Mathematical objects, equivalence

Physicist P. A. M. Dirac suggested an inspired notation for the Hilbert space of quantum mechanics [essentially, the Euclidean space of (9.20a, b) for / — oo, which introduces some subtleties not required for the finite-dimensional thermodynamic geometry]. Dirac s notation applies equally well to matrix equations [such as (9.7)-(9.19)] and to differential equations [such as Schrodinger s equation] that relate operators (mathematical objects that change functions or vectors of the space) and wavefunctions in quantum theory. Dirac s notation shows explicitly that the disparate-looking matrix mechanical vs. wave mechanical representations of quantum theory are actually equivalent, by exhibiting them in unified symbols that are free of the extraneous details of a particular mathematical representation. Dirac s notation can also help us to recognize such commonality in alternative mathematical representations of equilibrium thermodynamics. 

First we will focus attention on selected topics relating to the equivalence between benzenoid hydrocarbons, and special types of graphs and other mathematical objects that we can associate with benzenoids- In particular we will explore relations involving caterpillar trees [3] associated with catacondensed benzenoids and their line graphs [17] called, as already mentioned, Clar graphs [4]. Also relations involving "boards" (known technically as polyominos) of special properties such as those associated with "king" and "rook" pieces of chess... 

In many cases, instead of the PDF in Eq. (5.127), an equivalent formulation in terms of a corresponding conditional NDF Al(Vp, p Vp, p) is used. The corresponding mathematical object is such that the integral over phase space now yields... 

The clarity comes from the real form given by D. Hestenes to the electron wave function that replaces, in a strict equivalence, the Dirac spinor. This form is directly inscribed in the frame of the geometry of the Minkwoski space in which the experiments are necessarily placed. The simplicity derives from the unification of the language used to describe the mathematical objects of the theory and the data of the experiments. 

El-Basil S, Randic M (1992) Equivalence of Mathematical Objects of Interest in Chemistry and Physics. Adv Quant Chem 24 239-290 Graovac A, Gutman I, Randic M, Trinajstic N (1973) Kekule Index for Valence Bond Structures of Conjugated Polycyclic System. J Am Chem Soc 95 6267-6273... 

Common conventions of graph theory are followed here (see, e.g.. Refs. 8-10), but a rigid distinction is not always maintained between chemical and mathematical contexts. So the term graph is used interchangeably in both its strict sense as a mathematical object and as a shorthand term for its realization as an actual (usually carbon) molecule. Some equivalent pairs of terms are treated as being synonymous atom = vertex, bond = edge, valency = degree, and so on. 

S. El-Basil and M. Randic, Equivalence of mathematical objects of interest in chemistry and physics, Adv. Quantum Ghem. 24 (1992) 239-290. 

From a mathematical point of view, conformations are special subsets of phase space a) invariant sets of MD systems, which correspond to infinite durations of stay (or relaxation times) and contain all subsets associated with different conformations, b) almost invariant sets, which correspond to finite relaxation times and consist of conformational subsets. In order to characterize the dynamics of a system, these subsets are the interesting objects. As already mentioned above, invariant measures are fixed points of the Frobenius-Perron operator or, equivalently, eigenmodes of the Frobenius-Perron operator associated with eigenvalue exactly 1. In view of this property, almost invariant sets will be understood to be connected with eigenmodes associated with (real) eigenvalues close (but not equal) to 1 - an idea recently developed in [6]. 

The Object Constraint Language (OCL), a standard part of UML 1.1, is a specification language used in conjunction with UML models. It is an expression-based, side-effect-free language that eschews mathematical symbols (V, 3, and so on) for textual equivalents (forAII, exists). It uses a syntax more usual in object-oriented languages V x T, p(x) becomes T->forAII (x x.p). 

In summary, microdosimetry is the study and quantification of the spatial and temporal distributions of absorbed energy in irradiated matter [15,17,22,23]. One makes a distinction between regional microdosimetry [the object of which is the study of micro-dosimetric distributions /(z)j and structural microdosimetry (a mathematically more advanced approach, which is concerned with characterizing the spatial distribution of individual energy deposition events, i.e., ionizations and/or excitations). Regional microdosimetry asserts that the effect is entirely determined by the amount of specific energy deposited in the relevant site (typically, a cell nucleus). The two kinds of microdosimetry, regional and structural, were shown to be in fact mathematically equivalent—once the sensitive site is judiciously determined [16]. 

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