Strip out harmful code but allow Iframe
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Kaplansky's theorem of infinitely many right inverses in monoids? There's a theorem of Kaplansky that states that if an element $u$ of a ring has more than one right inverse, then it in fact has infinitely many. I could prove this by assuming $v$ is a right inverse, and then showing that the elements $v+(1-vu)u^n$ are right inverses for all $n$ and distinct.  To see they're distinct, I suppose $v+(1-vu)u^n=v+(1-vu)u^m$ for distinct $n$ and $m$. I suppose $n&gt;m$. Since $u$ is cancellable on the right, this implies $(1-vu)u^{n-m}=1-vu$. Then $(1-vu)u^{n-m-1}u+vu=((1-vu)u^{n-m-1}+v)u=1$, so $u$ has a left inverse, but then $u$ would be a unit, and hence have only one right inverse.  Does the same theorem hold in monoids, or is there some counterexample? I would like to convert a large LaTeX project (i.e. spanning multiple files) into plain text. The purpose is generation of statistics, so representing mathematics is not an issue. In fact, all mathematics is ideally ignored.  I have found  but could not get it to run. It seems unfinished, but is exactly what I am looking for otherwise. How to remove headers and footers for pages between chapters? What are the differences in targets, and when should I switch to one? So I get the 4 options for target selection are:   Attackers Badges Randoms K.O.s   What I don't understand is the pros/cons for selecting each one, and what situations would make each one useful.  Things like Badges and K.O. makes no sense to start off with because nobody starts off with badges, and nobody is going to start off near death unless they are purposely trying to lose (by hard dropping int he same position). I've tried the strategy then of just sticking with Attackers, but then I all of sudden get wave after wave of garbage late into the game.    So it begs the question -- what are the differences in target choices, and how can I use that to my advantage to become a Tetris 99 champion? Math people:  The title is the question. I am convinced that uncountable sets exist, thanks to Cantor's diagonal proof.  It is not intuitively clear to me that uncountable ordinals or cardinals should exist.  What axioms of set theory are needed to accept their existence?  I think I have read here that the Axiom of Choice is not necessary.  I am reluctant to accept as "true" any theorems that rely on the Axiom of Choice.  EDIT: This question may be a duplicate: I just found  .  I apologize if this is so. Array slices in C# How do you do it? Given a byte array:  byte[] foo = new byte[4096];   How would I get the first x bytes of the array as a separate array? (Specifically, I need it as an IEnumerable&lt;byte&gt;)  This is for working with Sockets. I figure the easiest way would be array slicing, similar to Perls syntax:  @bar = @foo[0..40];   Which would return the first 41 elements into the @bar array. Is there something in C# that I'm just missing, or is there some other thing I should be doing?  LINQ is an option for me (.NET 3.5), if that helps any. If $x_1,x_2,\ldots,x_n$ are the roots for $1+x+x^2+\ldots+x^n=0$, find the value of $\frac{1}{x_1-1}+\frac{1}{x_2-1}+\ldots+\frac{1}{x_n-1}$ Let $x_1,x_2,\ldots,x_n$ be the roots for $1+x+x^2+\ldots+x^n=0$. Find the value of   $$P(1)=\frac{1}{x_1-1}+\frac{1}{x_2-1}+\ldots+\frac{1}{x_n-1}$$   Source: IME entrance exam (Military Institute of Engineering, Brazil), date not provided (possibly from the 1950s)  My attempt: Developing expression $P(1)$, replacing the 1 by $x$, follows $$P(x)=\frac{(x_2-x)\cdots (x_n-x)+\ldots+(x_1-x)\cdots (x_{n-1}-x)}{(x_1-x)(x_2-x)\cdots (x_n-x)}$$    As $x_1,x_2,\ldots,x_n$ are the roots, it must be true that   $$Q(x)=(x-x_1)\cdots(x-x_n)=1+x+x^2+\ldots+x^n$$ and  $$Q(1)=(1-x_1)\cdots(1-x_n)=n+1$$ Therefore the denominator of $P(1)$ is  $$(-1)^{n} (n+1).$$ But I could not find a way to simplify the numerator.  Another fact that is probably useful is that  $$1+x^{n+1}=(1-x)(x^n+x^{n-1}+\ldots+x+1)$$ with roots that are 1 in addition of the given roots $x_1,x_2,\ldots,x_n$ for the original equation, that is  $$x_k=\text{cis}(\frac{2k\pi}{n+1}),\ \ k=1,\ldots,n.$$  This is as far as I could go...    Hints and full answers are welcomed. Determine the Galois group of $\mathbb{Q}(\sqrt{a+b\sqrt{d}})$ Suppose $L=\mathbb{Q}(\sqrt{a+b\sqrt{d}})$,($d$ and $a+b\sqrt{d}$ are square free algebraic integers) when is $L/\mathbb{Q}$ a normal extension? When does $Aut(L/\mathbb{Q})=\mathbb{Z}/4\mathbb{Z}$ or $Aut(L/\mathbb{Q})=\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$? Cursor is set to move by increments. how to reset to normal? My cursor is moving by every selection by large increments!  Normally when I use Blender, I am able to select and move objects to any coordinate. Now, all of the sudden, when I select an object or mesh the selection moves by an inch or so in each direction.  How do I reset it to normal? Oddly wide spaces in subsection title I have a problem, this is what it looks like:     And this is the code:  \subsection{Bakterienst\"amme und Anzuchtbedingungen}   Where does the huge whitespace come from? Is there a way to spellcheck the word under the cursor with a keyboard shortcut? Mirror modifier mirrors on different axis/axes from the selected one(s) I have 2 objects a face and an ear. Mirror modifier works fine for the face. But when I use mirror modifier to duplicate the ear on X axis, it duplicates on Z axis. And when I use mirror modifier on Z axis it mirrors on X axis. How to correct the axis orientation? I was thinking about writing a partial derivative macro that has a similar syntax to the . ()  The syntax could be like this:   \D{f}{x} expands to \frac{\partial f}{\partial x} \D{f}{x,y} expands to \frac{\partial^2 f}{\partial x\,\partial y} \D{f}{{x,2},y} expands to  \frac{\partial^3 f}{\partial x^2\,\partial y} et cetera for any number of derivatives.   Bonus points for an optional argument so that for instance   \D[subscript]{f}{{x,2},y} expands to f_{xxy}   I looked at the  package but the syntax is a little tedious.  For instance, to get \frac{\partial^3 f}{\partial x^2\,\partial y} with that package I would need to do \md{f}{3}{x}{2}{y}{}.  And it doesn't generalize to three variables.  Before I dive in I thought I'd ask the hive and see what's been done. Difference between pre-increment and post-increment in a loop? Is there a difference in ++i and i++ in a for loop? Is it simply a syntax thing? Consider a particle in a potential well. Let’s assume it’s a simple harmonic oscillator potential and the particle is in its ground state with energy E0 = (1/2) &#8463;&omega;0. We measure its position (measurement-1) with a high degree of accuracy which localises the particle, corresponding to a superposition of momentum (and therefore energy) states.  Now we measure the particle’s energy (measurement-2) and happen to find that it’s E10 = (21/2) &#8463;&omega;0. Where did the extra energy come from?  In the textbooks it’s claimed that the extra energy comes from the act of observation but I wonder how that could work. Measurement-1 which probed the position of the particle can’t have delivered to it a precise amount of energy, while measurement-2 might just have been passive. No doubt there is entanglement here between the particle state and the measuring device but where, and which measurement? I want to show for $x\in\mathbb{R}$, the following inequality holds. $$ \left|\sin x\right|+\left|\sin (x+1)\right|+\left|\sin (x+2)\right|&gt;8/5. $$ This trigonometric inequality has been verified by Mathematica using the Plot commend. However, using the sum formula for the sine function seems doesn't work, and I cannot give a rigorous proof of it. Any suggestion, idea, or comment is welcome, thanks! I have been stuck in a problem, that has thrown my brain out of the coding. This problem is at very high priority and I need the solution as early as possible. Problem is as :  There are exactly N advertising boards on the highway. Now a company want to advertise on some of these advertising boards(each advertising board costs some money) .  Company strategy is that, they want at least 'K' advertisement should be there among M consecutive advertising boards. But at the same time Company want to pay minimum for its advertisement .  Now, what is the total number of ways, different ways Company can advertise meeting its minimum cost strategy.  As for Example: N= 3, M = 2, K=1 ==> there is only one way for minimum cost, ie. 0C0 , where '0' denotes No company advertisement, and 'C' denotes company advertisement board.  Similarly, for N =4, M =2, K =1 ==> there is 3 possible ways, ie. C0C0, 0C0C, 0CC0.  Hope to get a reply from someone soon. Thanks. Can I stay overnight [15-hour layover] in Haneda airport without a Japan visa? I’m a Filipino citizen. My flight goes from Vancouver to Manila via Tokyo Haneda. I will arrive in Japan at 6pm then depart at 9am the next day. I'm not planning to go out of the airport during my layover, so I don't see a need to have a transit visa. However, I called the Consulate-General of Japan here in Canada, and they asked me to get a transit visa.  Then I called the Ministry of Foreign Affairs in Japan and asked them the same question.. at first, they told me I don’t need a visa as long as I stay inside the airport, but when I told them that the Consulate-General of Japan in Canada asked me to get a visa, they also asked me to get one.  I'm so confused. I would like to know your knowledge or experience regarding this. Is it possible to repeat my track object along a path? I tried using the method shown here  using an array and then a curve modifier, but this happened when I added the curve modifier and chose the path as the curve object. Show by induction that for all natural numbers n>3  $(1+\frac{1}{n})^n&lt;n$  Let $(1+\frac{1}{n})^n&lt;n$ be true !   We show that $(1+\frac{1}{n+1})^{n+1})&lt;n+1$  $(1+\frac{1}{n})^n(1+\frac{1}{n})&lt;n(1+\frac{1}{n})$   $(1+\frac{1}{n})^{n+1}&lt;n+1$  So now we just need to show that   $(1+\frac{1}{n+1})^{n+1}&lt;(1+\frac{1}{n})^{n+1}$  So here , If I show that :   $(1+\frac{1}{n+1})&lt;(1+\frac{1}{n})$   Does that mean im done with exercise ?