Re: Bbc Compacta Answer Key Class 8
Nichelle Gruger
I would like to study/understand the (complete) classification of compact lie groups. I know there are a lot of books on this subject, but I'd like to hear what's the best route I can follow (in your opinion, obviously), since there are a lot of different ideas involved.
For references, I'd check out Fulton and Harris' book "Representation Theory". I'm not sure if it actually does 4., but that's a fairly easy exercise afterwards (except for perhaps the exceptional groups).
Of course, this is a much longer route to the punchline, and I am not recommending it as a good way to learn the classification of compact Lie groups in terms of root data (though it would not be circular to do so). But there is something remarkable about the direct link between compact Lie groups and algebraic groups (allowing disconnectedness as well, and no specified maximal torus), not defined by going through the crutch of root data and Lie algebras. Historically the case of compact groups was a very important guide for Borel and Tits and others when developing the structure theory for connected reductive groups, and the above result "explains" a posteriori why this case was such an excellent guide to the general case.
I don't know if it's appropriate to link to self-advertise here. So at the risk of a minor faux pas, my edited notes from the Lie Groups class taught by Prof. Mark Haiman are available here (Wayback Machine). The bulk of the notes is the classification of complex semisimple Lie groups. For compact ones, follow the same argument, but add one fact: a simple group over R is compact iff the Killing form is negative definite.
The answers given are useful. Naturallysome care always has to be taken with the connectedness question, sincefinite groups might be regarded as compact Lie groups or as algebraic groups.Then a full classification becomes unreasonable. On the other hand, somecompact groups or algebraic groups occur most naturally as disconnected groups withan interesting component group.
Unfortunately, root data only ever seem to be discussed in the context of algebraic groups; I've never seen an elementary treatment which talks about root data for compact Lie groups without reference to the algebraic group case.
Classification seems hard, and revolves around extension problems, which are often intractable. Years ago, I was able to prove that there are only countably many compact Lie groups. This is immediate from structure theory in the connected case, but took me some effort for disconnected compact Lie groups. I've always wondered if there's an easy way to prove this.
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I've a VirtualBox VM which configured a very large hard disk size (bigger than host). By my mistake, a program on the VM generated lots of log files and the VDI file size keeps growing until there is no space on the host.
This isn't a direct answer per se, as I'm addressing the problem, not the question. Instead of periodically compacting the image, this solution uses discard to automatically remove unused blocks in the host's VM disk image.
FDE and cryptoroot are specifically not covered, as there are security concerns and none of the other solutions to this question would allow compacting either. The Arch Linux wiki has information about TRIM and dm-crypt devices.
With VBox exited and no VMs running, add discard support to your disks by setting both discard and nonrotational for each disk in the config file for the VM. At this time discard is not in the GUI, but nonrotational is exposed as the "Solid-state Drive" checkbox. (ref: vbox forums, discard support)
Manually trim free blocks now with fstrim. fstrim uses the mounted filesystem, not the block device backing it. Instead of setting continuous discard in fstab, this could be done on a weekly cron. (The weekly cron is recommended for physical SSDs which may have questionable support for TRIM, but this is not relevant here since underlying SSDs are handled by the host OS. see: ssd trim warning).
For a period of some years-- perhaps 1997-2007 or so-- 32-bit operating systems were still the norm, but hard disks larger than 2GB were already in use. As a result, when attempting to consume all free space by writing a file of zeroes (which should always be done as root, to include root's privileged free space, which no one else can touch), you may see:
If this occurs, you have most likely hit a 2GB file size limitation. This was common at the time because many file operations returned results in signed 32-bit integers, so that negative values could report error codes. This effectively meant that offset results were limited to 2^31 bytes without special measures.
I don't want to enable TRIM support in OS, because every data deletion will force data compacting in VDI file, making guest system unusable when VDI file is on classic rotational disc. For me better is to perform compacting by hand e.g. once per month.
A very neat trick to supplement the accepted answer is that you can get away without doing any compacting at all after zeroing guest space, by using a compressed file system on the host (e.g. selecting to compress the folder of virtual drives on NTFS properties on a Windows host). This in fact has the benefit to save a lot more space because operating systems tend to hold a lot of repetitive text or binary files (e.g. a 30GB guest drive that had 15GB of space zeroed can turn to 4GB on the host drive).
Just to add an alternative to the accepted answer: vboxmanage clonehd not only clones but also compacts virtual disks, so if you also need to clone it you can use a similar process and do it in one go (in my case I was moving the VM from an external disk to an internal disk with less space so I needed clone+compact):
N2 - Strategy-proof social choice functions are characterized for societies where the space of alternatives is any full dimensional compact subset of a Euclidean space and all voters have generalized single-peaked preferences. Our results build upon and extend those obtained for cartesian product ranges by Border and Jordan (1983). By admitting a large set of non-Cartesian ranges, we give a partial answer to the major open question left unresolved in this pioneering article. We prove that our class is composed by generalized median voter schemes which satisfy an additional condition, called the intersection property [Barbera, Masso, and Neme (1997)]. Journal Of Economic Literature Classification Number: D71. (C) 1998 Academic Press.
AB - Strategy-proof social choice functions are characterized for societies where the space of alternatives is any full dimensional compact subset of a Euclidean space and all voters have generalized single-peaked preferences. Our results build upon and extend those obtained for cartesian product ranges by Border and Jordan (1983). By admitting a large set of non-Cartesian ranges, we give a partial answer to the major open question left unresolved in this pioneering article. We prove that our class is composed by generalized median voter schemes which satisfy an additional condition, called the intersection property [Barbera, Masso, and Neme (1997)]. Journal Of Economic Literature Classification Number: D71. (C) 1998 Academic Press.
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