Prove that if a fraction is broken up into two the resulting two fractions cannot both have a larger value
Given 4 integers, $a, b, c, d > 0$, does $\frac{a}{b} < \frac{c}{d}$ imply $\frac{a}{b} < \frac{a+c}{b+d} < \frac{c}{d}$?
Groups in derived series are normal in the whole group I am considering a finitely generated, solvable group $G$ which has subexponential growth. Let $G=G_0\vartriangleright G_1\vartriangleright....\vartriangleright G_d\vartriangleright G_{d+1}=1$ denote the derived series of G.  It should be the case that $G_d$ is normal in $G$. Does anyone know why this is true? I don't know whether all of the assumptions above are necessary for the result. I saw this movie a while back. I don't remember how it starts, but from where I remember watching it, it may have not been too far from the start. Apparently, Earth has been conquered by some alien empire or something and humanity are basically cavemen being used as slaves. From what I remember, the main antagonist alien is seen taking one of the slaves to a room where a machine is used on him to make him smart; this is apparently a big no-no so the main antagonist and his sidekick have to keep it quiet. The idea was that by making one of the slaves smart, it could increase the work output of the others. Apparently the antagonist is using the slaves on Earth to gather gold bullion  which is apparently still very valuable to the aliens but he's doing it illegally (not the slave labor; that's apparently ok). The protagonist who had the machine used on him becomes the intermediary between the slaves and the aliens and begins teaching the other slaves. I remember him teaching them mathematics and I remember him saying that it was universally accepted (I think he may have been referring to algebra or trigonometry). At some point he becomes aware that they are slaves and organizes a plan to free not only his fellow slaves but the Earth. Some escape and find an old museum which has old flight simulators, and shows that humanity wasn't always a slave race and that it may have been modern times in which Earth was invaded (i.e. no lasers or starships). I think they also find Fort Knox as there is a huge collection of gold bricks and weapons. The protagonist comes up with a plan to destroy some sort of teleporter the aliens use to transport stuff to and from Earth, however that alone wont be enough as they could fight back, which will be bad for humanity, which is still undeveloped. So the plan also includes using the teleporter before destroying it to to transport and detonate a nuclear warhead on the alien home world which apparently cripples any attempt to immediately send a fleet but also as a show of strength. After executing their plan, the main antagonist is captured by the humans and as an insult is locked in a cage surrounded by the gold he sought. If I recall correctly the place is set to detonate as to destroy the precocious gold and that will be used against the aliens should they mobilize a fleet in the future (i.e. attack us and we destroy Earth's supply of gold). The antagonist's side kick for some reason I remember as a big black man from whom the antagonist had removed his hand at some point and also apparently the antagonist was cruel to him. After the antagonist is locked up, he is trying to appeal to his sidekick's loyalty, however the sidekick takes the humans' side, I'm also getting an image that apart from treating him kindly the sidekick's missing hand was restored. Invest money and get 120% back - a site says Create a virtual file that is actually a command Is there any way to create a virtual file, such that reading from the file actually reads from the stdout of a command; writing to the file is acually writing to the stdin of a command?  So far I have kludged this with an inotifywait on a file, which calls a command when the file is modified, taking it's input from the file and writing back to it.  I don't like that the inotifywait has to be constantly restarted though (and I have to ensure that it is always running). I only use this file perhaps twice a week. We usually say that the ultimate source of energy for earth is the Sun. So this means that sun creates energy but according to the law of the conservation of energy energy can neither be created nor destroyed, so how does the energy come into the sun? That is just an example. We know that energy is transformed from one form to another but is not created and destroyed. But that is confusing because there must be an ultimate source of energy. How is  energy created in this source? If it is related to the atomic level, then how can atoms possess energy? Is the law of the conservation of energy untrue? Are there wizarding schools in the rest of the "magical" world? Are there other schools for witches and wizards in the rest of the world? In the books they only speak of 3 schools, but what about Asian or American people?  And even more, were they affected by the events happening in Europe? Why does autoboxing make some calls ambiguous in Java? I noticed today that auto-boxing can sometimes cause ambiguity in method overload resolution. The simplest example appears to be this:  public class Test {     static void f(Object a, boolean b) {}     static void f(Object a, Object b) {}      static void m(int a, boolean b) { f(a,b); } }   When compiled, it causes the following error:  Test.java:5: reference to f is ambiguous, both method     f(java.lang.Object,boolean) in Test and method     f(java.lang.Object,java.lang.Object) in Test match  static void m(int a, boolean b) { f(a, b); }                                   ^   The fix to this error is trivial: just use explicit auto-boxing:  static void m(int a, boolean b) { f((Object)a, b); }   Which correctly calls the first overload as expected.  So why did the overload resolution fail? Why didn't the compiler auto-box the first argument, and accept the second argument normally? Why did I have to request auto-boxing explicitly? What's the source of "shipped" in a romantic sense? What type of buildings offer what level of jobs? There are three tiers of workers (as is everything else); tier 1, 2 and 3. What I am having a hard time with is figuring out what I can build to target workers from each of the tiers. I gather what I am sure everyone else has, low tech industrial (regardless of density really) provides mostly tier 1 jobs.  But what are my other building options for tier 1 jobs? If I want to target tier 2 or 3 what types of buildings should I consider? Proof by induction that $2 + 4 + 6 + \cdots + 2n = n(n+1)$ "Speed" of Gravity and Speed of Light I need to upload my signature in a PDF document, but I do not know how to edit the background of this image.   I wish to remove the bluish-tint background, and make it pure white.   Can anyone help me with this, please? Macbook Air - how to install a dual bootable Ubuntu 14.04 LTS On my Macbook Air 10.9.2 without crashing the current setup, how can i install as dual boot: Ubuntu 14.04 LTS ? (can anyone please advise some step by step please) Baking with oven without bottom heating element My mom wants to do baking but her oven doesn't have bottom heating element. It only has an element on top, with a fan at the back.  She told me that she can't make me a good pizza or cake without it.  Is there any way around? Suppose $f:X\rightarrow Y$ is a surjection and you want to show that there exists $g:Y\rightarrow X$ s.t. $f\circ g=\mathrm{id}_Y$. You need the AC to show this.  However, suppose $f$ is a injection and you want show there is $g$ s.t. $g\circ f=\mathrm{id}_X$. Then, according to my textbook, you don't need the AC to show this.  This is counterintuitive to me, because it's like you need a special axiom to claim that an infinite product of big sets is nonempty, while you don't need one to claim that an infinite product of singleton sets is nonempty, which seems smaller than the former.  So why don't you need the AC to show the latter?  EDIT: $X$ should be nonempty.  EDIT 2: I realized (after asking this) that my question mostly concerns whether the AC is needed to say that an infinite product of finite sets is nonempty, and why. Using same modulus for RSA I know that there exist some attack when using same modulus.      But with a little modification, $m$ is the plain-text $N$ is the RSA modulus $r_1, r_2$ is the random padding $e, s$ is the public exponent $C_e, C_s$ is the cipher-text  encrypt as follow $$С_e = (m + r_1)^e   \bmod N$$ $$С_s = (m + r_2)^s   \bmod N$$  If attacker only knows $C_e, C_s, r_1, r_2, e, s, N$. Is it possible to know $m$?  It seems the  does not work? Homomorphism between free groups Let $F_a$ be the group which is freely generated by $a$ elements. How to show that there is a homomorphism from $F_a$ onto $F_b$ if and only if $b\le a$?  I was thinking one possibility is if $F_a$ generated freely by $\{x_1, \dots , x_a\}$ and $F_b$ generated freely by $\{y_1, \dots , y_b\}$ then consider map which says $\phi(x_i)=y_i$ for $1\le i \le b$ and $y_1$ otherwise. I do not know if this works or even how to proceed. How to show this fact? How can I set a dark theme in TeXstudio? After perusing , I'm considering switching from TeXmaker (in which I use the preset "dark theme") to its fork, TeXstudio.  What I mean by "dark theme" is a dark background with suitable text colours, which, in my experience, is easier on the eyes than a bright background is. Here is an example:    TeXstudio doesn't seem to offer any preset dark theme, though, which means I have to customise all the colours/syntax-highlighting settings from scratch to get a dark theme even remotely usable. This can prove very time-consuming.   I'm aware this might sound a bit cheeky but,   if you are a long-time TeXstudio user, and you have fine-tuned the editor colours to get a visually pleasant dark theme,   please consider posting the relevant lines from your settings (.txsprofile or .tmxprofile) file as an answer below.  TeXstudio users who would like to use a dark theme could then build on your settings to fine-tune their own. Setting nautilus-open-terminal to launch Terminator rather than gnome-terminal When and where was “portfolio career“ first used? The expression  meaning:  the fact of having several part-time jobs at once, rather than one full-time job:    A portfolio career is suitable for people who want to have a variety in their work life.   (Cambridge Dictionary) Etymonline has no information about the origin of the above expression, can anyone tell me when and where it was first used?