I don't care much about itunes upgrades, but I was interested to see the actual release on Windows, and thought I might see how it compares to Opera/FF/IE. The license agreement has blunted my interest, though:
"4. Consent to Use of Data. You agree that Apple and its subsidiaries may collect and use technical and related information, including but not limited to technical information about your computer, system and application software, and peripherals, that is gathered periodically to facilitate the provision of software updates, product support and other services to you (if any) related to the Apple Software, and to verify compliance with the terms of this License. Apple may use this information, as long as it is in a form that does not personally identify you, to improve our products or to provide services or technologies to you."
You seem to be under the impression that undercover officers target only "victimless" crimes like drug crimes and prostitution.
Do you believe that undercover agents never work to expose actual violent criminals? (Say, for example, the mob, street gangs, biker gangs, etc.)
So I should assume you are against the different levels of murder and manslaughter? That you advocate that any wrongful death should be punished exactly as any other?
This is really a straw man. The difference between degrees of murder and manslaughter is the level of intent: did you plan ahead of time to kill him, decide to kill him on the spot, or not even mean to kill him at all, etc. That distinction is quite different from asking "why did you intend to kill him?" The difference between intentionally and unintentionally causing death is not the same as intentionally killing someone because he was an [epithet] or because he slept with your wife or whatever.
Decisions of a court are only precedent in the formal sense in courts below that which made the ruling. One might of course cite a favorable ruling in legal arguments, but it is not binding on other courts.
The US does have hate speech laws, but they are very limited in scope by the application of the First Amendment. See for example R.A.V. v. City of St. Paul where SCOTUS overturned a hate speech law because it amounted to viewpoint discrimination. The classes of speech which can be constitutionally restricted remain quite small.
Similarly, US libel laws are restricted in any number of ways. Truth is always an absolute defense. Strict culpability is no allowed (some level of mens rea must be proven).
From TFA: "The judge denied the motion to quash the RIAA's subpoenas in November, ordering OSU to provide the identities of the students it believed were behind the IP addresses flagged by MediaSentry."
Or, to be more careful: Obviously, actually talking about "sqrt(-1)" (or "-sqrt(-1)") doesn't mean anything at all. Literally saying "add sqrt(-1) to the reals and call it i" is nonsense. However, it is a natural way of expressing the perfectly sound field extension construction I described above, wherein we produce an element whose square is -1 and then extend the reals with it.
Referring to "the" square root of a number is ambiguous. For convenience, we define the "principal" square root of a real number to be the positive one, and this is what is meant by "sqrt(x)". This definition doesn't work in the complexes because there are no positive or negative imaginary numbers. If you wish to be able to identify a principal root in C, the definition has to be extended, and one could equally well choose either square root of -1 to be the principal one. This is true in any construction. In the ordered pairs construction, I think almost anyone would choose (0,1) to be the principal root (because the second component is positive) but the choice is really quite arbitrary; (0,-1) works fine. Obviously if two texts extend the definition of the principal root differently, then they will diverge in places, but that doesn't invalidate the underlying construction of C.
Well, I wouldn't, cause that sounds like a pain in the ass.
More seriously, http://en.wikipedia.org/wiki/Field_extension provides a description of how you would actually go about constructing new elements to add to a field (see the Examples section). If you believe in polynomials,ring theory, and field theory, then we can look at the polynomials with real coeffiecients (these form a ring). Then look at some polynomial of interest (x^2 + 1) and look at the ideal it generates (meaning the things of the form p (x^2 + 1) where p is some other polynomial). Then we take the quotient of the polynomial ring by the ideal (the elements in the quotient are the sets {a + i | i is an element of I} for every polynomial a (where I is the ideal)). These are all well defined things and you could sit down and write all the elements out (except for the you-die-first thing). The quotient here turns out to be a field, it embeds the reals, and the equivalance class x + I is the square root of -1. If you know the theory it's easy to show that, if you don't the computations will be meaningless (it's just a long series of simple manipulations). Note that I kinda lied; this field doesn't literally contain the reals in the usual sense, just a field isomorphic to them (namely, the sets {r + i | i in I} for each real r in R, I believe). The point is that all these things are well defined and you could do the construction and figure out exactly how to write i if you cared.
That said, if you really want to write the elements down and have them be readable, the ordered-pairs construction is probably nicer. I just wanted to demonstrate that the extending-with-a-root construction is sound. Also, while it's perhaps harder to express from first principles, I think it's more elegant in some ways (particularly because the idea of the construction can be used to add roots for any irreducible polynomial).
The distinction you draw between the two roots is artificial; they have all the same properties (the technical term is that they are "algebraically indistinguishible"). In the real numbers, there is an important difference between the two square roots of a number, because there's an important difference between the positive and negative numbers (the negatives don't have square roots themselves, for example). The same is not true when we start talking about imaginary numbers; the choice of sign is really quite arbitrary.
In particular, the mapping a+bi |-> a-bi is an automorphism on the complex numbers; it preserves all the structure (easily shown by direct computation). Any structural statement you could make about a+bi (having factors, roots, etc) you can also make about a-bi (so everything true about i is true about -i and vice versa).
What this means is that if I pick a square root of -1, call it i, and extend the reals with it, and you pick the "other" square root of -1, call it i, and extend the reals with it, we get exactly the same field; no renaming is necessary
There is nothing unsound about the sqrt(-1) definition.
Your definition is also valid, although it's worth noting that under your definition the complex numbers don't actually contain the reals (they just embed an isomorphic copy, namely (x,0) for every x) and the usual convenient notation (a+bi) becomes a defined shortand rather than the actual sytax. Additionally, you have this weird multiplication rule with no particular justification other than "it works if we do it this way", which is really inelegant.
The sqrt(-1) approach is nicer in those regards. The reals are a field (R) in which there is no square root of -1. We introduce an element i that has the property that i^2 = -1. Then we look at the field extension R(i), that is, the smallest superfield of R containing i. Algorithmically, think of adding i and then adding additional terms as needed until you have something closed under addition/multiplication/inverse. (This can all be defined formally and proved to exist, but this is/., not a field theory class). The result is precisely the complex numbers in the usual sense, with the nice properties that R is a subfield of C, the usual notation is exactly the correct notation, and the multiplication operation is perfectly natural.
First, segments have nothing to do with physical addresses. A logical address specifies a segment and a 32-bit offset; after a bunch of gymnastics, this gets converted to a 32-bit physical address (assuming PAE is not in use). So even though there are a lot more than 2^32 logical addresses, you only get 2^32 physical addresses, which limits the amount of RAM.
It's actually even worse than that, though; the virtual address space is still only 2^32. You are probably confused by memories of real mode, where the point of segmentation was to increase the available address space. In protected mode, though, segementation exists only to carve up the address space in funny ways. A bunch of checks are done against the segment (checking privilege level, comparing the offset to the segment limit, etc.), and then if it's all ok the 32-bit offset is added to the segment base to produce a 32-bit linear address (which then goes through the usual paging to produce the physical address). So you are still limited to a 32-bit linear address space for virtual addresses; segments just allow you to access that address space in different ways.
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I don't care much about itunes upgrades, but I was interested to see the actual release on Windows, and thought I might see how it compares to Opera/FF/IE. The license agreement has blunted my interest, though: "4. Consent to Use of Data. You agree that Apple and its subsidiaries may collect and use technical and related information, including but not limited to technical information about your computer, system and application software, and peripherals, that is gathered periodically to facilitate the provision of software updates, product support and other services to you (if any) related to the Apple Software, and to verify compliance with the terms of this License. Apple may use this information, as long as it is in a form that does not personally identify you, to improve our products or to provide services or technologies to you."
You seem to be under the impression that undercover officers target only "victimless" crimes like drug crimes and prostitution. Do you believe that undercover agents never work to expose actual violent criminals? (Say, for example, the mob, street gangs, biker gangs, etc.)
This is really a straw man. The difference between degrees of murder and manslaughter is the level of intent: did you plan ahead of time to kill him, decide to kill him on the spot, or not even mean to kill him at all, etc. That distinction is quite different from asking "why did you intend to kill him?" The difference between intentionally and unintentionally causing death is not the same as intentionally killing someone because he was an [epithet] or because he slept with your wife or whatever.
Decisions of a court are only precedent in the formal sense in courts below that which made the ruling. One might of course cite a favorable ruling in legal arguments, but it is not binding on other courts.
The US does have hate speech laws, but they are very limited in scope by the application of the First Amendment. See for example R.A.V. v. City of St. Paul where SCOTUS overturned a hate speech law because it amounted to viewpoint discrimination. The classes of speech which can be constitutionally restricted remain quite small.
Similarly, US libel laws are restricted in any number of ways. Truth is always an absolute defense. Strict culpability is no allowed (some level of mens rea must be proven).
From TFA: "The judge denied the motion to quash the RIAA's subpoenas in November, ordering OSU to provide the identities of the students it believed were behind the IP addresses flagged by MediaSentry."
The school has received a court order.
Or, to be more careful: Obviously, actually talking about "sqrt(-1)" (or "-sqrt(-1)") doesn't mean anything at all. Literally saying "add sqrt(-1) to the reals and call it i" is nonsense. However, it is a natural way of expressing the perfectly sound field extension construction I described above, wherein we produce an element whose square is -1 and then extend the reals with it.
That argument is just an abuse of notation.
Referring to "the" square root of a number is ambiguous. For convenience, we define the "principal" square root of a real number to be the positive one, and this is what is meant by "sqrt(x)". This definition doesn't work in the complexes because there are no positive or negative imaginary numbers. If you wish to be able to identify a principal root in C, the definition has to be extended, and one could equally well choose either square root of -1 to be the principal one. This is true in any construction. In the ordered pairs construction, I think almost anyone would choose (0,1) to be the principal root (because the second component is positive) but the choice is really quite arbitrary; (0,-1) works fine. Obviously if two texts extend the definition of the principal root differently, then they will diverge in places, but that doesn't invalidate the underlying construction of C.
Well, I wouldn't, cause that sounds like a pain in the ass.
More seriously, http://en.wikipedia.org/wiki/Field_extension provides a description of how you would actually go about constructing new elements to add to a field (see the Examples section). If you believe in polynomials,ring theory, and field theory, then we can look at the polynomials with real coeffiecients (these form a ring). Then look at some polynomial of interest (x^2 + 1) and look at the ideal it generates (meaning the things of the form p (x^2 + 1) where p is some other polynomial). Then we take the quotient of the polynomial ring by the ideal (the elements in the quotient are the sets {a + i | i is an element of I} for every polynomial a (where I is the ideal)). These are all well defined things and you could sit down and write all the elements out (except for the you-die-first thing). The quotient here turns out to be a field, it embeds the reals, and the equivalance class x + I is the square root of -1. If you know the theory it's easy to show that, if you don't the computations will be meaningless (it's just a long series of simple manipulations). Note that I kinda lied; this field doesn't literally contain the reals in the usual sense, just a field isomorphic to them (namely, the sets {r + i | i in I} for each real r in R, I believe). The point is that all these things are well defined and you could do the construction and figure out exactly how to write i if you cared.
That said, if you really want to write the elements down and have them be readable, the ordered-pairs construction is probably nicer. I just wanted to demonstrate that the extending-with-a-root construction is sound. Also, while it's perhaps harder to express from first principles, I think it's more elegant in some ways (particularly because the idea of the construction can be used to add roots for any irreducible polynomial).
You are indeed mistaken.
The distinction you draw between the two roots is artificial; they have all the same properties (the technical term is that they are "algebraically indistinguishible"). In the real numbers, there is an important difference between the two square roots of a number, because there's an important difference between the positive and negative numbers (the negatives don't have square roots themselves, for example). The same is not true when we start talking about imaginary numbers; the choice of sign is really quite arbitrary.
In particular, the mapping a+bi |-> a-bi is an automorphism on the complex numbers; it preserves all the structure (easily shown by direct computation). Any structural statement you could make about a+bi (having factors, roots, etc) you can also make about a-bi (so everything true about i is true about -i and vice versa).
What this means is that if I pick a square root of -1, call it i, and extend the reals with it, and you pick the "other" square root of -1, call it i, and extend the reals with it, we get exactly the same field; no renaming is necessary
There is nothing unsound about the sqrt(-1) definition. Your definition is also valid, although it's worth noting that under your definition the complex numbers don't actually contain the reals (they just embed an isomorphic copy, namely (x,0) for every x) and the usual convenient notation (a+bi) becomes a defined shortand rather than the actual sytax. Additionally, you have this weird multiplication rule with no particular justification other than "it works if we do it this way", which is really inelegant. The sqrt(-1) approach is nicer in those regards. The reals are a field (R) in which there is no square root of -1. We introduce an element i that has the property that i^2 = -1. Then we look at the field extension R(i), that is, the smallest superfield of R containing i. Algorithmically, think of adding i and then adding additional terms as needed until you have something closed under addition/multiplication/inverse. (This can all be defined formally and proved to exist, but this is /., not a field theory class). The result is precisely the complex numbers in the usual sense, with the nice properties that R is a subfield of C, the usual notation is exactly the correct notation, and the multiplication operation is perfectly natural.
There are actually two reasons this doesn't work.
First, segments have nothing to do with physical addresses. A logical address specifies a segment and a 32-bit offset; after a bunch of gymnastics, this gets converted to a 32-bit physical address (assuming PAE is not in use). So even though there are a lot more than 2^32 logical addresses, you only get 2^32 physical addresses, which limits the amount of RAM.
It's actually even worse than that, though; the virtual address space is still only 2^32. You are probably confused by memories of real mode, where the point of segmentation was to increase the available address space. In protected mode, though, segementation exists only to carve up the address space in funny ways. A bunch of checks are done against the segment (checking privilege level, comparing the offset to the segment limit, etc.), and then if it's all ok the 32-bit offset is added to the segment base to produce a 32-bit linear address (which then goes through the usual paging to produce the physical address). So you are still limited to a 32-bit linear address space for virtual addresses; segments just allow you to access that address space in different ways.
http://en.wikipedia.org/wiki/Segment_selector