Space axis

Statistical properties of a data set can be preserved only if the statistical distribution of the data is assumed. PCA assumes the multivariate data are described by a Gaussian distribution, and then PCA is calculated considering only the second moment of the probability distribution of the data (covariance matrix). Indeed, for normally distributed data the covariance matrix (XTX) completely describes the data, once they are zero-centered. From a geometric point of view, any covariance matrix, since it is a symmetric matrix, is associated with a hyper-ellipsoid in N dimensional space. PCA corresponds to a coordinate rotation from the natural sensor space axis to a novel axis basis formed by the principal... 

This is an extremely important result. It says that the standard deviation of the onedimensional spatial distribution of objects moving randomly along a space axis grows as the square root of time and that this growth is controlled by the product of the mean free path and the mean velocity, (uxX). Note that this quantity has the dimension [L2rR ]. Its meaning will become clear below. 

The symmetry cf a helical structure is defined by an space axis, which combines a rotation of2 77/n radiusabout an axis followed by a translation cfnvh cfthe repeat distance. 

The subperiodicity of the axial projection within the unit cell in real space of course has an effect in diffraction space. Recall the reciprocal relationship between the two. If a distance is halved in real space, it is doubled in reciprocal space if a periodicity is quartered in real space, it is multiplied fourfold in reciprocal space, and so on. Thus, along the corresponding reciprocal space axis c, or the 00/ line of diffraction intensities, reflection spacing must correspond to the increased periodicity, or smaller repeat distance in real space. In reciprocal space, reflections will appear less frequently, at greater intervals. For a 2i axis, the appearance of reflections will have double the normal periodic interval. For a 2i axis, reflections will occur along the 00/ line of reciprocal lattice points only for / = 2,4,6, 8.. 

The truncation error is usually proportional to a power of the grid spacing Axi and/or the time step At. If the most important error term is proportional to (Z xj) or (Zlf) we call the method an nth-order approximation, often simply indicated by 0 Ax ) or 0 At ), respectively, n > 0 is required to... 

The measured responses to the combinations of multi-frequency selective pulse excitation can be unscrambled for each volume element by transformation with a super-Hadamard matrix. The dimension of this matrix equals the product of the dimensions of the Hadamard matrices used for encoding each space axis. 

Typically, a NMR measurement consists of two steps. At first the contrast, in dependence on the effect under investigation, is prepared. In a second step, the frequency axis is converted into a space axis by applying of the gradient field in direction of the desired profile. For a three-dimensional image, this has to be done in each local direction x, y, and z (y and z analogous) ... 

The common two-dimensional representation is done in terms of a time axis and one space axis, which is interpreted as three space directions at the same time. The so-called time cone, and by implication the complementary space cone, extends into a further undefined dimension, perpendicular to the x,t plane of the diagram. To get the complete picture it is necessary to superimpose three mutually perpendicular Minkowski bodies of this type, which is only posssible in four dimensions. In this superposition the time axis does not remain fixed and becomes entangled with the space directions as seen in three-dimensional space. The three-dimensional surface of the generalized hght cone of Figure 4.2 becomes a surface in four-dimensional space-time that separates all space into two equivalent regions. 

To check the correctness of the obtained solutions we used the semi-discrete method of numerical solving of two differential equations in specific derivatives. The point of the method lies in modeling the finite-difference scheme for the space axis, the variable t being considered as continuous. In this case, the 2N ordinary differential equation system, which can be solved, for example, by the Runge-Kutta-Felberg s method with automatic choice of step along t, is obtained ... 

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