have a directory, dir1 which contains many files whose names end in either .jpg or .png. I want to copy all the .png files to dir2 which is empty.  This command works:  find dir1 -name '*.png' -exec cp {} dir2 \;   but this command doesn't:  find dir1 -name '*.png' -exec cp {} dir2 + find: missing argument to `-exec'   I also tried:  find dir1 -name '*.png' -exec cp {} -t dir2 + find: missing argument to `-exec'   and:  find dir1 -name '*.png' -exec cp {} dir2 \+ find: missing argument to `-exec'   After looking at , I even tried:  find dir1 -name '*.png' -exec cp {} dir2 {} + find: Only one instance of {} is supported with -exec ... +    says that:     -exec {} + was added in [version] 4.2.12 in 2005   My version of find is 4.4.2.  What am I doing wrong? TikZ: What EXACTLY does the the |- notation for arrows do? I am currently trying to work out how the perpendicular arrows  work, and I fail.  Taking:  \draw [arrow] (node1) -- node[sloped, above] {Caption} (node1 |- node2.north west);   I suspected that the notation meant: Draw an arrow from node 1, to the north-west corner of  node 2. Make the arrow perpendicular to node 1.  Further attempts to draw arrows told me: No, that's not it. I have sadly failed to figure out what exactly the notation is supposed to be doing though! Is there any "explanation for dummies" that will lead me to understand what exactly    node1 |-   Is supposed to do?  Thanks! Batch fit images into ratio with blurred background I need to resize a lot of images into a 4:3 ratio with the desired effect.  Is there a way to do this using photoshop actions? (the background can be scaled or two copies moved to the sides, either way works)     image credits: Prove that $\left (\frac{1}{a}+1 \right)\left (\frac{1}{b}+1 \right)\left (\frac{1}{c}+1 \right) \geq 64.$ Let $a,b,$ and $c$ be positive numbers with $a+b+c = 1$. Prove that $$\left (\dfrac{1}{a}+1 \right)\left (\dfrac{1}{b}+1 \right)\left (\dfrac{1}{c}+1 \right) \geq 64.$$   Attempt  Expanding the LHS we obtain $\left (\dfrac{1+a}{a} \right)\left (\dfrac{1+b}{b} \right)\left (\dfrac{1+c}{c} \right)$. We are given that $a+b+c = 1$, so substituting that in we get $\left (\dfrac{b+c+2a}{a} \right)\left (\dfrac{a+c+2b}{b} \right)\left (\dfrac{a+b+2c}{c} \right)$. Then do I say $\left (\dfrac{b+c+2a}{a} \right)\left (\dfrac{a+c+2b}{b} \right)\left (\dfrac{a+b+2c}{c} \right) \geq 64$ and see if I can get a true statement from this? Do you need line of sight to cast spells on someone? Is there any rule for when it is better to use genitive case or noun as adjective? I'm not sure if there is any difference in meaning in this example:     The department of accounting       The accounting department   Can I also use both forms in situations like:     The restaurants of Paris / The Paris restaurants      The room window / The window of the room if $f^{(n)}(x_0),g^{(n)}(x_0)$ exist and $\lim_{x\to x_0} \dfrac{f(x)-g(x)}{(x-x_0)^n}=0$ then $f^{(r)}(x_0)=g^{(r)}(x_0 )$ for all $0\leq r\leq n.$ Show:   if $f^{(n)}(x_0),g^{(n)}(x_0)$ exist and   $$\lim_{x\to x_0} \dfrac{f(x)-g(x)}{(x-x_0)^n}=0$$   then $f^{(r)}(x_0)=g^{(r)}(x_0 )$ for all $0\leq r\leq n.$   My attempt:  let n = 1,  since $f'(x_0),g'(x_0)$ exists then The taylor polynomial exists: $$T_{f,1}(x)=f(x_0)+f'(x_0)(x-x_0), \lim_{x\to x_0} \dfrac{f(x)-T_{f,1}(x)}{(x-x_0)}=0$$  $$T_{g,1}(x)=g(x_0)+g'(x_0)(x-x_0), \lim_{x\to x_0} \dfrac{g(x)-T_{g,1}(x)}{(x-x_0)}=0$$  Then, \begin{align} \lim_{x\to x_0} \dfrac{f(x)-g(x)}{(x-x_0)^n} &amp; =\lim_{x\to x_0} \dfrac{f(x)-g(x)+T_{f,1}(x)-T_{f,1}(x)+T_{g,1}(x)-T_{g,1}(x)}{(x-x_0)^n}\\ &amp;=\lim_{x\to x_0} \dfrac{f(x)-g(x)+T_{f,1}(x)-T_{f,1}(x)+T_{g,1}(x)-T_{g,1}(x)}{(x-x_0)^n}\\ &amp;=\lim_{x\to x_0} \dfrac{f(x)-T_{f,1}(x)}{x-x_0}-\lim_{x\to x_0} \dfrac{g(x)-T_{g,1}(x)}{x-x_0}+\lim_{x\to x_0} \dfrac{T_{f,1}(x)-T_{g,1}(x)}{x-x_0}\\ &amp;=\lim_{x\to x_0}\dfrac{T_{f,1}(x)-T_{g,1}(x)}{x-x_0}\\ &amp;=0 \end{align}  Then, \begin{align} \lim_{x\to x_0}\dfrac{T_{f,1}(x)-T_{g,1}(x)}{x-x_0} &amp;=\lim_{x\to x_0}\dfrac{f(x_0)+f'(x_0)(x-x_0)-g(x_0)-g'(x_0)(x-x_0)}{x-x_0}\\ &amp;=\left(f(x_0)-g(x_0)\right)\lim_{x\to x_0}\dfrac{1}{x-x_0}+f'(x_0)-g'(x_0) \\&amp;=0 \end{align} I am confused because  $\lim_{x\to x_0}\dfrac{1}{x-x_0}$ does not exist. For the recently reported production (January 2017) of metallic hydrogen in the laboratory - what is the evidence exactly? My google news page has burst forth the news that metallic hydrogen has been produced in a laboratory using a diamond anvil cell. Only  mentions (at this particular moment) that it has been called into question at - at least - premature. That can be read about in Nature:   Here I only want to ask about the evidence. Is it only the sudden change in visual appearance change as recorded in a camera on top of the microscope, at a specific temperature and pressure? That it "looks shiny?" Or is there actually more direct data than that?  note: I am not asking if the data is sufficient or not, nor if it is convincing, or metallic or not. Just a narrowly defined question about the extent of the observations that have lead to the proposal that a metallic state was achieved.     above: potentially three states of hydrogen as viewed in a diamond anvil cell, from . Analogy: "as if" vs. "as though" This is something that confuses me from time to time. When making an analogy in literature, is it better to use the phrase "as if" or "as though", or is it completely a style thing?  E.g.      She looked frazzled, as if she had just gotten off of a roller coaster.   or     She looked frazzled, as though she had just gotten off of a roller coaster. How to activate hyphenation in \ttfamily without explictly setting a font? I am suprised that in the code below no hyphenation is done. And I can not find something about that on the web. Maybe I used the wrong words for searching.  \documentclass{scrartcl} \usepackage{xltxtra} \usepackage{polyglossia} \setdefaultlanguage[spelling=new]{german} \usepackage{tabu}  \begin{document} \begin{ttfamily} \begin{tabu} spread \textwidth {X[l]|X[l]|X[l]|X[l]}     A&amp;     B&amp;     Sichtweise, Haltung, Akzeptanz, useability, Anwenderfreundlichkeit&amp;     D\\ \end{tabu} \end{ttfamily} \end{document}