I'm a mathematics student in my junior year and I'm interested in computational complexity and specially geometric complexity theory. I'm going to learn algebraic geometry and representation theory but I want to consider the parts that are related to geometric complexity theory so I wonder What are the topics that should be mastered by someone who wants to understand geometric computational complexity?  Surely, a lot of algebraic geometry and representation theory are needed, but which topics? and representation theory of what? finite groups? lie algebras(probably not) etc? and which topics in algebraic geometry are needed? It would be great if it is possible to name some topics that are required to be well understood in algebraic geometry and representation theory to before tackling  geometric  complexity theory. naming good resources ( texts etc ) that cover this background will be highly appreciated too.  I have asked the same question on math stackexchange but got no answer, so I thought I should ask it here.
It seems that Geometric Complexity Theory requires much knowledge of pure maths such as algebraic geometry, representation theory.   While I am a CS student and do NOT have classes of very abstract and pure mathematics, I am interested in this program.  Is there a list of "minimum knowledge" for learning this theory ?     This list includes lecture notes of CS or math departments, surveys of any journal or conferences, and text books of pure math.  [EDIT: Added later] Thank you for your comments.  General theory of computing: I read Sipser's book with the title "Introduction to the Theory of Computation"  Complexity Theory :In particular, I am interested in concrete models for lower bounds of complexity. Thus I read the part of the "concrete lower bounds" in Arora-Barak's text book. I also have basic knowlede in the several chapers of the communication complexity book written by Nisan.  Basic Mathematics : I have learnt about proof-based linear algebra such as general definition of the vector space etc and precicely argument of calculas based on epsilon-delta argument.   Algebra: I have learned about definition and examples of group, ring, and field. I had a class for cs students, and have not learned about general thoery of this algebraic systems.
What happens 'internally' on a unstable/overclocked/undervolted CPU? When you undervolt or overlock an CPU, when it passes a certain limit it starts to make the system unstable, reboot, etc. But these are the effects of it.  What I'm curious about is what happens on the CPU. What happens? It gives wrong answers to instructions? Bits on cache flip? Where did the "wheel" group get its name? The wheel group on *nix computers typically refers to the group with some sort of root-like access. I've heard that on some *nixes it's the group of users with the right to run su, but on Linux that seems to be anyone (although you need the root password, naturally). On Linux distributions I've used it seems to be the group that by default has the right to use sudo; there's an entry in sudoers for them:  %wheel ALL=(ALL) ALL     But that's all tangential; my actual question is: Why is this group called wheel? I've heard miscellaneous explanations for it before, but don't know if any of them are correct. Does anyone know the actual history of the term? Can you help me resolve this paradox for a geodesic orbit? Does it maximize proper time to sit still rather than orbit for 1 year? Prove that for all sets A and B, A ∩ B = ∅ implies ( A ∪ B ) - B = A In the next proof we avail ourselves of the next lemma: For all sets A and B, ( A ∪ B ) - B = A. Proof: Let A and B be arbitrary sets and let x ∈ ( A ∪ B ) - B. x ∈ ( A ∪ B ) - B ⇔ x ∈ ( A ∪ B ) ∧ x ∉ B ⇔ ( x ∈ A ∨ x ∈ B ) ∧ x ∉ B ⇔ ( x ∈ A ∧ x ∉ B ) ∨ ( x ∈ B ∨ x ∉ B) For p ∧ ~ p ≡ F and p ∧ q ⇒ p where p and q are prepositions, ( x ∈ A ∧ x ∉ B ) ∨ ( x ∈ B ∨ x ∉ B ) ⇔ x ∈ A ∨ F By modus tollendo ponens, x ∈ A ∨ F ⇔ x ∈ A So then, x ∈ ( A ∪ B ) - B ⇔ x ∈ A And ( A ∪ B ) - B = A Therefore, the statement for all sets A and B, A ∩ B = ∅  implies ( A ∪ B ) - B = A is true trivially. Is this proof right? How can I insert a newline in a framebox? I have several frameboxes in my document, using the following command:  \framebox(115,115){ R-Sq: \newline For example}    I want to force a new line, but neither \newline or \\ appear to work. Is there any other way of doing this? " could " vs. " could have " As I know we can use both structures to talk about past ability , but is there any difference between them?  A. He could climb the mountain  B. He could have climbed the mountain Compare files that are in directory 1 but not directory 2? I intend to use fsck for checking my Linux partition of the main disk, because its file system is suspicious of being not unmounted cleanly by shutting down the system. I have searched in my environment variables: none of them except of PATH should control the fsck program. But the program apparently needs to be used during a boot: :~$ fsck -V fsck from util-linux 2.20.1 Checking all file systems. [/sbin/fsck.ext4 (1) -- /] fsck.ext4 /dev/sda6  e2fsck 1.42 (29-Nov-2011) /dev/sda6 is mounted. WARNING!!!  The filesystem is mounted.   If you continue you WILL cause SEVERE filesystem damage. Do you really want to continue? no check aborted.  (There is no /forcefsck file in my root according to  . My etc/fstab file contains    # /etc/fstab: static file system information. # # Use 'blkid' to print the universally unique identifier for a # device; this may be used with UUID= as a more robust way to name devices # that works even if disks are added and removed. See fstab(5). # #                 proc            /proc           proc    nodev,noexec,nosuid 0       0 # / was on /dev/sda6 during installation UUID=1ac55d8d-c112-4bc7-9e79-921d196f9f79 /               ext4    errors=remount-ro 0       1 # swap was on /dev/sda7 during installation UUID=54f7e314-50e2-419b-a45d-47c3058ecc00 none            swap    sw              0       0 /dev/fd0        /media/floppy0  auto    rw,user,noauto,exec,utf8 0       0   ). According to an answer to  I have tried tune2fs hoping that I'll get the present value of the max_mounts_count parameter to be able to set it to 1 for checking the partition during the next boot. But after sudo rm /var/lib/update-notifier/fsck-at-reboot and a restart I have only got a response  :~$ tune2fs -l /dev/sda6 tune2fs 1.42 (29-Nov-2011) tune2fs: Permission denied while trying to open /dev/sda6 Couldn't find valid filesystem superblock. Can I use sudo tune2fs -l /dev/sda6 safely? Thanks. How is offside determined in football? The offside rule on the pitch is often unclear and called suddenly. How do I know if a player is offside? Evaluate $\lim_{n \to \infty} \int_{0}^1 [x^n + (1-x)^n ]^{1/n} \ \mathrm{d}x$ What is the standard error of the sample standard deviation? I read from  that the standard error of the sample variance is  $$SE_{s^2} = \sqrt{\frac{2 \sigma^4}{N-1}}$$  What is the standard error of the sample standard deviation?  I'd be tempted to guess and say that $SE_{s} = \sqrt{SE_{s^2}}$ but I am not sure. How to show subsubsections and paragraphs in TOC? So we have something like:  \documentclass{article} \setcounter{secnumdepth}{2} \setcounter{tocdepth}{2} \usepackage{amsmath}  \begin{document}  \tableofcontents  \section{Brushless Motor Fundamentals} \subsection{Brushless Motor Operation} \subsubsection{DC Motor Operation} Torque is generated in DC motors from the magnetic force, also known as the Lorentz force, which is produced when an electric current is passed through a coil in a magnetic field. This force is given by $F=q[E+(v\times B)]$, where $F$ is the force perpendicular to the coil, $E$ is the electric field in the coil, $B$ is the magnetic field, and $v$ is the velocity of the charged particles in the coil. From mechanics, torque is $\tau=F\times r$.  \section{Brushless Motor Fundamentals 2} \subsection{Brushless Motor Operation} \subsubsection{DC Motor Operation 2} If the electrical force is ignored and the remaining magnetic force is used, with the assumption that $v$ is perpendicular to $B$, we find that $\tau=qvBrsin\theta$.  \end{document}   How to show subsubsections and paragraphs in TOC? Does anyone have a recommendation for a book to use for the self study of real analysis? Several years ago when I completed about half a semester of Real Analysis I, the instructor used "Introduction to Analysis" by Gaughan.   While it's a good book, I'm not sure it's suited for self study by itself.  I know it's a rigorous subject, but I'd like to try and find something that "dumbs down" the material a bit, then between the two books I might be able to make some headway. Spoilers from the Season 7 finale of Game of Thrones, "The Dragon and the Wolf".  In one of the final scenes, Jaime and Cersei are discussing Jaime's future with them, at which point       Cersei nods her head, signaling the kill order for 'The Mountain', yet 'The Mountain' pulled out his sword but why didn't he kill Jaime? User suggested great corrective edit to my answer, but the edit was rejected. What can I do to approve it? Cantor's diagonal argument can be used to show that a set $S$ is always smaller than its power set $\wp(S)$.  The proof works by showing that no function $f : S \rightarrow \wp(S)$ can be surjective by constructing the explicit set $D = \{ x \in S | x \notin f(s) \}$ from a function $f$ and showing that no element of $S$ maps to $D$.  This proof works because all bijections are surjections, so if no surjection from $S$ to $\wp(S)$ exists, then there cannot be a bijection between $S$ and $\wp(S)$.  My question is whether it is possible to run a "reverse diagonalization" that works by instead showing that there is no injection from $\wp(S)$ to $S$.  I am curious about this because I have never seen Cantor's theorem proved this way (or, more generally, any diagonal argument structured like this).  Is it possible to take Cantor's diagonal argument and "reverse" it to show that there cannot be an injection from $\wp(S)$ to $S$, rather than showing that there can be no surjection from $S$ to $\wp(S)$?  Thanks! Occasionally, the Unicode character sequence U+200C U+200B (ZWNJ ZWSP) is inserted into comments In     ... the number "5000" does not paste as such but instead as  5 0 U+200C U+200B 0 0   The HTML source looks like this:  perhaps the easiest would be &lt;code&gt;\edef\tmp{\everypar{\widowpenalty\the\widowpenalty\relax}}\tmp\widowpenalty=-50&amp;zwnj;&amp;#8203;00&lt;/code&gt; that may reset   Or (smaller snippet):  \widowpenalty=-50&amp;zwnj;&amp;#8203;00&lt;/code&gt; that   So in the HTML, we have extraneous &amp;zwnj; and &amp;#8203; in between the two middle digits of 5000.  Upon pasting and trying out this code, this first led to a mysterious compiler error. The same issue happened within a (now deleted) comment thread below ; there, these or similar two characters were inserted after the letter "r" within the string \ref ("\ref").  Note that not all browsers wrap the same way. The above screenshot was for Firefox 19.0.2, in which it breaks between \relax}} and \tmp, with 5000 insidiously left intact on the second line but with the two special characters remaining in the middle. So that is why this took me completely by surprise. Other browsers might indeed only break on the invisible Unicode characters, such as Chrome 25:    Or on a smaller screen with the mobile site, multiple line breaks might show, possibly still leaving the Unicode characters unused: Money from my parents from abroad to my bank account in UK I am from Japan. I have moved to UK recently on a spouse visa.   I still have my Japanese bank account which I have been using even before I came to UK. I have saving in that account and also my parents transfer me some money per month in my bank account in Japan. I have an International Debit Card on this account.  I am withdrawing this Money in UK via ATM / Debit Card.  In this case, do I have to pay taxes in UK or declare this as income in some form. Pretty sure that my edit was correct but it was rejected? Normality test with large data set I think I am dealing with this issue   I have large data set(10k data points) that slightly diverges from normal, and I get p-value of 0. I am interested in having perhaps a more crude test that tells me if data is extremely divergent from normal, versus looking somewhat normally distributed. I am currently trying Kolmogorov-Smirnoff, and in both cases I just get p-value of 0. Any alternatives?  I also looked at this:   So, taking all this into consideration, is there any kind of test I can perform that distinguishes between data that's roughly normal, and not normally distributed at all?  I am using scipy.