Random chord

Various researchers have pointed out that the process of randomly selecting chords on a circle to solve this problem leads to some famous paradoxes. The term random chord is ambiguous unless the exact procedure for producing it is defined, as in my program code... 

The scanning infrared laser method sends an oscillating laser beam into a suspension and detects back-scattered infrared light. The random chords within particles are thus measured and this information is related to the particle size distribution of the suspension. This method can be used to... 

Basel, U. (2012b). Random chords and point distances in regular polygons. arXivpreprint arXiv 1204.2707v2. 

Based on the curvature behaviour of the SAS correlation function y(r), the distribution laws of random distances between the interfaces were analyzed [12]. This is examplified for a sequence of 4 pores and 3 walls in Fig. 5. Furthermore, the appearance of all minima and maxima in the chord-length distribution, see Fig. 4, at d= 2.1 nm, 2.5 nm, 2.9 nm,... is traced back to the mean chord lengths <1> and . 

Apb is the scattering length density difference, Q is Porod s invariant, and Y the mean chord length. For the calculation of Yo(r) we approximated I(q) hy a cubic spline. The equations used for the calculation of " pore and " soUd are to be found in [8,30,39-41,47]. Analytical expressions for the descriptors of RES were published in [10,11,13,42,43]. In its most simple variant, the stochastic optimization procedure evolves the two-point probability S2 (r) of a binary representation of the sample towards S2(r) by randomly excWiging binary ceUs of different phases, starting from a random configuration which meets the preset volume fractions. After each exchange the objective function... 

Nitrogen adsorption (NA) experiments yield diameter distribution densities V d) of the random pore diameter d. Based on the assumed particle shape (circular cylinder, random diameter d), the function V d) can be transformed into a chord length distribution density (CLD) A 1). CLDs are connected with the porosity p of the material. The direct transformation step,... 

Lasentec Labtec 1000 is a laboratory instrument that covers the size range from 0.7 to 250 pm in 28 size channels. The data are generated as scanned counts an empirical frequency distribution created from classification of chords from randomly oriented particles. Software can convert these chords to a spherical equivalent distribution on the assumption that the chords were generated from an assembly of spherical particles this software contains a filter system to reject improbable data that would tend to skew the distribution to a coarser size. A discrimination... 

FIGURE 5.1. One hundred randomly selected chords in a circle. 

Appendix 1). Would our results be different if we were to randomly choose chords that were all parallel to a given direction ... 

Modern image analyzers permit fine scrutiny of individual particles such that a shape function can be analytically determined. Assuming that the group of particles analyzed have a random orientation on the microscope slide, a series of horizontal chords through the particle can be related to their distances from the center of the particle. This type of relationship can be utilized with statistical techniques of stere-ology to account for nonspherical shapes of particles in various analyses. 

It can be shown from the theorems of fractal geometry that the sets of intercepts created in a line search examination of a dispersed field of view constitutes a Cantorian set in one dimensional space. If limited to the points where the search tracks, enter and leave the profiles, revealed is a Cantorian dust that can be linked to the richness of the ore [31,32]. It can be shown that the fraction of the random lines falling within the profiles is the same as area fraction occupied by the dispersed fineparticles in the field of view. It can also be shown that the average size of the chord tracking across the dispersed fineparticles is a function of the average size of the dispersed fineparticle. These relationships were first developed by Rosiwal [33, 34, 35]-... 

Ceramicists and powder metallurgists have also made extensive studies of techniques for studying the structure of holes and dispersed species in items made from powder compacts and these technologists refer to the measurement technology either by the term quantitative stereology or quantitative microscopy [30]. It has been shown that if a dispersion such as that shown in Figure 2.16(a), is randomly dispersed then the sets of chords drawn in a parallel line scan search such as that of Figure 2.16(c) (i) has a log-normal distribution. Whereas an )omerated field of view, or a non randomized... 

The most recently developed method for the analysis of SAXS patterns of TPEs in the strained state is utilizing the multidimensional chord distribution function (CDF) [16]. The CDF is defined in the physical space, rather than in the reciprocal space. It has peaks with both positive and negative values. In the third row of Figure 3, the elevations of the CDF are shown the depressions are depicted in the fourth row. If the hard domains were distributed at random, no negative values in the CDF would be observed, because depressions describe... 

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