$\newcommand{\tr}{\mathrm{Tr }}$ Let be $\mathcal{M}_n(\mathbb{C})$ the set of complex-valued square matrices of order $n$ and $[A, B] = AB - BA$ a commutator.  If, for some $A, B \in \mathcal{M}_n(\mathbb{C})$, we have $[A, [A, B]] = 0$, then $[A, B]$ is nilpotent.  So: $[A, [A, B]] = 0$ says that $A$ and $[A, B]$ commutes.  Let be $\lambda \in \mathbb{C}$ and $X \in \mathbb{C}^n$ such that $[A, B]X = \lambda X$ (an eigenvalue/eigenvector pair), such values always exist as $[A, B] \in \mathcal{M}_n(\mathbb{C})$ and must have $n$ eigenvalues.  Left-multiplication of the previous relation gives: $A[A, B]X = \lambda AX$.  By hypothesis, we have: $[A, B](AX) = \lambda (AX)$.  If we denote $E_{\lambda}$ the set of eigenvectors of $[A, B]$ for the $\lambda$ value, we have: $X \in E_{\lambda} \implies AX \in E_{\lambda}$.  As a result, for all $k \in \mathbb{N}, A^k X \in E_{\lambda}$.  Now, we also have $\tr [A, B] = 0$ because of properties on the trace.  If $A$ is not nilpotent or nilpotent for $k \geq n$, then: $(X, AX, \ldots, A^{n - 1} X)$ is linearly independent, as $\dim E_{\lambda} \leq n$ and $\dim E_{\lambda} \geq n$.  We conclude that $\dim E_{\lambda} = n$, finally: $n \lambda = 0$, now: $\lambda = 0$.  We conclude that $[A, B]$ is nilpotent.  If $A$ is nilpotent for $k \leq n - 1$, then $\dim E_{\lambda} \geq k$.  Here, I don't really know how to proceed, I'd like to show we cannot have any other value than $\lambda$ and it must be $0$ due to the trace.
Let $A,B \in \mathbb{M}_n (\mathbb{C})$. If $A^2B + BA^2 = 2ABA$ then exist $k \in \mathbb{N}$ where $(AB-BA)^k = 0$.  I tried solve with minimal polynomial, but I did not have much effect.
Limit of ${a_n}^{1/n}$ is equal to $\lim_{n\to\infty} a_{n+1}/a_n$ Proof for Property of Complex Numbers Is the inequality $\lvert z_1 + z_2 \rvert \ge \lvert z_1 \rvert - \lvert z_2 \rvert$ incorrect, where $z_1$ and $z_2$ are any two complex numbers? I need an example to prove that it is. And in case it is correct, can you please give the proof? Thanks for any help. Android: Overlay on Android Camera Preview I want to install the latest Python tarball on Ubuntu, downloaded from .  Is this is a correct way to install?  ./configure make make install   If not, how do I do that? Why does the version of Java verified by Oracle differ from the version shown at the command line? Questions  Why do versions differ?   Should we trust Oracle's verification routine?    In some cases, the  is different from the version shown at the command line.   Example    That example, in detail  According to Oracle in the browser:    Java 7 Update 9   According to the Java Console:   Java Plug-in 10.9.2.05 Using JRE version 1.7.0_09-b05 Java HotSpot(TM) 64-Bit Server VM   According to the java command:   sh-3.2$ java -version java version "1.8.0-ea" Java(TM) SE Runtime Environment (build 1.8.0-ea-b58) Java HotSpot(TM) 64-Bit Server VM (build 25.0-b02, mixed mode) Subsample of a random sample: random sample? My mother is given a UK tourist visa for 2 years with a duration of stay as 180 days. It is a c type multiple entry visa. Is this 180 days is the total stay she can take or is it duration of stay per visit.  Thanks in advance for your reply. Group of positive rationals under multiplication not isomorphic to group of rationals How do I prove if a set of operations is complete? For example $\{\land, \lnot\}$ apparently forms a functionally complete set as it can form any logical expression.  But I don't really know what this means or how you show it. Do we just show that some set is complete and then in the future if we have a new set, show that we can replicate the known operations with the new ones?  Do you have to find these strategically or is there a systematic way? How do I find the files installed by a particular package? Where is the .deb file to look at? I have installed pulseaudio-module-bluetooth using apt.  $ type pulseaudio-module-bluetooth bash: type: pulseaudio-module-bluetooth: not found $ which pulseaudio-module-bluetooth $ whereis pulseaudio-module-bluetooth pulseaudio-module-bluetooth:   Clearly, I'm looking for the wrong thing.  The package does not simply install a command that is the same name as the package.  Alright, then.  I want to find out what all commands (or executables) this package installed, and their locations.  The answers to "" tell me how to find the files installed by the package if I have a .deb file.  I'm not installing directly from a .deb file, though.  I'm using apt.  Where is the .deb file that that used?  Is there a copy on my system somewhere that I can query with the commands in the answers to that question?  Where?  If there isn't a local copy on my system, can I get one with apt?  How?  Is there some handy apt (or similar) command that wraps this up for me, so that I do not have to run dpkg-deb directly?  What is it?  Can I find the package's file list entirely on-line, without explicitly downloading any .deb files and before installing anything with apt?  How? How is the notation $X\sim N(\mu,\sigma^2)$ read? Is it $X$ follows a normal distribution? Or $X$ is a normal distribution? Or perhaps $X$ is approximately normal..  What if there are several variables that follow (or whatever the words is) the same distribution? How is it written? Delay answer downvote rep loss to voter to help hide voter's identity A while back, I posted an answer to SO, which was downvoted.  I wasn't sure who did it, but had a suspicion because I thought I had seen a particular user lose a reputation point.  When I deleted the answer, my suspicion was confirmed by a point gain in reputation by the particular user.  Now, I wasn't going to attempt any sort of retaliation (and did deserve the downvote) but this seems like it could lead to abuse.  My suggestion is to apply answer downvote rep loss/gain at a particular time of day, and not immediately after the vote or answer deletion. Does an unbalanced sample matter when doing logistic regression? Okay, so I think I have a decent enough sample, taking into account the 20:1 rule of thumb: a fairly large sample (N=374) for a total of 7 candidate predictor variables.   My problem is the following: whatever set of predictor variables I use, the classifications never get better than a specificity of 100% and a sensitivity of 0%. However unsatisfactory, this could actually be the best possible result, given the set of candidate predictor variables (from which I can't deviate).   But, I couldn't help but think I could do better, so I noticed that the categories of the dependent variable were quite unevenly balanced, almost 4:1. Could a more balanced subsample improve classifications? How to reduce space between image and its caption? What does first authorship really mean? In my field (theoretical computer science), authors of any paper are always listed alphabetically; our papers .  (Well... hardly ever.)  In most other disciplines, at least within science and engineering, the ordering of authors is a signal about their relative contributions to the paper, with the first author indicating the most significant contributor.  Hiring and promotion committees do give extra weight to "first-author papers" (and sometimes have to be reminded that not all areas have them).  As an outsider, I find this practice confusing.  What does first authorship actually mean in your discipline? I understand vaguely that the first author is supposed to be the one who "", but what counts as "work" in this comparison?  Does "most" mean "more than all the other coauthors together" or just "more than any other coauthor"?  What happens when the comparison is unclear?   How often is "did the most work" the actual truth, versus a cover story for a more complex political decision?  I realize that the precise answer is different for every paper.  I'm looking for general guidelines for how an outsider (like me) should interpret first authorship in your field.  Pointers to guidelines from journals or professional societies would be especially helpful.  Please give only one answer per discipline. Multivariate differentiability verification I have tried an attempt on the following:  Let $F:U\in \Bbb{R}^n\to \Bbb{R}$ and $f:\Bbb{R}\to \Bbb{R}$ where $f$ is an even function. Now $F(\mathbf{x})=f(|\mathbf{x}|)$, where $|\ . |$ is the Euclidean norm. Given $f$ is $C^r$, I have to show that $F$ is $C^r$.  My attempt was to argument by induction. I showed that $F$ is $C^1$, basically using univariate chain rule on $\frac{\partial f\left(\sqrt{x_1^2+\dots+x_n^2}\right)}{\partial x_1}$ and arguing that it would be continuous using the fact that $f$ is $C^r$. Since the argument works for all $x_i$, it establishes first-differntiability for $F$ and continuity of the operator $DF(\mathbf{x})$. Then I moved on to show $F$ is $C^r$ assuming $F$ is $C^{r-1}$ and $f$ is $C^r$. Now what I did was partially differentiate:  $$ {\partial \over \partial x_2} \left[\frac{\partial ^{k-1}f}{\partial x_1^{k-1}}\right] = \underbrace{{\partial \over \partial |\mathbf{x}|} \left[\frac{\partial ^{k-1}f}{\partial x_1^{k-1}}\right]}_{(1)} . \underbrace{{\partial |\mathbf{x}| \over \partial x_2}}_{(2)} $$  My argument runs as follows: this partial derivative is continuous since $f$ is $C^r$ and hence $(1)$ is continuous. Continuity of $(2)$ is establishd in a similar manner, since $\sqrt{\mathbf{x}}$ is $C^r$ if $\mathbf{x} \neq \mathbf{0}$. Then the product would be continuous. I am uneasy on multivariate arguments and dont know if my approach is completely off track. I didnt use the even function property, so I am sure I might me missing something. Any hints and references would be welcome. Thanks! Lets say an object is tied to a string and it is being swung in a circle:    In this case, the tension of the string acts as a centrifugal force that keeps the object moving in a circle. However, if centrifugal force is the only force acting on the object, why doesn't it collapse into the center? I am pretty sure this has to do with the object's inertia (or the centrifugal pseudo-force), where it moves slightly forward in a tangent before falling, restoring the orbit.  Also, what keeps the string taut during the swinging? There has to be some force countering the centripetal force that keeps the string tight. I think this might again have to do with the intertia/centrifugal pseudo-force but can a pseudo-force balance a real force? Also, if centripetal were to be balanced by the pseudo-force, wouldn't the object not move at all (since the net force would be 0)? i have created a schedulable class which has to trigger a batch job. The run method for the batch class will get list of data to process from a webservice callout. Now, i got the exception below. Any suggestions please.     System.CalloutException: Callout from scheduled Apex not supported.   global class scheduledDeltaBatchable implements Schedulable {    global void execute(SchedulableContext sc) {     ID batchprocessid = BatchUpsertSAPCustomerMasterBPSync.run(SAPServices.GetSAPCustomerMasterBPSync('Delta'));    } } How to assign many variables the same value in a smart way? Should methods in a Java interface be declared with or without a public access modifier? Should methods in a Java interface be declared with or without the public access modifier?  Technically it doesn't matter, of course. A class method that implements an interface is always public. But what is a better convention?  Java itself is not consistent in this. See for instance Collection vs. Comparable, or Future vs. ScriptEngine.