Surely there can be no real collision resistant Hash function since hash values are of finite size which means there are finitely many of them, but there are an infinite number of messages we could hash, which means (using the pigeon-hole principle) that some messages must lead to the same hash value.
What is "preimage resistance", and how can the lack thereof be exploited? How is this different from collision resistance, and are there any known preimage attacks that would be considered feasible?
Prove that $$\nabla_{\mathrm X} \mbox{tr} (\mathrm A \mathrm X^{-1} \mathrm B) = - \mathrm X^{-\top} \mathrm A^\top \mathrm B^\top \mathrm X^{-\top}$$   My proof is below. I am interested in other proofs.    My proof  Let   $$f (\mathrm X) := \mbox{tr} (\mathrm A \mathrm X^{-1} \mathrm B)$$  Hence,  $$\begin{array}{rl} f (\mathrm X + h \mathrm V) &amp;= \mbox{tr} (\mathrm A (\mathrm X + h \mathrm V)^{-1} \mathrm B)\\ &amp;= \mbox{tr} (\mathrm A (\mathrm X ( \mathrm I + h \mathrm X^{-1} \mathrm V))^{-1} \mathrm B)\\ &amp;= \mbox{tr} (\mathrm A ( \mathrm I + h \mathrm X^{-1} \mathrm V)^{-1} \mathrm X^{-1} \mathrm B)\\ &amp;= \mbox{tr} (\mathrm A ( \mathrm I - h \mathrm X^{-1} \mathrm V + O (h^2)) \mathrm X^{-1} \mathrm B)\\ &amp;= \mbox{tr} (\mathrm A \mathrm X^{-1} \mathrm B) - h \, \mbox{tr} ( \mathrm A \mathrm X^{-1} \mathrm V \mathrm X^{-1} \mathrm B) + O (h^2)\\ &amp;= f (\mathrm X) - h \, \mbox{tr} ( \mathrm X^{-1} \mathrm B \mathrm A \mathrm X^{-1} \mathrm V ) + O (h^2)\end{array}$$  Thus, the directional derivative of $f$ in the direction of $\mathrm V$ at $\mathrm X$ is  $$D_{\mathrm V} f (\mathrm X) = - \mbox{tr} ( \mathrm X^{-1} \mathrm B \mathrm A \mathrm X^{-1} \mathrm V ) = - \mbox{tr} ( (\mathrm X^{-\top} \mathrm A^\top \mathrm B^\top \mathrm X^{-\top})^\top \mathrm V ) = - \langle \mathrm X^{-\top} \mathrm A^\top \mathrm B^\top \mathrm X^{-\top}, \mathrm V \rangle$$  and, lastly,  $$\nabla_{\mathrm X} \mbox{tr} (\mathrm A \mathrm X^{-1} \mathrm B) = - \mathrm X^{-\top} \mathrm A^\top \mathrm B^\top \mathrm X^{-\top}$$ Universal extractors With an  of archive/compression file formats, is there a single free/open-source command line tool to rule them all? Perhaps something with a consistent / unified set of flags? (note my  friendly )  I once run into a set of aliases meant to largely simplify the task of compressing/de-compressing files with bindings to tar and other utils, but I can't find this anymore.  Update: How can I configure something like atool to not use unzip to extract zip files (which apparently can't handle files larger than 4GB) and to use gunzip instead?  $ aunpack large_file.zip error:  Zip file too big (greater than 4294959102 bytes) Archive:  large_file.zip warning [large_file.zip]:  1491344848 extra bytes at beginning or within zipfile   (attempting to process anyway) error [large_file.zip]:  start of central directory not found;   zipfile corrupt.   (please check that you have transferred or created the zipfile in the   appropriate BINARY mode and that you have compiled UnZip properly) aunpack: unzip ...: non-zero return-code Copy files from a list to a folder How to stop Mathematica 12.1 from chopping off the axes arrows? Solve for $A,B$: $\mathrm{LCM}(A,B)=168$, $\mathrm{HCF(A,B)}=12$ Is it possible to know the (public) IP address of a computer where TeamViewer is ready for a remote control session - without connecting to the computer? In what situations is it more appropriate to use an HTML IMG tag to display an image, as opposed to a CSS background-image, and vice-versa?  Factors may include accessibility, browser support, dynamic content, or any kind of technical limits or usability principles. How do you get the footer to stay at the bottom of a Web page? I have a simple 2-column layout with a footer that clears both the right and left div in my markup.  My problem is that I can't get the footer to stay at the bottom of the page in all browsers.  It works if the content pushes the footer down, but that's not always the case. The following MWE compiles (for me)  \documentclass{article}  \usepackage{pgfplots} \pgfplotsset{compat=newest} \usetikzlibrary{external} \tikzexternalize[prefix=figures/externalized/]   \begin{document} \begin{figure}     \centering     \begin{tikzpicture}          \begin{axis}  %           \addplot3[surf,mesh/rows=129,mesh/cols=129,mesh/ordering=y varies] table[x=X,y=Y,z=psi] {figures/data/LidDrivenCavity/Lid_SPIKE_128x128_Re1000_psi.dat};              \addplot3[contour gnuplot,mesh/rows=129,mesh/cols=129,mesh/ordering=y varies] table[x=X,y=Y,z=psi] {figures/data/LidDrivenCavity/Lid_SPIKE_128x128_Re1000_psi.dat};          \end{axis}      \end{tikzpicture}     \caption{ciao} \end{figure} \end{document}   where the data file is . If I de-comment the first \addplot3 and comment the second one, then I get an error. The final part of the log file (the interesting one, I think) is  ===== 'mode=convert with system call': Invoking 'pdflatex -shell-escape -halt-o n-error -interaction=batchmode -jobname "figures/externalized/prova-figure0" "\ def\tikzexternalrealjob{prova}\input{prova}"' ======== \openout3 = `prova.auxlock'.  runsystem(pdflatex -shell-escape -halt-on-error -interaction=batchmode -jobname  "figures/externalized/prova-figure0" "\def\tikzexternalrealjob{prova}\input{pr ova}")...executed.  \openout3 = `prova.auxlock'.   ./prova.tex:22: Package tikz Error: Sorry, the system call 'pdflatex -shell-esc ape -halt-on-error -interaction=batchmode -jobname "figures/externalized/prova- figure0" "\def\tikzexternalrealjob{prova}\input{prova}"' did NOT result in a us able output file 'figures/externalized/prova-figure0' (expected one of .pdf:.jp g:.jpeg:.png:). Please verify that you have enabled system calls. For pdflatex,  this is 'pdflatex -shell-escape'. Sometimes it is also named 'write 18' or som ething like that. Or maybe the command simply failed? Error messages can be fou nd in 'figures/externalized/prova-figure0.log'. If you continue now, I'll try t o typeset the picture.  See the tikz package documentation for explanation. Type  H &lt;return&gt;  for immediate help.  ...                                                l.22    \end{tikzpicture}  ?  PGFPlots: reading {figures/data/LidDrivenCavity/Lid_SPIKE_128x128_Re1000_psi.da t} Runaway argument? {\pgfkeyssetvalue {/pgfplots/execute at begin plot visualization}{}\pgfkeyssetv alue \ETC. ./prova.tex:22: TeX capacity exceeded, sorry [main memory size=5000000]. &lt;argument&gt; ...5e-1],2Y7.753628e-3]}{2Y8.20894\ETC.  l.22    \end{tikzpicture}  If you really absolutely need more capacity, you can ask a wizard to enlarge me.   Here is how much of TeX's memory you used:  20761 strings out of 493054  547383 string characters out of 6134761  5000001 words of memory out of 5000000  23958 multiletter control sequences out of 15000+600000  3640 words of font info for 14 fonts, out of 8000000 for 9000  1141 hyphenation exceptions out of 8191  62i,4n,84p,728b,1519s stack positions out of 5000i,500n,10000p,200000b,80000s ./prova.tex:22:  ==&gt; Fatal error occurred, no output PDF file produced!   similar (or the same) error appears if externalization is not enabled/used.  My TeX distribution is TeX Live on Mac OS X 10.9.5 and I use TeXShop as editor.  Once I (partially) solved a  on Windows+MikTeX+TeXniccenter by modifying/adding options to the command executed when compile button is pressed. I say partially since I actually solved the problem for some data file. 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Then prove that  $\displaystyle \frac{\partial x}{\partial y} \frac{\partial y}{\partial z} \frac{\partial z}{\partial x} = -1$  Verify this is the case for the ideal gas equation $pv =RT$ where (where $p,v,T$ are the three variables and $R$ is the constant). Cool Text Highlighting in LaTeX How to get Drupal 7 installed in a sub-folder / sub-directory properly? Currently i'm trying to installing a drupal 7 installation in a subfolder of another drupal public_html folder, but it doesnt seem to work. I am getting a redirect loop error in chrome. I already modified the .htaccess in the parent drupal installation to ignore the sub-folder and i also set the $base_url in the sub-directory drupal installation. I accidentally killed the orphanage lady before I started the quest. She just pissed me off, sadly it will not acknowledge that I have finished the quest and I haven't been contacted. Is there another way to join the Dark Brotherhood? How to force the browser to reload cached CSS and JavaScript files I have noticed that some browsers (in particular, Firefox and ) are very zealous in using cached copies of .css and .js files, even between browser sessions. This leads to a problem when you update one of these files, but the user's browser keeps on using the cached copy. What is the most elegant way of forcing the user's browser to reload the file when it has changed? Ideally, the solution would not force the browser to reload the file on every visit to the page.  I have found  and  suggestion to be useful. It turns out there is a term for this: auto-versioning. I have posted a new answer below which is a combination of my original solution and John's suggestion. Another idea that was suggested by SCdF would be to append a bogus query string to the file. (Some Python code, to automatically use the timestamp as a bogus query string, was .) 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Grep: The asterisk (*) doesn't always work If I grep a document that contains the following:  ThisExampleString  ...for the expression This*String or *String, nothing is returned. However, This* returns the above line as expected.  Whether the expression is enclosed in quotes makes no difference.  I thought the asterisk indicated any number of unknown characters? Why does it only work if it's at the start of the expression? If this is intended behavior, what do I use instead of the expressions This*String and *String?