I need to prove that the Lindenbaum-Tarski algebra is dense in the sense that: if     $ [\phi] \leqslant [\psi] $ then there is an $\chi $ such that  $ [\phi] \leqslant [\chi] $ and $[\chi] \leqslant [\psi]  $ I know this is only true if there are infinite  sentential letters. I started by writting that If $ \psi= p_0p_1..p_n \space \space  \psi=q_0q_1...q_m$   then $ \chi= \psi \land q $  for some propositional variable q  that does not appear  $ \psi $ satisfies that $ [\chi ] \leqslant [\psi]  $ that is, by definition,  $  \chi \models \psi $. But i still have to proof that  $  [\phi] \leqslant [\chi] $ I guess it has something to do with the initial  hypotesis   $ [\phi] \leqslant [\psi] $  but i can not find out how.
Suppose we define the following relationship of ordering between propositions in PROP: $$ \phi &lt; \psi := (\models \phi \rightarrow \psi \land \lnot(\models \psi \rightarrow \phi) ) $$ Show that if exists $\phi$ and $\psi$ such that: $$ \phi &lt; \psi $$ Then there is a $\sigma$ such that: $$ \phi &lt; \sigma &lt; \psi $$ I have tried to use valuation to find what $\sigma$ could possibly be, so for all valuations v:  If $v(\phi)=0$ and $v(\psi)=0$ then, since $\models \sigma \rightarrow \psi$, $v(\sigma)=0$  If $v(\phi)=1$ and $v(\psi)=1$ then, since $\models \phi \rightarrow \sigma$, $v(\sigma)=1$  It is impossible that $v(\phi)=1$ and $v(\psi)=0$ since $\models \phi \rightarrow \psi$  If $v(\phi)=1$ and $v(\psi)=0$ then $v(\sigma)$ could be either 0 or 1, and it would satisfy the tautologies, but if:  $v(\sigma)=0$, then the valuation of $\sigma$ would be equal to valuation of $\psi$ for all valuations and thus $\models \psi \rightarrow \sigma$, which is a contraditcion; $v(\sigma)=1$, then the valuation of $\sigma$ would be equal to valuation of $\phi$ for all valuations and thus $\models \sigma \rightarrow \phi$, which is a contraditcion.
Integral $\int^{ \pi /2}_{0} \ln (\sin x)\ dx$ $$\int^{ \pi /2}_{0} \ln (\sin x)\  dx$$  The answer is $- \frac{\pi}{2} \ln 2$.  I have changed it into $$\frac{1}{2} \int^{1}_{0} t d \ln t^{2}$$  But I didn't get the answer with it. Organic yet accurate modeling with the golden spiral Foreword  First off, let me say that I am profoundly amazed at the orderliness in nature. If you haven't heard about the Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 13, ...), you should check it out.  This mathematical concept of the Fibonacci numbers can be shaped into a set of squares, where the lengths of the sides of the squares follow the Fibonacci sequence:     This is sometimes called a golden ratio. It can be formed into a (golden) spiral. (a.k.a. logarithmic spiral):      Application  I am looking for ways on how to utilize this concept in Blender. Primarily in organic modeling. The golden spiral is found all throughout nature. From flower seed heads, sea shells, and ram horns to hurricanes and spiral galaxies.  My question is: how can I model based off of one or multiple golden spirals? Specifically, I'm interested in modeling a sunflower head like the one in the following image. I'm mostly interested in the point of origin of the individual seeds (not worried about the shape). Does Blender have any features related to this? Short, Politically Correct word for Native Americans No more than four syllables, more PC than Indians.  EDIT: I arbitrarily chose four syllables because any more seemed like a mouthful. I like to be PC and not have to stumble over 6+ syllables. I can't seem to get this hat on straight! I carefully positioned my hat to fit my head just right:     But it seems to be shifted to the left a few pixels elsewhere (same in Firefox and Chrome):   Stack Overfow   Skeptics   The top bar is also affected on some sites:   Stack Overfow   Meta SE   InfoSec   However, it is in the correct position in the top bar on some sites somehow:   Skeptics   Bricks   It's not a huge deal of course, it just bothers perfectionists like me. How do I use the Natural Earth SQLite DB with QGIS? I just downloaded the Natural Earth data in SQLite format from . I had assumed that this would be a SpatiaLite database, but it appears not to be! QGIS is failing to recognize it as a spatial database. OGR supposedly supports reading geometries even when stored in a plain SQLite database, but maybe QGIS doesn't use OGR for SpatiaLite?  The SQLite database does have geometry_columns and spatial_ref_sys tables. Is there some way to convert it to a full SpatiaLite database? Array versus List: When to use which? I get this error whenever I try to install programs using the terminal:  home@ubuntu:~$ apt-get install myunity E: Could not open lock file /var/lib/dpkg/lock - open (13: Permission denied) E: Unable to lock the administration directory (/var/lib/dpkg/), are you root?   Also I'm unable to install updates using the terminal. How do I get a gnome-terminal window to open from a crontab?  I have tried this:      * * * * * /usr/bin/gnome-terminal       * * * * * source /home/user_name/.bashrc ; /usr/bin/gnome-terminal      * * * * * source /home/user_name/.profile ; /usr/bin/gnome-terminal      * * * * * source /home/user_name/.bashrc ; /home/user_name/bin/opengnometerminal.sh   None of these seem to work. i install ubuntu and followed a guide to put 20gb in '/' and the rest (150gb) in '/home'. Now i just found out that a lot of the things installed in ubuntu usually goes to '/' instead of '/home'. I have a lot more things to install.  So what is the point of '/home' ? How do I fix this ? Should I transfer 150gb from '/home' to / ?  Filesystem      Size  Used Avail Use% Mounted on udev            7.8G     0  7.8G   0% /dev tmpfs           1.6G  9.5M  1.6G   1% /run /dev/sda6        19G   16G  1.8G  91% / tmpfs           7.8G  676K  7.8G   1% /dev/shm tmpfs           5.0M  4.0K  5.0M   1% /run/lock tmpfs           7.8G     0  7.8G   0% /sys/fs/cgroup /dev/sda2        96M   29M   68M  31% /boot/efi /dev/sda8       159G  1.1G  150G   1% /home tmpfs           1.6G   56K  1.6G   1% /run/user/1000   what is /boot ? Its not '/' or '/home' Fatal error occurred, no output PDF file produced in php file We are using following code in .php script for generate .pdf file.  $result1 = exec("R CMD Sweave /var/www/html/r/r_latex_sweave.Rnw");  $result2 = exec("pdflatex /var/www/html/r/r_latex_sweave.tex");   but, we are getting this errors when we are executing .Rnw and .tex files.  ! I can't write on file `r_latex_sweave.log'. Please type another transcript file name:  ! Emergency stop !  ==> Fatal error occurred, no output PDF file produced!  Note: We have .tex and .Rnw folder in /var/www/html folder? Purpose of hole in stove leading to oven I'm staying in a hotel and have access to a "electric coil stove" for one of the first times.  I accidentally turned on the oven light and saw that the light was shining up through only one of the stoves:    There's only a port from the oven into the top right stove, and note that the stove is not on.  Here's a view from within the oven:    Is there a hole here as a vent for the oven for some reason, or is this for the stove? I typically use a glass "flat-top stove" that doesn't have an oven underneath it. I've never seen a direct hole from a stove to the oven before, and have to wonder what the purpose is. What are valid values for the id attribute in HTML? Need help understanding "getbits()" method in Chapter 2 of K&R C In chapter 2, the section on bitwise operators (section 2.9), I'm having trouble understanding how one of the sample methods works.  Here's the method provided:  unsigned int getbits(unsigned int x, int p, int n) {     return (x &gt;&gt; (p + 1 - n)) &amp; ~(~0 &lt;&lt; n); }   The idea is that, for the given number x, it will return the n bits starting at position p, counting from the right (with the farthest right bit being position 0). Given the following main() method:  int main(void) {     int x = 0xF994, p = 4, n = 3;     int z = getbits(x, p, n);     printf("getbits(%u (%x), %d, %d) = %u (%X)\n", x, x, p, n, z, z);      return 0; }   The output is:     getbits(63892 (f994), 4, 3) = 5 (5)   I get portions of this, but am having trouble with the "big picture," mostly because of the bits (no pun intended) that I don't understand.  The part I'm specifically having issues with is the complements piece: ~(~0 &lt;&lt; n). I think I get the first part, dealing with x; it's this part (and then the mask) that I'm struggling with -- and how it all comes together to actually retrieve those bits. (Which I've verified it is doing, both with code and checking my results using calc.exe -- thank God it has a binary view!)  Any help? What are the alternatives to OS X's Spotlight? Boolean expression (grammar) parser in c++ I recently upgraded to 12.04 from 11.10 and since upgrading have been unable to log in to my user profile. The upgrade went okay and I can login to a guest session fine but whenever I try to login to my profile, after entering my password, I just return to the login screen.  I've changed my password in Root (passwd &lt;username&gt;) and can log in to tty1 with no issues, it's just in GUI I'm having problems.  I'm using a HP dv7 laptop, 32-bit Ubuntu install, Intel® Core™2 Duo CPU P7350 @ 2.00GHz × 2, Nvidia graphics.  Any help would be appreciated. I see questions/information online about how to get rid of corpses but I'm not sure what the consequences to having them lying around are.  I know that I can use a morgue to store them and that they will eventually be taken away in a hearse. However, I've always just left them lying around. This doesn't seem to have any obvious consequences.  Do they upset prisoners or staff (e.g. increasing the danger level beyond what death normally does)? Make them sick? Slow down pathing? Is this something that's not implemented yet or is leaving dead bodies all over the place meant to be completely fine?  The wiki's pages for  and  don't shed any light on this. All I'm finding is the how of getting rid of corpses/glitches related to it but not why it matters in the first place. All I've seen that's related is that apparently prisoners can loot guard corpses for keys, but there's no obvious consequences for prisoner corpses which are what I tend to end up with.  So why not just leave them in the corridors? PHP conditionals, brackets needed? I was just browsing a forum and someone asked about a PHP file they had found on the web.  It has several spots like this in the code:  if ($REMOTE_ADDR == "") $ip = "no ip"; else $ip = getHostByAddr($REMOTE_ADDR);  I have always thought brackets are needed to enclose what you want to do if the condition is true.  Is there some other alternative, such as if it is on the same line you don't?  There is also another line like this: if ($action != ""): mail("$adminaddress","Visitor Comment from YOUR SITE",  My instinct is to say this wouldn't work, but I also don't know if it is an outdated PHP file and it used to work? Better yet, are there any Spacewalk-like software packages that work for both RedHat-based and Debian-based systems? Calculate total sum of all numbers in c:forEach loop I have a Java bean like this:  class Person {   int age;   String name; }   I'd like to iterate over a collection of these beans in a JSP, showing each person in a HTML table row, and in the last row of the table I'd like to show the total of all the ages.  The code to generate the table rows will look something like this:  &lt;c:forEach var="person" items="${personList}"&gt;   &lt;tr&gt;&lt;td&gt;${person.name}&lt;td&gt;&lt;td&gt;${person.age}&lt;/td&gt;&lt;/tr&gt; &lt;/c:forEach&gt;   However, I'm struggling to find a way to calculate the age total that will be shown in the final row without resorting to scriptlet code, any suggestions?