Topology measurements

Nowadays SPM is one of the most widely used technique for topology measurements of nanodimensional objects. Relatively little effort has been devoted currently to standardizing scanning probe microscopy as quantitative method for chemical and physical analysis [1]. 

First, we note that the standard photodetection is a local measurement of the field variables (intensities). At the same time, the Aharonov-Bohm effect represents a topological measurement referred to the properties of vector potential along some loop. In the usual form, the Aharonov-Bohm effect deals with static or slowly time-varying magnetic fields [101]. The effect consists in the appearance of a persistent current in a metallic loop over which the magnetic flux passes. This current is a periodic function of magnetic flux with the period of flux quantum hc/e. Besides that, certain resistance oscillations in the loop incorporated into an external circuit with the same period can occur. 

According to the proposed prediction calculation procedure, it is clear that the actual distance between the two elements 5 and t, which satisfies the prediction conditions for any unknown element u, is crucial, and the larger the distance the larger the potential uncertainty in the prediction. Thus, a first topological measure of the prediction precision is provided by the connectivity operator C(s,t) previously defined the precision decreases for increased C(s,t). 

On the other hand, TRE was also proposed as a quantitative topological measure of aromaticity [68,70,72]. Since the aromaticity of a conjugated molecule depends on many factors - and not only on cyclic conjugation - it is hardly surprising that this second application of TRE led to many controversies. In the recent chemical literature TRE has been criticized by some authors [102-110] and defended by others [111-113]. 

The word shape is loaded with strong personal intuitions of what each of us thinks defines an object in three-dimensional (3D) space. Sometimes, we may view an object as a distribution of points in space and conceive its shape in terms of distances and angles, that is, local geometrical measures. More often, we associate the notion of shape with a solid object defined by an envelope surface. In this case, a precise geometrical characterization may lose its relevance, and it may be sufficient to characterize shape in a more qualitative fashion. Global topological measures are better suited here. Therefore, we... 

A range of physicochemical properties such as partial atomic charges [9] or measures of the polarizabihty [10] can be calculated, for example with the program package PETRA [11]. The topological autocorrelation vector is invariant with respect to translation, rotation, and the conformer of the molecule considered. An alignment of molecules is not necessary for the calculation of their autocorrelation vectors. 

The physical techniques used in IC analysis all employ some type of primary analytical beam to irradiate a substrate and interact with the substrate s physical or chemical properties, producing a secondary effect that is measured and interpreted. The three most commonly used analytical beams are electron, ion, and photon x-ray beams. Each combination of primary irradiation and secondary effect defines a specific analytical technique. The IC substrate properties that are most frequendy analyzed include size, elemental and compositional identification, topology, morphology, lateral and depth resolution of surface features or implantation profiles, and film thickness and conformance. A summary of commonly used analytical techniques for VLSI technology can be found in Table 3. 

That Dfractai givcs the expected result for simple sets in Euclidean space is easy to see. If A consists of a single point, for example, we have N A, e) = 1, Ve, and thus that / fractal = 0. Similarly, if A is a line segment of length L, then N A,e) L/e so that Dfraciai = 1. In fact, for the usual n dinien.sional Euclidean sets, the fractal dimension equals the topological dimension. There are nrore complicated sets, however, for wliic h the two measures differ. 

Since particular choices for intermediate link additions will generally yield nonisomorphic sequences of topologies, the typical intermediate dynamical behaviors are obtained by averaging each measure over the number Ng of such collections of graphs gi,g2,-.-,gr, w+i -... 

Except for occasional bursts of irregularity when specific measures are particularly sensitive to the topology and may show unexpectedly large absolute deviations, range independent class-2 rules generally possess regular, predictable profiles which are relatively insensitive to exact g topologies. 

There is a strong dependence on the exact lattice structure intermediate topologies typically have large measure fluctuations. 

---