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. [1]
Sturmian words are a special case of cutting sequences where the curves are straight lines of irrational slope. [2]
Graph paper, coordinate paper, grid paper, or squared paper is writing paper that is printed with fine lines making up a regular grid. The lines are often used as guides for plotting graphs of functions or experimental data and drawing curves. It is commonly found in mathematics and engineering education settings and in laboratory notebooks. Graph paper is available either as loose leaf paper or bound in notebooks.
In mathematics, the Thue–Morse sequence, or Prouhet–Thue–Morse sequence, is the binary sequence obtained by starting with 0 and successively appending the Boolean complement of the sequence obtained thus far. The first few steps of this procedure yield the strings 0 then 01, 0110, 01101001, 0110100110010110, and so on, which are prefixes of the Thue–Morse sequence. The full sequence begins:
In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero elements, often called the empty string and denoted by ε or λ, as the identity element. The free monoid on a set A is usually denoted A∗. The free semigroup on A is the subsemigroup of A∗ containing all elements except the empty string. It is usually denoted A+.
In mathematics, the Prouhet–Thue–Morse constant, named for Eugène Prouhet, Axel Thue, and Marston Morse, is the number—denoted by —whose binary expansion .01101001100101101001011001101001... is given by the Thue–Morse sequence. That is,
In mathematics, the Minkowski question-mark function denoted by ?(x), is a function possessing various unusual fractal properties, defined by Hermann Minkowski. It maps quadratic irrationals to rational numbers on the unit interval, via an expression relating the continued fraction expansions of the quadratics to the binary expansions of the rationals, given by Arnaud Denjoy in 1938. In addition, it maps rational numbers to dyadic rationals, as can be seen by a recursive definition closely related to the Stern–Brocot tree.
In mathematics, a Sturmian word, 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.
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.
Superimposition is the placement of one thing over another, typically so that both are still evident.
Thermodynamic diagrams are diagrams used to represent the thermodynamic states of a material and the consequences of manipulating this material. For instance, a temperature–entropy diagram may be used to demonstrate the behavior of a fluid as it is changed by a compressor.
In mathematics and theoretical computer science, an automatic sequence is an infinite sequence of terms characterized by a finite automaton. The n-th term of an automatic sequence a(n) is a mapping of the final state reached in a finite automaton accepting the digits of the number n in some fixed base k.
A Markov partition 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.
In mathematics, the Rudin–Shapiro sequence, also known as the Golay–Rudin–Shapiro sequence, is an infinite automatic sequence named after Marcel Golay, Walter Rudin, and Harold S. Shapiro, who independently investigated its properties.
In computer science, the complexity function of a string, a finite or infinite sequence of letters from some alphabet, is the function that counts the number of distinct factors from that string. More generally, the complexity function of a language, a set of finite words over an alphabet, counts the number of distinct words of given length.
In mathematics, the Rauzy fractal is a fractal set associated with the Tribonacci substitution
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 and theoretical computer science, an unavoidable pattern is a pattern of symbols that must occur in any sufficiently long string over an alphabet. An avoidable pattern is one for which there are infinitely many words no part of which match the pattern.
In mathematics, a recurrent word or sequence is an infinite word over a finite alphabet in which every factor occurs infinitely often. 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.
 | This combinatorics-related article is a stub. You can help Wikipedia by expanding it. |