Aperiodic (other)
Aperiodic means non-periodic. Typically it refers to aperiodic function. Aperiodic may also refer to: * Aperiodic finite state automaton * Aperiodic frequency * Aperiodic graph * Aperiodic semigroup * Aperiodic set of prototiles * Aperiodic tiling See also * Periodic (other) * Strange attractor In the mathematics, mathematical field of dynamical systems, an 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 va ...

Aperiodic Function
A periodic function, also called a periodic waveform (or simply periodic wave), is a Function (mathematics), function that repeats its values at regular intervals or period (physics), periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit Frequency, periodicity. Any function that is not periodic is called ''aperiodic''. Definition A function is said to be periodic if, for some nonzero constant , it is the case that :f(x+P) = f(x) for all values of in the domain. A nonzero constant for which this is the case is called a period of the function. If there exists a least positive constant with this property, it is called the fundamental period (also primitive period, basic period, or prime period.) Often, "the" period of a function ...
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Aperiodic Finite State Automaton
An aperiodic finite-state automaton (also called a counter-free automaton) is a finite-state automaton whose transition monoid is aperiodic. Properties A regular language is star-free if and only if it is accepted by an automaton with a finite and aperiodic transition monoid. This result of algebraic automata theory is due to Marcel-Paul Schützenberger. In particular, the minimum automaton of a star-free language is always counter-free (however, a star-free language may also be recognized by other automata that are not aperiodic). A counter-free language is a regular language for which there is an integer ''n'' such that for all words ''x'', ''y'', ''z'' and integers ''m'' ≥ ''n'' we have ''xy''''m''''z'' in ''L'' if and only if ''xy''''n''''z'' in ''L''. For these languages, when a string contains enough repetitions of any substring (at least ''n'' repetitions), changing the number of repetitions to another number that is at least ''n'' cannot change membership in the lan ...
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Aperiodic Frequency
Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio signals (sound), radio waves, and light. The interval of time between events is called the period. It is the reciprocal of the frequency. For example, if a heart beats at a frequency of 120 times per minute (2 hertz), its period is one half of a second. Special definitions of frequency are used in certain contexts, such as the angular frequency in rotational or cyclical properties, when the rate of angular progress is measured. Spatial frequency is defined for properties that vary or cccur repeatedly in geometry or space. The unit of measurement of frequency in the International System of Units (SI) is the hertz, having the symbol Hz. Definitions and units For cyclical phenomena such as oscillations, waves, or for examples ...
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Aperiodic Graph
In the mathematical area of graph theory, a directed graph is said to be aperiodic if there is no integer ''k'' > 1 that divides the length of every cycle of the graph. Equivalently, a graph is aperiodic if the greatest common divisor of the lengths of its cycles is one; this greatest common divisor for a graph ''G'' is called the ''period'' of ''G''. Graphs that cannot be aperiodic In any directed bipartite graph, all cycle lengths are even. Therefore, no directed bipartite graph can be aperiodic. In any directed acyclic graph, it is a vacuous truth that every ''k'' divides all cycles (because there are no directed cycles to divide) so no directed acyclic graph can be aperiodic. And in any directed cycle graph, there is only one cycle, so every cycle's length is divisible by ''n'', the length of that cycle. Testing for aperiodicity Suppose that ''G'' is strongly connected and that ''k'' divides the lengths of all cycles in ''G''. Consider the results of per ...
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Aperiodic Semigroup
In mathematics, an aperiodic semigroup is a semigroup ''S'' such that every element is aperiodic, that is, for each ''x'' in ''S'' there exists a positive integer ''n'' such that ''xn'' = ''x''''n''+1. An aperiodic monoid is an aperiodic semigroup which is a monoid. Finite aperiodic semigroups A finite semigroup is aperiodic if and only if it contains no nontrivial subgroups, so a synonym used (only?) in such contexts is group-free semigroup. In terms of Green's relations, a finite semigroup is aperiodic if and only if its ''H''-relation is trivial. These two characterizations extend to group-bound semigroups. A celebrated result of algebraic automata theory due to Marcel-Paul Schützenberger asserts that a language is star-free if and only if its syntactic monoid is finite and aperiodic.Schützenberger, Marcel-Paul, "On finite monoids having only trivial subgroups," ''Information and Control'', Vol 8 No. 2, pp. 190–194, 1965. A consequence of the Krohn–Rhodes theorem is ...
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Aperiodic Set Of Prototiles
A set of prototiles is aperiodic tiling, aperiodic if copies of the prototiles can be assembled to create Tessellation, tilings, such that all possible tessellation patterns are non-periodic tiling, periodic. The ''aperiodicity'' referred to is a property of the particular set of prototiles; the various resulting tilings themselves are just non-periodic. A given set of tiles, in the Euclidean plane or some other geometric setting, ''admits a tiling'' if non-overlapping copies of the tiles in the set can be fitted together to cover the entire space. A given set of tiles might admit periodic tilings — that is, tilings that remain invariant after being shifted by a Translation (geometry), translation (for example, a lattice of square tiles is periodic). It is not difficult to design a set of tiles that admits non-periodic tilings as well as periodic tilings. (For example, randomly arranged tilings using a 2×2 square and 2×1 rectangle are typically non-periodic.) However, an ...
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Aperiodic Tiling
An aperiodic tiling is a non-periodic Tessellation, tiling with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types (or prototiles) is aperiodic set of prototiles, aperiodic if copies of these tiles can form only non-periodic tiling, periodic tilings. The Penrose tilings are a well-known example of aperiodic tilings. In March 2023, four researchers, David Smith (amateur mathematician), David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss, announced the proof that the tile discovered by David Smith is an Einstein problem, aperiodic monotile, i.e., a solution to the einstein problem, a problem that seeks the existence of any single shape aperiodic tile. In May 2023 the same authors published a chiral aperiodic monotile with similar but stronger constraints. Aperiodic tilings serve as mathematical models for quasicrystals, physical solids that were discovered in 1982 by Dan Shechtman who subs ...
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Periodic (other)
Periodicity or periodic may refer to: Mathematics * Bott periodicity theorem, addresses Bott periodicity: a modulo-8 recurrence relation in the homotopy groups of classical groups * Periodic function, a function whose output contains values that repeat periodically * Periodic mapping Physical sciences * Periodic table of chemical elements * Periodic trends, relative characteristics of chemical elements observed * Redshift periodicity, astronomical term for redshift quantization Other uses * Fokker periodicity blocks, which mathematically relate musical intervals * Periodic acid, a compound of iodine * Principle of periodicity, a concept in generally accepted accounting principles * Quasiperiodicity, property of a system that displays irregular periodicity See also * Aperiodic (other) * Cycle (other) * Frequency (other) * Period (other) * Periodical * Seasonality In time series data, seasonality refers to the trends that occur at spec ...
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