Snippets (math and computing)
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Three-gap theorem
In mathematics, the three-gap theorem, three-distance theorem, or Steinhaus conjecture states that if one places n points on a circle, at angles of θ, 2θ, 3θ ... from the starting point, then there will be at most three distinct distances between pairs of points in adjacent positions around the circle. When there are three distances, the largest of the three always equals the sum of the other two.[1] Unless θ is a rational multiple of π, there will also be at least two distinct distances.
This result was conjectured by Hugo Steinhaus, and proved in the 1950s by Vera T. Sós, János Surányi (hu), and Stanisław Świerczkowski. Its applications include the study of plant growth and musical tuning systems, and the theory of Sturmian words.
Lonely runner conjecture
Siemion Fajtlowicz
A mathematical conjecture is more than a formula. Usually, it is also an expression of personal opinion concerning the significance, nontriviality, and correctness of this formula. When the act of "making a conjecture" is attributed to a machine, the author of the program should be expected to clearly explain how the program, as opposed to the users, reached these conclusions.
https://www.math.uh.edu/~siemion/postscript.pdf
https://en.wikipedia.org/wiki/Siemion_Fajtlowicz
Versine
The versine or versed sine is a trigonometric function already appearing in some of the earliest trigonometric tables. The versine of an angle equals 1 minus its cosine.
There are several related functions, most notably the coversine and haversine. The latter, half a versine, is of particular importance in the haversine formula of navigation.
https://en.wikipedia.org/wiki/Versine
See also:
- Trigonometric identities
- Exsecant and excosecant
- Versiera (Witch of Agnesi)
- Exponential minus 1
- Natural logarithm plus 1
Graph coloring game
The graph coloring game is a mathematical game related to graph theory. Coloring game problems arose as game-theoretic versions of well-known graph coloring problems. In a coloring game, two players use a given set of colors to construct a coloring of a graph, following specific rules depending on the game we consider. One player tries to successfully complete the coloring of the graph, when the other one tries to prevent him from achieving it.
https://en.wikipedia.org/wiki/Graph_coloring_game
Lotka–Volterra equations
The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order nonlinear differential equations, frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. The populations change through time according to the pair of equations.
https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations
Klee–Minty cube
The Klee–Minty cube or Klee–Minty polytope (named after Victor Klee and George J. Minty (de)) is a unit hypercube of variable dimension whose corners have been perturbed. Klee and Minty demonstrated that George Dantzig's simplex algorithm has poor worst-case performance when initialized at one corner of their "squashed cube".
https://en.wikipedia.org/wiki/Klee%E2%80%93Minty_cube
Browser machine learning
Divergent series
Les séries divergentes sont en général quelque chose de bien fatal et c’est une honte qu’on ose y fonder aucune démonstration. ("Divergent series are in general something fatal, and it is a disgrace to base any proof on them." Often translated as "Divergent series are an invention of the devil …") N. H. Abel, letter to Holmboe, January 1826, reprinted in volume 2 of his collected papers.
https://en.wikipedia.org/wiki/Divergent_series
Fuzzing
Fuzzing or fuzz testing is an automated software testing technique that involves providing invalid, unexpected, or random data as inputs to a computer program.
https://en.wikipedia.org/wiki/Fuzzing
Sturmian word
In mathematics, a Sturmian word (Sturmian sequence or billiard sequence), named after Jacques Charles François Sturm, is a certain kind of infinitely long sequence of characters. Such a sequence can be generated by considering a game of English billiards on a square table. The struck ball will successively hit the vertical and horizontal edges labelled 0 and 1 generating a sequence of letters. This sequence is a Sturmian word.
Bathtub curve
The bathtub curve is widely used in reliability engineering. It describes a particular form of the hazard function which comprises three parts:
- The first part is a decreasing failure rate, known as early failures.
- The second part is a constant failure rate, known as random failures.
- The third part is an increasing failure rate, known as wear-out failures.
The name is derived from the cross-sectional shape of a bathtub: steep sides and a flat bottom.
https://en.wikipedia.org/wiki/Bathtub_curve
Slack variable
Slack variable: In an optimization problem, a slack variable is a variable that is added to an inequality constraint to transform it into an equality. Introducing a slack variable replaces an inequality constraint with an equality constraint and a non-negativity constraint on the slack variable.