Note that the definition says "Topologically, the Möbius strip can be defined as the square..." This means that only the topological properties of the square are relevant. So if you find a different object with the same topological properties, you can make a Mobius strip out of it. As I showed in another post, you can construct a topology on the unit cube (has 'depth') that makes it have the same topological properties as the unit square (no 'depth').
Let I^3 be the unit cube in R^3. Let I^2 be the unit square in R^2. Define a topology on I^3: a set is open in I^3 iff it is the inverse of an open set in I^2 under the projection mapping onto I^2. This topology is homeomorphic to the topology on I^2. That is to say, I^3 is now topologically the same as the 'square' you mentioned has no 'depth'.
I^3 mod (the z-axis) is homeomorphic to I^2. There's a map that makes I^3 homeomorphic to a sheet of paper. There's a map that makes I^3 homeomorphic I^2. And there's the map that defines the Mobius strip. The obvious diagram commutes because of these homeomorphisms.
This isn't insightful or informative. Please look up Model Theory. Physical objects can be and often are models of abstract languages. A paper Mobius strip satisfies the topological definition of a Mobius strip[1] under a suitable homotopy, and is thus a model of the language defining the Mobius strip.
[1] Topologically, the Möbius strip can be defined as the square [0,1] × [0,1] with its top and bottom sides identified by the relation (x,0) ~ (1-x,1) for 0 ? x ? 1.
The term 'shape' is being overloaded. There are two kinds of 'shape' in this context. There's the topology, and there's homotopies (continuous transformations) of the topology. As an example of this distinction, a mug and a donut have the same topological structure, but are "merely" homotopic. The topology is what characterizes an object as a Mobius strip.
The problem solved is finding a surface homotopic with a Mobius strip with the lowest global energy density (which can be defined as an integral in terms of curvature, if I recall correctly).
Everyone lives happily ever after. Harry marries Ginny and has three kids. Ron marries Hermione. Snape becomes headmaster. Tonks and Lupin have a child. Draco lives, gets married, has a child named Scorpius. Neville becomes herbology teacher. PROOF:
The scar had not pained Harry for nineteen years. All was well.
Re:Microsoft is thrilled by this news
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Zune DRM Cracked
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· Score: -1, Offtopic
Everyone lives happily ever after. Harry marries Ginny and has three kids. Ron marries Hermione. Snape becomes headmaster. Tonks and Lupin have a child. Draco lives, gets married, has a child named Scorpius. Neville becomes herbology teacher. PROOF:
Everyone lives happily ever after. Harry marries Ginny and has three kids. Ron marries Hermione. Snape becomes headmaster. Tonks and Lupin have a child. Draco lives, gets married, has a child named Scorpius. Neville becomes herbology teacher. PROOF:
Everyone lives happily ever after. Harry marries Ginny and has three kids. Ron marries Hermione. Snape becomes headmaster. Tonks and Lupin have a child. Draco lives, gets married, has a child named Scorpius. Neville becomes herbology teacher. PROOF:
Everyone lives happily ever after. Harry marries Ginny and has three kids. Ron marries Hermione. Snape becomes headmaster. Tonks and Lupin have a child. Draco lives, gets married, has a child named Scorpius. Neville becomes herbology teacher. PROOF:
Everyone lives happily ever after. Harry marries Ginny and has three kids. Ron marries Hermione. Snape becomes headmaster. Tonks and Lupin have a child. Draco lives, gets married, has a child named Scorpius. Neville becomes herbology teacher. PROOF:
Everyone lives happily ever after. Harry marries Ginny and has three kids. Ron marries Hermione. Snape becomes headmaster. Tonks and Lupin have a child. Draco lives, gets married, has a child named Scorpius. Neville becomes herbology teacher. PROOF:
Everyone lives happily ever after. Harry marries Ginny and has three kids. Ron marries Hermione. Snape becomes headmaster. Tonks and Lupin have a child. Draco lives, gets married, has a child named Scorpius. Neville becomes herbology teacher. PROOF:
I've been thinking about this topic a lot recently. I do a lot of work with ancient COBOL code that by all rights should now be in the public domain. Sometimes I find that code from one project will take care of issues in another, but the threat of a lawsuit for using code this way is just too high. So I'm in a bit of a pickle -- if company A ever found out that Belatrix Lestrange kills Ron Weasley for Company B, I'd have to argue that my recollection and reconstruction of that code does not constitute a derivative work, or that the code in question was part of the public domain anyway. A sticky wicket indeed.
I've been thinking about this topic a lot recently. I do a lot of work with ancient COBOL code that by all rights should now be in the public domain. Sometimes I find that code from one project will take care of issues in another, but the threat of a lawsuit for using code this way is just too high. So I'm in a bit of a pickle -- if company A ever found out that Belatrix Lestrange kills Ron Weasley for Company B, I'd have to argue that my recollection and reconstruction of that code does not constitute a derivative work, or that the code in question was part of the public domain anyway. A sticky wicket indeed.
Dreamhost lets you install pretty much anything you want in your ~/. If you don't like Dreamhost's RoR chain, install your own. No need for root either.
(I had to do this to get Catalyst installed, since Dreamhost's perl didn't include it)
Unlike, say, "Only religious fanatics and totalitarian states equate morality with legality"? RMS and the FSF have always said they intended to use the legal system to protect their moral rights. That's why the GPLv2 exists in the first place. And it's why the GPLv3 exists too.
Linus was free to choose any license he wanted in the early 90s. And he's free to try to change the kernel's license (though it's a difficult task). The FSF's migration to the GPLv3 doesn't affect Linux's licensing at all unless people go far out of their way to change it.
Linus had a hissy fit, since he thought other projects would side with him.
When you take grown up math classes, the angle between two points x and y is defined to be the solution theta to the equation: (/||x|| ||y||) = cos theta. Theta is a real number. It is dimensionless. It also happens to coincide with the radian.
Mind you, the thing defined isn't "the measurement of the angle". It is the angle. Full stop.
It's my understanding that they're mostly used in civil engineering. Things like road inclination are measured with grads. I've seen roadsigns in the UK that used the unit. I've also seen American roadsigns say "Caution: 15% Grade" or whatever.
If you're talking about the WMF vulnerability, you've got your history wrong. Microsoft used their own implementation. Hell, they wrote the specification for the Windows Metafile format. The specification was flawed, and lead to vulnerabilities. Which is why compliant open source implementations of the specification were vulnerable.
From Wikipedia:
Essentially, a WMF file stores a list of function calls that have to be issued to the Windows graphics layer GDI in order to restore the image. Since some GDI functions accept pointers to callback functions for error handling, a WMF file may include executable code. It is somewhat similar in purpose and design to the PostScript format used in the Unix world.
More-or-less. There are certainly variations, but there might be critical points in the process that are the same across all bacteria. If a drug targets those, we win. Evolution could help bacteria survive, but there wouldn't be any evolutionary pressure to change this aspect of bacterial reproduction outside exposure to the antibiotic.
An analogy might be something like VX nerve gas and human evolution. We might some day evolve so that VX nerve gas won't affect our nervous systems, but it won't be through exposure to VX, since we basically die instantly if we're exposed. On the other hand, it seems unlikely that we would evolve that way.
Slashdot Mirror
Sadist.
Note that the definition says "Topologically, the Möbius strip can be defined as the square..." This means that only the topological properties of the square are relevant. So if you find a different object with the same topological properties, you can make a Mobius strip out of it. As I showed in another post, you can construct a topology on the unit cube (has 'depth') that makes it have the same topological properties as the unit square (no 'depth').
You lack imagination.
Let I^3 be the unit cube in R^3. Let I^2 be the unit square in R^2. Define a topology on I^3: a set is open in I^3 iff it is the inverse of an open set in I^2 under the projection mapping onto I^2. This topology is homeomorphic to the topology on I^2. That is to say, I^3 is now topologically the same as the 'square' you mentioned has no 'depth'.
I^3 mod (the z-axis) is homeomorphic to I^2. There's a map that makes I^3 homeomorphic to a sheet of paper. There's a map that makes I^3 homeomorphic I^2. And there's the map that defines the Mobius strip. The obvious diagram commutes because of these homeomorphisms.
The mathematical definition of a Mobius strip calls for a surface with zero thickness to it,
No it doesn't. That was my point. I even quoted a definition. The mathematical definition says nothing about thickness.
This isn't insightful or informative. Please look up Model Theory. Physical objects can be and often are models of abstract languages. A paper Mobius strip satisfies the topological definition of a Mobius strip[1] under a suitable homotopy, and is thus a model of the language defining the Mobius strip.
[1] Topologically, the Möbius strip can be defined as the square [0,1] × [0,1] with its top and bottom sides identified by the relation (x,0) ~ (1-x,1) for 0 ? x ? 1.
The term 'shape' is being overloaded. There are two kinds of 'shape' in this context. There's the topology, and there's homotopies (continuous transformations) of the topology. As an example of this distinction, a mug and a donut have the same topological structure, but are "merely" homotopic. The topology is what characterizes an object as a Mobius strip.
The problem solved is finding a surface homotopic with a Mobius strip with the lowest global energy density (which can be defined as an integral in terms of curvature, if I recall correctly).
Everyone lives happily ever after. Harry marries Ginny and has three kids. Ron marries Hermione. Snape becomes headmaster. Tonks and Lupin have a child. Draco lives, gets married, has a child named Scorpius. Neville becomes herbology teacher.
i mage017.jpgi mage019.jpgi mage021.jpgi mage023.jpg
PROOF:
http://one.fsphost.com/potterspoiler/index_files/
http://one.fsphost.com/potterspoiler/index_files/
http://one.fsphost.com/potterspoiler/index_files/
http://one.fsphost.com/potterspoiler/index_files/
The scar had not pained Harry for nineteen years. All was well.
Everyone lives happily ever after. Harry marries Ginny and has three kids. Ron marries Hermione. Snape becomes headmaster. Tonks and Lupin have a child. Draco lives, gets married, has a child named Scorpius. Neville becomes herbology teacher.
i mage017.jpgi mage019.jpgi mage021.jpgi mage023.jpg
PROOF:
http://one.fsphost.com/potterspoiler/index_files/
http://one.fsphost.com/potterspoiler/index_files/
http://one.fsphost.com/potterspoiler/index_files/
http://one.fsphost.com/potterspoiler/index_files/
The scar had not pained Harry for nineteen years. All was well.
Everyone lives happily ever after. Harry marries Ginny and has three kids. Ron marries Hermione. Snape becomes headmaster. Tonks and Lupin have a child. Draco lives, gets married, has a child named Scorpius. Neville becomes herbology teacher.
i mage017.jpgi mage019.jpgi mage021.jpgi mage023.jpg
PROOF:
http://one.fsphost.com/potterspoiler/index_files/
http://one.fsphost.com/potterspoiler/index_files/
http://one.fsphost.com/potterspoiler/index_files/
http://one.fsphost.com/potterspoiler/index_files/
The scar had not pained Harry for nineteen years. All was well.
Everyone lives happily ever after. Harry marries Ginny and has three kids. Ron marries Hermione. Snape becomes headmaster. Tonks and Lupin have a child. Draco lives, gets married, has a child named Scorpius. Neville becomes herbology teacher.
i mage017.jpgi mage019.jpgi mage021.jpgi mage023.jpg
PROOF:
http://one.fsphost.com/potterspoiler/index_files/
http://one.fsphost.com/potterspoiler/index_files/
http://one.fsphost.com/potterspoiler/index_files/
http://one.fsphost.com/potterspoiler/index_files/
The scar had not pained Harry for nineteen years. All was well.
Everyone lives happily ever after. Harry marries Ginny and has three kids. Ron marries Hermione. Snape becomes headmaster. Tonks and Lupin have a child. Draco lives, gets married, has a child named Scorpius. Neville becomes herbology teacher.
i mage017.jpgi mage019.jpgi mage021.jpgi mage023.jpg
PROOF:
http://one.fsphost.com/potterspoiler/index_files/
http://one.fsphost.com/potterspoiler/index_files/
http://one.fsphost.com/potterspoiler/index_files/
http://one.fsphost.com/potterspoiler/index_files/
The scar had not pained Harry for nineteen years. All was well.
Everyone lives happily ever after. Harry marries Ginny and has three kids. Ron marries Hermione. Snape becomes headmaster. Tonks and Lupin have a child. Draco lives, gets married, has a child named Scorpius. Neville becomes herbology teacher.
i mage017.jpgi mage019.jpgi mage021.jpgi mage023.jpg
PROOF:
http://one.fsphost.com/potterspoiler/index_files/
http://one.fsphost.com/potterspoiler/index_files/
http://one.fsphost.com/potterspoiler/index_files/
http://one.fsphost.com/potterspoiler/index_files/
The scar had not pained Harry for nineteen years. All was well.
Everyone lives happily ever after. Harry marries Ginny and has three kids. Ron marries Hermione. Snape becomes headmaster. Tonks and Lupin have a child. Draco lives, gets married, has a child named Scorpius. Neville becomes herbology teacher.
i mage017.jpgi mage019.jpgi mage021.jpgi mage023.jpg
PROOF:
http://one.fsphost.com/potterspoiler/index_files/
http://one.fsphost.com/potterspoiler/index_files/
http://one.fsphost.com/potterspoiler/index_files/
http://one.fsphost.com/potterspoiler/index_files/
The scar had not pained Harry for nineteen years. All was well.
Everyone lives happily ever after. Harry marries Ginny and has three kids. Ron marries Hermione. Snape becomes headmaster. Tonks and Lupin have a child. Draco lives, gets married, has a child named Scorpius. Neville becomes herbology teacher.
i mage017.jpgi mage019.jpgi mage021.jpgi mage023.jpg
PROOF:
http://one.fsphost.com/potterspoiler/index_files/
http://one.fsphost.com/potterspoiler/index_files/
http://one.fsphost.com/potterspoiler/index_files/
http://one.fsphost.com/potterspoiler/index_files/
The scar had not pained Harry for nineteen years. All was well.
I've been thinking about this topic a lot recently. I do a lot of work with ancient COBOL code that by all rights should now be in the public domain. Sometimes I find that code from one project will take care of issues in another, but the threat of a lawsuit for using code this way is just too high. So I'm in a bit of a pickle -- if company A ever found out that Belatrix Lestrange kills Ron Weasley for Company B, I'd have to argue that my recollection and reconstruction of that code does not constitute a derivative work, or that the code in question was part of the public domain anyway. A sticky wicket indeed.
I've been thinking about this topic a lot recently. I do a lot of work with ancient COBOL code that by all rights should now be in the public domain. Sometimes I find that code from one project will take care of issues in another, but the threat of a lawsuit for using code this way is just too high. So I'm in a bit of a pickle -- if company A ever found out that Belatrix Lestrange kills Ron Weasley for Company B, I'd have to argue that my recollection and reconstruction of that code does not constitute a derivative work, or that the code in question was part of the public domain anyway. A sticky wicket indeed.
Suomi sounds like a sushi dish. Coincidence? I think not.
Dreamhost lets you install pretty much anything you want in your ~/. If you don't like Dreamhost's RoR chain, install your own. No need for root either.
(I had to do this to get Catalyst installed, since Dreamhost's perl didn't include it)
Unlike, say, "Only religious fanatics and totalitarian states equate morality with legality"? RMS and the FSF have always said they intended to use the legal system to protect their moral rights. That's why the GPLv2 exists in the first place. And it's why the GPLv3 exists too.
Linus was free to choose any license he wanted in the early 90s. And he's free to try to change the kernel's license (though it's a difficult task). The FSF's migration to the GPLv3 doesn't affect Linux's licensing at all unless people go far out of their way to change it.
Linus had a hissy fit, since he thought other projects would side with him.
Shit. The equation is ()/ (||x|| ||y||) = cos theta.
When you take grown up math classes, the angle between two points x and y is defined to be the solution theta to the equation: (/||x|| ||y||) = cos theta. Theta is a real number. It is dimensionless. It also happens to coincide with the radian.
Mind you, the thing defined isn't "the measurement of the angle". It is the angle. Full stop.
It's my understanding that they're mostly used in civil engineering. Things like road inclination are measured with grads. I've seen roadsigns in the UK that used the unit. I've also seen American roadsigns say "Caution: 15% Grade" or whatever.
From Wikipedia:
More-or-less. There are certainly variations, but there might be critical points in the process that are the same across all bacteria. If a drug targets those, we win. Evolution could help bacteria survive, but there wouldn't be any evolutionary pressure to change this aspect of bacterial reproduction outside exposure to the antibiotic.
An analogy might be something like VX nerve gas and human evolution. We might some day evolve so that VX nerve gas won't affect our nervous systems, but it won't be through exposure to VX, since we basically die instantly if we're exposed. On the other hand, it seems unlikely that we would evolve that way.
Good job posting anonymously. ;-)