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Most are about five minutes (some longer, some shorter) and cover topics like chemistry, physics, calculus, geometry, biology, Algebra, trigonometry, grammar, ACT prep, and SAT prep. ” Kudos to their ability to achieve that goal. They are professionally recorded and presented by expert teachers with a class screen or whiteboard.
For example, as transportation networks play a key role in moving goods and materials from suppliers to customers, Zach hopes to integrate models of global transportation networks into his models of global supply chain networks. There are many branches of maths, including algebra, geometry, calculus and statistics.
And in Mathematica and the Wolfram Language that’s achieved with Integrate. And over the years that’s exactly what we’ve achieved—for integrals, sums, differential equations, etc. It’s the end of a long journey, and a satisfying achievement in the quest to make as much mathematical knowledge as possible automatically computable.
How do we achieve this? Let’s say that we’re trying to achieve the objective of having an efficient transportation system for carrying people around. The same is true of axioms for areas of abstract algebra like group theory—as well as basic Euclidean geometry (at least for integers). One person wants to get a cookie.
Any integral of an algebraic function can in principle be done in terms of our general DifferentialRoot objects. Turning from calculus to algebra, we’ve added the function PolynomialSumOfSquaresList that provides a kind of “certificate of positivity” for a multivariate polynomial. And a third of a century later—in Version 13.0—we’re
For example, we know (as I discovered in 2000) that (( b · c ) · a ) · ( b · (( b · a ) · b )) = a is the minimal axiom system for Boolean algebra , because FindEquationalProof finds a path that proves it. Still, finding such paths is what automated theorem provers do. It’s not simple to do this.
Part of what this achieves is to generalize beyond traditional mathematics the kind of constructs that can appear in models. To say something more global requires the whole knitting together of “economic space” achieved by all the local transactions in the network. It’s very much like in the emergence of physical space.
Part of what this achieves is to generalize beyond traditional mathematics the kind of constructs that can appear in models. To say something more global requires the whole knitting together of “economic space” achieved by all the local transactions in the network. It’s very much like in the emergence of physical space.
An idea that was someone’s great achievement had been buried and lost to the world. A test example coming soon is whether I can easily explain math ideas like algebra and calculus this way.) And I’m then usually left with a strong sense of responsibility. But now I have found it again, and it rests on me to bring it into the future.
Ed was never officially a “test pilot”, but he told me stories about figuring out how to take his plane higher than anyone else—and achieving weightlessness by flying his plane in a perfect free-fall trajectory by maintaining an eraser floating in midair in front of him. Then McCarthy started to explain ways a computer could do algebra.
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