How do they film an actor talking to themselves?
How are dual-role films shot?
A particular big Delta I would like to insert the below mathematical symbol, it looks like a big Delta but different. Is there command to obtain a similar shape? How to disprove this fallacy that derivatives of $x^2$ and $x+x+x+\dots\quad(x\text{ times})$ are not same. Possible Duplicate:          \begin{align*} x^2 &amp;= \underbrace{x + x + x + \dots + x}_{x \text{ times}}, \\ \therefore \frac{\mathrm{d}}{\mathrm{d}x} (x^2) &amp;= \frac{\mathrm{d}}{\mathrm{d}x} (\underbrace{x + x + x + \dots + x}_{x \text{ times}}) \\ &amp;= \underbrace{1 + 1 + 1 + \dots + 1}_{x \text{ times}} \\ &amp;= x. \end{align*}  But we know that $$ \frac{\mathrm{d}}{\mathrm{d}x} (x^2) = 2x. $$  So what is the problem?  My take is that we cannot differentiate both sides because $\underbrace{{x+x+x+\cdots+x}}_{x \text{  times}}$ is not fixed and thus $1$ is not equal to $2$. What tool can I use to manage the configuration of my Windows Server environments I help manage environments for an application that uses cloud-based Windows server VMs. The application stack consists of Windows Server 2008 and Windows Server 2012, SQL Server 2008, IIS, and SharePoint 2010. I have several environments Dev/Test/Stage/Prod. I am looking for a way to do configuration management of those environments to make sure any environmental changes are applied in a consisent way and there is a way to verify that no changes have been made to one environment that have not been made to other environments.  I have been reading up on Puppet, Chef, etc and I like the idea of doing delcarative configuration management. What is a good method or tool to use to help manage the configuration of these environments. I know a little  about SCCM but it is too expensive for this particular situation although I would still be interested in knowing if it has the capabiltiy to do those things. How do you subtract Dates in Java? The bottom bracket of my bike was completely stripped to the point that when you push the left side crank arm, the right crank moves and you can see the bottom bracket lock move out the frame. The bike shop tried putting Teflon tape but it's temporary. It's a 26' xc hard tail. For all I know both sides of the bottom bracket(square tapered) are stripped clean hence the Teflon tape put by the previous owner. Can I fix the problem by changing into a hallow tech cranks because it works like a thread less bb, or should I just buy a thread less square tapered bb? Any suggestion on what to do? Where is the New Oxford American Dictionary stored? Which site deals with career development questions for programmers? Is there any site in the network where career development questions about the programming profession can be asked ? I thought Programmer's was the site for it, but that doesn't appear to be the case. I came upon an interesting way to relatively quickly compute modular exponentiation with large numbers. However, I do not fully understand it and was hoping for a better explanation.  The method basically states if you have $x^y \bmod N$, it can be computed by repeatedly multiplying by $x$ modulo $N$:  $$x \bmod N \rightarrow x^2 \bmod N \rightarrow x^4 \bmod N \rightarrow x^8 \bmod N \rightarrow \ldots \rightarrow x^{2[\log y]} \bmod N$$  This is supposed to calculate in $\mathcal{O}(\log_2 N)$ time.  An example is given:  $$x^{25} = x^{11001_2} = x^{10000_2} * x^{1000_2} * x^{1_2} = x^{16} * x^8 * x^1$$  I don't really understand how this is faster. Aren't you just doing the same thing? You're just breaking down the exponents into representations of base $2$. Also where does $x^{2[\log y]}$ come from? If you're breaking it down into exponents of two where does a logarithm come from? (Aren't logs representations of numbers in exponents of $10$?)  I'm sorry if this appears to be a really stupid question - I have minimal knowledge in math.  Also, I wasn't sure if this belongs here or on stack exchange, so give me the word and I'll move it. what are the meanings of &quot;treated observation&quot; and &quot;non-treated observation&quot; in this context ?  Further, propensity score matching is a more robust method of matching that quantitative accounting researchers frequently use (Rosenbaum &amp; Rubin, 1983). With this technique, the researcher first estimates a logit or probit regression with the treatment effect as the dependent variable (i.e., S&amp;P 500 membership, continuing with our earlier example) and independent variables that are known determinants of the treatment (possibly economic performance, firm size, age, industry, etc. in our example). From this first-stage regression, the researcher obtains propensity scores. 6 The researcher then matches each treated observation to a non-treated observation based on the distance between their propensity scores (i.e., those observations with the closest propensity scores are matched). 7 Hoi et al. (2013) utilize propensity score matching in their study that links irresponsible CSR activities with tax aggressiveness. They first generate predicted propensity scores for each firm-year using a model of the determinants of a firm engaging in a high level of negative, irresponsible CSR activity. Then, they match a treatment firm (one with a high level of negative CSR activities) to a non-treatment firm (one with a low level of, or no, negative CSR activities) by matching observations with the closest propensity scores. Hoi et al. (2013) then re- test the relation between negative CSR activities and tax aggressiveness using this matched sample. These matching techniques help mitigate the researcher’s potential failure to consider certain firm characteristics that differ between treatment and non-treatment firms (Dhaliwal et al., 2012). Baire: Show that $f\colon \mathbb{R}\to\mathbb{R}$ is a polynomial in an open bounded set Let $f\colon\mathbb{R}\to\mathbb{R}$ be an infinitely differentiable function, and suppose that for each $x\in\mathbb{R},$ $\exists\  n=n(x)\in\mathbb{N}$ such that $f^{(n)}(x)=0$. For each fixed $n\in\mathbb{N}$, consider the sets $F_n=\{x\in\mathbb{R} : f^{(n)}(x)=0\}$ and let $I=[a,b]$ a finite closed interval.  Show that $$I=\bigcup_{n=0}^\infty(I\cap F_n).$$ Show that there is a non empty open set $A\subset I$ where $f$ is a polynomial.   My attempt:  If $x\in I$ then by hypotesis there is $n$ such that $f^{(n)}(x)=0$ then $x\in F_n$ so then $\bigcup F_n\supset I$ then $$I=I\cap\bigcup_{n=0}^\infty F_n=\bigcup_{n=0}^\infty(I\cap F_n).$$  Let $$B_k=\bigcap_{n=k}^\infty F_n.$$ I want to show that $I\cap B_k$ has non-empty interior for some $k\in\mathbb{N}$. Then there is $A\subset B_k$ and by Taylor's Theorem $f$ will be a polynomial. But I didn't know how to prove this. I tried using the following bigger set: $$B=I\cap\bigcup_{k=0}^\infty B_k=I\cap\bigcup_{k=0}^\infty \bigcap_{n=k}^\infty F_n.$$ I think that it suffices to show that $B^o\neq\emptyset$ maybe using the Baire's Theorem proving that $B$ is a not meagre set.   More general Problem  Let $f$ be an infinitely differentiable function on $[0,1]$ and suppose that for each $x \in [0,1]$ there is an integer $n \in \mathbb{N}$ such that $f^{(n)}(x)=0$. Then does $f$ coincide on $[0,1]$ with some polynomial? If yes then how. I thought of using Weierstrass approximation theorem, but couldn't succeed. Answer: The proof is by contradiction. Assume $f$ is not a polynomial. Consider the following closed sets: $$ S_n = \{x: f^{(n)}(x) = 0\} $$ and $$ X = \{x: \forall (a,b)\ni x: f\restriction_{(a,b)}\text{ is not a polynomial} \}. $$ It is clear that $X$ is a non-empty closed set without isolated points. Applying Baire category theorem to the covering $\{X\cap S_n\}$ of $X$ we get that there exists an interval $(a,b)$ such that $(a,b)\cap X$ is non-empty and $$ (a,b)\cap X\subset S_n $$ for some $n$. Since every $x\in (a,b)\cap X$ is an accumulation point we also have that $x\in S_m$ for all $m\ge n$ and $x\in (a,b)\cap X$. Now consider any maximal interval $(c,e)\subset ((a,b)-X)$. Recall that $f$ is a polynomial of some degree $d$ on $(c,e)$. Therefore $f^{(d)}=\mathrm{const}\neq 0$ on $[c,e]$. Hence $d&lt; n$. (Since either $c$ or $e$ is in $X$.) So we get that $f^{(n)}=0$ on $(a,b)$ which is in contradiction with $(a,b)\cap X$ being non-empty. Fundamentally, what does a friction force do for a rolling object? I am very confused by the way my textbook explains it. In my textbook , it says:  A wheel rolling on a horizontal flat surface at a constant velocity experiences no friction force. Why?   On the same surface, there is an acceleration of the wheel pointing to the right (probably caused by a force), so the ball is angularly accelerating in the clockwise direction. In this case, a friction force appears, and it is also pointing to the right. How come? What does the friction do for this wheel?  On an inclined plane, a ball freely rolls down the surface. The direction of friction is up the ramp, which confuses me because in the previous example the friction force was in the direction of the wheel's acceleration.   And there is a difference if a wheel is freely rolling and if there is a torque acting on the ball's center of mass. WHY???  What does friction do in these cases? How does it cause an object to roll? Can fresh pasta sheets be used to assemble a lasagne, then left in the fridge overnight before baking? Or will this make them mushy or otherwise degrade them?  I have recently gotten into making fresh pasta, and I want to prep a lasagne the night before. I have done this before with store-bough dried noodles, but I'm not sure if doing this will have a negative effect on fresh, more delicate pasta. How do I get:  $$\cos{\theta} \lt \frac{\sin{\theta}}{\theta} \lt 1$$ Find a bijection to prove a fact about integer partitions Prove that for fixed positive integers $k$ and $n$, the number of partitions of $n$ is equal to the number of partitions of $2n + k$ into $n + k$ parts.  I tried this with different values of n but can't seem to get a bijection between the two partitions. There are no interfaces on which a capture can be done Looking for book: contact with aliens, five different races, mismatched claws, biological imperative to breed Why there is "the" before some names but not others Is there a rule beyond the common "no the with proper nouns and names" for the following problem?     I saw the Empire State Building.      We went to the White House.      We saw the Golden Gate Bridge.   but     I went to Death Valley National Park.      I crossed Brooklyn Bridge. When can you switch the order of limits? How to create custom config section in app.config? How to get vertex normals from a shape-key in Python? I'm trying to create an exporter for a format, which includes shape key support. This format requires both vertex coordinates and normals to be specified. This should be logical, because the vertex selected on the image below won't have the same normals:     Comparing it to this one:    But I actually haven't found any way to export shape key normals. Both Bmesh Vertex and ShapeKey Vertex only store the coordinates.  Is there any way to get them?