Complex value

By defining a norm defining the distance between two signals, one can easily spot its variations. This distance provides an image of the signal evolution. Signals are typically Lissajous ( orbits ), i.e. arrays 2 of successive complex-valued points... 

Referring to figure Bl.8.5 the radii of the tliree circles are the magnitudes of the observed structure amplitudes of a reflection from the native protein, and of the same reflection from two heavy-atom derivatives, dl and d2- We assume that we have been able to detemiine the heavy-atom positions in the derivatives and hl and h2 are the calculated heavy-atom contributions to the structure amplitudes of the derivatives. The centres of the derivative circles are at points - hl and - h2 in the complex plane, and the three circles intersect at one point, which is therefore the complex value of The phases for as many reflections as possible can then be... 

Figure Bl.8.5. Pp Pdl and Fdl are the measured stnicture amplitudes of a reflection from a native protein and from two heavy-atom derivatives. and are the heavy atom contributions. The pomt at which the tliree circles intersect is the complex value of F. ...

What is addressed by these sources is the ontology of quantal description. Wave functions (and other related quantities, like Green functions or density matrices), far from being mere compendia or short-hand listings of observational data, obtained in the domain of real numbers, possess an actuality of tbeir own. From a knowledge of the wave functions for real values of the variables and by relying on their analytical behavior for complex values, new properties come to the open, in a way that one can perhaps view, echoing the quotations above, as miraculous. ... 

To see that this phase has no relation to the number of ci s encircled (if this statement is not already obvious), we note that this last result is true no matter what the values of the coefficients k, X, and so on are provided only that the latter is nonzero. In contrast, the number of ci s depends on their values for example, for some values of the parameters the vanishing of the off-diagonal matrix elements occurs for complex values of q, and these do not represent physical ci s. The model used in [270] represents a special case, in which it was possible to derive a relation between the number of ci s and the Berry phase acquired upon circling about them. We are concerned with more general situations. For these it is not warranted, for example, to count up the total number of ci s by circling with a large radius. 

In the sequel, we assume that the quantum subsystem has been truncated to a finite-dimensional system by an appropriate spatial discretization and a corresponding representation of the wave function by a complex-valued vector Ip C. The discretized quantum operators T, V and H are denoted by T e V(q) E and H q) e respectively. In the following... 

Two important facets of these functions should be recognized. First, the sin z is unbounded and, second, e takes all complex values except 0. 

Complex—Value stored as two words, one representing the real part of the number and the other representing the imaginary part. 

Complex valued random variables will be discussed in Section 3.8. For the time being it is sufficient to state that the expectation of a complex function is defined by E[fa = E[fa] + iE[fa] where fa and fa denote the real and imaginary parts, respectively, of fa... 

We conclude this section with some remarks concerning complexvalued random variables. A complex-valued random variable can always be written in the form = r + iff> where T and t are realvalued random variables. This means that all averages involving ... 

Other postulates required to complete the definition of will not be listed here they are concerned with the existence of a basis set of vectors and we shall discuss that question in some detail in the next section. For the present we may summarize the above defining properties of Hilbert space by saying that it is a linear space with a complex-valued scalar product. 

We shall in this chapter be most concerned with the following example of Hilbert space. Each element /> is a complex-valued numerical function f(x) of one or more continuous variables represented collectively by the symbol x, such that the integral of its square modulus exists ... 

Complex conjugation, 492 Complex-valued random variables, 144 Compound distribution, 270 Conditional distribution functions, 148, 152... 

Random variables, 114,117 completely correlated, 147 complex-valued, 144 discrete, 196... 

Monochromatic Waves (1.14) A monochromatic e.m. wave Vcj r,t) can be decomposed into the product of a time-independent, complex-valued term Ucj r) and a purely time-dependent complex factor expjojt with unity magnitude. The time-independent term is a solution of the Helmholtz equation. Sets of base functions which are solutions of the Helmholtz equation are plane waves (constant wave vector k and spherical waves whose amplitude varies with the inverse of the distance of their centers. 

Usually we are only interested in mutual intensity suitably normalised to account for the magnitude of the helds, which is called the complex degree of coherence 712 (r). This quantity is complex valued with a magnitude between 0 and 1, and describes the degree of likeness of two e. m. waves at positions ri and C2 in space separated by a time difference r. A value of 0 represents complete decorrelation ( incoherence ) and a value of 1 represents complete eorrelation ( perfect coherence ) while the complex argument represents a difference in optical phase of the helds. Special cases are the complex degree of self coherence 7n(r) where a held is compared with itself at the same position but different times, and the complex coherence factor pi2 = 712(0) which refers to the case where a held is correlated at two posihons at the same time. 

First we define the linear map that produces the densities from N-particle states. It is a map from the space of A -particle Trace Class operators into the space of complex valued absolute integrable functions of space-spin variables... 

Eh Nonlinear energy functional based on the Hamiltonian H-, it acts on the space of absolutely integrable complex valued functions of the variables y. 

The real wave packet (RWP) method, developed by Gray and Bahnt-Kuiti [ 1], is an approach for obtaining accurate quantum dynamics information. Unlike most wave packet methods [2] it utilizes only the real part of the generally complex-valued, time-evolving wave packet, and the effective Hamiltonian operator generating the dynamics is a certain function of the actual Hamiltonian operator of interest. Time steps in the RWP method are accomphshed by a simple three-term Chebyshev... 

In the same way as described above, we can formulate the multidimensional theory without relying on the complex-valued Lagrange manifold that constitutes one of the main obstacles of the conventional multidimensional WKB theory [62,63,77,78]. Another crucial point is that the theory should not depend on any local coordinates, which gives a cumbersome problem in practical applications. Below, a general formulation is described, which is free from these difficuluties and applicable to vertually any multidimensional systems [30]. 

When Ip / 0, rs takes complex values and accordingly, we define the excursion time n = Re(rf). Notice that the classical simulations do not provide correct r s [20],... 

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