Transitive relation

Re-examining the four conditions, one notices that the congruence of graphs as defined here establishes a reflexive, symmetric, and transitive relation between graphs. Observe that (IV) does not involve an arbitrary set but a group of permutations. 

A low-spin to high-spin transition relates to the crystal field splitting of the d-orbitals in an octahedral or tetrahedral crystal field. However, even in cases where the energy difference between two spin states is much larger, electronic transitions are observed. An atom with total spin quantum number S has (22 + 1) orientations. In a magnetic field the atom will have a number of discrete energy levels with... 

When using these concepts one has to keep in mind that homochirality by chemical similarity is not always a transitive relation and is thus not suitable for establishing equivalence. 

A polymer may be amorphous, crystalline, or a combination of both. Many polymers actually have both crystalline and amorphous regions, i.e., a semicrystalline polymer. The Tg is a transition related to the motion in the amorphous regions of the polymer [3,8,9], Below the Tg, an amorphous polymer can be said... 

Fig. 30. Diagram of transitions related to photoemission into solution 1—photoexcitation of an electron and photoemission, 2—thermalization of the photoemitted electron in the solution, 3—solvation of the thermalized electron, and 4—trapping of the solvated electron by acceptor A in the solution. de)oc is the lower edge of the band of delocalized states in the solution, solv is the energy level of the solvated electron, and EA is the acceptor energy level.

The nature of the electronic transition related to the 4.9 eV absorption band (the C state) of oxy radical is still unknown. It was experimentally found [65] that the quantum yield of the red luminescence (1.95 eV) is equal to 0.5 + 0.2. Consequently, the nature of this excited state should be so that the transition to the B state will be possible with high probability from this state. This means that the terms corresponding to the C and B states are crossed or converged (come closer to each other) at any arbitrary point of the configurational space. 

The polymethines rely on the n=>7i electronic transition to achieve their long wavelength absorption characteristics, a transition which is uniquely related to the presence of the polar end-groups. The polyenes must rely on the larger energy transitions related to the conjugated chain alone, n n, etc. transitions usually occur in the near... 

Trinitrophenyl-nucleotides constitute a unique class of fluorescent ATP-ana-logs, since they bind at the catalytic site of the phosphoenzy me of the Ca2+-ATPase after departure of ADP and give off a tremendous fluorescence signal upon conversion of the phosphoenzyme intermediate from ADP-sensitive to ADP-insen-sitive (c.f., Figure 6), possibly reflecting a hydrophilic-hydrophobic transition related to closure of the catalytic site (Andersen et al., 1985 Seebregts and McIntosh, 1989). 

Consider the excitation operator k (k corresponding to the elementary transition relating k (root state) to a target state k) F,f acting from the right to commutator [H, k)(k ] yields ... 

As suggested by several other chapters in this volume, the machinery of visuospatial reasoning may provide an economical basis for reasoning about transitive relations. Our visual systems are adept at computing spatial relations—such as above( ), larger-than( )—and many of these relations are transitive If object A is above object B and B is above C, then A will be above C. Importantly, the visual machinery that computes these relations from the information in a visual image must have this... 

Figure 1-24 Glass Transition Temperature (Tg) for Maltose, Maltose Polymers, and Extrapolated Value for Starch. M indicates molecular weight. Source Reprinted with permission from Y.H. Roos, Glass Transition-Related Physico-Chemical Changes in Foods, Food Technology, Vol. 49, No. 10, p. 98, 1995, Institute of Food Technologists.

Roos, Y.H. 1995. Glass transition-related physicochemical changes in foods. Food Technol. 49, no. 10 97-102. 

In algebra the equal sign stands between two algebraic expressions and indicates that two expressions are related by a reflexive, symmetric and transitive relation. The mathematical expressions on either side of the = sign are mathematically identical and interchangeable in equations. 

Spectroscopic methods, such as FT-infrared (FTIR) and Raman spectroscopy detect changes in molecular vibrational characteristics in noncrystalline solid and supercooled liquid states. Various nuclear magnetic resonance (NMR) techniques and electron spin resonance (ESR) spectroscopy, however, are more commonly used, detecting transition-related changes in molecular rotation and diffusion (Champion et al. 2000). These methods have been used for studies of the amorphous state of a number of sugars in dehydrated and freeze-concentrated systems (Roudaut et al. 2004). 

Figure 5.5. State diagram of sucrose with typical experimental data at high concentrations and in the maximally freeze-concentrated state. Dynamic mechanical and dielectric measurements show glass transition-related relaxations.

Magnetic phase transitions related to structural phase transitions similar to those described in neutral radical systems based on l,3,5-trithia-2,4,6-triazapentalenyl (ttta)" ... 

Finally, we have not observed a spontaneous transition from a low flux to a high flux state (Figure 24.7 A) with our previous MR-based membranes [3,5]. The fact whether this transition is observed depends on the feed concentration suggests that the transition is a transport-related phenomenon. It is possible that this transition relates to the concept of cooperative (high flux) vs. noncooperative (low flux) dehybridization (Figure 24.9), but further studies, both experimental and modeling, will be required before a definitive mechanism for this transition can be proposed. 

Our results also show interesting matrix-element effects for features 4 and 5. In particular, the shallowest component of peak 4 at 6 eV is almost resolution limited at /iv 50 eV (Fig. 2) but is only a weak shoulder at Av—65 eV (Fig. 1). Feature 5 is clear from Av—50 to 80 eV but was not discernible at other energies, despite the fact that the experimental resolution was not varied. Again, these effects demonstrate the importance of dipole selection rules for these highly symmetric molecules. In contrast, solids rarely exhibit modulation in structure at such high photon energies except for transitions related to critical points, i.e., primary Mahan cone emis-... 

Kolb et al. [3.155, 3.224, 3.225] studied experimentally the system Au(lll)/Cu, H, so/. Along with observed discontinuous-like q E) isotherms (Fig. 3.50), non-monotonous current transients were potentiostatically measured (Fig. 3.51). These findings were explained in terms of first order phase transitions related to the peaks in the cyclic voltammograms. 

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