Proving Modified Taylor expansion of $e^x$ (negative sign for odd powers) is always positive when n is EVEN.
$P_n(x):=1+ \sum_{m=1}^n\dfrac{x^m}{m!}$ has no real root for even $n$ and exactly one real root for odd $n$
jQuery appended table adds closing tag at the end of the text automatically. Why? $ not defined using jQuery in WordPress Any javascript event occurring when user clicks Stop load button? 3 questions about Algebraic Geometry and Zariski topology I've read this exercise on my book, but I don't know how to prove it... Can someone help me?  Let X $\subseteq \mathbb{A}^n$ be an algebraic set, and $f, g:X\rightarrow k$ two regular functions.  a) If we see $f$ as an application $X \rightarrow \mathbb{A}^1(k)$, prove that $f$ is continuous in the Zariski topology.  b) If $X$ is irreducible and an open $U \subseteq X$ exists such that $f_{|U} = g_{|U}$, prove that $f=g$.  c) Give an example of $X, f$ and $g$ such that $f \neq g$ but $f_{|U} = g_{|U}$ in an open $U \subseteq X$.  Thank you!  For b) I have been able to prove that:  Let E be the subset where $f = g$ so E is closed (Why? I know that $E = (f,g)^{-1}(\Delta_{k})$ but... why is $(\Delta_{k})$ closed?). So $U \subseteq E \Rightarrow \bar{U} \subseteq \bar{E}=E$ (beacuse E is closed) $\Rightarrow X \subseteq E$, because $\bar{U}=X \Rightarrow E = X$ and $f=g$ in X. How does the Eco-Cooler air conditioner really work? describes a device for developing countries which apparently cools the air by 5 °C without electricity. According to , it works by lowering the pressure to drag in the ambient air. Wouldn't the ambient air be at the same temperature?  Does the constriction compress the air and temporarily increase the pressure and so temperature, which then dissipates heat because it is hotter than surroundings; then after passing the constriction, the pressure returns to almost ambient but the temperature drops to below ambient?  How does the Eco-Cooler air conditioner actually work, if at all? How to terminate a script? I am aware of the die() command in PHP which exits a script early.  How can I do this in Python? How to derive variance-covariance matrix of coefficients in linear regression I am reading a book on linear regression and have some trouble understanding the variance-covariance matrix of $\mathbf{b}$:    The diagonal items are easy enough, but the off-diagonal ones are a bit more difficult, what puzzles me is that  $$ \sigma(b_0, b_1) = E(b_0 b_1) - E(b_0)E(b_1) = E(b_0 b_1) - \beta_0 \beta_1 $$  but there is no trace of $\beta_0$ and $\beta_1$ here. Two domains (when SSL) on same directory How to write a function (piecewise) with bracket outside? How to write this Piecewise function using LaTeX?  I tried $ \begin{array}{cc}   \{ &amp;      \begin{array}{cc}       0 &amp; x\leq 0 \\       \frac{100-x}{100} &amp; 0\leq x\leq 100 \\       0 &amp; 100\leq x     \end{array} \end{array} $  but it didn't work perfectly, the bracket &quot;{&quot; looks small, see: Selecting text in profile activity comments causes unexpected clipping Ownership of content in Community Wiki posts I am relatively new to Java, and often find that I need to sort a Map&lt;Key, Value&gt; on the values.  Since the values are not unique, I find myself converting the keySet into an array, and sorting that array through array sort with a custom comparator that sorts on the value associated with the key.  Is there an easier way? Every finite set contains its supremum: proof improvement. Every finite subset of $\mathbb R$ contains its supremum (and its infimum)   Proof Let $A=\{a_1,...,a_n\}$ be a finite subset of $\mathbb{R}$. Since it is non-empty and it is bounded ($\max A$ is an upper bound), it has supremum, that is $\exists \sup A$ and by definition $\forall a \in A \;\, a \leq \sup A$. Let's suppose that $\sup A  \not\in A$ then, since $\max A \in A$ we have that $\max A &lt; \sup A$. But considering that $\mathbb Q$ is dense in $\mathbb R$ we can conclude that $\exists r \in \mathbb Q$ s.t. $\max A &lt; r &lt; \sup A$, but this is absurd since $r$ is an upper bound of $A$ and it is lower than the supremum. Necessarily, $\sup A\in A$.  Is there anything wrong? Is there any way to prove this without using density of $\mathbb Q$ or another property? Thanks in advance. How do I use '&' literally in LaTeX? The question is:     Are there square matrices $A,B$ over $\mathbb{C}$ s.t. $A^2+B^2=AB$ and $BA-AB$ is non-singular?   From $A^2+B^2=AB$ one could obtain $A^3+B^3=0$. Can we get something from this?  Edit:  $$A^2+B^2=AB\implies\\ A(A^2+B^2)=A^2B \implies\\ A^3+AB^2=A^2B \implies \\ A^3+(A^2+B^2)B=A^2B \implies \\  A^3+A^2B+B^3=A^2B \implies \\ A^3+B^3=0 $$ Why images are blury when extracting image sequence from video clip? I have an issue when generating an image sequence from a video clip. Pictures 1,2,and 3 are three consecutive images from a video clip. Picture 4 is a close-up of the heel of one of them. The camera is on a tripod. As you can see there is no issue with the brick wall, but with any moving part there is. The camera is a Panasonic HDC-SD100. The clip is AVCHD 1920/1080. If I use VLC to extract the image sequence there is no issue. I use all the default settings except for the rendering resolution which I set to 100% and image compression to 0%. I have the issue with Blender 2.79b and 2.80. With this image issue it is impossible to do camera tracking. Any ideas what I need to do in order to solve this. Best regards. Feature selection with Random Forests Is expression for law of equipartion of energy valid for real gas? I know that according to law of equipartion of energy ,energy per molecule per degree of freedom is 1/2 Kb T I wanted to know if it is true for a real gas ? input to pic with 12V and resistors I have 12V, Use these 12V to power a device (that has a 12V output signal) and to power a 7805. After the 7805 there is a pic microcontroller where I would have to connect the output from my device. I was thinking of having 2 resistors in a 3/1 ratio in order to have about 4V at the pin of the pic. Can this be done, given that the ground is common, or will it give some sort of problem with the lm7805? Cancel misclicked flags I'm wondering (since it just happened to me), is it possible to "cancel" ("undo", "delete", "retract", whatever) a flag I just raised if I misclicked it, so the mods won't waste time?  I looked everywhere, but I can't find such a feature.