If I apply the relations given in  and  to the values for the mass of the observable universe (obtained from Wolfram alpha) as M=3.4 10^54 kg, then I get a diameter of 1.1 10^28 m, about 10 times the estimated diameter of the obs. universe at 8.8 10^26 m. Clearly in the same ballpark. Tinkering with the mass, and excluding dark energy (for lack of 'rest mass'?) reduces the input to 28% of M, and that produces a black hole diameter only 3 times too big. Now, is this coincidence? Reading , I find no other remark addressing this 'coincidence' than Johannes's (quote) The radius of the observable universe equals the Schwarzschild radius of the total mass of the observable universe (unquote). Is Johannes' statement correct and generally accepted? If so, what is the logic here?  Edit: from"": "... we'd have shown that the Schwarzschild radius of the mass of the observable universe is equal to it's radius. Sadly it doesn't quite work. The dimension R is the current size of the observable universe, which is around 46.6 billion light years, while the size used in Hubble's law, re, is the current apparent size 13.7 billion light years." So we are to conclude that the 'close fit' (apart from a factor 3 or so) between the Schwarzschild radius and the estimated mass (apart from dark energy) is indeed a numerical coincidence? Please note that John Rennie's argument introduces the 'critical density' and Hubble's Law, while the relation between mass and radius of a black hole allow us to input a mass and obtain a radius without that.
So i was reading about  on Wiki and I found a interesting thing written there .    It says that the S. radius of the universe is as big as the size of the universe?   How is this possible?  Since most the universe is empty space shouldn't the S. radius of our universe be significantly smaller then 13.7 light years?
Video editing. Audio and Video strip have different length Tool to confirm Gaussian fit I have a series of (x1,y1) points.  I'm using a 3rd party software tool to which I feed these points. The tool then provides a mechanism for me to get back a series of (x2,y2) points that are on a Gaussian curve that has been fit to the data.  I'd like to confirm that the points I'm getting back are correct (because, frankly, sometimes they don't look like they are.)  What would be an easy on-line or Excel procedure that would allow me to enter the X1, Y1 series and then get back a series of X2, Y2 points, plot both series, and maybe even see the s.d.?  It's been 20+ years since I did any stats, so I'm just "cook-booking" this with no understanding of what I'm trying to do.  EDIT  The is a plot by wavelength, of intensity.  X-axis is the colors of a rainbow, in the familiar ROYGBIV order. The Y-axis is measured intensity of that particular color.   The help page from the vendor is here:    I call the library with each x1, y1 data point.  I then call the method above. I then call the library again with a series of x values (or maybe the x1 series itself), retrieving the y2 values on the fitted curve. I also retrieve the sd of the fit.  EDIT 2  Below is a link to an image of the dataset and Gaussian fit that has me concerned.  I think that the peak of the fitted curve is too low. What rules say that summoning undead is evil, and what rules say why? Geomagnetic reversal? I have always been told that the geomagnetic field acts "as if" there were a bar magnet inside the Earth. I now this is not true, but I feel that knowing what is actually going on would help me understand the answer to the question that I'm about to ask:  Why does geomagnetic reversal happen? Is it something gradual or relatively fast?  (What is the origin of the geomagnetic field? Can we answer my questions starting from this mechanism? How to assign the result of the previous expression to a variable? I have got a specific folder located in "/home/.../reboot/".  I have a series of users that require read-only access to the ever changing files within.  My problem is that the same users are not authorized to view any of the parent directory files.  Right now, the only thing I have in place is that their starting folder is at "/home/.../reboot/" by changing the etc/passwd file; and I need to block all access to other folders.  How could I only give them access to this directory? Apply a modifier as a shapekey into existing shapekey or merge shapekeys? I am making a shapekey of a model shrinkwrapped to another model. The shrinkmap modifier is too inaccurate to wrap to the model alone. So, my first step was to roughly hand position the geometry around of the first model around the second, then use shrinkwrap. The rough positioning was done as a shape key. I can then apply shrinkwrap, but its as a separate shapekey. The two do not work well together. I need them to be the same shapekey. How do I apply a modifier to an existing shapekey, or merge/combine/join the two shapekeys? How do I create an Ubuntu live USB using a Mac? I'd like to create an Ubuntu live USB stick on a Mac to use to install Ubuntu on another machine. How do I accomplish this? My colleague and I were discussing about certain types of customers in the IT industry. You have to work extremely hard to convince them so that they think about opening their wallets .  However, they should not be mistaken to be stingy (carries a negative connotation). Cost-conscious is a closely related term but is a bit wordy. Another word that I considered is shrewd but I don't know if it fits my context well.  Usage Sentence:     Nowadays, IT customers are becoming increasingly ________. They demand more for less and probably think a zillion times before reaching for their pockets. If they are not convinced 100%, they simply won't open their wallets.    Is there is a noun or an adjective to describe such people? I am also open to using idioms, if there is one that conveys the intentions well.  N.B:  I am looking for words with neutral to positive connotations. Convergence of exponential Brownian martingale to zero almost surely Define the exponential Brownian martingale as $N_t = \exp\left\{a W_t - \frac12 a^2 t \right\}$ which is a martingale with respect to the natural filtration of $W$ which stands for a standard Brownian motion.  I now want to prove that $N_t \rightarrow 0$ a.s. for $t \rightarrow \infty$. As $N_t$ is a martingale it is by definition in $L^1$ hence Doob's martingale convergence theorem tells us that there exists a $N_\infty \in L^1$ such that $N_t \rightarrow N_\infty$ a.s., but how could I prove that $N_\infty \equiv 0$? Any help is greatly appreciated! Proof Verification: Converse of Intermediate Value Theorem A function $f$ is increasing on $A$ if $f(x)\leq f(y)$ for all $x&lt;y$ in $A$. Show that the Intermediate Value Theorem does have a converse if we assume $f$ is increasing on $[a,b]$.  The converse of IVT along with the additional hypothesis would look like this: Lef $f:[a,b] \rightarrow \mathbb{R}$ be an increasing function on $[a,b]$ which satisfies: if $L$ is a real number such that $f(a)&lt;L&lt;f(b)$ (or $f(a)&gt;L&gt;f(b)$), then there exists a point $c\in (a,b)$ where $f(c)=L.$ If the preceding conditions are met, then $f$ is continuous on $[a,b]$.  The proof (if it's correct) I'm going to show is only for the $c \in (a,b)$, but I think for end points ($a$ or $b$), the method should be similar.  We want to show that given $c \in (a,b)$, for all $\epsilon&gt;0$, there exists a $\delta&gt;0$ such that $|x-c|&lt;\delta \implies |f(x)-f(c)|&lt;\epsilon$.  Since $f$ is increasing, we know that $f(a)\leq f(c)$. If $f(c)-\epsilon&lt;f(a)$, then set $x_1=a$, if $f(c)-\epsilon\geq f(a)$, then we know by hypothesis that a $x_1&lt;c$ exists such that $f(x_1)=f(c)-\epsilon.$ In either case, we have for $x \in (x_1,c]$ $$ f(c)-\epsilon \leq f(x_1) \leq f(x) \leq f(c) .$$  In a similar way, since we know that $f(c)\leq f(b)$, we can deduce that a $x_2&gt;c$ exists such that, for $x \in [c,x_2)$ $$f(c)\leq f(x)\leq f(x_2)\leq f(c)+\epsilon .$$  Pick $\delta = \min \{c-x_1,x_2-c \}$, we then have $$|x-c|&lt;\delta \implies |f(x)-f(c)|&lt;\epsilon .$$  Is this correct? In the proof that I saw, they originally used $\frac{\epsilon}{2}$, instead of just $\epsilon$, but I think this one works just as well, what do you guys think? Find the all possible real solutions of $x^y=y^x$ Find the all possible real solutions of $$x^y=y^x$$ $x,y$ both are real numbers.  My attempt:I observed the following solutions  $x=2,y=4$  $x=4,y=2$  $y=x$  Is there any other possible solutions? Fastest way to clamp a real (fixed/floating point) value? Is there a more efficient way to clamp real numbers than using if statements or ternary operators? I want to do this both for doubles and for a 32-bit fixpoint implementation (16.16). I'm not asking for code that can handle both cases; they will be handled in separate functions.  Obviously, I can do something like:  double clampedA; double a = calculate(); clampedA = a &gt; MY_MAX ? MY_MAX : a; clampedA = a &lt; MY_MIN ? MY_MIN : a;   or  double a = calculate(); double clampedA = a; if(clampedA &gt; MY_MAX)     clampedA = MY_MAX; else if(clampedA &lt; MY_MIN)     clampedA = MY_MIN;   The fixpoint version would use functions/macros for comparisons.  This is done in a performance-critical part of the code, so I'm looking for an as efficient way to do it as possible (which I suspect would involve bit-manipulation)   EDIT: It has to be standard/portable C, platform-specific functionality is not of any interest here. Also, MY_MIN and MY_MAX are the same type as the value I want clamped (doubles in the examples above). Backup GNOME-Terminal nginx support for .htaccess / rewrite rules? Differences from Apache? When does a Square Matrix have an LU Decomposition? Are global variables bad? Correct usage of 'but for'? Does 'but for' mean:  'If we had X (but we didn't), Y would have been the consequence'?  Or can it also mean; because we had X, as a result Y happened?   Some different examples of but for:  (Case 1)     But for a cup coffee, it was to be a productive day of writing.   Unfortunately, I'd run out, and spent the day craving.    (Case 2)     But for the glass of wine I had dinner, the house work didn't get   done.   (Case 3)     But for the glass of wine I had for dinner, I would have got the   house work done   What is the correct usage of "but for"? Combinations of selecting $n$ objects with $k$ different types Suppose that I am buying cakes for a party. There are $k$ different types and I intend to buy a total of $n$ cakes. How many different combinations of cakes could I possibly bring to the party? How to trigger workflow from button in O365 I created a script editor webpart. I added a button which start a js function in the onClick event. I would like to start a workflow which is implemented on some list. Is this possible?