In most modern Calculus courses, the history behind the useful mathematical results are often left ignored. Though the pragmatic uses for Calculus are numerous, without a fundamental understanding of the origins of its methods, the student is left applying memorized techniques–often lacking an understanding of why those techniques work. It is our intent to explore the historical path, in significant mathematical detail, to the elementary methods of the Calculus.
When researchers in mathematics write grant requests, they are compelled to justify their work to taxpayers and speculate on possible future applications of their research.In the traditional dance of researchers and grant givers, this is well understood: the mathematicians know they are allowed some wishful leeway, and the reviewers generally expect and accept that. Besides, math research is dirt cheap.
But when we talk to the general public, I feel that this overselling is damaging. It misrepresents many of the reasons behind the mathematical endeavor: curiosity, a search of beauty, and a sense of awe which are wholly decoupled from applications…
In the 2018 collection of articles Feferman on Foundations (about which more in a follow-up post), we learn that shortly before his death Sol Feferman proposed to OUP a sequel to his terrific volume of papers In the Light of Logic. He wanted to collect together some more of his later papers of broader philosophical interest, under the suggested title Logic, Mathematics, and Conceptual Structuralism.
There are many circumstances in which uncertainty is warranted.1 Gas temperature measurements, weather forecasts, horse races, coin flips, and clinical trials all have some uncertainty involved. Probability theory is the science that finds commonality among these seemingly disconnected phenomena. We can observe, for example, that the summation of many “repeatable” random events, properly normalized, begins to look like a gaussian distribution (aka the central limit theorem). We can notice common shapes in the histograms of these repeatable experiments, such as “fat-tailed” or “power law” distributions. And if the event is not repeatable, we can at least apply the rules of probability theory to avoid inconsistencies in our thinking (which would allow a savvy adversary to take advantage of us when gambling).
However, I believe there are tasteful and distasteful applications of probability theory. This is because the application of probability to a particular event requires a suspension of disbelief. To consider an event as repeatable and iid is to accept that the causal factors driving the outcome are (practically) unobservable and therefore ignorable. In effect, it means giving up on deeply understanding a causal explanation of the phenomena and instead sweeping the details under the rug of “the distribution.”
This makes probability theory the science of last resort. Only after truly exhausting your ability to investigate causal factors and processes should you indulge in probabilistic thinking. Doing otherwise is a cop-out, one that dangerously feels “scientific.”
A fundamental relation in celestial mechanics is Kepler’s equation, linking an orbit’s mean anomaly to its eccentric anomaly and eccentricity. Being transcendental, the equation cannot be directly solved for eccentric anomaly by conventional treatments; much work has been devoted to approximate methods. Here, we give an explicit integral solution, utilizing methods recently applied to the ‘geometric goat problem’ and to the dynamics of spherical collapse. The solution is given as a ratio of contour integrals; these can be efficiently computed via numerical integration for arbitrary eccentricities. The method is found to be highly accurate in practice, with our C++ implementation outperforming conventional root-finding and series approaches by a factor greater than two.
This post is about a neat trick that is certainly not of my invention, but that should really be more well-known; at least, I haven’t heard of it till I stumbled across it while reading Box2D sources.
There are a lot of bounding volumes out there; the most widespread are certainly spheres and boxes, which come in two flavors - axis-aligned bounding boxes (AABB) with faces parallel to the coordinate planes, and oriented bounding boxes (OBB), which is essentially a AABB and an orientation matrix.
It’s common to use AABB in spatial subdivision structures, like octrees, kD-trees, ABT and so on - the intersection test between two AABB is pretty straightforward. However, when dealing with dynamic meshes, it is needed to recalculate the AABB of the mesh when the mesh transformation changes.
I’ve been curious about Gottfried Leibniz for years, not least because he seems to have wanted to build something like Mathematica and Wolfram|Alpha, and perhaps A New Kind of Science as well—though three centuries too early. So when I took a trip recently to Germany, I was excited to be able to visit his archive in Hanover.
Leafing through his yellowed (but still robust enough for me to touch) pages of notes, I felt a certain connection—as I tried to imagine what he was thinking when he wrote them, and tried to relate what I saw in them to what we now know after three more centuries:
Despite finding no specific examples, researchers have proved the existence of a pervasive kind of prime number so delicate that changing any of its infinite digits renders it composite.
How many people would you need at a party to guarantee that at least three individuals know each other, or that at least three do not know each other? Working this out with pen and paper may take a while, but many mathematicians will readily tell you the answer is six. This party scenario, also called the “friends and strangers” theorem, is based on a concept known as Ramsey numbers, named after early 20th-century British Mathematician Frank Ramsey.
Now, imagine that more people are invited to this hypothetical party. How many would be required to ensure that at least five people know each other, or, conversely, that at least five are strangers? The answer is not clear. Indeed, mathematicians only know that the number of required people would be at least 43 and no more than 48. The actual answer falls somewhere in this range but is unknown. Add even more people to the party, and the uncertainty in the problem quickly becomes enormous.
Now, for the first time in decades, Caltech professor of mathematics David Conlon and his colleague Asaf Ferber of UC Irvine have shrunk that uncertainty by exponential amounts, for a special category of Ramsey numbers known as multicolor Ramsey numbers. The researchers describe their work in a study appearing in the journal Advances in Mathematics.