Where does the 10 km of neutrons stars comes from? Neutrons stars are known to have a radius of about 10 km to 15 km.  Where that size comes from ?  How to theoretically derive that number, using Newton's theory of gravity ?  I know the magnetic flux conservation : $\Phi = \pi R_0^2 \, B_0 \approx \pi R^2 \, B$, but this requires that we know the initial and final magnetic field at the surface of the star, and the star's size before the supernova.  So this isn't satisfying.  Conservation of angular momentum alone isn't telling the radius neither : $S = \frac{2}{5} \, M R_0^2 \, \omega_0 = \frac{2}{5} \, M R^2 \, \omega$, since this requires that we know the star size and angular velocity before the supernova.  Using Newton's theory of gravitation and conservation of energy (or another method ?), how can we derive the theoretical size of NS ?  The only input number that I could accept in the derivation is the NS angular velocity $\omega$ (and mass $M \approx 1.44 \, M_{\odot}$), since this can be found from conservation of angular momentum and simple models of supernova (if we already know the NS radius !).  Currently, the only rough argument that I know is the following :  assuming an uniform sphere rotating at its maximal value to support gravity, we should have a balance relation like this : \begin{equation}\tag{1} F_{\text{centripetal}} \approx M \, \omega^2 R = F_{\text{grav}} \approx \frac{G M^2}{R^2}. \end{equation} Isolating $R$ gives \begin{equation}\tag{2} R \approx \Big( \frac{G M}{\omega^2} \Big)^{\frac{1}{3}}. \end{equation} Inserting mass $M \approx 1.44 M_{\odot}$ and period $T \approx 1 \text{ms}$ give something interesting : \begin{equation}\tag{3} R_{\text{NS}} \approx 16.9 \text{km}. \end{equation} I strongly suspect this could be improved or made more rigorous (even if it's less precise), using the conservation of energy.    EDIT : Another argument comes from the density.  A neutron star has a density comparable to a nucleus or a neutron, so \begin{equation}\tag{4} \rho = \frac{3 M}{4 \pi R^3} \approx \rho_{\text{neutron}} = \frac{3 m_N}{4 \pi r_N^3}. \end{equation} This gives the star's radius (using $M \approx 1.44 \, M_{\odot}$ and $r_N \approx 10^{-15} \, \mathrm{m}$) : \begin{equation}\tag{5} R \approx \Big( \frac{M}{m_N} \Big)^{\frac{1}{3}} \, r_N \approx 12 \mathrm{km}. \end{equation}
Size and density of neutron stars Most of the books which I looked at give approximately 10 km as the radius of a neutron star. Just yesterday I looked at a book by Dave Goldberg titled The Universe In the Rearview Mirror (2013) which says that they're "only about 5 km in radius" [p.225]. Is this true, [i.e., is there some recent evidence for this], or did he make a mistake here?
What is a primary decomposition of the ideal $I = \langle xy, x - yz \rangle$? Compute $\mathbb{P}\{ W_t < 0 \, \, \text{for all} \, \, 1 < t < 2\}$ for a Brownian motion $(W_t)_{t \geq 0}$ Let $(W_t)_{t \ge 0}$ be a standard one-dimensional Brownian motion. Calculate $$\mathbb{P}\{ W_t &lt; 0 \, \, \text{for all} \, \, 1 &lt; t &lt; 2\}.$$ I can only think that this will be conditional on $W_1$. Please tell how to proceed? How do I calculate CRC32 of a string How do I calculate the CRC32 (Cyclic Redundancy Checksum) of a string in .NET? Who decides which half of the universe should be destroyed? When I open a window to air out the room, how does the smell disperse? I have a bit of a unique problem. Well, maybe not a problem because I'm really just curious about it, but...  I have a simple 8 digit calculator. It has +, -, x, /, and a constant operation function. It has no usable memory slot.  I need (lit. would like to know how) to calculate ln(x) on this calculator. Any series I have found lose a lot of precision since whatever is outside the 8 digit range is truncated without rounding. I have used Padé approximants with trig functions, and a continued fraction for exp(), but none of these two methods are adequate for ln(x), either because there is no way around the truncation error or it requires writing down many intermediate values, or it requires memorizing and punching in many constants.  Does anyone know any good way to calculate ln(x) (or any other base, since memorizing one constant is OK) to as many digits as possible as easily as possible on this calculator (bonus points to methods where the value on the screen is immediately, meaning you don't need to write it down)?  For that matter, if you happen to know some good ways to calculate other special functions, I am interested in that as well.  Thanks! Show that one-sided limits always exist for a monotone function (on an interval) Show that one-sided limits always exist for a monotone function on an interval $[a,b]$.  Me attempt:  1) If a function is monotone on an interval $[a,b]$, then $f(a)\le f(x) \le f(b)$ for $x\in[a,b]$. Therefore if there exists left-hand (right-hand) limit of this function at a given point, then it must be finite. Now we must show that a both left-hand and right=hand limits exist. We use the fact that if we take a sequence $(x_n)\rightarrow c^{+} \in [a,b]$ then $f(x_n)$ is bounded and monotone and therefore it is convergent. How to determine which form has been submited and to validate them in a servlet The cube of any number not a multiple of $7$, will equal one more or one less than a multiple of $7$ Yeah so I'm kind of stuck on this problem, and I have two questions.   Is there a way to define a number mathematically so that it cannot be a multiple of $7$? I know $7k+1,\ 7k+2,\ 7k+3,\ \cdots$, but that will take ages to prove for each case.  Is this a proof? $$(7k + a)^3 \equiv (0\cdot k + a)^3 \equiv a^3 \bmod 7$$    Thank you. Testing stored procedure scalability I have an email application that will be called upon to deliver to the UI the number of new messages for a given user on each page load. I have a few variations of things I am testing on the DB level but all are abstracted by the stored proc call.  I'm trying to slam the DB to see what the breaking point (# of requests per second) would be.   In a nutshell, I have a table such as this userId, newMsgCount with a clustered index on userId. SQL should be able to server hundreds or thousands of these responses per second. I think the laggard is my .NET app.  How can I make this a good test to achieve the test results based on SQL performance?  Is there a tool for this that i can give it a stored proc name and param for it to pund my  DB?  I want to see if the DB can return a min. of 250 responses per second. Lost my APFS partition after using EaseUS partition manager on Bootcamp I used macOS mojave(APFS partition) and Windows10(Bootcamp). And I tried to expand Bootcamp partition with moving Windows OEM partition by EaseUS partition master in windows. After this, I can’t find APFS partition installed macOS.  I use MacBook Pro 2016.  I think “disk0s2” was APFS partition, but now it is “Windows Recovery” partition.  Do you have any ideas to solve this? (I am Japanese, so I'm sorry for my poor English)  I tried “gpt show” and “diskutil list” on macOS Recovery   Additional information  I tried dd if=/dev/disk0s2 count=1 bs=512 command in a Terminal app of macOS Utility from network boot, and the terminal says,  -bash-3.2# dd if=/dev/disk0s2 count=1 bs=512 | vis -c 1+0 records in 1+0 records out 512 bytes transferred in 0.005472 secs (93568 bytes/sec) )us??-ҵ?\^A\0\0\0\0\0\0\0?\M^F\^Z\0\0\0\0\0\^A\0\0\M^@\0\0\0\0NXSB\0\^P\0\0?\M^B?\^B\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\^B\0\0\0\0\0\0\0"\^R???\M^_@}\M^Bvy57 ?(h?8\0\0\0\0\0?\M^F\^Z\0\0\0\0\0\^X\^A\0\0 l\0\0?\M^J\0\0\0\0\0\0\^R\r\0\0\0\0\0\0\a\0\0\0004*\0\0\^E\0\0\0\^B\0\0\0\^S*\0\0!\0\0\0\M^Qr8\0\0\0\0\0\M^U?\b\0\0\0\0\0\^A\^D\0\0\0\0\0\0\0\0\0\0d\0\0\0\^C\^D\0\0\0\0\0\0\M^T\M^N\^A\0\0\0\0\0\M^V\M^N\^A\0\0\0\0\0?0\^B\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0-bash-3.2#   . Google map only loading half of itself I've only seen The Force Awakens once, but I'm as far as I can remember, Phasma doesn't really do much, except:      get taken hostage that one time and disable the shields around the Starkiller after being asked twice   Was she originally supposed to play a bigger role?  I was really expecting to see much more of her, or at least see her in combat. How to execute a Matlab script from C++ Changing an element's ID with jQuery Exact value for $\cos 36°$ Good morning! I'm having trouble with this problem... It's just taking me forever and I'm worn out and I'm lost on how to use a double angle identity for $72=2⋅36$  The problem reads as follows  An exact value for $\cos36°$ can be found using the following procedure. Begin by considering $\sin108°$. Note that $108=72+36$ and use the sine sum identity. Also note that $72=2⋅36$ and use double angle identities. If there are any common factors in each term, factor them out and cancel them if they are not equal to zero. You should eventually obtain a quadratic equation containing $\cos36°$. Use the quadratic formula to obtain the exact value for $\cos36°$. (Note that the quadratic formula should give two solutions. One can be disregarded - why?  Thank you in advance to anyone who can help. The word that always gets me mixed up : JUST I had just arrived at the hotel, checked in and gotten into my room. Just as I was about to unpack, I noticed that one of my bags was missing. At first I thought I had left it back at the airport, but it turned out that I had left it on the taxi.  Here, what's the meaning and role of "just"?  Why do you guys input the word "just" here?  What exactly does the word do grammatically?   Could you help me? Does an unbalanced sample matter when doing logistic regression? Okay, so I think I have a decent enough sample, taking into account the 20:1 rule of thumb: a fairly large sample (N=374) for a total of 7 candidate predictor variables.   My problem is the following: whatever set of predictor variables I use, the classifications never get better than a specificity of 100% and a sensitivity of 0%. However unsatisfactory, this could actually be the best possible result, given the set of candidate predictor variables (from which I can't deviate).   But, I couldn't help but think I could do better, so I noticed that the categories of the dependent variable were quite unevenly balanced, almost 4:1. Could a more balanced subsample improve classifications? Is rejecting the hypothesis using p-value equivalent to hypothesis not belonging to the confidence interval? While formally deriving the confidence interval of an estimate, I ended up with a formula that resembles very closely the way $p$-value is computed.  Thus the question: are they formally equivalent? I.e. is rejecting an hypotheses $H_0 = 0$ with a critical value $\alpha$ equivalent to $0$ not belonging to the confidence interval with critical value $\alpha$? I'm trying to find the name of a lower-budget science fiction film that was available for viewing on demands on MIT student cable in the late 1990s.  I suspect the film was made earlier in that decade, and it definitely seemed to have been influenced by RoboCop.  The plot involves characters is a somewhat dystopian near future.  One of the characters (named Hawkins, I think) is initially in a wheelchair, and that is interfering with his relationship with the woman he loves.  (He watches virtual reality simulations of himself having sex with her when he's alone at home.)  Maybe halfway through the movie, he decides there's no hope for their relationship, so he lets the bad guys turn him into a RoboCop-like cyborg.  I think he ends up fighting the other main characters, but I don't remember how the movie ends.  The film was clearly shot cheaply but competently.  The color palette was rather drab, I think, and there were few indoor sets.  It's possible that the film never got a normal release of any kind, since I don't know where the students at MIT cable got ahold of it.  So does anybody know what this film was?