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Spectrum continuation analysis (SCA) is a generalization of the concept of Fourier series to non-periodic functions of which only a fragment has been sampled in the time domain. Recall that a Fourier series is only suitable to the analysis of periodic (or finite-domain) functions f ( x ) with period 2π.
There are several ways to mathematically define quasicrystalline patterns. One definition, the "cut and project" construction, is based on the work of Harald Bohr (mathematician brother of Niels Bohr). The concept of an almost periodic function (also called a quasiperiodic function) was studied by Bohr, including work of Bohl and Escanglon. [47]
This means periodic functions such as the sine and cosine functions cannot exist in Hardy fields. This avoidance of periodic functions also means that every element in a Hardy field has a (possibly infinite) limit at infinity, so if f is an element of H , then
The presence of the screw symmetry resulted in a reevaluation of the requirements for non-periodicity. [4] Chaim Goodman-Strauss suggested that a tiling be considered strongly aperiodic if it admits no infinite cyclic group of Euclidean motions as symmetries, and that only tile sets which enforce strong aperiodicity be called strongly aperiodic ...
A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a cycle . [ 1 ]
Fourier discovered that every continuous, periodic function could be described as an infinite sum of trigonometric functions. Even non-periodic functions can be represented as an integral of sines and cosines through the Fourier transform. This has applications to quantum mechanics [64] and communications, [65] among other fields.
Also called infinitesimal calculus A foundation of calculus, first developed in the 17th century, that makes use of infinitesimal numbers. Calculus of moving surfaces an extension of the theory of tensor calculus to include deforming manifolds. Calculus of variations the field dedicated to maximizing or minimizing functionals. It used to be called functional calculus. Catastrophe theory a ...
The existence of the thermodynamic limit for the free energy and spin correlations were proved by Ginibre, extending to this case the Griffiths inequality. [3]Using the Griffiths inequality in the formulation of Ginibre, Aizenman and Simon [4] proved that the two point spin correlation of the ferromagnetics XY model in dimension D, coupling J > 0 and inverse temperature β is dominated by (i.e ...