How can I solve the following second order differential equation   $$ \frac{d^2\theta \:}{dt^2\:}\:=-\frac{g\theta \:\:}{L} $$ with initial conditions $$θ(0) = θ_0 , v(0) = v_0  $$  I know that I can rewrite the equation as  $$ \frac{d^2\theta \:}{dt^2\:}\:+\frac{g\theta \:\:}{L} = 0 $$    , find characteristic polynomial and solve the equation. I am just confused on how to solve this problem because of all the constants.  Thanks for help in advance
The differential equation $$ {d^2\alpha \over dt^2} + {g \over L} \cdot \alpha = 0 $$  describes a 1-dimensional mathematical pendulum, where $\alpha $ is the angle, $ g = 9.82 $, and $ L = 0.2 $ is the lentgh of the string. What is the position of the pendulum after 1 second if the velocity at time equals zero is $ 0 $ m/s and the angle at time equals zero is $ {\pi \over 60} $ radians.  To my understanding this is a homogenous equation, ergo  $$ y'' + Cy = 0$$  And with the help of the characteristic equation I get the complex roots $ Ci \, $ and $ -Ci $.  This is a farily simple differential equation, but I have a feeling my calculations so far are wrong (mainly because I didn't expect to see complex roots for this equation).
Polygon intersection using turf js I have a laptop running 11.10 with 2 network interfaces, wired and wireless.  Wireless interface connects to the router with DHCP and connects to the internet. Wired interface is connected to a switch. I want the computers on the switch to get their IP addresses from the router not my laptop.     How would I go about setting this up? Bridging the interfaces? Bash script copy file to user's (wildcard) home dir How would one go about proving that the rationals are not the countable intersection of open sets? I'm trying to prove that the rationals are not the countable intersection of open sets, but I still can't understand why   $$\bigcap_{n \in \mathbf{N}} \left\{\left(q - \frac 1n, q + \frac 1n\right) : q \in \mathbf{Q}\right\}$$   isn't a counter-example. Any ideas? Thanks! I`m using Firefox Sync on a bunch of computers.   I would like to manage my account, by removing some computers. I can't find the place to do this, it should be on the Firefox Sync page, but the official page only has the privacy and download information. If you have a single atom in a box, what state of matter would it be in? If the walls of the box were made of a completely inert material, what state of matter would the atom be in? Or would it not have one? I realized this is a duplicate of several other questions. Lightning Components Basics: Attributes and Expressions Trail The error :      The Packed field is either not using the lightning input component,   the checked value of the item's Packed__c attribute or the correct   type of input.   My Code:  &lt;aura:component &gt;     &lt;aura:attribute name="item" type="Camping_Item__c"                      required="true"                     default="{Name:'Tent', Price__c:100, Quantity__c:1, Packed__c:true}"                     /&gt;         &lt;p&gt; The Item is &lt;ui:outputText value ="{!v.item}"&gt;&lt;/ui:outputText&gt;&lt;/p&gt;          &lt;ui:outputText value="{!v.item.Name}"/&gt;     &lt;lightning:input type="toggle" label="Packed" name="togglevalue" value="{!v.item.Packed__c}" /&gt;     &lt;lightning:formattedNumber value="{!v.item.Price__c}" style="currency" currencyCode="USD" currencyDisplayAs="symbol"/&gt;      &lt;lightning:formattedNumber  value="{!v.item.Quantity__c}"/&gt; &lt;/aura:component&gt; Deleting array elements in JavaScript - delete vs splice What was the origin of Star Trek's Borg? How many days a year is a U.S. citizen allowed to stay in Morocco? I arrived in Morocco in October of 2014 and renewed my visa by traveling to Spain by ship for a day and getting my passport stamped again in January of 2015. My question is, am I allowed to leave the country and repeat this process this year by traveling again to another country, other than my own in order to return to Morocco for another 90 days? And if so, how many times a year is this allowed? I have read 180 days on some websites but there is not any clear answers anywhere about this that I can find. I am trying to encrypt an entire USB drive with Truecrypt 7.1a. But as long as there is a partition it will only encrypt that. This will make it appear like the USB is full instead of completely empty, which isnt what Im aiming for.   I've tried with DISKPART, USB Partition Tool, and the built-in storage manager. When I delete all partitions, Windows still assigns the drive a letter and assumes that it has a partition covering the whole space.  I know that it is possible as I have done it with another USB, but I just don't remember how I did it.  Ideas? I am trying to retrieve VersionData with a standard REST API call from a VisualForce Page a then upload it to a S3 bucket but end up with corrupted files. I believe this is an encoding problem on the client side as plain text or CSVs work like a charm. I can upload VersionData from Apex but there are heap and request size limitations so I need to send the data from the client.   This is the call and returned string, I tried changing mimeType and encoding on both sides (SF call and S3 upload) with no luck.   Any ideas how to properly process images or pdf files? Thanks a lot  THE PROBLEM IS NOW SOLVED  Files should not be treated as strings. I solved the problem by leaving jQuery's ajax as I did not succeed to request a response type. By switching to plain JS and making a GET call (XMLHttpRequest) with response type set to blob I received an Object, instead of a string. Base64 string can be then retreived using JS FileReader. The boundary of union is the union of boundaries when the sets have disjoint closures How to Expand , Resize "extended partition"? I have a USB drive that I can see/access/modify/etc in Linux/Ubuntu but it doesn't show up on my Mac OSX.  I did sudo fdisk /dev/sdb (where my USB is) and then g to create a new GPT partition table, n to make a new partition, used all default settings, then w to save and exit.  Then I did sudo mkfs.fat -F32 /dev/sdb1 to format the partition with a FAT32 filesystem.  I ejected the USB and put it in my Mac and it doesn't show up / get detected.  What did I miss? Let $R= \mathbb{Q}[x^2 ,x^3 ]$, the set of all polynomials over $\mathbb{Q}$ with no $x$ term. Show that a gcd of $x^5$ and $x^6$ does not exist in $R$.   I am reasoning for the absurd and I am assuming that $gcd (x ^ 5, x ^ 6) = ax ^ 2 + bx ^ 3\in\mathbb{Q}[x^2 ,x^3 ] $, then, for the Bezout identity there exists $p (x) = cx ^ 2 + dx^3$ and  $q(x)= ex ^ 2 + fx ^ 3$ in $\mathbb{Q}[x^2 ,x^3 ]$ such that $x ^ 5p (x) + x ^ 6q (x) = ax ^ 2 + bx ^ 3$, which is absurd since $$9 = deg (x ^ 5p (x) + x ^ 6q (x)) \neq  deg (ax ^ 2 + bx ^ 3) = 3.$$  Is it okay what I did? Thank you very much. Here's a question I've been asked;  Given distinct primes $p,q,r$, show that any group $G$ of order $pqr$ is not simple.   So far, my idea has been to individually check each possible proper subgroup, since by Lagrange there are only $$\dbinom{3}{2} + \dbinom{3}{1} + 1$$ of them, and show that at least one of them other than the trivial group is normal. Surely there is a smarter way, since generalizing this approach wouldn't work so well if I wanted to show similar results for groups who's order is a product of $n$ distinct primes.  Is there some other approach? Thanks for any insights PostgreSQL Shapefile import failed I am very new to PostGIS/PostgreSQL. In fact, I downloaded it because of the answers I got to this question:   I installed both PostgreSQL and PostGIS. But I am not able to import a shapefile to the database? It gives an error, "Shapefile import failed". What am I doing wrong? Does Wasserstein distance require the source and target distributions to have the same mass? If we minimize the Wasserstein loss, $$W_1 (P_S, P_T) = \underset{\gamma \in \Pi}{\text{min }} \sum_{x,y} |x-y|\gamma(x,y)$$ which means we are looking for the coupling $\gamma(x,y)$ that minimizes the cost of moving mass from the source distribution $P_S$ to the target distribution $P_T$,  Do the masses of the source and target distributions have to be identical? If so or not, how are the masses of $P_T$ and $P_S$ accounted for mathematically in, or when minimizing, the shown formula, or what might be missing/overlooked by the Wasserstein approach overall due to the mass differences? Infinite algebraic extension of a finite field I have recently started studying algebraic field extensions and I got to know that algebraic closures $\overline{F}$ of finite fields $F$ are infinite. Therefore, I've asked myself the following question and couldn't come up with an answer.  Suppose $F$ was a finite field, i.e. $|F| &lt; \infty$. Is it possible to construct an algebraic extension $F'$ of $F$ sucht that $|F'| = \infty$ and $F' \subsetneq \overline{F}$? Meaning, that $F'$ is not algebraically closed.