I'm just looking for a simple reminder when it comes to aperture, such as: when there's a lot of light in the atmosphere where a photo is exposed, the aperture should be low, or something. The more light, the lower the aperture - something like that, but in the correct sense because I'm still not sure about it.
What is the exposure triangle? How do the "sides" affect my photographs?
Is there a way to set up a default folder for Windows Explorer? I often use Win + E to pop up Windows Explorer, and then I usually have to navigate down into a few network folders to find my primary directory. Is there a way to set this folder as the default so that Windows Explorer always opens to this folder? Residue integral: $\int_{- \infty}^{+ \infty} \frac{e^{ax}}{1+e^x} dx$ with $0 \lt a \lt 1$. I'm self studying complex analysis. I've encountered the following integral:  $$\int_{- \infty}^{+ \infty} \frac{e^{ax}}{1+e^x} dx \text{ with } a \in \mathbb{R},\  0 \lt a \lt 1. $$  I've done the substitution $e^x = y$. What kind of contour can I use in this case ? In the 3rd and 4th editions of D&amp;D, there were explicit rules for determining if a character knew anything about a monster before them.  In 3rd edition, for example, use of the  skill with a general DC of 10 + the monster's HD allowed for determining one fact, plus one fact per 5 points over the check.  Looking at the Intelligence section of the 5e PHB, I don't see any similar notation.  Does 5e provide any guidance as to when a player could use their out-of-game knowledge about a monster, or when the player might be told things their character would probably know in-game? I was able to read and write NTFS filesystems normally in Mountain Lion, but after the upgrade this stopped to work.  I installed ntfs-3g and fuse4x using homebrew in Mountain Lion and followed the instructions displayed by homebrew to finish the installation of fuse4x kernel extension.  To try to solve this problem I removed ntfs-3g, fuse4x and fuse4x-kext and reinstalled them. But this didn't solve my problem.  Does anyone know how to solve this? Thanks. Proof of $\frac{(n-1)S^2}{\sigma^2} \sim \chi^2_{n-1}$ It's a standard result that given $X_1,\cdots ,X_n $ random sample from $N(\mu,\sigma^2)$, the random variable $$\frac{(n-1)S^2}{\sigma^2}$$ has a chi-square distribution with $(n-1)$ degrees of freedom, where $$S^2=\frac{1}{n-1}\sum^{n}_{i=1}(X_i-\bar{X})^2.$$ I would like help in proving the above result. Thanks. Strip whitespace from jsp output Is there a proof from the first principle that for the Lagrangian $L$,    $$L = T\text{(kinetic energy)} - V\text{(potential energy)}$$  in classical mechanics? Assume that Cartesian coordinates are used. Among the combinations, $L = T - nV$, only $n=1$ works. Is there a fundamental reason for it?  On the other hand, the variational principle used in deriving the equations of motion, Euler-Lagrange equation, is general enough (can be used to to find the optimum of any parametrized integral) and does not specify the form of Lagrangian. I appreciate for anyone who gives the  answer, and if possible, the primary source (who published the answer first in the literature).      Notes added on Sept 22: - Both answers are correct as far as I can find. Both answerers were not sure about what I meant by the term I used: 'first principle'. I like to elaborate what I was thinking, not meant to be condescending or anything near to that. Please have a little understanding if the words I use are not well-thought of. - We do science by collecting facts, forming empirical laws, building a theory which generalizes the laws, then we go back to the lab and find if the generalization part can stand up to the verification. Newton's laws are close to the end of empirical laws, meaning that they are easily verified in the lab. These laws are not limited to gravity, but are used mostly under the condition of gravity. When we generalize and express them in Lagrangian or Hamiltonian, they can be used where Newton's laws cannot, for example, on electromagnetism, or any other forces unknown to us. Lagrangian or Hamiltonian and the derived equations of motion are generalizations and more on the theory side, relatively speaking; at least those are a little more theoretical than Newton's laws. We still go to lab to verify these generalizations, but it's somewhat harder to do so, like we have to use Large Hadron Collider. - But here is a new problem, as @Jerry Schirmer pointed out in his comment and I agreed. Lagrangian is great tool if we know its expression. If we don't, then we are at lost. Lagrangian is almost as useless as Newton's laws for a new mysterious force. It's almost as useless but not quite, because we can try and error. We have much better luck to try and error on Lagrangian than on equations of motion. - Oh, variational principle is a 'first principle' in my mind and is used to derive Euler-Lagrange equation. But variational principle does not give a clue about the explicit expression of Lagrangian. This is the point I'm driving at. This is why I'm looking for help, say, in Physics SE. If someone knew the reason why n=1 in L=T-nV, then we could use this reasoning to find out about a mysterious force. It looks like that someone is in the future. What are the various physical mechanisms for energy transfer to the photon during blackbody emission? Finding a 2x2 Matrix raised to the power of 1000 Randomly place and scale objects? Maximize $\mathrm{tr}(Q^TCQ)$ subject to $Q^TQ=I$ Let $C \in \mathbb{R}^{d \times d}$ be symmetric, and   $$Q = \begin{bmatrix}       \vert &amp; \vert &amp;    &amp; \vert \\       q_1   &amp; q_2   &amp; \dots &amp; q_K  \\       \vert &amp; \vert &amp;    &amp; \vert       \end{bmatrix} \in \mathbb{R}^{d\times K}$$  where $d \geq K$. Using Lagrange multipliers,   $$\begin{array}{ll} \text{maximize} &amp; \mbox{tr} \left( Q^T C Q \right)\\ \text{subject to} &amp; Q^T Q = I\end{array}$$    I am unfamiliar with these kind of constraints with this method, and after reading  I believe the same specific and simple result given is also applicable, and therefore the lagrangian would be:  $$\mathcal{L}(Q,\lambda)=\mathrm{tr}(Q^TCQ)-\left&lt;\lambda,Q^TQ-I\right&gt;$$  where $\lambda\in\mathbb{R}^{K\times K}$, and $\left&lt;\cdot,\cdot\right&gt;$ is the element wise inner product (what kind of makes sense to me since we're actually adding as many constraints as there are elements in these matrices.  In attempting to do that I start taking $\frac{\partial \mathcal{L}}{\partial Q}=O\in\mathbb{R}^{d\times K}$, and compute that LHS element by element; for the $(l,m)$ one:  \begin{equation} 0=\frac{\partial \mathcal{L}}{\partial Q_{lm}}=(CQ+C^TQ)_{lm}-\underbrace{\frac{\partial}{\partial Q_{lm}}\sum_{i,j}\lambda_{i,j}(Q^TQ-I)_{ij}}_{=\lambda_{lm}\frac{\partial (Q^TQ)_{lm}}{\partial Q_{lm}}}=2(CQ)_{lm}-\lambda_{lm}\frac{\partial (q_l^Tq_m)}{\partial q_m(l)} \tag{1}\end{equation}  where in the last step I've used the definition I made at the beginning for $Q$, and $q_m(l)$ denotes the $l$-th component of the column vector $q_m$.  In trying to compute the very last term:  $$\frac{\partial (q_l^Tq_m)}{\partial q_m(l)}=\frac{\partial \left[q_l(1)q_m(1)+ \ldots + q_l(d)q_m(d)\right]}{\partial q_m(l)}= \begin{cases}     q_l(l)\equiv Q_{ll}    &amp; \text{if } l\neq m\\     2q_l(l)\equiv 2Q_{ll}  &amp; \text{if} l=m \end{cases}$$  The whole equality (1) then can be written:  $$0=2(CQ)_{lm}-\lambda_{lm}Q_{ll}(1+\delta_{lm})$$  where $\delta_{lm}$ is the Kronecker delta.  The equation for the other stationary point of the lagrangian, $\frac{\partial \mathcal{L}}{\partial \lambda}=O\in\mathbb{R}^{K\times K}$, for the $(l,m)$ element as well:  $$ 0=\frac{\partial \mathcal L}{\partial \lambda_{lm}}= \frac{\partial }{\partial \lambda_{lm}}\sum_{i,j}\lambda_{i,j}(Q^TQ-I)_{ij}=(Q^TQ-I)_{lm}\tag{2}$$  what obviously leads to $(Q^TQ)_{lm}=\delta_{lm}$.  All this should tell that the columns of $Q$ are eventually the $K$ first eigenvectors of $C$, but I don't know how to continue from here to prove that, supposing I didn't make a mistake. Please I would sincerely appreciate any help.    Edit:  I have rewritten the inner product as a trace of a product of matrices (after seeing ):  $$\left&lt;\lambda,Q^TQ-I\right&gt;=\sum_{i,j}\lambda_{i,j}(Q^TQ-I)_{ij}=\mathrm{tr}(\lambda^TQ^TQ) $$  and have thus managed to do the derivative without losing the matrix format (using formulas from the ):  \begin{align} O=&amp;\frac{\partial \mathcal{L}}{\partial Q}=\frac{\partial}{\partial Q}\mathrm{tr}(Q^TCQ)-\frac{\partial}{\partial Q}\underbrace{\mathrm{tr}(\lambda^T(Q^TQ-I))}_{\mathrm{tr}(\lambda^TQ^TQ)-\mathrm{tr}(\lambda^T)}\\=&amp;(CQ+C^TQ)-(Q(\lambda^T)^T+Q\lambda^T)=2CQ+Q(\lambda+\lambda^T) \end{align}  And this leads to:  $$CQ=Q\underbrace{\left(-\frac{\lambda+\lambda^T}{2}\right)}_{:=\widetilde{\lambda}};\quad CQ=Q$$  If the defined matrix $\widetilde{\lambda}=Q^TCQ$ were diagonal we would already have the result. How does "curved space" explain gravitational attraction? They say that gravity is technically not a real force and that it's caused by objects traveling a straight path through curved space, and that space becomes curved by mass, giving the illusion of a force of gravity.  That makes perfect sense for planetary orbits, but a lot less sense for the expression of gravity that we are the most familiar with in our day-to-day lives: "what goes up must come down."  Imagine that I hold a ball in my hand, several feet off the ground, with my fingers curled around it, and my hand is above the ball.  Then I open my fingers, releasing my grip on it, being very careful to not impart any momentum to the ball from my hand as I do so.  An object at rest remains at rest unless acted upon by an outside force.  If the ball is not moving (relative to my inertial reference frame), it has no path to travel that's any different from the Earth's path through space.  It should remain at rest, hanging there in the air.  And yet it falls, demonstrating that an outside force (gravity) did indeed act upon it.  How does curved space explain this? Why would Louise telling the Chinese commander about his wife's last words change his mind? In the future in Arrival, the commander says that he changed his mind about going to war with the aliens because Louise called his private number and told him his wife's dying words. My question is what did Louise say to him (like in English) and why did it change his mind? What is the best way to globally change the amount of vertical space between items in list environments? How to describe "object" arguments in jsdoc? How to synchronize systemd stop order from one service? Suppose I have a.service which I cannot modify, and b.service which I can modify, I want to see this stop order:  Stopping b.service ... Stopped b.service Stopping a.service ... Stopped a.service   instead of this:  Stopping b.service ... Stopping a.service ... Stopped b.service Stopped a.service   Assume that b.service may take at least 20 seconds to finish. If I'm logged in to a system via SSH, is there a way to copy a file back to my local system without firing up another terminal or screen session and doing scp or something similar or without doing SSH from the remote system back to the local system? HP laptop F1-12 keys inverted with Fn I recently bought an HP laptop. I'm pleased with it, save for one thing.  All the F1-12 keys are inverted. I mean the following:  All the F-keys have a symbol on it. Normally you would press Fn + your to use it. On this laptop, it's reversed. The symbol is what you get when you press the button normally and you have to press Fn + F-key to get the usual action.  So, if I want to refresh a webpage, instead of just F5, I have to do Fn + F5  I want to change this to regular behavior. Anybody know how I can do this?  Running Win7 Home premium, 64-bit. The equivalence of open-set definition of continuous mappings to the $\epsilon-\delta$ definition A function $f: M\rightarrow N$ is defined to be continuous iff $\forall$ open set $U\subseteq N$, the preimage of $U$ is also open.  I was trying to prove that such a definition would be equivalent to the $\epsilon -\delta$ definition in a metric space, and I could only do so if I assume $f$ is surjective, or otherwise a counterexample would be $g: \mathbb{R}\rightarrow\mathbb{R}$ where $g(x)=x^2$. As if I take $U=(-1,1)$, then its preimage is $[0,1)$, which is neither open or closed. In fact, any $U$ containing elements that cannot be achieved by $g$ would face the same problem.  So I was wondering that have I remembered the definition of preimage wrong, or do I need to make sure that $U$ must be chosen from the image set of $f$? Should you pick a baby up everytime it cries? My husband and i have differing opinions on this. I don't believe in the term 'spoilt'. I would be very happy if my daughter was 'spoilt' in the sense that she expects to be treated with the highest possible respect and love and affection in all her relationships.   Also it would not bother me if she think she is special and unique (that's exactly what I want her to think). Mother's face is a mirror for the baby - through the mother's reactions and attention, the child learns that she is a lovable and valuable human being.   I spend all my time with my baby when she is awake. I play games with her, I smile, and laugh, and I never ever let her cry. I rush to pick her up immediately, even if it's the middle of the night, I force myself to get up, no matter how tired I am (I admit it's hard sometimes, and I do let her cry for 10-20 seconds sometimes, but never more).