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A human visual system model (HVS model) is used by image processing, video processing and computer vision experts to deal with biological and psychological processes that are not yet fully understood.
Irrational rotations form a fundamental example in the theory of dynamical systems. According to the Denjoy theorem, every orientation-preserving C 2-diffeomorphism of the circle with an irrational rotation number θ is topologically conjugate to T θ. An irrational rotation is a measure-preserving ergodic transformation, but it is not mixing.
Rational numbers have irrationality exponent 1, while (as a consequence of Dirichlet's approximation theorem) every irrational number has irrationality exponent at least 2. On the other hand, an application of Borel-Cantelli lemma shows that almost all numbers, including all algebraic irrational numbers , have an irrationality exponent exactly ...
Irrational rotation on a 2-torus For a linear flow on the torus, all orbits are either periodic or dense on a subset of the n {\displaystyle n} -torus, which is a k {\displaystyle k} -torus. When the components of ω {\displaystyle \omega } are rationally independent all the orbits are dense on the whole space.
The rotation number of f is a rational number p/q (in the lowest terms). Then f has a periodic orbit, every periodic orbit has period q, and the order of the points on each such orbit coincides with the order of the points for a rotation by p/q. Moreover, every forward orbit of f converges to a periodic orbit.
The theorem appears as Corollary 3.12 in Niven's book on irrational numbers. [2] The theorem extends to the other trigonometric functions as well. [2] For rational values of θ, the only rational values of the sine or cosine are 0, ±1/2, and ±1; the only rational values of the secant or cosecant are ±1 and ±2; and the only rational values ...
The only assumption we made was that log 2 3 is rational (and so expressible as a quotient of integers m/n with n ≠ 0). The contradiction means that this assumption must be false, i.e. log 2 3 is irrational, and can never be expressed as a quotient of integers m/n with n ≠ 0. Cases such as log 10 2 can be treated similarly.
In mathematics, the Farey sequence of order n is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, [a] which when in lowest terms have denominators less than or equal to n, arranged in order of increasing size.