Euclidean object

Minimizing this function is equivalent to finding a low-dimensional configuration of points which has Euclidean object-to-object distances by as close as possible to some transformation/(.) of the original distances or dissimilarities, dy. Thus, the model distances by are not necessarily fitted to the original dy, as in classical MDS, but to some admissible transformation of the measured distances. For example, when the transformation is a general monotonic transformation it preserves the... 

Note 2 For a Euclidean object of constant density, d 3, but for a fractal object, d <3, such that its density decreases as the object gets larger. 

Another method used for data analysis of non-Euclidean objects is fractals. A true fractal object is scale invariant (i.e., exhibits seU-similarity) thus a fractal dimension is obtained ITom the outline of an object by varying the scale of analysis. The ITactal dimension (FD) of an irregular geometry is a measure of the space-filling... 

In Figure 10.12 the curves a(t), calculated according to Equation (10.8) under the condition B = constant for D = 1.5, 1.8 and 2.1 and also for d = 3 are given. It is easy to see that in accordance with the previously-stated treatment the rate of reaction increases in process of increase D and reaches the greatest value in Euclidean space at d = 3. It should be noted that in reactions of fractal objects according to relationship (10.1), at D = d = 3, a = constant, and in view of a boundary condition a = 0 at t = 0, it means that such reactions for three-dimensional Euclidean objects do not proceed at all. 

Fractals are self-similar objects invariant with respect to local dilatations, i.e., objects that reproduce the same shape during observation at various magnifications. The concept of fractals as self-similar ensembles was introduced by Mandelbrot [1], who defined a fractal as a set for which the Hausdorff-Besicovitch dimension always exceeds the topological dimension. The fractal dimension D of an object inserted in a d-dimensional Euclidean space varies from 1 to d. Fractal objects are the natural filling of the gap between known Euclidean objects with integer dimensions 0,1, 2, 3. .. The majority of naturally existing objects proved to be fractal, which is the main reason for the vigorous development of the methods of fractal analysis. 

For Euclidean objects (smooth curves or regular lattices in a plane or in a bulk, etc), the following identity holds [16] ... 

Yet another important property of fractals which distinguishes them from traditional Euclidean objects is that at least three dimensions have to be determined, namely, d, the dimension of the enveloping Euclidean space, df, the fractal (Hausdorff) dimension, and d the spectral (fraction) dimension, which characterises the object connectivity. [For Euclidean spaces, d = d = d this allows Euclidean objects to be regarded as a specific ( degenerate ) case of fractal objects. Below we shall repeatedly encounter this statement] [27]. This means that two fractal dimensions, d( and d are needed to describe the structure of a fractal object (for example, a polymer) even when the d value is fixed. This situation corresponds to the statement of non-equilibrium thermodynamics according to which at least two parameters of order are required to describe thermodynamically nonequilibrium solids (polymers), for which the Prigogine-Defay criterion is not met [28, 29]. 

An important difference between Equations (11.62) and (11.63) is that in the former, X depends on one parameter, which is typical of equilibrium Euclidean objects, while in the latter, it is a function of two parameters, which is typical of thermodynamically nonequlibrium fractal objects. Therefore, the use of Equation (11.62) in the latter case is improper. 

As it follows from the Eq. (140), for compact (Euclidean) object (Z) =d) the value p=l (in relative units). One from the most important properties... 

For Euclidean object Dj.= d and pfr = const = pEnc. For fractal object increasing (or MM and N) in polycondensation process at a=const results to pfr reduction in virtne of the condition Dj,< d [52]. Therefore there are critical values MMc(Nc) and tc, above which synthesis reaction ceases [118, 119, 121]. This process is simulated within the frameworks of irreversible aggregation models according to the mechanism cluster-cluster. 

In Fig. 23, the comparison of the calculated with exponent 0.847 instead of approx. 0.6 [46] application dependences In (1-Q) on t according to the Eq. (88) of Chapter 1 with experimental data is also adduced. As one can see, at the coefficients A and B proper choice the good correspondence of theory (the Eq. (88) of Chapter 1) and experiment is obtained. Let us note, that exponent increasing in the Eq. (88) of Chapter 1 from 0.6 for Euclidean objects up to 0.847 for fractal ones means more rapid decay of monomers contents with time and, hence, more rapid polymerization reaction realization at other equal conditions. 

It is obvious, that in case of Euclidean object D=d and p=const. 

Euclidean objects (dense spherical particles) are most likely to form in systems (e.g., aqueous silicates) in which the particle is slightly soluble in the solvent. In this case, monomers can dissolve and reprecipitate until the equilibrium structure (having a minimum surface area) is obtained. In nonaqueous systems (e.g., silicon alkoxide-alcohol-water solutions), the solubility of the solid phase is so limited that condensation reactions are virtually irreversible. Bonds form at random and cannot convert to the equilibrium configuration, thereby leading to fractal polymeric clusters. 

Liu IS (2004) On Euclidean objectivity and the principle of material frame-indifference. Continuum Mech Thermodyn 16 177-183... 

The important argument in favor of fractal approach application is the usage of two order parameter values, which are necessary for correct description of polymer mediums structure and properties features. As it is known, solid phase polymers are thermodynamically nonequilibrium mediums, for which Prigogine-Defay criterion is not fulfilled, and therefore, two order parameters are required, as a minimum, for their structure description. In its turn, one order parameter is required for Euclidean object characterization (its Euclidean dimension d). In general case three parameters (dimensions) are necessary for fractal object correct description dimension of Euclidean space d, fractal (Hausdorff) object dimension d and its spectral (fraction)... 

A > a and k > 1, is realized in the case, when the indicated chain part loses its fractal properties and becomes Euclidean object, that is, when its fractal dimension becomes equal to topological dimension, in other words, at the condition =1.0 fiilfillment [9],... 

Thus, the stated above results demonstrated, that volume changes availability or absence in uniaxial tension process is due to structure type. If the structure is Euclidean object (dimension J = 3, v = 0.5 [8]), then volume changes are absent, if it is fractal object (2 < < 3.0 < v < 0.5 [7]), then... 

Hence, a solid-phase polymers deformation process is realized in fractal space with the dimension, which is equal to structure dimension d. In such space the deformation process can be presented schematically as the devil s staircase [39]. Its horizontal sections correspond to temporal intervals, where deformation is absent. In this case deformation process is described with using of fractal time t, which belongs to the points of Cantor s set [30]. If Euclidean object deformation is considered then time belongs to real numbers set. 

Another important property of an attractor is its dimension. It is, loosely speaking, the number of independent degrees of freedom relevant to the dynamical behavior. There are several different definitions that differ mainly in the measure used [35, 39] these definitions all yield the Euclidean value of dimension for Euclidean objects, but for strange attractors the dimension is in general fractional ("fractal [40]). As an example of computing the dimension from experimental data, we will describe a procedure for computing the information dimension, d [41]. Let N(e) be the number of points in a ball of radius e about a point x on an attractor. For a uniform density of points one would have... 

When a polyfunctional (/ > 2) monomer forms bonds at random, or when a particulate sol aggregates, it is common to form fractal structures. (An example is the polymeric cluster in Fig. 3.) A mass fractal is distinguished from a conventional Euclidean object by the fact that the mass (m) of the fractal increases with its radius (r) according to... 

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