Complex numbers, interpretation

Some studies of more complicated systems have been attempted but, as the number of variables increases, the complexities of interpretation and the difficulties in obtaining meaningful kinetic data become intractable. 

A final example is the concept of QM state. It is often stated that the wave function must be square integrable because the modulus square of the wave function is a probability distribution. States in QM are rays in Hilbert space, which are equivalence classes of wave functions. The equivalence relation between two wave functions is that one wave function is equal to the other multiplied by a complex number. The space of QM states is then a projective space, which by an infinite stereographic projection is isomorphic to a sphere in Hilbert space with any radius, conventionally chosen as one. Hence states can be identified with normalized wave functions as representatives from each equivalence class. This fact is important for the probability interpretation, but it is not a consequence of the probability interpretation. 

I assume that the reader interprets the complex numbers j/ (... 

A potential limitation in the application of MS to near-surface measurements is the tremendous number of compounds in the atmosphere, particularly organics, and hence the increased complexity of interpretation of the single mass spectrum. In the MS ion source, the use of particular ion-molecule reactions to form the ions of interest or the ionization of one selected com-... 

The position of the label should be away from sites chemically instable or from sites of metabolic attack to ensure that the label is kept in the main metabolic fragments. Of course, this is difficult for complex molecules, especially when the metabolic attack takes part in the center of the drag molecule. Then it can become necessary to introduce a second label and to repeat the set of radiokinetic studies. A double-labeling strategy to minimize the number of radiokinetic studies has normally to be refused due to complexity of interpretation of the data. 

In the quasi-static case, effective frequency dependent moduli and loss factors may be calculated from Equation 8. With respect to Equation 29, a lossy matrix material implies that k is now a complex number. The new expressions for c and a differ from Equations 31 and 32, but follow straightforwardly. Equation 30 is usually cited only for elastic matrix materials, but, of course, it need not be used to interpret a. The potential problem (also with viscoelastic inclusions) is that the derivation of Equation 30 is based on homogeneous stress waves, whereas in viscoelastic materials one should, strictly speaking, consider inhomogeneous waves. The results obtained from Equation 29 are reasonable in the sense of yielding the expected superposition of scattering and dissipation effects. 

It is possible to use the idea of evaluating the polynomial at a real or complex value as an aid to proving that one polynomial is a factor of another, by showing that all the roots of the first are also roots of the second. In fact in the Laplace transform the domain is definitely that of complex numbers, and [CDM91] uses this interpretation very fluently and to good effect. 

It may be seen (Table I and Figure 1) that the methyl radical becomes relatively more stable, and more are produced for a given dosage of radiation, as the number of molecules of water of crystallization increases. The spectra of the anhydrous acetates are too complex to interpret in detail, but both the absolute concentration and the relative amount of the methyl radical compared with other radical species must be small. 

Usually, atomic mass spectra are considerably simpler and easier to interpret than optical emission spectra. This property is particularly important for samples that contain rare earth elements and other heavy metals. such as iron, that yield complex emission spectra. Figure 11-15b illustrates this advantage. This spectral simplicity is further illustrated in Figure 11-16. which is the atontic mass spectrum for a mixture of fourteen rare earth elements that range in atomic mass number from 1.39 to 175. T he optical emission spectrum for such a mixture would be so complex that interpretation would be tedious, time-consuming, and perhaps impo.ssible. 

A complex number z can be interpreted as a 2-vector in an xy-coordinate system, z=x + iy, where i is the symbol for and x and y are real numbers. Here x and y are referred to as the real and imaginary parts of z. Alternatively, the complex number can be associated with polar coordinates, i.e. the distance r from the origin and the angle 0 relative to the real axis. 

The length (or norm) of a vector follows directly from the interpretation of a vector as a directional line from the origin to a point in space, and is defined as the square root of the dot product of the vector with itselt If the vector components are complex numbers the transposition is replaced by the adjoint instead. 

The interpretation is the same as for the Fourier transform defined in section 10.1.6 For each required frequency value, we multiply the signal by a complex exponential waveform of that frequency, and sum the result over the time period. The result is a complex number which describes the magnitude and phase at that frequency. The process is repeated until the magnitude and phase is found for every fi-equency. 

While the FT of the even cosine function is real, the result for sin(27rvot) is imaginary (S(v — vo)- --b Vo))/(2i). Since the sine function is 90° out of phase to the corresponding cosine, it is clear that the imaginary axis is used to keep track of phase shifts, consistent with the polar representation of a complex number x + iy = re with r = Jx - -y and 4> = tan (y/x). In this phasor picture, positive and negative frequencies can be interpreted as clockwise and counterclockwise rotations in the complex plane. 

The mechanism of complex formation of metals with chitosan is manifold and is probably dominated by different processes such as adsorption, ion exchange, and chelation under different conditions. In Fig. 15.6, it can be seen that the sorption capacity of chromium and cadmium is similar, while zinc has approximately double the affinity of these two metal ions for chitosan. Chitosan has almost three times the removal capacity for copper sorption than that for cadmium or chromium. These effects have proved complex to interpret but are a function of a number of parameters ionic radii ionic charge electron structure and possibly some hydration capacity of the metal ions solution pH and nature and availability of sites for chitosan. 

Although the / s obey the sum rule (39), they are complex numbers which impede their interpretation in terms of probabilities. If we keep the assumption that the effective Hamiltonian does not depend on the energy, the inverse Laplace transformation allows to determine the dynamics for any initial state belonging to the model space. Using (12) the evolution operator (3) projected in the model space reads... 

There are many way to implement FFTs, and different ways of reporting the results. The FFT value is generally a complex number - so the FFT of 7 samples will contain J complex values. The real and imaginary components may be treated as two separate values, and the values numbered from 1 to 27, or 0 to 27- 1. The FFT routine described by Press et al. (2007) - Mathematical Recipes - uses 7 = 1 for the index of the first value, and it is the value corresponding to the DC value . Other FFT routines may provide these results in a different order, with different indices. The simplest way to ensure that you understand the convention is to perform the FFT on a sample set for a textbook case - like a square wave, or saw-tooth - and verify that your interpretation of the results agrees with the textbook results. 

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