... linearly1.1
The relation depends upon the initial energy distribution.
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... grasp1.2
Some idea of the rate of progress of this technology can be gained from such quotes as Such a machine in the hands of a competent operator can produce 400 full-length products or 1,000 sums during an 8-hour working day''[42], referring to an electromechanical desk calculator typical for scientific use in the 1950s. The diagonalization of an 8-by-8 matrix was a weekend-long task[125].
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... profile2.1
Note that I will also refer to as the band profile, since it differs only by a constant factor.
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... formula3.1
This formula was first found (in ) by Wallace Clement Sabine in 1898, in his studies of Harvard's own Jefferson Laboratory main auditorium, room 250. Sabine used this expression for the mean free path to give a formula for the reverberation time of a hall [21,173].
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... vector3.2
The cross-product form used here for and is strictly valid in 2D and 3D only. For the higher-dimensional generalization of a general rotation should be used.
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... components4.1
The condition that a deformation not move the center of mass' (centroid of the cavity volume) is . This is in general different from the condition for having zero overlap with translations, namely .
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... Vergini 6.1
Personal communication.
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... volume6.2
For an example of appearance of surface waves in an example problem in linear spaces see [77].
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... state6.3
as also found by E. Vergini, personal communication.
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... do6.4
This is an elaboration of a suggestion of Vergini (personal communication).
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... profile7.1
By non-adiabatic, we mean that even at a QPC's narrowest region the transverse profile is changing rapidly. Clearly every QPC becomes non-adiabatic' at the coupling to infinite-width reservoirs: this type of non-adiabaticity we do not include because it does not cause significant impedance mismatch, as explained in Ref. [206].
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... region7.2
Of course, throughout this chapter we could imagine the incident wave on the right-hand side, and the same conductance would result (since we are in linear response); see Section 7.6.
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... profile7.3
We could equally well imagine that the QPC can be closed off' (no transmission) by varying a parameter (this is often true experimentally), and define as the full wavefunction in this closed-off state. Thus would be the sum of an incident plane wave and a more complicated outgoing wave. This alternative definition may be better in systems where the wall has disorder, or where there is more complicated structure on the left-hand side than shown in Fig. 7.1a. The two definitions are equivalent as far as Sections 7.3 and 7.4 are concerned.
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...sakurai) 7.4
Presentations of two-dimensional scattering theory are rare; see [132,3,43].
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... wall7.5
This argument can also be verified in the more specific case of the left region being a rectangular Dirichlet box, in which case the exact eigenfunctions are known and can be written explicitly in terms of a sum of for incidences and . However the phase-space presentation is more general, and applies to the real situation where the left region is chaotic (diffusive elastic scattering).
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... regime7.6
The only requirement for this 1D argument to hold is that the effective longitudinal potential tend to a constant asymptotically. This allows a 1D density of states to be defined. It precludes for instance the Mathieu basis mentioned in Section 7.3, unless a reparametrization of the radial' coordinate is performed.
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... imply7.7
With , the conductance is still symmetric under swapping the leads. This results from the 2-terminal special case of unitarity sum rules[55], namely that the rows and columns of the matrix of absolute-value-squared -matrix elements must all sum to 1. Thus the reciprocity derived here is preserved for . How the classical argument from the previous paragraph generalizes for is not known by the authors.
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... beam7.8
Because we wish to consider general illumination and general , our definition of conductance' coincides with that of Ref. [191] only in the case of isotropic illumination . The beam brightness per unit range, that is, its phase-space density, is assumed uniform in position space, and is proportional to . This is also proportional to defined in Ref. [191].
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... defined8.1
Our definitions of and correspond to those of C. S. Adams et al. [2]
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... frequency8.2
Even though the frequencies will actually differ by a couple of percent because of their detunings, and cause a difference in decay lengths of the correct sign, this effect alone is not large enough to create large values. Future designs with larger detunings, possibly much larger than the fine splitting, may be able to use this effect alone
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... direction8.3
Following the notation of Fernandez & Lu[139] for rectangular guides, mode has () antinodes in the () direction, and overall polarization axis . However as the symmetry line is approached (see Figure 8.2), mixing of polarizations occurs for the case and the notation becomes more arbitrary.
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... constant8.4
We made the assumption that the positive and negative detunings caused negligible shifts in the wavelength used for the bound mode calculation. For the largest detunings used in this chapter the fractional shift is 1.8%.
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... workstation8.5
Silicon Graphics 75MHz R8000 processor commercial compilers, with a SPEC FP95 rating [186] of between 10 and 12.
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... estimated8.6
For some of the earliest bending loss estimation in a rectangular fiber see [142] and for the case of a cylindrical fiber see [143]. Most of the work done since then has been on the more complicated situation of fibers with cladding.
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... complicatedA.1
Doron Cohen, personal communication.
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... belowD.1
I have found other examples, for instance the deformation field is special for the potential . Such a deformation is dilation-like, but not a linear dilation given by the symmetric part of .
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... sequenceH.1
This will not be given, but briefly it involves expanding the second term, use of Gauss' theorem and the identity . Meanwhile the first term is compared to the volume integral of and its expansion, using the identity . Both these identities require Einstein summation notation to prove. Bypassing such a tricky maze of formulae is one purpose of this appendix.
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`