What is the easiest explanation for how the symmetry of laws under translation in time relates to energy conservation in physics?
I'd like to understand  and its contents as to what it implies in a bit simpler terms. I am familiar with mathematics unto Calculus 1,2,3 and some linear algebra and group theory. I am familiar with that for each symmetry there is a conservation law.    Now is there a one-one correspondence between these two always? What about laws like conservation of charge?  And conservation of energy?  And conservation of mass?  Considering our view of nature was changed as Special Relativity was introduced and thus, what earlier the theorem would have asserted as mass and energy to be originating from different symmetries then how did we reconcile these two as just one symmetry?
Why doesn't `sudo cd /var/named` work? Word for phrase Good for you, bad for us There is a phrase I have heard quite a bit recently dealing with people that are leaving the department for promotions, etc. When a person leaves, it is good for them, but bad for us.  It is good for them because they will be pursuing a new opportunity, but not so good for the department since they will have to be replaced, and others will have to pick up the slack until someone else can be hired and get up to speed.  Is there a word that means that a deal is good for one party, but not necessarily the other? I am looking for a device that can track my location (lat,long) and speed all day long. Perhaps, I need GPS and accelerometer data bundled together.  What I want to do is to log my daily activities. I spend time walking, running, driving, taking a bus, staying in a building, etc., just like any normal human-being :) I am willing to carry the device with me at all times and would like to see every place I go into and my speed at certain positions.  My requirements are that the location and speed data should be recorded at least every ~20 seconds. I would also like the device to timestamp (date+time) each record. It would be enough if timestamp is precise to nearest minute and the location can go off about a maximum of 10-15m. (30-50ft.)  What I want is a portable device so I can carry all day long for about 2-3 months. I will only use it during casual, daily activities.  I don't want real-time tracking. I can collect the data from a storage later on. which app triggers the "Windows features" dialog? I keep getting this window titles "Windows Features" It then says that:     An app on your PC needs the following Windows feature:      .NET Framework 3.5 (Includes .NET 2.0 and 3.0)   and I can choose to install or skip (being warned that my apps might not work correctly).  There is also a "telll me more about this feature" link but it explains about the .NET framework.  I want to know if there is a way for me to know which application is it that asks for this feature as the window just keeps popping out of the context of activating a new app...  This all started just a few days ago after some unattended update on my windows 10. I've been working with Corel Draw previously and now i use Illustrator more.Still getting used to some of their features though. I want to insert an image into a circle I've drawn but i don't know how to go about it.It is called power clipping in Corel. Any answers would be appreciated. TikZ: Having circle borders only grown to the outside What is the !! (not not) operator in JavaScript? What is a "locked" post? A post can be locked.   What does it mean to lock a post? Who can lock a post? When should a post be locked? Evaluating $\int^\pi_{-\pi} \cos^3(x) \cos(ax)~dx$ I need to evaluate $I = \int^\pi_{-\pi} \cos^3(x) \cos(ax)~dx$ where $a$ is some integer.   I get: $\dfrac{2a(a^2-7)\sin(\pi a)}{a^4 - 10a^2 + 9}$. However $\sin(\pi a)$ is $0$ for all $a$ so $I=0$. But as noted by an answer, there are answers for $a = 1,3,-1,-3$. How do you pass a function as a parameter in C? Inline image alignment issue for app and mobile web What is an example of Gâteaux differentiable but not Fréchet differentiable at a point in a finite-dimensional space? Let $V,W$ be nonzero normed spaces over $\mathbb{K}$ such that $V$ is finite-dimensional.  Let $E$ open in $\mathbb{K}$ and $p\in E$.  Let $f:E\rightarrow W$ be Gâteaux-differentiable at $p$.  Is $f$ necessarily Fréchet-differentiable at $p$ in this case?  I think this is not true in general, but cannot find a counterexample. What would be a counterexample? I'm taking a complex analysis course and doing some practice computing residues &amp; evaluating integrals. I pulled out an old book called     On page 196-197, there are some interesting integrals to evaluate. I'm at 5.4.3.10.: I was able to do question 1, but was stumped at how to even begin with question 2:     Evaluate the integral   $$\int_0^{2 \pi} e^{\cos x} \cos (nx - \sin x) \,dx ,$$   where $n$ is an natural number.   The answer is simply $\frac{2 \pi}{n}$. Any hints? Prove without using graphing calculators that $f: \mathbb R\to \mathbb R,\,f(x)=x+\sin x$ is both one-to-one, onto (bijective) function. Prove that the function $f:\mathbb R\to \mathbb R$ defined by $f(x)=x+\sin x$ for $x\in \mathbb R$ is a bijective function.    The codomain of the $f(x)=x+\sin x$ is $\mathbb R$ and the range is also $\mathbb R$. So this function is an onto function. But I am confused in proving this function is one-to-one. I know about its graph and I know that if a function passes the  horizontal line test (i.e horizontal lines should not cut the function at more than one point), then it is a one-to-one function. The graph of this function looks like the graph of $y=x$ with sinusoids going along the $y=x$ line. If I use  a graphing calculator at hand, then I can tell that it is a one-to-one function and $f(x)=\frac{x}{2}+\sin x$ or $\frac{x}{3}+\sin x$ functions are not, but in the examination I need to prove this function is one-to-one theoritically, without graphing calculators. I tried the method which we generally use to prove a function is one-to-one but no success. Let $f(x_1)=f(x_2)$ and we have to prove that $x_1=x_2$ in order fot the function  to be one-to-one. Let $x_1+\sin x_1=x_2+\sin x_2$ But I am stuck here and could not proceed further. Mean Squared Error (MSE) of parameter $p$ in binomial distribution: intuition Use Yaml for MongoMapper Config It seems that containing lava in solid glass, stone and iron blocks may not be sufficient to prevent wool from burning, even if the lava is completely encased.         After some delay, it seems that fire will jump onto the wool above the iron.   Are solid blocks more effective than airspace, or not? If so, Does it matter whether the corner block is filled in? What are the exact rules on distance that I need, and how can I prevent flammable blocks from being eliminated by nearby lava? Bonus: Does a flowing lava block behave differently than a lava source block when it comes to fire creation? I studied propositional logic, and everyday I see applications of what I learned on the internet, in mathematical books and miscelaneous resources.        One particular case is sentences in the form $a \rightarrow b$ , that is, material implications. While I fully understand the mathematical logic approach to the study of those kinds of formulas, sometimes I get confused about what someone ( using implication sentences in the ordinary language ) is trying to say.        In ordinary language, I often see affirmations in the form of an implication.  One example would be someone affirming that "the moon is made of cheese" $\rightarrow 2+2=4$. [ Kleene, Mathematical Logic ].      I perfectly understand that if we choose the interpretation that gives the antecedent as false ( and the consequent as true ) we get that the whole implication is also true, but I would like to understand EXACTLY what he is trying to say when he AFFIRMS a sentence in the form of an implication. Is he saying that under the commonly agreed interpretation of that sentence ( antecedent = false, consequent = true ), that implication becomes true?      Another example, totally different, would be someone affirming that $x \in \mathbb{N} \rightarrow x \in \mathbb{N}$ or $x \in \mathbb{Z}_{&lt;0}$. Is he now affirming the same kind of sentence or saying something totally different?      When he affirms that sentence, is he saying that he finds the sentence to be a tautology?   Why didn't he say that?          Then, with these two examples we have an affirmation of the same kind of sentence (implication) meaning completely different things, in one case he affirms to say that under the commonly agreed interpretation of the sentence, it is true. In another case, he affirms to say that there's no interpretation that makes the sentence false.       Am I missing something? There is really that ambiguity in ordinary language, when people are affirming sentences?  Can't I just take some general instance of the affirmation of a sentence in the form $a \rightarrow b$ in ordinary language and know exactly what he's trying to say, regardless of the propositions he chooses the substitute $a$ and $b$ for?       Thanks a bunch. Prove that $$\nabla_{\mathrm X} \mbox{tr} (\mathrm A \mathrm X^{-1} \mathrm B) = - \mathrm X^{-\top} \mathrm A^\top \mathrm B^\top \mathrm X^{-\top}$$   My proof is below. I am interested in other proofs.    My proof  Let   $$f (\mathrm X) := \mbox{tr} (\mathrm A \mathrm X^{-1} \mathrm B)$$  Hence,  $$\begin{array}{rl} f (\mathrm X + h \mathrm V) &amp;= \mbox{tr} (\mathrm A (\mathrm X + h \mathrm V)^{-1} \mathrm B)\\ &amp;= \mbox{tr} (\mathrm A (\mathrm X ( \mathrm I + h \mathrm X^{-1} \mathrm V))^{-1} \mathrm B)\\ &amp;= \mbox{tr} (\mathrm A ( \mathrm I + h \mathrm X^{-1} \mathrm V)^{-1} \mathrm X^{-1} \mathrm B)\\ &amp;= \mbox{tr} (\mathrm A ( \mathrm I - h \mathrm X^{-1} \mathrm V + O (h^2)) \mathrm X^{-1} \mathrm B)\\ &amp;= \mbox{tr} (\mathrm A \mathrm X^{-1} \mathrm B) - h \, \mbox{tr} ( \mathrm A \mathrm X^{-1} \mathrm V \mathrm X^{-1} \mathrm B) + O (h^2)\\ &amp;= f (\mathrm X) - h \, \mbox{tr} ( \mathrm X^{-1} \mathrm B \mathrm A \mathrm X^{-1} \mathrm V ) + O (h^2)\end{array}$$  Thus, the directional derivative of $f$ in the direction of $\mathrm V$ at $\mathrm X$ is  $$D_{\mathrm V} f (\mathrm X) = - \mbox{tr} ( \mathrm X^{-1} \mathrm B \mathrm A \mathrm X^{-1} \mathrm V ) = - \mbox{tr} ( (\mathrm X^{-\top} \mathrm A^\top \mathrm B^\top \mathrm X^{-\top})^\top \mathrm V ) = - \langle \mathrm X^{-\top} \mathrm A^\top \mathrm B^\top \mathrm X^{-\top}, \mathrm V \rangle$$  and, lastly,  $$\nabla_{\mathrm X} \mbox{tr} (\mathrm A \mathrm X^{-1} \mathrm B) = - \mathrm X^{-\top} \mathrm A^\top \mathrm B^\top \mathrm X^{-\top}$$ Images can be pushed outside the boundaries of a post by using nested lists A   B   C   D   E   F             I don't even know. It just seemed weird.  Yes, I understand I could just avoid doing something like that. But the problem already exists with a single level.   Consider having 2 images in a post. Both exceed the maximum width of a post. One is nested in a list. Key to this issue is that my original images were much larger. They were automatically resized.  Image    List      I think it looks weird.