Sort map of maps i.e. nested maps in java
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For symmetric operators one has: $$A\text{ symmetric}:\quad\mathcal{R}(A\pm\imath)=\mathcal{H}\implies A^*=A$$ How to prove this in an unveiling way? Why is the company email still "at stackoverflow" I have a speed-reading program I purchased with uses dotnet2. However, dotnet2 doesn't run on 64 bit wine. How would I install dotnet2? Where can I find a free, very quick, and reliable implementation of FFT in C#?  That can be used in a product? Or are there any restrictions? Function whose image of every open interval is $(-\infty,\infty)$ How to find a function from reals to reals such that the image of every open interval is the whole of R?  Is there one which maps rationals to rationals? How do I draw ovals in TikZ? How can I draw this graph in the pic via tikz. Let $A, B \in \mathcal M_n(\mathbb C)$, we want to know if there's any relation between their spectral radius and the spectral radius of $A + B$.  The first part of this exercise is giving examples of the three posibilities: $\rho(A) + \rho(B) &lt; \rho(A+B)$ ; $\rho(A) + \rho(B) &gt; \rho(A+B)$ and $\rho(A)+ \rho(B) =\rho(A+B)$. This part was easy because there are no conditions about A or B.  The second part is proving that if A and B are normal matrixes, (i.e. $ AA^* =A^*A$, where $A^*$ is its conjugate transpose), then $ \rho(A+B) \le \rho(A) + \rho(B) $.   So my first attempt was using the Schur's decomposition, so $A=U^{-1}D_AU$ where $D_A$ is a diagonal matrix with the eigenvalues of $A$ in its diagonal and U is an unitary matrix. Similarly, B is similar to a diagonal matrix $D_B$ with the eigenvalues of B in its diagonal. That way I have $\rho(A) = \rho(D_A)$ and $\rho(B) = \rho(D_B)$ so now since $D_A + D_B$ is a diagonal matrix with $\lambda_i + \mu_i$ for $ 1\le i \le n $, where $\lambda_i$ and $\mu_i$ are the eigenvalues of $A$ and $B$ respectively. From here it's easy to see that $ \rho(D_A+D_B) \le \rho(D_A) + \rho(D_B) $, since $ |\lambda_i+\mu_i|\le|\lambda_i|+|\mu_i| \ \forall i$. But I don't see how to include $\rho(A+B)$ in this.  And the last part of the exercise is proving that if $ \{X_i\}$ is a basis of eigenvectors, which is common to A and B, then $ \rho(A+B) \le \rho(A) + \rho(B) $.  Here's my attempt, but I would like to know if it's correct:  Let $\lambda_i$ and $\mu_i$ be eigenvalues of A and B associated to the vector $X_i$, then $AX_i=\lambda_i X_i$ and $BX_i = \mu_iX_i$, and if we sum both equalities, we have $AX_i + BX_i = \lambda_iX_i + \mu_iXi$ so $(A+B)X_i=(\lambda_i+\mu_i)X_i$ so $(\lambda_i+\mu_i)$ is an eigenvalor of $(A+B)$ associated to the eigenvector $X_i$. Then is again clear that $|\lambda_i+\mu_i|\le|\lambda_i|+|\mu_i| \le \rho(A) + \rho(B)\ \forall i$, and particularly, for $\max_i |\lambda_i+\mu_i| = \rho(A+B)$ so $ \rho(A+B)\le \rho(A) + \rho(B)$  Any kind of help is appreciated and thank you very much in advance! Sequences in $C(X)$ converging uniformly implies uniform boundedness and equicontinuity Let $X$ be a metric space and let $\{ f_n \}_{n \geq 1}$ be a sequence in $C(X)$. Show that if $\{ f_n \}_{n \geq 1}$ converges uniformly on $X$ then $\{ f_n \}_{n \geq 1}$ is both uniformly bounded and equicontinuous.  Could I get a hint? What is the 'Badlock Bug'? A user on the Ask Ubuntu General Room posted a link to . After some googling around, all I can find is that it is a mysterious security bug, that uses the same website template as Heartbleed.  I manage Linux Servers, a mysterious security bug does not sit well with me. What exactly is it, and how can I protect my servers from it? Is 60 minutes enough layover time at Los Angeles International Airport? I am flying in to LAX on American Airlines and have a 60 min layover before I fly out with American Airlines (same ticket).  Will this be enough time? When applying dropout in artificial neural networks, one needs to compensate for the fact that at training time a portion of the neurons were deactivated. To do so, there exist two common strategies:   scaling the activation at test time inverting the dropout during the training phase   The two strategies are summarized in the  below, taken from .  Which strategy is preferable, and why?    Scaling the activation at test time:    Inverting the dropout during the training phase: Converting mxd to earlier version using ArcPy? I tried to use GNOME Nightlight in Ubuntu 17.04. The settings only allow me to turn it on and adjust the time. Is it possible to adjust the intensity or temperature of the hue? Currently, it's a bit too much. Help us test the alpha version of our Android app We are in the early stages of development of the official Stack Exchange Android app and we need your help with testing. To set a few expectations: there will be bugs, crashes, and missing features, all of which will be fixed in .    We are initially targeting Android devices running  (Android 4.1 or higher) starting with phones (up to ~5 inch) and eventually moving our way to tablets.    To register for alpha testing .  After you register we will send you an invite to a private Google community where you can find a link to download our app directly from the  store. The invitations will be sent out in waves, so if you don’t get an invite immediately, be patient.  Crash reports will be automatically reported back to us using . Bugs or feature requests should be reported here on meta and tagged . Please do not use the Google community to report bugs.  As always, we welcome your feedback and are committed to building the best mobile app possible for browsing the Stack Exchange sites. Why doesn't the Related question list show that questions are closed? I saw a question on Meta that was obviously going to be a duplicate:        And indeed, looking at the "Related" questions list for that question showed a number of duplicates:    Being the good citizen I am, I tried to vote to close (technically Flag as duplicate as I've not got the rep here to vote to close) it as an Exact Duplicate only to find myself falling into a rabbit hole of Closed, Duplicate questions. Is there any way to flag a closed question as such in the Related list?  Mostly because the first one I picked was closed as a duplicate of the second one, which was closed as a duplicate of the one that is now listed as the duplicate/linked answer.   Edit to respond to comment:  I take the point about "not voting to close on title alone", and normally wouldn't, however as the questions had identical titles, and it's pretty clear what they should be asking, I clicked the first one:    which lead to:  which leads to:  - but I'd got bored following links and didn't check that one as the first two were identical, so assumed this was too and didn't want to see how deep the rabbit hole led.   In the related list, there's only one other question in there that is an exact duplicate, but hasn't (currently) been closed as such which is:      Reasoning for highlighting closed questions:  The "Questions with similar titles" does display the closed titles when asking the question, so it should be fairly possible to include.  If there are clearly duplicate titles, then as a reader/reviewer would be helpful to know which ones are already closed as duplicates so you can either:   Use the remaining open questions to focus on which of those is the canonical answer to close this with. Use the closed questions to find the canonical answer (which failed for me with my route of clicks). Taylor's Remainder $x-\frac{x^2}{2}+\frac{x^3}{3(1+x)}<\log(1+x) <x-\frac{x^2}{2}+\frac{x^3}{3}$ Prove that   $\displaystyle x-\frac{x^2}{2}+\frac{x^3}{3(1+x)}&lt;\log(1+x) &lt;x-\frac{x^2}{2}+\frac{x^3}{3}$  My attempt: I can prove this by taking one side at a time and assuming $\displaystyle f(x) = \log (1+x)-x+\frac{x^2}{2}-\frac{x^3}{3}$   and then proving that it is a decreasing function and then the same for the other side.  But I am looking for a better solution using Lagrange's Mean Value Theorem or using Taylor's remainder. Nuclear Binding energy How do I use Greek letters to enumerate a list? how do you change the numbering of a question so that the subparts are numbered with greek symbols like alpha and beta instead of the usual roman numerals? How do I summon mobs with block heads on Minecraft? Is "I am sat" bad English? I believe it is incorrect and instead either the present continuous     I am sitting   or the predicate adjective     I am seated   should be used.  I hear this quite often, however, and ultimately usage overrules formality.  Does anybody know where this originated or how to describe it grammatically?