How do I boot into console mode?
How do I disable X at boot time so that the system boots in text mode?
Effective Refractive Index in Paper I am reading the paper, entitled "Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings" by Eric B. Grann, M. G. Moharam, and Drew A. Pommet.   In the paper they mention this formula:        Do you know how it can be derived and where it comes from ? When does a Square Matrix have an LU Decomposition? How to get next/previous record in MySQL? I have come across a question about determining possible minimal polynomials for a rank one linear operator and I am wondering if I am using the correct proof method.  I think that the facts needed to solve this problem come from the section on Nilpotent operators from Hoffman and Kunze's "Linear Algebra".     Question: Let $V$ be a vector space of dimension $n$ over the field $F$ and consider a linear operator $T$ on $V$ such that $\mathrm{rank}(T) = 1$.  List all possible minimal polynomials for $T$.   Sketch of Proof:   If $T$ is nilpotent then the minimal polynomial of $T$ is $x^k$ for some $k\leq n$.  So suppose $T$ is not nilpotent, then we can argue that $T$ is diagonalizable based on the fact that $T$ must have one nonzero eigenvalue otherwise it would be nilpotent (I am leaving details of the proof of diagonalization but it is the observation that the characteristic space of the nonzero eigenvalue is the range of T and has dimension $1$).  Thus the minimal polynomial of $T$ is just a linear term $x-c$.    Did I make a mistake assuming that $T$ can have only one nonzero eigenvalue?  Thanks for your help Copy material to another object? I saw a video where the person copied a material from one object to another (Cycles).  How can I accomplish this? Proof by induction on a rooted tree with even number of vertices Let $T = (V, E)$ be a rooted tree with an even number of vertices. Prove by induction that there exists a set of edges $S \subseteq E$ such that every $v \in V$ is incident with an odd number of edges in $S$.  I was wondering if my argument is sufficient to prove the statement above:  Base case: $|V|=2$, there is only 1 edge and both vertices are incident with 1 edge $-&gt;$ $S \subseteq E$.  Inductive step:  Let $|V|&gt;2$, implying that $T$ has at least 2 external vertices. Let us choose two arbitrary external vertices $w,u \in V$. Such vertices are only incident to 1 edge each, $e_w$ and $e_u$. We can say that $e_w,e_u \in S$.   By induction, there exists a set $S$ in $T'=T-\{w,u\}=(V',E')$ such that every $v \in V'$ is incident with an odd number of edges in $S$ because removing 2 vertices from a tree with an even number of vertices maintains the property $|V|=2n$. Gapless superconductivity I don't understand what is meant by &quot;gapless superconductors&quot;. As far as I know, the superconducting gap is necessairy for both zero resistance and the Meissner effect, the two hallmarks of superconductivity. So, what does it mean to have gapless superconductivity and how does it work? Thanks in advance, Greetings I don't have complex analysis at my beck and call, and I only have a low level of knowledge in topology, but I need to prove that this metric space (for any real $r$ and $R$ with $r &lt; R$)$$ X = \{ (x, y) \in \mathbb{R}^2 \ | \ r \leq x^2 + y^2 \leq R \}$$  with the Manhattan metric $d((x_1, y_1), (x_2, y_2)) = |x_1-x_2| + |y_1-y_2|$ is not simply connected.  I've already prooven that it's path connected, and now I need to show there are some points $P$ and $Q$ with two paths between them such that one cannot be continuously 'morphed' into the other.  $ \ $  What I have so far is as follows:  I take $P = (0, r)$ and $Q = (0, -r)$, with $f_0$ being a path from $P$ to $Q$ going clockwise around the circle radius $r$ and $f_1$ being much the same but going counterclockwise.  Now I assume there is a function $g : [0, 1]^2 \to X$ such that:   $g(s, 0) = f_0 (s)$ $g(s, 1) = f_1 (s)$ $g(0, t) = P$ $g(1, t) = Q$   To get the final result, I need to show that this function cannot be continuous, but for the life of me I cannot.  For some context, these are the topics which have been visited during the course, roughly in order of recentness.   Multiple connectedness Simple connectedness Pathwise connectedness Interior points Boundary points Open sets Compactness Complete metric spaces Bounded metric spaces Totally bounded metric spaces Closed sets Closure of a metric space Limit points Cauchy sequences Convergence Continuity Metric space quivalence Metric equivalence   Edit: I might have an argument that works, though it's far from rigourous. We can shrink $R$ to be as close to $r$ as we want, so we can essentially constrain the annulus down to a circle and thus force any path from the left side of the circle to the right side to go through $P$ or $Q$.  So holding $s \in (0, 1)$ constant and varying $t$ must produce a path through $P$ or $Q$ for any $s$. If for some $s$ it passes through $P$ and for some other $s$ it passes through $Q$, then there must be $s_0$ such that $\forall \epsilon &gt;0  \ \exists \delta \leq \epsilon$ st. $s_0$ produces a path through $P$ and $s_0 + \delta$ produces a path through $Q$.  Now we can consider the path $g(s, \frac{1}{2})$, and note that it must have a discontinuity at $s_0$.  Now consider the case where all $s$ produce paths through only one of $P$ or $Q$. WOLOG: $P$. Now, at $t = \frac{1}{2}$, $s$ arbitrarily close to $1$ are mapped away from $Q$, but $1$ is always mapped to $Q$ by definition, so $g(s, \frac{1}{2})$ has a discontinuity at $s=1$. Therefore $g(s, t)$ is not continuous.  This argument is definitely iffy to me, if no one has their own argument (using sufficiently low level concepts), then criticism on the above would be appreciated. What would be a good way to go about attacking a Quagmire 3 cipher?  I understand that it is polyalphabetic with a key and an indicator. I have started by taking the cipher text, taking every n-th letter and putting that in a string, computing the index of coincidence for the n strings I get and then averaging that. I would assume that whatever n value corresponds with the closest average index of coincident to English (0.066) would be the number of alphabets, i.e. the length of the indicator.  Where do I go from there? Or is my approach incorrect? function a () {     return "foo"; }  a.b = function () {     return "bar"; }  function c () { }; c.prototype = a;  var d = new c(); d.b(); // returns "bar" d(); // throws exception, d is not a function   Is there some way for d to be a function, and yet still inherit properties from a? Apache on CentOS 5.9 VM serves modified files incorrectly UV map / model turned black on saving - how to recover? In the middle of touching up a new texture, I pressed save and suddenly my textures all turned black on the model and UV texture - just vanished.  I tried reopening the file in another instance of Blender, I tried to CtrlZ too and it just did this weird patchy thing on my texture.   How do I recover it and what did I do wrong? Why quantum mechanics? Imagine you're teaching a first course on quantum mechanics in which your students are well-versed in classical mechanics, but have never seen any quantum before.  How would you motivate the subject and convince your students that in fact classical mechanics cannot explain the real world and that quantum mechanics, given your knowledge of classical mechanics, is the most obvious alternative to try?  If you sit down and think about it, the idea that the state of a system, instead of being specified by the finitely many particles' position and momentum, is now described by an element of some abstract (rigged) Hilbert space and that the observables correspond to self-adjoint operators on the space of states is not at all obvious.  Why should this be the case, or at least, why might we expect this to be the case?   Then there is the issue of measurement which is even more difficult to motivate.  In the usual formulation of quantum mechanics, we assume that, given a state $|\psi \rangle$ and an observable $A$, the probability of measuring a value between $a$ and $a+da$ is given by $|\langle a|\psi \rangle |^2da$ (and furthermore, if $a$ is not an eigenvalue of $A$, then the probability of measuring a value in this interval is $0$).  How would you convince your students that this had to be the case?  I have thought about this question of motivation for a couple of years now, and so far, the only answers I've come up with are incomplete, not entirely satisfactory, and seem to be much more non-trivial than I feel they should be.  So, what do you guys think?  Can you motivate the usual formulation of quantum mechanics using only classical mechanics and minimal appeal to experimental results?  Note that, at some point, you will have to make reference to experiment.  After all, this is the reason why we needed to develop quantum mechanics.  In principle, we could just say "The Born Rule is true because its experimentally verified.", but I find this particularly unsatisfying.  I think we can do better.  Thus, I would ask that when you do invoke the results of an experiment, you do so to only justify fundamental truths, by which I mean something that can not itself just be explained in terms of more theory.  You might say that my conjecture is that the Born Rule is not a fundamental truth in this sense, but can instead be explained by more fundamental theory, which itself is justified via experiment.  Edit:  To clarify, I will try to make use of a much simpler example.  In an ideal gas, if you fix the volume, then the temperature is proportional to pressure.  So we may ask "Why?".  You could say "Well, because experiment.", or alternatively you could say "It is a trivial corollary of the ideal gas law.".  If you choose the latter, you can then ask why that is true.  Once again, you can just say "Because experiment." or you could try to prove it using more fundamental physical truths (using the kinetic theory of gases, for example).  The objective, then, is to come up with the most fundamental physical truths, prove everything else we know in terms of those, and then verify the fundamental physical truths via experiment.  And in this particular case, the objective is to do this with quantum mechanics. 3-manifold examples of homomorphisms between fundamental groups that are not induced by continuous maps I am looking for some examples of closed, orientable 3-manifolds $M$ and $N$ and a homomorphism $\phi : \pi_1(M) \to \pi_1(N)$ such that $\phi$ is not induced by any continuous map $f : M \to N$.  Are there examples of this occurring for lens spaces?  I do not believe that this phenomenon can occur for maps between surfaces. No updates since a long time Using Ubuntu, I have a habit of running: sudo apt update sudo apt upgrade   on every startup. But for a month, I guess, there have been no updates whatsoever. The message shows: All packages are up to date  Now I am starting to worry. Can there be a fault in my update manager? As far as I know, I've made no changes to the package manager settings. But no updates for almost a month seems kind of weird. Any feedback would be helpful. Thanks. Evaluate definite interval of normal distribution How to typeset subscript in usual text mode? When I create a folder or copy files, after the operation finishes I have to manually refresh the Explorer window (F5). For example, when I create a folder, I don't see the "New Folder" until I hit F5.  Does anyone have any idea how to fix this? We work heavily with serialization and having to specify Serializable tag on every object we use is kind of a burden. Especially when it's a 3rd-party class that we can't really change.  The question is: since Serializable is an empty interface and Java provides robust serialization once you add  implements Serializable - why didn't they make everything serializable and that's it?    What am I missing? Can anyone recommend a simple API that will allow me to use read a CSV input file, do some simple transformations, and then write it.  A quick google has found  which looks promising.  I just wanted to check what others are using before I couple myself to this API.