Applications of Baire's Threom
Your favourite application of the Baire Category Theorem
Does the shape of a containter affect the apparent mass of liquid inside it when measured by a scale? For both containers it is true that:  $p_{bottom}=p_{atm}+\rho gh$ and $F_{bottom}=a*p_{bottom} \iff F_{bottom}=a*p_{atm} + a\rho gh$  I do not know how a scale functions, but I guess that the only force it can count is the force by the liquid on the bottom of each container (the only surface that is in touch with the scale is the bottom of the container). Thus, if we measured the mass of these two masses of water (lets assume that the mass of the containers is negligible) it would appear to be the same in each case. Am I right? Bridged adapter in Virtual Box - cannot obtain IP I have virtual box with RHEL installed. My virtual box is running on windows 7 64 -bit. I configured bridged connection, but when I am restarting network interfaces, I receive following error:  Determining IP information for eth1... failed   On Windows XP it works without problems. I disabled Windows Firewall and installed latest Virtual Box, still without success.   NAT is working without any problems. What did I missed?  UPDATE  Ouput of ifconfig -a is:  eth0      Link encap:Ethernet  HWaddr 08:00:27:B6:FC:69           inet addr:10.0.2.15  Bcast:10.0.2.255  Mask:255.255.255.0           inet6 addr: fe80::a00:27ff:feb6:fc69/64 Scope:Link           UP BROADCAST RUNNING MULTICAST  MTU:1500  Metric:1           RX packets:430 errors:0 dropped:0 overruns:0 frame:0           TX packets:346 errors:0 dropped:0 overruns:0 carrier:0           collisions:0 txqueuelen:1000           RX bytes:36520 (35.6 KiB)  TX bytes:31455 (30.7 KiB)  eth1      Link encap:Ethernet  HWaddr 08:00:27:C3:84:4A           inet6 addr: fe80::a00:27ff:fec3:844a/64 Scope:Link           UP BROADCAST RUNNING MULTICAST  MTU:1500  Metric:1           RX packets:1464 errors:0 dropped:0 overruns:0 frame:0           TX packets:17 errors:0 dropped:0 overruns:0 carrier:0           collisions:0 txqueuelen:1000           RX bytes:135419 (132.2 KiB)  TX bytes:4230 (4.1 KiB)  lo        Link encap:Local Loopback           inet addr:127.0.0.1  Mask:255.0.0.0           inet6 addr: ::1/128 Scope:Host           UP LOOPBACK RUNNING  MTU:16436  Metric:1           RX packets:0 errors:0 dropped:0 overruns:0 frame:0           TX packets:0 errors:0 dropped:0 overruns:0 carrier:0           collisions:0 txqueuelen:0           RX bytes:0 (0.0 b)  TX bytes:0 (0.0 b)   Output of route -n  Kernel IP routing table Destination     Gateway         Genmask         Flags Metric Ref    Use Iface 10.0.2.0        0.0.0.0         255.255.255.0   U     0      0        0 eth0 169.254.0.0     0.0.0.0         255.255.0.0     U     1002   0        0 eth0 0.0.0.0         10.0.2.2        0.0.0.0         UG    0      0        0 eth0   Output of dhclient -v eth1 is:  Listening on LPF/eth1/08:00:27:c3:84:4a Sending on   LPF/eth1/08:00:27:c3:84:4a Sending on   Socket/fallback DHCPDISCOVER on eth1 to 255.255.255.255 port 67 interval 6 (xid=0x6afb6c4c) DHCPDISCOVER on eth1 to 255.255.255.255 port 67 interval 9 (xid=0x6afb6c4c) DHCPDISCOVER on eth1 to 255.255.255.255 port 67 interval 15 (xid=0x6afb6c4c) DHCPDISCOVER on eth1 to 255.255.255.255 port 67 interval 18 (xid=0x6afb6c4c) DHCPDISCOVER on eth1 to 255.255.255.255 port 67 interval 13 (xid=0x6afb6c4c) No DHCPOFFERS received. No working leases in persistent database - sleeping.   Moreover the same machine with the same configuration works well on Windows XP. The problem is only on Windows 7. Maximum table size for a MySQL database What is the maximum size for a MySQL table? Is it 2 million at 50GB? 5 million at 80GB?  At the higher end of the size scale, do I need to think about compressing the data? Or perhaps splitting the table if it grew too big? I have a div 200 x 200 px. I want to place a 50 x 50 px image right in the middle of the div.   How can it be done?  I am able to get it centered horizontally by using text-align: center for the div. But vertical alignment is the issue.. Can people PLEASE spell out their slang acronyms Is it normal for focal length to change while focusing? When I change the focus distance on my Pentax 55-300mm lens, the focal length appears to change.  More specifically, the lens appears to zoom in as I focus towards the minimum focusing distance.  This is most noticeable at the 55mm end of the lens.  Is this normal? Homomorphism between compact Hausdorff spaces. Student missing exam for “medical reasons” Can I use 'in' instead of 'at'? -- "I have more books in my house." I am learning English using the Duolingo English course, one of correct solution was      I have more books at my house.   Can I use      I have more books in my house.   If it is OK, what is the difference between both? How to make a jquery click event fire only on first click What's a concise way to check that environment variables are set in a Unix shell script? How can I tell how long (more or less) it will take me to complete a game? Possible Duplicate:          I often want to know how long a game is so that I can decide whether or not to buy it, or even to decide if I've got enough free time available in the near-future to make it worth trying to play and complete over a weekend. Sometimes I even want to know just because I'm getting a little tired of the game and I want to have a sense of whether or not I'm nearing the end, so I can decide whether to tough it out or pack it in.  What are some good ways of determining how long a given game takes to complete, on average? $f$ is irreducible $\iff$ $G$ act transitively on the roots Let $F$ be a field and $f\in F[X]$ a separable polynomial. Let $K_f$ be the splitting field of $f$ and $G_f=\text{Gal}(K_f/F)$ its Galois group. Show that $f$ is irreducible $\iff$ $G_f$ acts transitively on the roots on $f$. My work $\implies :$ Let $f\in F[x]$ irreducible, $\alpha \in K_f$ a root and $\alpha _1,...,\alpha _n$ the other roots of $f$ (so $K_f=F(\alpha ,\alpha _1,...,\alpha _n)$). What I have to show (I think) is that for all $i$ there is a $\sigma \in G_f$ s.t. $\sigma \alpha =\alpha _i.$ Q1) Is it right ? Q2) Is $f$ the minimal polynomial of $\alpha $ ? (To me it is, but my teacher said that it can be not the case, but I don't understand why). Since $\alpha $ and $\alpha _1$ have the same minimal polynomial, there is an isomorphism $\sigma :\alpha \longmapsto \alpha _1$. We can observe that $\sigma : F(\alpha )\longrightarrow K_f$. Now $F(\alpha ,\alpha _1)/F(\alpha )$ is algebraic, then we can extend $\sigma $ to a morphism $F(\alpha ,\alpha _1)\longrightarrow K_f$. Now, we can continue like that and extend $\sigma $ as $\psi: K_f\longrightarrow K_f$ s.t. $\psi(\alpha )=\alpha _1$. Q3) The fact that $\psi(\alpha )=\alpha _1$ comes from the fact that $\psi|_{F(\alpha )}=\sigma $, right ? The bijectivity is obvious since it's a field homomorphism and that $\sigma (K_f)=K_f$. Q4) How to conclude that $G_f$ act transitively on the roots ? I don't have particular problem for the converse. Convert .sdf database to .mdf database Difference between force.com and salesforce.com? What the difference between the two?  This description seems to be a bit out of date and underspecified. As the title says, I've heard two possible meanings for turning down the air conditioning: It could mean set the target temperature lower (i.e. colder) or make it work less (i.e. warmer).  Turning down the heat makes it colder, but both meanings are in agreement there. How to get Steam to recognize existing game files from a different source? Let's say I have a game bought on retail/other digital download and installed on my hard drive. I buy the same game (or a minor variant of it, say the GOTY edition) on Steam. It should be possible to use the existing installation to avoid the generally big downloads. This is very useful as it saves both bandwidth and time.   Previously, as far as I understood, Steam downloaded the game files directly to steam\steamapps\common\&lt;game_name&gt;. Since this folder would be created on starting the download, pausing and closing steam, copying the game files to this folder and then validating game files would make Steam absorb as much as possible and just download the missing files.  Currently, Steam downloads the files to steam\steamapps\downloading\&lt;game_id&gt; and then moves them over to the aforementioned folder once the download finishes. So first of all, what the final folder would be is not clear. So what would be the correct way of doing the same thing, that is, make steam use as much of the existing files as possible? Should the files be copied to downloading\*\, or should I figure out the actual folder (say via googling) and copy the files to steamapps\common\*\?  More specifically, I had a retail copy of Arkham City installed and I just bought the GOTY version on Steam. Now I tried both of the above things, in either case, the validating does not seem to do anything. Assuming I have the correct folder name steamapps\common\Batman Arkham City GOTY, according to , if I copy the files there, Steam seems to just continue with its 17GB download. If I copy it to steamapps\downloading\200260 then the download keeps stopping every few seconds with "Disk Write Error" after modifying 1 or 2 files, and I have to keep hitting resume. HSA - Continue to Invest? I am trying to create a map in both Leaflet and OpenLayers 3 which displays the  layer EMAP8 from the WMS http://wms.nlsc.gov.tw/wms. I can't get it to work. I can open the layer in QGIS, but when I try to use it in Leaflet like this:  var overlay_taiwanemapen0 = L.tileLayer.wms('http://wms.nlsc.gov.tw/wms', {     layers: 'EMAP8',     format: 'image/png',     transparent: true,     continuousWorld : true, });   or in OpenLayers 3 like this:  var lyr_taiwanemapen = new ol.layer.Tile({     source: new ol.source.TileWMS(({         url: "http://wms.nlsc.gov.tw/wms",         params: {"LAYERS": "EMAP8", "TILED": "true"},     })),     title: "taiwane-map(en)", });   I get nothing displayed. I can see layers being requested, and data being transferred, but nothing on the map.  What am I doing wrong? Finding the limit of $\left(\frac{n}{n+1}\right)^n$ Find the limit of: $$\lim_{n\to\infty}\left(\frac{n}{n+1}\right)^n$$   I'm pretty sure it goes to zero since $(n+1)^n &gt; n^n$ but when I input large numbers it goes to $0.36$.  Also, when factoring: $$n^{1/n}\left(\frac{1}{1+\frac1n}\right)^n$$ it looks like it goes to $1$.   So how can I find this limit?