I suspect that the Jacobson radical of the ring  $$A = \begin{bmatrix} R &amp; M \\ 0 &amp; S \\ \end{bmatrix}$$  should be   $$\mathfrak{J}(A) = \begin{bmatrix} \mathfrak{J}(R) &amp; M \\ 0 &amp; \mathfrak{J}(S) \\ \end{bmatrix}$$  I know how to classify the left/right/two-sided ideals of this ring (T.Y. Lam, Chapter 1, Page 18) but I don't know how to find the maximal ones!  I'd like to see a proof of my conjecture if it's indeed true.  EDIT: I believe that this is not a duplicate of the linked question because first of all, this question is more general. Secondly, it requests the proof for something that is worth having its own separate question and thirdly, there's no proof in that linked question.
Let $A$ be the $\mathbb{C}$-subalgebra of $M_4(\mathbb C)$ consisting of matrices of the form   \begin{pmatrix} * &amp; * &amp; * &amp; *\\ * &amp; * &amp; * &amp;*\\ 0 &amp; 0 &amp; *&amp;0 \\ 0 &amp; 0&amp;* &amp;* \end{pmatrix}   Then how would I find the Jacobson radical $J(A)$ of $A$?    I don't really understand finding the Jacobson radical of $\mathbb{C}$-subalgebras.  Any help on this would be greatly appreciated, thank you!
Is there a destructor for Java? Why is '.' a hard link in Unix? I have a atmega8 that is working with 3.6V dc, I need to control some 12V relays with a not chip (7404), what is the simplest way to do this? I have connected a 12V dc to the relays, so when I connect the relays to the chip and then the atmega8, the current returns to the atmega through the not chip. Is this solution correct? There a better way to do this? I was writing a program in C++ to find all solutions of ab = c, where a, b and c together use all the digits 0-9 exactly once. The program looped over values of a and b, and it ran a digit-counting routine each time on a, b and ab to check if the digits condition was satisfied.  However, spurious solutions can be generated when ab overflows the integer limit. I ended up checking for this using code like:  unsigned long b, c, c_test; ... c_test=c*b;         // Possible overflow if (c_test/b != c) {/* There has been an overflow*/} else c=c_test;      // No overflow   Is there a better way of testing for overflow? I know that some chips have an internal flag that is set when overflow occurs, but I've never seen it accessed through C or C++.    Beware that signed int overflow is undefined behaviour in C and C++, and thus you have to detect it without actually causing it. For signed int overflow before addition, see . I am not sure of the meaning of this sentence:     The significance of culture and identity in development has to do not so much with the cultural factor in the process of development as with abandoning Eurocentric development thinking.   I'm not sure what the use of as is supposed to convey. Does it mean "development has to do not so much ... as with abandoning", i.e. not as much as abandoning? Does the abandoning have more to do with it?  And the second question: Why does it use "has to do not so much". Why not "has not so much to do"? Is "have to" a verb? I'm trying to insert a figure into my LaTeX file using the following syntax:  \begin{figure}   \centering   \includegraphics [width=3in]{fig1.png}   \caption{Calculation of shortest path betweenness}   \label{Fig.3:Shortest path betweenness} \end{figure}   It worked for me before but this time I'm getting the following error:   LaTeX Error: \begin{figure} on input line xxx ended by \end{document}   Also, hovering over the red cross symbol indicating error in Eclipse, I see this message    paragraph ended before Gin@iii was completed.   It's obvious that I have the \end{figure} to end my \begin{figure} so I don't understand how I still get that error.  Can someone please tell me what the problem is?  Edit: Add more details for debugging. The problem went away if I remove '[width=3in]'. Here is the detail error:  pdflatex&gt; Runaway argument? pdflatex&gt; width=3in]{fig1.png} \caption {Caption} \label {Fig.1}  pdflatex&gt; ! Paragraph ended before \Gin@iii was complete. pdflatex&gt; &lt;to be read again&gt;  pdflatex&gt;                    \par  pdflatex&gt; l.40  pdflatex&gt;        pdflatex&gt;  pdflatex&gt; Underfull \vbox (badness 10000) has occurred while \output is active pdflatex&gt; (./document.bbl pdflatex&gt;  pdflatex&gt; Package natbib Warning: Empty `thebibliography' environment on input line 3.   Added sample code:  \documentclass[10pt,psfig,letterpaper,twocolumn]{article} \usepackage{geometry} \usepackage{graphics} \usepackage{setspace} \usepackage{natbib} \usepackage{verbatim} \usepackage[scaled=0.9]{helvet} \singlespacing \paperwidth 8.5in \paperheight 11in \oddsidemargin 0in \headsep 1.3cm  \geometry{left=0.75in,top=0.75in,right=0.75in,bottom=1in} \textwidth 7in  \textheight 9.25in \columnsep 0.4in \footskip 0in  \renewcommand{\bibname}{REFERENCE} \pagestyle{empty}  \begin{document} \bibliographystyle{acm}   \title{\fontfamily{phv}\selectfont{\huge{\bfseries{Title here}}}} \author{ {\fontfamily{ptm}\selectfont{\large{\bfseries{author name}}}}\thanks{contact info }}  \maketitle \thispagestyle{empty}  \section*{\fontfamily{phv}\selectfont{\normalsize{\bfseries{Section name}}}}  \begin{figure} \centering \includegraphics[width=3in]{fig1.png} \caption{Caption} \label{Fig.1} \end{figure}  \pagebreak[4] \vspace*{5.52in} \bibliography{ref} \end{document} If you activated a free upgrade to Windows 10 or bought and activated   Windows 10 from the Microsoft Store, you have a digital license for   your device. This means you can reinstall the same edition of Windows   10 that your device has a digital license for without entering a   product key. During reinstallation, if you’re asked to enter a product   key, you can skip this step. Windows 10 will automatically activate   online after the installation is complete.   source:   Does MS digital license 'fingerprint' hardware information  during the free upgrade / activation and store it (at Microsoft) , so that the next re-install queries MS for the fingerprint and activates Windows 10?  Thought the disk drive may change, I expect that none of the other fingerprinted hardware will change. I am stuck with the identity  $$ F_{2n} = \sum_{k=1}^n \binom{n}{k} F_k, $$  which happens to be formula . I am using induction, but so far without too much result.  $$ \sum_{k=1}^{n+1} \binom{n+1}{k} F_k = \\ F_{n+1} + \sum_{k=1}^n \binom{n+1}{k} F_k = \\ F_{n+1} + \sum_{k=1}^n \binom{n}{k} F_k + \sum_{k=1}^n \binom{n}{k-1}F_k =\\ F_{n+1} + F_{2n} + \sum_{k=1}^n \binom{n}{k-1}F_k, $$   I am stuck with the sum $ \sum_{k=1}^n \binom{n}{k-1}F_k$, because, on the other hand   $$ F_{n+1}+ \sum_{k=1}^n \binom{n}{k-1}F_k = F_{2n+1} $$  Which would conclude the proof. What am I missing? jQuery: Uncheck other checkbox on one checked Visit to US before F1 using the Visa Waiver Program I am going to start a phd at the US with an F1 visa. University told me that I could enter the US only 30 days before the date on I-20. Since English is not my first language (and this would be a big change in my life) I would like to go there by June.  The thing is that I have an European passport with a valid ESTA. My plan would be to arrive with this passport, stay 4-6 weeks , go to Canada for a short vacation (maybe flight maybe by land) and come back to the USA with F1 visa.  I have two questions about this:  1) is this possible?   2) If yes, how could I justify knowing I will lose my come-back flight during my first entry? (I would purchase round trip because one-way are too expensive, probably with less than 90 days difference) I often find myself adding a lot of \textsl{} around inside \emph{} or \textit{} blocks because I find italicized digits to look too "showoffy" and "baroque" and to hinder legibility (I'm using $$ when the number has a mathematical sense, but this situation occurs in less technical texts like CVs, and such.).  Could someone help me write a renewcommand to automate this, or better, a single low level command that makes \emph, \itshape, \textit and friends choose slanted over italic for numbers? I'm not too familiar with the low level latex font selection system nor with a way to detect digits from regular letters.  Edit: MWE of what I need and what I want to avoid  \documentclass{article} \begin{document} He thought: '\emph{Why 1234567890 apples?}' then he realized '\emph{ah yes, he has \textsl{1234567890} childs}' \end{document}   gives:   I find the italicized version (first one) of 2,3,4 and 7 to look especially bad. Mail on Big Sur does not reliably find dollar values Slow leak on front cruiser tire I ride my cruiser to the supermarket and back, using either a trailer or a basket over the front wheel for carrying groceries. The bike has developed a persistent slow leak on the front wheel, and I've replaced the tube without it helping. I can't see anything puncturing the tire, and the spokes aren't poking through the rim tape that I can see.   These are 26" wheels with low pressure (around 40 - 50 psi). What else can I look for that might be causing this? TypeError: 'str' does not support the buffer interface My actual question is to find the number of transversal given a collection of set ...  After a little bit of study it has come down to:  How can we count the number of matchings in a bipartite graph with parts of size $m$ and $n$ such that it covers all $m$ vertices of the first part, $m \le n$?  I already know that there is a way to count the number of perfect matchings in a bipartite graph with $m=n$ using the permanent of its (square) incidence matrix. How to send variable to an inline shell-script? Local explanation of the Aharonov-Bohm effect in terms of force fields Here is an interesting paper for the Physics SE community:      On the role of potentials in the Aharonov-Bohm effect. Lev Vaidman. .  [quant-ph].   You should check it out because it's an amusing read, but I will summarise the argument to whet your appetite, assuming you have basic familiarity with the . In the traditional AB setup, one considers an electron in a superposition of paths, taking it in two opposite directions around a solenoid treated as a classical source of the electromagnetic field. The observable relative phase acquired between the electron's paths is attributed to the influence of the magnetic vector potential on the electron, which cannot be globally gauged away - despite the absence of a physical field anywhere along the electron's path(s) - due to a topological obstruction.  Instead, Vaidman considers the effect of the electron on the sources of the field, treating the latter as quantum particles. He shows that the relative phase between the two branches of the wavefunction can be considered as arising from the action of the physical field of the electron, which is not zero at the position of the sources. However, Vaidman uses highly contrived gedankenexperiments and completely semi-classical arguments, which begs a pair of concrete and related questions.  1) Can Vaidman's first, electric AB effect gedankenexperiment be described in a fully quantum manner, by solving (at least approximately) the three-particle Schroedinger equation? If not, why not?  2) Is it possible to explain within this formalism the experiments of Tonomura et al. (), who used a superconductor to completely shield the magnetic field of the source? How to factor a fourth degree polynomial I'm working on a math problem but I am having a hard time figuring out the method used by my textbook to make this factorization:  $$x^4 + 10x^3 + 39x^2 + 70x + 50 = (x^2 + 4x + 5)(x^2 + 6x + 10)$$  I've tried to see if this equation can be factored by grouping or by long division to no avail. Any help would be greatly appreciated. Now, that Adobe has stopped issuing new releases for Linux desktops, the only option for users is to use Google Chrome, for those who want to use the latest Flash updates. I use Chromium. Is there any way to get Chromium use the Flash from Google Chrome? The reason I as this question is that I have noticed both these browsers run in the background, so technically, the browsers' plugins are available to use. Why do video cables (DVI, VGA) still screw in?