What is the word for a person who acts super nice to you to impress others and build his image, but in real life, he doesn't actually mean it? For example, what would you call Cindrella's stepmom who treats her well in front of her dad, however, when he dies she makes Cindrella do chores? I need this word to describe a ruler who is just pretending to protect his people for the sake of his image and doesn't really care about whatever happens to them.
I'm looking for a word that can describe a person who does nice things for other people (e.g holding the door open, carrying someone's things) but only for self gain; this person only does nice things to make themselves look good in the eyes of others, but doesn't actually care about the well-being of the people they're helping.
Is floating point math broken? Consider the following code:   0.1 + 0.2 == 0.3  -&gt;  false   0.1 + 0.2         -&gt;  0.30000000000000004   Why do these inaccuracies happen? I want to prove, that $GL(n,K)$ is open in $M(n,K)$, where ($K=\{\mathbb{R},\mathbb{C},\mathbb{H}\}$.  For the prove I don't want to use that the determinant is continous. Alternativly I assume a submultiplicative norm.  First step: Define $U_A:=\{B\in M(n,K):\Vert A-B\Vert&lt;\frac{1}{\Vert A^{-1}\Vert}\}$ and show that it is a neighbourhod of $A$ in $M(n,K)$.  Second step: I show that the series $\sum_{k\ge0}(1-A^{-1}B)^kA^{-1}$ converges absolutely and the limit is inverse to $B$.  For the second step I should use the Neumann series, right? But I have some problems with the first step. Any help? How to compute the integral $\int_{0}^{\infty} \frac{x^{1/3}}{1+x^{2}} \ dx$ I want to compute this integral $$\displaystyle\int_{0}^{\infty} \frac{x^{1/3}}{1+x^{2}} \ dx$$  What I did was the following. I substituted $x=t^{6}$, so that my $dx= 6t^{5} \ dt$ and so the integral changes to $$\int_{0}^{\infty} \frac{t^2}{1+t^{12}} \cdot 6 t^{5} \ dt =6 \cdot \int_{0}^{\infty} \frac{t^7}{1+t^{12}} \: dt$$  Now If I substitute $t^{4}=v$ then what I will be having is the following integral $$\frac{6}{4}\cdot \int_{0}^{\infty} \frac{v}{1+v^{3}} \ dv$$  Now I can write $1+v^{3} = (1+v) \cdot (1-v+v^{2})$ and so I have   \begin{align*} \int_{0}^{\infty} \frac{v}{1+v^{3}}\: dv &amp;= \int_{0}^{\infty}\biggl[\frac{1}{3}\cdot \frac{v+1}{1-v+v^{2}} - \frac{1}{3} \cdot \frac{1}{1+v}\biggr]\: dv \end{align*}  Now the point is that the integral of $1/(1+v) \to \infty$, so I am not sure if this is the right way to do. Can anyone suggest anything? I was writing a toString() for a class in Java the other day by manually writing out each element of the class to a String and it occurred to me that using reflection it might be possible to create a generic toString() method that could work on ALL classes. I.E. it would figure out the field names and values and send them out to a String.  Getting the field names is fairly simple, here is what a co-worker came up with:  public static List initFieldArray(String className) throws ClassNotFoundException {      Class c = Class.forName(className);     Field field[] = c.getFields();     List&lt;String&gt; classFields = new ArrayList(field.length);      for (int i = 0; i &lt; field.length; i++) {         String cf = field[i].toString();         classFields.add(cf.substring(cf.lastIndexOf(".") + 1));     }      return classFields; }   Using a factory I could reduce the performance overhead by storing the fields once, the first time the toString() is called. However finding the values could be a lot more expensive.  Due to the performance of reflection this may be more hypothetical then practical. But I am interested in the idea of reflection and how I can use it to improve my everyday programming. Self-promotion in chat We know from the  that it is accepted only in a limited way in answers and frowned upon if found to be spammy. We know from  that it can be deleted at any time from comments if the comment is not relevant or spammy.  But what about the chat? This leaves chat to be the only legitimate place to promote products or services that the user is affiliated to (albeit in an ad-like way).  What is your opinion? Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$ Problem:   If $f(x)$ is continous at $x=0$, and $\lim\limits_{x\to 0} \dfrac{f(ax)-f(x)}{x}=b$, $a, b$ are constants and $|a|&gt;1$, prove that $f'(0)$ exists and $f'(0)=\dfrac{b}{a-1}$.   This approach is definitely wrong:    \begin{align} b&amp;=\lim_{x\to 0} \frac{f(ax)-f(x)}{x}\\ &amp;=\lim_{x\to 0} \frac{f(ax)-f(0)-(f(x)-f(0))}{x}\\ &amp;=af'(0)-f'(0)\\ &amp;=(a-1)f'(0) \end{align}  I will show you a case why this approach is wrong:     \[f(x)= \begin{cases} 1,&amp;x\neq0\\ 0,&amp;x=0 \end{cases}\]   $\lim_{x\to0}\dfrac{f(3x)-f(x)}{x}=\lim_{x\to0} \dfrac{1-1}{x}=0$   but $\lim_{x\to0}\dfrac{f(3x)}{x}=\infty$,$\lim_{x\to0}\dfrac{f(x)}{x}=\infty$   Does anyone know how to prove it? Thanks in advance! Resistor in series needed with LED string? Proving the geometric sum formula by induction Inequality $\left(a-1+\frac{1}b\right)\left(b-1+\frac{1}c\right)\left(c-1+\frac{1}a\right)\leq1$ Define path for packages in the LaTeX file (analog of \input@path or \graphicspath for \usepackage) I execute quite a lot of sudo commands.  I noticed that if I execute one and then another one within few minutes, the 2nd time I don't get the message:     [sudo] password for my_username:   but I do get it when there is more time between the execution of the two commands.  What's the period of time in which the message is not appearing? How can we check it / update it? $$ A $$  Light is travelling away from the absolute center of the Schwarzchild black hole (i.e. none of that spinning/stretching field) at $\ C $, and so shouldn't $\ \Delta C$ need to be greater than $\ C $ in order for the light to collapse into the black hole, therefore violating the relativistic limit of $\ C $?    It seems like light would just 'hover' right in the center of a black hole then. Because if it goes away from it, the event horizon pulls it back, and if it goes into it, the velocity change must be greater than $\ C $ which violates general relativity; so technically there's no way for the light to move away, and so it would just be still.  The thing is that the light must move at the speed of light or general relativity doesn't work. It's massless... But in the center of a black hole, if the above is true (it can't move or escape), then it is not travelling at the speed of light. So it cannot be still.  It cannot be spinning in a constant orbit either, because that would require velocity in an outward angle. And it cannot move in an outwardly fashion, only in an inwardly one.  Is it spinning in an infinitely decaying spiral towards the center?  If so what happens to it if and when it reaches the direct and complete center, i.e. The case of it being 'shot' from the center.    $$ B $$  If you shot light at an angle towards a black hole, but not directly to the center of it  then that light from the outside will eventually become the same as the light that I said was shot from the center to the outside; because in both cases, they end up in an infinitely decaying spiral.    If so it would end up exactly like the light in the above paragraph, so then we need to figure out what happens to that light to figure this one out, or vice versa.    $$ C $$  If you shot light directly towards the center of a black hole it would have no spiral to decay into, so therefore it must stop at some point in the center of a black hole. If it stops though, it loses it's momentum and violates general relativity again because it's energy "disappears".    If so then it violates conservation laws. So I'm assuming this simply is not true.    Is the light travelling into a black hole somehow still travelling at the speed of light, in every direction possible where that direction keeps it facing towards the center?  If so, why is there not a point at which the only thing left for the light to do is stay still, since all other paths would increase the distance between it and the black hole's center, and therefore become Case A. And if there isn't, what would happen if there was?  Also, if it's a time and observation answer where it supposedly takes and infinite amount of time for C to hit the center, wouldn't it be the case that the black hole is 'eating' all of the available light in the universe and keeping it in stasis?  Also, what happens in a Kerr black hole then? Does it just rip the field of time/gravity at an angle such that the light would never get closer than the center? Find the maximum value of $72\int\limits_{0}^{y}\sqrt{x^4+(y-y^2)^2}dx$ Find the maximum value of $72\int\limits_{0}^{y}\sqrt{x^4+(y-y^2)^2}dx  $ for $y\in[0,1].$  I tried to differentiate the given function by using DUIS leibnitz rule but the calculations are messy and I tried to solve directly by integrating it but that also is not working.Can someone please help me in solving this question? I have a PHP script that can encode a PNG image to a Base64 string.  I'd like to do the same thing using JavaScript. I know how to open files, but I'm not sure how to do the encoding. I'm not used to working with binary data. Software to synchronize two directories (local/remote) I need software that would detect changes in directory (A) every 5 seconds or so and upload/delete changed files from/at (B) remote directory.  I found non-commercial software, . I am wondering if there are any commercial alternatives? Convert LaTeX to HTML Gradient and Swiftest Ascent I want to understand intuitively why it is that the gradient gives the direction of steepest ascent. (I will consider the case of $f:\mathbb{R}^2\to\mathbb{R}$)  The standard proof is to note that the directional derivative is $$D_vf=v\cdot \nabla f=|\nabla f|\,\cos\theta$$ which is maximized at $\theta=0$. This is a good verification, but it doesn't really help me understand the result. is there a way to add hyperlinks in a Illustrator document? I am creating a CV in Illustrator and exporting it to a PDF. I just read an article which stated that it is a good idea that we use hyperlinks in the CV so that the employer can just click and not have to copy the address. I like this idea, but I don't know how to add a hyperlink to text in Illustrator. Is it possible? Can “due to” and “because of ” be used interchangeably? Is it fine to use due to in place of because of ? How about the other way around? Are any of these sentences ungrammatical?   He was lost because of the storm. He was lost due to the storm. He lost his way due to the storm. He  lost his way because of the storm. Is there a way to get the count of rows in all tables in a MySQL database without running a SELECT count() on each table?