I know what trace is in matrix algebra, but does it have any intuitive physical significance when acted on a quantum mechanical operator? What aspect of an operator $A$ does ${\rm Tr}(A)$ capture? Or is it just a convenient mathematical tool and nothing more?
I was recently thinking what information do we get from a matrix. So if we say the columns (or rows) of a matrix define the basis of a system, say vectors of 3 dimensional space. Then the determinant will tell about the volume of the space enclosed by those vectors. Does trace give any information about the space?
Is the total variation function uniform continuous or continuous? How to work the BibTeX? I have searched the tutorials but have not been successful in understand them. The tutorials always mensionam concepts that I do not know. I would like to understand the basics of BibTeX.  By basics I mean:   How do I install BiBTeX? How to build a file in bib? For this question assume that I already have the bibtex format each reference provided by databases like MathSciNet. See observation 1 below. What I write commands in the file format. Tex to call call the format file bib? How to exploit the format of the bibliography that comes out in PDF? The commands are made in the preamble of the file. Arquifo bib or format. Tex? What must be the BiBTeX with biblatex? Is there any tutorial (in pdf or video) that teaches step by step how to operate BiBTeX?   Observation 1 I already have the bibtex format each reference provided by databases like MathSciNet:   @article {MR2373353,  AUTHOR = {Avila, Artur and Lyubich, Mikhail},  TITLE = {Hausdorff dimension and conformal measures of {F}eigenbaum {J}ulia sets},  JOURNAL = {J. Amer. Math. Soc.},  FJOURNAL = {Journal of the American Mathematical Society},  VOLUME = {21},  YEAR = {2008}, NUMBER = {2},  PAGES = {305--363},  ISSN = {0894-0347},  MRCLASS = {37F35 (37F10 37F25)},  MRNUMBER = {2373353 (2009i:37112)},  MRREVIEWER = {Feliks Przytycki},  DOI = {10.1090/S0894-0347-07-00583-8},  URL = {http://dx.doi.org/10.1090/S0894-0347-07-00583-8}, } Why is looping over find's output bad practice? Divisibility tests and remainder mod $9,11,99,101,999,1001\,$ (casting out nines and elevens) Now I have proved the following two problems:      (1) Prove that a number and the sum of its digits have the same remainder upon division by 9.    (2) Given an integer, consider the difference between the sum of the digits in odd positions (counting from the right) and those in even positions. Show that the difference has the same remainder as the number itself when divided by 11.    Then, I need help on the following problem:      Partition the digits of a number into groups of two and three, respectively, starting from the right. Treat these as two- and three-digit numbers, respectively, and proceed similarly as in the two problems above. What divisibility rules arise?    In fact, I did not quite get what this problem asks for and not clear about this process. Could anyone give me some hints on this, please?  Thanks in advance. How to apply multiple styles in WPF In WPF, how would I apply multiple styles to a FrameworkElement?  For instance, I have a control which already has a style.  I also have a separate style which I would like to add to it without blowing away the first one.  The styles have different TargetTypes, so I can't just extend one with the other. Running programs in the background from terminal How do I run a program in the background of a shell, with the ability to close the shell while leaving the program running? Lets say my UI is having problems or for some reason, I need to boot up a program from the terminal window, say, nm-applet:  nm-applet   When it's started, it occupies the foreground of the terminal window.  Is there any simple way to run the program in the background without needing to leave the terminal open or have it occupy the whole terminal?  On that note, I did find a way to run programs from the terminal and have it allow for other inputs, by appending an ampersand (&amp;) to the command as such:  nm-applet &amp;   But this isn't much use as any processes started in the terminal are killed once the terminal is closed. Flexible table with calculating column width fails starting from division by 10 Story where the protagonist is turned off from an automated system of professional attribution and instant learning Is there something like python's interactive REPL mode, but for Java? Show $$\lim_{h \to \ 0} \frac{f(x + 2h) - 2f(x+h) + f(x)}{h^{2}} = f''(x)$$   Proof:  By definition:    $$f'(x) = \lim_{h \to \ 0} \frac{f(x + h) - f(x)}{h}$$  Using this idea it would imply:   $$1)\ \ f''(x) = \lim_{h \to \ 0} \frac{f'(x + h) - f'(x)}{h}$$  As such it is required that I find an expression for $f'(x+h)$. This is where I'm not sure if the step I took is legitimate.  An expression for $f'(x + h)$ is:  $$f'(x+h) =  \lim_{h \to \ 0} \frac{f(x + 2h) - f(x + h)}{h}$$  Combining this with the definition of $f'(x)$ and inserting it into 1) you arrive at: $$\lim_{h \to \ 0} \frac{f(x + 2h) - 2f(x+h) + f(x)}{h^{2}} = f''(x)$$   As required.    Concern: I feel a discomfort with this solution. Even though "mechanically" it worked out, if I am taking the limit as $h \rightarrow 0$ that would mean $x + 2h$ and $x + h$ both go to $x$. But I am attempting to use the idea that $x +2h$ goes to $x + h$. Perhaps it is a notation idea that I need to communicate better, but I feel it is larger than just that. Is it possible to detect 32 bit vs 64 bit in a bash script? I am writing a bash script to deal with some installations in an automated way... I have the possibility of getting one such program in 32 or 64 bit binary... is it possible to detect the machine architecture from bash so I can select the correct binary?  This will be for Ubuntu machines. How do I go about tracking down IO usage in Linux and other *nix OS'? Occasionally, something on my system starts hammering the disk. I would like to log data that would let me know about I/O usage per process, then I could go back to it later and see what keeps doing this. How should I go about solving this problem? Bonus points for solving this for arbitrary block devices. Placement of the asterisk in pointer declarations I've recently decided that I just have to finally learn C/C++, and there is one thing I do not really understand about pointers or more precisely, their definition.  How about these examples:   int* test; int *test; int * test; int* test,test2; int *test,test2; int * test,test2;   Now, to my understanding, the first three cases are all doing the same: Test is not an int, but a pointer to one.  The second set of examples is a bit more tricky. In case 4, both test and test2 will be pointers to an int, whereas in case 5, only test is a pointer, whereas test2 is a "real" int. What about case 6? Same as case 5? When does the set enter set theory? I wonder about the foundations of set theory and my question can be stated in some related forms:   If we base Zermelo–Fraenkel set theory on first order logic, does that mean first order logic is not allowed to contain the notion of sets? The axioms of Zermelo–Fraenkel set theory seem to already expect the notion of a set to be defined. Is there are pre-definition of what we are dealing with? And where? In set theory, if a function is defined as a set using tuples, why or how does first order logic and the axioms of Zermelo–Fraenkel set theory contain parameter dependend properties $\psi(u_1,u_2,q,...)$, which basically are functions? I recently installed Ubuntu 12.10 and it requires a passphrase to boot up (I installed it with an encrypted file system).  Do I have to reinstall to change to a standard unencrypted file system? Output for Print Is there a way to output for Print. I want to save in a higher DPI than the default 72dpi say 300dpi that's a good quality for printed pieces. I tried downloading an add-on but don't know if I am installing it correctly it isn't showing up but the bar in python is green. I'm using the 2.69 version of Blender. If no such feature exists what size have any of you found best for print quality. Thank you Over the past couple of days there have been a few questions about overlapping proposals and questions that could belong on multiple sites:        What is the best practice before / after asking a question if you think your question might be better on another site?  or  If I think my question might belong on multiple sites as per the above posts? is exactly what I'm looking for. I want my description environment to produce this:  Label is long   text  lab2            blalballalaa label3          possible second rule label four      hello world   Someone offered this:  \newenvironment{mydescription}[1]                                                   {\begin{list}{}%    {\renewcommand\makelabel[1]{##1:\hfill}%    \settowidth\labelwidth{\makelabel{#1}}%    \setlength\leftmargin{\labelwidth}    \addtolength\leftmargin{\labelsep}}}   {\end{list}}   Which works great (the parameter is the longest label), except I think the enumitem should be able to do this. Unfortunately, the documentation for it is extremely confusing to me. Could someone show me how to achieve the same result with enumitem? I'm working in C, and I have to concatenate a few things.  Right now I have this:  message = strcat("TEXT ", var);  message2 = strcat(strcat("TEXT ", foo), strcat(" TEXT ", bar));   Now if you have experience in C I'm sure you realize that this gives you a segmentation fault when you try to run it. So how do I work around that? Multi-core vs Multi-CPU I am requesting a new server from our Server team. Which is better for SQL Server performance: 2 CPUs with 8 cores per CPU or 1 CPU with 16 cores?