Monty Hall Problem

Monty Hall problem is a brain teaser, in the form of a probability puzzle. The problem was originally posed (and solved) in a letter by Steve Selvin to the American Statistician in 1975. The Monty Hall problem is mathematically closely related to the earlier Three Prisoners problem and to the much older Bertrand's box paradox. …

Aleph Number

Aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. The cardinality of the natural numbers is ℵ0. Examples of such sets are: the set of all integers,any infinite subset of the integers, such as the set of all square numbers or the set of all prime numbers,the set of all rational numbers,the set of …

Knot Theory

Knot theory studies the mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be undone, the simplest knot being a ring (or "unknot"). In mathematical language, a knot is an embedding of a …

Multifractal System

Multifractal system is a generalization of a fractal system in which a single exponent (the fractal dimension) is not enough to describe its dynamics; instead, a continuous spectrum of exponents (the so-called singularity spectrum) is needed. Multifractal systems are often modeled by stochastic processes such as multiplicative cascades. Modelling as a multiplicative cascade also leads …

Random walk

Random walk is a mathematical object, that describes a path that consists of a succession of random steps on some mathematical space such as the integers. An elementary example of a random walk is the random walk on the integer number line, Z, which starts at 0 and at each step moves +1 or −1 …

Game theory

Game theory is the study of mathematical models of strategic interactions among people that always aims to perform optimal actions based on given premises and information. It has applications in all fields of social science, as well as in logic, systems science and computer science. ©Dreamstime In the 21st century, game theory applies to a wide range …

Robot kinematics

Robot kinematics applies geometry to the study of the movement of multi-degree of freedom kinematic chains that form the structure of robotic systems. The emphasis on geometry means that the links of the robot are modeled as rigid bodies and its joints are assumed to provide pure rotation or translation. A fundamental tool in robot …

Space in mathematics

Space is a set with some added structure. While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of "space" itself. ©Wikipedia A space consists of selected mathematical objects that are treated as points, and selected relationships between these points. The nature of the points can vary …

Attractor

Attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain close even if slightly disturbed. Attractors are portions or subsets of the phase space of a dynamical system. Until the 1960s, …

Quasicrystal

Quasicrystal is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. While crystals, according to the classical crystallographic restriction theorem, can possess only two-, three-, four-, and six-fold rotational symmetries, the Bragg diffraction pattern of quasicrystals shows sharp peaks with other symmetry …