I am learning measure theory and i know there are some reason for introducing measure theory. I just want to know the reason behind its invention.
Is the motivation behind the introduction of measure theory the Lebesgue integral? In order to evaluate such an integral we need the length of each of the horizontal strip of width $h$. I have a question here. Why do we need such a general concept of measure? Length is a kind of measure, but why did Lebesgue think of so general a concept of "measure" when he just needed the length to evaluate his integrals? Is there any other motivation behind this ?
I have 8 individual SQL Server 2008 R2 machines, each hosting 1 database. Each database has an identical table structure and schema, and entirely unique data.  I would like to establish a reporting server (may be 2008 or 2012), that consolidates the rows  from selected tables across the 8 source servers into a single instance of those tables on the reporting server. This is one-way replication (no changes will be made to the reporting server). I will need to replicate changes from the source databases with relatively low latency (say 20-30 seconds).  Furthermore, I'd like to find methods to achieve this with as little impact to the source servers as is practical.  3rd-party agents, triggers, or schema mods to those servers are difficult in my environment.  My questions:   What are promising architectures and technologies for  achieving this goal?  I've looked at SQL Server Merge Replication, but I am concerned about latency. Is this an appropriate technology for this goal? Are there many-to-one architectures for transactional replication?   Should I be looking at 1-to-1 replication into 8 databases on my reporting server, followed by some custom merge function (a 2-step replication)?   Thanks, John $Q_8$ is isomorphic to a subgroup of $S_8$, but not isomorphic to a subgroup of $S_n$ for $n\leq 7$. Question is to prove that :     $Q_8$ is isomorphic to a subgroup of $S_8$, but not isomorphic to a subgroup of $S_n$ for $n\leq 7$.   I see that $Q_8$ is isomorphic to subgroup of $S_8$ by left multiplication action.  Hint given was to prove that stabilizer of any point contains $\{\pm 1\}$.  To prove Cayley's theorem, stating any group is isomorphic to a subgroup of $S_n$ we take action of given group on a set $A$ having same cardinality.  with that motivation I want to check if there is a Isomorphism then there is a map from $G\times A \rightarrow A$. i.e., $G$ gives a permutation group $S_A$.  I tried in same manner. Suppose $Q_8$ is isomorphic to subgroup of $S_n$ with $n\leq 7$  then it should come from a group action of $Q_8$ on a set of cardinality atmost 7.   Suppose $Q_8$ acts on a set $A$ with possible cardinality at most 7.  $stab(a)=\{g\in Q_8 : g.a=g \forall a\in A \}$  $cl(a)=\{g.a : g\in Q_8\}$  I know number of elements in class of $a$ equals to index of stabilizer.  As $cl(a)=\{g.a : g\in Q_8\}\in A$ i.e., $cl(a)\subseteq A$ and as $|A|\leq 7$ i see that $|cl(a)|\leq 7$.  But, $|cl(a)|=|Q_8:stab(a)|$ for any element $a\in A$.  So, $|Q_8:stab(a)|=|cl(a)|\leq 7$ for all $a\in A$.   So, $stab(a)$ should be non trivial subgroup of $Q_8$ if not then $|Q_8:stab(a)|=8$  non trivial subgroup (proper) of $Q_8$ contains $\{\pm1\}$.  So, In the worst case, $\{\pm 1\}\subseteq stab(a)$ for all $a\in A$.  As $Ker(\eta)=\cap_{a\in A}stab(a)$ (where $\eta$ is the action of $Q_8$ on $A$)   we see that $\{\pm 1\}\subseteq Ket(\eta)$ which means that $Ker(\eta)$ is non trivial.  thus there is no isomorphism (coming from $\eta$) between $Q_8$ and any subgroup of $S_7$.  I would be thankful if someone can check whether my approach is correct or if there is any other simple possible way.  P.S : Usually what i do to see whether two groups are isomorphic or not is to check for cardinality, abelian property, no of elements with same order and so on. But I was having no idea when i fail in all these ways. With this Group actions i could see possibility for getting a precise conclusion on Isomorphisms.I would like to Thank Mr. Jyrki Lahtonen (a user of Math.SE) who made me to get used to Group actions.  P.S $2$: If any thing is wrong in my idea, it is entirely my fault, and if anything is correct in this whole credit should go to Mr. Jyrki Lahtonen In film reviews/essays about a book or film, does the title of the movie/book always have to be highlighted/put in italics? Err 80 appears when I turn on my Canon EOS 5D Mark II Recently when on holiday in Italy my 5DII stopped powering up. Turn the switch on and - nada. Pulling the battery and popping it back in brings the camera to life, however as soon as the camera goes in to sleep mode, or I turn the power off, the problem returns. Sometimes the camera shows Error 80 at this point:     Err 80      Shooting is not possible due to an error.    Turn the camera off and on again or re-install the battery.   I've seen various posts on the interwebs about pulling the clock battery and leaving the camera for a while - have tried this and also replaced the clock battery to no avail.   Has anyone else had this problem - did the body have to go back to Canon for repair or did you manage to fix it? Where can I find the source code of Ubuntu? I would like to know where to find the source code of Ubuntu. I'd like to see how far it is "open source". Why does the amount of energy transferred depend on distance rather than time? The change in energy of an object can be determined by the work equation, where work is the change in energy:  $$ W = F \cdot d $$  I conceptualize the transfer of energy as simply a series of small "packets" of energy being transferred at every Planck length. These small "packets" of energy add up to the total energy transferred (i.e. work). I'm not sure if this conceptualization is correct, so correct me if I am wrong.  However, it makes me wonder why the amount of energy transferred is dependent on distance and not time.  $$ m_1 = 10~kg \\ m_2 = 20~kg \\ W_1 = (10~N)\cdot(5~m) = 50~J \\ W_2 = (10~N)\cdot(5~m) = 50~J \\ W_1 = W_2 \\ t_1 \neq t_2 $$  If I apply a constant force on an object, why isn't the energy transferred at a constant rate with respect to time? The energy transfer rate varies dependent on how long it takes to cover the set distance.  In other words: why is the energy transferred consistent per unit of distance, and not per unit of time? I was trying to install Teamviewer, but I was getting a dependency error. I tried to install the required packages, but with no luck.  I'm getting this error:  Unpacking teamviewer (from teamviewer_linux_x64.deb) ... dpkg: dependency problems prevent configuration of teamviewer:  teamviewer depends on lib32asound2; however:   Package lib32asound2 is not installed.  teamviewer depends on lib32z1; however:   Package lib32z1 is not installed.  teamviewer depends on ia32-libs; however:   Package ia32-libs is not installed.  dpkg: error processing teamviewer (--install):  dependency problems - leaving unconfigured Errors were encountered while processing:  teamviewer   I tried   sudo apt-get -f install   Getting   Package ia32-libs is not available, but is referred to by another package. This may mean that the package is missing, has been obsoleted, or is only available from another source However the following packages replace it:   lib32z1 lib32ncurses5 lib32bz2-1.0  Package lib32asound2 is not available, but is referred to by another package. This may mean that the package is missing, has been obsoleted, or is only available from another source  E: Package 'lib32asound2' has no installation candidate E: Package 'ia32-libs' has no installation candidate   I can't even get to the   sudo dpkg -i teamviewer_linux_x64.deb   If I force installation  sudo dpkg --force-depends -i teamviewer_linux_x64.deb   Although it's "Setting up Temviewer" it gives me this:    How can I fix this problem? In the inbox of the app, there is no way to differentiate between comments and answers as is in the global inbox of the website.  Could this be added? Where is the icon used in this dialogue box? Where on my disk can I find the icon used in this dialogue box in OS X？    It may be Security.icns in OS X 10.7.3    I cannot find it in:   /System/Library/CoreServices/CoreTypes/Contents/Resources There are only LockedIcon.icns and UnlockedIcon.icns in this folder who looks like the icon in the dialogue box,the size of the icon in the dialogue box is 128x128px, but LockedIcon.icns and UnlockedIcon.icns is 32x32px.   lockedIcon.icns    So they are not the same icon.   /System/Library/CoreServices/Finder.app/Contents/Resources/ /System/Library/CoreServices/Problem Reporter.app/Contents/Resources /Applications/System Preferences.app/Contents/Resources /Applications/Utilities/Keychain Access.app/Contents/Resources /System/Library/CoreServices/Certificate Assistant.app/Contents/Resources/ Should black holes be considered two-dimensional or three-dimensional? I know enough amount of set theory and logic to study grad-level math. However, I don't know more advanced set theory and logic, such as the ones on Kunen's or Shoenfield's texts. Although it's good to learn such advanced topics, I may not have enough time to do so, since I will probably put my emphasis on algebra, analysis or geometry as well as mathematical physics.   Although how much set theory and logic I should learn is totally up to what sort of analysis, algebra or geometry I will study, I want to know how much of them typical competitive algebraists/analysts/geometers know. If you are one of them, how much do you know? When will data.SE get updated data? promised that data.SE would &quot;soon&quot; get live data:     I realize it's a long weekend, but given that data.SE is now over 2 months out of date, it would be nice if this was finally rolled out. Plural or singular verb for collective term I would like to get a USB token to secure my keys. My use case is protection of 3 GnuPG keys that I will be using 10 times per day at least. I plan to create a new key ring from scratch. Because ECC seems more future-oriented than RSA, this is what I chose to use. I'm wondering which USB token to choose as well as which curve.  On  it is said that many people think curves recommended by the NIST or defined by Brainpool have a doubtful origin; therefore, they recommend the non-standardized Bernstein’s Curve 25519. On  one poster says that    He is not aware of any profound criticism of Brainpool curves, and Bernstein’s Curve 25519 is hard to protect against side-channel attacks when implemented in embedded devices.   As a result, I'm a bit lost as to which key/curve to choose. Evaluate the limit of $\lim_{n\to\infty}\frac{1}{n^2}\left(\frac{2}{1}+\frac{9}{2}+\frac{64}{9}+\cdots+\frac{(n+1)^{n}}{n^{n-1}}\right)$ How to input Traditional Chinese in pdfLaTeX I'm trying to write Chinese using pdfLaTeX. But currently I can only write Simplified Chinese Characters, If I try writing Traditional Chinese then the text wont show up.  This is what I have:  \documentclass{article} \usepackage{CJKutf8} \begin{document} \begin{CJK*}{UTF8}{gbsn} 文章内容。 \clearpage\end{CJK*} \end{document}   What would I need to do to get Traditional Chinese? Making a (simple) directory tree Here's my source:  #!/bin/bash  echo "Running script to free general cached memory!" echo ""; echo "Script must be run as root!"; echo ""; echo "Clearing swap!"; swapoff -a &amp;&amp; swapon -a; echo ""; echo "Clear inodes and page file!"; echo 1 &gt; /proc/sys/vm/drop_caches; echo "";   It clears caches and stuff, and it echoes that it needs to be run as root in the terminal. I basically just want the script to cease running if it detects it's not being executed as root.   Example:  "Running script to free general cached memory!" "Warning: script must be run as root or with elevated privileges!" "Error: script not running as root or with sudo! Exiting..."   If run with elevated privileges, it just runs as normal. Any ideas? Thanks! Should I sign, watermark or otherwise label a framed print? And if so how? I have an opportunity to show some of my work in a small local coffee shop.  Are there any guidelines/practices to signing, labeling or watermarking a photo which is to be on display?  Or should a photo be signed, labelled or marked at all? Is signing the back of the photo and showing my name on a separate card below the frame itself the "better" way to do this?   I noticed many people on the web put a very noticeable watermark with their logo, copyright and name right on the photo.  Personally, I find this distracting as it often ruins the photograph itself often turning what could be art to kitsch. Painters most often sign the painting itself but this to me is not distracting because it is done with the painter's own hand and it often "works" with the painting. What is the purpose of using a FIFO vs a temporary file or a pipe? From APUE     FIFOs can be used to duplicate an output stream in a series of shell   commands. This prevents writing the data to an intermediate disk   ﬁle (similar to using pipes to avoid intermediate  disk  ﬁles).       But  whereas  pipes  can  be  used  only  for  linear  connections   between processes, a FIFO has a name, so it can be used for   nonlinear connections.      Consider  a  procedure that  needs  to  process  a  ﬁltered  input    stream  twice.   mkfifo fifo1 prog3 &lt; fifo1 &amp; prog1 &lt; infile | tee fifo1 | prog2       We create the FIFO and then start prog3 in the background, reading   from the FIFO. We then  start prog1 and  use tee to  send  its  input    to  both  the  FIFO  and prog2.    How does a FIFO "duplicate an output stream in a series of shell commands"? Isn't this done by tee instead of a FIFO? In the example, mkfifo fifo1 creates a file in the current directory, and fifo1 seems replaceable with a regular file . So what is the point of a FIFO "prevent writing the data to an intermediate disk file"? What do "linear connections" and "nonlinear connections" between processes mean? What does it mean that a FIFO can be used for nonlinear connections, while a pipe can be only used for linear connections between processes?   Thanks.