Let $A$ be a symmetric, positive definite matrix.  Is is true that the det$ A$(which is the product of eigenvalues) is smaller or equal to the product of diagonal elements of $A$ ?  I could not prove it, and not even sure if the inequality above is true.
In the Wikipedia article on positive definite matrices they claim that if $M$ is positive definite, then the determinant of $M$ is bounded by the product of its diagonal entries. How might we show this?  Ideas: The sum of the diagonal entries of $M$ is the trace, which is also the sum of the eigenvalues. The product of the eigenvalues is the determinant. All the diagonal entries of $A$ must be real and strictly positive. Also we have $|m_{ij}|\leq \sqrt{m_{ii}m_{jj}}\leq \frac{m_{ii}+m_{jj}}{2}$. So far nothing is working...  We might try to write the Gram matrix $m_{ij} = \langle v_i, v_j \rangle$, and then $\prod_i m_{ii} = \prod_i \|v_i\|^2$. Now we'd need to show that the determinant, which is a polynomial in the entries of $v_i$, is less than $\prod_i \|v_i\|^2$ somehow.
Grandi's Series; tends to $1/2$, but why is this considered a valid sum? Landscape printing from HTML I have a HTML report, which needs to be printed landscape because of the many columns. It there a way to do this, without the user having to change the document settings?  And what are the options amongst browsers. From a collection of short stories I read at least 30 years ago. :/  This is going to be incredibly vague, so apologies in advance.  A human and alien (blob/octopus-like...I think) from warring races are captured by an unidentified higher intelligence and put in an enclosure (more like a zoo habitat rather than a prison cell). They are separated by an invisible barrier and can see each other.  I think they were both trying to escape or trying to figure out how they could kill the other first. There was a bulldozer/loader type vehicle they could use at one point to build a shelter (or something).  I believe they were crafting makeshift weapons as well. There was no communication between the two as far as I remember.  The main tension in the story seemed to be was the race to get to the other first.  This has been stuck in my head forever. I know its a very common theme.  The short stories may have included Asimov. There has been some recent behavior change on SE (around July 2015).  I have two accounts I use, one Google openId based for one set of sites (call it Account-A) and another SE login based for a different set of sites (Account-B).  Used to be that I could be logged into one set of sites with the first user and a different set with the other users.    But now I am logged in with the Account-A on Site-1.  When I go to one of the sites I use the other account for and click "log in" under the SE menu, I go to a page where I only have an option to confirm creation of an account based on my Google login.  It does not present a way to log in with an SE account.    How can I log in with my other account here?  (and before you flag this as a duplicate, please ensure the other question is recent enough and that the answer is relevant.  This behavior has definitely morphed over the years... and I am having trouble finding something that is relevant.  For instance this question  simply points out how to merge the accounts which is NOT what I want  .  This question  talks about presenting different user info on each site but that's not what I want -- I want totally separate accounts). Why do mirrorless cameras have quicker autofocus than SLRs using live view? How do i show that $w=z^2$ maps $|z-1|\leq 1$ onto the cardioid $R=2(1+\cos \theta)$? If $G$ is a compact topological group, how to show that a finite index subgroup of $G$ is open ? I really don't know where/how to start...  PS : I precise that by "compact" I mean that it is hausdorff and that any recovering of $G$ by open sets has a finite sub-recovering (sorry if I wasn't unclear) What are the different kinds of boxes in (La)TeX? I've seen hbox, vbox, mbox, parbox and others used in various code samples. What are the different kinds of box in (La)Tex? Which are suitable for use in LaTeX, as opposed to plain TeX? Ubuntu 20.04 on DualBoot fail 90% of tries - Initramfs unpacking failed: Decoding failed I just installed Ubuntun 20.04 on a Dell Latitude 7400 on DualBoot. I had to remove the RST as mentioned on this post :  It works but now I need to test 5 or more times to boot. After the grub screen, most of the time i have a black screen. Sometimes I have this error : Initramfs unpacking failed: Decoding failed And then it starts.  here is a file called initramfs.conf cat /etc/initramfs-tools/initramfs.conf MODULES=most  BUSYBOX=auto  COMPCACHE_SIZE=&quot;&quot;  COMPRESS=lz4   DEVICE=  NFSROOT=auto RUNSIZE=10% Could it be that Goldbach conjecture is undecidable? Maximum weight an average bicycle can carry? A few years ago, I've bought  (see edit) bike. At that time I was around 90 kgs and I didn't have any problems.  Today I'm around 120 kgs and I was considering (among other things) starting cycling again. Is this bike "good" enough to carry me? What should I expect? Any tips?  EDIT! Apparently, Drag have renamed some of their bikes in the last few years, so the above information is a bit wrong.  seems to be my bike. At first I didn't recognize much difference, but this one has suspension on the front which mine also has. Does this change anything Divisibility is transitive: $\ a\mid b\mid c\,\Rightarrow\ a\mid c$ As the title says, if a number is divisible by a number, is it always divisible by that number's factors?  An example being that $100$ is divisible by $20$, it is also divisible by $10, 5, 4, 2$ as well?  Does this always apply? What is a 'rollback'? What is a 'rollback' on Stack Exchange sites? I have seen this mentioned but I do not understand the term. Please note: This post is not a sandbox for performing rollbacks. Please do not perform rollbacks on this post without valid reason. If you want to try out a few rollbacks, do it  instead.   See also   in the Help Center $\sigma$-algebra vs. $\sigma$-field: is there any difference? The subject says it all: is there any difference between the two concepts of $\sigma$-algebra and $\sigma$-field? In probability theory, they seem to be used more or less interchangeably. If there is no difference, is there any historical reason why some people/schools use the term $\sigma$-algebra, while others use the term $\sigma-$field? Is there a "sentence stretch" that acts horizontally in a similar way as \setstretch acts vertically? On Latin.SE, I am getting a 404 error when clicking any of the three "history" links on the .  That is, this:        Gives this:        All three editable sections have a revision history with two revisions or more, and all three give an error.  The actual links are , , and . Explanation for "emails"? Enumerated list with square brackets I want to make an enumerated list in the following format:  [1] foo foo [2] foo foo ...    How I can do that using enumerated lists? I have tried the following:  \begin{enumerate}[\left[ 1 \right]]   but it does not work. Random quadratic equation I would like to produce quadratic equation with integer coefficients.  Here's my code using pgfmath.  \documentclass{article} \usepackage{tikz} \pgfmathsetseed{\pdfuniformdeviate 10000000}  \begin{document} $\pgfmathdeclarerandomlist{signe}{{+}{-}}% \pgfmathrandomitem{\s}{signe}% \pgfmathparse{random(2)}% \pgfmathparse{ifthenelse(\pgfmathresult==1,"-"," ")}\pgfmathresult \pgfmathrandominteger{\a}{1}{6} \pgfmathparse{ifthenelse(\a==1," x^{2} ","\a x^{2}")}\pgfmathresult % coeff a \pgfmathdeclarerandomlist{lincoeff}{{}{2}{3}{4}{5}{6}} \pgfmathrandomitem{\b}{lincoeff}% \pgfmathrandomitem{\s'}{signe}% \pgfmathsetmacro{\e}{random(0,6)} \pgfmathparse{ifthenelse(\e==0," ", " \s' \b x ")}\pgfmathresult % coeff b \pgfmathrandominteger{\c}{0}{6} \pgfmathrandomitem{\s''}{signe}% \pgfmathparse{ifthenelse(\c==0,"","\s'' \c")}\pgfmathresult % coeff c =0$ \end{document}   It works but it seems to be not a good way, I had some pain with the x-coeficient and my solution looks complicated. I tried to use an ifthenelse inside an other ifthenelse but without success. Is there a better way ? Recover scaling factor in TikZ Does it exist some tikz register (let say \zoomfactor) that provides the curent scale factor. For example can I do something like :   \begin{tikzpicture}[scale=2]     \begin{scope}[scale=1.5]         \draw (0,0) circle (\zoomfactor);     \end{scope} \end{tikzpicture}   and to obtain circle of size 3=2*1.5 ?