Numbers imaginary

If an nth degree polynomial does indeed have a total of n roots, then we must accept roots containing square roots of negative numbers—imaginary... 

Draw a two-dimensional graph with the horizontal axis representing all real numbers and the vertical axis representing all imaginary numbers (an Argand... 

A eomplex number has a real part and an imaginary part and is usually denoted ... 

Fig. 25-3. Bubble concept. This pollution control concept places an imaginary bubble over an entire industrial plant, evaluating emissions from the facility as a whole instead of requiring control point-by-point on emission sources. Numbers represent emissions from individual sources, some of which can be fugitive sources, and from the entire industrial plant. Source Drawing courtesy of the Chemical Manufacturers Association.

Imaginary frequencies are listed in the output of a frequency calculation as negative numbers. By definition, a structure which has n imaginary frequencies is an n order saddle point. Thus, ordinary transition structures are usually characterized by one imaginary frequency since they are first-order saddle points. 

All structures Number of imaginary frequencies = 1. For ab initio calculations Relative energies are ZPE-corrected (scaUiig factor, 0.89). 

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

AIMAG Returns the imaginary part of a complex number... 

The flux ( J ) is a common measure of the rate of mass transport at a fixed point. It is defined as the number of molecules penetrating a unit area of an imaginary plane in a unit of time, and has the units of mol cm 2 s-1. The flux to the electrode is described mathematically by a differential equation, known as the Nemst-Planck equation, given here for one dimension ... 

Ma is defined by Equation 5.1.8 and / is defined by Equation 5.1.7. Typical curves for fhe imaginary part of the transfer frmction, lm[Tr], are plotted in Figure 5.1.13. These curves are calculated for a flame speed of 0.3 m/s, the other parameters in the coefficients A, B, C, and D are appropriate for a lean mefhane flame. The response is shown for three typical dimensionless wave numbers, kS = 0.01,0.03, and 0.1, which correspond to dimensional... 

From these results it is possible to make another estimate of a property of the solution system. It is known that the freezing point of a solvent is lowered by approximately 1.86°C for every mole of the solute present. From the estimates of the temperature of the solvent and the solution modeled above, the decrease in the temperature can be estimated. From this value, the number of cells comprising a mole of solute may be reckoned. Thus, a value may be stated for an imaginary molecular weight of the cells used in the study. 

The next step after apodization of the t time-domain data is Fourier transformation and phase correction. As a result of the Fourier transformations of the t2 time domain, a number of different spectra are generated. Each spectrum corresponds to the behavior of the nuclear spins during the corresponding evolution period, with one spectrum resulting from each t value. A set of spectra is thus obtained, with the rows of the matrix now containing Areal and A imaginary data points. These real and imagi-... 

The matrix obtained after the F Fourier transformation and rearrangement of the data set contains a number of spectra. If we look down the columns of these spectra parallel to h, we can see the variation of signal intensities with different evolution periods. Subdivision of the data matrix parallel to gives columns of data containing both the real and the imaginary parts of each spectrum. An equal number of zeros is now added and the data sets subjected to Fourier transformation along I,. This Fourier transformation may be either a Redfield transform, if the h data are acquired alternately (as on the Bruker instruments), or a complex Fourier transform, if the <2 data are collected as simultaneous A and B quadrature pairs (as on the Varian instruments). Window multiplication for may be with the same function as that employed for (e.g., in COSY), or it may be with a different function (e.g., in 2D /-resolved or heteronuclear-shift-correlation experiments). 

Each zone is divided by an imaginary grid with units of approximately 4 meterA series of consecutive numbers is then assigned to the units of each grid. Through the use of a random numbers table, nj (for the Kth zone) of the grid units for sampling locations are selected, where... 

The most intense 826-cm band is broader than the other bands. The broadened band suggests a frequency distribution in the observed portion of the surface. Indeed, the symmetric peak in the imaginary part of the spectrum is fitted with a Gaussian function rather than with a Lorenz function. The bandwidth was estimated to be 56 cm by considering the instrumental resolution, 15 cm in this particular spectrum. This number is larger than the intrinsic bandwidth of the bulk modes [50]. 

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