$(\frac{1}{2},\frac{1}{2})$ and $(\frac{1}{2},0)\bigoplus (0,\frac{1}{2})$ I am confused about the notation.  What's the differences between $(\frac{1}{2},\frac{1}{2})$ and $(\frac{1}{2},0)\bigoplus  (0,\frac{1}{2})$, or maybe $(\frac{1}{2},0)\bigoplus  (\frac{1}{2},0)$ ?  (1) $so(1,3)=su(2) \bigoplus su(2)$ has different representations, I don't understand why  $(\frac{1}{2},\frac{1}{2})$ is irreducible, since we already have direct sum  $\bigoplus$ between two $su(2)$, and direct sum usually means reducible.   The Schwartz book (page 163) says $(\frac{n}{2},\frac{n}{2})$ These are each irreducible representations of the Lorentz algebra, but reducible representations of the $su(2)$ subalgebra corresponding to spin.  (2) The group elements are acting on some space, then $\bigoplus$ means the direct sum of these spaces, like $(\frac{1}{2},0)\bigoplus  (0,\frac{1}{2})$. Then what does $(\frac{1}{2},\frac{1}{2})$ means, a direction product $(\frac{1}{2},0)\bigotimes  (0,\frac{1}{2})$ ?
Vector spaces for the irreducible representations of the Lorentz Group EDIT:  The vector space for the $(\frac{1}{2},0)$ Representation is $\mathbb{C}^2$ as mentioned by Qmechanic in the comments to his answer below! The vector spaces for the other representations remain unanswered.  The definition of a representation is a map (a homomorphism) to the space of linear operators over a vector space. My question is: What are the corresponding vector spaces for the   $(0,0)$ Representation $(\frac{1}{2},0)$ Representation  $(0,\frac{1}{2})$ Representation $(\frac{1}{2},0) \oplus (0,\frac{1}{2}) $ Representation $(\frac{1}{2},\frac{1}{2})$ Representation infinite dimensional Representation?
I can't get the coordinates on my computer because it keeps changing the brightness.  I have a windows 8 Samsung, and I'm playing Minecraft version 1.6.4. Every once in a while after updating MiKTeX, the installation gets into an inconsistent state. This might be a problem with the packages (an example is described in ), but sometimes packages are split or renamed. The cases I remember are when  and when miktex-etex-base-2,9 was renamed to miktex-etex-base-2.9 (if I recall correctly; I am sure it was either that or something very similar). In each case the MiKTeX Update process suggested to remove the "old" packages but it didn't give the user any hints about him having to manually add the respective replacement packages back in the package manager to get one's MiKTeX installation back into a usable state.  What are best practices for keeping MiKTeX up to date? Note that I'm not asking about installing new packages or fonts in general; there are other questions covering this area.   In which order should one execute the Update process and the Synchronization of the repository? , it is recommended to check the MiKTeX homepage before updating. What else should one do? When and in which order (and why) should one execute the "Refresh FNDB" and "Update Formats" processes within the MiKTeX Options menu? What else can happen? Other advice? When do you use POST and when do you use GET? Polynomial passing through two points with specific tangents How can I calculate a Polynomial that passes through the origin and a given point (P) having given tangents at the origin (Ot) and the given point (Op)?  The given point will always be to the right of the origin (Px &gt; 0). The polynomial should be a function of x (i.e. it moves from left to right).  I believe a 3rd degree polynomial will always fit these constraints, but I'm not certain.  I am looking for a function f( x, Px, Py, Ot, Pt ) = ? which plots a polynomial for x, satisfying the four given values.  When x = 0,   f()   = 0 When x = Px,  f()   = Py When x = 0,   f'()  = Ot When x = Px,  f'()  = Pt Remove Indentation for a Single Paragraph Is there a way to remove the indentation at the beginning of a paragraph for a single paragraph.  I know I could do  \setlength{\parindent}{0cm} This is my paragraph.  Blah blah blah. \setlength{\parindent}{default}   Where default is whatever the default indentation is for the document class I am using (part of the problem is that I don't know what this default value is).  I feel as if there should be a better way to accomplish this than by the above method.  I am looking for a solution of the form:  \noindent This is my paragraph.  Blah blah blah.   Is there a way to do this, or will I have to resort to the former method?  If I have to resort to the former method, what is the default indentation value for the amsart document class? Why is it that scanf() needs the l in "%lf" when reading a double, when printf() can use "%f" regardless of whether its argument is a double or a float?  Example code:  double d; scanf("%lf", &amp;d); printf("%f", d); What did the general public know about Darth Vader? When Darth Vader first becomes Palpatine's apprentice, very few know of his existence and connection to the Emperor: the members of the Trade Federation (whom he dispatches) and Yoda, Kenobi, Organa, and their associates are all that seem to know at the end of Episode III.  Ultimately, the Resistance would know about him as they would be fighting him, and Organa could have been able to inform them about his identity and the fact that he is a Force user. The military of the Empire would know of his existence, but even high-ranking members likely wouldn't know his past or identity. However, they would know about the fact that he is a Force user due to his tendency to use his "sorcerer's ways" to choke people.  So, how much would the general public have known about Vader? Would they have known him as just another military leader, or would they have known he was Palpatine's right-hand man? Would they have known about his ability to use the Force, and therefore the possibility that he was a Sith? I've gone into the Nether and seemed to have lost my portal out of it, I also can't die in lava or by mobs attacking me, I'm appear to be invincible and can't die.  This is Single Player by the way.  Why is this happening? Book identification about boy who gets computer zapped into head Generate a log of all the installs on a system (whether apt-get/yum/pip/npm/...) Number of vertices in the Klein bottle triangulation given in "Topology and geometry for physicists" by Nash and Sen I stumbled upon the following triangulation of the Klein Bottle in page 76 of the book Topology and geometry for physicists by Charles Nash and Siddhartha Sen.    Below the figure, it says "the triangulation $K$ consists of eight $0$-simplexes[...]".  I think this is not an eight-vertex triangulation, but rather a nine-vertex triangulation of the Klein bottle. I also know, thanks to the great comments under  question, that an $8$-vertex triangulation of the Klein bottle is a minimal one, hence the one portrayed in the book isn't minimal.  Am I correct? How old is SpongeBob? I don't mean how old is the series, I mean what is SpongeBob's age if he were a real person. I ask because his age seems to be ambiguous in the show.   He has a full-time job He has his own house He is always referred to as a kid (especially in the movie) He acts childish and immature In one episode, he was pretending to be an adult/grown-up Does there exist a continuous function such that f(x) is rational for every irrational x, and f(x) is irrational for every rational x? Does there exist a continuous function such that f(x) is rational for every irrational x, and f(x) is irrational for every rational x? I can think of examples where this would be true, but I'm having a lot of trouble with the continuity. I have a requirement to create triggers dynamically after a managed package is installed. I have read that the tooling API can be used to create triggers dynamically however it cannot be used in production. Is it possible to create triggers from a managed package in an org? I want to be able to create the trigger based on some user inputs. How to change file encoding in NetBeans? Suppose I'm given the mean and one quantile (e.g. the 20% quantile) of a random variable $x$, and I want to find the parameters $\alpha$ and $\beta$ of a Beta distribution that has the same mean and quantile.  Is there an efficient way to do it?    Using just the mean, I know that since $\bar{x} = \frac{\alpha}{\alpha+\beta}$, we have $\beta = \frac{\alpha}{\bar{x}} - \alpha$.  So we only really have one parameter to estimate.  But I'm unsure how to use the quantile information to take the next step.  Maybe there's something I can do with the Incomplete Beta when I know the ratio $\frac{\beta}{\alpha} = \frac{1-\bar{x}}{\bar{x}}$?  I have access to R myself, so I could use a numerical optimizer for this, but ideally I need a method that can be carried out in Excel in someone else's environment.  Excel does have BETA.DIST() and BETA.INV() functions available.  A look-up table would be fine, but a closed-form formula would be better if it's possible. I am little confused with the applicability of reinterpret_cast vs static_cast. From what I have read the general rules are to use static cast when the types can be interpreted at compile time hence the word static. This is the cast the C++ compiler uses internally for implicit casts also.  reinterpret_casts are applicable in two scenarios:   convert integer types to pointer types and vice versa  convert one pointer type to another. The general idea I get is this is unportable and should be avoided.     Where I am a little confused is one usage which I need, I am calling C++ from C and the C code needs to hold on to the C++ object so basically it holds a void*. What cast should be used to convert between the void * and the Class type?  I have seen usage of both static_cast and reinterpret_cast? Though from what I have been reading it appears static is better as the cast can happen at compile time? Though it says to use reinterpret_cast to convert from one pointer type to another? Unable to acheive desired effect with ID mask I'm trying to make the render have an effect where the stars (which are exploded fragments) have a glowing glare-like effect and the other objects (i.e. film reel and cubes) are their natural colors.  However, despite having made the star objects all a pass index of 1 and the rest 0 by default, everything seems to render a green glowing effect that's supposed to be only for the objects with a pass index of 1.  To illustrate:  This is the render of the image naturally (I rendered at very low quality but it's just for demonstration purposes):    And this is with the composite nodes:    So in essence I want the objects in the the first render I showed to keep those colors, but the stars to have the colors of the green glowing effect.  As you can see from the 2nd picture everything has the green effect.  My node setup is pretty big so I will provide a link to download the  so you can see the nodes more clearly. Debt condition abuse? The rules states that you cannot have two of the same conditions.  You can take a debt condition to gain two additional successes on your influence test to gain assets from the reserve. Thus if you already have a debt, does that mean that you automatically gain two additional successes for each gain asset actions? How far can one get in analysis without leaving $\mathbb{Q}$? Suppose you're trying to teach analysis to a stubborn algebraist who refuses to acknowledge the existence of any characteristic $0$ field other than $\mathbb{Q}$.  How ugly are things going to get for him?  The algebraist argues that the real numbers are a silly construction because any real number can be approximated to arbitrarily high precision by the rational numbers - i.e., given any real number $r$ and any $\epsilon&gt;0$, the set $\left\{x\in\mathbb{Q}:\left|x-r\right|&lt;\epsilon\right\}$ is nonempty, thus sating the mad gods of algebra.  As @J.M. and @75064 pointed out to me in chat, we do start having some topology problems, for example that $f(x)=x^2$ and $g(x)=2$ are nonintersecting functions in $\mathbb{Q}$.  They do, however, come arbitrarily close to intersecting, i.e. given any $\epsilon&gt;0$ there exist rational solutions to $\left|2-x^2\right|&lt;\epsilon$.  The algebraist doesn't find this totally unsatisfying.     Where is this guy really going to start running into trouble?  Are there definitions in analysis which simply can't be reasonably formulated without leaving the rational numbers?  Which concepts would be particularly difficult to understand without the rest of the reals?