Representation Linking

G. Constraints on Off-Diagonal Elements from Other Positive-Definite Hamiltonians Linear Inequalities from the Spatial Representation Linking the Orbital and Spatial Representations... 

E and are the energy and the width of the useful part of the continuum (doorway state) [22, 33]. The two-dimensional non-Hermitian effective Hamiltonian (30) is the simplest matrix representation linking the microscopic level characterized by the complex energy E — iFc/2 to the macroscopic level of interest (the resonance). In Eq. (30), the energy of the resonance El is real. We will see below that if the resonance is weakly coupled to the microscopic level (AE F ), the complex part of energy can be uncovered by... 

We have alluded to the comrection between the molecular PES and the spectroscopic Hamiltonian. These are two very different representations of the molecular Hamiltonian, yet both are supposed to describe the same molecular dynamics. Furthemrore, the PES often is obtained via ab initio quairtum mechanical calculations while the spectroscopic Hamiltonian is most often obtained by an empirical fit to an experimental spectrum. Is there a direct link between these two seemingly very different ways of apprehending the molecular Hamiltonian and dynamics And if so, how consistent are these two distinct ways of viewing the molecule ... 

Figure B3.3.11. The classical ring polymer isomorphism, forA = 2 atoms, using/ = 5 beads. The wavy lines represent quantum spring bonds between different imaginary-time representations of the same atom. The dashed lines represent real pair-potential interactions, each diminished by a factor P, between the atoms, linking corresponding imaginary times.

The reason why complexity and symmetry are linked together is quite straightforward. Indeed, a representation of highly symmetrical systems requires fewer characteristics than that of objects having low symmetry because, if we know the characteristics of one object, we can employ them to represent all those which are symmetrical with the given one. 

Fig. 25. Schematic representation of imprinting (a) cross-linking polymerization ia the presence of a template (T) to obtain cavities of specific shape and a defined spatial arrangement of functional groups (binding sites. A—C) (b) cross-linked polymer prepared from the template monomer and ethylene...

Fig. 2. A representation of the cellulose chain ia solution, projected against three two-dimensional surfaces. The circles represent the oxygen atoms that link the iadividual glucose residues, and the lines take the place of the sugar residues. This result of a modeling study (39) iadicated a molecule somewhat more...

An important characterization parameter for ceUulose ethers, in addition to the chemical nature of the substituent, is the extent of substitution. As the Haworth representation of the ceUulose polymer shows, it is a linear, unbranched polysaccharide composed of glucopyranose (anhydroglucose) monosaccharide units linked through thek 1,4 positions by the P anomeric configuration. 

Fig. 3. Mechanisms for polymer degradation. The illustration is a schematic representation of three degradation mechanisms I, cleavage of cross-links II, hydrolysis, ionisa tion, or protonation of pendent groups III, backbone cleavage. Actual biodegradation may be a combination of these mechanisms.

It is possible to go beyond the SASA/PB approximation and develop better approximations to current implicit solvent representations with sophisticated statistical mechanical models based on distribution functions or integral equations (see Section V.A). An alternative intermediate approach consists in including a small number of explicit solvent molecules near the solute while the influence of the remain bulk solvent molecules is taken into account implicitly (see Section V.B). On the other hand, in some cases it is necessary to use a treatment that is markedly simpler than SASA/PB to carry out extensive conformational searches. In such situations, it possible to use empirical models that describe the entire solvation free energy on the basis of the SASA (see Section V.C). An even simpler class of approximations consists in using infonnation-based potentials constructed to mimic and reproduce the statistical trends observed in macromolecular structures (see Section V.D). Although the microscopic basis of these approximations is not yet formally linked to a statistical mechanical formulation of implicit solvent, full SASA models and empirical information-based potentials may be very effective for particular problems. 

For the sake of clarity, ten coordination sites are drawn a little further away from the surface of the particle in Fig. 15(a)-(c). These sites are real surface sites and the formal link is shown by a solid line. In this way the different C2 units are easily distinguished in the figure and the formation of six-membered rings is obvious. The planar tubule representations of Fig. 15(a )-(c ) are equivalent to those in Fig. 15(a)-(c), respectively. The former figures allow a better understanding of tubule growth. Arriving C, units are first coordinated to the catalyst coordination... 

IMAS has a facility called EXPLORE allows the analyst to specify which indicators (e.g., temperatures, pressures, valve settings) are present, and which are absent in a particular scenario. EXPLORE then traverses the various links in the mental model representation network and generates a report that simulates the worker s thinking processes. This form of simulation provides useful information to the analyst with regard to the worker s capability to achieve correct diagnoses. Embrey (1985) gives an example of these simulations for the mental model in Figure 4.13. 

The graphical representation of developing 1-dimensional systems, in which link additions must be shown as arcs to avoid overlap with existing links, is needlessly confusing and is not considered. 

Fig. 10.2 Schematic representation of connections in Rosenblatt s Photoreceptron [rosenbl58]. The synaptic connections from the S-units to A-units can be either excitatory or inhibitory connections between A-units and R-units may include inhibitory feedback loops. Response layer units are also linked to other R-units with inhibitory connections.

The Lewis structures encountered in Chapter 2 are two-dimensional representations of the links between atoms—their connectivity—and except in the simplest cases do not depict the arrangement of atoms in space. The valence-shell electron-pair repulsion model (VSEPR model) extends Lewis s theory of bonding to account for molecular shapes by adding rules that account for bond angles. The model starts from the idea that because electrons repel one another, the shapes of simple molecules correspond to arrangements in which pairs of bonding electrons lie as far apart as possible. Specifically ... 

FIGURE 1119 The lysozyme molecule is a typical enzyme molecule. Lysozyme is present in a number of places in the body, including tears and the mucus in the nose. One of its functions is to attack the cell walls of bacteria and destroy them. This "ribbon" representation shows only the general arrangement of the atoms to emphasize the overall shape of the molecule the ribbon actually consists of amino acids linked together (Section 19.13). 

A representation of the stratospheric system that shields terrestrial life from excessive solar ultraviolet radiation is presented in Figure 4. Our primary concern is the decrease of stratospheric ozone, most striking in the Antarctic, which has been linked to increases in CFCs from the troposphere, and the possible increased transport of these compounds between the stratosphere and the troposphere by increased temperature driven circulation. 

Our interpretation of the finding from the study is that even within a single level of representation, i.e. sub-microscopic, some students were unable to translate between different modes of representation. In this case, there was evidence that students were unable to translate their verbal mode of understanding into the diagrammatic mode (or visual mode). If the translation between different modes within the same level is problematic, it would be very challenging when students are required to smoothly mentally move about and link all the three - the macro, the sub-microscopic and symbolic levels - of chemical phenomena. 

If a concept map is regarded as a representation of knowledge and that a set of given concepts can be linked meaningfully but differently, it is surprising that how students make meaning from a given concept map has not been extensively studied. 

This provides a very strong tool for communicating explanations, as the teacher can move between discussing the bench phenomena and the (sub-microscopic) explanatory models readily. By presenting an equation that describes the reaction (a macroscopic phenomena that students can see etc.) in a form that directly links to the molecules or other quanticles (ions, etc.) considered to be present at the sub-microscopic level, the symbolic representation acts as a referent to both levels and so at a meta-level also represents the relationship and mapping between substances and quanticles. 

The sub-micro level is real, but is not visible and so it can be difficult to comprehend. As Kozma and Russell (1997) point out, understanding chemistry relies on making sense of the invisible and the untouchable (p. 949). Explaining chemical reactions demands that a mental picture is developed to represent the sub-micro particles in the substances being observed. Chemical diagrams are one form of representation that contributes to a mental model. It is not yet possible to see how the atoms interact, thus the chemist relies on the atomic theory of matter on which the sub-micro level is based. This is presented diagrammatically in Fig. 8.2. The links from the sub-micro level to the theory and representational level is shown with the dotted line. 

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