Canonical structure diagram

An interesting observation by David Weininger (personal communication) is that feeding a canonical SMILES to DEPICT yields what is termed a canonical structure diagram. ... 

The canonical jelly roll barrel is schematically illustrated in Figure 16.13. Superposition of the structures of coat proteins from different viruses show that the eight p strands of the jelly roll barrel form a conserved core. This is illustrated in Figure 16.14, which shows structural diagrams of three different coat proteins. These diagrams also show that the p strands are clearly arranged in two sheets of four strands each P strands 1, 8, 3, and 6 form one sheet and strands 2, 7, 4, and 5 form the second sheet. Hydrophobic residues from these sheets pack inside the barrel. 

Tetrasulfur tetranitride is prototypal for other cyclic S—N compounds in the sense that its electronic structure is not accounted for by any single classical bonding diagram. In terms of a valence bond/resonance approach, the following canonical structures all merit consideration ... 

The moiety that has the cationic centre in the carbonium ion adjacent to a carbon/carbon double bond is called an allylic cation. The diagram represents one of the canonical structures for this cation. 

We will now take a closer look at the adsorption transition in the phase diagram (Fig. 13.12) and we do this by a microcanonical analysis [307, 308]. As we have discussed in detail in Section 2,7, the microcanonical approach allows for a unique identification of transition points and a precise description of the energetic and entropic properties of structural transitions in finite systems. The transition bands in canonical pseudophase diagrams are replaced by transition lines. Figure 13.15 shows the microcanonical entropy per monomer s e)=N lng e) as a function of the energy per monomer e=EfN for a polymer with N=, 20 monomers and a surface attraction strength = 5, as obtained from multicanonical simulations of the model described in Section 13.6. 

To summarize we have reproduced the intricate structural properties of the Fe-Co, Fe-Ni and the Fe-Cu alloys by means of LMTO-ASA-CPA theory. We conclude that the phase diagram of especially the Fe-Ni alloys is heavily influenced by short range order effects. The general trend of a bcc-fcc phase transition at lower Fe concentrations is in accordance with simple band Ailing effects from canonical band theory. Due to this the structural stability of the Fe-Co alloys may be understood from VGA and canonical band calculations, since the common band model is appropriate below the Fermi energy for this system. However, for the Fe-Ni and the Fe-Cu system this simple picture breaks down. 

Fig. 1. The vector-bond diagrams for three structures of the canonical set of fourteen for n — 4, and some of their superposition patterns.

It should be clear that the structure represented by this diagram is far more systematic and symmetrical than the structure involved in the canonical interpretation. It can also answer one of the criticisms of Aristotle raised by Ackrill. Recall that Ackrill says ... 

The phenomenon of four singly-occupied active-space orbitals (AO or MO) arises in many seemingly-unrelated molecular situations. The singlet spin (S = 0) Rumer diagrams for these orbitals indicate that there are two linearly-independent or canonical spin-pairings schemes. These spin-pairings are well-exemplified by the 7i-electrons of butadiene, for which two canonical Lewis VB structures with different 7i-electron spin pairings are those of 1 and 2. 

The embedding of encoded watermark bits from u, in the jth structure Mj is considered. A block diagram of the embedding scheme is depicted in Fig. 3. First, a canonic representation of the structure is obtained as described above. Next, the host data vector Xj is extracted (see also Section 4.7). Here, it is assumed that L j elements are extracted from the structure Mj, and the elements of are scaled such that the watermark w,- can be embedded with a variance = l. 

First-Principles Approach to Guinier-Preston Zones. We have already seen that the combination of first-principles calculations with Monte Carlo methods is a powerful synthesis which allows for the accurate analysis of structural questions. In chap. 6 we noted that with effective Hamiltonians deduced from a lower-level microscopic analysis it is possible to explore the systematics of phase diagrams with an accuracy that mimics that of the host microscopic model. An even more challenging set of related questions concern the emergence of microstructure in two-phase systems. An age-old question of this type hinted at in the previous chapter is the development of precipitates in alloys, with the canonical example being that of the Al-Cu system. 

Let us fix I and S. By Lemma 2.43 and Lemma 2.55, we have various natural maps between functors on sheaves arising from the closed structures and the monoidal pairs, involving various J-diagrams of schemes, where J varies subcategories of I. In the sequel, many of the natural maps are referred as the canonical maps or the canonical isomorphisms without any explicit definitions. Many of them are defined in [26] and Chapter 1, and various commutativity theorems are proved there. 

In addition to three canonical morphologies of aggregates (S, C, and L), more complex associations of block copolymer molecules could be found in certain regions of the diagram. In particular, a recent theoretical study [24] predicts the existence of branched cylinders in the vicinity of the S-C binodal line. The latter occupy a narrow corridor and coexist with cylindrical and spherical micelles. Branched structures and networks of aggregates formed by diblock copolymer with quenched PE block were also considered in [17]. 

Binary Decision Diagrams (BDDs for short) form a heuristically efficient data structure to represent formulas of the propositional logic. Let P be a totally ordered finite set of boolean propositions. Let / be a boolean formula over P, bdd f) is the BDD representing /, and bdd f) is the size if this BDD. Bryant (1986) showed that BDDs axe a canonical representation two equivalent formulas are represented with the same BDD ... 

Figure 3 Stereo diagram showing the placement of counterions 6 A from the phosphorus along the OIP—P—02P bisector. The same canonical B-form structure of DNA used in Figure 1 was used here.

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