Surely, if you have 2 surfaces interlocking then a horizontal force would also result affect the force of friction. It seems unintuitive that friction is only proportional to normal force.  Is there an intuitive proof of $F=\mu N$?
It is a general concept that horizontal forces can only influence physical quantities along horizontal direction; similarly vertical force can produce acceleration etc. only along vertical direction. This is the same concept that helps us study projectile motion as a combination of two independent motions. If this relation is universally true, how come, then friction depend on a force which normal to it?
Why is $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$? Insert list of figures in the table of contents $\sigma$-algebra vs. $\sigma$-field: is there any difference? The subject says it all: is there any difference between the two concepts of $\sigma$-algebra and $\sigma$-field? In probability theory, they seem to be used more or less interchangeably. If there is no difference, is there any historical reason why some people/schools use the term $\sigma$-algebra, while others use the term $\sigma-$field? Finding solution of non linear DE $x\ddot{x}-\dot{x}^2=1$ I am looking for help on how to find the solution of the following differential equation,  $$x\ddot{x}-\dot{x}^2=1,$$  which comes from solving the Euler-Lagrange equations for the lagrangian $L=x\sqrt{\dot{x}^2+1}$. It is from an exercise in my field theory course. My professor provided me with the solution $x(t)=c_1\cosh{\left(\frac{t-c_2}{c_1}\right)}$, which is a solution when plugged into the equation, but does not give any derivation.   Is there a way to solve this DE other than an educated guess? If so how? Prove holomorphism of expression I have to prove that given $\gamma$ contour of $G \subset \mathbb{C}$  and $f$ continuous function on $G$. Then the function $g(z) = \int_{\gamma} \frac{f(w)}{w-z}dw$ is holomorphic on $\mathbb{C} - \bar{G}$ My attempt: I started by using that given any parametrization $\alpha :[a,b] \rightarrow \gamma$ we can rewrite $g(z)$ as: $g(z) = \int_a^b \frac{f(\alpha(t))}{\alpha(t)-z}\alpha'(t)dt$ Now using the fundamental theorem I derivate to see where $g'(z)$ is defined and I get: $g'(z) = \frac{f(\alpha(b))}{\alpha(b)-z}\alpha'(b) - \frac{f(\alpha(a))}{\alpha(a)-z}\alpha'(a) $ Here I get stuck, I fail so see how this links back to $g$ being holomorphic at $\mathbb{C} - \bar{G}$. Any hints on how to proceed? Thanks! Coloring a black and white PDF graphic I have a graphic in PDF format (originally converted from an SVG) that is monochromatic (black on white). Is it possible using TikZ or PSTricks but not (Illustrator or Inkscape) to include this image in a different color?  Example:    While it lasts (there seems to be no long term way to host a pdf), you can .  Edit: Simple was perhaps a bit of a misnomer - the pdfs are cropped and monochromatic, but to intricate to redraw on my own. The above example is a character from a font (which I can recolor), but it gives a good example of what shapes I am talking about. I'm desperatly confused by notations and formulations so if someone could clarify the following things a little Í would be deeply grateful. The Lie algebra $\mathfrak{so}(1,3)_+^{\uparrow}$ of the proper orthochronous Lorentz group $SO(1,3)_+^{\uparrow}$ is given by \begin{equation} [J_i,J_j]=i \epsilon_{ijk} J_k \end{equation} \begin{equation} [J_i,K_j]=i \epsilon_{ijk} K_k \end{equation} \begin{equation} [K_i,K_j]=- i \epsilon_{ijk} J_k \end{equation} We can now define new generators with the old ones $N^{\pm}_i= \frac{1}{2}(J_i \pm i K_i)$ which satisfy \begin{equation} [N^{+}_i,N^{+}_j] = i \epsilon_{ijk} N^{+}_k ,\end{equation} \begin{equation} [N^{-}_i,N^{-}_j] = i \epsilon_{ijk} N^{-}_k ,\end{equation} \begin{equation} [N^{+}_i,N^{-}_j] = 0. \end{equation} where we can see that $N^{+}_i$ and $N^{-}_i$ make up a copy of the Lie algebra $\mathfrak{su}(2)$ each. My problem is to get what is going one here mathematically precise. Are the following statements correct and if not why:  When we build the new operators from the old generators we complexified $\mathfrak{so}(1,3)_+^{\uparrow}$ \begin{equation}(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} = \mathfrak{so}(1,3)_+^{\uparrow}\otimes \mathbb{C} \end{equation} We saw that $\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C}$ is isomorph to two copies of the complexified Lie algebra of $\mathfrak{su(2)}$: $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq \mathfrak{su(2)}_{\mathbb{C}} \oplus \mathfrak{su(2)}_{\mathbb{C}} $. Where exactly did we need that $\mathfrak{su(2)}$ is complexified here? The Lie algebras defined by $N^{\pm}_i$ are exactly those of $\mathfrak{su(2)}$ and we never use complex linear combination of $N^{\pm}_i$ or am I wrong here? $\mathfrak{su(2)}_{\mathbb{C}}$ is isomorph to $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C}$: \begin{equation}\mathfrak{su(2)}_{\mathbb{C}} \simeq (\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C} \end{equation} Here $(\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C}$ denotes the complexified Lie algebra of $SL(2,\mathbb{C})$ Is $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{R}$ correct?  Here $(\mathfrak{sl}(2, \mathbb{C}))_\mathbb{R}$ denotes the real Lie algebra of $SL(2,\mathbb{C})$ Is $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C} \oplus (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C}$ correct?  I looked this topic up in different books and each seemed to state something different. One book even used three differrent versions of $\mathfrak{sl}(2,\mathbb{C}) $ namely: $\mathfrak{sl}(2,\mathbb{C}) $, $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C}$ and $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{R}$.  states simply that $\mathfrak{sl}(2,\mathbb{C}) $ is the complexification of $\mathfrak{su(2)}$ without making any reference to $SL(2,\mathbb{C})$ which does not help me either. Any help would be great. Capillary action and conservation of energy When I dip a paper towel in a cup of water the water gets drawn up due to . How is this reconciled with conservation of energy, as it would seem on the surface that the potential energy of the system has gone up with no work being done? Removing Firefox in Ubuntu with all add-ons like it never existed I have been using Firefox for a long time in Ubuntu, I downloaded some add-ons but it made firefox extremely slow, even if I remove them nothing will change.  I tried to remove Firefox also, but re-installing it will somehow recover the same Firefox I had- it seems it write some info somewhere that even if you remove Firefox, it will keep them.  Moreover, I am not able to use see any flash videos or website with Firefox(chrome works fine).. Honestly I tampered with plug-in settings in Firefox that this happened. But right now I can't do anything to overwrite the setting...  My question is how I can remove Firefox in a way to remove any previous traces and re-install a new version like for the first time? My internet connection used to be a direct LAN connection to my provider. Back then, everything would load fine on both Windows and Ubuntu (dual boot). However, a while ago they started needing me to dial (PPPoE) using a username and password. Gateway, subnet mask, IP, DNS servers all stayed the same. But since then, I haven't been able to browse certain websites on Ubuntu, even though there have been no such issues on Windows. Some example websites are -  (although share.ovi.com loads fine, and nokia.com loads fine),  (works on Chrome(ium) and Opera but not on Firefox (both 3.6 and 4)) Mozilla  and other random websites.  Some of the websites that don't load show timeout messages and for some websites (like the moz addons one), the browser will keep trying to load without an end (I've left it like that even for hours but not noticed anything different happen).  I have tried changing the DNS servers to public ones. I have even tried booting from a Fedora LiveCD and then changing the DNS to those (and even to the ones of OpenDNS), but the exact same thing happens. What could be inherently wrong with some config within Linux itself that is causing this problem?  Does anyone know why this is happening and how it can be fixed?  Note: This question has been  on SU, but not gotten any responses.  Update: Just saw  that someone else was having similar problem and solved it by putting a NetworkManager.conf file in /etc/NetworkManager. What needs to be in that file? Effective marketing strategies for independent game projects I would really like to hear how other independents generally approach marketing their game with limited resources. I know how critical marketing is since I've seen it first hand on several fronts. I've been part of teams that didn't put an ounce of marketing into their project and didn't come close to meeting expectations. I've also poured a lot of time into visiting forums to discuss our previous title which resulted in minor bumps over time.  I really think  is doing some amazing grass roots marketing with the way they are managing their community around their Blog. They've massed a following that is willing to pay (far in advance) them for their game before they finish it to help support them. That's awesome.  But how many people are really willing to divulge such information that early? Should that be a concern? Should I start talking about my next project before I really have anything implemented?   What has worked for you? I would be particularly interested to hear how people approach Apple's iTunes store. I believe we're looking at around 33,000 games on the store these days. Should I use 'or' or 'nor' after a negative statement? Starting eclipse from unity launcher What is the best way of defining white noise process so it is intuitive and easy to understand? Which is the simplest way to update a Label from another Thread?   I have a Form running on thread1, and from that I'm starting another thread (thread2).  While thread2 is processing some files I would like to update a Label on the Form with the current status of thread2's work.   How could I do that? How do I install an R package from source? I need to graph a rectangle on the Cartesian coordinate system. Is there an equation for a rectangle? I can't find it anywhere. TTY doesn't wait for password I upgraded from 16.04 to 18.04. When I press Ctrl+Alt+F3 I get the terminal which prompts for the username. When I type my username and press Enter, it doesn't wait for my password (its as if I pressed Enter without entering anything). Even if I enter something, the text shown in plaintext and then it keeps on saying my password is wrong. After few times, the screen is cleared and it prompts for my username again and the same loop goes on. Say I have the titanic kaggle competition, but I'm not interested in the competition for predicting survival for each individual. Instead I want the most accurate estimate of total survivors on the titanic. Would this be achieved by using a probabilistic model, then adding the probabilities for each individual? For example, if I have 3 people and 1 survived, but my model produced 0.4, 0.4, and 0.4 probabilities for each person to survive, I calculate 0 survived. But if I add 0.4 for each person, I get 1.2, which is closer to the actual. Does this make sense? Filter a Python list by predicate I would want to do something like:  &gt;&gt;&gt; lst = [1, 2, 3, 4, 5] &gt;&gt;&gt; lst.find(lambda x: x % 2 == 0) 2 &gt;&gt;&gt; lst.findall(lambda x: x % 2 == 0) [2, 4]   Is there anything nearing such behavior in Python's standard libraries?  I know it's very easy to roll-your-own here, but I'm looking for a more standard way.