Structure representation history

The examples of interwoven and knotted structures have prevailed in all walks of architecture, symbolism and art, and it is not our aim here to give an exhaustive list of where one may encounter different representations of knots. Be that as it may, this short journey through the history of knot representation would not be complete without mentioning the great Dutch graphical artist M.C. Escher, whose beautiful representations of the Mobius strip and the trefoil knot are renowned. They prove a great inspiration for chemists wishing to adorn the covers of chemistry journals. 

Mathematical chemistry, the new challenging discipline of chemistry has established itself in recent years. Its main goal is to develop formal (mathematical) methods for chemical theory and (to some extent) for data analysis. Its history may be traced back to Caley s attempt, more than 100 years ago, to use the graph theoretical representation and interpretation of the chemical constitution of molecules for the enumeration of acyclic chemical structures. Graph theory and related areas of discrete mathematics are the main tools of qualitative theoretical treatment of chemistry [1,2]. However, attempts to contemplate connections between mathematics and chemistry and to predict new chemical facts with the help of formal mathematics have been scarce throughout the entire history of chemistry. 

At the other end of the spectrum, our goal might be a set of differential equations that describe the behavior of each of the components and their interactions. These would likely attract anyone who aspired to reduce chemical phenomena to mechanics, whether classical (in Butlerov s time) or quantum (in our own). Butlerov explicitly rejected Gerhardt s option while expressing considerable doubt about when, if ever, the second option would be attainable. As it happened, the subsequent history of chemical structure has seen the development of a number of intermediate representations. 

Although the basic chemical features of coal can be qualitatively and in some instances semiquantitatively specified, average-structure models that purportedly reflect statistically preferred molecular structures of coal offer little that advances an understanding of coal. In part, this is due to a continuing paucity of relevant or reliable data and to the procedures used to formulate the constructs. But meaningful representations of molecular structure are currently also precluded by indications that the assumption that underlies average-structure models, namely, that there exists a more or less unique, systematic, rank-dependent, molecular chemistry of coal, is not sustained by the current evidence. Several examples, all drawn from the open literature, are presented to support the view that the chemistry of a coal is heavily influenced by its source materials and early formative history and that coals of similar rank may therefore be chemically much more diverse than is usually supposed. 

Ultimately, the test of the efficacy of the use of theory in explaining history lies in how much our understanding of the historical event in question is enhanced by the new treatment. The foregoing chapters have employed the rubric of discursive representations in explaining the Sino-American rapprochement in three ways. First, they identified the key subdiscourses of reconciliation with China and explored their individual rationales, achievements, and limitations. Second, they analyzed how these separate discursive strands related to each other in the case of conflicting discourses, how one eventually became dominant and in other cases, how one subdiscourse evolved into another, with what continuities and innovations. This included an examination of the mechanics of transition - in particular, the structural or external factors that influenced the efficacy of a particular discourse the bureaucratic politics that... 

In the history of mathematics, uncertainty was approached in the XVlP century by Pascal and Fermat who introduced the notion of probability. However, probabilities do not allow one to process subjective beliefs nor imprecise or vague knowledge, such as in computer modeling of three-dimensional structure. Subjectivity and imprecision were only considered from 1965, when Zadeh, known for his work in systems theory, introduced the notion of fuzzy set. The concept of fuzziness introduces partial membership to classes, admitting intermediary situations between no and full membership. Zadeh s theory of possibility, introduced in 1977, constitutes a framework allowing for the representation of such uncertain concepts of non-probabilistic nature (9). The concept of fuzzy set allows one to consider imprecision and uncertainty in a single formalism and to quantitatively measure the preference of one hypothesis versus another. Note, however, that Bayesian probabilities could have been used instead. 

Figure 4. Schematic representation of the shear rate program, structural history and accompanying stress response of a shear-rate-reduction flow. Shear stress is seen to undershoot immediately following the reduction in shear rate. Model parameters are identical to those used in previous figures.

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