Resultant vectors

This Cantor set may be explicitly visualized in the n-dimensional euclidean space 7 " by defining a mapping xp F —> 7 " ([grass83] and [packl]). The coordinate of the resulting vector xpid) is given by ... 

Upon terminating this process the resultant vector y(2p) is just the solution y = y(2p) on the layer t =... 

The normalization step prevents the elements in v from becoming either too large or too small during the numerical computation. The two operations above define the cycle of the powering algorithm, which can be iterated until convergence of the elements in the vector v within a predefined tolerance. It can be easily shown that after n iterations the resulting vector w can be expressed as ... 

Every minimization departs from an initial estimation for the vector field. The minimizations were carried out with a starting configuration obtained by randomizing the coefficients aj and a the resulting vector field has no preferential orientation and the distribution of curvature in the simulation box exhibits a long tail mainly due to abrupt changes in the direction of the stream lines (see Fig. 3.2A). 

The sum over all j can be denoted as BkGkf where Gfc Is the number of paths containing r, and Bk Is a resultant vector analogous to Bj. Unfortunately Bk cannot be determined from the Information at hand. It Is therefore necessary to assume that B Is the same for all k i.e., 6 and... 

If polarized light passes through a medium that exhibits optical rotation, the motion along one of the circular vectors is slower than that of the other. The resultant vector is thus displaced from the original vector by some angle, . Figure 16.5 shows the vector model in which the phase difference is [Pg.588]

The graphic demonstration thereof is self-explanatory. The linear(resultant) vector defines a fluctuating electromagnetic field. Each of the rotating vectors describes a helix around the time axis. 

The mathematics necessary to understand the diffraction of X rays by a crystal will not be discussed in any detail here. Chapter 4 of reference 10 contains an excellent discussion. The arrangement of unit cells in a crystal in a periodic manner leads to the Laue diffraction conditions shown in equations 3.3 where vectors a, b, and c as well as lattice indices h, k, and l have been defined in Figure 3.5 and S is a vector quantity equal to the difference between the resultant vector s after diffraction and the incident X-ray beam wave vector So so that S = s - So-... 

Figure 2.4). The second transformation scales the resulting vector UTx... 

Because of round-off errors, symmetry contamination is often present even when the initial vector is properly symmetrized. To circumvent this problem, an effective scheme to reinforce the symmetry at every Lanczos recursion step has been proposed independently by Chen and Guo100 and by Wang and Carrington.195 Specifically, the Lanczos recursion is executed with symmetry-adapted vectors, but the matrix-vector multiplication is performed at every Lanczos step with the unsymmetrized vector. In other words, the symmetrized vectors are combined just before the operation Hq, and the resultant vector is symmetrized using the projection operators ... 

Addition and subtraction of matrices are performed element-wise the matrices must have the same size (see Figure A.2.2). Multiplication of a vector or a matrix with a scalar is also performed element-wise in the case of a vector the resulting vector has the same direction but a different length. Vectors a and b = —a have the same length but reverse direction. 

ROOTR-RESULTANT VECTOR OF LENGTH M CONTAINING REAL ROOTS OF THE POLYNOMIAL... 

ROOTI-RESULTANT VECTOR OF LENGTH M CONTAINING THE CORRESPONDING IMAGINARY ROOTS OF THE POLYNOMIAL lER -ERROR CODE WHERE lERO NO ERROR lER 1 M LESS THAN ONE lER 2 M GREATER THAN 36... 

This equation tells us that the i-th column of a matrix, M, contains the result of operating on the i-th unit vector ej with the matrix. More specifically, the element Mk in the k-th row and i-th column is the component of Me in the direction of the ek unit vector. As a generalization, we can construct any matrix by first deciding how the matrix affects the elementary unit vectors and then placing the resulting vectors as the columns of the matrix. 

All gauge theory depends on the rotation of an -component vector whose 4-derivative does not transform covariantly as shown in Eq. (18). The reason is that i(x) and i(x + dx) are measured in different coordinate systems the field t has different values at different points, but /(x) and /(x) + d f are measured with respect to different coordinate axes. The quantity d i carries information about the nature of the field / itself, but also about the rotation of the axes in the internal gauge space on moving from x + dx. This leads to the concept of parallel transport in the internal gauge space and the resulting vector [6] is denoted i(x) + d i. The notion of parallel transport is at the root of all gauge theory and implies the introduction of g, defined by... 

Thus, if each water dipole is represented by a vector, the vectors are completely randomized in direction1 (Fig. 6.1). There is no net resultant vector, i.e., there is no alignment of the solvent dipoles in any preferred direction. Further, the positive and... 

For an assembly of nuclei with magnetic moment fi the resultant vector sum is the magnetization vector M. Its time dependence is also given by eq. (1.19). 

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