No shower water until sink turned on!
Why would turning off a sink faucet stop the shower?
Equivalence Relation problem Where are some good places to find royalty free stock images? I'm always on the hunt for free stock images to use while I design, what are some good resources for images that are under a a royalty free or creative commons licence? How to show that the commutator subgroup is a normal subgroup It is suggested as an exercise in Serge Lang's book "Algebra" to show that the commutator subgroup $G^c$ of a group $G$ is a normal subgroup.   I'd like to do that but I am afraid I need help,   I think the first thing I need to figure out is how a general element in the commutator subgroup looks like, so that I can check that the defining condition for normality is satisfied.  That is, supposing for a moment that a general element in $G^c$ is denoted by $g$, I need to show that $aga^{-1} \in G^c,$ for all $a \in G$.  But here I get stuck, first because I am unsure how to write a general element in $G^c$ - a simple product in $G^c$ is of the form $xyx^{-1}y^{-1}aba^{-1}b^{-1}$ where $a,b \in G$. I cannot see a way to simplify this - I am sure there is one, but somehow I am blind today.  The second thing then is, even if one tries out the conjugation of a simple element like $xyx^{-1}y^{-1}$ in $G^c$, again not simplification offers itself easily I think .. what am I missing ?  An alternative would be to find a homomorphism of $G$ whose kernel is precisely $G^c$ - here I tried to think of this as a map $G \times G \to G$ but whatever I cook up is not a homomorphism.  Thanks for your hints !! Epsilon delta proof for $\lim_\limits{x\to a} \sqrt{x} = \sqrt{a}$ where $a>0$ $f(x) = \sqrt{x}$. For right limit, $x-a &gt; 0$. Let $x-a &lt; \delta$ then $(\sqrt{x}-\sqrt{a}) \lt \frac{\delta}{\sqrt{x}+\sqrt{a}}$. Then chose we must choose $\epsilon $ so that $\epsilon &gt; \frac{\delta}{\sqrt{x}+\sqrt{a}}$. Is this correct? How would I conclude the proof?  Edit - Follow up  From answers, I try a proof for the same limit but this time Left side:  For left side limit, $x-a &lt; 0$ so $a-x &gt; 0$. Let $a-x &lt; \delta$ then $\sqrt{a}-\sqrt{x} \lt \frac{\delta}{\sqrt{a}+\sqrt{x}} \lt \frac{\delta}{\sqrt{a}}$.   So let me begin the proof by saying to choose $\delta = \epsilon \sqrt a$ then $a-x \lt \delta $ so $(\sqrt a - \sqrt x )(\sqrt a + \sqrt x) \lt \epsilon \sqrt a$ so $\sqrt a - \sqrt x \lt \epsilon$ Prove that if W is a subspace of a finite dimensional vector space V, then dim(W) ≤ dim(V). My attempt at a solution:  Proof  Given a vector space $V$, let $S={v_1,...,v_n}$ be a basis for $V$. Consequently $\dim(V)=n$. Now given $W$, a subspace of $V$, it will be shown that $\dim(W)\le \dim V$   First, consider the case where $W=${0}. Then $dim(W)=0 \le\ n$,   Subsequently, consider the case for $W=V$. Then it follows that $dim(W)=dim(V)=n$.  Finally, consider the case where $W \subset V$ and without loss of generality assume $v_n \notin W$. Let $T={v_1,...,v_k}$ where $k\lt n$ be a basis for the subspace W. Now, since S is a basis for V, for every $v_i \in S$,  $c_1v_1+...+c_nv_n=0$ has only the trivial solution, thus none of the vectors in S can be expressed as linear combinations of the other vectors in S. It follows that the set T is not a basis for V as $v_n$ cannot be generated from T. The $dim(W)$ is given by the number of vectors in the basis. Therefore, $dim(W)=k \lt n$  Therefore, it has been shown that if $W \subseteq V$ then $dim(W) \le dim(V)$ Ability to update profile from the Stack Exchange Android App It would be nice to update the profile page from within the app.  Perhaps not many people update their profiles more frequently, but when required to do so, I would prefer to do it from the app instead of the mobile browser. Determine if wire in switch box is safe to re-purpose? I need to pull a neutral wire to my light switch for a smart switch.  It's presently switching an outlet that I don't need switched, and the other side of the outlet already has unswitched power.  My plan is to use a smart switch, which requires neutral, to control a different device wirelessly.    I can't feasibly run a new wire, because the wires aren't run through conduit.  Instead, my plan is to repurpose the switched wire to get neutral from the outlet.    The switch, which controls an outlet has two wires, one black and one red.  The switched outlet has a white neutral wire, plus the red and black wires that I assume run directly to the switch.  I've completely disconnected the red and black wires from the outlet and the switch.  The black wire is measuring 120V AC when measured against ground using the box as ground, both at the outlet and at the switch. Using my DMM, the red wire is measuring 24V AC with respect to the box/ground at both sides.  I assume this is a  caused by induction from the charged black wire over to the floating red wire.    Our house is ~100 years old and this particular switch seems to be run using armored cable (note the paper wrap around each wire.)  There are literally only two wires in the switch box; no wire nuts.    If relevant, we do have some old knob and tube wiring active, but I'm pretty confident there's none on this circuit.  Is there some way I can determine with enough confidence that the red wire isn't connected elsewhere such that I can know it's safe to connect the red wire at the outlet to netural? Using Perl to change the url homepage for Internet Explorer Where does gravity get its energy from? I would like to know where gravity gets its energy to attract physical bodies?  I know that the law of conservation states that total energy of an isolated system cannot change. So gravity has to be getting its energy from somewhere, or else things like hydropower plants wouldn't be able to turn the power of the falling water into a spinning rotor.  Just to be clear, Lets create an example:  Lets say we have two objects with equal mass close to each other. So gravity does its job and it pulls each other closer, this gets turned into kinetic energy. This is where I'm lost. According to the law of conservation energy can't be created or destroyed and the kinetic energy comes from the gravitational pull so where does the gravitational pull gets its energy.  If that energy isn't being recycled from some where else then that means you have just created energy, therefore breaking the law of conservation. If $G$ has no proper subgroup, prove that $G$ is cyclic of order $p$, where $p$ is a prime number.     I know that since $G $is a group with no proper subgroups, $g \in G$ is not just the identity. I don't know where to go from there. Does a surjective ring homomorphism have to be surjective on the unit groups? There is no set of all sets.. So there are a few posts about this already, but they skip over the problem I have.  The proof of Cantor's theorem elegantly shows that if we consider $f:A\rightarrow \mathcal{P}(A)$, the set $B=\{x\in A : x\not\in f(x) \}$ by its definition cannot lie in the image of $f$, precluding it from being a bijection - hence power sets have strictly greater cardinality.  But now, I thought one (obvious) way to characterise the "set of all sets" would be $S=\mathcal{P}(S)$, as indeed, it contains all possible sets (perhaps here is the problem, but I don't immediately see how using this as a definition for the set of all sets is bad). Now consider simply taking the identity function $i:S\rightarrow S$. Then $B=\{x\in S : x\not\in x \}$ - the set of all sets which do not contain themselves. But this doesn't exist. It's exactly the set of Russell's paradox. If this isn't a set, and so is not in $S$, then the proof no longer offers an obstruction to $i$ being surjective.    Each other post I've seen on the matter uses Cantor's theorem to show such a set doesn't exist. My query is to find that something else that must be happening, since I think the proof has issues for this particular set. I'm expecting other issues to arise with the definition $S=\mathcal{P}(S)$... In my script, I have for loop over many cube objects (~1000) and the treatment is very slow. Looking more in details, I notice that in the same amount of loops:   if I use python operation or simple blender operation like  obj = bpy.data.objects[obj_name]   or   obj.select = True   it takes less than 0.08s (for all 1000 objects) but as soon as I start using blender operators like:  bpy.ops.object.select_pattern() # or bpy.ops.object.duplicate() # or bpy.ops.object.location_clear() # or bpy.ops.object.transform_apply() # etc.   then, the performance drops down deeply with more than 6s for the same amount of objects.   And, a last information, if I reduce the amount of objects from 1000 to 50, the same set of operations with bpy.ops is 0.03s - so extrapolating, it would be 0.6s for the 1000 objects and not 6s. It is as is with 1000 objects, we loose a factor 10 in speed in comparison than with 50 objects.  I tried reducing the complexity of my mesh changing the cubes in planes but it had no effect at whole on performance.  Is there something particular to know to improve these performance or a way to use the bpy.ops methods with better performances? Obviously, I am missing something important probably. I am confused about the usage of the words like and unlike in sentences. Like and unlike make me confused because I see them being used everywhere almost interchangeably, and to make matters worse I even saw commas making difference in meaning.  Which of the following sentences is most correct, and least ambiguous?      Unlike Xbox, Playstation can't play Halo.     Playstation can't play Halo unlike Xbox can.     Playstation can't play Halo, unlike Xbox can.     Like Xbox, Playstation can't play Halo.     Playstation can't play Halo like Xbox can.     Playstation can't play Halo, like Xbox can.     Playstation can't play Halo unlike Xbox.     Playstation can't play Halo, unlike Xbox.     Playstation can't play Halo like Xbox.     Playstation can't play Halo, like Xbox.    In case you don't know what I actually wanted to say in those sentences: I'm trying to say that Playstation can't run the Halo video game and Xbox can run that game. I probably should have tried sentences with no brands but I can't imagine any meaningful sentences at this moment which contains like/unlike. Given the case I made two independent changes in one file: eg. added a new method and changed another method.  I often don't want to commit both changes as one commit, but as two independent commits.  On a git repository I would use the Interactive Mode of  to split the hunk into smaller ones:   git add --patch   What's the easiest way to do this with Subversion? (Maybe even using an Eclipse plug-in)  Update: In , Ryan calls it: “The Tangled Working Copy Problem.” Using Zorn's lemma show that $\mathbb R^+$ is the disjoint union of two sets closed under addition. If I'm floating in space and I turn on a flashlight, will I accelerate? Photons have no mass but they can push things, as evidenced by .   Can photons push the source which is emitting them? If yes, will a more intense flashlight accelerate me more? Does the wavelength of the light matter? Is this practical for space propulsion? Doesn't it defy the law of momentum conservation?  Note: As John Rennie mentioned, all in all the wavelength doesn't matter, but for a more accurate answer regarding that, see the comments in  .  Related Wikipedia articles: , How to completely remove Garage Band from macOS? Remove the Yoast SEO Post Metabox Yoasts SEO plugin adds a metabox to the post edit screen. I'm trying to remove this for users who aren't editors or above.  I've tried putting a remove_meta_box call in on admin_init, trying to remove the action on $wpseo_metabox but to no avail.  How do I remove this metabox without requiring user intervention (the user should never know the metabox existed, so clicking on screen options is not an option ) Photoelectric Effect - Dependence of current on frequency