Tai's article was printed in February of 1994. An author comment printed in the October 1994 issue is titled "Tai's formula is the trapezoidal rule." I don't have full text access to either, but the title of the followup is not encouraging.
As for a calclusless physics course, I took one in high school. It was the most advanced in the school, sadly, and didn't include calculus--even though half the class was in AP Calc or had already taken it. Anyway, I digress. Mostly it focused on free body diagrams and solving simple kinetics problems. I remember the equations of kinematics were used a lot. They relate velocity, position, and time under constant acceleration and are all trivially derived with calculus, though for the purposes of that class they fell out of the sky. I vaguely recall some electricity and magnetism, though I can't imagine what could have been taught without differential equations.... Oh, I remember--we spent some time on Ohm's Law. And the teacher brought in a 1 Tesla magnet which screwed up the TV (CRT) from a dozen feet. He also shot a potato gun into the air, had us time how long it took to get down, and had us calculate the muzzle velocity assuming the path was purely vertical and the potato was under constant acceleration. That problem is solved with one of the kinematics equations I mentioned.
So, I suppose we basically did a little algebra, memorized some formulas (or derived them ourselves, as the case may have been), and screwed around a lot. I imagine a similar course in a university would be slightly more formal and stuff in more equations pulled from thin air. The students aren't interested in the derivation anyway, I suppose; why bother giving it to them? For that matter, why bother teaching physics to med students?
"Reinvented" is putting it a bit strongly, at least from the abstract of the paper (I, shockingly, don't have access to the Diabetes Care journal to see the full extent of the "discovery"). As well as I can gather, he noticed the area of a curve can be approximated by making a bunch of rectangles underneath it, and that you can be "clever" and add a triangle above the rectangles to get an even better answer. That's not even close to reinventing integration. To be honest, it's not even integration in a formal sense; no idea of limits seems to be used, for instance, or boundedness, infinite sums, or infimums/supremums.
Did he, say, find the fundamental theorem of calculus and derivatives, along with a few formulae like the binomial theorem which gives the usual power rule? Is he able to compute some integrals symbolically? If so, I'd be impressed. But, and without being able to read the article itself, he seems like a guy who got tired of counting cells on graph paper and noticed he could do a little better by drawing trapezoids.
I think it's a little more realistic than that. It seems reasonable to me for CS to worm its way into primary and secondary education more. Maybe the natural evolution of education will produce more people who can implement their own computer-y ideas.
Imagine being unable to program in a stereotypical sci-fi future--wouldn't you feel like the odd one out? As computers make up more daily interactions, maybe programming skill in the general population will increase.
You have already told me... that you don't really understand fully what goes in to the work they're doing [by saying "just"].
I'm not so sure. I'm a programmer, but have developed an intense dislike for coding. It's precisely because I know what goes in a lot of coding work that, given a great idea, I might think I "just" need a coder. They can write the ultimately simple code that makes up the bulk of the project while I do something more interesting.
Say I had an idea for a news site for nerds. What difficulty is there in coding that that a trained monkey shouldn't be able to overcome? Every difficulty I can imagine in making such a site is a solved problem--solved many times over, even. Of course, some projects would be more difficult--like your example of a new OS (though one person coding that seems far fetched, depending on the scale). For a certain level of project difficulty, I suppose I have to agree. But for many projects, programmers are a dime a dozen, doing the ultimately uncreative job of piecing together bits of technology to make something that will be obsolete in a few years anyway. I value their job-related uniqueness as much as my waitress', and calling them "just" a coder is appropriate.
(Incidentally, math is my thing instead of programming. Much more interesting, for me at least.)
Personally I have never wanted to view a PDF inside the browser so I'm not sure why so many people think it's important.
It's nice to have PDFs alongside my other tabs. It reduces the number of windows I have open by 1 per PDF, if I already have at least 1 other tab open. This is a small thing that wouldn't really bother me if it stopped existing one day. Also, some sites display PDFs mixed with HTML on the same page. It's probably not great style, but such is life.
If I were a troll with a bot net, what might I want to do? Maybe DoS WikiLeaks before a hyped release of classified documents, expecting lots of people to blame the US government. This seems much more plausible to me than your possibilities.
Any cardinality higher than aleph-zero is uncountable. I suppose aleph-one is by far the most popular example, though. Who knows, maybe the US has a cardinality between the two working for it? (Yes, that's a continuum hypothesis joke:).)
Sir Roger Penrose is one of the more prominent living physicists. Penrose tilings were named after him (in a nutshell, they generate infinitely complex mosaics with only a few tile types). These tilings later came up in quasicrystals. He also invented twistor theory in the 60's, which is another way to view spacetime. Ed Witten of string theory/M-theory fame--perhaps the second most famous living physicist behind Hawking (my opinion)--applied twistor theory to string theory in 2003. Penrose has controversial views on human consciousness and has suggested our brains must work by a quantum mechanical process. He's written several books on the subject including The Emporer's New Mind. He won the Dirac Medal and Prize in 1989 (Hawking won in 1987; Witten won a similarly-named award in 1985) and has won a laundry list of other awards for theoretical physics. He was knighted in 1994 for his contributions to physics, is an emeritus professor at the University of Oxford, and is 79.
Big-O notation can be used to refer to running time instead of the number of steps in an algorithm's solution. It's purely a notational convenience, saying that up to a multiplicative constant a function is eventually bounded above by another function. (Formally, there's a c so that for some X, x larger than X means f(x) is less than c*g(x).) In this generality it can be applied wherever you see fit.
There are 288 distinct 4x4 Sudoku puzzles (each grid is 2x2; the usual 9x9 case has 3x3 grids). There are ~4 billion 16-square grids of numbers 1-4, so these can be found easily with completely naive methods. A few hours is probably too generous:).
From the article, the source for this information is in a "U.S.-China Economic and Security Review Commission... report to Congress." Also, "A draft copy of the report was obtained on Tuesday by FoxNews.com. The final 2010 annual report to Congress will be released during a press conference in Washington on Wednesday." So, I suppose you can wait till Wednesday and find the report if you're really interested. I'd be sad and a bit surprised if the report's evidence was shoddy, but I wouldn't be at all surprised if the media misrepresented the report in some way. I wish they had simply quoted it.
The story you linked was posted on April 9th. The article in the summary says (twice) its redirect happened on April 18th. They couldn't be the same if these dates are accurate.
That said, this April 18th attack is discussed in a "316-page report to Congress," so it's pretty clear it wasn't *just* noticed.
I always think of Matlab as numeric and Mathematica as symbolic. Sure Mathematica can do some numeric things, but that never seemed to be its real purpose.
I'm not in the field, so I've never seen anyone use Mathematica. But math is a continuum from colloquial to formal. Any tool that allows folks to play around with math in an informal, yet computationally sound way is a good thing. There is nothing wrong with VB and C coexisting -- it is a good thing to have different tools and abstractions available to the laymen and the experts.
I don't think that analogy quite holds up. The free-form language allowed is effectively a shorthand for programming language-style use of Mathematica. I think a more accurate analogy is VB.NET to C#.NET. The VB syntax is friendlier (eg. "MustInherit" for "abstract") but the differences are only skin deep since they compile to the same thing anyway.
"Colloquial" math to me means "non-rigorous", like high school calculus, which is not at all a direct translation away from formal math. The free-form syntax, on the other hand, is pretty much a direct translation.
Try hanging a picture relative to such a frame of reference. You'd say "I measured it and the holes should be exactly 3 inches apart but they won't fit!" because of distance contraction. The grandparent just meant there's no preferred inertial reference frame inasmuch as the speed of light is the same in all of them. They didn't mean there was no reasonable frame, such as (approximately) the one we're in right now.
Slashdot Mirror
You can read the PDF of the source, for $38.00. If anyone does, could you tell us what you find?
Tai's article was printed in February of 1994. An author comment printed in the October 1994 issue is titled "Tai's formula is the trapezoidal rule." I don't have full text access to either, but the title of the followup is not encouraging.
The derivation of the trapezoidal rule is like 3 lines of algebra.
As for a calclusless physics course, I took one in high school. It was the most advanced in the school, sadly, and didn't include calculus--even though half the class was in AP Calc or had already taken it. Anyway, I digress. Mostly it focused on free body diagrams and solving simple kinetics problems. I remember the equations of kinematics were used a lot. They relate velocity, position, and time under constant acceleration and are all trivially derived with calculus, though for the purposes of that class they fell out of the sky. I vaguely recall some electricity and magnetism, though I can't imagine what could have been taught without differential equations.... Oh, I remember--we spent some time on Ohm's Law. And the teacher brought in a 1 Tesla magnet which screwed up the TV (CRT) from a dozen feet. He also shot a potato gun into the air, had us time how long it took to get down, and had us calculate the muzzle velocity assuming the path was purely vertical and the potato was under constant acceleration. That problem is solved with one of the kinematics equations I mentioned.
So, I suppose we basically did a little algebra, memorized some formulas (or derived them ourselves, as the case may have been), and screwed around a lot. I imagine a similar course in a university would be slightly more formal and stuff in more equations pulled from thin air. The students aren't interested in the derivation anyway, I suppose; why bother giving it to them? For that matter, why bother teaching physics to med students?
...he reinvented integration...
"Reinvented" is putting it a bit strongly, at least from the abstract of the paper (I, shockingly, don't have access to the Diabetes Care journal to see the full extent of the "discovery"). As well as I can gather, he noticed the area of a curve can be approximated by making a bunch of rectangles underneath it, and that you can be "clever" and add a triangle above the rectangles to get an even better answer. That's not even close to reinventing integration. To be honest, it's not even integration in a formal sense; no idea of limits seems to be used, for instance, or boundedness, infinite sums, or infimums/supremums.
Did he, say, find the fundamental theorem of calculus and derivatives, along with a few formulae like the binomial theorem which gives the usual power rule? Is he able to compute some integrals symbolically? If so, I'd be impressed. But, and without being able to read the article itself, he seems like a guy who got tired of counting cells on graph paper and noticed he could do a little better by drawing trapezoids.
I think it's a little more realistic than that. It seems reasonable to me for CS to worm its way into primary and secondary education more. Maybe the natural evolution of education will produce more people who can implement their own computer-y ideas.
Imagine being unable to program in a stereotypical sci-fi future--wouldn't you feel like the odd one out? As computers make up more daily interactions, maybe programming skill in the general population will increase.
You have already told me ... that you don't really understand fully what goes in to the work they're doing [by saying "just"].
I'm not so sure. I'm a programmer, but have developed an intense dislike for coding. It's precisely because I know what goes in a lot of coding work that, given a great idea, I might think I "just" need a coder. They can write the ultimately simple code that makes up the bulk of the project while I do something more interesting.
Say I had an idea for a news site for nerds. What difficulty is there in coding that that a trained monkey shouldn't be able to overcome? Every difficulty I can imagine in making such a site is a solved problem--solved many times over, even. Of course, some projects would be more difficult--like your example of a new OS (though one person coding that seems far fetched, depending on the scale). For a certain level of project difficulty, I suppose I have to agree. But for many projects, programmers are a dime a dozen, doing the ultimately uncreative job of piecing together bits of technology to make something that will be obsolete in a few years anyway. I value their job-related uniqueness as much as my waitress', and calling them "just" a coder is appropriate.
(Incidentally, math is my thing instead of programming. Much more interesting, for me at least.)
Windows has no SSH program installed by default that I'm aware of. Did you mean "on practically all machines running Cygwin"?
I bet Cygwin can do it. Maybe someone more knowledgeable will come along and clarify the possibilities.
Personally I have never wanted to view a PDF inside the browser so I'm not sure why so many people think it's important.
It's nice to have PDFs alongside my other tabs. It reduces the number of windows I have open by 1 per PDF, if I already have at least 1 other tab open. This is a small thing that wouldn't really bother me if it stopped existing one day. Also, some sites display PDFs mixed with HTML on the same page. It's probably not great style, but such is life.
It depends on how asinine security clearance reviewers are. If the guy reviewing you cares (or is required to care), then it's good advice regardless.
(Translation: I was abandoned as a child and raised by haxorers, yooz insensitive clod!)
If I were a troll with a bot net, what might I want to do? Maybe DoS WikiLeaks before a hyped release of classified documents, expecting lots of people to blame the US government. This seems much more plausible to me than your possibilities.
Any cardinality higher than aleph-zero is uncountable. I suppose aleph-one is by far the most popular example, though. Who knows, maybe the US has a cardinality between the two working for it? (Yes, that's a continuum hypothesis joke :).)
Sir Roger Penrose is one of the more prominent living physicists. Penrose tilings were named after him (in a nutshell, they generate infinitely complex mosaics with only a few tile types). These tilings later came up in quasicrystals. He also invented twistor theory in the 60's, which is another way to view spacetime. Ed Witten of string theory/M-theory fame--perhaps the second most famous living physicist behind Hawking (my opinion)--applied twistor theory to string theory in 2003. Penrose has controversial views on human consciousness and has suggested our brains must work by a quantum mechanical process. He's written several books on the subject including The Emporer's New Mind . He won the Dirac Medal and Prize in 1989 (Hawking won in 1987; Witten won a similarly-named award in 1985) and has won a laundry list of other awards for theoretical physics. He was knighted in 1994 for his contributions to physics, is an emeritus professor at the University of Oxford, and is 79.
Bob Barker?
(This is a reference to God's number, which is 20.)
Big-O notation can be used to refer to running time instead of the number of steps in an algorithm's solution. It's purely a notational convenience, saying that up to a multiplicative constant a function is eventually bounded above by another function. (Formally, there's a c so that for some X, x larger than X means f(x) is less than c*g(x).) In this generality it can be applied wherever you see fit.
There are 288 distinct 4x4 Sudoku puzzles (each grid is 2x2; the usual 9x9 case has 3x3 grids). There are ~4 billion 16-square grids of numbers 1-4, so these can be found easily with completely naive methods. A few hours is probably too generous :).
From the article, the source for this information is in a "U.S.-China Economic and Security Review Commission ... report to Congress." Also, "A draft copy of the report was obtained on Tuesday by FoxNews.com. The final 2010 annual report to Congress will be released during a press conference in Washington on Wednesday." So, I suppose you can wait till Wednesday and find the report if you're really interested. I'd be sad and a bit surprised if the report's evidence was shoddy, but I wouldn't be at all surprised if the media misrepresented the report in some way. I wish they had simply quoted it.
The crazy thing is that this happened months ago, and nobody noticed.
Odd, Slashdot reported the day afterward: Chinese ISP Hijacks the Internet (Again).
The story you linked was posted on April 9th. The article in the summary says (twice) its redirect happened on April 18th. They couldn't be the same if these dates are accurate.
That said, this April 18th attack is discussed in a "316-page report to Congress," so it's pretty clear it wasn't *just* noticed.
I always think of Matlab as numeric and Mathematica as symbolic. Sure Mathematica can do some numeric things, but that never seemed to be its real purpose.
I'm not in the field, so I've never seen anyone use Mathematica. But math is a continuum from colloquial to formal. Any tool that allows folks to play around with math in an informal, yet computationally sound way is a good thing. There is nothing wrong with VB and C coexisting -- it is a good thing to have different tools and abstractions available to the laymen and the experts.
I don't think that analogy quite holds up. The free-form language allowed is effectively a shorthand for programming language-style use of Mathematica. I think a more accurate analogy is VB.NET to C#.NET. The VB syntax is friendlier (eg. "MustInherit" for "abstract") but the differences are only skin deep since they compile to the same thing anyway.
"Colloquial" math to me means "non-rigorous", like high school calculus, which is not at all a direct translation away from formal math. The free-form syntax, on the other hand, is pretty much a direct translation.
This headline and "article" is another effing Slashdot sponsored advertisement.
I find it more interesting than the other current headlines. Slashvertisement and news aren't necessarily distinct, depending on the crowd.
Try hanging a picture relative to such a frame of reference. You'd say "I measured it and the holes should be exactly 3 inches apart but they won't fit!" because of distance contraction. The grandparent just meant there's no preferred inertial reference frame inasmuch as the speed of light is the same in all of them. They didn't mean there was no reasonable frame, such as (approximately) the one we're in right now.