Let $R$ be a finite ring of order $p^2$ with unity $e$ and characteristic $p$. This ring is commutative but I cannot get why it is.   I know that this ring looks as $\mathbb Z /p\mathbb Z \times Z /p\mathbb Z$. Could anyone help me to show the commutativity of this ring?
I would like to show that ring of order $p^2$ is commutative.   Taking $G=(R, +)$ as group, we have two possible isomorphism classes $\mathbb Z /p^2\mathbb Z$ and $\mathbb Z/ p\mathbb Z \times \mathbb Z /p\mathbb Z$.   Since characterstic must divide the size of the group then we have two possibilities $p$ and $p^2$.   Now IU don't understand how can I reason to say that the multiplication is commutative and how can I conclude for the case when characterstic is $p$?
I have a directory with numerous files. Part of the files have the 755 permissions and the other part have 644 permissions. I'd like to convert the files with 755 permissions to 644. I have tried the following line by running it from the directory itself:  find . -perm 755 -exec chmod 644 {}\;   However as a result, the permission changed only for the directory itself and after changing it back I found out that the files permissions remained unchanged. Do I miss something? Can I recover app data after the phone has been Factory Reset? Without remembering application data, I reset my Galaxy S3 to factory state. Now, I lost data for one expense manager app. Is there any way to recover the data? I was shooting the lighting of the city that is visible at the top of the trees if you shoot landscape.  No matter what I do, the image is blurry. What have i done:   ISO 100 Manual focus towards infinity Using a tripod Using really sharp prime &amp; good camera   Basically, everything I could. The only example i have is here, but it is not good, because image is underexposed:    The main question: Why can't night photographs be as sharp as daily photographs (with approperiate exposure times of course)? I read a story--the first part of one, at any rate--years ago, and haven't tracked it down since. It started with a man been sent to a planet under attack to pose as a monk--he arrived in the middle of an assault, but still had to do penance for being late. So he wore a hair shirt, that created an agonising reaction. He was captured by the attackers (who took him for the man he was posing as), who then stood off the planet in their starships and watched as a 'hell burner' weapon tumbled out of deep space and torched the planet. The attackers then explained that they weren't responsible for the hell burner, but every time they arrived at a planet, a hell burner turned up afterwards, at shorter and shorter intervals.  As the story developed, there was some kind of time travelling dimension, and a character turned out to have his organs reversed. I may have got some details wrong, but this was the shape of it.  Anyone know this story, and the author? I've added many PPAs using the add-apt-repository command. Is there a simple way to remove these PPAs? I've checked in /etc/apt/sources.list for the appropriate deb lines but they aren't there.   This is on a server system so a command line solution would be great! If you multiclass a Warlock and a Sorcerer, then you have a character with both the Spellcasting and Pact Magic features.  PHB Page 101, Sorcerer, Spellcasting, Spell Slots:     You regain all expended spell slots when you finish a long rest.   PHB Page 107, Warlock, Pact Magic, Spell Slots:     You regain all expended spell slots when you finish a short or long rest.   PHB Page 165, Customization Options, Multiclassing, Class Features:     Pact Magic. If you have both the Spellcasting class feature and the Pact Magic class feature from the warlock class, you can use the spell slots you gain from the Pact Magic feature to cast spells you know or have prepared from classes with the Spellcasting class feature, and you can use the spell slots you gain from the Spellcasting class feature to cast warlock spells you know.   Do the spell slots keep their previous method of recovery (sorcerer slots long rest, warlock slots any rest), or do they all act consistently as I am led to believe from the language of the multiclass section? Do they all recover only from long rests as per the sorcerer feature, or can they all be recovered with a short rest because the warlock feature says so? Prove that :  $$ \frac{\sqrt{\pi}}{2} e^{-\frac{a^2}{4} } =\int_0^{\infty} e^{-x^2} \cos( a x) \ \mathrm{d}x$$    the only thing I can think of is differentiating the RHS and trying to get :  $$ -2 f'(a) =a f(a) $$ But I couldn't do it.  Can anyone show me how to do this ? How can I get textures on edge of walls like in Super Metroid and Aquaria? Games like Super Metroid and Aquaria present the terrain with the other facing parts having rocks and stuff while deeper behind them (i.e. underground) there's different detail or just black.      I would like to do something similar using polygons. Terrain is created in my current level as a set of overlapping square boxes. I'm not sure if this rendering method will work such a system for creating terrain but if anyone has ideas I'd love to hear them. Otherwise I'd like to know how I should re-write the terrain rendering system so it actually works to draw terrain in this manner... If $\alpha_{1},\ldots,\alpha_{m}$ are vectors different from zero vector, then there is a linear functional $f$ such that $f(\alpha_{i})\neq 0$ Prove without using graphing calculators that $f: \mathbb R\to \mathbb R,\,f(x)=x+\sin x$ is both one-to-one, onto (bijective) function. Prove that the function $f:\mathbb R\to \mathbb R$ defined by $f(x)=x+\sin x$ for $x\in \mathbb R$ is a bijective function.    The codomain of the $f(x)=x+\sin x$ is $\mathbb R$ and the range is also $\mathbb R$. So this function is an onto function. But I am confused in proving this function is one-to-one. I know about its graph and I know that if a function passes the  horizontal line test (i.e horizontal lines should not cut the function at more than one point), then it is a one-to-one function. The graph of this function looks like the graph of $y=x$ with sinusoids going along the $y=x$ line. If I use  a graphing calculator at hand, then I can tell that it is a one-to-one function and $f(x)=\frac{x}{2}+\sin x$ or $\frac{x}{3}+\sin x$ functions are not, but in the examination I need to prove this function is one-to-one theoritically, without graphing calculators. I tried the method which we generally use to prove a function is one-to-one but no success. Let $f(x_1)=f(x_2)$ and we have to prove that $x_1=x_2$ in order fot the function  to be one-to-one. Let $x_1+\sin x_1=x_2+\sin x_2$ But I am stuck here and could not proceed further. Sometimes a question I ask will get answers that don't help me, or maybe it doesn't receive any answers at all.  How about making a notification when somebody links to my question?   It could be something like this:     New information regarding your question might be available at {link}   It might help me getting an answer from related questions that weren't there when I first asked the question. Using THE before some countries I have been taught that it's grammatically proper to use the before countries like "Sudan and Yemen" but I was never given a reason (if there exists any, of course) when I asked why, except for "it is one of those ambiguous parts of English language that we &ndash; as non-native speakers &ndash; should just accept without trying to look for a reason". Now, I'd like to know if there is any answer to this "why" or I should keep on believing that there are no hard and fast rules when it comes to questions like these. Who is sample No. 118 in The Case of Charles Dexter Ward? In H. P. Lovecraft's The Case of Charles Dexter Ward, Dr. Willett      Enters Joseph Curwen's dungeon and resurrects a dead person identified as No. 118 by reciting the incantation of the Ascending Node over his salts.   Dr. Willett is told by      The resurrected Joseph Curwen, impersonating Charles Dexter Ward, who he secretly murdered,   that   " 'twas Number 118, and I conceive you would have shook had you looked it up in my list in t'other room."  No. 118 himself gives Dr. Willett a note asking the doctor to      kill Joseph Curwen.   It soon becomes clear that No. 118 is opposed to the nefarious dealings that have been transpiring and desirous of being an ally of Dr. Willett.  Who is No. 118?  Earlier in the story,      The grave of Ezra Weeden, a primary instigator in Joseph Curwen's initial downfall and the romantic rival of Curwen, is robbed. There doesn't seem to be any other mention of Weeden in the 20th century occurrences, so without a further connection the passage seems superfluous.   The shared goals of these two people seem to indicate that No. 118 may, in fact, have been that person, but the      Note given by No. 118 to Dr. Willett in eighth century Latin seems to indicate someone much older than Ezra Weeden.   It also occurred to me that No. 118 could be      The recently murdered Charles Dexter Ward himself. Ward's reference to killing Dr. Allen (who is actually an alias of Curwen) and dissolving his body with acid and No. 118's mention of doing the same to Curwen seem to support the idea, but that still doesn't explain the archaic note. Ward is described as writing in a somewhat modern style, at least as opposed to Curwen. It also doesn't explain why Curwen would want to murder Ward if he was going to resurrect him later anyway. Down Arrow Key is not working Totally bounded, complete $\implies$ compact I'm trying to recall the name of a book I read, probably published after 1990, where alien AI infects satellites and a barrier is established, isolating earth from space that maybe slows down time inside of it, relative to the passage of time outside.  Mars or the moon is also colonized in an attempt to flee the now real possibility that the sun will become a red giant and wipe out humanity in the not-too-distant future (as perceived on earth), but the colony gets one of the time barriers as well.  I think the name of the novel was short, "The _______" (one word).  The story focused more on the human cost and followed one particular family, one member of which was somehow also partially infected by the alien AI, but I don't remember as much about them. Prove $ \sum \frac{\cos n} { \sqrt n}$ converges proof that function is constant I'm annoyed by quite a simple problem in calculus (I apologize in advance if I'm not using adequate terms in English, I don't take the course in English nor am I a native speaker):  Let $f:\mathbb R\longrightarrow \mathbb R$ be a function which is continuous at $0$ and satisfies $f(x)=f(2x)\,\forall x\in\mathbb R$.  My question is how do I formally prove that for every $x&lt;0$ $f(x)=c_1$, and for every $x&gt;0$ $f(x)=c_2$, where $c_1,c_2\in \mathbb R$? After proving those two statements (or at least one of them WLOG) it would be quite easy to prove that $f(0)=c_1=c_2$ by the definition of continuity of $f(0)$ and thus $f(x)$ is constant. Those two statements are just so obvious to see that I can't think of any formal way of proving so without "cheating". Thanks in advance! I am looking for an intuitive explanation as to why/how row rank of a matrix = column rank. I've read the proof at  and I understand the proof, but I don't "get it". Can someone help me out with this ?   I find it hard to wrap my head around the idea of how the column space and the row space is related at a fundamental level. Counting surjective functions