Claude Shannon, the father of Information Theory, died of Alzheimer's just a few years ago. He was certainly very well educated, and apparantly did indeed suffer quite a bit with the disease.
Although my first toy was a Commode 64, which was replaced by the 128, then the 386SX, then the (watch out) Pentium, my first computer was an SGI Indy. Sure, before that I called those toys computers, but this was a superior machine. The rear windows lept up with the touch of a button like frogs in a dynamite pond. Well, not the Windows, but you get the idea. That was my first taste of a real operating system, and hardware beefy enough to run it. I dont use SGI hardware anymore, but I've been hooked on *nix ever since.
Indeed. In the aftermath of hurricane andrew my father and I (both hams) went into the areas with heavy devistation to take messages from people with no communication and pass them on to thier friends and relatives across the country. We also sat by the radio at home and made phone calls on behalf of other hams who were in the field taking messages. I'm sure this is happening in LA as well. Why doesn't ham radio get more press in times like this? Because Big Media doesn't want to encourage encroachment on THEIR airwaves!
Cold fusion is a name for any nuclear fusion reaction that occurs well below the temperature required for thermonuclear reactions (which occur at millions of degrees Celsius).
Yes, time is the ultimate asset, the only real thing we have.
People say time is money, but really, time is life.
(On the other hand, money is choice.)
"Bob Wallace was a software pioneer, the ninth employee at Microsoft, the worlds top amateur neuroscientist, and a visionary philanthropist who laid the financial foundations of The Heffter Research Institute. He was also one of the most patient and caring people one would ever meet. When he died of pneumonia at an untimely 53, we lost a great and good friend."
-Heffter
Bob Wallace was indeed an incredible character. I was lucky enough to meet him in a USENET group focused on recreational chemicals. He replied to one of my first posts, and I immediately realized there were some amazing people lurking in the USENET. Indeed some great things have come from the micro$oft billion$. Rest in peace, my friend.
There's some nice calculus lurking in there. Something about approaching a number from either side. Oh wait! You might be saying there's an ideal amount of commenting and in large numbers programmers tend to converge on that amount. Interesting. This would make a good research topic.
A truly universal turing complete machine would need an infinite amount of memory. So no, there are no real physical implementations of universal turing machines.
Where are the comments about finding new rings around uranus? Surely some slashdotters out there have found a few of those tonight, eh? Or how about "moon" being defined, as I always thought it was, as my ass in a car window with pants pulled down.
Why would anyone hyphenate HP? It's always been HP. There's no hyphen there! I think M$ did this for some sinister reason, like screwing up search results. Hmm, I guess HP is the photo studio that took the picture. Are they the one's preinstalling linux on H-P's laptops?
Snipped? Like the Depends coupon you snipped out of the sunday paper? Or sniped, like I just sniped that P out of 'snipped' from the roof of a neighboring sentence?
Yes, all strings are finite, but the set of all strings will contain strings of length N for any N you like. It will also contain strings of length N+1, or even, for any N you pick, if I pick a K, it will contain strings of length N + K. So the set of all strings (or, if you prefer, "the set of all finite strings," which is equivalent) is not only an infinite set, but an uncountably infinite set at that. As for Skolem's Paradox, well, it's not really all that scary. It's just a neat effect. I think, in quite simple terms, the "paradox" is concerned with how a countable set can be a subset of an uncountable set. More specifically, how a function can map from an uncountable set to a countable set. Well, it's trivial that the naturals, N, are a subset of the reals, since by math induction we could show that for any natural number, there is a real-valued equivalent. Take this function, F, for example...
F: N -> R For any N in the naturals, append a decimal point to it and then for every K in the naturals append a zero to it. The inverse of this function, F', which has it's domain as a subset of the reals, maps onto one and only one natural number in N. Big deal, that's not so paradoxical. Consider F' the "whole part" function, which returns the whole part of any real number. Not surprisingly, the domain and range of the whole part function are subsets of the reals, even though one is uncountable and the other countable.
Richard's paradox is not a paradox at all. The literal meaning of "paradox" is "contrary to popular opinion". Literally, the concatenation of roots "para" and "doxa" which mean, respectively, opposite, and, popular opinion. It may have been that, in Richard's time, this was contrary to popular opinion. (For those reading, Richard's paradox is that the reals which are uncountable can be defined by a finite number of words.) Well, it's no surprise to anyone who has taken a formal logic class, that the reals are just that sort of thing, and that it shouldnt be so surprising.
Back in the day it was common courtesy to register the user/pass cipherpunks/cipherpunks on a site that needed reg. like these. So first you try the cipherpunks login, if it works, great, no reg needed. If not, you create it and let other cool punks in the know use it. Good to see this tradition is alive and well.
I'll consider your definition of real numbers: any that can be expressed as a string of UNICODE characters. I deny that the set of all such strings (of UNICODE characters) is countable, however. You say that it should be obvious, but provide no indication of where that intuition comes from. If you could explain more (because it is faaar from obvious to me) how that set can be counted, I would be very appreciative. To be sure, please show how to arrange such a set in one-to-one correspondence with the integers. I doubt you will be able to, though, as the following argument aims to demonstrate...
First, I would like to suggest a simple point: The number of elements in the set of UNICODE characters, call this U, is greater than 3. Since U > 3, the set of all strings of unicode characters should be at least as big as the set of all strings of characters from the set {a, b, c}. For any string of length N, there are more combinations of N-many UNICODE characters than there are combinations of N-many characters from {a, b, c}. This much should be obvious. For simplicity, if I could show that the set of all strings of characters in {a, b, c} is uncountably infinite, then the set of all strings of UNICODE characters should be uncountably infinite as well. This is simply because there must be at least as many strings of UNICODE characters as there are strings of {a, b, c}, as I said earlier.
I will now demonstrate that the set of all strings of characters from the set {a, b, c} is uncountably infinite. If you are familiar with the Cantor Diagonalization Process, you can probably guess where this is going...
1. Assume (to later get a contradiction) that the set, called S, of all strings of characters from {a, b, c} is countably infinite. Furthermore, that it is enumerated S1, S2, S3,...
2. For each n, the string Sn is composed of characters Sn(1), Sn(2), Sn(3),... (where n is a Natural number: 1, 2, 3,...)
The enumeration would go like this: S1(1), S1(2), S1(3),... S2(1), S2(2), S2(3),...... Sn(1), Sn(2),..., Sn(n),...
So Si(j) represents the j'th character of the i'th string.
3. Consider the string S1(1), S2(2), S3(3),..., Si(i),... Which, because it is constructed of only individual characters from strings of characters {a, b, c}, we know for sure that it too is a string of characters from {a, b, c}.
4. Take another string, call it S*, which we define as S*(1), S*(2), S*(3),..., S*(n),... where S*(n) = { a if Si(i) is in {b, c}; OR b if Si(i) is in {a} }
5. The string S*, which is constructed purely of characters from {a, b} (a subset of {a, b, c}) should be in our original enumeration of all strings of characters from {a, b, c} since it is constructed from a subset of {a, b, c}. In other words, the proposition "S* == Sn" should be true for some value of n (as usual, n is a Natural number: 1, 2, 3,...).
6. However, for each n in {1, 2, 3,...}, S* != Sn because it will NEVER be the true that "S*(n) == Sn(n)". That is, the whole string S* will never match a whole string Sn (for any n you like) because the n'th characters from each will not be equal (by stipulation, of course).
7. Therefore, a string called S* exists, which is composed only from characters in {a, b, c}, yet is not included in the set S of all such possible strings. Hence we have contradicted our original assumption that the set of all strings of characters {a, b, c} is countable.
8. The set of all strings of characters {a, b, c} is uncountably infinite (has strictly more members than there are natural numbers).
Since the set of all strings of UNICODE characters can't be smaller than the set of all strings of a subset of U
Someone, I think his name was Dedekind, might disagree... check it out.
Dedekind was a contemporary of Cantor, and proposed a clever definition of real numbers which conceives of each as a pair of sets. All members of S1 are less than any member of S2, and furthermore, S1 has no greatest member. This is a perfectly consistent (and interesting) formulation of the reals by an eminent 19th century mathematician; surely it can't be too silly to refer to real numbers as a set.
this could also help spammers. say a company publishes a list under this new recommendation. suppose their mail server is unsecured agains third party relaying. now the spammer has a list of valid domains they can use to bounce through the mail server. lets hope anyone smart enough to use this is also smart enough to secure their mail server.
Carriage before the horse?
on
BSA IDC FUD
·
· Score: 1
The BSA is very strict about auditing companies (or at least scaring them) in these countries specifically because they are the countries with the largest (most money msking/spending) IT sectors. They're just tooting their own horn.
Re:Does anyone find it odd...
on
Strike on Iraq
·
· Score: 1
the destruction that some of us implicitly (and explicitly) support?
What is implicit or explicit support? Sounds kinda fishy...
No, the point is, that whatever IP a person accesses google from once, hitting reload will access google from that same IP. Therefore, if Google responds specifically to an IP (or set of IPs) then one would get the same response each time.
Slashdot Mirror
Claude Shannon, the father of Information Theory, died of Alzheimer's just a few years ago. He was certainly very well educated, and apparantly did indeed suffer quite a bit with the disease.
Although my first toy was a Commode 64, which was replaced by the 128, then the 386SX, then the (watch out) Pentium, my first computer was an SGI Indy. Sure, before that I called those toys computers, but this was a superior machine. The rear windows lept up with the touch of a button like frogs in a dynamite pond. Well, not the Windows, but you get the idea. That was my first taste of a real operating system, and hardware beefy enough to run it. I dont use SGI hardware anymore, but I've been hooked on *nix ever since.
It's called brainfuck, and in case you didn't know, it's what taco is writing slash 3.0 in.
O.K. plagiarist, where is that from? For those of us who havent been enlightened by asimov... yet. :)
Indeed. In the aftermath of hurricane andrew my father and I (both hams) went into the areas with heavy devistation to take messages from people with no communication and pass them on to thier friends and relatives across the country. We also sat by the radio at home and made phone calls on behalf of other hams who were in the field taking messages. I'm sure this is happening in LA as well. Why doesn't ham radio get more press in times like this? Because Big Media doesn't want to encourage encroachment on THEIR airwaves!
Indeed, and King Cats ought to remember me as well. Hi to all! :) Hope you're hacking always happy.
NY Times Obit, A.D.P
There's some nice calculus lurking in there. Something about approaching a number from either side. Oh wait! You might be saying there's an ideal amount of commenting and in large numbers programmers tend to converge on that amount. Interesting. This would make a good research topic.
Could our PDAs swap business cards via a handshake? What will come next? VIRUSES?
A truly universal turing complete machine would need an infinite amount of memory. So no, there are no real physical implementations of universal turing machines.
Where are the comments about finding new rings around uranus? Surely some slashdotters out there have found a few of those tonight, eh? Or how about "moon" being defined, as I always thought it was, as my ass in a car window with pants pulled down.
Why would anyone hyphenate HP? It's always been HP. There's no hyphen there! I think M$ did this for some sinister reason, like screwing up search results. Hmm, I guess HP is the photo studio that took the picture. Are they the one's preinstalling linux on H-P's laptops?
If not, just flip the pane over. :)
Snipped? Like the Depends coupon you snipped out of the sunday paper? Or sniped, like I just sniped that P out of 'snipped' from the roof of a neighboring sentence?
Still forgetting to click Post Anonymously!
Yes, all strings are finite, but the set of all strings will contain strings of length N for any N you like. It will also contain strings of length N+1, or even, for any N you pick, if I pick a K, it will contain strings of length N + K. So the set of all strings (or, if you prefer, "the set of all finite strings," which is equivalent) is not only an infinite set, but an uncountably infinite set at that. As for Skolem's Paradox, well, it's not really all that scary. It's just a neat effect. I think, in quite simple terms, the "paradox" is concerned with how a countable set can be a subset of an uncountable set. More specifically, how a function can map from an uncountable set to a countable set. Well, it's trivial that the naturals, N, are a subset of the reals, since by math induction we could show that for any natural number, there is a real-valued equivalent. Take this function, F, for example...
F: N -> R
For any N in the naturals, append a decimal point to it and then for every K in the naturals append a zero to it. The inverse of this function, F', which has it's domain as a subset of the reals, maps onto one and only one natural number in N. Big deal, that's not so paradoxical. Consider F' the "whole part" function, which returns the whole part of any real number. Not surprisingly, the domain and range of the whole part function are subsets of the reals, even though one is uncountable and the other countable.
Richard's paradox is not a paradox at all. The literal meaning of "paradox" is "contrary to popular opinion". Literally, the concatenation of roots "para" and "doxa" which mean, respectively, opposite, and, popular opinion. It may have been that, in Richard's time, this was contrary to popular opinion. (For those reading, Richard's paradox is that the reals which are uncountable can be defined by a finite number of words.) Well, it's no surprise to anyone who has taken a formal logic class, that the reals are just that sort of thing, and that it shouldnt be so surprising.
Back in the day it was common courtesy to register the user/pass cipherpunks/cipherpunks on a site that needed reg. like these. So first you try the cipherpunks login, if it works, great, no reg needed. If not, you create it and let other cool punks in the know use it. Good to see this tradition is alive and well.
Shouts to all the cipherpunks reading this.
Interesting, but wrong...
...
... ...)
... ... ... ..., Sn(n), ...
..., Si(i), ...
..., S*(n), ...
...).
...}, S* != Sn because it will NEVER be the true that "S*(n) == Sn(n)". That is, the whole string S* will never match a whole string Sn (for any n you like) because the n'th characters from each will not be equal (by stipulation, of course).
I'll consider your definition of real numbers: any that can be expressed as a string of UNICODE characters. I deny that the set of all such strings (of UNICODE characters) is countable, however. You say that it should be obvious, but provide no indication of where that intuition comes from. If you could explain more (because it is faaar from obvious to me) how that set can be counted, I would be very appreciative. To be sure, please show how to arrange such a set in one-to-one correspondence with the integers. I doubt you will be able to, though, as the following argument aims to demonstrate...
First, I would like to suggest a simple point:
The number of elements in the set of UNICODE characters, call this U, is greater than 3. Since U > 3, the set of all strings of unicode characters should be at least as big as the set of all strings of characters from the set {a, b, c}. For any string of length N, there are more combinations of N-many UNICODE characters than there are combinations of N-many characters from {a, b, c}. This much should be obvious. For simplicity, if I could show that the set of all strings of characters in {a, b, c} is uncountably infinite, then the set of all strings of UNICODE characters should be uncountably infinite as well. This is simply because there must be at least as many strings of UNICODE characters as there are strings of {a, b, c}, as I said earlier.
I will now demonstrate that the set of all strings of characters from the set {a, b, c} is uncountably infinite. If you are familiar with the Cantor Diagonalization Process, you can probably guess where this is going...
1. Assume (to later get a contradiction) that the set, called S, of all strings of characters from {a, b, c} is countably infinite. Furthermore, that it is enumerated S1, S2, S3,
2. For each n, the string Sn is composed of characters Sn(1), Sn(2), Sn(3),
(where n is a Natural number: 1, 2, 3,
The enumeration would go like this:
S1(1), S1(2), S1(3),
S2(1), S2(2), S2(3),
Sn(1), Sn(2),
So Si(j) represents the j'th character of the i'th string.
3. Consider the string S1(1), S2(2), S3(3),
Which, because it is constructed of only individual characters from strings of characters {a, b, c}, we know for sure that it too is a string of characters from {a, b, c}.
4. Take another string, call it S*, which we define as S*(1), S*(2), S*(3),
where S*(n) = {
a if Si(i) is in {b, c}; OR
b if Si(i) is in {a}
}
5. The string S*, which is constructed purely of characters from {a, b} (a subset of {a, b, c}) should be in our original enumeration of all strings of characters from {a, b, c} since it is constructed from a subset of {a, b, c}. In other words, the proposition "S* == Sn" should be true for some value of n (as usual, n is a Natural number: 1, 2, 3,
6. However, for each n in {1, 2, 3,
7. Therefore, a string called S* exists, which is composed only from characters in {a, b, c}, yet is not included in the set S of all such possible strings. Hence we have contradicted our original assumption that the set of all strings of characters {a, b, c} is countable.
8. The set of all strings of characters {a, b, c} is uncountably infinite (has strictly more members than there are natural numbers).
Since the set of all strings of UNICODE characters can't be smaller than the set of all strings of a subset of U
Someone, I think his name was Dedekind, might disagree... check it out.
Dedekind was a contemporary of Cantor, and proposed a clever definition of real numbers which conceives of each as a pair of sets. All members of S1 are less than any member of S2, and furthermore, S1 has no greatest member. This is a perfectly consistent (and interesting) formulation of the reals by an eminent 19th century mathematician; surely it can't be too silly to refer to real numbers as a set.
Hey taco,
I'll be your second head! And between scenes, we can go get high behind the set.
this could also help spammers. say a company publishes a list under this new recommendation. suppose their mail server is unsecured agains third party relaying. now the spammer has a list of valid domains they can use to bounce through the mail server. lets hope anyone smart enough to use this is also smart enough to secure their mail server.
The BSA is very strict about auditing companies (or at least scaring them) in these countries specifically because they are the countries with the largest (most money msking/spending) IT sectors. They're just tooting their own horn.
the destruction that some of us implicitly (and explicitly) support?
What is implicit or explicit support? Sounds kinda fishy...
No, the point is, that whatever IP a person accesses google from once, hitting reload will access google from that same IP. Therefore, if Google responds specifically to an IP (or set of IPs) then one would get the same response each time.