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Rotating mathematics

Fig. 2. The axj and ejj carbon-carbon Cartesian symmetry displacements of benzene. The radial and tangential displacements and y, respectively, are positive at each atom in the direction indicated in and its clockwise rotated counterparts. The analogous angle bending and bond stretching displacements have been pictured in Fig. 3. A counterclockwise rotation of the molecule by 60 about the z -axis replaces by - (l/2) -j + (V3/2) 2, and by - V3j2)Sl ia — (1/2) 2j SC = ). [Such a physical rotation mathematically... Fig. 2. The axj and ejj carbon-carbon Cartesian symmetry displacements of benzene. The radial and tangential displacements and y, respectively, are positive at each atom in the direction indicated in and its clockwise rotated counterparts. The analogous angle bending and bond stretching displacements have been pictured in Fig. 3. A counterclockwise rotation of the molecule by 60 about the z -axis replaces by - (l/2) -j + (V3/2) 2, and by - V3j2)Sl ia — (1/2) 2j SC = ). [Such a physical rotation mathematically...
As this rotation mathematically interchanges time and space coordinates, it means that they are symmetry related and no longer separable in the usual way. It is therefore more appropriate to deal with four-dimensional space-time, rather than the traditional three-dimensional space and absolute time. To visualize Minkowski space, it is useful first to review some properties of the complex plane. [Pg.41]

Figure 1. shows the measured phase differenee derived using equation (6). A close match between the three sets of data points can be seen. Small jumps in the phase delay at 5tt, 3tt and most noticeably at tt are the result of the mathematical analysis used. As the cell is rotated such that tlie optical axis of the crystal structure runs parallel to the angle of polarisation, the cell acts as a phase-only modulator, and the voltage induced refractive index change no longer provides rotation of polarisation. This is desirable as ultimately the device is to be introduced to an interferometer, and any differing polarisations induced in the beams of such a device results in lower intensity modulation. [Pg.682]

Atoms have complete spherical synnnetry, and the angidar momentum states can be considered as different synnnetry classes of that spherical symmetry. The nuclear framework of a molecule has a much lower synnnetry. Synnnetry operations for the molecule are transfonnations such as rotations about an axis, reflection in a plane, or inversion tlnough a point at the centre of the molecule, which leave the molecule in an equivalent configuration. Every molecule has one such operation, the identity operation, which just leaves the molecule alone. Many molecules have one or more additional operations. The set of operations for a molecule fonn a mathematical group, and the methods of group theory provide a way to classify electronic and vibrational states according to whatever symmetry does exist. That classification leads to selection rules for transitions between those states. A complete discussion of the methods is beyond the scope of this chapter, but we will consider a few illustrative examples. Additional details will also be found in section A 1.4 on molecular symmetry. [Pg.1134]

Field variables identified by their magnitude and two associated directions are called second-order tensors (by analogy a scalar is said to be a zero-order tensor and a vector is a first-order tensor). An important example of a second-order tensor is the physical function stress which is a surface force identified by magnitude, direction and orientation of the surface upon which it is acting. Using a mathematical approach a second-order Cartesian tensor is defined as an entity having nine components T/j, i, j = 1, 2, 3, in the Cartesian coordinate system of ol23 which on rotation of the system to ol 2 3 become... [Pg.258]

An operator is a mathematical instruction. For example, the operator d/dx is the instruction to differentiate once with respect to a . Matrices in general, and the matrix R of Chapter 6 in particular, are operators. The matrix R is an instruction to rotate a part of the operand matrix through a certain angle, 0 as in Eq. (6-62). [Pg.207]

Thus the Jacobi procedure, by making many rotations of the elements of the operand matrix, ultimately arrives at the operator matrix that diagonalizes it. Mathematically, we can imagine one operator matr ix that would have diagonalized the operand matr ix R, all in one step... [Pg.207]

The mathematical model chosen for this analysis is that of a cylinder rotating about its axis (Fig. 2). Suitable end caps are assumed. The Hquid phase is introduced continuously at one end so that its angular velocity is identical everywhere with that of the cylinder. The dow is assumed to be uniform in the axial direction, forming a layer bound outwardly by the cylinder and inwardly by a free air—Hquid surface. Initially the continuous Hquid phase contains uniformly distributed spherical particles of a given size. The concentration of these particles is sufftcientiy low that thein interaction during sedimentation is neglected. [Pg.397]

Preprocessing methods of rotation shift the orientation of the data points with respect to the coordinate axes by some angle 9 (Fig. 5). The operation is performed mathematically by applying a rotation matrix R to the original data matrix X to obtain the coordinates of the points with respect to Y, the new axes ... [Pg.420]

X andjy are data matrices in row format, ie, the samples correspond to rows and the variables to columns. Some mathematical Hterature uses column vectors and matrices and thus would represent this equation as T = X. The purpose of rotation in general is to find an orientation of the points that results in enhanced understanding of the underlying chemical behavior of the system. [Pg.420]

For field-oriented controls, a mathematical model of the machine is developed in terms of rotating field to represent its operating parameters such as /V 4, 7, and 0 and all parameters that can inlluence the performance of the machine. The actual operating quantities arc then computed in terms of rotating field and corrected to the required level through open- or closed-loop control schemes to achieve very precise speed control. To make the model similar to that lor a d.c. machine, equation (6.2) is further resolved into two components, one direct axis and the other quadrature axis, as di.sciis.sed later. Now it is possible to monitor and vary these components individually, as with a d.c. machine. With this phasor control we can now achieve a high dynamic performance and accuracy of speed control in an a.c. machine, similar to a separately excited d.c. machine. A d.c. machine provides extremely accurate speed control due to the independent controls of its field and armature currents. [Pg.106]

Sometimes in practice the dial indicators are mounted on the couplings, but it is best to mount and fix the indicatttrs onto the shafts because the couplings may be eeeentrie to the shaft centerlines. Rotate the shafts and obtain the displacement readings. Project these readings graphically (tr mathematically to the motor base t(t determine the adjustments required, and the spacing shims under each foot. [Pg.145]

This method is most usefiil when only one of the shafts can be rotated for the alignment procedure, or when the two shaft ends are very close to each other. Obtain the displacement readings with the dial on the rim (OD) of the coupling and the coupling face. Project these readings mathematically or graphically to the motor base to determine the required adjustments and shims for each foot. This method is not as precise and may have a built-in error, if the coupling center is eccentric from the shaft centerline. [Pg.146]

With polarizers fidly crossed and the specimen rotated to maximum brightness, the sample thickness is determined with the aid of a calibrated eyepiece micrometer, and the polarization (retardation) color is noted. From these the birefringence may be determined mathematically or graphically with the aid of a Michel-L vy chart. [Pg.65]

Thus, the Tsai-Wu tensor failure criterion is obviously of more general character than the Tsai-Hill or Hoffman failure criteria. Specific advantages of the Tsai-Wu failure criterion include (1) invariance under rotation or redefinition of coordinates (2) transformation via known tensor-transformation laws (so data interpretation is eased) and (3) symmetry properties similar to those of the stiffnesses and compliances. Accordingly, the mathematical operations with this tensor failure criterion are well-known and relatively straightforward. [Pg.116]

In 1821 Michael Faraday sent Ampere details of his memoir on rotary effects, provoking Ampere to consider why linear conductors tended to follow circular paths. Ampere built a device where a conductor rotated around a permanent magnet, and in 1822 used electric currents to make a bar magnet spin. Ampere spent the years from 1821 to 1825 investigating the relationship between the phenomena and devising a mathematical model, publishing his results in 1827. Ampere described the laws of action of electric currents and presented a mathematical formula for the force between two currents. However, not everyone accepted the electrodynamic molecule theory for the electrodynamic molecule. Faraday felt there was no evidence for Ampere s assumptions and even in France the electrodynamic molecule was viewed with skepticism. It was accepted, however, by Wilhelm Weber and became the basis of his theory of electromagnetism. [Pg.71]

Recently a mathematical model has been presented [135] that enables one to analyze and plan deflection tool runs. The model is a set of equations that relate the original hole inclination angle (a), new hole inclination angle (a, ), overall angle change (P), change of direction (Ae), and tool-face rotation from original course direction (7). [Pg.1088]

Jean-Baptiste Biot (1774-1862) was born in Paris, France, and was educated there at the Ecole Polytechnique. In 1800. he was appointed professor of mathematical physics atthe College de France. His work on determining the optica rotation of naturally occurring molecules included an experiment on turpentine, which caught fire and nearly burned down the church building he was using for his experiments. [Pg.295]

By starting with this partition function and going through considerable mathematical manipulation, one arrives at the following equations for calculating the corrections to the rigid rotator and harmonic oscillator values calculated from Table 10,4, U... [Pg.560]

Because non-adiabatic collisions induce transitions between rotational levels, these levels do not participate in the relaxation process independently as in (1.11), but are correlated with each other. The degree of correlation is determined by the kernel of Eq. (1.3). A one-parameter model for such a kernel adopted in Eq. (1.6) meets the requirement formulated in (1.2). Mathematically it is suitable to solve integral equation (1.2) in a general way. The form of the kernel in Eq. (1.6) was first proposed by Keilson and Storer to describe the relaxation of the translational velocity [10]. Later it was employed in a number of other problems [24, 25], including the one under discussion [26, 27]. [Pg.17]

From a mathematical perspective either of the two cases (correlated or non-correlated) considerably simplifies the situation [26]. Thus, it is not surprising that all non-adiabatic theories of rotational and orientational relaxation in gases are subdivided into two classes according to the type of collisions. Sack s model A [26], referred to as Langevin model in subsequent papers, falls into the first class (correlated or weak collisions process) [29, 30, 12]. The second class includes Gordon s extended diffusion model [8], [22] and Sack s model B [26], later considered as a non-correlated or strong collision process [29, 31, 32],... [Pg.19]


See other pages where Rotating mathematics is mentioned: [Pg.381]    [Pg.177]    [Pg.381]    [Pg.177]    [Pg.477]    [Pg.478]    [Pg.658]    [Pg.78]    [Pg.2354]    [Pg.2810]    [Pg.280]    [Pg.610]    [Pg.91]    [Pg.164]    [Pg.512]    [Pg.513]    [Pg.280]    [Pg.38]    [Pg.420]    [Pg.444]    [Pg.9]    [Pg.696]    [Pg.66]    [Pg.696]    [Pg.1035]    [Pg.568]    [Pg.660]    [Pg.99]    [Pg.213]    [Pg.226]   
See also in sourсe #XX -- [ Pg.182 ]




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