Concepts inNoise vs computational intractability in dynamics
Computational complexity theory
Computational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other. In this context, a computational problem is understood to be a task that is in principle amenable to being solved by a computer (which basically means that the problem can be stated by a set of mathematical instructions).
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Noise
In common use, the word noise means any unwanted sound. In physics and analog electronics, noise is a mostly unwanted random addition to a signal; it is called noise as a generalisation of the acoustic noise ("static") heard when listening to a weak radio transmission with significant electrical noise. Signal noise is heard as acoustic noise if the signal is converted into sound (e.g. , played through a loudspeaker); it manifests as "snow" on a television or video image.
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Computability theory
Computability theory, also called recursion theory, is a branch of mathematical logic and computer science that originated in the 1930s with the study of computable functions and Turing degrees. The field has since grown to include the study of generalized computability and definability. In these areas, recursion theory overlaps with proof theory and effective descriptive set theory.
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Statistical mechanics
Statistical mechanics or statistical thermodynamics is a branch of physics that applies probability theory, which contains mathematical tools for dealing with large populations, to the study of the thermodynamic behavior of systems composed of a large number of particles.
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Invariant measure
In mathematics, an invariant measure is a measure that is preserved by some function. Ergodic theory is the study of invariant measures in dynamical systems. The Krylov¿Bogolyubov theorem proves the existence of invariant measures under certain conditions on the function and space under consideration.
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Halting problem
In computability theory, the halting problem can be stated as follows: Given a description of an arbitrary computer program, decide whether the program finishes running or continues to run forever. This is equivalent to the problem of deciding, given a program and an input, whether the program will eventually halt when run with that input, or will run forever. Alan Turing proved in 1936 that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist.
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Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, engineering, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the butterfly effect.
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Undecidable problem
In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is impossible to construct a single algorithm that always leads to a correct yes-or-no answer. A decision problem is any arbitrary yes-or-no question on an infinite set of inputs. Because of this, it is traditional to define the decision problem equivalently as the set of inputs for which the problem returns yes.
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