Black Hole gravity pull Please I want to Know that if the radius of the black hole = 0 then how it have its surface and how it absorbs things?  $$\begin{align}  g &amp;= \frac{Gm}{0^2} \\    &amp;= \infty \end{align}$$  then without surface and volume how it absorbs things?
Can a black hole be explained by Newtonian gravity? In the simple explanation that a black hole appears when a big star collapses under missing internal pressure and huge gravity, I can't see  any need to invoke relativity. Is this correct?
How can I create a custom terminal command (to run a script)? Prove by induction that $n! > 2^n$ Suppose that when $n=k$  $(k\geq4)$, we have that $k!&gt;2^k$.  Now, we have to prove that $(k+1)!\geq2^{k+1}$ when $n=(k+1)$ $(k\geq4)$.  $$(k+1)! = (k+1)k! &gt; (k+1)2^k  \text{ (since }k!&gt;2^k)$$  That implies $(k+1)!&gt;2^k2$, since $(k+1)&gt;2$ because of $k$ is greater than or equal to 4.  Therefore $(k+1)!&gt;2^{k+1}$  Finally  we may conclude that $n!&gt;2^n$ for all integers $n\geq4$    Solved, thank you! Is it possible to scale up an image to increase the the pixel density? create multilingual node programmatically i'm trying to create a multilingual node using a php script.  I'm doing this:   $node = new stdClass();  $node-&gt;title = "Test ".mktime();  $node-&gt;type = "job";  $node-&gt;language = 'de'; // Or e.g. 'en' if locale is enabled  $node-&gt;uid = $user-&gt;uid;   $node-&gt;status = 1; //(1 or 0): published or not  $node-&gt;promote = 0; //(1 or 0): promoted to front page  $node-&gt;comment = 1; //2 = comments on, 1 = comments off  $node-&gt;path = array('alias' =&gt; 'das-ist-ein-test-'.mktime());  $node-&gt;body[$node-&gt;language][0]['value']   = "Body Body Body";  $node-&gt;body[$node-&gt;language] = text_summary("Body Body Body");  $node-&gt;body[$node-&gt;language]  = 'filtered_html';  node_save($node);   As long as i set $node->language to 'und' everything is working fine. But as soon as I set the language to 'de' or 'en' (both languages are activated in my Drupal installation) the body text doesn't get saved.  So my question is: can anybody provide me with a working example of how to store a multlingual node (de,en) into Drupal using program code.  Thank you Display each sub-directory size in a list format using one line command in Bash? I want to get a list of the directories and their sizes in a list format like how you get when you do a ls -l.  The thing is that is there a one line command that can do this? I see others have long commands just to output this. That's just too long.  What command can do this or combination of commands that can be easily typed? du -h gives it, but it displays all of the sub-folders which is not what I want. just the current directories folders. What's the difference between lists and tuples? What's the difference?  What are the advantages / disadvantages of tuples / lists? Which is correct, "you and I" or "you and me"? Let $f: \Bbb R \to \Bbb R$ be a differentiable function such that $\sup_{x \in \Bbb R}|f'(x)| \lt \infty$. Then (UGC CSIR-2015, DECEMEMBER, MATHEMATICAL SCIENCES)        $f$ maps a bounded sequence to a bounded sequence.   $f$ maps a Cauchy sequence to a Cauchy sequence.   $f$ maps a convergent sequence to a convergent sequence.   $f$ is uniformly continuous.      I choose all of the options as possible answers because the condition $\sup_{x\in \Bbb R}|f'(x)| \lt \infty$ forces $f$ to be uniformly continuous.(Because $f$ becomes Lipschitz and Lipschitz condition implies uniform continuity)  i.e. $\frac {|f(x)-f(y)|}{|x-y|} \le \sup_{x\in \Bbb R}|f'(x)|$ $ \forall x,y$.  Hence all other options are bound to be true.  Am I correct? How to copy text out of a PDF without losing formatting? I am trying to classify using randomForest classification in GEE . However, I always get the error User memory limit exceeded, so I assume the number  of the feature are too large. Nevertheless, I saw many developers can classify even bigger dataset, so I am wondering if there is a better option there? The code can be accessed via  var fire= ee.FeatureCollection('users/spatola/recovery_rf_1712').map(function(feature){     var num = ee.Number.parse(feature.get('CLC_311'));     return feature.set('CLC_311', num);   }) var indices= ['EVI_mean','Dnbr','NBR2_mean','NBR2_min','NBR2_max','EVI_min','EVI_max']; var fire1=  fire.filter(ee.Filter.notNull(indices)); var fire2= fire1.select('Class_R','Dnbr','EVI_max','NBR2_max','imageId') var bands= ['NBR2_max', 'EVI_max']; var classProperty= 'Class_R'; var random= fire2.randomColumn('random'); var split_train= 0.8; var split_test= 0.2; var trainPartition= random.filter(ee.Filter.lt('random',split_train)); var testPartition= random.filter(ee.Filter.gte('random',split_test));  var RFtrained_c= ee.Classifier.smileRandomForest({  numberOfTrees: 100,  minLeafPopulation:2,  seed: 123,  }) .setOutputMode('CLASSIFICATION') .train({     features: trainPartition,     inputProperties: bands,     classProperty: classProperty,     }) print(RFtrained_c); var dict =  RFtrained_c.explain(); var test = testPartition.classify(RFtrained_c) print(test.first()) Reputation has been changed but not reflected on tooltip. Is there a term for an issue or activity that occurs only until the moment that you ask someone else to watch? Is there a term for the coincidence that occurs when an issue ONLY occurs until the moment that you ask someone else to watch or help you resolve it?    For example, having an issue with software until you try to show someone, then it stops being problematic.  Or an issue with the computer/phone that resolves as soon as a help ticket is put in.    The final example is when you have a problem with your car (funny noise, etc) - you hesitate taking it in for weeks because you know it will disappear as soon as the mechanic drives it...then it does.    This is different than a , where the action of applying a debugger, for example, changes the behavior of the activity being watched.  It's also different from a 'works on my machine' scenario, where only the original user has the problem.   What is it called when a problem resolves itself just by the mention of it?  Jinxing the issue? :)  I see this every day supporting my software product -- the users are experienced and are not making mistakes at one time then doing something different when I watch.  The problems also truly resolve (coincidentally) as soon as they ask for help - they do not reoccur.  I guess it's a case of synchronicity? Summation of series $\sum_{n=1}^\infty \frac{n^a}{b^n}$? How can we evaluate this series  $$\sum_{n=1}^\infty \frac{n^a}{b^n}?$$  Here $a$ and $b$ are positive integers. If $b=1$ then series will be diverging, in other cases, it will be converging, but how to find this sum?  Isn't there a simple solution which doesn't involve Stirling numbers? The Ubuntu installer does not allow me to install on NTFS partitions, but certain circumstances requires me to do so. Is it possible? FaceTime not working on iPhone 6 Plus purchased from Dubai I bought my iPhone 6 Plus in Dubai. I am using the phone in Sri Lanka and I cannot use the FaceTime. How can I use FaceTime while in Sri Lanka? How do rich people guarantee the safety of their money, when savings exceed the FDIC limit? I am curious how a millionaire would guarantee the safety of his money, given that the FDIC only insures up to $250K of an individual's deposits at a bank.  If someone had $3 million that they wanted to put into the bank, would they have to open up 12 different bank accounts and deposit $250K into each one, so that all of his money is insured by the FDIC?  Or is there an easier way to guarantee the safety of all his money? If $\gcd(a,b)=1$, then there exists integers $x$ and $y$ such that $xa + yb = 1$ Are global variables bad? In C/C++, are global variables as bad as my professor thinks they are? I need a two word phrase to refer to a community of people (academia + non-academia) who come together to share/exchange their knowledge/ideas openly on topics of their interests/ specializations.  I would appreciate the use of small and easier/common words for the phrase. Single words are also acceptable. Feel free to throw out your creative ideas and newly invented phrases. Trying to plot the sin(x) function, I get this result:    The MWE is here:  \documentclass{standalone} \usepackage{ pgfplots} \begin{document} \begin{tikzpicture} \begin{axis} \addplot[samples=500]{sin(x)}; \end{axis} \end{tikzpicture} \end{document}   cos(x) looks like an inverted parabola and tan looks just like sin.  I'm using pdflatex and MiKTeX. Any ideas?