The scatter plot shows two data series (x,y) the are placed somehow around a tilted ellipse. This is in fact the way I have drawn this example plot. Which statistics of the data sets are the most suitable to estimate back the geometrical parameters of the underlying ellipse? The objective is to have an ellipse that deviates the least from the points (perhaps considering distances in direction normal to the ellipse?). Such parameters could be the location of the centre, the lengths of the major and minor axes and, interestingly, the tilt with respect to the x axis.  There are many mathematical representations for an  -- using any that is amenable to a statistical interpretation would do.
Question: Best way to derive an ellipse formula when given bunch of points (say 60 points which when connected draws an ellipse)?  Background: Using R Momocs library function  which returns confidence ellipse coordinates. Would be nice to abstract those points into a formula.  You can see my ellipse points in the image below;    Any suggestions much appreciated, Thanks
Does any particle ever reach any singularity inside the black hole? Is there some rule against ending a sentence with the contraction "it's"? I heard this lyric in a song the other day and it just sounded so wrong that I assumed it must be incorrect grammar, but I can't find any specific prohibition that applies.     That's what it's.    That rolls off your tongue with the grace of a moose in a tutu, but I can't figure out why.  There is clearly no problem with ending other sentences with a contraction. These sound fine.     I thought I could, but I can't.   Stop touching that, it will fall off if you don't.   You say that the sky is green, but it isn't.   Also, it sounds just fine if you remove the contraction:     That's what it is.   So what's up with this construction? Should it be avoided? Backup of clients' directory to Windows Server 2008 Is this an instance of Priori Incantatem? During the climax of Fantastic Beasts and Where to Find Them, Tina briefly duels with Graves and their wands appear to lock with each other (see picture). I believe this also happens some other times in the later Harry Potter films other than Harry and Voldemort's duel in the graveyard during Goblet of Fire.    We know that this is definitely a case of Priori Incantatem for the graveyard duel in Goblet of Fire, as the wands are locked for a sufficient amount of time for past spells to begin showing themselves (significantly, showing the people Voldemort had murdered), but is this the case for the other instances of wands locking? If so, does that mean there are a lot more wands out there with shared cores, as it was my understanding this could only happen if that criteria was satisfied? Break a long command to multiple lines I got a shell script and inside I got the command as below #!/bin/sh sh ../jboss/bin/standalone.sh --server-config=standalone-full.xml -Djboss.node.name=node1 -b 0.0.0.0 -bmanagement 0.0.0.0  How can I break the line to multiple lines for better understanding and management #!/bin/sh sh ../jboss/bin/standalone.sh    --server-config=standalone-full.xml   -Djboss.node.name=node1   -b 0.0.0.0   -bmanagement 0.0.0.0  Tried with backslash at the end of each line and it doesnt work. How do I install drivers for the RTL8821CE Network Controller on Ubuntu 20.04? My Laptop (HP Laptop 14-ck1xxx) came with the RTL8821CE Network Controller. When I run lspci | grep Network I get: 03:00.0 Network controller: Realtek Semiconductor Co., Ltd. RTL8821CE 802.11ac PCIe Wireless Network Adapter  I‘m unable to get the Wi-Fi to work. What I did so far:  Download the repo at . Unzip the file and rename the folder from rtl8821ce-master to rtl8821ce. Run the following commands in the terminal:  cd Downloads/rtl8821ce/ sudo apt install bc module-assistant build-essential dkms sudo m-a prepare sudo ./dkms-install.sh    Everything seemed to install and execute successfully. Then, I rebooted my laptop and made sure secure boot is disabled. However, Ubuntu still displays: No Wi-Fi Adapter Found - make sure you have a Wi-Fi adapter plugged in and turned on  The command uname -r returns: 5.4.0-7634-generic  The command sudo lshw -C network returns: *-network                         description: Ethernet interface        product: RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller        vendor: Realtek Semiconductor Co., Ltd.        physical id: 0        bus info: pci@0000:02:00.0        logical name: eno1        version: 15        serial: c4:65:16:08:52:bf        size: 1Gbit/s        capacity: 1Gbit/s        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress msix bus_master cap_list ethernet physical tp mii 10bt 10bt-fd 100bt 100bt-fd 1000bt-fd autonegotiation        configuration: autonegotiation=on broadcast=yes driver=r8169 duplex=full firmware=rtl8168h-2_0.0.2 02/26/15 ip=192.168.178.37 latency=0 link=yes multicast=yes port=MII speed=1Gbit/s        resources: irq:16 ioport:4000(size=256) memory:a1204000-a1204fff memory:a1200000-a1203fff   *-network UNCLAIMED        description: Network controller        product: RTL8821CE 802.11ac PCIe Wireless Network Adapter        vendor: Realtek Semiconductor Co., Ltd.        physical id: 0        bus info: pci@0000:03:00.0        version: 00        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress cap_list        configuration: latency=0        resources: ioport:3000(size=256) memory:a1100000-a110ffff  What am I doing wrong? How can this be solved? Showing $\sum_{i = 1}^n\frac1{i(i+1)} = 1-\frac1{n+1}$ without induction? I oversaw a high-school mathematics test the other day, and one of the problems was the following     Show, using induction or other means, that $$\sum_{i = 1}^n\frac1{i(i+1)} = 1-\frac1{n+1}$$   The induction proof is very standard, where the induction step relies on the fact that $\frac{1}{n+1} + \frac{1}{n(n+1)} = \frac{1}{n}$, and I'm sure it's been answered on this site before. However, I got intrigued by the "or other means" part of the question.  I don't know whether the teacher who wrote the test even considered any alternative solutions (he may just have written it so that if anyone has a crazy idea that works out, then they should get a full score for it), but I tried to find one. For instance, we may do the following telescope-ish argument: $$ \frac{1}{1(1+1)} + \frac{1}{2(2+1)} + \cdots + \frac{1}{(n-1)n} + \frac{1}{n(n+1)} + \frac{1}{n+1}\\ = \frac{1}{1(1+1)} + \frac{1}{2(2+1)} + \cdots + \frac{1}{(n-1)n} + \frac1n\\ \vdots\\ = \frac{1}{1(1+1)} + \frac12\\ = 1 $$ However, I feel that this is just an induction proof in disguise (hidden in the vertical dots). If one uses the mechanics of the induction proof to check whether the formula is true for a specific $n$, then one certainly does the exact same calculations as I have done here.  Is there a proof of this fact that clearly does not use induction (or at least hides it better)? The more elementary the better, and the ultimate goal would be to do it within the syllabus of the students taking the test (or at least not far from it). For reference, they should be familiar with the summation formula of arithmetic and geometric series and their derivations (so techniques resembling those would be well within specifications).  If there is a solution using calculus, then the students should be able to integrate elementary trigonometric functions, as well as exponential functions, logarithms and rational functions. They are familiar with integration by parts, substitution and partial fractions.  I welcome more advanced solutions as well, of course. I am trying very hard to install Canon LBP2900b Printer in Ubuntu 13.10.  I have searched and googled a lot for the solution over fortnight but none of the site / link gave me the simple solution for me. How can accomplish my goal? How to change value of ArrayList element in java So this is something I just found out about that I find odd:   Opportunity attacks—trigger off of movement that uses the creatures movement (action, move action, bonus action, reaction.)  Sentinel—Whenever a creature within 5 ft of you makes an attack at a creature other than you.   Booming blade—If the target willingly moves.   Mental Prison spell—if it moves, attacks through it, or reaches any part of body out of it.   So the “willingly” is different than other rulings for how effects related to movement works. However what exactly fits willing?   Cause fear—nope magic is having them move. Dissonant whispers—same thing. They have to immediately move.   So the creature has to want to move completely of its own accord.   Crown of Madness—Humanoid creature you choose must make a wisdom saving throw. On a fail it is charmed by you. (Next is flavor text so skipping.) While charmed in this way the target must make an attack on a creature other than itself that you mentally choose on each of its turns before moving.  It can act normally if you choose no creature or if none are within reach.   Crown of madness isn't making them move but doesnt specifically say willing.  And if the ruling for willing is movement not caused by magic what about magic causing situations where you take damage if you don't move. Would a creature shimmering with evocation energy be spared because he moved out of the magical blender that is cloud of daggers? If no, then what is the go-to rule for willing movement that I should be using?  In short, what movement counts as willing, and where can it be found? Is moving out of obvious danger caused by magic, willing? Is moving away from an ally so you don't hit them, willing?   If there isn't any actual ruling on willing, that's fine; I'll go back to running it with my interpretation, so that it has same wording as AOO. If there is an actual 5e term or explanation for willing (that isn't Sage Advice), please let me know. I just don't see why it was necessary to put a term in the spell that isn't used anywhere else. I understand it could have been overlooked since it was in a splatbook or has just been ignored since most people probably assume that there is "Forced " and "Willing" movement, with the willing being defined as: any movement from an action, move action, bonus action, or reaction. In same way AOO work. Long names cut off in questions and answers The Unhandled Exception's name is cut off in this  and in the answer too. The last 'n' looks like an 'r'.  Using Firefox 3.6.6.  Examples: The exercise is: Prove that $-(-v)=v$ for every $v \in V$  Proof Suppose  $v \in V$ and $V$ is a vector space. Then $-(-v) \in V$ as result of the scalar multiplication property and $-(-v)=-(-1\cdot v)=-1 \cdot(-1\cdot v) =(-1 \cdot-1)\cdot v = 1 \cdot v = v $ The desired result $-(-v)=v$ holds.  The solution manual gives the proof:  Proof   I just wanted to make sure that the way I did the proof isn't missing anything?Thanks This question might be wrong, I am not sure about it. Because I am not clear on this.  I know add_action it is used to to hook our function to the specified function. For example add_action('wp_head'.'myfunc'); now what ever code in myfunc will be executed in the wp_head(). This is clear but i am having doubt in do_action what it does?  I think it is used to create our own hook like already available hooks(wp_head,wp_footer,..etc) If i am correct can anyone show me a simple understandable answer with simple example.  I have tried the difference in internet but all are pointing to difference between add_action and add_filter. I don't want to go there because first i want to clarify this and then I’ll move there.  Can anybody help me?  EDIT after Question POST  function custom_register() {     echo '&lt;script&gt;jQuery(document).ready(function(){alert("Learning Hooks");});&lt;/script&gt;';  } do_action('custom');  add_action('custom','custom_register');   I tried this in plugin but i didn't get the alert message.  But when i hook the same function with wp_head then it is working fine  /******************working****************/ add_action('wp_head','custom_register'); every planar graph can be expressed as a union of five edge-disjoint forests How can I prove that every planar graph can be expressed as a union of (maximum) five edge-disjoint forests?  Only thing I can think about is going somehow like the $5$ color theorem but I get stuck in the middle. Or with induction, but still something doesn't fit. Can you help? What's wrong with this Wheatstone bridge? I was trying to solve this unbalanced Wheatstone bridge and found that the current $i$ in the mid wire is 9A .This result was achieved by using node voltage method.    Then I rearranged the same circuit to the form shown in the second image.  and at this point I flipped the right half of the circuit so as to obtain the equal resistors on the same side.    now I again rearranged the circuit as shown.    Here, the final circuit  is a balanced Wheatstone bridge and thus the current i must be 0.  So clearly something went wrong in there.  I saw a post  before this one just to make sure that there is no mistake in rearranging the circuit in the way I have done.   It was answered by the user PhillipWood as being correct and was unfortunately put on hold.  So can someone please point out what exactly went wrong with the above method. How do I remove the LAMP stack so I can start over? I'd like to start over with my webserver and I'd like to reinstall all the LAMP packages, how do I do that? jQuery 1.9 checkbox not checked second time after once it unchecked Definition of textbook RSA What is the definition of textbook or "raw" RSA?  What are some of the properties of textbook RSA?  How does it differ from other schemes based on RSA? Why would spacetime curvature cause gravity? Show that every uncountable set of real numbers has a point of accumulation.