Matrix inversion is continuous
Prove the map sending a matrix to it's inverse is continuous.
In mathematics, we like talking about idempotent matrices. In a masters program that I am currently enrolled in, I have heard one professor pronounce it one way, and another professor pronounce it another way. The  explains it better than I could:  Is there a correct pronunciation for idempotent? Is it like omnipotent (om-nip-o-tent), so id-EMP-o-tent, or is it more like the two seperate words idem + potent I'd say EYE-dm-POT-nt, but then I speak Brit. Charles Matthews 07:57, 30 Jun 2004 (UTC) Ditto, but then I speak Australian ;) Dysprosia 08:01, 30 Jun 2004 (UTC) Actually the British is more like EYE-dm-PO-tnt, I guess. Charles Matthews I'd just like to pedantically point out that it would be pronounced i-DEM-po-tent because because syllable onsets are maximized. daesotho 20:33, 21 Oct 2004 (UTC) The American pronunciation puts primary stress on the second syllable ('dem') and secondary stress on the fourth syllable ('tence'). Could somebody more familiar with IPA than I am please add this to the article as an accepted pronunciation? My American dictionary says |ˈīdemˌpōtənt| for the adjective. I speak American (non-natively) and I wouldn't stress the second syllable for itempotence. My only question is, right now it says /ˌaɪdɪmˈpoʊtəns/ -- but can't it be stressed on the first syllable too? Should we add /'aɪdɪmˌpoʊtəns/ as an alternative? -- 87.160.141.177 (talk) 14:50, 3 June 2011 (UTC) And by the way I think it's silly giving a pronounciation guide as this is just imposing a particular accent. (For what it's worth I pronounce idempotent with a short 'i', as in the word 'id'). Alex Selby I agree with Alex Selby - the pronunciation guide should be extended to show the variation, or removed. Strictly, there's no reason for the first i to be long (neuter idem has a short first syllable in Latin), and I can't find any authoritative source which advocates the pronunciation given. Ms7821 (talk) 22:59, 15 July 2015 (UTC)  Is there a &quot;correct&quot; pronunciation for this word? Solving linear congruence (modular inverse or fraction) via gcd Bezout equation Isn't finding the inverse of $a$, that is, $a'$ in $aa'\equiv1\pmod{m}$ equivalent to solving the diophantine equation $aa'-mb=1$, where the unknowns are $a'$ and $b$? I have seem some answers on this site (where the extended Euclidean Algorithm is mentioned mainly) as well as looked up some books but there is no mention of this. Am I going wrong somewhere or is this a correct method of finding modular inverses? Also can't we find the Bézout's coefficients by solving the corresponding diophantine equation instead of using the extended Euclidean Algorithm? Stack Overflow in ES, PT, RU, JP have mixed-language descriptions I own a Windows 10 K or N Edition which doesn't include the media player package. My problem is: I want to have preview thumbnails in the explorer for my MP3s and videos.  I had the issue before when I updated to Win10 but there  saved me. Since the Anniversary update they are gone again.   Now I can install the old update but I still don't get the media player nor the preview images.  I tried most of the things Microsoft offers me but they are either not installable or don't work.  Any ideas what might be the correct update package?  Thx!  Ron The axiom of extensionality is:  ${\forall}a({\forall}b({\forall}c(((c{\in}a){\iff}(c{\in}b)){\iff}(a=b))))$  The definition of $(P{\iff}Q)$ is:  ${\lnot}((P{\implies}Q){\implies}({\lnot}(Q{\implies}P)))$  or, in other words  $(P{\land}Q){\lor}({\lnot}P{\land}{\lnot}Q)$  In the case of the statement $((c{\in}a){\iff}(c{\in}b))$, this leads us to two options: $((c{\in}a){\land}(c{\in}b))$ and $({\lnot}(c{\in}a){\land}{\lnot}(c{\in}b))$. So far nothing weird. Now, let's see all the possible results of the axiom.  1.$((c{\in}a){\land}(c{\in}b)){\land}(a=b)$ 2.${\lnot}((c{\in}a){\land}(c{\in}b)){\land}{\lnot}(a=b)$ 3.$({\lnot}(c{\in}a){\land}{\lnot}(c{\in}b)){\land}(a=b)$ 4.${\lnot}({\lnot}(c{\in}a){\land}{\lnot}(c{\in}b)){\land}{\lnot}(a=b)$  Coincidentally, they are also all possible statements regarding the statements $((c{\in}a){\land}(c{\in}b))$ and $(a=b)$: both parts are true, the first part is true and the other is false, the second part is true and the first is false and neither is true. The axiom seems not to state anything new, because rephrased it's just produces a list of all possible options regarding the two parts of the statement. Worse, it doesn't seem to say anything about equality, since it allows for $a=b$ with both $(c{\in}a){\land}(c{\in}b)$ and ${\lnot}((c{\in}a){\land}(c{\in}b))$. So, what is the point of this axiom? And in what way does it say anything about equality? Proving that $\mathbb{Z}_m\oplus \mathbb{Z}_n \cong \mathbb{Z}_d\oplus \mathbb{Z}_l $ as groups, where $l=\mathrm{lcm}(m,n)$ and $d=\gcd(m,n)$ How would one go about proving that $\mathbb{Z}_m\oplus \mathbb{Z}_n \cong \mathbb{Z}_d\oplus \mathbb{Z}_l $ as groups, where  $l=\mathrm{lcm}(m,n)$ and $d=\gcd(m,n)$?    I am attempting to use the fundamental theorem of finitely generated abelian groups but am struggling. In the interest of honesty, this is a past exam question that I am attempting for which the solutions are not available. EDIT: As the question was asked in the paper, the Chinese Remainder Theorem would not be permitted since this is only proved in the follow-up course. Return row only if value doesn't exist Why doesn't a \foreach variable work in \draw in an \axis? I'm finding some weird behavior with TikZ's \foreach within an \begin{axis} environment.  This works:  \foreach \x in {-2,-1,...,2} {   \addplot[thick, domain=0.05+(\x*2-1)*pi/2:+(\x*2+1)*pi/2-0.05] (x, {tan(deg(x))});  }   but this doesn't:  \foreach \x in {-2,-1,...,2} {   \draw (\x,-5) -- (\x,5); }   It gives the error  ! Undefined control sequence. &lt;argument&gt; \x                ,-5 l.771     \end{axis}   Here's the really weird part: it only fails when the axis has defined bounds. For example, here's a MWE that works (but doesn't draw anything):  \documentclass{standalone} \usepackage{pgfplots} \begin{document} \begin{tikzpicture}   \begin{axis} % [ ymin=-3, ymax=3, xmin=-5, xmax=5]     \foreach \q in {-2,-1,...,2} {       \draw (axis cs: \q,-5) -- (axis cs: \q,5);     }   \end{axis} \end{tikzpicture} \end{document}   but removing the comment causes it to fail.  Why is this and how can I prevent it?  (My end goal is to draw five segments of the tangent function and their verical symptotes.) I am using natbib and agsm for harvard-style referencing. My related code is  \usepackage{natbib} \bibliographystyle{agsm}   Then I am using \citep{} for citations. My problem is that in text I am getting     (Author et al. 2012)   but and I want     (Author et al. 2012)   Does anyone know how to change this and get 'et al' in italics in the text?  I am using TexPad and BibDesk and I have not been able to find the .bst files in Terminal.  I know that it is often correct to have the 'et al.' unitalicised but my university regulations requires italics. How to avoid pagenumering with text in scrlttr2 Have a Letter in scrlttr2. I cannot avoid/exclude the text before the page number 'Seite' (see print-screen). The  recommends to add \pagenumbering{arabic}. But for my document, it has no effect.    %--------------------------------------------------------------------------- \documentclass%% %---------------------------------------------------------------------------   [fontsize=12pt,%%          Schriftgroesse %--------------------------------------------------------------------------- % Satzspiegel    paper=a4,%%               Papierformat    enlargefirstpage=on,%%    Erste Seite anders    pagenumber=footcenter,%%   Seitenzahl oben mittig %--------------------------------------------------------------------------- % Layout    headsepline=on,%%         Linie unter der Seitenzahl    parskip=half,%%           Abstand zwischen Absaetzen %--------------------------------------------------------------------------- % Formatierung    draft=on%%                Entwurfsmodus ]{scrlttr2} %--------------------------------------------------------------------------- \usepackage{german} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc}  %--------------------------------------------------------------------------- \begin{document} %--------------------------------------------------------------------------- % Briefstil und Position des Briefkopfs \LoadLetterOption{DIN} %% oder: DINmtext, SN, SNleft, KOMAold. \makeatletter \pagenumbering{arabic}  %--------------------------------------------------------------------------- % Seitenstil \pagestyle{plain}%% keine Header in der Kopfzeile %--------------------------------------------------------------------------- \begin{letter}{Herr\\                 Mustermann\\                Straße 1\\                12345 Stadt} %--------------------------------------------------------------------------- \setkomavar{subject}{Beispielschreiben} %--------------------------------------------------------------------------- \opening{Liebe Herr Mustermann,}  here is some Text.  \pagebreak  here is some Text. %--------------------------------------------------------------------------- \end{letter} %--------------------------------------------------------------------------- \end{document} %--------------------------------------------------------------------------- Different results for integration using Mathematica and MATLAB Posts flagged for attention but there are none The comment with the Flagged Posts that seems to be new as of today states there are posts with flags awaiting my attention but the counter shows zero and when I go into flagged posts there is nothing waiting for me. Don't replicators make the Ferengi rather pointless? Ferengis are all about acquisition and making a profit. But, in the age of replicators, where everything you want can be produced instantly, what would be the point of their lifestyle? How do you find out which program is using too much memory? Ubuntu has been crashing on me recently. I think its because it runs out of memory so I ran the free -m command and found that my memory usage was really high. So then, i ran top to find the culprit, but the displayed processes were using less than 1.5% of memory. How do I know which program is making ubuntu crash/run out of memory? Below is the output:  shafee@shafee-pc:~$ free -m              total       used       free     shared    buffers     cached Mem:          3824       3714        110          0        978       1954 -/+ buffers/cache:        780       3044 Swap:           99          0         99  shafee@shafee-pc:~$ top top - 02:12:14 up  1:24,  2 users,  load average: 0.16, 0.42, 1.49 Tasks: 182 total,   1 running, 181 sleeping,   0 stopped,   0 zombie Cpu(s):  2.9%us,  1.9%sy,  0.3%ni, 79.3%id, 15.5%wa,  0.0%hi,  0.2%si,  0.0%st Mem:   3916708k total,  3803848k used,   112860k free,  1002308k buffers Swap:   102396k total,        0k used,   102396k free,  2001852k cached    PID USER      PR  NI  VIRT  RES  SHR S %CPU %MEM    TIME+  COMMAND              4200 root      20   0  289m  53m  38m S    2  1.4   1:06.45 Xorg                 5590 shafee    20   0 19348 1368  956 R    2  0.0   0:00.01 top                     1 root      20   0 24124 2136 1264 S    0  0.1   0:02.05 init                    2 root      20   0     0    0    0 S    0  0.0   0:00.00 kthreadd                3 root      20   0     0    0    0 S    0  0.0   0:24.23 ksoftirqd/0             6 root      RT   0     0    0    0 S    0  0.0   0:00.00 migration/0             7 root      RT   0     0    0    0 S    0  0.0   0:00.00 migration/1             9 root      20   0     0    0    0 S    0  0.0   0:00.11 ksoftirqd/1            11 root      RT   0     0    0    0 S    0  0.0   0:00.00 migration/2            13 root      20   0     0    0    0 S    0  0.0   0:03.89 ksoftirqd/2            14 root      RT   0     0    0    0 S    0  0.0   0:00.00 migration/3            16 root      20   0     0    0    0 S    0  0.0   0:00.16 ksoftirqd/3            17 root       0 -20     0    0    0 S    0  0.0   0:00.00 cpuset                 18 root       0 -20     0    0    0 S    0  0.0   0:00.00 khelper                19 root       0 -20     0    0    0 S    0  0.0   0:00.00 netns                  21 root      20   0     0    0    0 S    0  0.0   0:00.01 sync_supers         shafee@shafee-pc:~$   Or is it normal to have 110mb of free memory and is my issue related to something else? I am planning to visit London for 6 days from the US. I have given my biometrics and when I was preparing to mail my documents to the UK visa processing center in the US, I used this link from the UK government website to cross-check the documents to be sent to the processing center.     (Guide to supporting documents).  Something that really surprised me was in Section 4 (documents that should NOT be sent along with the package) was hotel bookings and flights bookings.     Section 4: documents you should not send unless specifically requested      This page provides guidance on the types of documents that are not required to consider your application.         hotel bookings   flight bookings      Typically I have always been asked to provide my travel plans, accommodation plans and proof of them. This is the first time that travel plans and accommodation plans are not asked to be sent. What might be the reason behind it? This really confuses me. Any help in clearing my confusion would be greatly welcome. I am trying to ask a question on SE, have some mock data in Excel I've copied, and am simply trying to paste in to my question.  I've done this before dozens of times, but suddenly am having this issue.  However, when I do so, I get prompted to upload an image. (This happens with Ctrl + V and also right click --> Paste):    Am I overlooking something?  Edit: Playing around, I can copy from Firefox/browsers and Word, and paste in to the box as usual. Excel seems to be the only one giving the issue so far.  Edit2:  are still having this issue. How can I migrate a Windows 10 from BIOS/MBR boot to UEFI/GPT without reinstalling? Simulating group_concat MySQL function in Microsoft SQL Server 2005? Capturing console output from a .NET application (C#) How do I invoke a console application from my .NET application and capture all the output generated in the console?   (Remember, I don't want to save the information first in a file and then relist as I would love to receive it as live.) A continuous bijection from a compact space to a $T_2$ space is always a homeomorphism This problem is bugging me for some time.     $f:(X,\mathcal{T}) \to (Y,\mathcal{T}')$ is a continuous bijection where $(X,\mathcal{T})$ is compact and $(Y,\mathcal{T}')$ is $T_2$ (i.e. Hausdorff and $T_1$). Then show that $f$ must be a homeomorphism.   So far I have failed to understand any relation between the compactness of domain and $T_2$-ness of the range. I tried in vain to prove that $f(U)$ is open in $(Y,\mathcal{T}')$ whenever $U$ is so in $(X,\mathcal{T})$. I know $f$ makes $X$ and $Y$ respectively $T_2$ and compact and hence makes both of them $T_4$ spaces. But that is not coming to any use. I would be glad if someone helps me with a solution. Thank you.