Fourier transforms on multidimensional lattices and their exploitation in physics

该程序包括以下项目：*1.分解矩阵。*在本研究中，我们发展了由李群定义的有限格片段上的采样数据的傅立叶分解理论。分解序列包含的项数恰好与采样数据的离散点的数量相同。这意味着，研究的成果和应用应该在计算速度、内存和硬件要求方面显著提高多维数字数据处理。可以使用分解矩阵来计算n维上的加速傅立叶展开。*2.紧致半单李群上数据的中心分裂。*具有阶中心z的半单李群上的任何数据都可以分裂成z个分量，这些分量加起来构成数据函数。每个组件都可以独立处理。这种可能性从来没有被利用过，除非在[30]中指出了3个涉及低组排名的例子，即数据的低维。在该项目中，我们将系统地探索将数据分解为任意阶群的可能性。*3.多维Nyquist-Shannon定理。*Nyquist-Shannon定理描述了信号被采样的点数与信号傅立叶展开中的谐波数之间的关系。这个定理对于一维信号和一维谐函数是众所周知的。*目标是将Nyquist-Shannon定理推广到在所有可能的对称性和密度的格点上采样的N维信号。当‘信号’和‘谐波’被发现为任意密度和任意对称性的格点时，将描述2D和3D情况(以现成的形式)。*4.使用李群对各向同性材料的胡克定律近似。*多面体通过改变其参数的变形可以被用来确定它们的一些动态行为。假设参数为拉格朗日力学中的广义坐标，则可以找到相应的振动和运动。*5.彩色数字数据的分解。*我们打算描述如何用N种不同的颜色给晶格上色。它包括按照某种代数规则将0到N之间的一个整数附加到任意格点上。我们将给出由半单紧李群定义的格的描述，它是单色点集的并。*6.傅立叶空间中数字数据的加密和解密。*我们对该方法的总体想法是在保持展开函数不变的情况下排列分解系数。为了进行短期扩展，可以尝试所有的排列。在现实情况下，当展开可以有数百个项时，它是安全的，因为所有的排列都不能被探索。在我们的方法中，系数的排列是由准晶映射规则提供的，该规则可以应用于扩展到任意有限长度。

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.**