Physics

Cluster (spacecraft)

Fourier analysis

Mathematics

Musical

Fourier transform

Statistical physics

Basis (linear algebra)

Visual arts

Mathematical analysis

Artificial intelligence

Wavelet transform

Programming language

Geometry

Cluster expansion

Computer science

Quantum mechanics

Formalism (music)

Applied mathematics

Wavelet

Discrete Fourier transform (general)

Fractional Fourier transform

Physical Review B

A rigorous mathematical foundation for the cluster-expansion method is presented. It is shown that the cluster basis developed by Sanchez et al. [Physica A 128, 334 (1984)] is a multidimensional discrete Fourier transform while the general formalism of Sanchez [Phys. Rev. B 48, 14013 (1993)] corresponds to a multidimensional discrete wavelet transform. For functions that depend nonlinearly on the concentration, it is shown that the cluster basis corresponding to a multidimensional discrete Fourier transform does not converge, as it is usually assumed, to a finite cluster expansion or to an Ising-type model representation of the energy of formation of alloys. The multidimensional wavelet transform, based on a variable basis cluster expansion, is shown to provide a satisfactory solution to the deficiencies of the discrete Fourier-transform approach. Several examples aimed at illustrating the main findings and conclusions of this work are given.

Cluster expansion and the configurational theory of alloys