I can find the Noether current for space time translation symmetry by demanding that the first order correction to the Lagrangian vanishes upon infinitesimal translations of coordinates. But in cases like Maxwell Lagrangian I get a non symmetric energy momentum tensor when I do this. I have seen places that mention that a symmetric energy-momentum tensor can be obtained if one takes the derivative wrt the metric. But if I am given a Lagrangian without the metric explicitly written in, how do I rewrite the Lagrangian with the correct metric dependence?  For example say I am thinking of the the Free boson Lagrangian or the Maxwell Lagrangian? Is there a unique way to write the Lagrangian with the space time metric? I presume there has to be metric written in such a way that the action is an integral of a d+1 form. Is that correct?
What is the correspondence between the conserved canonical energy-momentum tensor, which is $$ T^{\mu\nu}_{can} := \sum_{i=1}^N\frac{\delta\mathcal{L}_{Matter}}{\delta(\partial_\mu f_i)}\partial^\nu f_i - \eta^{\mu\nu}\mathcal{L}$$ (the four conserved Noether currents corresponding to four possible spacetime translations)  where $\{f_i\}_{i=1}^N$ are the $N$ matter fields in the theory, and we assume $f_i\mapsto\alpha^\nu\partial_\nu f_i$ for translations,  and stress-energy tensor from the Einstein-Hilbert action, which is: $$ T_{\mu\nu}\equiv-\frac{2}{\sqrt{-g}}\frac{\delta\mathcal{L}_{Matter}}{\delta g^{\mu\nu}} $$   In particular, how do you get that the two are equal (are they?) for Minkowski space, for which there is no variation in the metric?
Does opacity:0 have exactly the same effect as visibility:hidden Cross-thread operation not valid: Control accessed from a thread other than the thread it was created on I have a scenario. (Windows Forms, C#, .NET)   There is a main form which hosts some user control. The user control does some heavy data operation, such that if I directly call the UserControl_Load method the UI become nonresponsive for the duration for load method execution. To overcome this I load data on different thread (trying to change existing code as little as I can)   I used a background worker thread which will be loading the data and when done will notify the application that it has done its work.   Now came a real problem. All the UI (main form and its child usercontrols) was created on the primary main thread. In the LOAD method of the usercontrol I'm fetching data based on the values of some control (like textbox) on userControl.    The pseudocode would look like this:  CODE 1  UserContrl1_LoadDataMethod() {     if (textbox1.text == "MyName") // This gives exception     {         //Load data corresponding to "MyName".         //Populate a globale variable List&lt;string&gt; which will be binded to grid at some later stage.     } }   The Exception it gave was     Cross-thread operation not valid: Control accessed from a thread other than the thread it was created on.   To know more about this I did some googling and a suggestion came up like using the following code  CODE 2  UserContrl1_LoadDataMethod() {     if (InvokeRequired) // Line #1     {         this.Invoke(new MethodInvoker(UserContrl1_LoadDataMethod));         return;     }      if (textbox1.text == "MyName") // Now it wont give an exception     {     //Load data correspondin to "MyName"         //Populate a globale variable List&lt;string&gt; which will be binded to grid at some later stage     } }   BUT BUT BUT... it seems I'm back to square one. The Application again  become nonresponsive. It seems to be due to the execution of line #1 if condition. The loading task is again done by the parent thread and not the third that I spawned.  I don't know whether I perceived this right or wrong. I'm new to threading.  How do I resolve this and also what is the effect of execution of Line#1 if block?  The situation is this: I want to load data into a global variable based on the value of a control. I don't want to change the value of a control from the child thread. I'm not going to do it ever from a child thread.   So only accessing the value so that the corresponding data can be fetched from the database. The pdf of $\frac{X_1-\bar{X}}{S}$ Suppose $X_1, X_2,...,X_n$ be i.i.d from $N(\mu,\sigma^2)$ with unknown $\mu \in \mathcal R$ and $\sigma^2&gt;0$  Let $Z=\frac{X_1-\bar{X}}{S},$ S is the standard deviation here.  It is can be shown that  $Z$ has the Lebesgue p.d.f.  $$f(z)=\frac{\sqrt{n} \Gamma\left(\frac{n-1}{2}\right)}{\sqrt{\pi}(n-1)\Gamma\left(\frac{n-2}{2}\right)}\left[1-\frac{nz^2}{(n-1)^2}\right]^{n/2-2}I_{(0,(n-1)/\sqrt{n})}(|Z|)$$  My question is then how to get this pdf?  The question is from  in example 3.3.4  to find the UMVUE of $P(X_1 \le c)$. I can understand the logic and procedures to find the UMVUE but don't know how to get the pdf.  I think this question also relate to this    Thank you very much for help or point to any related references will be also appropriated. Why does a cordless circular saw have the blade on the left? At the moment considering the purchase of a circular saw for light work. I'm looking at a corded model and a cordless model. But the question I have is not about which one is better.   My question today is why the corded one has the blade on the right of my hand and the cordless one has the blade on the left of my hand. See both models below.     Now, I know some manufacturers make left- and right-handed versions of their circular saw. And I'm also aware that 'worm drive circular saws' usually have the blade to the left of my hand. But it seems odd to me. Especially as it seems to apply to most manufacturers too.   Was wondering if somebody has an explanation for this. Evaluation of $ \sum_{k=0}^n \cos k\theta $ I just wanted to evaluate  $$ \sum_{k=0}^n \cos k\theta $$  and I know that it should give  $$  \cos\left(\frac{n\theta}{2}\right)\frac{\sin\left(\frac{(n+1)\theta}{2}\right)}{\sin(\theta / 2)}   $$  I tried to start by writing the sum as  $$ 1 + \cos\theta + \cos 2\theta + \cdots  + \cos n\theta $$  and expand each cosine by its series representation. But this soon looked not very helpful so I need some clue about how this partial sum is calculated more efficiently ... Ubuntu 17.10 and later: can't change the input switching shortcut to alt+shift When I try to set the default combo for switching to different language layouts in my keyboard from Shift+Super+Space to Alt+Shift I can't.  I used to have this combo since Ubuntu 14 and even before that when I used to have Windows. Now for some stupid reason it does not allow (actually nothing happens its not like I get an error or the combo is used elsewhere). How could I bypass this bug? BM01 2011/12 Question 6 Geometry Problem Let $ABC$ be an acute-angled triangle. The feet of the altitudes from $A,B$ and $C$ are $D, E$ and $F$ respectively. Prove that $DE +DF \le BC$ and determine the triangles for which equality holds. The altitude from A is the line through A which is perpendicular to BC. The foot of this altitude is the point D where it meets BC. The other altitudes are similarly defined.   Thanks in advance for any contributions. In "Seeing like a state" of James C.Scott there is a sentence at the beginning of a paragraph (Acknowledgements xi):     There are a good many scholars whose writings opened up new   perspectives for me or provided outstanding analyses of issues that I   could not have hoped to study so comprehensively on my own.   What worries me is a good many scholars. Is this some type of inversion or ancient English? Is this grammatically right? Why is a here? I thought that appropriate version would be There are many good scholars. Am I wrong? How to make traditional output for derivatives Would a fast interstellar spaceship benefit from an aerodynamic shape? Some (generous) assumptions:   We have a spaceship that can reach a reasonable fraction of light speed. The ship is able to withstand the high energies of matter impacting at that speed.   Given the amount of matter in inter-stellar space, at high speed, would it encounter enough of it and frequently enough that an aerodynamic shape would significantly reduce its drag (and thus save fuel)? Does the force of kinetic friction increase with the relative speed of the objects involved? If not, why not? Does the force of kinetic friction increase with the relative speed of the objects involved?    I have heard and read that the answer is no.  This is counter intuitive, and is a big part of why the "Airplane on a Treadmill" question is so interesting.  What phenomena are at work with kinetic friction, and why does it not increase with relative speed? Why doesn't a bike/bicycle fall if going with a high speed? When choosing cards for a deck, one of the most important things to consider is the number of land you put in. The biggest issue is mana curve, of course. What shortcuts are best for determining the proper amount of land with a given mana curve? Finding Zero Divisors and Units of Rings $\mathbb{Z}_n \times \mathbb{Z}_m$ NSDate Format outputting wrong date R's randomForest can not handle more than 32 levels. What is workaround? R's randomForest package can not handle factor with more than 32 levels. When it is given more than 32 levels, it emits an error message:     Can not handle categorical predictors with more than 32 categories.   But the data I have has several factors. Some of them have 1000+ levels and some of them have 100+. It even has 'state' of united states which is 52.   So, here's my question.   Why is there such limitation? randomForest refuse to run even for the simple case.  &gt; d &lt;- data.frame(x=factor(1:50), y=1:50) &gt; randomForest(y ~ x, data=d)   Error in randomForest.default(m, y, ...) :    Can not handle categorical predictors with more than 32 categories.   If it is simply due to memory limitation, how can scikit learn's randomForeestRegressor run with more than 32 levels? What is the best way to handle this problem? Suppose that I have X1, X2, ..., X50 independent variables and Y is dependent variable. And suppose that X1, X2 and X3 has more than 32 levels. What should I do?  What I'm thinking of is running clustering algorithm for each of X1, X2 and X3 where distance is defined as difference in Y. I'll run three clusterings as there are three problematic variables. And in each clustering, I wish I can find similar levels. And I'll merge them.  How does this sound? Does existence of $\gcd$ implies PID? iPhone App › Add voice recognition? Probability of selecting an even natural number from the set $\Bbb N$. I confirmed on  thread that there are as many as even natural numbers as there are natural numbers.   Question : Suppose I have selected a number $n \in \mathbb N$; what is the probability that $n$ is even?  My Thought : $\text{Probability} = \dfrac{\text{n(E)}}{\text{n(S)}}$ Here $\text{n(S)}$ is the set of all natural numbers i.e. $\mathbb N$, and $\text{n(E)}$ is set of all even natural numbers. Since it is proved that number of elements is the set $\mathbb N$ is exactly the same as the number of elements in the set of natural numbers (it’s very easy to put the set of natural numbers, $\Bbb N=\{0,1,2,3,\dots\}$, into one-to-one correspondence with the set $\text{E}=\{0,2,4,6,\dots\}$ of even natural numbers; the map $\Bbb N\to \text{E}:n\mapsto 2n$ is clearly a bijection.) ; Thus, Probability $= \boxed 1$ I know this is definitely wrong.Probability must be $0.5$. But where am I wrong? Can anyone explain ? Thanks! Which is faster/best? SELECT * or SELECT column1, colum2, column3, etc