Imaginary space

Bond, N. (1950) The Scientific Pioneer Returns, in Lancelot Biggs Spaceman by Nelson S. Bond. New York Doubleday. (Story originally published in 1940.) A ship accelerates into imaginary space that turns out to be a parallel universe. Einstein and Planck fiddled around with hyper-spatial mechanics and discovered that mass is altered when it travels at high velocity. The gadget worked better than you expected. ... 

The imaginary character of the waves in the wave particle model becomes clearer when the theory of two particles which repel each other, like two electrons, is considered. It is then found necessary to consider a set of waves in an imaginary space of six dimensions. Such waves are certainly imaginary, so that there is little doubt that if there is anything at all like reality in the wave particle model, it is the particles, and not the waves. 

We can imagine that xi, x2, x3> and x4 are the distances of the event from the origin measured along four mutually perpendicular lines as coordinate axes. This is impossible in actual space, but we can imagine a space in which four mutually perpendicular lines can be drawn through a point. Such a space is said to be of four dimensions. In this imaginary space of four dimensions we suppose we have a set of four perpendicular coordinate axes, and that the two events are represented by two points in this space. The first event is represented by the point at the origin O and the other by a point P with coordinates x3, x2, x3, and x4. 

Differentiation, specialization, maturation, and transformation are generally accompanied by small changes in physical intrinsic properties of cells. Differences in size, density, and surface charge position each cell type in a rather unique part of a size—density—surface charge continuum. It is the purpose of cell separation to transport by some physical means dissimilar, closely neighboring cells from this imaginary space into separate fractions that are obtained after one-dimensional migration of the cells. 

Looking at the numerator of Equation 3.6 as a polynomial in Z, there will be N values of Z that make the numerator equal to zero, and the transfer function will be zero at these values. These values that make the polynomial equal to zero are potentially complex numbers Re + jim, where Re and Im are called the real and imaginary parts, and j = V( 1). The zero values are called zeroes of the filter because they cause the gain of the transfer function to be zero. The two-dimensional (real and imaginary) space of possible values of Z is called the z-plane. 

The computational advantage of using the Hankel functions is that they contain the argument fir p. Thus, as we enter the imaginary space, would become imaginary (see Section 3.4), leading to modified Hankel functions that faU off exponentially as /o or n increase (see Fig. 3.7). Thus, we could obtain the mutual impedances between arrays with elements of arbitrary orientation. Once these... 

Fig. 4.2 Typical scan impedance for an infinite and a finite array at the fixed frequency 8 GHz (i.e., below resonance) as a function of scan angle. Note By feeding the elements with actual voltage generators like a phased array, we can scan the beam" beyond endfire into the imaginary space where Is l > 7 and only evanescent waves are possible, provided that the interelement spacing Dx is < 0.5 k.

The interelement spacings Dx were not less than X/2 that is, grating lobes could occur as we move into imaginary space. As discussed in Section 4.9.3, that automatically rules out free surface waves. 

We shall investigate the F-plane case analogous to the H-plane case above, namely by plotting the scan impedance from broadside (Sx = 0) and aU the way into the end of imaginary space and back into real space. 

A typical example of an F-plane scan impedance for an infinite array is shown in Fig. 4.33. We start at broadside at rj = 0° for = 0 and proceed to grazing at T] = 90° for = 1, which marks our entrance into the imaginary space. As Sx becomes larger than 1, we see that the scan impedance moves downward along the imaginary axis until it reaches its lowest level for = 112 from where it starts moving upward and evenmaUy crosses the real axis for Sx = 1.65. For... 

Fig. 4.33 The scan impedance atf = 9 GHz when the array in Fig. 4.32c is scanned in the E piane. Broadside starts at s> = 0 and grazing is at Sx= 1.0. Higher vaiues of s correspond to scan in the imaginary space where the scan impedance is purely Imaginary. It gets to the "end of imaginary space for Sx = 2.08 from where it goes right back down the way it got in.

The overall phenomenon effectively looks like the so-called deep tim-neling. [46, 140, 149, 151, 269] However, of course, they are independent phenomena. In the quantmn deep tmmeling, the paths go over the barrier through imaginary space and/or time, as in the instanton theory. [46] On the other hand, all the non-Born-Oppenheimer paths in this study are supposed to run in the real space, although the present theory of path branching can be generalized so that the paths can penetrate into complex spaces. 

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