Reapply for UK PLAB visa after refusal
How soon can I re-apply for a UK visa after getting a refusal?
How to execute a script just by double clicking like .EXE files in Windows? How can I make a bash script executable by double clicking just like .exe files in Windows? I tried , but there are two consequences:   the terminal twinkles, disappears, and nothing is done. you must specify to run in terminal in order to work.   I have a script that installs tomcat on an offline PC, including all dependencies of tomcat in the script. I need to make the script work on double clicking like windows since most who use the script will not be familiar with Ubuntu.  Forget the above explanation. I want to make a script that can be run by double-clicking on it, without using the terminal. Anybody knows how? Compactness of $\operatorname{Spec}(A)$ In an exercise in Atiyah-Macdonald it asks to prove that the prime spectrum $\operatorname{Spec}(A)$ of a commutative ring $A$ as a topological space $X$ (with the Zariski Topology) is compact.   Now because the basic open sets $X_f = \{\mathfrak{p} \in \operatorname{Spec} (A) : \{f\} \not\subseteq \mathfrak{p} \}$ form a basis for the Zariski Topology it suffices to consider the case when   $$X = \bigcup_{i \in I} X_{f_i}$$   where $I$ is some index set. Then taking the complement on both sides we get that   $$\emptyset = \bigcap_{i \in I} X_{f_i}^c$$  so there is no prime ideal $\mathfrak{p}$ of $A$ such that all the $f_i$'s are in $\mathfrak{p}$. Now from here I am able to show that the ideal generated by the $f_i$'s is the whole ring as follows. Since there is no prime ideal $\mathfrak{p}$ such that all the $f_i \in \mathfrak{p}$, it is clear that there is no $\mathfrak{p}$ such that $(f_i) \subseteq \mathfrak{p}$ for all $i \in I.$ Taking a sum over all the $i$ then gives $$\sum_{i \in I} (f_i) = (1).$$  Now here's the problem:          How do I show from here that there is an equation of the form $1 = \sum_{i \in J} f_ig_i,$ where $g_i \in A$ and $J$ some finite subset of $I$?       This part has been giving me a headache. I am not sure if the finiteness bit has to do with algebra, topology or the fact that we are dealing with prime ideals.  This is not a homework problem but rather for self-study.  $\textbf{Edit:}$ I have posted my answer below after the discussion with Dylan and Pierre. I am trying to understand the Georgi chapter of tensor methods in $SU(3)$ representations, and I don't know how to resolve the tensor product of 2 matrices in a 2 heavy quark + 2 light antiquark system. Namely:  Let the tetraquark system be:           $QQ\bar q\bar{q}$  So, those $Q$ heavy quarks interact and can be understood in the $3\bigotimes 3$ representations for $SU(3)$, and $\bar q\bar q$ can be understood as the $\bar 3\bigotimes\bar 3=3\bigoplus\bar 6$ representations for $SU(3)$.  The interaction diagram for the $Q$ heavy quarks is shown in the following scheme, which is an analogy to the Feynman Diagram where a gluon is understood to be exchanged between upper and lower arrows:  $i'\xrightarrow{(T_{a})^{i'}_{i}}i ,  {}^iQ$  $j'\xrightarrow{(T_{a})^{j'}_{j}}j ,  {}^jQ$  So, as usuall we consider the elements of $3$ representation as $u^i$, and therefore the elements of $\bar 3$ representation as $v_{j}$.  I must understand how does the following expression transform:  $(T_{a})^{i'}_{i}(T_{a})^{j'}_{j}\cdot(1/2)\cdot (w^iv^j+w^jv^i)=\Xi\cdot(1/2)\cdot(w^{i'}v^{j'}+w^{j'}v^{i'})$  In other words, how can one give the correct value of $\Xi$?  Note1: $T_a$ is understood to be the $a-ith$ generator of the associated Lie Algebra, which come from the definition of the Gell-Mann Matrices.  $(T_{a})^{i}_{j}\propto(\lambda)_{ij}$, in matrix notation.  Note2: I understand the Young Tableaux and they are not intended to be used here, so please don't try to explain this situation through the tables.   Note3: I assume this is a purely solvable mathematical problem and doesn't need a deep insight into the physical concepts...  Also requesting information about how to understand the tensor products in these cases, since I have been checking many sources but the definitions seem not clear enough. Anyone ever seen a "wok adapter" for induction cooktops? In theory a cheap pressed steel wok would work on an , but the "contact patch" for a traditional round bottomed wok would be so small it would be ineffective.  Has anyone seen a steel or cast-iron "wok adapter" with a flat bottom and a concave "bowl" on top to sit the wok in, serving dual purposes of supporting the wok on the stovetop and transferring the induction heat to the wok?  Could anyone speculate if this would this even work? Or wouldn't you be able to get the wok hot enough? How to rotate the curve but not the axes? Definition of $\mathbb{R}^\infty$ I have a question on my assignment asking me to prove that if $A$ is an $n\times n$ orthogonal matrix, then $\det(A) = \pm1$. What I did so far is:  We know that $A\cdot A^t = I$ (since it is orthogonal), and that would mean that if $A$ is a matrix with elements $a,b,c,d$ then :  \begin{cases}a^2 + b^2 = 1\\ ac + bd = 0\\ c^2 + d^2 = 1\end{cases}  which implies that \begin{align}\det(A)&amp;= ad - bc\\ &amp;= -\frac{bd^2}{c} - cb\\ &amp;= -\frac{bd^2}{c} - \frac{c^2b}{c}\\ &amp;= -\frac{bd^2 + bc^2}{c}\\ &amp;= -\frac{b(d^2 + c^2)}{c}\\ &amp;= -\frac{b(1)}{c}\\ &amp;= -b/c\end{align}  I might be doing something wrong here, could someone please help me out understand this? Massless charged particles Are there any massless (zero invariant mass) particles carrying electric charge?  If not, why not? Do we expect to see any or are they a theoretical impossibility? How to use articles with scientific words, such as mass, force, impulse, field and others? How are articles used with scientific quantities, such as mass, force, impulse, field and others? How to re-download Lion to create a VMWare Fusion 4 virtual machine? I have a MacBook with Lion installed.  I bought VMWare Fusion 4 because I want to create a Lion VM.  As far as I can tell I need to point VMWare to the Lion installer that is downloaded from the App Store, but that file deleted itself after Lion was installed.  How do I re-download the Lion installer (and keep it around) to create a VMWare Fusion 4 virtual machine? How do you list the primary key of a SQL Server table? Why do the Localization of a Ring This question may be a bit vague, but neverthless, i would like to see an answer. Wikipedia tells me that:   In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring.   It is clear that for integral domains, we have the Field of Fractions, and we work on it. It's obvious that a Field has multiplicative inverses. Now if we consider any arbitrary ring $R$, by the definition of localization, it means that we are adding multiplicative inverses to $R$ thereby wanting $R$ to be a division ring or a field. My question, why is it so important to look at this concept of Localization. Or how would the theory look like if we never had the concept of Localization. $R$ is commutative, $I$,$J$ are ideals, $I+J=R$, then $IJ=I\cap J$ If $R$ is a commutative ring and $I$ and $J$ are ideals s.t. $I+J=R$ then show that $IJ=I\cap J$.  I've already shown that $IJ \subset I\cap J$, now I need to show the reverse inclusion.   I'm a bit lost, so far i'm just figuring out what pieces I have to work with.  Such as:  $\forall r\in R$ $\exists i\in I ,j\in J$ s.t. $i+j=r$  $\forall ij\in IJ$, $ij=i_1$ and $ij=j_1$ for some $i_1\in I$, $j_1\in J$.  Also, if I let $x\in I\cap J$, then $x=i_2=j_2=i+j$ for some $i_2\in I$, $j_2\in J$  Anyone, having problem getting to the conclusion here, thanks in advance Spy from Earth studies Galactic Empire nobility and prevents plot to depose Emperor Add folder name to beginning of filename How to perform entity validation on code level? I have an entity that has a field attached to it and the values that user types into the field has to be globally(website) unique(compared to other entities).  I have added validation to the entity form but what if I create this entity programmatically? How can I prevent the entity to be saved into the storage?  I've noticed there is a validate method in ContentEntityBase, though not in ConfigEntityBase. Is this the place for it? If so, I could use some examples. How to output a counter with leading zeros? I can output the value of a counter in a LaTeX document via:  \setcounter{mycounter}{1} \arabic{mycounter}   Which inserts a 1. But how do I format the output by, for example, adding a leading zero if mycounter is less than 10 in a way similar to what  lets you do by specifying %02d as your format string? Root filling up although it should have some space This has been perplexing me for some time. I'm on Ubuntu 11.04. My root partition is filling up and I can't figure out what's causing it. Even stranger, when I run df -h, this is the output:  Filesystem            Size  Used Avail Use% Mounted on /dev/sda2             9.2G  8.8G     0 100% /   As you see, there should be about 400 MB free, but Avail shows 0. What could be causing this? I've even uninstalled some programs to free the space and it fills up again. Suggestions on how to find out which, if any, files are filling up the space are also welcome. Why am I getting no power when I pedal? Yesterday when I was biking my mountain bike on level paved path, I would pedal but get no power to the wheel.  To use a car analogy, it was as if I had the clutch in and was apply gas.  It seemed to happen most often after I coasted and then went back to pedalling.  I would pedal and realize that no power was going to the wheel.  The chain was definitely on the sprockets and moving. I could change gears no problem while I had no power.  Sometimes stopping the bike and starting again would work, or just trying again.  Needless to say, I'm taking it in as I'm not mechanically inclined, but I'm hoping to learn something.  Perhaps a link to a site that explains the problem?  So my questions.   What is going on?  I have heard of something called the "freehub".  I really don't know how bikes work though, so I'm hoping for an explanation of how one can pedal and give power to the wheel and then stop pedalling and the wheel just keeps moving (without the pedals moving).   If you are on a ride, is there any emergency adjustment to make?   For what it's worth, it's an old bike.  Circa 1993 Performance Mtn Bike.  So it's not too surprising that it's falling apart.  I haven't used it a whole lot, and have tried to wash and oil it, but I probably haven't taken the care that a biking aficionado would.  Also, besides losing power after coasting, one time I was pedaling and suddenly the pedals weren't powering the wheel.  I almost went over the handlebars!  So, I definitely want to get it fixed.  I'd just like to know a little bit about what is going on.  Thanks,  Dave.  Bonus question:  If there is enough info to make a diagnosis and you have ideas on costs to fix, perhaps you could comment on whether it's worth fixing.  Like I said, it's circa 1993 and cost about $450 then.  No shocks or anything.  Just 21 speed Mtn. Bike. How to show (or hide) a hidden file I wanted to add classpath to my JDK. While searching a way to do it I noticed everyone open .profile file using terminal. But when I want to open and edit it with a text editor I couldn't see that file in Home folder.  Can someone tell me why is this. And also if possible how can I access .profile in home folder the GUI way.