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Oh, if you think mathworld is awesome, this is going to blow your mind.

Not to be a pain but maybe if we knew what you were interested in and what level of science you're into it would be helpful.

Like me for instance: I'm far from being an astrophysicist but I consider the Discovery Channel version of science insulting. I normally read the dumbed down news and go to other sources to find out more about the elements of the story to get me more familiar with the concepts. Normally it comes full circle to some better articles relating to the original subject. Like for math concepts I normally first turn to Wolfram Mathworld.

A quick Google, oops I mean a quick search using the web site www.google.com, on the Vancouver Stock Exchange turned up this.

I still remember the time the prof in our numies class talked about rounding down if the trailing digit was 1-4, rounding up if it was 6-9, and randomly choosing up or down if it was 5. We were all thinking...Wha? None of the packages I have used since then (eg NASTRAN) employ such a strategy as far as I know, although it may be buried deep inside the guts somewhere.

But it's always been in the back of my mind, doesn't rounding up on 5 introduce a bias?

I'm referring to intrinsic curvature (as opposed to extrinsic curvature) - the idea that it's a property of the surface itself, and not how it is bent in a higher dimension. See http://mathworld.wolfram.com/Curvature.html or http://www.geom.uiuc.edu/docs/doyle/mpls/handouts/ node21.html .

Ummm... How do you use a slide rule? There's no buttons and I can't figure out where to put in the batteries. My Mathematica teacher referred to them and several people laughed, but I didn't get the joke.

- Tash
Hybrids

P.S.: Behold, thy name is sarcasm.

A mathematical analysis can be found here with some cool pictures, but it doesn't explain the rules, unfortunately.

I'll try to make this as simple as possible so your primitive mind can comprehend it:

Given Windows NT is in the set POSIX.
Suppose POSIX is a subset of UNIX.
Then by the definiton of subset, Windows is in the set UNIX.

Reducto ad absurdum => POSIX is not a subset of UNIX. There's a chance that you meant that posix defines a subset of traditional unix features, which is true. It's too bad you lack the basic language skills required to make this simple statement.

You're probably right, but I thought I'd mention that one of the author's names I recognized as a top-notch mathematician: Andrew M. Odlyzko. I read about him in a book about the race to prove the Riemann Hypothesis.

I'd say he's a pretty smart guy - I don't about practical or "street" smarts - but some smart people don't value money so highly.

But the such attack still takes longer than what we have left of fuel in our Sun (~5Gy).

Let's not be too hasty. A 320-bit key has about 97 decimal digits. This is less digits than the RSA-129 key that was broken in 1994 in 6-months by a grid of several thousand computers followed by a couple days of super computer time.

A 512-bit key has about 154 decimal digits, which is less than the RSA 193 key that was cracked last November.

See this for a fairly good reference. If you really need your data secured for the next 20 or so years, you had better be using a 1024-bit or better key.

Gauss's Law says that the gravitational acceleration of a body anywhere in an enclosed sphere is 0. At L4, L5 Earth and Sun graviational forces are balanced. The only accelerations that don't cancel out are the two body accelerations of interest. It is surprising to me that the bodies orbit as fast as 10 times per day. I wonder why they don't use heavier Uranium as the mass. It is an interesting side note that a body can stably orbit one of these points. They orbit with no body (!) at the focus. The Genesis Probe and WMAP missions have already taken advantage of this.

Interestingly, Google's search toolbar will be available only when Shockwave is downloaded for use with Internet Explorer on Windows.

otherwise known as a steinmetz solid, which is often used as a demonstration for engineering drawing or architecture classes to show that a 3-d drawing of an object is not sufficient to determine its actual shape. A mouhefanggai in 3-D drawings looks like a sphere, but is actually a ridged object with a surface consisting entirely of flat-wrapped curves, rather than compound curves.

I prefer the Mathworld web page to Wikipedia. The information is more reliable, and if you're unclear on the definitions, the terms are all hyperlinked so you can see what stuff means. http://mathworld.wolfram.com/PoincareConjecture.ht ml

It's a true 4 dimensional puzzle in the sense that this is what you could build as a rubik's cube equivalent if we lived in a 4d universe rather than a 3d universe.

The green cubes that appear and disappear as you make moves are from the 'hidden' face of the hypercube, which has 8 faces. Their projection is using a base unfolding, to understand what they've done consider the parallel from unfolding a 3d cube into 2d. Imagine you are staring precisely face on at a cube:

      XXX
      XXX
      XXX

Now unfold all the sides connected to the X's so you can see them straight on:

      OOO
      OOO
      OOO
AAAXXXBBB
AAAXXXBBB
AAAXXXBBB
      MMM
      MMM
      MMM

If you started playing a game of rubik's cube on this, you'd soon see another letter show up whenever you made a move, let's call it G for green. Where do the G's come from? From the sixth face of the cube that wasn't visible due to the choice of unfolding. The face exactly opposite of the X's ... the 'rear' of the cube if you will.

Same thing in the 4d case. There are 8 faces, only 7 of which are visible due to their poor choice of unfolding technique.

Here's wolfram's hypercube page for more info:
http://mathworld.wolfram.com/Hypercube.html

He's referring to this fantastic book, where dialogues between Achilles and the tortoise (c.f. Zeno's Paradoxes) are sometimes used to dramatise various concepts.

Considering MS has pretty much said they intend to kill Google as the dominant search engine, the competition is pretty obvious. Though to be more specific it's really more like MSN vs Google competing for web supremacy.

Ever since I saw question #5 on the Google Labs Aptitude Test, "What's wrong with Unix? How would you fix it?", I've always wondered if Google was working on an OS of their own on the sly. If I was Microsoft, I'd be extremely worried about this prospect, since pretty much every Google offering has randomly appeared on the Google Labs site, for the most part with very little fanfare.

Not to say that it is or is not likely (that's a question I'm in no position to answer), but imagine what it would do to MS if a free Google-flavored Linux distro popped up without warning two weeks before Vista shipped? If there's one company out there that could/would concievably try to make such a thing and get it idiot-proof enough to let the average non-tech person use it effectively (this isn't a bait, but none of the current distributions are there, yet), it's Google. And I think the company has enough goodwill stored up (not to mention the media darling status it has attained) that people would actually pay attention to it and give it a try, even if it was bundled with Google Pack or some other way for Google to monetize. Needless to say, this would all but cement Linux as the operating system of choice for the concievable future, since there would finally be an incentive for everyone to create Linux versions of their programs instead of (or along with) Mac and Windows ones if a reasonable percentage of people were using it.

So if I was Bill Gates, I'd be wetting myself over the Google problem. It's not that Google has indicated any desire to destroy Microsoft, it's that they would stand a fighting chance if they decided to give it a go. No company has ever had that power before, so it's quite rational that MS wants to squash them before the tables turn.

The research i've been doing in P2P networks (due to my involvement in the okopipi project) has shocked me. In file sharing, we're living in the STONE AGE. Yes, even with bittorrent (which depends on centralized servers, and there's practically no privacy. And anonymous bittorrent like mutorrent is closed source, who knows if they got a backdoor in there).

EDonkey uses MD4 for hashing, it depends on central servers, and has no anonymity at all. And without mentioning queue # 4892 for a popular file.

Unfortunately for filesharers, file sharing networks based on modern P2P architectures is very scarse. The supernodes / ultrapeers approach is obsolete, easy to disrupt both denial of service and eavesdropping attacks.

The future of P2P is Overlay Networks.

From an architectural point of view, I would recommend the KAD p2p network, which bases its architecture on the relatively-new kadelmia network (See Technical paper on Kadlemia, 2002).

Even then, Kadelmia could be improved because it's based on a Pastry network topology - compared to other topologies like De Bruijn Graphs, proposed by a recent paper in 2003.

And more research is being done dealing with load balancing, anonymity, trust, reputation, etc.

As I said, current peer to peer networks are in the stone age. Someone needs to design a file sharing network based on the latest research, and publish it.

No.

I find graphing in Microsoft Excel to be completely unacceptable. I know that Excel will export data to the applications listed here and I wonder if Open Office's Calc program will do the same.

Of course maybe Open Office Org could see if they cannot find a means by which they could create a competitor to something like Wolfram's Mathematica .

I thought you were making that word up! You weren't.

Menger sponge is the more common term.

If you don't slow it up, you will have cargo turning around the sun about in the same way as the Appolo asteroids (http://scienceworld.wolfram.com/astronomy/ApolloA steroid.html). Unless slowed down properly, their orbits will always stay close to earth's orbit. And guess what, some of those could come back to hit earth as a radioactive meteorites.

Not good.

Simple numerical integrators destroy these identities
at order dt^n for some small but finite n. Run the code
forwards and one can find finite time blow ups due to
the stepping algorithm- however even after a single
time step the numerical solution has unphysical aspects

Finding /constraint conserving/ algorithms is tricky
http://www.ima.umn.edu/nr/abstracts/6-24abs.html

In a sense, the Online Encyclopedia of Integer Sequences, hosted by AT&T Research, does this job already.

With over 100,000 web pages, searchable, with posters' email addresses given, and both internal and external hotlinks and citations to hardcopy literature, this has been the leading collaborationware in Mathematics. The Online Encyclopedia of Integer Sequences (or OEIS) recently faced a problem with increasing numbers of clueless postings.

The distinguished panel of editors, under Dr. Neil J. A. Sloane, first added a keyword of "probation." Submissions so tagged, unless okayed by an editor, are deleted after a reasonable time. At my urging, citing the history of Slashdot, they even more recently adopted the keyword "less" -- meaning less than interesting, but better than probation. "Less" sequences stay in the database, but are given minimum priority in searches.

Similarly, MathWorld is a form of collaborationware or pseudowiki. Although edited by Dr. Eric W. Weisstein and his staff, it encourages submission by form from anyone, and posts attribution to such submissions, and lists of contributors.

I contend that web-based systems have substantially affected the practice of Mathematics. Social mechanisms such as pioneered by Slashdot contribute to weeding out useless from interesting contributions. As with Wikipedia, one's academic credentials mean nothing here. What matters is the quality of one's submissions, as evaluated by one's online peers.

There also many fine Math blogs, but that's another topic.

-- Jonathan Vos Post

The Earth is more of an oblate spheroid so technically by your definition it's not a planet. ;)

Hint: it is one of most common applications of linear programming:

Linear programming, sometimes known as linear optimization, is the problem of maximizing or minimizing a linear function over a convex polyhedron specified by linear and non-negativity constraints. Linear programming theory falls within convex optimization theory and is also considered to be an important part of operations research. Linear programming is extensively used in business and economics, but may also be used to solve certain engineering problems.

Examples from economics include Leontief's input-output model, the determination of shadow prices, etc., an example of a business application would be maximizing profit in a factory that manufactures a number of different products from the same raw material using the same resources, and example engineering applications include Chebyshev approximation and the design of structures (e.g., limit analysis of a planar truss).

(from here)

This subject is heavy on math, requires fluency in matrix (linear) algebra and is usually taught around 4th semester in universities. I dare you to solve such a task using, for example, the simplex method, just in your head :-)

There isn't any particularly better definition of randomness than "unpredicability".

That's true not just as a rule of thumb, but in a more formal sense as well. The word "random" is pretty hard to come up with a mathematically formal definition for, and "pretty hard" may mean "impossible" depending on your definition of "definition" (more on that later). To make things simple, let's just talk about sequences of ones and zeros. Take for example the sequence 01101110010111011110001001101010111100110111101111 ... Definitions of randomness from statistics and probability just require a potentially random sequence to have all possible subsequences of a given length appear with the same frequency. That is, 0 appears exactly as often as 1; 00 appears exactly as often as 01, 10, and 11; 000 as often as 001, 010, 011, 100, 101, 110, and 111; and so on. The sequence I gave above passes those tests with flying colors. But it's not random at all. I'll put some spaces in it, and you'll see the pattern: 0 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111... It's simply counting in binary. The longer you extend the sequence, the better it does in statistical randomness tests--the first few dozen bits have a pretty strong bias for 1 over 0, but that ends up as noise in the long run.

The relatively young field of information theory introduces the concept of "algorithmic randomness." The randomness of a sequence of bits is defined to be the length of the shortest Universal Turing Machine program which ouputs that sequence. In pseudocode, our example sequence is output by the program:

let i = 0
while (true) do
output i
let i = i + 1
end while

That's a comically short program to generate an arbitrarily long sequence. So the example fails tests for algorithmic randomness miserably. The fun part is that the problem of finding the shortest UTM program to generate a given sequence is provably intractable. Thanks to the the Halting Problem, you can't always tell if a given UTM program will halt or loop infinitely. All you could ever know is whether or not the program has output the desired sequence yet--if it's still running, it may do so eventually and then halt, it may output something else and then halt, or it may keep running forever. So algorithmic randomness plugs the holes in statistical randomness by trading an unreliably solvable problem for a reliably unsolvable one. You can't ever be sure a sequence is random, but you can sometimes be sure it isn't.

I got off on a bit of a tangent there about information theory, but my point is that algorithmic randomness captures what we mean by "random" much better than statistical randomness. And algorithmic randomness is just a mathematically formal way of saying "unpredictable."

1 used to be considered a prime number some time ago (a century maybe?). There was also a time when 2 wasn't considered a prime number (possibly those two times overlapped).

The reason for dropping those considerations was that they learned that it only served to complicate otherwise simple mathematicas, because of forcing people to introduce special cases in definitions and proofs to account for those numbers.

You can read more details about this here.

Saying that Fourier analysis becomes, when you go to the discrete domain, "simple linear algebra" shows very clearly that you do not understand the complexities of discrete fourier analysis. That something is in the end reduced to simple operations does not mean that the subject has been trivialized. Most of number theory "reduces" to finite, discrete computation with integer numbers; the representation theory of semisimple Lie groups (and that of the symetic groups) "reduces" to "mere" handling of Ferrers diagrams and Young tableaux. No one who knows anything about these subjects would ever use the word "simple"...

In any case, if you think that one could do with just the discrete version of Fourier theory, and that somehow the original, 'continuous' theory becomes irrelevant when dealing with iPods and what not, well, you need to go back and read up on those subjects...

Saying that Fourier analysis becomes, when you go to the discrete domain, "simple linear algebra" shows very clearly that you do not understand the complexities of discrete fourier analysis. That something is in the end reduced to simple operations does not mean that the subject has been trivialized. Most of number theory "reduces" to finite, discrete computation with integer numbers; the representation theory of semisimple Lie groups (and that of the symetic groups) "reduces" to "mere" handling of Ferrers diagrams and Young tableaux. No one who knows anything about these subjects would ever use the word "simple"...

In any case, if you think that one could do with just the discrete version of Fourier theory, and that somehow the original, 'continuous' theory becomes irrelevant when dealing with iPods and what not, well, you need to go back and read up on those subjects...

The article gives a good overview for the casual reader--if you're interested in the Riemann Zeta Function itself, look here (Zeta Funciton) or here (Zeroes)

I love reading about this stuff, but the actual relation between the zeroes and the prime number theorem must have passed right over my head. Anyone else get it?

The article gives a good overview for the casual reader--if you're interested in the Riemann Zeta Function itself, look here (Zeta Funciton) or here (Zeroes)

I love reading about this stuff, but the actual relation between the zeroes and the prime number theorem must have passed right over my head. Anyone else get it?

Sorry but that's just plain wrong.

When considering the risk of adding one stock to a particular portfolio (say, the S&P 500), the key determining factor is not the individual variation in the Google stock, but the covariance (http://mathworld.wolfram.com/Covariance.html) it has with all the other stocks in the portfolio already. When it comes to building a portfolio, we primarily care about the return of each stock + covariance among the stocks.

It's really easy to check this just by looking at how the variance of multiple variables (in this case, stocks) is calculated (http://mathworld.wolfram.com/Variance.html).

Sorry but that's just plain wrong.

When considering the risk of adding one stock to a particular portfolio (say, the S&P 500), the key determining factor is not the individual variation in the Google stock, but the covariance (http://mathworld.wolfram.com/Covariance.html) it has with all the other stocks in the portfolio already. When it comes to building a portfolio, we primarily care about the return of each stock + covariance among the stocks.

It's really easy to check this just by looking at how the variance of multiple variables (in this case, stocks) is calculated (http://mathworld.wolfram.com/Variance.html).

>For a game, the best way to solve ODEs is numerically. Since you don't need the >precision of the exact solution, the solutions are considerably simpler >computationally once you've linearized them. Doing RK4 on the fly is precisely >the best solution to the problem.


While RK4 is a good general purpose integrator, it's better to use embedded methods that will speed up the runtime due to adaptive stepsizing.

http://beige.ucs.indiana.edu/B673/node54.html


Also, I would speculate that most games might involve functions that evolve with a steady stiffness. For such methods, it's faster better to do a Jacobian based Rosenbrock Method (Kaps Wanner basically) on large intervals and then do a Bulirsch-Stoer rational extrpolation.

http://en.wikipedia.org/wiki/Numerical_integration

I can't find any good references to this online, but I'm just parrotting Numerical Recipes Chapter 16.


>solving a linearized ODE is just plain ol' ordinary matrix math, very >parallelizeable

Well linearization will only work well near a fixed point of the phase space. Generic ODEs are not as simple to parallelize it as one may think. Data parallelization is trivial (provided your libraries/subroutines are thread-safe). However, really solid function parallelization of the integration of a particular set of IC's across one time interval is very difficult as a differential equation evolves causally through time. Thus, in order to know what's what at time T, you'll need to know all the values at earlier times t<T so a parallel thread has to wait for those other parallel threads to finish, making it pointless.


What you CAN do of course is parallelize the step incrementation process itself, however you run into the same problem during the calculation of each term in a particular RK step increment. If you look at the formulae (http://mathworld.wolfram.com/Runge-KuttaMethod.ht ml)

You need k1 to calculate k2, k2 to calculate k3 & so on. Thus, the k2 calculating thread has to wait for the k1 thread to finish etc etc.

So I don't think parallelizing this will help much.

I'll admit that I have never done any functional parallelization of ODE's myself, as I use GSL/numerical recipes that are not multithreaded (though thread-safe) , so I do not have any first hand info on how much it's going to speed up the program .

There has been some talk of parallelizing GSL itself

( http://www2.imm.dtu.dk/~jw/para04/Abstracts/enriqu e_s_quintana/enrique_s_quintana.html
)

though again I don't know if there has been any concrete progress.

The halting problem is not the same thing as self awareness, unless you have a very strange definition of "self aware". So if you, as a human, can't answer whether the Collatz function terminates on an arbitrary finite integer, does that mean that you are not self aware? Do you think a Chimpanzee is self-aware? How many theorems have they proved?

This is exactly why Turing proposed the Turing test. It's the only objective way to gauge human-like intelligence proposed thus far. You have to bypass a human's bias about our uniqueness.

I am doing research into the pondoromotive force at the moment work goes slow because my Uni has nothing that can produce the effect. Hell, we can't even make our damn buckyballs.

Current electrical motors/generators are up to 99% efficient, and the loss is mostly in resistive loss in wire.

There is no room for any meaningful improvement unless you claim to have more than 100% efficiency, and they do. Lunatic bin right here!

Current electrical motors/generators are up to 99% efficient, and the loss is mostly in resistive loss in wire.

There is no room for any meaningful improvement unless you claim to have more than 100% efficiency, and they do. Lunatic bin right here.!

I was curious as to what they based their claim on?

First, go to http://www.flynnresearch.net/ to se some details on this.

The answer is:
Just doctor up formulaes: Force is proportional to magnetic flux. Se http://scienceworld.wolfram.com/physics/ Look up 'amperes law', 'magnetic force' and 'Lorentz force'. As you can see they are all _linear_. I.e. F=B*k. (Force = Field times some constant. Flynn makes the relationship quadratic: F=B^2*A/2u.

To translate for /. readers: You have one C++ programmer, and you need more work done. Just hire one more programmer, and to your surprise, you get 4 times as much done.

This is a result of Snell's law. When light travels from a material with a high index of refraction to one with a low index, its angle with the plane gets smaller. See http://scienceworld.wolfram.com/physics/SnellsLaw. html where they show the light going in the opposite direction.

If you go at a low enough angle, you get reflection. Inside a fiber optic cable, all the light rays are going at a low enough angle to have reflection (and they reflect at the same angle). But if you bend the cable a bit, some rays will have a high enough angle that they escape. Note that my 'high' and 'low' are with respect to the plane, while usually when doing calculations the angle is measured against the normal.

Mass ~2xvolume tons: ~300,000,000,000,000,000,000 t

It appears that Earth's mass is about 5,976,000,000,000,000,000,000,000 kg. If we hauled 1000 kg per day into orbit, we'd be shipping about 0.00000000000000000596% of Earth's mass into orbit per year!

And, in a wonderful "pot-kettle-black" moment, you say TFEstimate is off by about 14 orders of magnitude...

For those of us who studied maths to get degrees in computers, there's really only about a half dozen orders of magnitude; for example O(1), O(log n), O(n), O(n log n), O(n ^ k), O(2 ^ log(n)), O(2 ^ n), and O(2 ^ n ^ n ^ n ^ ...(n times)).

Ackermann's function is somewhere in the next order of magnitude above the last one there... Some theorize that the halting problem and other uncomputable problems are bounded by the next order of magnitude beyond that.

And we'd need to go another four orders of magnitude to get your "14 orders of magnitude" , even though we exausted the number of atoms in the universe before we even got here.

IOW, just making the point that different readers read different things in the same words.

Highschool flunky here, that has an intuitive, but apparently mostly useless understanding of basic physics.

Easy. Einsteins theories only say that traveling as fast as light is impossible (unless mass = 0)... but from someones vantage point, particles that are always traveling faster than light could exist...

Actually, for technical issues Wikipedia articles seem to be ok -- people who write articles are people who care, and thus they usually have reasonable expertise. As a mathematician I can say that the Math articles are quite reasonable. Still can replace MathWorld, but if you need a definition you can look it up. Physics articles are not quite as good, mostly because of popular influences (tend to discuss popular controversies), but are rather reliable. Politics is a different kettle of fish -- because people have a stake and are rather more biased.

When someone says they are proud to drve an SUV I imagine them in a klein bottle shaped SUV driving it up their own ass. YMMV but if your are driving an SUV it will be low.

The problem with ID is that its proponents are making a passive/aggressive attack upon (the godless) theory of evolution bc they claim that Evolution has "gaps" or fails to explain everything about the existence of organisms. They are going after what they perceive is a weakness and in there ignorance running into is strength. Like all theories, Evolution evolves. It is subjected to the scientific method of test and observation. If it is found lacking, corrections are made to improve its accuracy. And this is the crux of the Creationists argument. According to them, since Evolution is subject to corrections then it doesn't have to be accepted as gospel truth. Therefore, *ANYTHING* is possible including their competing "theory" which is based on intuition and faith instead of science. Let me be clear here. The creationist are using a variation of the Fallacy of the Stolen Concept. The concept they are stealing is what constitutes the basis of proof. Since reason, logic, and observation are used to enhance or invalidate prior scientific knowledge that was based on reason, logic, and observation then this invalidate reason, logic, and scientific observation(!) In other words they want to USE logic to make a logical argument that logic doesn't work therefore we should turn to faith instead. What they are throwing out with the bathwater is that when science overturns previously held truths it replaces it with something better based on reason, not faith. This is really an attack on how we acquire knowledge as a species. Do we assume the awesome responsibility of thinking for ourselves and learn from our errors? Or, do we abdicate our brains, stop thinking and let mystic faith provide the answers. Human knowledge is not omnipotent, nor is completely impotent either. Just because I can't prove Fermat's Last Theoremhttp://mathworld.wolfram.com/FermatsLastThe orem.html does not mean that i dont know 1+1=2. Human knowledge is a PROCESS of discovery and the scientific community should not be afraid to admit it.

The IPCC Report is not really a scientific study, it's a meta study, a study of studies.

The problem with this approach is it tends towards argumentum ad populum which basically means if enough people believe in something it must be true which is not at all how science works.

Wein's displacement law gives Earth's radation peak in the infrared region at about 10 micrometers. H20 makes up 2% (50 times more than CO2) of the amosphere and has a much higher reflectivity in the 10 micromenter region.

As water vapor, H2O has a positive feedback causing further "warming" but when it forms clouds it has a negative "cooling" affect. So there's a least one model than suggests CO2 will cool the Earth. Also, more clouds means more rain which means more plants which means less CO2. So it's quite possible for the Earth to self regulate itself.

I'm not saying CO2 isn't a problem but what the IPCC has done is to take the worse possible senario out of a whole bunch of other options.

Don't forget, CO2 makes up only 0.04% of the atmosphere and over 90% of CO2 came from natural, non anthropogenic sources.

There's also some evidence that about 30% of the 8 gigatons of annual CO2 can be accounted for by forest fires

Let's not even get into volcanic activity.

You're wrong -- GPS satellites are in 12 hour orbits, not 24: http://documents.wolfram.com/applications/astronom er/Notebooks/GPSSatellites.html

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.