We know that on increasing the kinetic energy of a gas its temperature increases: $$\text{Kinetic Energy} = \frac32 kT$$ where $k$ is the Boltzmann constant. This equation is derived from the .   When we switch on a fan we increase the kinetic energy of the gas molecules increasing their temperature, but we generally feel cold. This happens even if we are not sweating. What am I missing here?
So I was thinking... If heat I feel is just lots of particles going wild and transferring their energy to other bodies, why am I not burned by the wind?  When I thought about it more I figured out that wind usually carries some humidity, and since particles of liquid are moving same speed as the wind, they are basically static relative to each other, so no energy is transferred between them (wind and water particles). And if that water sticks to my skin and wind blows, it'll evaporate thus taking energy from my skin and make me feel cold.  Thing is, I don't think that's really the case but even if it is, if I somehow dry out the wind, will it burn me if it's strong enough? And winds can reach some pretty high velocities (though I must admit I'm not sure if they are comparable to movement of atoms in warm bodies etc...).  So. Bottom line. Can I be burned by wind in some perfect scenario?
Computing the Galois group of $x^4+ax^2+b \in \mathbb{Q}[x] $ I want to compute the Galois group of some polynomials, but I want to see some examples first. For example this proposition could be helpful. I don't know how to prove it &lt;.&lt;  Let's consider a polynomial $f(x)= x^4+ax^2+b \in \mathbb{Z} $. Let $ \pm\alpha , \pm\beta$ denote the roots of $f(x)$. Note that $f$ is irreducible iff $\alpha^2, \alpha\pm\beta \notin \mathbb{Q}$. So let's suppose that $f(x)$ is irreducible. Let's denote $G$ the galois group of $f$ (the Galois group of the splitting field $\mathbb{Q}(\alpha,\beta)/\mathbb{Q}$)  $i)$ $G\cong V$ the Klein 4-group iff $\alpha\beta \in \mathbb{Q}$  $ii)$ $G\cong \mathbb{Z}_4$ iff $\mathbb{Q}(\alpha\beta)=\mathbb{Q}(\alpha^2)$  $iii)$ $G\cong D_8$ the dihedral group of order 8, iff $\alpha\beta \notin \mathbb{Q}(\alpha^2)$  I need help with the proofs of this propositions... Well I'm very lost with the computation of Galois group, some care must be exercised, I don't have it. For example I know that Galois group acts transitively, in the sense that $u\in K/F$ a galois extension, the other roots of the minimal polynomial of $m_u(x)\in F[x]$ are precisely the elements $ \sigma(u)$ , where $\sigma \in Gal(K/F)$. And clearly any automorphism, is determined by only the value of a basis. In this case we are working on $\mathbb{Q}(\alpha,\beta)$, so I have to know the values of both. But I have to respect the algebraic relations involving $\alpha , \beta$ I proved that in $i),ii)$ $\mathbb{Q}(\alpha,\beta) = \mathbb{Q}(\alpha) $ So I have to know the values on $\alpha$.   My first question is: If in both cases $i),ii)$, I have an extension of degree 4 (so $|G|=4$) I have to map $\alpha$ to determine the automorphism, but I have also to map $\alpha$ to all of it's four roots. Why the galois group in this case are different?  My second question is about $iii)$ I have no idea How to attack it. Limit of the sequence $a_n=\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}$ While studying I read about the principle of underdetermination of scientific theories. I made some researches online but I am more confused than before. I read about the Quine-Duhem holistic thesis, but I do not know if is the same as underdetermination. Parametric Interpolation of 2D data Poll on best VNC / remote desktop software for assisting others on Windows/Mac machines from Ubuntu?    I've heard good things about TeamViewer and Fog Creek Copilot, but I'm wondering if the included GNOME Vinaigre VNC client is good enough for this.  To specify, I'm looking for best option based on:   SIMPLEST ease-of-use for client to download/use on their end. See #1. Works cross-platform I am able to control client's mouse and/or keyboard from remote machine. Why is the cone measurable? Problem Denote by $\lambda_n$ the Lebesgue measure on $\mathbb {R}^n$. Let $h \in \mathbb {R}_{&gt;0}$. Let $A \subset \mathbb {R}^{n-1}$ be Lebesgue-measurable with finite measure. Define the cone $C(A,h)$ as the union of all straight lines connecting a point in $A \times\{0\}$ with the point $(0, h) \in \mathbb {R}^{n-1} \times \mathbb{R}$. Prove that $\lambda_n (C(A,h))=\frac{h}{n}\cdot \lambda_{n-1}(A)$. Where I struggle  Assuming that $C(A,h)$ is (Lebesgue-)measurable, this is a simple application of Fubini. How Fubini is used is all clear to me.  However, why is $C(A,h)$ measurable? It seems one could roughly argue as follows: As $A$ and $[0,h]$  are measurable so is $A \times [0,h]$. As $A \times [0,h]$ is measurable it is the union of a null set and a Borel set. $C(A,h)$ is the image of $A \times [0,h]$ under a certain diffeomorphism (c.f. ). Diffeomorphisms map null sets to null sets, and Borel sets to Borel sets. Hence, $C(A,h)$ is (as a finite union of measurable sets) measurable.  Unfortunately, I am not allowed to use the above properties of diffeomorphisms. What are other ways to prove the claim? Please only give a hint to set me thinking. Excel destroying special character when saved as CSV TeX support works fine in question and answer bodies, and comments can be rendered manually by tapping and selecting Render MathJax. See related question .  However, the titles are not rendered which makes the browsing experience less than perfect. See screenshot below.   Are there any plans to implement MathJax support in titles?  (I'm running Stack Exhange iOS App Version 1.0.0 on an iPhone 4S with iOS 7.1.1)    EDIT: I was just informed about  asking for the same functionality for the Android app. I guess I shouldn't hope too much then... How to configure blender so that it uses only 50% of total CPU when rendering? Most common C# bitwise operations on enums For the life of me, I can't remember how to set, delete, toggle or test a bit in a bitfield. Either I'm unsure or I mix them up because I rarely need these. So a "bit-cheat-sheet" would be nice to have.  For example:   flags = flags | FlagsEnum.Bit4;  // Set bit 4.   or  if ((flags &amp; FlagsEnum.Bit4)) == FlagsEnum.Bit4) // Is there a less verbose way?   Can you give examples of all the other common operations, preferably in C# syntax using a [Flags] enum? Can sudo be reinstalled after being removed? I am trying to mimic the interesting proof from  to give a generating set of the ideal correspond to the $d$-uple embedding: We fix the notation $\Delta = \big\{ (\nu_0, \dots, \nu_n) \in {\mathbb{N}}^{n+1}_0 ~:~ \nu_0 + \dots + \nu_n = d \big\}$. The polynomial map is $\theta : k[y_\nu : \nu \in \Delta] \rightarrow k[x_0, \dotsc, x_n]$ given by $$ \theta ( y_\nu ) := x^{\nu_0}_0 \cdots x^{\nu_n}_n $$ and we know that ${\mathrm{ker}}~\theta$ is the ideal of the $d$-uple embedding, i.e., $\rho_d({\mathbb{P}}^n) = Z({\mathrm{ker}}~\theta)$ (cf. Exercise I.2.12 Hartshorne). Now let $W \leq k[y_\nu : \nu \in \Delta]$ denote the ideal $$ W := \Big\langle y_{\tau_1} y_{\tau_2} - y_{\tau^{\prime}_1} y_{\tau^{\prime}_2} ~:~ \tau_1, \tau_2, \tau^{\prime}_1, \tau^{\prime}_2 \in \Delta, \tau_1 + \tau_2 = \tau^{\prime}_1 + \tau^{\prime}_2 \Big\rangle $$ Clearly, $W \subseteq {\mathrm{ker}}~\theta$. Using the lemma in the above link, we need to produce a $k$-subspace $T \subseteq k[y_\nu : \nu \in \Delta]$ with the conditions $$ T + W = k[y_\nu : \nu \in \Delta], \hspace{.2in} \theta \lvert_{T} = {\mathrm{injective}} $$ After a few thoughts it seems reasonable to think of the following subspace as a candidate for $T$: First define the equivalence relation on $\Delta \times \Delta$ given by $(\tau_1, \tau_2) \sim (\tau^{\prime}_1, \tau^{\prime}_2)$ if $\tau_1 + \tau_2 = \tau^{\prime}_1 + \tau^{\prime}_2$. Let $\Omega \subseteq \Delta \times \Delta$ be a subset with a unique representative from each class of $\sim$. Now let $T$ be the $k$-span of the monomials $y^{\alpha_1}_{\tau_1} \dotsc y^{\alpha_r}_{\tau_r}$ (with the distinct elements $\tau_1, \dotsc, \tau_r \in \Delta$ and $\alpha_1, \dotsc, \alpha_r \geq 1$) such that $(\tau_i, \tau_j) \in \Omega$ for each $i \neq j$. Clearly, $\theta$ is injective if restricted to $T$. I got lost in the step : Each monomial $M$ of $k[y_\nu : \nu \in \Delta]$ satisfy $M \equiv b$ mod $W$ for some $b \in T$. Did I do anything wrong? Compiling with either XeLaTeX or LuaLaTeX, \emph{} no longer displays text as expected, i.e. in italics. \textit{} works fine. Here is a MWE demonstrating the issue.  \documentclass{article}  \usepackage{fontspec}  \begin{document}      \emph{lorem ipsum}      \textit{lorem ipsum}  \end{document}   Here is the output I get :   I have TeXLive 2016 (MacTeX) installed. I updated it an hour ago with sudo tlmgr update --self --all.  I guess there is something wrong but what ? I'd be happy to have some help. Thank you. What does this depiction of a black hole in the movie Interstellar mean? I was expecting a whirlpool in 3D and the matter glowing from friction as it nears the center, as I expected a event horizon to be negligible visually.  How does this depiction work? How big is the central sphere? I am puzzled by the perpendicular circles. Are they the event horizons if they are both visible? What would be a path of a particle as it gets swallowed into the singularity? How can I echo a newline in a batch file? How can you you insert a newline from your batch file output?  I want to do something like:  echo hello\nworld   Which would output:  hello world Path connected space that is not locally connected Record Animation In BGE I am having trouble getting the Record Animation setting to work in the BGE. Is it possible to record animated characters that are controlled using BGE's logic bricks with some script? In my script I change the positions of the characters bones. I've been reading up on STL containers in my book on C++, specifically the section on the STL and its containers. Now I do understand each and every one of them have their own specific properties, and I'm close to memorizing all of them... But what I do not yet grasp is in which scenario each of them is used.  What is the explanation? Example code is much prefered. Motion Tracking objects that go out of frame and re-enter later I am motion tracking a video in which the camera turns away from one scene and then shortly returns to it. However, the segments that went off-screen lose their track and do not continue tracking once the camera returns. I am wondering if there is a manual way to tell a segment where and when to continue looking for it's tracking target. Is this possible to do? Find the subgroups of A4 I had a question I was hoping for some help on:     Find all of the subgroups of $A_4$   Here is what I know:  $A_4$ is the alternating group on 4 letters. That is it is the set of all even permutations. The elements are:  $(1), (12)(34), (13)(24), (14)(23), (123), (132), (124), (142), (134), (143), (234), (243)$   which totals to 12 elements. Which means, the subgroups should have order 1,2,3,4,6 and 12. The groups of order 1 and order 12 are trivial. I also know the groups of order 2 which are:  $[1,(12)(34)], [1, (13)(24)], [1, (14)(23)]$  but would someone be able to help me find the subgroups of order 3, 4 and 6 and be able to provide a good, easy explanation of how you got to them for me? I'm pretty sure there is no such groups of order 6, but I don't know how to explain why.  Thank you so much for your help in advance, I really appreciate it!