Mathematical objects, equivalence relations

Recall from Section 1.7 that the standard mathematical way to deal with irrelevant ambiguity is to define an equivalence relation and work with equivalence classes. In this case of tensors, the irrelevant ambiguity arises from the different ways of writing the same object as a linear combination of products. We will use this insight to define tensor products. Suppose V and W are complex vector spaces. Consider the complex vector space V W generated by the set... 

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... 

The ensemble of all equivalent positions for a space group is unique and may be considered the mathematical definition of the space group. It provides the basis for manipulating objects and points related by symmetry in a digital computer. Equivalent positions are another way of stating both the space group and the Bravais lattice of a crystal. 

The word "geometry" features in most languages as a concept to relate the sizes of different objects. Of the smaller number of individuals with more advanced mathematical skills the overwhelming majority would consider geometry as equivalent to the ancient discipline, described more correctly as Euclidean geometry. That is the version of geometry codified by Euclid in... 

Calorimeters that are inertial objects comprise the most numerous group of calorimeters. Let us assume that a calorimeter has only the integrating or proportional properties of the object, as a certain idealization. Of course, this idealization is well-founded in many cases. The relation describing the dynamic properties of the object is equivalent to the mathematical model of the calorimeter. It is expressed as a function of time, frequency or a complex variable domain. 

The comparative testing type of technical transfer generates results that need to be compared using one or more statistical tools. Although data can be evaluated subjectively, the use of statistics will build objectivity into the data analysis and allow unbiased comparison of the data sets. Because comparative testing involves destructive testing of individual samples, a successful transfer involves proving mathematically that the data sets are equivalent. The complexity of the data treatment is directly related to that of the method transfer ... 

The Clausius statement is It is impossible for a process to occur that has the sole effect of removing a quantity of heat from an object at a lower temperature and transferring this quantity of heat to an object at a higher temperature. In other words, heat cannot flow spontaneously from a cooler to a hotter object if nothing else happens. The Clausius statement of the second law is closely related to ordinary experience. The Kelvin statement is less closely related, and it is remarkable that the statements are equivalent to each other and to the mathematical statement of the second law, which establishes that the entropy is a state function. 

The mathematical interpretation of the problem is given below. If a condition F(xq) describes the belonging of an object to a certain class, and Ai(xi),. .., Aji(Xn) are some initial descriptors, then assignment of a pair (Q,Level), were Q - statistics of the relation measure between F(xq) and some expression G containing variables Aj(xj ) and Level -present threshold numbers, formalizes the criterion which makes terms "the best classification rule" and "an allowed rule" sensible. The best rule maximizes Q, while an allowed G satisfies the inequality Q (F,G) > Level and thus defines the empirical interpretation of such statements as "F(xo) is equivalent to G(A,(xi... 

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