Empty intersection in chain rule for probability
Unary intersection of the empty set
How to use the keyboard to drag and drop? Seeking block (sub-district) level boundaries for India? I'm looking for block (sub-district) level shapefiles for India. The GADM database only has files for the 2300 Tehsils, whereas I'm looking for the about 5500 Census blocks (sub-districts) for any year possible. How to run a program on a remote pc via command line? How can I run a program on a remote Windows PC in our network without the need of killing an existing remote-desktop-connection? I only need a command line for running the program and getting its results. Thank you! I installed Oneiric as a guest OS in VirtualBox using the Mini CD. I installed the ubuntu-desktop package and everything seems to work fine - well, except for a font issue with the terminal:    As you can see, some of the letters overlap and it's difficult to read. I checked the settings and it the checkbox "Use the system fixed width font" is checked with "Monospace | 12" selected underneath it.    Edit: running the commands mentioned in the comments:  $ fc-match "Monospace" DejaVuSansMono.ttf: "DejaVu Sans Mono" "Book" $ gconftool --get /desktop/gnome/interface/monospace_font_name Monospace 10 $ debsums ttf-dejavu-core /usr/share/doc/ttf-dejavu-core/AUTHORS                                        OK /usr/share/doc/ttf-dejavu-core/BUGS                                           OK /usr/share/doc/ttf-dejavu-core/NEWS.gz                                        OK /usr/share/doc/ttf-dejavu-core/README                                         OK /usr/share/doc/ttf-dejavu-core/changelog.Debian.gz                            OK /usr/share/doc/ttf-dejavu-core/copyright                                      OK /usr/share/doc/ttf-dejavu-core/langcover.txt.gz                               OK /usr/share/doc/ttf-dejavu-core/status.txt.gz                                  OK /usr/share/doc/ttf-dejavu-core/unicover.txt.gz                                OK /usr/share/fonts/truetype/ttf-dejavu/DejaVuSans-Bold.ttf                      OK /usr/share/fonts/truetype/ttf-dejavu/DejaVuSans.ttf                           OK /usr/share/fonts/truetype/ttf-dejavu/DejaVuSansMono-Bold.ttf                  OK /usr/share/fonts/truetype/ttf-dejavu/DejaVuSansMono.ttf                       OK /usr/share/fonts/truetype/ttf-dejavu/DejaVuSerif-Bold.ttf                     OK /usr/share/fonts/truetype/ttf-dejavu/DejaVuSerif.ttf                          OK Choice of $q$ in Baby Rudin's Example 1.1 First, my apologies if this has already been asked/answered.  I wasn't able to find this question via search.  My question comes from Rudin's "Principles of Mathematical Analysis," or "Baby Rudin," Ch 1, Example 1.1 on p. 2.  In the second version of the proof, showing that sets A and B do not have greatest or lowest elements respectively, he presents a seemingly arbitrary assignment of a number $q$ that satisfies equations (3) and (4), plus other conditions needed to show that $q$ is the right number for the proof.  As an exercise, I tried to derive his choice of $q$ so that I may learn more about the problem.  If we write equations (3) as $q = p - (p^2 - 2)x$, we can write (4) as   $$ q^2 - 2 = (p^2 - 2)[1 - 2px + (p^2 - 2)x^2]. $$  Here, we need a rational $x &gt; 0$, chosen such that the expression in $[...]$ is positive.  Using the quadratic formula and the sign of $(p^2 - 2)$, it can be shown that we need   $$ x \in \left(0, \frac{1}{p + \sqrt{2}}\right) \mbox{ for } p \in A, $$  or, for $p \in B$, $x &lt; 1/\left(p + \sqrt{2}\right)$ or $x &gt; 1/\left(p - \sqrt{2}\right)$.  Notice that there are MANY solutions to these equations! The easiest to see, perhaps, is letting $x = 1/(p + n)$ for $n \geq 2$.  Notice that Rudin chooses $n = 2$ for his answer, but it checks out easily for other $n$.  The Question:  Why does Rudin choose $x = 1/(p + 2)$ specifically?  Is it just to make the expressions work out clearly algebraically?  Why doesn't he comment on his particular choice or the nature of the set of solutions that will work for the proof?  Is there a simpler derivation for the number $q$ that I am missing? Total combinations from number of unique combination? How can you find count of the total combinations possible if you have the count of unique combinations? Suppose I need to form a sum of $3$ using $\{1,2\}$ Unique combn. $(1+1+1)$, $(1+2)$ hence '$2$' unique but total is '$3$' i.e, $(1+1+1)$, $(1+2)$, $(2+1)$ I know the elements used to make the combination Note: Number of unique combinations (ie. 2 here) is known. How can I find the total number of combination given the unique combinations? If $a$ divides $bc$ and $\gcd(a,b) = d$ then $\frac a d$ divides c I'm trying to prove that if $a$ divides $bc$ and $\gcd(a,b) = d$ then $\frac a d$ divides c. I tried using Bezout identity but couldn't get anywhere. How can I install updates and shut down from the command line? I know that there is a simple way to shutdown the computer in command prompt, but is it possible to do system updates and then shutdown the computer in command prompt?   For example, when Vista needs to install updates, it will let me install them and shut down by clicking the Shut Down button (while relegating the normal shut down button into the menu). So is there a command to do this in command prompt? I am confused whether drift velocity depends upon area or not . I checked answers for this question but all of them were cancelling each other out . from the equation $$I = neAv$$ It is clear that drift velocity should depend upon area but at the same time don't you think that by increasing the area of cross section of a conductor should have resulted in increase in drift velocity please help me out with it in detailed answer of how should i go with all the possible cases and explanation ,thanks :) Shining a laser onto a mirror Theoretically, if I shined a laser at a mirror at an angle of 0 degrees so that the light was perfectly reflected back to the light source, then I should not be able to see the light because it is not reflected to my eyes.  | |  me | |============ laser | | |   However this is evidently not the case, as I am able to see the laser point on the mirror. Why is this the case? My initial assumption is that it is due to diffuse reflection on the imperfect mirror's surface, but I am not sure. What characters are allowed for the display name? I tried editing my display name and for some names it gives me this error:     Display Name can only contain a-z, 0-9, spaces, apostrophes or hyphens and must start with a letter or number   That is clearly not the case, because it will take the following names:   NullUserException -龍- NullUserException Ω    NullUserException ఠ_ఠ NullUserException ಠ_ಠ NullUserException இ_இ   But it won't let me do:   NullUserException  (╯°o°)╯ NullUserException ٩๏̯͡๏)۶ NullUserException ☜(ﾟヮﾟ☜) NullUserException อ้_อ้   So what's the regex used (assuming that's what's being used), so I can try out allowed names without actually changing it? Does an increasing sequence of reals converge if the difference of consecutive terms approaches zero? If $a_n$ is a sequence such that   $$a_1 \leq a_2 \leq a_3 \leq \dotsb$$   and has the property that $a_{n+1}-a_n \to 0$, then can we conclude that $a_n$ is convergent?   I know that without the condition that the sequence is increasing, this is not true, as we could consider the sequence given in  to a similar question that does not require the sequence to be increasing.  $$0, 1, \frac12, 0, \frac13, \frac23, 1, \frac34, \frac12, \frac14, 0, \frac15, \frac25, \frac35, \frac45, 1, \dotsc$$  This oscillates between $0$ and $1$, while the difference of consecutive terms approaches $0$ since the difference is always of the form $\pm\frac1m$ and $m$ increases the further we go in this sequence.    So how can we use the condition that $a_n$ is increasing to show that $a_n$ must converge?  Or is this still not sufficient? I'm cooking two different dishes in the oven at the same time that each require different temperatures. One is a roasted vegetable dish that says to cook at 475 for 35-40 minutes and the other is parmesan crusted pork chops at 350 for 40-45 minutes. What do you suggest I do? How to keep a constant baselineskip when using minipages (or \parboxes)? When a minipage is used to box a text paragraph, the first line of the minipaged text and the last line of the previous text paragraph are closer than two consecutive lines in a standard text paragraph; analogously, the last line of text inside the minipage and the first line of text after the minipage are also closer.  The following MWE illustrates this:  \documentclass{article} \usepackage{lipsum}  \newcommand\Text{Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Ut purus elit, vestibulum ut, placerat ac, adipiscing vitae, felis. Curabitur dictum gravida mauris. Nam arcu libero, nonummy eget, consectetuer id, vulputate a, magna.\textbullet}  \begin{document}  \Text  \noindent\begin{minipage}{\linewidth} \Text \end{minipage}  \Text  \end{document}   How can I guarantee a constant baseline-skip even if minipages (or \parboxes) are used? How to use Mathematica functions in Python programs? How to run a fresh version of AIX in a Virtual Machine with a Linux host? Find the derived subgroup of $A_4$ Find the derived subgroup of $A_4$.   Since it is $A_4$, for a permutation $\sigma$ to be in $A_4$, $\sigma$ must have a cycle structure of $2$ cycles. Therefore, $\sigma=(ab)(cd)$.  The commutator of such elements, would be obviously another permutation of $2$-cycle (of length $2$).  Does it mean that $A'_4=A_4$? I've been trying to figure out how to change the chart from scientific notation to just show the 5 digit numbers. I've tried scale and scale ticks with not success.  How can I change the y-axis and data values to show the decimals not in scientific notation?  \begin{tikzpicture} \centering \begin{axis}[     width=300pt,     height=250pt,     axis lines*=left, % Don't display the top and right lines     ybar, % Display it as a bargraph     enlarge y limits={upper,value=0.2},     legend style={at={(0.5,-0.20)}, % -.20 y-axis     anchor=north,legend columns=-1},     ylabel={Densities},     nodes near coords,     every node near coord/.append style={ anchor=mid west, rotate=90},     symbolic x coords={Class1,Class3,Class5,Class7,Class9,Class11,Class13,Class15,Class17,Class19,Class21,Class23},     xtick=data,     xticklabel style={ inner sep=0pt, anchor=north east, rotate=70 },     nodes near coords align={vertical}, ] \addplot coordinates { (Class1,0.0968) (Class3,0.0887) (Class5,0.1452) (Class7,0.0726)     (Class9,0.0403) (Class11,0.0081) (Class13,0.0161) (Class15,0.0000)     (Class17,0.0161) (Class19,0.0081) (Class21,0.0000) (Class23,0.0081) }; \end{axis} \end{tikzpicture} Can we have a notification when the answer to any of my question got edited? If a user asked a question and got an answer which was not helpful.  After sometime the answerer edits the answer with some good content. How the OP will get to know that?  Is it necessary to add comment check my edited answer?  We can say that the OP should refresh the page to find the helpful answer.  Consider if the OP finds the answer as helpful and accepted that. After sometime if the answerer/anyother user edit the answer to change the content which may not solve the issue.   How will it be notified to OP?  currently we are notifying the OP when there is any comment to the queation or any new answers or any comments with the name of OP with @ to any answer.  This will lead the future visitors to confuse and reduce the quality of post.  Can we have a notification to the original poster when the answer to any of his/her question got edited? Subgroups of finite abelian groups. For every subgroup $H$ of a finite abelian group $G,$ there exists a subgroup $N$ of $G$ such that $G/N \cong H.$ I need to prove this or give a counter example.  I am aware of isomorphism theorems and classification of abelian groups, direct products etc.