Invariant definition

Thus, it should be stressed that the mathematical topological theory investigates, as a rule, the problems of classification of knots and links, the construction of topological invariants, definitions of topological classes, etc. whereas the fundamental physical problem in the theory of topological properties of polymer chains is the determination of the entropy, S = In Z with the fixed topological state of chains. Both these problems are very difficult, but important. 

Reaction paths are a widely used concept in theoretical chemistry. It is evident that the invariance problem, which was mathematically solved a long time ago (cf. the report given in Ref. [1 ]), penetrates again and again the discussions in this field (see Ref. [2]). We give both the non-invariant and the invariant definitions with respect to the choice of the particular coordinate system for two important kinds of chemical reaction pathways (RP), namely, steepest descent lines (SDP) and gradient extremal (GE) curves. 

Before we start our analysis of the quality of the energy approximation by graph invariants, it is reasonable to look at their total numbers for different systems. Table 1 shows these numbers for some ice nanotubes INT . In particular, the dependence of the numbers of distinct graph invariants on the order of the invariant and the indices m and n is revealed. The values in the table can be found numerically by applying the graph invariant definition to a specific water cluster. Alternatively, one can determine these numbers analytically on the basis of symmetry considerations. Transformation of bond variables defines a (reducible) representation T of the symmetry group G of the cluster. The number of the h order invariants is simply the number of the totally symmetric representations in the h symmetric power of T. Thus, only characters of T are necessary to determine numbers in Table 1. 

After these preliminaries we are now ready for a mathematically precise definition of an almost invariant set. Let p M he any probability measure. Wc say that the set B is 5-almost invariant with respect to p if... 

Similar invariance concepts for anisotropic materials were also developed by Tsai and Pagano [2-7]. For anisotropy, the following definitions... 

In the case of non-metallic materials, the term corrosion invariably refers to their-deterioration from chemical causes, but a similar concept is not necessarily applicable to metals. Many authorities consider that the term metallic corrosion embraces all interactions of a metal or alloy (solid or liquid) with its environment, irrespective of whether this is deliberate and beneficial or adventitious and deleterious. Thus this definition of corrosion, which for convenience will be referred to as the transformation definition. 

Lorentz-Invariance on a Lattice One of the most obvious shortcomings of a CA-based microphysics has to do with the lack of conventional symmetries. A lattice, by definition, has preferred directions and so is structurally anisotropic. How can we hope to generate symmetries where none fundamentally exist A strong hint comes from our discussion of lattice gases in chapter 9, where we saw that symmetries that do not exist on the microscopic lattice level often emerge on the macroscopic dyneimical level. For example, an appropriate set of microscopic LG rules can spawn circular wavefronts on anisotropic lattices. 

One of the principal reasons for the usefulness of these definitions lies in the fact that they are invariant under a change of representation. One verifies that under a change of representation wherein... 

The choice of ijr 2 — 1, together with the antiunitary character of U(it), guarantees the invariance of the equal time commutation rules under U(it). With these definitions of the transformation properties of the spin field operators one verifies that... 

The ri fiiatrix, due to the tune ordering operator in its definition is not invariant under time inversion. The invariance of the theory under tahi ihversidn has the following important consequence for the S-matrix since this operator s matrix elements axe given by ... 

It is well to remember that in the past, the unit of electrical current—the international ampere—was defined as the strength of an invariant current which, when sent throngh a silver nitrate solution, would deposit l.lllSOOmg silver at the cathode. Today, another definition of the ampere as an SI unit is valid. 

Remarkably, only one nuclear constant, Q, is needed in (4.17) to describe the quadrupole moment of the nucleus, whereas the full quadrupole tensor Q has five independent invariants. The simplification is possible because the nucleus has a definite angular momentum (7) which, in classical terms, imposes cylindrical symmetry of the charge distribution. Choosing x, = z as symmetry axis, the off-diagonal elements Qij are zero and the energy change caused by nuclear... 

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