From Stephen Diehl’s Exotic Programming Ideas: Part 2 (Term Rewriting). “A very general function intended to include most of the known special functions as particular cases”. This is particularly useful for automatic integration.
At the core of Mathematica’s integration engine is transformations involving this approach along with a variety of other heuristics that work quite well. For example the MeijerGReduce function can be used to transform many clases of special functions into the G-function equiveleants and then FullSimplify and other identities can reduce these expressions further.
the majority of the established special functions can be represented in terms of the Meijer G-function
A notable property is the closure of the set of all G-functions not only under differentiation but also under indefinite integration
the closure implies that whenever a function is expressible as a G-function of a constant multiple of some constant power of the function argument, f(x) = G(cxÎł), the derivative and the antiderivative of this function are expressible so too
A review of Where Is My Flying Car? by J. Storrs Hall
a good explanation for technological stagnation is that the only technological revolution of the last 50 years, computing, was the only one that didn’t need more power than could be provided by the technology of the 1970s.
I was shocked to learn that certain designs of nuclear batteries were actually manufactured decades ago and used safely in implantable pacemakers.
just over a pound of enriched uranium has as much energy as 10,000 gallons of gasoline or over 100,000 pounds of anthracite coal
True nanotech, he says, was killed by federal funding. Well, not by federal funding directly, but by a storm of academic politics that followed predictably from the $500 million National Nanotech Initiative kicked off under President Clinton. With a new pot of money on the table, and with academic funding being largely a zero-sum game, researchers in adjacent fields responded in two ways. First, they rebranded whatever they were doing “nanotech”, even projects such as nanoscale materials science that are unrelated to the original vision of atomically precise manufacturing. Second, they aggressively attacked that original vision. The result was that all the funding and credibility for true nanotech evaporated.
The Machiavelli Effect: Hall cites a passage from Machiavelli written in the 1500s that describes how politically dangerous it is to attempt to introduce an innovation: all of those who will be the losers if you succeed are galvanized against you, whereas those who would be the winners are much less motivated, given how speculative and uncertain the new innovation is.
Through maybe the 1950s, visions of the future, although varied, were optimistic. People believed in progress and saw technology as taking us forward to a better world. In the span of a generation, that changed, with the shift becoming prominent by the late 1960s. A “counterculture” arose which did not believe in technology or progress: indeed, a major element of the counterculture was the environmentalist movement, much of which saw technology and industry as actively destroying the Earth.
Let us consider the optimization problem in low dimensions vs high dimensions. In low dimensions, it is true that there exists lots of local minima. However in high dimensions, local minima are not really the critical points that are the most prevalent in points of interest. When we optimize neural networks or any high dimensional function, for most of the trajectory we optimize, the critical points(the points where the derivative is zero or close to zero) are saddle points. Saddle points, unlike local minima, are easily escapable.
Lots of great responses, but to be honest I don’t see almost any of them as worth re-reading every year. Best response is this list which includes about a hundred books, which I assume would take all year to re-read, but I’ll allow it anyway.
(I don’t understand the x-axis so I’m not sure if this really makes sense)
So many great points about the process and funding of scientific research in this thread.