Snippets (math and computing)

Things to use or delete. See Snippets.

Beam and Warming scheme

In numerical mathematics, Beam and Warming scheme or Beam–Warming implicit scheme introduced in 1978 by Richard M. Beam and R. F. Warming,[1][2] is a second order accurate implicit scheme, mainly used for solving non-linear hyperbolic equation. It is not used much nowadays.

https://en.wikipedia.org/wiki/Beam_and_Warming_scheme

Beam and Warming scheme

Trash can shortcut on Mac desktop

http://osxdaily.com/2010/07/29/add-a-trash-icon-to-the-desktop/

Launch the Terminal app (found in /Applications/Utilities) and enter the following syntax exactly:

ln -s ~/.Trash ~/Desktop/Trash

Hilbert's paradox of the Grand Hotel

Hilbert's paradox of the Grand Hotel (colloquial: Infinite Hotel Paradox or Hilbert's Hotel) is a thought experiment which illustrates a counterintuitive property of infinite sets. It is demonstrated that a fully occupied hotel with infinitely many rooms may still accommodate additional guests, even infinitely many of them, and this process may be repeated infinitely often. The idea was introduced by David Hilbert in a 1924 lecture "Über das Unendliche", reprinted in (Hilbert 2013, p.730), and was popularized through George Gamow's 1947 book One Two Three... Infinity.[1][2]

https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel

This and Thus

https://en.wikipedia.org/wiki/This_(computer_programming)

Hysteresis

Hysteresis is the dependence of the state of a system on its history. For example, a magnet may have more than one possible magnetic moment in a given magnetic field, depending on how the field changed in the past. Plots of a single component of the moment often form a loop or hysteresis curve, where there are different values of one variable depending on the direction of change of another variable. This history dependence is the basis of memory in a hard disk drive and the remanence that retains a record of the Earth's magnetic field magnitude in the past. Hysteresis occurs in ferromagnetic and ferroelectric materials, as well as in the deformation of rubber bands and shape-memory alloys and many other natural phenomena. In natural systems it is often associated with irreversible thermodynamic change such as phase transitions and with internal friction; and dissipation is a common side effect.

Illumination problem

The illumination problem is a resolved mathematical problem first posed by Ernst Straus in the 1950s. Straus asked if a room with mirrored walls can always be illuminated by a single point light source, allowing for repeated reflection of light off the mirrored walls. Alternatively, the question can be stated as asking that if a billiard table can be constructed in any required shape, is there a shape possible such that there is a point where it is impossible to pot the billiard ball in a pocket at another point, assuming the ball is point-like and continues infinitely rather than stopping due to friction.

The problem was first solved in 1958 by Roger Penrose using ellipses to form the Penrose unilluminable room. He showed there exists a room with curved walls that must always have dark regions if lit only by a single point source.

https://en.wikipedia.org/wiki/Illumination_problem

Casting out nines

The expression "casting out nines" may refer to any one of three arithmetical procedures:

• Adding the decimal digits of a positive whole number, while optionally ignoring any 9s or digits which sum to a multiple of 9. The result of this procedure is a number which is smaller than the original whenever the original has more than one digit, leaves the same remainder as the original after division by nine, and may be obtained from the original by subtracting a multiple of 9 from it. The name of the procedure derives from this latter property.

• Repeated application of this procedure to the results obtained from previous applications until a single-digit number is obtained. This single-digit number is called the "digital root" of the original. If a number is divisible by 9, its digital root is 9. Otherwise, its digital root is the remainder it leaves after being divided by 9.

• A sanity test in which the above-mentioned procedures are used to check for errors in arithmetical calculations. The test is carried out by applying the same sequence of arithmetical operations to the digital roots of the operands as are applied to the operands themselves. If no mistakes are made in the calculations, the digital roots of the two resultants should be the same. If they are different, therefore, one or more mistakes must have been made in the calculations.

https://en.wikipedia.org/wiki/Casting_out_nines

The Bottle Imp

https://boingboing.net/2019/02/07/the-paradox-of-the-bottle-imp.html

Particle systems

• Particle system (nonfiction)

Rogue AI written in Plankalkül

"Somewhere out there is a rogue AI written in Plankalkül."

-- User FGD135 @ Boing Boing comments in response to "Trump signs ‘American AI Initiative’ executive order to prioritize federal funding for artificial intelligence research".

Baby-step giant-step

In group theory, the baby-step giant-step is a meet-in-the-middle algorithm for computing the discrete logarithm. The discrete log problem is of fundamental importance to the area of public key cryptography. Many of the most commonly used cryptography systems are based on the assumption that the discrete log is extremely difficult to compute; the more difficult it is, the more security it provides a data transfer. One way to increase the difficulty of the discrete log problem is to base the cryptosystem on a larger group.

https://en.wikipedia.org/wiki/Baby-step_giant-step

Nontransitive dice

A set of dice is nontransitive if it contains three dice, A, B, and C, with the property that A rolls higher than B more than half the time, and B rolls higher than C more than half the time, but it is not true that A rolls higher than C more than half the time. In other words, a set of dice is nontransitive if the binary relation – X rolls a higher number than Y more than half the time – on its elements is not transitive.

It is possible to find sets of dice with the even stronger property that, for each dice in the set, there is another die that rolls a higher number than it more than half the time. Using such a set of dice, one can invent games which are biased in ways that people unused to nontransitive dice might not expect.

Efron's dice are a set of four nontransitive dice invented by Bradley Efron.

https://en.wikipedia.org/wiki/Nontransitive_dice

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. 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.

• https://en.wikipedia.org/wiki/Three-gap_theorem

Lonely runner conjecture

• https://en.wikipedia.org/wiki/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

http://www.infoworld.com/article/3212884/machine-learning/machine-learning-comes-to-your-browser-via-javascript.html

https://tenso.rs/

http://www.infoworld.com/article/3216464/machine-learning/googles-deeplearnjs-brings-machine-learning-to-the-browser.html

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.

Impossible Puzzle

• https://en.wikipedia.org/wiki/Sum_and_Product_Puzzle

Sorting algorithms

• https://www.youtube.com/watch?v=kPRA0W1kECg

"The Talk" (quantum computing)

• https://www.smbc-comics.com/comic/the-talk-3