Cosine definition

I4bis. Cosines definition between states vectors... 

The hyperbolic sine, hyperbolic cosine, etc. of any number x are functions related to the exponential function e . Their definitions and properties are very similar to the trigonometric functions and are given in Table 1-5. 

In powder EPR simulators we use the orientation of the static-field vector B with respect to the molecular xyz-axes system as the definition of molecular orientation. The orientation is defined in terms of the polar angles 0, direction cosines as defined previously in Equation 5.3. To solve Equation 8.17 we have to define the direction cosines k, of B, in terms of the direction cosines li of B. 

Periodicity in space means that it repeats at regular intervals, known as the wavelength, A. Periodicity in time means that it moves past a fixed point at a steady rate characterised by the period r, which counts the crests passing per unit time. By definition, the velocity v = A/r. It is custom to use the reciprocals of wavelength 1/X — (k/2-ir) or 9, known as the wavenumber (k = wave vector) and 1/t — v, the frequency, or angular frequency u = 2itv. Since a sine or cosine (harmonic) wave repeats at intervals of 2n, it can be described in terms of the function... 

Figure 4.21 Definition of direction cosines g and. (a) Lane case, (b) Bragg case...

Problem 8-13. This analogy between inner products of functions and of vectors can be pursued further. Suggest reasonable definitions for (a) the norm of a function, /, (b) a normalized function, and (c) the cosine of the angle between two functions. 

The terms displacement and true power factor, are widely mentioned in power factor studies. Displacement power factor is the cosine of the angle between the fundamental voltage and current waveforms. The fundamental waveforms are by definition pure sinusoids. But, if the waveform distortion is due to harmonics (which is very often the case), the power factor angles are different than what would be for the fundamental waves alone. The presence of harmonics introduces additional phase shift between the voltage and the current. True power factor is calculated as the ratio between the total active power used in a circuit (including harmonics) and the total apparent power (including harmonics) supplied from the source ... 

FIGURE 1.1 Definition of the sine and cosine function, in terms of positions on a circle with radius 1. 

Don t let the definition based on tracking around a circle fool you—sine and cosine waves appear in many problems in chemistry and physics. The motion of a mass... 

Shortly after the Cosmic Microwave Background (CMB) was discovered, the first anisotropy in the CMB was seen the dipole pattern due to the motion of the observer relative to the rest of the Universe (Conklin, 1969). After confirmation by Henry, 1971 and by Corey and Wilkinson, 1976 the fourth discovery of the dipole (Smoot et al., 1977) showed a very definite cosine pattern as expected for a Doppler effect, and placed an upper limit on any further variations in Tcmb Further improvements in the measurement of the dipole anisotropy were made by the Differential Microwave Radiometers (DMR) experiment on COBE (Bennett et al., 1996 and by the Wilkinson Microwave... 

The manner in which sample-to-sample resemblance is defined is a key difference between the various hierarchical clustering techniques. Sample analyses may be similar to one another in a variety of ways and reflect interest in drawing attention to different underlying processes or properties. The selection of an appropriate measure of similarity is dependent, therefore, on the objectives of the research as set forth in the problem definition. Examples of different similarity measures or coefficients that have been used in compositional studies are average Euclidean distance, correlation, and cosine. Many others that could be applied are discussed in the literature dealing with cluster analysis (15, 18, 19, 36, 37). 

Let a be the angle (less than two right angles) that the direction of the force F makes with the direction MM of the displacement of the material point by the definition of the cosine of an angle, the work we have just described will be represented in magnitude and direction by the formula... 

Analysis of molecular similarity is based on the quantitative determination of the overlap between fingerprints of the query structure and all database members. As descriptors of a given molecule can be considered as a vector of real or binary attributes, most of the similarity measures are derived as vectorial distances. Tanimoto and Cosine coefficients are the most popular measures of similarity.Definitions of similarity metrics are collected in Table 3. 

The electron density in a crystal precisely fits the definition of a periodic function in which an exact repeat occurs at regularly fixed intervals in any direction (the crystal lattice translations). Therefore the electron density in a crystal with a periodicity d can be described by a Fourier synthesis in which each component cosine wave (which we will call an electron-density wave) has a periodicity (i.e., wavelength) d/n, and the amplitude of the rath-order Bragg reflection. 

Fourier synthesis The summation of sine and cosine waves to give a periodic function an example is the computation of an electron density map from waves of known phase, frequency, and amplitude F (. See also the definition in Chapter 1. 

On account of the boundary conditions /f(0) = 0 and ip l) = 0 (string of length I with ends fixed) the cosine vibration drops out at once as being inconsistent with the first condition. But even the sine vibration is not a solution of the boundary value problem, unless l /X is an exact integral multiple of tt, so that ifj vanishes when x = 1. It is only for definite values of A (the proper values) that we obtain possible forms of vibration these are given in fact by... 

The reference frames chosen for the definition of the direction cosines m, of stress depend on the sub-family of the centres considered. The general piezospectroscopic parameters for the trigonal and orthorhombic I centres are given in Tables 8.10 and 8.11. 

This equation may be compared with the (36.19) of [43], Vol. 4), for example, by taking into account that here there is a change of sign in (10.5), an exchange between the real and the imaginary part of S due to our exchange between cosine and sinus in the asymptotic definition of g and /, that the normalization is made on the energy scale e (see Sect. 10.1.3), and also that the equation cited in reference contains an useless factor 2. 

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