Why is $SU(3)$ and not $U(3)$ the correct gauge symmetry? If quarks come in three colours $r$, $g$ and $b$ than (neglecting all other quantum numbers and spacial freedom for now) a state of a quark would be a vector in $\mathbb{C}^3$.  If we are now looking for a global symmetry redefining the colours than I would be left with the unitary group $U(3)$. However the correct gauge group is $SU(3)$.
Why $SU(3)$ and not $U(3)$? Is there a good reason not to pick $U(3)$ as the colour group? Is there any experiment or intrinsic reason that would ruled out $U(3)$ as colour group instead?
I'm looking to do a third-party assessment of the false-positive rate of a video classification algorithm. Since I have a lot of video I'm trying to do a power analysis to figure out exactly how much video I need to look through so that it is representative of all the video data at a given confidence interval.  The algorithm flags video sequences that have at least one cat in it, and I'm looking to evaluate the frequency of false positives on a new unlabeled test set. So I have tagged all the video that my algorithm has identified a cat in and now want to sample the tagged video sections and look through them manually to validate my model since looking through all of it would take too long! Note, I'm not looking to refine the model at this point, just assess it.  My null hypothesis is that the FP rate of the sample of video I watch is equal to the FP rate of all the video.   I think I can use this formula to determine the number of video sequences to view:    Here is my question: am I thinking through this formulation correctly? Since my model has a CV false positive rate of ~0.96, I figure I can use that as a reference. Can I use that for the null hypothesis proportion, p0? Or will that be p, the true proportion?    I've been using this online calculator:   I ask because when setting the parameters I have, I'm getting very small sample sizes, like less than 10 sequences to view. That can't be right. When we canonically quantize the scalar field in QFT, we use a Lorentz invariant integration measure given by $$\widetilde{dk} \equiv \frac{d^3k}{(2\pi)^3 2\omega(\textbf{k})}.$$  How can I show that it is Lorentz invariant? If I run a randomForest model, I can then make predictions based on the model.  Is there a way to get a prediction interval of each of the predictions such that I know how "sure" the model is of its answer.  If this is possible is it simply based on the variability of the dependent variable for the whole model or will it have wider and narrower intervals depending on the particular decision tree that was followed for a particular prediction? Dependency is not satisfiable: libtiff4 when trying to install Lightworks on Ubuntu 14.04 I was trying to install Lightworks 11.5.1 (amd64) on Ubuntu 14.04(amd64) but there's a notice: Dependency is not satisfiable: libtiff4. Is there anyway to solve this dependency issue?  Thanks in advance. A positive integer $n$ is called a balancing number if  $$1 + 2 + \cdots + (n − 1) = (n + 1) + (n + 2) + \cdots + (n + r) \tag{1}$$  for some positive integer $r$.     Problem: Find the general solution (closed-form expression) of equation $(1)$, in other words, find $f_1(m)=n, f_2(m)=r$ where $f_1, f_2$ are solution of $(1)$ for any integer $m \in \mathbb Z$.   I have asked a  with a constrain involving Pell's equation. Proving $\frac{\sin x}{x} =\left(1-\frac{x^2}{\pi^2}\right)\left(1-\frac{x^2}{2^2\pi^2}\right) \left(1-\frac{x^2}{3^2\pi^2}\right)\cdots$ How to prove the following product? $$\frac{\sin(x)}{x}= \left(1+\frac{x}{\pi}\right) \left(1-\frac{x}{\pi}\right) \left(1+\frac{x}{2\pi}\right) \left(1-\frac{x}{2\pi}\right) \left(1+\frac{x}{3\pi}\right) \left(1-\frac{x}{3\pi}\right)\cdots$$ Showing that $\frac{z^n}{1+z^{2n}}$ is real for $|z|=1$. An .mov video that is upside down in Windows appears right side up in OSX No border-spacing before first item and after last item Call Servlet and invoke Java code from JavaScript along with parameters When is a function satisfying the Cauchy-Riemann equations holomorphic? It is, of course, one of the first results in basic complex analysis that a holomorphic function satisfies the Cauchy-Riemann equations when considered as a differentiable two-variable real function. I have always seen the converse as: if $f$ is continuously differentiable as a function from $U \subset \mathbb{R}^2$ to $\mathbb{R}^2$ and satisfies the Cauchy-Riemann equations, then it is holomorphic (see e.g. Stein and Shakarchi, or ). Why is the $C^1$ condition necessary? I don't see where this comes in to the proof below.   Assume that $u(x,y)$ and $v(x,y)$ are continuously differentiable and satisfy the Cauchy-Riemann equations. Let $h=h_1 + h_2i$. Then \begin{equation*} u(x+h_1, y+h_2) - u(x,y) = \frac{\partial u}{\partial x} h_1 + \frac{\partial u}{\partial y}h_2 + o(|h|) \end{equation*} and  \begin{equation*} v(x+h_1, y+h_2) - v(x,y) = \frac{\partial v}{\partial x} h_1 + \frac{\partial v}{\partial y} h_2 + o(|h|). \end{equation*} Multiplying the second equation by $i$ and adding the two together gives  \begin{align*} (u+iv)(z+h)-(u+iv)(z) &amp;= \frac{\partial u}{\partial x} h_1 + i \frac{\partial v}{\partial x} h_1 + \frac{\partial u}{\partial y} h_2 + i \frac{\partial v}{\partial y} h_2 + o(|h|)\\\  &amp;= \left( \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} \right) (h_1+i h_2) + o(|h|). \end{align*}  Now dividing by $h$ gives us the desired result.   Does there exist a differentiable but not $C^1$ function $f: U \rightarrow \mathbb{R}^2$ which satisfies the Cauchy-Riemann equations and does NOT correspond to a complex-differentiable function? Multiple tags search broken when searching as anonymous user? When I input [asdf][lisp] into the search field on , without being logged in, only one tag remains. Is it broken? Use the definition of derivative to prove $\lim_{x\to 0}\ln(x+1)/x =1$ How would I use the definition of derivative to prove  $$\lim_{x\to 0} \frac{\ln(1+x)}{x} = 1$$  I got to $$\frac{\frac{\ln(1+x+h)}{(x+h)} - \frac{\ln(1+x)}{x}}{h}$$ but have no idea where to go from here.   On another site I found someones answer where they stated the following: $$ \lim_{x\to 0} \frac{\ln(1+x) - \ln(1+0)}{x-0}  = [\ln(1+x)]'\rvert_x = 0 $$ but I am unsure why the $x$ in the $x-0$ is removed. Can someone please explain? Number of reflexive relations defined on a set A with n elements What is the difference between an Intel i7 and a Xeon quad core processor? I know the  processors have been around for a long time and are mostly used in servers, but I am curious, why do people not use the Xeons in a high performance desktop?    As far as I know about the best desktop processor out there now is the  line. The i7's and Xeons are both quad-core processors, what is the main difference in these? I just saw that the MacBook Pro's are using the quad core Xeons instead of the i7's. How to get Drupal 7 installed in a sub-folder / sub-directory properly? Currently i'm trying to installing a drupal 7 installation in a subfolder of another drupal public_html folder, but it doesnt seem to work. I am getting a redirect loop error in chrome. I already modified the .htaccess in the parent drupal installation to ignore the sub-folder and i also set the $base_url in the sub-directory drupal installation. Name for the center amplitude of a sine wave I am trying to find the name for the value at the center of a sine wave $$ y(t) = A\sin (kx \pm \omega t+\varphi) + D $$ i.e. the offset $D$ of the wave in the $y$-direction.  calls it Center Amplitude, but further googling this name is difficult and does not provide any further sources. I also found the expression , but I'm not sure that it refers to the same thing. Is there a commonly used name for the $y$-direction-offset of a sine wave? I need to prove that the center of $A_n$ is trivial for $n \geq 4$.  $Z(A_3) = A_3$, since $A_3 = \mathbb{Z}/3\mathbb{Z}$ is commutative.  One idea is two use "counting" technique. First of, all we count cojugacy classes of even permutations in $S_n$. It's the classes correspodning to the types $[\lambda_1, ..., \lambda_r]$ where $n \equiv r \mod 2$  Now, the conjugacy class in $S_n$ correspoding to the type $[\lambda_1, ..., \lambda_r]$ splits into two equal-sized classes in $A_n$ if $\lambda_1, ..., \lambda_r$ are distinct odd numbers. If they aren't then it is preserved in $A_n$.  Then we can use "the counting formula"(  ) of elements in each "even" class of $S_n$ and divide one by $2$ if needed.  Still, I'm not sure if it can be done. Seems to be a lot of work there. Maybe there are other, he easier ways? Or, maybe, it is the way, in this case, I wouild appreciate any advices. If $\sum_{n_0}^{\infty} a_n$ diverges prove that $\sum_{n_0}^{\infty} \frac{a_n}{a_1+a_2+...+a_n} = +\infty $ I've tried to use the limit test: $\lim_{n \rightarrow \infty} \frac{a_n}{b_n} = C$ in the series of $a_n$ and $\frac{a_n}{a_1+a_2+...+a_n}$ but it was inconclusive.  The question does not especify the sign of $a_n$ but proving it for positives would already help.  I think the sign can change as long as the partial sums never get to be zero. Like if the series has only negative numbers the result seems to stil hold. Merge files based on matching of first column I have two files: File1  ARS-BFGL-BAC-10975 0.9303 688423261 1 01/04/2015 0.9983763305 ARS-BFGL-BAC-11025 0.9092 688423261 1 01/04/2015 0.9983763305 ARS-BFGL-BAC-11044 0.9626 688423261 2 01/04/2015 0.9983763305 ARS-BFGL-BAC-11193 0.9544 688423261 1 01/04/2015 0.9983763305 ARS-BFGL-BAC-10975 0.9303 688423263 1 01/04/2015 0.9983763305 ARS-BFGL-BAC-11025 0.9092 688423263 1 01/04/2015 0.9983763305 ARS-BFGL-BAC-11044 0.9626 688423263 2 01/04/2015 0.9983763305 ARS-BFGL-BAC-11193 0.9544 688423263 1 01/04/2015 0.9983763305   File2:  ARS-BFGL-BAC-10975 10 21225382 ARS-BFGL-BAC-11025 10 84516867 ARS-BFGL-BAC-11193 1 29303546   Desired output  ARS-BFGL-BAC-10975 0.9303 688423261 1 01/04/2015 0.9983763305 10 21225382 ARS-BFGL-BAC-11025 0.9092 688423261 1 01/04/2015 0.9983763305 10 84516867 ARS-BFGL-BAC-11193 0.9544 688423261 1 01/04/2015 0.9983763305 1 29303546 ARS-BFGL-BAC-10975 0.9303 688423263 1 01/04/2015 0.9983763305 10 21225382 ARS-BFGL-BAC-11025 0.9092 688423263 1 01/04/2015 0.9983763305 10 84516867 ARS-BFGL-BAC-11193 0.9544 688423263 1 01/04/2015 0.9983763305 1 29303546   So file 1 has many more rows than file 2. I only want to keep rows in the output that are in file 2 based on column1.   I have tried join but I cant get it to work right-it will tell me my files are not sorted  join -j 1 -o 1.1,1.2,1.3,1.4,1.5,1.6,2.2,2.3 &lt;(sort -k1 file1) &lt;(sort -k1 file2)   Preferably I would prefer an awk command. File 1 will be very large. I have tried   awk 'FNR==NR{a[$1]=$2 FS $3;next}{ print $0, a[$1]}' file2 file1 &gt; output   Any help would be much appreciated. Thanks  Sorry I cant comment below but just to clarify file not all rows in column1 in file 1 will be in file2.  The awk command  awk 'FNR==NR{a[$1]=$2 FS $3;next} $1 in a {print $0, a[$1]}'    will only keep the amount of rows there is in file 2. But ideally what I want is were for expample ARS-BFGL-10975 is repeated twice (realistically way more) to appear twice in my output.  Thanks for the help so far