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Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science.
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures.
In mathematics, the Prouhet–Thue–Morse constant, named for Eugène Prouhet, Axel Thue, and Marston Morse, is the number—denoted by τ—whose binary expansion 0.01101001100101101001011001101001... is given by the Prouhet–Thue–Morse sequence. That is,
Grigory Aleksandrovich Margulis is a Russian-American mathematician known for his work on lattices in Lie groups, and the introduction of methods from ergodic theory into diophantine approximation. He was awarded a Fields Medal in 1978, a Wolf Prize in Mathematics in 2005, and an Abel Prize in 2020, becoming the fifth mathematician to receive the three prizes. In 1991, he joined the faculty of Yale University, where he is currently the Erastus L. De Forest Professor of Mathematics.
In mathematics, subshifts of finite type are used to model dynamical systems, and in particular are the objects of study in symbolic dynamics and ergodic theory. They also describe the set of all possible sequences executed by a finite state machine. The most widely studied shift spaces are the subshifts of finite type.
In geometry, a tile substitution is a method for constructing highly ordered tilings. Most importantly, some tile substitutions generate aperiodic tilings, which are tilings whose prototiles do not admit any tiling with translational symmetry. The most famous of these are the Penrose tilings. Substitution tilings are special cases of finite subdivision rules, which do not require the tiles to be geometrically rigid.
A Markov partition in mathematics is a tool used in dynamical systems theory, allowing the methods of symbolic dynamics to be applied to the study of hyperbolic dynamics. By using a Markov partition, the system can be made to resemble a discrete-time Markov process, with the long-term dynamical characteristics of the system represented as a Markov shift. The appellation 'Markov' is appropriate because the resulting dynamics of the system obeys the Markov property. The Markov partition thus allows standard techniques from symbolic dynamics to be applied, including the computation of expectation values, correlations, topological entropy, topological zeta functions, Fredholm determinants and the like.
Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra.
Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Classically, discrete dynamics refers to the study of the iteration of self-maps of the complex plane or real line. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, p-adic, and/or algebraic points under repeated application of a polynomial or rational function. A fundamental goal is to describe arithmetic properties in terms of underlying geometric structures.
In mathematics, the Kolakoski sequence, sometimes also known as the Oldenburger–Kolakoski sequence, is an infinite sequence of symbols {1,2} that is the sequence of run lengths in its own run-length encoding. It is named after the recreational mathematician William Kolakoski (1944–97) who described it in 1965, but it was previously discussed by Rufus Oldenburger in 1939.
In computer science, the complexity function of a word or string is the function that counts the number of distinct factors of that string. More generally, the complexity function of a formal language counts the number of distinct words of given length.
In mathematics, the Rauzy fractal is a fractal set associated with the Tribonacci substitution
Christopher Deninger is a German mathematician at the University of Münster. Deninger's research focuses on arithmetic geometry, including applications to L-functions.
In digital geometry, a cutting sequence is a sequence of symbols whose elements correspond to the individual grid lines crossed ("cut") as a curve crosses a square grid.
Mathematics is a broad subject that is commonly divided in many areas that may be defined by their objects of study, by the used methods, or by both. For example, analytic number theory is a subarea of number theory devoted to the use of methods of analysis for the study of natural numbers.
In mathematics, a sesquipower or Zimin word is a string over an alphabet with identical prefix and suffix. Sesquipowers are unavoidable patterns, in the sense that all sufficiently long strings contain one.
In mathematics, a recurrent word or sequence is an infinite word over a finite alphabet in which every factor occurs infinitely many times. An infinite word is recurrent if and only if it is a sesquipower.
In mathematics and computer science, the critical exponent of a finite or infinite sequence of symbols over a finite alphabet describes the largest number of times a contiguous subsequence can be repeated. For example, the critical exponent of "Mississippi" is 7/3, as it contains the string "ississi", which is of length 7 and period 3.
In mathematics, Ostrowski numeration, named after Alexander Ostrowski, is either of two related numeration systems based on continued fractions: a non-standard positional numeral system for integers and a non-integer representation of real numbers.
Valérie Berthé is a French mathematician who works as a director of research for the Centre national de la recherche scientifique (CNRS) at the Institut de Recherche en Informatique Fondamentale (IRIF), a joint project between CNRS and Paris Diderot University. Her research involves symbolic dynamics, combinatorics on words, discrete geometry, numeral systems, tessellations, and fractals.
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