Complex number, representation

The time dependence of any periodic function can always be expressed in terms of sine and cosine functions in this sense, it is very convenient to apply the complex number representation using the following equation [135] ... 

With an alternating applied voltage, mathematically described with the help of the complex number representation, and using the following equation... 

It is often convenient to use a complex number representation for sound waves [2]. The harmonic wave is represented by the complex exponential. 

In order to make the mathematics easier, the differential equation is solved using a complex number representation of the sinusoidally varying quantities (see appendix A). Thus, the field E is written as... 

Ellipticity can be induced by reflection because of the fact that the reflection coefficients for the two components in the p and s directions are different. A reflection coefficient comprises two parts, an amplitude term and a phase term, and for this reason complex number representation is used to describe it. The "phasor notation is a convention that provides a compact way of representing reflection coefficients... 

The concept of electrical impedance was first introduced by Oliver Heaviside in the 1880s and was soon after developed in terms of vector diagrams and complex number representation by A. E. Keimelly and C. P. Steinmetz [1]. Since then the technique gained in exposure and popularity, propelled by a series of scientific advancements in the... 

Making use of the polar representation of a complex number, the nuclear wave function can be written as a product of a real amplitude, A, and a real phase, S,... 

Ru 2,2 -bipyridine complexes can form a large number of colored compounds upon successive reduction, with the formal Ru oxidation state from +2 to -4. In the case of highly reduced complexes, proper representation of the electrochromic reaction is actually the reduction of the hgand, not that of the metal center. 

A Hopf algebra emerges by a proper redefinition of the antilinear characteristics of TFD. Consider g = giti = 1,2,3,.. be an associative algebra defined on the field of the complex numbers and let g be equipped with a Lie algebra structure specified by giOgj = C gk, where 0 is the Lie product and Cfj are the structure constants (we are assuming the rule of sum over repeated indeces). Now we take g first realized by C = Ai,i = 1,2,3,.. such that the commutator [Ai,Aj is the Lie product of elements Ai,Aj G C. Consider tp and (p two representations of C, such that ip (A) (linear operators defined on a representation vector space As a consequence,... 

So far we have seen that if we begin with the Boltzmann superposition integral and include in that expression a mathematical representation for the stress or strain we apply, it is possible to derive a relationship between the instrumental response and the properties of the material. For an oscillating strain the problem can be solved either using complex number theory or simple trigonometric functions for the deformation applied. Suppose we apply a strain described by a sine wave ... 

These equations show that the structure factors in general are complex numbers. Another representation of equation 3 is given by ... 

This character is the same as the character of the representation on by matrix multiplication in fact, these two representations are isomorphic, as the reader may show in Exercise 4.36. This is an example of the general phenomenon that will help us to classify representations finite-dimensional representations are isomorphic if and only if their characters are equal. See Proposition 6.12. Note that while a representation is a relatively complicated object, a character is simply a function from a group to the complex numbers it is remarkable that so much information about the complicated object is encapsulated in the simpler object. 

Now suppose that the representations Vi and V2 are indeed isomorphic. Let T and f denote isomorphisms (of representations) from V to 72- It suffices to show that T must be a scalar multiple of T. Consider the linear transformation T o V2 V2- By Exercise 4.19, this linear transformation is an isomorphism of representations. Hence by Proposition 6.3, there must be a complex number A such that T o 7 = kl, and hence t = XT. Note that because 7 is an isomorphism, A 0. ... 

Exercise 8.10 In this exercise we construct infinite-dimensional irreducible representations of the Lie algebra su (2). Suppose k. is a complex number such that L in for any nonnegative integer n. Consider a countable set S = vo, Vi, 172,... and let V denote the complex vector space of finite linear combinations of elements of S. Show that V can be made into a complex... 

The reader familiar with the presentation of the state space of a spin-1/2 particle as S /T (i.e., the set of normalized pairs of complex numbers modulo a phase factor) may wonder why we even bother to introduce P(C2). One reason is that complex projective spaces are familiar to many mathematicians in the interest of interdisciplinary communication, it is useful to know that the state space of a spin-1/2 particle (and other spin particles, as we will see in Section 10.4) are complex projective spaces. Another reason is that in order to apply the powerful machinery of representation theory (including eigenvalues and superposition), there must be a linear space somewhere in the background by considering a projective space, we make the role of the linear space explicit. Finally, as we discuss in the next section, the effects of the complex scalar product on a linear space linger usefully in the projective space. 

In this particular example we could have avoided some of the labour involved in finding the combinations of hybrid orbitals which are equal to px and ptf, by using the 8 point group (to which the molecule also belongs). For this point group, the two-dimensional representation, the cause of all the trouble, can be expressed as two complex one-dimensionl representations. The orbitals p, and py are then just as easy to obtain as the s-orbital. Any complex numbers which result are eliminated at the end of the treatment by addition and subtraction of the orbitals formed. This is the technique which was used in 10-7 to find the 7T-molecular orbitals of the trivinylmethyl radical. It is, however, of no avail when dealing with point groups which have three-dimensional irreducible representations as in our next example, CH4. 

For certain point groups, we have one-dimensional (irreducible) representations with complex characters. Suppose that the normal coordinate Qx transforms according to the one-dimensional (irreducible) representation T some of whose characters are complex numbers. We then have for any symmetry operation R... 

In order to illuminate both the phase problem and its solution, I will represent structure factors as vectors on a two-dimensional plane of complex numbers of the form a + ib, where i is the imaginary number (—1)1/2. This allows me to show geometrically how to compute phases. I will begin by introducing complex numbers and their representation as points having coordinates (a,b) on the complex plane. Then I will show how to represent structure factors as vectors on the same plane. Because we will now start thinking of the structure factor as a vector, I will hereafter write it in boldface (FM,Z) instead of the italics used for simple variables and functions. Finally, I will use the vector representation of structure factors to explain a few common methods of obtaining phases. 

A representation of structure factors on this plane must include the two properties we need in order to construct p(x,y,z) amplitude and phase. Crystallog-raphers represent each structure factor as a complex vector, that is, a vector (not a point) on the plane of complex numbers. The length of this vector represents the amplitude of the structure factor. Thus the length of the vector representing structure factor Fhkl is proportional to The second prop-... 

Figure 12.1. Argand diagram for the representation of complex numbers in the complex plane C.

The PFs gi gy] are a set ofg2 complex numbers, which by convention are all chosen to be square roots of unity. (For vector representations the PFs are all unity.) PFs have the following properties (Altmann (1977)) ... 

The vector model of a single spin is the vector representation of the complex number in the individual density matrix of a single nucleus. This density matrix consists of only one complex number thus there is only one vector in the model. In the case of more than one nuclei, the density matrix is larger, there are more single quantum coherences and more vectors belong to one spin set in the model. Moreover, in case of a strongly coupled spin system, the density matrix has different numerical form for different basis sets of the vector space of the simulation (the basis can be one of the ) and

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Any continuous sequence of data h(t) in the time domain can also be described as a continuous sequence in the frequency domain, where the sequence is specified by giving its amplitude as a function of frequency, H(f). For a real sequence h(t) (the case for any physical process), H(f) is series of complex numbers. It is useful to regard h(t) and 11(f) as two representations of the same sequence, with h(t) representing the sequence in the time domain and H( f) representing the sequence in the frequency domain. These two representations are called transform pairs. The frequency and time domains are related through the Fourier transform equations... 

Sluyters or Cole-Cole plot), in a form similar to the representation of complex numbers (Argand diagram). 

Complex plane plot — The complex number Z = Z + iZ", where i = v/-i, can be represented by a point in the Cartesian plane whose abscissa is the real part of Z and ordinate the imaginary part of Z. In this representation the abscissa is called the real axis (or the axis of reals) and the ordinate the imaginary axis (the axis of imaginaries), the plane OZ Z" itself being referred to as the complex plane [i]. The representing point of a complex number Z is referred to as the point Z. 

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