Aren't they teaching basic boolean algebra to kids anymore?
I presume you mean basic Boolean ring theory. Though a Boolean algebra and a Boolean ring generate the same structures, they are distinct concepts. A Boolean algebra is a lattice that satisfies certain axioms. No other function symbols outside of the meet and join are necessary to characterize a Boolean algebra. A Boolean ring is a ring with some constraints on multiplication.
Unless you mean the very rudiments of the Boolean Ring operations, they have never taught Boolean Algebra to kids. Lattices are hard.
Re:One recipe this book doesn't cover
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Rails Recipes
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Catalyst (the elegant MVC framework) is written by many of the same people who work on Class::DBI. It used to integrate very nicely, but CDBI has been deprecated in favor of DBIx::Class (also written by the Catalyst core, iirc), which is mostly backwards compatible with CDBI but adds a lot of new code generation facilities.
Wow, I haven't met an overbearing Christian since I got out of high school. I almost fell for the troll aspect of your post.
Life is full of risks. To paraphrase Benjamin Franklin, those who give up an important freedom for temporary safety deserve neither. Marriage is not the solution to the problems you raised. Effective birth control is. Indeed, many married couples don't want to have children.
Yes. I do want to gamble with my life. I do it every day. I shudder to think how boring a life I would have if I didn't.
Interesting. From the sound of your description (and assuming you really work for MS), it sounds like Microsoft is trying to implement Apple's OS X release strategy. They put up a fairly significant effort to get the first OS X release, but ended up with a highly modular, easily maintained codebase. Now they work on implementing the "next big thing" and release just about every 18 months.
Any idea if significant amounts of BSD code are going to make it into the next Windows?
WHOA, the Tomorrow People. I haven't thought about that show in years. Thanks for reminding me. It was one of my favorites as a kid.
Them and Alex Mack. She was hot when I was 12.
You're talking about the "One Point Compactification" of the Real Line., which produces the "extended Reals". Yes, that works. But 1/x still isn't continuous on that space. For a proof, note that the limit as x->0 from the left is -infinity, but the limit from the right is infinity.
A similar question is whether 1/x^2 is continuous at x = 0. I would continue to say no, but the proof I gave above wouldn't work in this case.
Even barring using the one point compactification (or just using it implicity), one can give meaning to the use of infinity in limit notation. And they would be pronounced just the same way as regular old limits. Heck, they even (kind of) mean the same thing intuitively. But they are not limits in the strict sense. It's a helpful abuse of notation. Unfortunely, abuses of notation tend to bite mathematicians in the ass when people (like the GGP) think that the meaning and truth of a mathematical statement depends on how it's annotated.
Thank you for explaining all of that to me! I had no idea any of this worked, and I've been a professional mathematician for years.
You might want to look at the definition of a limit. Which says, (I'm quoting Rudin, but my explanatory additions are in parentheses):
Let f be defined on (a set E). Let x be a limit point of E (that is, a point contained in the closure of E). We write f(t) -> A as t -> x, or lim t->x f(t) = A, if there exists a *number* A with the following property:
for every epsilon > 0, there exists a delta > 0 such that |f(t) - A| epsilon for all points t of E for which 0|t - x| delta.
Using the limit notation for infinite "limits" is an abuse of notation. It's not egregious, and it's useful. But infinity is not a limit, because it isn't even a number.
As Nintendo has shown with the Wii, you don't need to have cutting edge graphics or processing speed, you need an innovative idea.
This is certainly true, and your observation almost touches upon an interesting parallel.
In some respects, Nintendo is the Apple of the console world. They produce quality hardware in an attractive package. They rely on interesting, well integrated features to sell their hardware. The Wii even looks like a MacMini on its side.
I wouldn't suggest that Apple couldn't do as good a job as Nintendo. But is there really room for both in the market? Especially when on considers Nintendo's (and presumably Apple's) target audience? If Apple made a compelling feature, Nintendo would be forced to retaliate with another. Ideas are a scarce resource, and I doubt Nintendo or Apple has a large enough cache of them to avoid lame gimmicks. Kids might be fond of gimmicks, but grown ups usually aren't.
Oh come on. I was just suggesting that making a distributed "ROFL" printing system using the ROFL-inducing vulnerability would be funny. In a geek way. I could have just said "using a functional equivalent of your code" to get the point across too.
And yet somehow we have no problems tranporting oil to non-oil producing regions.
We don't?
Wait, we do. And that's the prime economic reason developing alternative energy strategies is in the US's (and everyone else's) best interests, despite our reliance on our current profits in the energy market.
I think the best use for this technology would be to put it on every roof in in America (and Europe and eventually the world), and use nuclear power as a method to buffer against periods of low sunlight.
While the major volcano/meteor event you mentioned could deplete the nuclear buffer, it would do that (and worse) now.
At the very least, considering the effects on the economy that nearly free energy would have, we could build enough nuclear power plants to completely handle our energy needs in case such an unfortunate turn of events occured. Hell, we could sell of the surplus nuclear energy to subsidize projects like the complete mechanization of food production, -- obviously using our nearly free energy. Or just lower taxes (though I would prefer the former)
I suppose it could be bad globally, but global weather effects would get lost in the inherent "randomness" of weather. The butterfly effect you alluded to shows that all events are (in a fairly abstract sense equally) weather changing, not just the big ones.
I'm thinking 260 or so square miles of solar cells would severely alter the local ecosystem. But who cares? There's a chance for more life in the area with lowered ground temperatures.
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Aren't they teaching basic boolean algebra to kids anymore?
I presume you mean basic Boolean ring theory. Though a Boolean algebra and a Boolean ring generate the same structures, they are distinct concepts. A Boolean algebra is a lattice that satisfies certain axioms. No other function symbols outside of the meet and join are necessary to characterize a Boolean algebra. A Boolean ring is a ring with some constraints on multiplication.
Unless you mean the very rudiments of the Boolean Ring operations, they have never taught Boolean Algebra to kids. Lattices are hard.
Sounds like a good chance for PFY and I to take care of those numerous denied expenses from my last expense report all in one swoop. ;)
Oh boy! Another TLA I had to GGL!
Are you, directly or indirectly, in the employ of CBS, its subsidiaries, or owners?
You are both tedious turds.
Catalyst (the elegant MVC framework) is written by many of the same people who work on Class::DBI. It used to integrate very nicely, but CDBI has been deprecated in favor of DBIx::Class (also written by the Catalyst core, iirc), which is mostly backwards compatible with CDBI but adds a lot of new code generation facilities.
Thank you for explaining the obvious.
Wow, I haven't met an overbearing Christian since I got out of high school. I almost fell for the troll aspect of your post.
Life is full of risks. To paraphrase Benjamin Franklin, those who give up an important freedom for temporary safety deserve neither. Marriage is not the solution to the problems you raised. Effective birth control is. Indeed, many married couples don't want to have children.
Yes. I do want to gamble with my life. I do it every day. I shudder to think how boring a life I would have if I didn't.
Because I'm horny, and while I know there are very real risks, I know they can be mitigated.
Yet again, Tool becomes a spring of insight.
Interesting. From the sound of your description (and assuming you really work for MS), it sounds like Microsoft is trying to implement Apple's OS X release strategy. They put up a fairly significant effort to get the first OS X release, but ended up with a highly modular, easily maintained codebase. Now they work on implementing the "next big thing" and release just about every 18 months. Any idea if significant amounts of BSD code are going to make it into the next Windows?
A tedious troll at that. The best trolls post once, maybe twice, and get a 200 comment shitstorm in response.
You are a tedious little turd. Aspie much?
Interesting. But note that the Romans fell to Germanic tribes when they could no longer support their military.
Has anybody studied late Roman history and modern Middle Eastern history enough to intelligently compare and contrast the situations? (I haven't.)
WHOA, the Tomorrow People. I haven't thought about that show in years. Thanks for reminding me. It was one of my favorites as a kid. Them and Alex Mack. She was hot when I was 12.
Nice, a new set of blogs to troll.
Practice what you preach and SHUT THE FUCK UP.
This is a fair summary of what's going on. I would like to point out, however, that in some branches of mathematics (Set Theory, in particular), there are "infinite numbers." Very interesting stuff: See http://en.wikipedia.org/wiki/Ordinal_Number and http://en.wikipedia.org/wiki/Cardinal_Number
:-)
This is neither here nor there in the context of calculus, however. Just thought I'd share.
You're talking about the "One Point Compactification" of the Real Line., which produces the "extended Reals". Yes, that works. But 1/x still isn't continuous on that space. For a proof, note that the limit as x->0 from the left is -infinity, but the limit from the right is infinity. A similar question is whether 1/x^2 is continuous at x = 0. I would continue to say no, but the proof I gave above wouldn't work in this case. Even barring using the one point compactification (or just using it implicity), one can give meaning to the use of infinity in limit notation. And they would be pronounced just the same way as regular old limits. Heck, they even (kind of) mean the same thing intuitively. But they are not limits in the strict sense. It's a helpful abuse of notation. Unfortunely, abuses of notation tend to bite mathematicians in the ass when people (like the GGP) think that the meaning and truth of a mathematical statement depends on how it's annotated.
Thank you for explaining all of that to me! I had no idea any of this worked, and I've been a professional mathematician for years. You might want to look at the definition of a limit. Which says, (I'm quoting Rudin, but my explanatory additions are in parentheses): Let f be defined on (a set E). Let x be a limit point of E (that is, a point contained in the closure of E). We write f(t) -> A as t -> x, or lim t->x f(t) = A, if there exists a *number* A with the following property: for every epsilon > 0, there exists a delta > 0 such that |f(t) - A| epsilon for all points t of E for which 0|t - x| delta. Using the limit notation for infinite "limits" is an abuse of notation. It's not egregious, and it's useful. But infinity is not a limit, because it isn't even a number.
Infinity isn't a real number. Ergo, it cannot be the limit of a sequence, as the definition of a limit include the priviso that it is a real number.
You can only perform the substitution lim x->a f(x) = f(a) when f is continuous at a. f(x) = 1/x is (very trivially) not continous at a = 0.
Damnit, why is this sort of thing spilling over from sci.math now?
As Nintendo has shown with the Wii, you don't need to have cutting edge graphics or processing speed, you need an innovative idea.
This is certainly true, and your observation almost touches upon an interesting parallel.
In some respects, Nintendo is the Apple of the console world. They produce quality hardware in an attractive package. They rely on interesting, well integrated features to sell their hardware. The Wii even looks like a MacMini on its side.
I wouldn't suggest that Apple couldn't do as good a job as Nintendo. But is there really room for both in the market? Especially when on considers Nintendo's (and presumably Apple's) target audience? If Apple made a compelling feature, Nintendo would be forced to retaliate with another. Ideas are a scarce resource, and I doubt Nintendo or Apple has a large enough cache of them to avoid lame gimmicks. Kids might be fond of gimmicks, but grown ups usually aren't.
In the end, this would erode both brands.
Oh come on. I was just suggesting that making a distributed "ROFL" printing system using the ROFL-inducing vulnerability would be funny. In a geek way. I could have just said "using a functional equivalent of your code" to get the point across too.
We don't?
Wait, we do. And that's the prime economic reason developing alternative energy strategies is in the US's (and everyone else's) best interests, despite our reliance on our current profits in the energy market.
I think the best use for this technology would be to put it on every roof in in America (and Europe and eventually the world), and use nuclear power as a method to buffer against periods of low sunlight.
While the major volcano/meteor event you mentioned could deplete the nuclear buffer, it would do that (and worse) now.
At the very least, considering the effects on the economy that nearly free energy would have, we could build enough nuclear power plants to completely handle our energy needs in case such an unfortunate turn of events occured. Hell, we could sell of the surplus nuclear energy to subsidize projects like the complete mechanization of food production, -- obviously using our nearly free energy. Or just lower taxes (though I would prefer the former)
I suppose it could be bad globally, but global weather effects would get lost in the inherent "randomness" of weather. The butterfly effect you alluded to shows that all events are (in a fairly abstract sense equally) weather changing, not just the big ones.
I'm thinking 260 or so square miles of solar cells would severely alter the local ecosystem. But who cares? There's a chance for more life in the area with lowered ground temperatures.