Despite all the talk about innovation, the cultural landscape still defaults to stovepiping. For example, in the Wikipedia article on sunk costs, there is no mention of Markov processes, which, foregrounding memorylessness, model the rational situation perfectly, and in the Wolfram MathWorld article on Markov processes, there is no mention of memorylessness, which is the main point of a Markov process. (Hence the need for an independent and autonomous enterprise to create a ‘mathematics mind map’ – to make up for the myriad lack of such pointers.)
keywords: communication, interconnectedness, see also, relatedness
Contending with large logistics is the default learning mode, and is known as ‘the school of hard knocks’.
keywords: education, asymptotics
A closed set is one that contains all the points that ride with it. (cf: “If you ride with outlaws, you’re an outlaw.”)
For example, 0 is not a positive number, but it ‘rides with’ the set of positive numbers.
keywords: point-set topology
The admonition not to cry over spilt milk is an early recognition of the sunk-cost fallacy.
This is a favorite saying of one of my favorite authors: P. G. Wodehouse.
Here is the Wikipedia article on him.
keywords: farce, Jeeves, episodes
This striking formula occurs here and there in mathematics. I’ve never seen it referenced explicitly as I’m doing now, much less named, so I am going to give it the name ‘immolation formula’, because it kind of looks like what results is a butchered form of f(x). Actual use of the immolation formula might occur in a kind of spread-out stepwise manner that disguises the fact that the immolation formula is at work, for example, at a certain juncture in the derivation of the computational formula for the logarithm that Tom Apostol provides in volume 1 of his 2-volume Calculus textbook.
Just as group theory can be use additive notation (typically, for abelian groups) or multiplicative notation, so also the immolation formula can be stated in multiplicative form as well: f(x)/(f(1/x).
A formula usually means a statement of some kind, whereas what we are dealing with here is not a statement, but merely a quantity given as a symbolic expression. So, we are using ‘formula’ in the wide sense to mean ‘symbolic expression’. It is convenient to have this wide sense. For example, if we are dealing with an equation of the form f(x) = g(x), it would be convenient to be able to (validly) refer to ‘the formula on the left side’ and / or ‘the formula on the right side’ of the equation, as well as having available the more sweeping traditional terms of LHS and RHS.
keywords: Mathematics, subtraction, additive inverse
Here is the Wolfram MathWorld article on this topic.
keywords: nonsingular matrix, invertible matrix, square matrix, identity matrix