Fourier transforms on multidimensional lattices and their exploitation in physics

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The program includes the following projects:***1. Decomposition matrices.***In this research we develop the theory of Fourier decomposition of the data sampled on finite fragments of the lattice defined by Lie group. The decomposition series contains exactly as many terms as the number of discrete points where the data is sampled. This implies that the outcomes and applications of the research should significantly improve multi-dimensional digital data processing in terms of computational speed, and memory and hardware requirements. It will be possible to compute accelerated Fourier expansions in n-dimensions using the decomposition matrices.***2. Central splitting of data on compact semisimple Lie groups.***Any data on a semisimple Lie group with the center of order z can be split into z components that add up to the data function. Each component can be processed independently. This possibility has never never been exploited except in [30] where 3 examples involving low group ranks, i.e. low dimensions of data are pointed out. In the project we will systematically explore the possibility of splitting the data for groups of any rank.***3. Multidimensional Nyquist-Shannon theorem.***The Nyquist-Shannon theorem describes the relation between the number of points at which a signal is sampled and the number of harmonics in the signal's Fourier expansion. The theorem is well known for 1-dimensional signals and 1-dimensional harmonics.***The goal is to generalize the Nyquist-Shannon theorem to N-dimensional signals sampled on points of lattices of all possible symmetries and densities. The 2D and 3D cases will be described (in a ready- to-use form) when the 'signals' and 'harmonics' are found as lattice points of any density and any symmetry.***4. Hooke's law approximation for isotropic material using Lie groups.***Deformation of polytopes via changes of their parameters can be exploited to determine some of their dynamic behaviour. Assuming the parameters as generalized coordinates in Lagrange mechanics, corresponding oscillations and movements can be found. ***5. Decomposition of colored digital data.***We intend to describe how to color a lattice by N different colors. It consists of attaching an integer between 0 and N to any lattice point according to some algebraic rule. We will present a description of a lattice, defined by the semisimple compact Lie groups, as a union of monochromatic sets of points. ***6. Encryption and decryption of digital data in the Fourier space.***Our general idea of the approach is to permute the decomposition coefficients while keeping the expansion functions in their place. For short expansion all permutations can be tried. In realistic cases when the expansion can have hundreds of terms it is secure because all the permutations cannot be explored. In our approach the permutation of the coefficients is provided by the quasicrystal mapping rule which can be applied to expansion to any finite length.**