Representations faithful

It is useful to represent the polyelectronic wave function of a compound by a valence bond (VB) structure that represents the bonding between the atoms. Frequently, a single VB structure suffices, sometimes it is necessary to use several. We assume for simplicity that a single VB stiucture provides a faithful representation. A common way to write down a VB structure is by the spin-paired determinant, that ensures the compliance with Pauli s principle (It is assumed that there are 2n paired electrons in the system)... 

Also neural networks were proposed to solve the problem of faithful representation of multidimensional data in representation spaces of lower dimensions [17]. 

Requirement 2. Self-consistency The peak lists must be faithful representations of the NOESY spectra, and the chemical shift positions of the NOESY cross peaks must be correctly calibrated to fit the chemical shift lists within the chemical shift tolerances. The range of allowed chemical shift variations ( tolerances ) for 1H should not exceed 0.02 ppm when working with homonuclear [1H,1H]-NOESY spectra, or 0.03 ppm when work-... 

It would be difficult, at this stage, to present a properly argued preference between schemes A and B in the Introduction as the more faithful representation for the disrotatory electrocyclic ring-opening of cyclohexadiene. Chemical intuition (at least that of the authors) would suggest scheme B, but scheme A is on the textbook pages and, as has been shown by the SC description of the 1,3-dipolar addition of fulminic acid to ethyne, a heterolytic scheme remains a definite possibility. 

If each group element corresponds to a different matrix, the representation is said to ht faithful. A faithful representation is a matrix group that is isomorphic to the group being represented. 

Using advances in computer reconstruction methods (see e.g. Kikkinides and Burganos, 2000 Torquato, 2001) and past experience with discrete particle deposit simulations (Konstandopoulos, 2000), we have developed algorithmic as well as process-based reconstruction techniques to generate three-dimensional (3D) digital materials that are faithful representations of DPF microstructures. We refer to this approach as DPF microflow simulation (MicroFlowS). MicroFlowS is thus a short name for a computational approach, which combines... 

It is not clear how improvements can be made without real progress on the difficult fundamental problems of diffusion in media with obstacles and cooperation of large-scale motions between interpenetrating chains which do not violate chain connectivity. The DeGennes reptation model (225) makes a significant contribution to the first problem, although in an admittedly simplified system. Rigorous calculations or computer simulations on well-defined models which relate to the second problem would be extremely valuable, even if the models themselves were not completely faithful representations of the assumed physical situation. It is not obvious how even to pose solvable problems, simplified or not, which relate to interchain cooperation. 

The set of components of the vector r1 in eq. (13) is the Jones symbol or Jones faithful representation of the symmetry operator R, and is usually written as (x / /) or x / z. For example, from eq. (15) the Jones symbol of the operator R (n/2 z) is (yxz) or yxz. In order to save space, particularly in tables, we will usually present Jones symbols without parentheses. A faithful representation is one which obeys the same multiplication table as the group elements (symmetry operators). 

If all the matrices L (A) are different, however, then T is isomorphous with G and it is called a true or faithful representation. 

This shows that the representation of the group on the nuclear position vectors referred to the laboratory fjxed coordinate system is not a faithful representation of the isometric group ( ) = r[Pg.25]

Although the H nuclei of the OH groups of glycol lay in the symmetry plane of the Cs tops, it is convenient to introduce nuclei in general site in a SRM associated to a particular molecule to be sure that a faithful representation of sr on the distances of that set is generated. 

The Cournot model is a standard oligopoly model, and it is often used in competition policy as a first approximation of how competition works (Martin, 1993). For our modelling purposes, the Cournot oligopoly model offered the best combination - faithful representation of market cost structure and behaviour, and flexibility to incorporate a mixture of profit-maximizing and sales-maximizing objectives - as well as tractable conversion into a spreadsheet modelling application. 

The RRK (after Rice, Ramsperger, and Kassel) theory is, like the Slater theory, a model for a unimolecular reaction rather than a faithful representation. The molecule is again represented by a collection of s uncoupled harmonic oscillators, which is an exact representation close to a stationary point on the potential energy surface. One of these... 

Techniques for the production of individual cDNAs, cloned in bacteria at the rate of one cDNA per bacterium, were multiplexed to produce cDNA libraries , large collections of cDNAs, simply by repeating the cloning procedure with complex cDNA mixtures. These complex cDNA mixtures, now in bacteria, can be surveyed by retrieving and sequencing each individual cDNA, giving a faithful representation of the abundance of each cDNA in the original mixture if sufficient cDNAs are sequenced. 

Before we leave the methods of this chapter, however, we should prove one more result all representations can be derived from a single faithful representation. 

The discrete Fourier transform provides a useful and efficient means of extracting information on the frequency components present in a time-varying signal and displaying the amplitudes of these components as a spectrum. However, a number of potential artifacts must be avoided if we are to obtain a faithful representation of the information actually present in the time domain signal. 

The minimum sampling rate of the ADC is critical to faithful representation of the signal frequencies. This rate, the number of measurements per second, is often also expressed as the dwell time of the ADC, that is, the time interval between measurements, At=t —, . It is well established in information theory that... 

The faithful representation of the shape of lines broadened greatly by dipolar and, especially, quadrupolar interactions often requires special experimental techniques. Because the FID lasts for only a very short time, a significant portion may be distorted as the spectrometer recovers from the short, powerful rf pulse. We saw in Section 2.9 that in liquids a 90°, t, 180° pulse sequence essentially recreates the FID in a spin echo, which is removed by 2r from the pulse. As we saw, such a pulse sequence refocuses the dephasing that results from magnetic field inhomogeneity but it does not refocus dephasing from natural relaxation processes such as dipolar interactions. However, a somewhat different pulse sequence can be used to create an echo in a solid—a dipolar echo or a quadrupolar echo—and this method is widely employed in obtaining solid state line shapes (for example, that in Fig. 7.10).The formation of these echoes cannot readily be explained in terms of the vector picture, but we use the formation of a dipolar echo as an example of the use of the product operator formalism in Section 11.6. 

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