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1. Mathematics does have hypotheses. Some eventually get proven (Fermat's Last Theorem), while others accumulate statstical support (Riemann Hypothesis). If mathematics were simply a matter of proving things, without doing numerical experiments, then mathematical hypotheses would not exist.
2. Mathematicians don't actually work in the manner you describe. Proof is the last, not first, step in mathematical research. Mathematicians look instead for patterns and connections -- the same kind of inductive reasoning that scientists use, in fact. The major difference between math and science is the degree of confidence at the end of the process.
3. The axioms in math aren't proven. In fact, it's unclear exactly what their status is. Some, like the axioms of set theory, seem to be fundamental features of our thought. Others, like the parallel postulate are just a matter of convention. In principle, an axiom could be "falsified" either by constructing a consistent math system that denies it, or else by showing that the axiom leads to inconsistencies.

1. Mathematics does have hypotheses. Some eventually get proven (Fermat's Last Theorem), while others accumulate statstical support (Riemann Hypothesis). If mathematics were simply a matter of proving things, without doing numerical experiments, then mathematical hypotheses would not exist.
2. Mathematicians don't actually work in the manner you describe. Proof is the last, not first, step in mathematical research. Mathematicians look instead for patterns and connections -- the same kind of inductive reasoning that scientists use, in fact. The major difference between math and science is the degree of confidence at the end of the process.
3. The axioms in math aren't proven. In fact, it's unclear exactly what their status is. Some, like the axioms of set theory, seem to be fundamental features of our thought. Others, like the parallel postulate are just a matter of convention. In principle, an axiom could be "falsified" either by constructing a consistent math system that denies it, or else by showing that the axiom leads to inconsistencies.

I will agree that mathematical models that we produce will be flawed and will have problems, but (with some exceptions) we can generally get "close enough" (such as in statistical applications, where a 95% confidence interval is enough, as you mentioned). However, I do not believe that math in general is flawed -- although our understanding may be.

Math, like physics or chemistry, is imuteable. We do not change it by understanding of it. What we consider math is just our understanding of what actually happens.

"But it's also useless to us except as it helps us understand the world around us, or model something we've obvserved, etc. There are still things it is wholely unconcerned with (but can be applied to in various situations)."

That is where I think you have something to learn. While math may seem, at first, to be unconnected except for use in models, it is not. Goedel's incompleteness theorem seems like it is not useful to someone with your point of view. However, Dr. Jurgensen has the CS POV where we see that not only do we have axiomatic systems about number theory, we see that when we organize problems into classes of ones we can solve in finite steps, we run across similar odd issues.

He has lecture slides up somewhere. His major findings are also published. I can't find the specific reference at the moment, but there is a string of publications up until 2005 that are mentioned in his most recent work.

From my own notes on his lecture slides:
"What does this mean?

* Independence is a wide-spread phenomenon.
* Independence seems to be a consequence of a deficit of information (we cannot gain information by deduction).
* The measure of information must be relativized ( instead of H).
* Proving (computing) cannot generate information, but can only make hidden information explicit. GIGO -- you only work with what is given.
* True and independent statements are no artifacts; independence is essentially independent of the particular formalization of the theory.

The question of whether there are many "interesting" true and independent statements remains unanswered.

Conjecture: It could be possible to prove that statements are correct merely in terms of their size and length related to the work."

So, as you see, an artifact of math affects information theory (indeed, they are almost the same once you get into the details). This affects the kinds of systems of logic we can build in computer science, and also what kinds of brain activity can exist if the human mind is at all like a Turing machine in terms of its ultimate implementation. I think that's important.

Math can seem to be unconnected, but the relationships are there.

Speaking of the UK/OZ/NZ "maths", I have an Aussie client who refers to any mathematics I do (for him) as "sums" (doesn't matter if it's addition or what). Granted, he's not a mathematician (neither am, i'm a programmer and rely on mathworld to remind me of all the math I didn't pay attention to in school while busy oogling over girls I couldn't get with), but he is quite a shrewd businessman.

Speaking of school and math... I was in highschool less than 10 years ago, and we were allowed to use graphing/programmable calculators for our math classes (including tests). I consistantly failed math tests. Not because I would get the answers wrong, quite the contrary -- I would get all the answers right. Using the programming capabilities of my calculator, I would write small applications to essentially do my homework and complete tests with speed. The teachers failed me due to the fact that I didn't show my work. Then treated me like a wise guy when I started to print out the application code prior to tests and hand it in along with the completed tests. Oddly, it turns out... I do math and programming every day in the "real world", so it's too bad that they didn't embrace the method I had for completing my tasks. (Later on I found out that, other kids would use the programming cabilities of their calculators just to make crib sheets for science courses... so I really think it was a good way to apply my calculator in class in the long run, and it was overlooked by my teachers).

Also... Another point is, if you're a street smart person who maybe doesn't directly use mathematics... You DO rely on others for your mathematics needs at some point. If it's just the cash register at 7/11 or if you hire a team of programmers, somehow, somewhere you DO rely on math and it's a part of your everyday life. I believe mathematics were developed by mankind out of sheer necessity and to describe things in everyday life ("How many units of grain will we need for the winter given a population of X people?", "How far is it from point A to point B?", etc etc) and you are doing some amount of math in your head throughout the day.

Lastly... Turns out that the word "sums" is being used properly by my client according to merriam-webster, check out Sums, defintition 5b

http://mathworld.wolfram.com/GoedelsIncompleteness Theorem.html

Many of today's technologies wouldn't be possible without modern mathematical topics like Fourier Analysis, the Shroedinger equation, and Symbolic Logic just to name a few.

Most of us use these technologies on our ipods, cars, and computers without even thinking about them.

Yay, Math!

there are statements in math that we know we can neither prove nor disprove

There called Axioms, and they are needed in all formal logic. If you really don't understand this concept visit:

http://mathworld.wolfram.com/Axiom.html

"This is the first time a computer program could simulate a phase transition because the computers would always bog down at what's known as the 'critical slowdown.' We figured out a way to perform a kind of end-run around that critical point slowdown and the results allow us to calculate certain critical point properties for the first time."

For example, in the Ising Model or the Potts Model, one can examine system parameters arbitrarily close to the critical point, in finite time, using a Cluster Algorithm. This page gives some information on how the cluster algorithm. The page has a java applet graphically depicting the system for a variety of algorithms.

Just for completeness, here's an Ising model applet that I wrote, which doesn't just have a system animation, but allows you to calculate and plot data (specific heat, magnetization, etc) as the system passes through the critical point. This applet uses the Metropolis algorithm for time advancement, hence it is subject to critical slowdown. In that respect, the applet is flawed because close to the critical point I don't generate enough Metropolis iterations to ensure the subsequent frame is sufficiently thermally indepdent from the previous state. However, the cluster algorithm would remove these limitations. This applet has actually been used in graduate physics classes at Johns Hopkins to demonstrate magnetic phase transitions.

And also for completeness, here's a Potts model applet, but it doesn't acquire data for plotting like the Ising model. The Potts applet actually uses the Microcanonical ensemble, whereby the energy of the system is conserved, but the Ising applet uses the Canonical ensemble, where the system is in contact with a heat bath at some settable temperature.

And in case anyone's curious, these applets (except for the first one) are part of the Java Virtual Physics Lab , which contains a few different physics java simulations I wrote to help with conceptual understanding.

"This is the first time a computer program could simulate a phase transition because the computers would always bog down at what's known as the 'critical slowdown.' We figured out a way to perform a kind of end-run around that critical point slowdown and the results allow us to calculate certain critical point properties for the first time."

For example, in the Ising Model or the Potts Model, one can examine system parameters arbitrarily close to the critical point, in finite time, using a Cluster Algorithm. This page gives some information on how the cluster algorithm. The page has a java applet graphically depicting the system for a variety of algorithms.

Just for completeness, here's an Ising model applet that I wrote, which doesn't just have a system animation, but allows you to calculate and plot data (specific heat, magnetization, etc) as the system passes through the critical point. This applet uses the Metropolis algorithm for time advancement, hence it is subject to critical slowdown. In that respect, the applet is flawed because close to the critical point I don't generate enough Metropolis iterations to ensure the subsequent frame is sufficiently thermally indepdent from the previous state. However, the cluster algorithm would remove these limitations. This applet has actually been used in graduate physics classes at Johns Hopkins to demonstrate magnetic phase transitions.

And also for completeness, here's a Potts model applet, but it doesn't acquire data for plotting like the Ising model. The Potts applet actually uses the Microcanonical ensemble, whereby the energy of the system is conserved, but the Ising applet uses the Canonical ensemble, where the system is in contact with a heat bath at some settable temperature.

And in case anyone's curious, these applets (except for the first one) are part of the Java Virtual Physics Lab , which contains a few different physics java simulations I wrote to help with conceptual understanding.

p.s.: Here is a link to the "mass, charge and spin" properties of black holes:
http://scienceworld.wolfram.com/physics/BlackHole. html

Its ok if the reynolds number (which is merely a ratio of the inertial forces to the viscous forces) is off a little bit... since the size of the model compared to the real craft is probably only an order of magnitude smaller, compared to many orders of magnitude larger than an atom, its inconsequential ... And actually it won't be off at all, in a modern wind tunnel it is calculated as a function of dynamic pressure , which will not vary.

Using the Pi theorem we can find nondimensional quantities. The quantities we measure in the test case will be the same for the real case.

-everphilski-

Classification of Finite Simple Groups and here

This "Theorem" completely categorizes finite simple groups - in effect the "building blocks" of Group Theory. It is one of the great triumphs of 20th century mathematics. It's also in the area of 15000 pages long, and represents the combined efforts of scores of mathematicians who worked on it. It is confidently believed to be correct, but seeing as very few people really understand the majority of this "theorem" in detail, it's their word that it "works".

"Computer memory is one-dimensional"

Took me a while to try to see this from your perspective... but no I think you're absolutely wrong.

One bit of computer memory in abstract needs at least two dimensions:
- time/statechange
- something for time/statechange to act upon

I can see why one would think of someething like a bit as purely on/off and thus only one dimension, but one can't have time/statechange completely on its own. And you won't be able to do/express much (like the simulation of further dimensions) with one bit of computer memory. If you want more bits it would make an array (or linear list or whatever really) which is a concept that is at least 2-dimensional in itself (and which can easily represent more). With time/statechanges that adds up to at least three dimensions needed (bit, linear address, statechange).

"capable of accurately modeling three-dimensional phenomena"

Yes and an even better example would be to have it model stuff like a pentatope (the simplest regular figure in four dimensions) and then even add POV statechanges (which takes yet another dimension which makes for a total of at least 5 simulated dimensions as you have the four "solid" ones as well as time/statechange) to get the interactive graphic at the linked page. Now you could have a large (actually infinite) number of possible statechanges available through rotation and translation (not implemented at the page for obvious reasons) at any lenght and any angle but that in itself will not enable you to create a simulated sixth dimension. For that you need to introduce a simulation/definition/variables representing a sixth dimension into the code. And even if you do that the simulation is still a 3-dimensional (x,y-coordinate + time) representation: you haven't actually created for real a fifth or sixth dimension.

So the real reason why any of these surveys into abstract higher dimensions are possible has to do with the word model which implies that someone figured out how to simulate additional dimensions. As such it's not an argument in the way you seem to portray it.

"...a zero-dimensional system (a single point)..."

A zero-dimentional system would not be dimensionless if it has statechanges, in fact a zero-dimensional system would be the same as absolute nothingness.

Those misconceptions aside I think you're totally missing the point of the grandparent because you dislike the reference to God. So let's take God out of the picture (even though I personally think the entity belongs there) because the example by the grandparent applies equally to the philosophical question of "free will vs. causality" (aka "How can one have free will in a world ruled by causality?") which such a fifth dimension (or multiverse if you will) solves in just the same way by removing any contradiction between the two.

Not quite. That would be what's called a hidden variables system: the unit still does have a real location, which is tracked by the program, even if it's inaccessible to an observer within the system. However, that doesn't appear to be the way our universe works; the Bell inequalities show that hidden variables are incompatible with locality.

Not quite. That would be what's called a hidden variables system: the unit still does have a real location, which is tracked by the program, even if it's inaccessible to an observer within the system. However, that doesn't appear to be the way our universe works; the Bell inequalities show that hidden variables are incompatible with locality.

exp(pi*sqrt(163)) is only a near integer, not an exact one. See Ramanujam constant.

The Gauss-Bonnet theorem asserts that the integral of the curvature of a (compact, oriented) surface equals 2 pi times its Euler characteristic, giving an extraordinary beautiful and deep formula.

(This is just one instance of what's called an index-theorem, which usually provide über-beautiful, über-general, über-deep formulas, but tend to be, well, less accessible to the masses...)

There is a semi-ugly rendition of Gauss-Bonnet'd formula into a GIF (Wolfram does GIFs...) here.

Check out Bernar Venet. The web site is a bit crap, a flash plugin or something. But click on 'paintings' and explore. Make sure you find the commutative diagrams the size of a house.

Anybody want to run a PRP test on this number? It will probably take at least 2 CPU-months.

Yes, it's called the Fundamental Theorem of Arithmetic http://mathworld.wolfram.com/FundamentalTheoremofA rithmetic.html

Graham and Rothschild (1971) also provided a lower limit by showing that N* must be at least 6. More recently, Exoo (2003) has shown that N* must be at least 11 and provides experimental evidence suggesting that it is actually even larger.

No. Because of Bertrand's postulate (a.k.a. Chebychev's Theorem) it is known that for every integer n > 1 there is a prime p such that n p 2n.

Why only probably? Just divide it by 2 and see if it comes out even.

Why only probably? Just divide it by 2 and see if it comes out even.

The Prime Number Theorem tells us that in a region of d around a large number n, there are approximately d/ln n primes. For a 9 million digit n, ln n is about 21 million, so one expects to see one prime every 21 million numbers. For example, between 10^9000000 and 1.1*10^9000000, one expects 5*10^8999991 prime numbers.

Bah, show me a new non-Mersenne prime and I'll be impressed...

I'm not a physicist, and I haven't had enough time to really look over the paper thoroughly, but I am a statistician.

My reading of the paper is that the Cauchy distribution is mentioned only to partially define a distribution that is used in an example. That is, there is nothing about the Cauchy distribution that is necessary for their results to hold. The Cauchy distribution is only relevant in an example, and only to partly define a density. Note, furthermore, that nowhere in the paper do they discuss the expectation of a Cauchy density, only the expectation of a score statistic. They do mention in the example that the Cauchy density is "centered" at a point E_0, but that's possible, as the central tendency of a Cauchy can be defined by the median of the distribution.

So you may be right, but I think that their discussion of the Cauchy doesn't detract from the rest of the paper.

Yes, but IMO the non-Euclidian solution is rather trivial: you have a wheel rotating around and following a straight cirumferential (great or small circle) path on a sphere. Being on a sphere, the wheel's path can be "straight" and also a closed loop. You could also have a solution on the curved surface of a cylinder

Whether or not that's an acceptable solution to you depends on how you define "straight" in the problem originally. If you take it to be something like 'a path whose coordinates remain constant except in one dimension' then by using a spherical or cylindrical coordinate system these are good, albeit simple, solutions.

However if you stick to Euclidian geometry in the Cartesian coordinate system then I don't see that there can ever be a real solution.

Sorry to correct, but as a physicist, your jargon is a little bit off. Probabilities deal with future events, the outcomes of which are unknown. What we are discussing here is the likelihood.

Followed the ultimate realization of _really_ astute students: "Why am I pondering such a stupid question? What a complete waste of my time!"...

Good point! You have successfully recapitulated a core part of the mystic traditions (many branches of buddhism, zennist thought, many of the pre- Christian European religions, probably all of the shammanist schools, etc). Lots of people say that there is a core Mystery to existence that is beyond human understanding.

For the geek, the most accessable expression of this is probably the "Copenhagen Interpretation" of post classical physics, which can be generalized to something like this:

Reality is not only stranger than we think; it is stranger than our ability to think can handle. We can't deal with it. We can, however, construct models like quantum mechanics that we can fool around with, and if we do it right, we'll end up with a model that is predictive, useful, and fun. But no matter how successful we make that model, we can be sure that reality itself is different from it in some important qualitative ways.

More on this here and there.

...This thought then leading to inner peace & happiness.

Uh, yeah, one path to Nirvanna is by this route, but first you have to visit the despair of recognizing that you can attach no meaning to your personal existence. Nowadays, most americans who take this route can get themselves labelled "depressed" and go on to some fun experiences with antidepressant medications and the satisfaction of having a blanket medical excuse for having botched up their lives, their relationships, and everything. That seems to me to be an unnecessary and dangerous detour, but wtf, the point is in the travelling, not the destination, right?

Assuming that this "_really_ astute" student successfully navigates around the MAO inhibitors and the SSRIs, and the various other problems en route, there is the inner peace and happiness of knowing that what is important is not who you are or what happens to you, but how you behave.

Here's a neat thought to end with:

Quantum mechanics: the dreams stuff is made of.

Wolfram Research has some interesting explication on historical methods of solving the quintic: http://library.wolfram.com/examples/quintic/main.h tml

"Calibrating" or training these devices must be fun. I have worked on modeling data with mathematical models for scientific projects. While studying the methods I found articles about psychologists using the same methods to model people's behavior. Of course, psychologists in the US would get their samples in the US. That was quite popular in the 60's. Later the psychologists applied their models with the coefficients from the US in Europe. It turned out that the Europeans are all crazy. They just would not match the hyperplanes.

At least the Europeans of today are not using lie detectors to protect air traffic, objects important for national security, check police officers, or innocent people like you and me.

we are 'due for one'.

Mutation of the flu virus into something seriously dangerous, like the 1918 variety, certainly qualifies as a Poisson process. The time between events in the Poisson distribution follows an Exponential distribution. The exponential distribution is "memoryless", that is, the probability that an event will occur in the first n years of a time interval is the same as the probability that, after any number of years in which an event has not occured, an event will occur in the next n years.

Shortly, the fact that we haven't had a flu epidemic recently has absolutely no bearing on whether or not one is coming soon. Even more shortly, we are not 'due for one'. This is known as the gambler's fallacy.

we are 'due for one'.

Mutation of the flu virus into something seriously dangerous, like the 1918 variety, certainly qualifies as a Poisson process. The time between events in the Poisson distribution follows an Exponential distribution. The exponential distribution is "memoryless", that is, the probability that an event will occur in the first n years of a time interval is the same as the probability that, after any number of years in which an event has not occured, an event will occur in the next n years.

Shortly, the fact that we haven't had a flu epidemic recently has absolutely no bearing on whether or not one is coming soon. Even more shortly, we are not 'due for one'. This is known as the gambler's fallacy.

we are 'due for one'.

Mutation of the flu virus into something seriously dangerous, like the 1918 variety, certainly qualifies as a Poisson process. The time between events in the Poisson distribution follows an Exponential distribution. The exponential distribution is "memoryless", that is, the probability that an event will occur in the first n years of a time interval is the same as the probability that, after any number of years in which an event has not occured, an event will occur in the next n years.

Shortly, the fact that we haven't had a flu epidemic recently has absolutely no bearing on whether or not one is coming soon. Even more shortly, we are not 'due for one'. This is known as the gambler's fallacy.

we are 'due for one'.

Mutation of the flu virus into something seriously dangerous, like the 1918 variety, certainly qualifies as a Poisson process. The time between events in the Poisson distribution follows an Exponential distribution. The exponential distribution is "memoryless", that is, the probability that an event will occur in the first n years of a time interval is the same as the probability that, after any number of years in which an event has not occured, an event will occur in the next n years.

Shortly, the fact that we haven't had a flu epidemic recently has absolutely no bearing on whether or not one is coming soon. Even more shortly, we are not 'due for one'. This is known as the gambler's fallacy.

Now let's hook this up with WolframTones and see how long it takes to generate a Billboard hit.

The answer might be t -> infinity

Turing machines have no problem factorising large primes. This is because turing machines are theoretical devices that have an infinite storage and can process an infinite number of instructions. Desktop computers are not turing machines, and their finite speed means that whilst they can factorise large numbers or invert the various crypto functions they can't do it quickly enough to be useful.

Now to say that we can precisely reproduce the effects any deterministic law using classical physics is plain wrong. For example a black hole is deterministic (as is all of GR) yet that doesn't mean you can study black holes in Newtonian physics. In the case of quantum computation it is not the lack indeterminism that gets in the way of standard computers, it is the lack of a tensor direct product state space, along with a dynamical law which acts on all of this state space.

Now this is where I get speculative...
The reason quantum computing can do certain exponential-time calulations in linear time is that it does the calculation on an exponentially large state space, all of these calculations being done at the same time. Consider this analogy...

Suppose you wish to calculate the cosine of an angle. What you could do is take a unit length stick, rotate it by said angle, project it back down to the starting orientation and measure that. If you wanted to calculate the cosines of two angles, alpha and beta, and were prepared to build a three dimensional machine then you could take your unit stick lying in the x-axis, rotate it by alpha in the z-axis by beta in the y-axis, then project it down to the xy-plane to find cos(alpha) and project it down to the xz-plane to find cos(beta). But what is the point in doing this, you might ask? Why not just rotate by alpha in x-axis and measure, then reset and rotate by beta in x-axis and measure? The reason is that it is possible to do the two rotations in just one rotation*, so long as you orientate your rotating device appropriately**. What this gives you is two calculations in just one operation. Now you can probably imagine some n-dimensional device that can calculate numerous*** cosines by doing a single rotation of a vector in its higher dimensional space, then reading off the components in each dimension. The problem with this scheme is firstly that there are only three spatial dimensions to play with in the real world, and secondly that even if there were more dimensions, the scaling is only polynomial which might not be worthwhile. You can probably now see how quantum computation is going to work - you find a 2^n state space, set it up in some simple initial condition, do a rotation**** of this huge space, thereby doing 2^n calculations, then read off the answer. The theoretical problem for QC is trying to work out exactly which rotation to do*****. The practical problem is to try not to read off the answer too soon.

* This is because 3D rotations form a group.

** It may seem like i've actually assumed the answer of the question in the method used to calculate it. In practice, it may be that the particular layout of the machine makes it easy to do this composition without actually having to calculate the cosines.

*** In fact, the 'computational power' of this device is quadratic, not just linear as one might think.

**** Actually it's the complex-number version of a rotation, a unitary transformation.

***** Deutsch, Grover, etc.

Turing machines have no problem factorising large primes. This is because turing machines are theoretical devices that have an infinite storage and can process an infinite number of instructions. Desktop computers are not turing machines, and their finite speed means that whilst they can factorise large numbers or invert the various crypto functions they can't do it quickly enough to be useful.

Now to say that we can precisely reproduce the effects any deterministic law using classical physics is plain wrong. For example a black hole is deterministic (as is all of GR) yet that doesn't mean you can study black holes in Newtonian physics. In the case of quantum computation it is not the lack indeterminism that gets in the way of standard computers, it is the lack of a tensor direct product state space, along with a dynamical law which acts on all of this state space.

Now this is where I get speculative...
The reason quantum computing can do certain exponential-time calulations in linear time is that it does the calculation on an exponentially large state space, all of these calculations being done at the same time. Consider this analogy...

Suppose you wish to calculate the cosine of an angle. What you could do is take a unit length stick, rotate it by said angle, project it back down to the starting orientation and measure that. If you wanted to calculate the cosines of two angles, alpha and beta, and were prepared to build a three dimensional machine then you could take your unit stick lying in the x-axis, rotate it by alpha in the z-axis by beta in the y-axis, then project it down to the xy-plane to find cos(alpha) and project it down to the xz-plane to find cos(beta). But what is the point in doing this, you might ask? Why not just rotate by alpha in x-axis and measure, then reset and rotate by beta in x-axis and measure? The reason is that it is possible to do the two rotations in just one rotation*, so long as you orientate your rotating device appropriately**. What this gives you is two calculations in just one operation. Now you can probably imagine some n-dimensional device that can calculate numerous*** cosines by doing a single rotation of a vector in its higher dimensional space, then reading off the components in each dimension. The problem with this scheme is firstly that there are only three spatial dimensions to play with in the real world, and secondly that even if there were more dimensions, the scaling is only polynomial which might not be worthwhile. You can probably now see how quantum computation is going to work - you find a 2^n state space, set it up in some simple initial condition, do a rotation**** of this huge space, thereby doing 2^n calculations, then read off the answer. The theoretical problem for QC is trying to work out exactly which rotation to do*****. The practical problem is to try not to read off the answer too soon.

* This is because 3D rotations form a group.

** It may seem like i've actually assumed the answer of the question in the method used to calculate it. In practice, it may be that the particular layout of the machine makes it easy to do this composition without actually having to calculate the cosines.

*** In fact, the 'computational power' of this device is quadratic, not just linear as one might think.

**** Actually it's the complex-number version of a rotation, a unitary transformation.

***** Deutsch, Grover, etc.

That the perjorative of "US Scientists" is applied whole cloth, when in fact the "scientists" in question are limited to the telecommunications industry.

But my absolute favorite passage was:

"and to allow yourself to get to the stage where you're a whole hour out of synchronisation with the Sun seems to be mad. Why can't we just leave things the way they are?"

US non-scientists, in conjuction with the FRENCH, have been doing this for well over a century, and seem to be just fine. And lest we forget, the English drug their heels on adopting the Gregorian calendar for 170 years, resulting in an inaccuracy of 11 days.

Stonehenge, however, keeps perfect "time" no matter how slowly the earth moves. My solution? Lighten the load. Let's have fewer Englishmen.

I originally worked out the solution to the cube when the Scientific American article by Douglas Hosfstader appeared. I never got my speed much below one minute. I did manage to win a T-Shirt at a Cube contest though - a contents with several hundreds of participants...

I swear, the first thing some /.ers think when they see anything related to MS is "FUD" or "EEE" or "world domination". Just like that Avalanche research paper. *sigh*

Why would I want to normalize normal vectors?

The word normal is used in two senses in the original sentence. A normal vector is a vector that is perpendicular to (or normal to) a surface. To normalize a vector is to set its length to one by dividing by its length.

Mathworld explains it here where they say:

... unit norm vectors might be called "normalized normal vectors" without redundancy.

Well, I did a little rough analysis to see what the worse case scenario might be. I didn't have much time to pull stats on wind generators, so I just pulled the numbers from the article and extrapolated from there. From the article: The municipal-grade 1 MW turbine would be about 220 feet high, half the size of a comparable propeller system.

Comparable propeller system: 440 feet high from base to propeller tip. Base height + propeller radius = 440 feet. I made some guesses about the size of the propellers based on what I have seen in real life.

Propeller radius = (1/4) base height

Which gives us:

Base height = 352.0 feet
Propeller radius = 88.0 feet

From the article: ... while the propeller designs typically can only generate energy into the low 50's. and The tips of the blades spin much faster than the wind speed, chopping through the air sometimes at speeds of 200 mph.

Propeller max wind speed: 55 mph ~ 80.7 feet/sec. Propeller max tip speed: 200 mph ~ 293.3 feet/sec.

Find the rotational velocity:

v = r * w
293.3 = 88.0 * w
w = 3.3 rad/sec

Let's assume that if the bird is hit by a portion of the propeller that is traveling less than 40 mph, they will only be grazed and not killed. So let's find the radius at which that occurs:

40 mph ~ 58.7 feet/sec
v = r * w
58.7 = r * 3.3
r = 17.8 feet

Next let's define the bird kill envelope as a cylinder 2 feet long and 2 feet in radius (a normal sized flock-type bird). And we will say that if the propeller hits the bird anywhere in this envelope, the result is a kill (tail feathers and such can hang outside this envelope). Since the birds are in a 55 mph tail wind (max conditions for the propeller), we will say they are traveling at a conservative 65 mph (wrt ground).

65 mph ~ 95.3 feet/sec

Time it takes the bird to travel through the propeller plane (idealized to be a plane with no thickness):
2 feet / (95.3 feet/sec) = 21 msec

Angle sweeped out by a single propeller in that time:
3.3 rad/sec * 0.021 sec = 0.069 rad * (180 deg / pi rad) = 4.0 deg

Typical 3 propeller design: 4.0 deg * 3 = 12.0 deg

Given enough time, total possible propeller sweep area:
total_A = pi * 88.0^2 = 24 328.5 ft^2

Given enough time, total possible kill zone area (tangential speed over 40 mph):
total_kill_A = total_A - pi * 17.8^2 = 23 333.1 ft^2

Over our actual time, actual possible kill zone area:
kill_A = total_kill_A * (12.0 deg / 360.0 deg) = 777.8 ft^2

Ratio of our actual possible kill zone area to total possible propeller sweep area:
ratio = kill_A / total_A = 0.032

Assuming a solid wall of birds flies directly into total_A, we'll use a standard packing density of 0.907 (more details).
birds_killed_A = 0.907 * kill_A = 705.5 ft^2

Single bird area:
bird_A = pi * 1^2 = pi ft^2

Number of birds killed in that area:
birds_killed = birds_killed_A / bird_A = 224

Number of birds that made it through alive:
alive_A = total_A - kill_A
birds_alive_A = 0.907 * alive_A
birds_alive = birds_alive_A / bird_A = 6799

So there you have it, 7023 total birds at risk, 6799 survived, 224 died. And that's just the birds that could have possibly been killed. Assuming they are flying uniformly through risk areas (propellers over 40 mph) and non-risk areas (open air), the percentage that die drops dramatically.

However, this estimation does not take into account things like migration paths, nesting areas, non-uniform bird attractions towards the towers, birds flying repeatedly through risk areas, etc, etc, etc. I think careful studies should be done whenever putting up a wind farm to ensure the local avian populations are not adversely affected. Especially when smaller groups of unique birds are considered, such as birds of prey.

Ok, so are these gravitational "waves" real or just a construct to explain gravity?

Essentially the trigger for this question is the whole sound/EM difference. EM is acutally the emission of "stuff" whilst sound is the propogation of energy through a medium and without the medium there is no sound just the vibration of the original source.

Waves are a term used to describe the solutions to the wave equation. While sound waves are a propagation in a medium similar expressions describe effects that have no transmission medium.

That is somewhat negative. He could do very well. Think of musical prodigys including W.A Mozart. In more recent times; Stephen Wolfram, creator of Mathematica http://www.stephenwolfram.com/about-sw/ "Born in London in 1959, Wolfram was educated at Eton, Oxford, and Caltech. He published his first scientific paper at the age of 15, and had received his Ph.D. in theoretical physics from Caltech by the age of 20. --" Went on to create the computer algebra system http://www.wolfram.com/

First off, I must say that IAAP; however, as with all science that shouldn't be your only reason to believe me, it just gives some credence to what I am about to say. One of the most important things to point out with both your critique of the article, and TFA itself, is with regards to the temperature said to be obtained.

The basic problem with describing the temperature of the fusion elements is that there is no clear temperature. To describe something as having a temp, it must be in some form of thermodynamic equilibrium. If you relate the temperature, for example, to the kinetic energy of a particle, then there is a certain distribution of energy of the particles in thermodynamic equilibrium, the Maxwell-Boltzmann distribution.

If, however, you take only a small number of particles at a very specific velocity in the distribution, you may have an exceedingly large temperature, but it cannot be said to be in thermodynamic equilibrium! So calling it eleventy-billion degrees Kelvin [sic], does not make it so.

By way of another example, we may use that same distribution to describe other forms of the energy of the particles, such as their atomic/molecular energy levels. If we to now preferentially prepare the particles in the higher energy states, and compare this energy distribution to a Fermi-Dirac Distribution or the Classical Limit, we see that in order to accomplish this distribution, we need to have either k (Boltzmann's Constant) or T (the ABSOLUTE temperature) to be negative. As k is a constant, we have the absolute temperature, which can never be less than zero Kelvin or Rankine, less than zero.

Then surely such a system could never exist, you cry. But it does, for this "population inversion" as it is called, is what drives nearly all forms of lasing with the exception of Free Electron LASERs and their ilk. Most everyone has seen a laser, and while they may be cool, they certainly are not that cool.

Basically, my major beef is that, unless the system is in thermodynamic equilibrium, you must be very careful about throwing out anything about raising the temperature or similar ideas.

So, why is it still called temperature? Because it's a nice handy word that we're all familiar with. And that is fine, as long as you don't take the analogy of temperature too far and try to apply ideas to it that can't be applied given your assumptions.

I'm certainly not saying you're wrong in being skeptical, nor am I saying TFA is wrong, I am merely suggesting that thermodynamics (particularly the first law) does not successfully deny the claims made.

pfft, as if that could ever happen. I mean, come on, who's ever heard of a scientist becoming a smug bastard because he was rushed through highschool and allowed to enter college early. Seriously, it's not like you can just be a jerk and still be treated with respect. You're certainly not gunna found any multimillion dollar companies and publish your own book because everyone with half a brain thinks you've lost it -- whilst the other half of the scienfic community think you might be onto something if only they could figure out what.

I'd like to hit you with a Dirac Delta, slap you with a sampling theorem, and then have Shannon kick your ass. Fourier will be here to escort your butt out of the room.

"remove all sounds below 20Hz before going onto CD, as that increases the dynamic range of remaining frequencies"

Uh, ok...show your math please.

... an "identify all the test cases" exercise with a simple problem: a function which classifies 3 side lengths with whichever type of triangle they could form.
... MAX_INT ...

Eh... I've read your story and are left wondering; are you sure the exercise function used *integer* parameters rather than reals/floats? Few sets of 3 integers will form a triangle, and you'd have to implement a test for integer triangles. This seems to be a bit much for an excercise about limiting cases...

That said, if the exercise used reals, I suppose the solution still should have tested for overfolows; like checking for NaN:s and INFs.