Why isn't execute permission needed to source a file?
Why does Bash's source not need the execution bit?
I installed an alpha or beta, am I up to date with the final release if I keep upgrading? Conditional expectation of $X$ given $\sin(X)$ The following is a rigorous proof of De Moivre's theorem by means of mathematical induction. The theorem put simply is that:  Any complex number, $z = a+bi$, on a cartesian plane can be expressed in polar form, where $a=r\cos\theta$ and $b=r\sin\theta$ and $r$ is the absolute distance from the origin to the point $z$. In light of this, the expression for the complex number $z$ in polar form is $z=r(\cos\theta+i\sin\theta)$ In order to find the $n^{th}$ power of $z$, the following rule applies: $$\displaystyle z^n=(r(\cos\theta+i\sin\theta))^n=r^n(\cos n\theta+i\sin n\theta)$$ where $n\in\mathbb N$ (I have confined $n$ to the natural numbers given that I intend on using mathematical induction to prove the theorem.  Base step: We must prove that what is stated is true for $n=1$, hence $$z^1=r^1(\cos 1\cdot\theta+i\sin1\cdot \theta)$$ $$=r(\cos\theta+i\sin\theta)$$ which is true. Next we assume it is true for $n=k$, and therefore proceed to the inductive step. Inductive step: When calculating $z^{n+1}$, it's the same as calculating $z^nz$, hence $$z^{n+1}=(r^n(\cos n\theta + i\sin n\theta))(r(\cos \theta+i\sin \theta))$$ $$=r^nr(\cos n\theta+i\sin n\theta)(\cos\theta +i\sin \theta)$$ $$=r^{n+1}(\cos n\theta\cdot\cos\theta+\cos n\theta \cdot i\sin\theta+i\sin n\theta\cdot\cos\theta+i^2\sin n\theta\cdot\sin\theta)$$ from here, $i^2=-1$, and therefore $+i^2\sin n\theta\cdot\sin\theta$ becomes $-\sin n\theta\cdot\sin\theta$. In addition to this, the following trigonometric identities will be used: $$\sin(n\theta + \theta) = \sin n\theta\cdot\cos\theta+\cos n\theta\cdot\sin\theta$$ $$or$$ $$k\sin(n \theta+\theta)=k\sin n \theta\cdot\cos \theta + \cos n\theta\cdot k\sin\theta$$ $$and$$ $$\cos(n \theta + \theta)=\cos n\theta\cdot\cos\theta-\sin n\theta\cdot\sin\theta$$ by using these, the original equation has now become $$=r^{n+1}(\cos(n \theta + \theta) + i\sin(n \theta + \theta))$$ then, after factoring out $\theta$, the theorem becomes $$=r^{n+1}(\cos((n+1)\theta)+i\sin((n+1)\theta))$$ which completes the proof. Question: is this proof correct all the way through? Or have I missed any conditions? If all is well in the case of this proof, could somebody aid me in proving it for $\{n|n\in\mathbb R,n\geq1\}$ What would be the sum of the following series?  $$\sum_{n=1}^{\infty} \frac{1}{n \cdot 2^n}$$  Thanks Vortex in liquid collects particles in center At xmas, I had a cup of tea with some debris at the bottom from the leaves.  With less than an inch of tea left, I'd shake the cup to get a little vortex going, then stop shaking and watch it spin.  At first, the particles were dispersed fairly evenly throughout the liquid, but as time went on (and the vortex slowed, although I don't know if it's relevant) the particles would collect in the middle, until, by the time the liquid appeared to almost no longer be turning, all the little bits were collected in this nice neat pile in the center.  What's the physical explanation of the accumulation of particles in the middle?  My guess is that it's something to do with a larger radius costing the particles more work through friction... What is the correct way to show tiered possession? When there are multiple places where possession is shown, how are the words structured?  A few examples:   Greg's son's laptop is on fire. Mary's friends' boat is sinking. Thomas's neighbor's cat ate a mouse. Apple's Stock App's stats are useful. Microsoft's Xbox 360's games are fun.   Are these the correct ways to express possession? Fixing scrolling in nano running in tmux in mate-terminal The problem:   I open a terminal (in Linux Mint, so mate-terminal) zsh is the shell Then I run tmux Edit a file with nano Scroll up and down that file with the cursor Issue: When scrolling down in nano, only the bottom half of the terminal window gets refreshed Issue: When scrolling up in nano, only the top half of the terminal windo gets refreshed   The complete nano view of file does not get refreshed in my terminal window when scrolling. Any tips?  Edit: my .tmux.conf  It seems that this line specifically is the culprit (as commenting it out fixes the problem):  set -g default-terminal "xterm-256color"   I'm pretty sure I added that line because I have issues even running nano during an SSH session.  Here is the full file:  set-option -g default-shell /bin/zsh  # Make sure tmux knows we're using 256 colours, for # correct colourised output set -g default-terminal "xterm-256color"  # The following were marked as "unknown", so # I do know what I'm doing wrong. #set -g mode-mouse on #setw -g mouse-select-window on #setw -g mouse-select-pane on  # Attempting to stop "alert" sound upon startup # but none of these are working... set-option bell-on-alert off set-option bell-action none set-option visual-bell off What is the Fastest a Character Can Move in One Turn? At level 20 with any magic items in the DMG, and any spells, what is the fastest a DnD 5e character could move using their movement?  (i.e., not including teleportation) For all $a, m, n \in \mathbb{Z}^+$,   $$\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$$ How to create an extruded halftone effect in illustrator or photoshop The boundary of union is the union of boundaries when the sets have disjoint closures "Lunch" vs. "dinner" vs. "supper" — times and meanings? I've seen cases where a noon-time meal is referred to as dinner, and the evening meal is called supper. There's also lunch around noon followed by dinner in the evening. Is there a particular difference between dinner and supper, or a circumstance where lunch becomes dinner? I have a list of dictionaries and want each item to be sorted by a specific value. Take into consideration the list: [{'name':'Homer', 'age':39}, {'name':'Bart', 'age':10}]  When sorted by name, it should become: [{'name':'Bart', 'age':10}, {'name':'Homer', 'age':39}] Multiplexing UART I'd like to interface my main microcontroller's hardware serial port to more than one device at the same time. My understanding is that UART on my microcontrollers is based on an active-low 5V TTL.  (I know what I'm explaining here can be technically achieved by using more suited protocols, such as SPI and I2C, but my project requires to be only implemented using UART.)  Here's a pseudo-schematics chart of what I'm trying to achieve:         &ndash; Schematic created using   Please note that:   I don't want to use a second UART interface on the master, or any of the slaves. Only one slave at a time is going to transmit or send data on the line to the master. (e.g. If the slave 1 is transmitting data, slave 2 and 3 should remain idle.) Whatever the master is transmitting can, and should be received by all of the slaves. All of the slaves should be hot-pluggable, meaning they could be removed or inserted at any time. The RX and TX wires to the master are parallel between each slave, so no daisy chaining.   So, my questions are as follows:   Assuming the correct impedance is considered, I shouldn't need an additional circuit to transmit data from the master's TX line to any of the slave's RX line simultaneously. Is this correct? I understand that without current limiting resistors between slaves' TX line and the master's RX line, if any of the devices begin to broadcast data, the line between the devices will short circuit and I risk damaging them. Is this correct? Assuming the items 1 and 2 are correct:  a) Would just some current limiting resistors be enough to use?  b) Can I use 74-series logic gates instead of resistors? Between options a and b above, what route should I choose? For a reliable 115200 baud rate connection, will the switching frequency of a 74HC08 AND gate suffice to be incorporated?   EDIT: I'm well aware of other well-suited protocols such as SPI and I2C. There's a reason I'm using UART, and that's because the optiboot bootloader that is used on the master's mcu communicates over UART, albeit UART's obvious limitations.  Also, while daisy-chaining is definitely a clever approach, due to breaking the 4th requirement, it also breaks bootloader functionality.  The devices could also be USB-to-Serial chips, and using SPI-to-USB or I2C-to-USB would not be feasible in this project.  So, I'd appreciate if you could focus on the 6 questions listed here instead, and share your UART knowledge. From the limit definition of differentiability, determine where the function $f(x) = \sqrt{x}$ is differentiable? For the function $f(x) = \sqrt{x}, \hspace{2 mm} f: [0, \infty) \rightarrow \mathbb{R}$, I wish to find all points $a$ at which $f(x)$ is differentiable, from the definition $$ f'(x) = \lim_{x \rightarrow a}\frac{f(x)-f(a)}{x-a} $$  Clearly, for our function $f(x) = \sqrt{x}$, this gives us $$ f'(x) = \lim_{x\to a}\frac{\sqrt{x} - \sqrt{a}}{x-a} $$  So the question boils down to showing for what values of $a$ this limit exists. How would this be done? Intuitively, I know that the answer is that the limit exists for all $a \in [0, \infty)$ and is given by $\frac{1}{2 \sqrt{a}}$, but I'm unsure of how to prove this from the limit definition of differentiability. I got a situation where i need to invoke future method from batch class. And i know that it is not possible to do. But there might be some workarounds to achieve it, right? And i dont want to invoke future method using shedule apex as my requirement will hit governor limits of "Total number of schedule jobs exceed". So i tried in below ways, but still fighting with same issue.  My design is  Scenario 1 :   Schedule Apex ===> 2. BatchClass (Runs every hour) (Generates a map with "N" records for every batch context, and passes it to Future method in an apex class)  ===> 3. Future method ( then it will login to a salesforce account and generates an attachment and sends an email to a corresponding email address) .   Scenario 2 :  Since, It is not possible to invoke future method from batch, i tried implementing an apex class with a normal static method  invoking future method and in a batch class, for every context i.e., execute() method, it will invoke a normal method which in turn invokes future method, but still got the same error      FATAL ERROR: Future method cant be called from future or Batch   ie.,   global class BatchClass implements Database.batachable&lt;Sobject&gt; {    global Iterable&lt;ScheduledReport__c&gt; start(Database.BatchableContext BC)  {         query=[];        return query;  }  global void execute(Database.BatchableContext BC, List&lt;ScheduledReport__c&gt; scope) {      FutureClass.normalMethod(mapRecords);   }    global void finish(Database.BatchableContext BC){   }  }   Future Class:   global class FutureClass {  global static void normalMethod(Map&lt;&gt; mapR) {      futureMethod(mapR);  }  @future(callout=true)  global static void futureMethod(Map&lt;&gt; m) {      //some code;   }  }   Scenario 3 :  Instead of invoking future class method from batch i tried updating list of records and done checkbox value to true and i've implemented after update event trigger to get the all the records in which checkbox values are true and from trigger i tried invoking future method , still didnt work , i got an error      FATAL ERROR: Future method cant be called from future or Batch   Please help me out.   Thanks ! Getting "Authentication is required" when trying to access the Play Store Over the past week, whenever I try  to open the Play Store/Market app on my phone, the app thinks for a while and then displays this error message:     Authentication is required. You need to sign into your Google Account.   But I am indeed signed into my Google account, as seen in other Google applications running on the phone, and as seen if I open the menu in the Market app and click the "Accounts" button.  I suspect this might be related to the fact that over the past few months I've had two different Google accounts synced to this device, and I have recently removed one of them from the phone. No other Google application has had any problems with removing one account.  Does anyone know how I can solve this issue? I can still install applications by using the market website (now "Google Play"), but this is still a big inconvenience. I also think I stopped getting updates. Do we still need to avoid using frames and iframes for SEO? I thought frame and iframe were bad for SEO, but today I was doing a search for "numismatica" on Google.it.  The 3rd site is lemonete.com which is all made with frames. A competitor site (nummus.com) that has no frames ranks much lower when searching for the same word: "numismatica".  So can we use frames without taking an SEO hit now?  Any explanation (possibly simple) would be appreciated also about the fact the 2nd site ranks much lower. :) BeanFactory vs ApplicationContext I'm pretty new to the Spring Framework, I've been playing around with it and putting a few samples apps together for the purposes of evaluating Spring MVC for use in an upcoming company project. So far I really like what I see in Spring MVC, seems very easy to use and encourages you to write classes that are very unit test-friendly.  Just as an exercise, I'm writing a main method for one of my sample/test projects. One thing I'm unclear about is the exact differences between BeanFactory and ApplicationContext - which is appropriate to use in which conditions?  I understand that ApplicationContext extends BeanFactory, but if I'm just writing a simple main method, do I need the extra functionality that ApplicationContext provides? And just exactly what kind of extra functionality does ApplicationContext provide?  In addition to answering "which should I use in a main() method", are there any standards or guidelines as far as which implementation I should use in such a scenario? Should my main() method be written to depend on the bean/application configuration to be in XML format - is that a safe assumption, or am I locking the user into something specific?  And does this answer change in a web environment - if any of my classes needed to be aware of Spring, are they more likely to need ApplicationContext?  Thanks for any help. I know a lot of these questions are probably answered in the reference manual, but I'm having a hard time finding a clear breakdown of these two interfaces and the pros/cons of each without reading thru the manual with a fine-tooth comb. What is a good introductory book on quantum mechanics? I'm really interested in quantum theory and would like to learn all that I can about it. I've followed a few tutorials and read a few books but none satisfied me completely. I'm looking for introductions for beginners which do not depend heavily on linear algebra or calculus, or which provide a soft introduction for the requisite mathematics as they go along.  What are good introductory guides to QM along these lines?