Probability: NEED HELP to Understand with the follow
[Probability]need help to understand the following expression
Is there a way to simply splice in an additional length of 12/2 NM-B cable? Is there a way without using junction boxes to simply splice in additional length of 12/2 NM-B cable? Can I use twist caps?  I'm putting in a dog door and as it so happens, there is some 12/2 right in the way of the location I've chosen. There's no additional slack in the cable, and I don't want to rip up more drywall than I have to. I'm tempted to just cut it and splice in an additional foot or two with twist caps to get it out of the way. I don't know if this is safe or legal. Do I need to get junction boxes? Attached some pictures for fun.  Also, there is a 1-gang electrical outlet box that is going to the outside. I'm moving that up a bit. Show that $\ker \hat{T} = \text{ann}(\text{range } T)$ This is an old exam problem: Let $V$ and $W$ be finite dimensional vector spaces over a field $F$ and let $T: V \to W$ be a linear transformation. Define $\hat{T}: W^* \to V^*$ by $(\hat{T}(f))(v)=f(T(v))$. Here, $U^*$ denotes the dual vector space $U$. Since $\hat{T}$ is a linear transformation prove $\ker \hat{T} = \text{ann}(\text{range } T)$ where $\text{ann}(Y) = \{g \in U^* : g(y)=0 \text{ for all } y \in Y \}$ for a subspace $Y$ of vector space $U$.  I'm really not sure how to start this problem. Any advice would be great! Why does my minecraft server disconnect the internet when it crashes? SharePoint Current User View using JavaScript CSS transition shorthand with multiple properties? let $a,b,c,d,e$ are positive real numbers,show that $$\dfrac{a-b}{b+c}+\dfrac{b-c}{c+d}+\dfrac{c-d}{d+e}+\dfrac{d-e}{e+a}+\dfrac{e-a}{a+b}\ge 0$$  My try:  I have solved follow Four-variable inequality:  let $a,b,c,d$ are positive real numbers,show that $$\dfrac{a-b}{b+c}+\dfrac{b-c}{c+d}+\dfrac{c-d}{d+a}+\dfrac{d-a}{a+b}\ge 0$$ poof:since \begin{align*} &amp;\dfrac{a-b}{b+c}+\dfrac{b-c}{c+d}+\dfrac{c-d}{d+a}+\dfrac{d-a}{a+b}=\dfrac{a+c}{b+c}+\dfrac{b+d}{c+d}+\dfrac{c+a}{d+a}+\dfrac{d+b}{a+b}-4\\ &amp;=(a+c)\left(\dfrac{1}{b+c}+\dfrac{1}{d+a}\right)+(b+d)\left(\dfrac{1}{c+d}+\dfrac{1}{a+b}\right)-4 \end{align*} By Cauchy-Schwarz inequality we have $$\dfrac{1}{b+c}+\dfrac{1}{d+a}\ge\dfrac{4}{(b+c)+(d+a)},\dfrac{1}{c+d}+\dfrac{1}{a+b}\ge\dfrac{4}{(c+d)+(a+b)}$$ so we get $$\dfrac{a-b}{b+c}+\dfrac{b-c}{c+d}+\dfrac{c-d}{d+a}+\dfrac{d-a}{a+b}\ge\dfrac{4(a+c)}{(b+c)+(d+a)}+\dfrac{4(b+d)}{(c+d)+(a+b)}-4=0$$ Equality holds for $a=c$ and $b=d$  By done!  But for  Five-variable inequality,I can't prove it.Thank you If two coins are flipped and one gets head, what is the probability that both get head? Has there ever been a warp / non-warp collision? Like something going at warp hits a planet, star, another ship, moon, satellite, alien, tribble, etc.... Basically, can stuff in subspace (warped stuff) hit stuff in real space (not warp)?  Star Trek: The Voyage Home shows us clearly that entering warp speed within Earth's atmosphere has no adverse consequences for the starship or the environment.  HMS Bounty went through miles of atmosphere, dust, birds, and clouds and nothing entered  to hit her. Enterprise did this again by dropping out of warp in Titan's atmosphere.  which explains some tech used, but now we have to ask if there is even a hazard of collision at warp. Has there ever been a collision in the franchise - any normal object being hit at warp speed? (This obviously doesn't include things traveling at warp together, like photon torpedoes from one ship to another).  Closest example I found: In  Enterprise attempts warp before balancing their warp drives and creates a wormhole instead. They nearly collide with an asteroid which was already inside the wormhole, but neither the ship not the asteroid were traveling at warp.  Compressing Space Real-world (non-canon) attempts to explain ST warp technology rely on compressing space itself in front of the vessel. The image sequence below from Star Trek: Into Darkness (canon) shows Enterprise warping past the USS Vengeance and confirms that Enterprise actually gets smaller (or compresses the space around it). The light on Vengeance shows that at the instant the two ships are side-by-side Enterprise looks like a toy beside it. (Large image needed for detail - sorry):    Thus objects at warp occupy less real space - possibly none at - because , which according to Memory-Alpha is implied to be the medium which FTL travel happens. Normally subspace and real space do not interact. When they do it is called a .  Also, objects inside the warp bubble are not moved by the warp drive:     The observer(spaceship) is still immersed in the interior of the warp bubble and this bubble is carried   out by the spacetime ”stream” at faster than light velocities with the observer at the rest with respect to   its local neighborhoods  (p. 25)   It is my belief that to be consistent with warp bubble physics derived from the Einstein Field Equations in general relativity, the ST interpretation of warp speed travel should not allow collisions between n-space objects and objects within a warp bubble and remove the phenomenon from any plotlines such that warp travel is intrinsically safe for the traveller, but possibly greatly impactful for the normal world outside.  But has it happened?  Points of clarification raised by comments  Deflector Shields: Yes, ships have them. Gene Roddenberry envisioned Navigational Shields which would deflect anything in front of the ship, from a single hydrogen atom to an asteroid (see his notes below).    It's important to note that much of what he had in this guide was changed in final production. In fact, based on this guide we could write an entire episode on a simple deflector sabotage resulting in them hitting a hydrogen atom. Real science calculations tell us that even hitting a photon of light would release incalculable energy.  These notes very much explain the many questions asking why ships don't go to warp inside a solar system. Solar systems are FULL of particles and debris, which would require a great deal of energy to deflect. Interstellar space simply draws less power because the shields work less. I am installing an overhead automatic garage door opener. My only power source close to where I will be hanging the motor is an overhead light socket that is operated by a switch on the wall next to the house entry door.   I wanted to know if it is possible to rewire the light socket with a standard electrical plug outlet AND to remove the switch &amp; wire it to provide constant power to the new outlet. The overhead light is on my 15A garage GFI circuit &amp; the opener lists 120V/6A as its power requirements.  I live in the US. Prove that $f$ has a fixed point . For $f:[a,b]\rightarrow [a,b]$ is a continuous . Prove that $f$ has a fixed point . Is that true if we change $[a,b]$ by $[a,b)$ or $(a,b)$. Per the kernel documentation:  This control is used to define how aggressive the kernel will swap memory pages.  Higher values will increase aggressiveness, lower values decrease the amount of swap.   However this is kind of vague. I'm trying to figure out exactly what the parameter ultimately controls. I know it adjusts how aggressively the kernel tries to swap out pages, but what decision making process in the kernel code does it affect?  Does it adjust how long ago a page must have been accessed before the kernel swaps it out? If so what does the default value of 60 represent? And how much is changed by a increment/decrement of 1 (a formula would be nice)? Or does it swap out pages based on their access frequency? Or something else? How to change/rewrite the lost password url? I used iThemes Security (formerly Better WP Security) hide back-end functionality to change the login url to /signin however clicking the lost password link /wp-login.php?action=lostpassword now generate a 404  Manually going to /signin?action=lostpassword works.  I came across the following code below which is suppose to make the change to however after some testing it doesn't seem to work  //******************************************************* //Function to change to lost password url //******************************************************** add_filter('site_url',  'wplogin_filter', 10, 3); function wplogin_filter( $url, $path, $orig_scheme ){     $old  = array( "/(wp-login\.php)/");     $new  = array( "signin/"); //this can be change to login or whatever or may remain there     return preg_replace( $old, $new, $url, 1); }   add to .htaccess  RewriteRule ^signin$ wp-login.php   How can i modify the above code in order to change the url for the lost password? Can toddlers be taught to safely navigate stairs? I've used Ubuntu before, but I'm still very new. I just installed 16.10 on a new computer yesterday, and the Wi-Fi has been considerably slower than my Windows laptop. I put the computer together mainly to play Dota 2, and I experience decent ping but constant 5%-10% packet loss while playing. Occasionally, after the computer has been left on for a while, the Wi-Fi connection will drop completely- I can still see networks in the drop down menu, including my home network, but my computer will not reconnect without a reboot. I've looked all over for a solution, but my inexperience has prevented me from getting anything to work. My Wi-Fi card is a Ralink RT2561/RT61, here's some info  The output of sudo lshw -C network   *-network                          description: Wireless interface         product: RT2561/RT61 802.11g PCI         vendor: Ralink corp.         physical id: 6         bus info: pci@0000:01:06.0         logical name: wlp1s6         version: 00         serial: 00:0e:2e:de:df:c0         width: 32 bits         clock: 33MHz         capabilities: pm bus_master cap_list ethernet physical wireless         configuration: broadcast=yes driver=rt61pci driverversion=4.8.0-22-generic firmware=0.8 ip=192.168.1.107  latency=64 link=yes multicast=yes wireless=IEEE 802.11         resources: irq:17 memory:dfef8000-dfefffff   Here is the output of sudo iwconfig:   *-network                          description: Wireless interface         product: RT2561/RT61 802.11g PCI         vendor: Ralink corp.         physical id: 6         bus info: pci@0000:01:06.0         logical name: wlp1s6         version: 00         serial: 00:0e:2e:de:df:c0         width: 32 bits         clock: 33MHz         capabilities: pm bus_master cap_list ethernet physical wireless         configuration: broadcast=yes driver=rt61pci driverversion=4.8.0-22-generic firmware=0.8 ip=192.168.1.107  latency=64 link=yes multicast=yes wireless=IEEE 802.11         resources: irq:17 memory:dfef8000-dfefffff   I'm at a loss, I'm very bad at this. If any additional info is needed I can provide it Vertex groups can not be named in blender 2.70 For some reason the name text field doesn't appear for vertex groups in blender 2.70. I tried in 2.69 and it's there and works fine. Was this feature removed in 2.70 or am I missing something? How do I check the integrity of a storage medium (hard disk or flash drive)? I've run across (personally or by proxy) a bunch of storage media that exhibited physical problems during their life. Hard disks, solid state drives and other flash storage media all fail after some time. The situation is worse with USB flash drives and flash cards because of the large amount of suppliers with inferior flash cell and flash controller quality.  How can I reliably detect if a storage drive suffers from physical damage?  Note that this question isn't about file system integrity (which can be checked with ). How to typeset the symbol “^” (caret/circumflex/hat) Add 'Page' above page numbers in Table of Contents I am using my school's latex . Unfortunately, since this package was last updated, we are now required to add the word 'Page' above the topmost page number on each page of the Table of Contents. This cls file includes the following bit of code, which puts 'Page' at the top of the first page, but not any subsequent pages (see attached)  \newcommand\chapteruaf{ \if@arabic\relax\else\if@chapterone\setcounter{page}{1}\else\relax\fi\fi \@chapteronefalse                     \if@openright\cleardoublepage\else\clearpage\fi                     \thispagestyle{myheadings}%                                                                                                    \global\@topnum\z@                     \@afterindentfalse                     \secdef\@chapter\@schapteruaf} \def\@schapteruaf#1{\if@twocolumn                    \@topnewpage[\@makeschapterheaduaf{#1}]%                                                                                     \else                    \@makeschapterheaduaf{#1}%                                                                                                     \@afterheading                  \fi} \def\@makeschapterheaduaf#1{%                                                                                                    {\parindent \z@ \centering     \reset@font     \normalsize \bfseries  #1\par                                                                                                  \vspace{-18.3pt}\hspace{143mm} Page                                                                                            \nobreak   }}   First page of the TOC:   second page of the TOC:   Any thoughts here? I have played with this a bit, and can't get it to work. Thanks. Show that $\sin(z) = z$ has infinitely many solutions in complex numbers. Little Picard theorem should help, but using big Picard theorem is undesirable. Thanks a lot! I can successfully tether my Mac (10.7.3) to my iPhone4 (5.0.1) via wifi, no problem, been doing it for months.  However, when I try to tether via bluetooth instead, I can't seem to see the internet from my mac.  Network Settings reports a green light for Bluetooth PAN, but my browser can't connect to anything at all.  My Mac reports very similar IP addresses for both, starting with 172.20.10 and differing only in the last octet.  Any idea why I can't successfully tether via bluetooth?