Symmetry Measure Definition

We define the continuous symmetry measure (CSM) as a quantifier of the minimum effort required to turn a given shape into a symmetric shape. This effort is measured by the sum of the square distances each point is moved from its location in the original shape to its location in the symmetric shape. Note that no a priori symmetric reference shape is assumed. 

Denote by Q the space of all shapes of a given dimension, where each shape P is represented by a sequence of n points We define a metric d on this space 

This metric defines a distance function between every two shapes in Q. 

We define the symmetry transform (ST) as the symmetric shape P closest to P in terms of the metric d. 

The CSM of a shape is now defined as the distance to the closest symmetric shape  

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Following the definition of the continuous symmetry measure (CSM) in Section II, the CSM values are limited to the range 0. .. 1 (where 1 is the normalization scale). The lower bound of the CSM is obvious from the fact that the average of the square of the distances moved by the object points, is necessarily non-negative. The upper bound of the average is limited to 1 since the object is previously... 

Two general classes of chirality measures have been recognized in the first, the degree of chirality expresses the extent to which a chiral object differs from an achiral reference object, while in the second it expresses the extent to which two enantio-morphs differ from each other [Buda et al., 1992]. The continuous chirality measure (CCM) recently proposed [Zabrodsky and Avnir, 1995] is an example of chirality measure belonging to the first class and is based on the general definition of continuous symmetry measure defined as ... 

The continuous chirality measure is an example of first class chirality measure based on the general definition of continuous symmetry measure it is defined as [Zabrodsky and Avnir, 1995]... 

The dimensionless spatial coordinate rj is measured in the thinnest dimension of rectangular catalysts. For cylindrical and spherical catalysts, r] is measured in the radial direction. The characteristic length L which appears in the intrapellet Damkohler number and is required to make the spatial coordinate dimensionless (i.e., rj = spatial coordinate/L) is one-half the thickness of catalysts with rectangular symmetry, measured in the thinnest dimension the radius of long cylindrical catalysts or the radius of spherical catalysts. q A is the molar density of reactant A divided by its value in the vicinity of the external surface of the catalyst, CAsurf- Hence, by definition, q A = 1 at r = 1. 

The proposed continuous symmetry measure (CSM) method which follows these guidelines is based on the following definition. Given a shape composed of rip points P, (/ = l...rip) and a symmetry group G, the symmetry measure 5(G) is a function of the minimal displacement the points P, of the shape must undergo in order to acquire G symmetry. The CSM method identifies the points P, of the nearest shape having the desired symmetry. Once the nearest P,- values are calculated, a continuous symmetry measure is evaluated as ... 

As far as the definition of a as a measure of the symmetry of the activation barrier is concerned, as shown in Figure 15, let us focus on the apical region of the intersection between the free energy curves illustrated in Figure 14. 

The problem of the structural multidomaining below Ta makes it difficult to reach a definite conclusion, as shown in Figs. 5 and 13. Measurements for defect-free and stress-free STO samples are indispensable for a definite conclusion about the symmetry of the ferroelectric phase of STO 18. Finally, we can conclude that STO 18 may be a typical soft mode ferroelectric. 

The observation of natural ORD or CD requires lack of symmetry in the molecule, but any molecule may exhibit magnetic circular dichroism (MCD). It constitutes a molecular analogy for the Zeeman effect in atomic spectra. Measurements in this area may well reveal substituent interactions which are masked in normal UV spectra. Extensive definitive papers of great interest which well illustrate this have appeared on MCD of pyridine derivatives, measured in cyclohexane, acetonitrile, and alcohol or aqueous acidic solutions for protonated... 

Deep state experiments measure carrier capture or emission rates, processes that are not sensitive to the microscopic structure (such as chemical composition, symmetry, or spin) of the defect. Therefore, the various techniques for analysis of deep states can at best only show a correlation with a particular impurity when used in conjunction with doping experiments. A definitive, unambiguous assignment is impossible without the aid of other experiments, such as high-resolution absorption or luminescence spectroscopy, or electron paramagnetic resonance (EPR). Unfortunately, these techniques are usually inapplicable to most deep levels. However, when absorption or luminescence lines are detectable and sharp, the symmetry of a defect can be deduced from Zeeman or stress experiments (see, for example, Ozeki et al. 1979b). In certain cases the energy of a transition is sensitive to the isotopic mass of an impurity, and use of isotopically enriched dopants can yield a positive chemical identification of a level. 

G. D. Birkhoff tried to give a mathematical theory of aesthetics by recognizing the complexity, C, and the order, O (i.e., its harmony and symmetry) that an object could be said to have. He then restated Hemsterhuis definition of the beautiful as that which gives us the greatest number of ideas in the shortest time as the relation aesthetic measure... 

In his interesting paper Professor Nicolis raises the question whether models can be envisioned which lead to a spontaneous spatial symmetry breaking in a chemical system, leading, for example, to the production of a polymer of definite chirality. It would be even more interesting if such a model would arise as a result of a measure preserving process that could mimic a Hamiltonian flow. Although we do not have such an example of a chiral process, which imbeds an axial vector into the polymer chain, several years ago we came across a stochastic process that appears to imbed a polar vector into a growing infinite chain. 

Figure 10.3 Definition of geometrical parameters for a CMA shown for the case of off-axis focusing of a point source (Q). Rt and Ra are the radii of the inner and outer field cylinders, respectively 4, is the radial image distance to the inner cylinder (in analogy, one can introduce a radial source distance ds, in the present case one has ds = R,) z is the total distance between source and image measured along the symmetry axis of the analyser zf is the corresponding distance for the field region is the entrance angle into the analyser (due to symmetry properties this is equal to the exit angle).

Based on the experimental frequencies and isotope shifts, a Quantum-Chemistry Assisted Normal Coordinate Analysis (QCA-NCA) has been performed. Details of the QCA-NCA procedure of I, including the f-matrix and the definition of the symmetry coordinates, have been described previously (12a). The NCA is based on model I (vide supra). Assignments of the experimentally observed vibrations and frequencies obtained with the QCA-NCA procedure are presented in Table II. The symbolic F-matrix for model I is shown in Scheme 3. Table III collects the force constants of the central N-N-M-N-N unit of I resulting from QCA-NCA. As evident from Table II, good agreement between measured and calculated frequencies is achieved, demonstrating the success of this method. 

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