Connectivity, degree

K results to substantial growth of reactive medium connectivity degree characterized by dimension ds from 0,42 up to 1,68. Let s mention, that such ds increase occurs without reactive mixture composition change. This means, that the energetic restrictions result to the... 

In case of reaction course in the Euclidean spaces the value D is equal to the dimension of this space d and for fractal spaces D is accepted equal to spectral dimension ds [6], By plotting p i=( 1 -O) (where O is conversion degree) as a function of t in log-log coordinates the value D from the slope of these plots can be determined. It was found, that the mentioned plots fall apart on two linear parts at t<100 min with small slope and at PT00 min the slope essentially increases. In this case the value ds varies within the limits 0,069-3,06. Since the considered reactions are proceed in Euclidean space, that is pointed by a linearity of kinetic curves Q-t, this means, that the reesterefication reaction proceeds in specific medium with Euclidean dimension d, but with connectivity degree, characterized by spectral dimension ds, typical for fractal spaces [5],... 

In summary, the activity level in the MB depends essentially on the connectivity degree and the firing threshold of KCs as expected. More surprising is the extremely wide distribution of possible activity levels which may introduce the necessity of gain control mechanisms. 

In summary, we have seen that if parameters (connectivity degree and firing threshold) are chosen wisely, fully random connections allow an almost always (in the loose sense of a very small failure probability) one-to-one projection of activity patterns from the AL to the MB, a necessary requirement for successful odor classification. At the same time, the activity level in the MB can remain reasonably low even though the absolute minimum for the confusion probability is attained at very high activity levels. 

We have seen in this subsection once again that one of the determining factors in making a system successful in the information processing framework with disordered (random) connections is the correct balance of system size, connectivity degrees and firing thresholds. Other factors like learning rates and output redundancy may play equally important roles. 

And in conclusion of the Chapter 1, let us consider the branched polymers behavior in restricted configurations, such as poses and slits. If for the linear polymers this situation is simple enough and is studied well [7], then for the branched polymers several restrictions exist, due to their larger connectivity degree [57]. The main difficulty in branched polymers swelling and conformational behavior treatment consists in the fact that... 

Hence, the main conclusion from the adduced above estimations is the fact that higher connectivity degree d> 1.0) of branehed polymers imposes more strict restrictions on conformation and behavior of such polymers in restricted geometries [57],... 

The spectral (finction) dimension r/, which characterize macromolecular coil connectivity degree, are its stmcture important characteristic (particularly for the branched polymers). For linear polymer chains r/=1.0 and for the branched ones J value is varied within the limits of 1.0-1.33 [45], The dimension J are a decisive one at calculation for fiactals of different types... 

Thus, the fractal analysis methods were used above for treatment of comb-like poly(sodiumoxi) methylsylseskvioxanes behavior in solution. It has been shown that the intrinsic viscosity reduction at transition from a linear analog to a branched one is due to the sole factor, namely, to a macromolecule connectivity degree enhancement, characterized by spectral dimension. This conclusion is confirmed by a good correspondence of the experimental and calculated according to Mark-Kuhn-Houwink equation fiactal variant intrinsic viscosity values. It has been shown that qualitative transition of the stmcture of branched polymer macromolecular coil from a good solvent to 0-solvent can be reached by a solvent change. 

This technique uses as maintained degrees of freedom the natural modes calculated with the connection nodes blocked. Additirmal deformation modes that have to be included in this method consist of the displacement field obtained when each one of the connection degrees of freedom is submitted to a unitary displacement while the rest is blocked (Craig and Bampton 1968). 

The reactive medium connectivity degree, controlled by the parameters c or and characterized by the spectral dimension d, defines polymerization reaction course, which follows directly from the Eq. (6). So, in Fig. 8 the dependence of polymerization rate (in relative units according to the data [1]) on the value d is shown, which is approximated by a straight line, passing through coordinates origin. The last cueumstance points out, that at d = 0 or = 0 polymerization reaction cannot be realized and its rate is equal to zero. 

As follows from the data of Figure 1, all the four adduced plots are linear, that allows to determine the value of spectral dimension J. The estimations have shown, that the imidization temperature T raising within the range 423-523 K results to J increase from 0.42 up to 1.68, i.e. to essential growth of reactionary system connectivity degree. In Fig. 2 the similar dependences for various Na -montmorillonite contents at fixed T = 473 K are shown. As one can see, nanofiller introduction exercises much weaker influence on d value than the imidization temperature raising [9]. 

Let us consider physical principles of reactionary system connectivity degree change, characterized by effective spectral dimension, at imidization temperature... 

Therefore, the data considered above demonstrated that the main parameter, controlling solid-state imidization rate, is the reactionary system connectivity degree characterized by its effective spectral dimension. In its turn, this dimension is a function of macromolecular coil structure that is polymeric reaction specific feature. Imidization temperature raising defines reactionary medium heterogeneity reduction and corresponding increase of its cormectivity degree [8,9]. 

SECTION I A NOTE ON MACROMOLECULAR COIL CONNECTIVITY DEGREE... 

Section I A Note on Macromolecular Coil Connectivity Degree... 

As it is known, autohesion strength (coupling of the identical material surfaces) depends on interactions between some groups of polymers and treats usually in purely chemical terms on a qualitative level [1, 2], In addition, the structure of neither polymer in volume nor its elements (for the example, macromolecular coil) is taken into consideration. The authors [3] showed that shear strength of autohesive joint depended on macromolecular coils contacts number A on the boundary of division polymer-polymer. This means, that value is defined by the macromolecular coil structure, which can be described within the frameworks of fiactal analysis with the help of three dimensions fractal (Hausdorff) spectral (fraction) J and the dimension of Euclidean space d, in which ifactal is considered [4]. As it is known [5], the dimension characterizes macromolecular coil connectivity degree and varies from 1.0 for linear chain up to 1.33 for very branched macromolecules. In connection with this the question arises, how the value influences on autohesive joint strength x or, in other words, what polymers are more preferable for the indicated joint formation - linear or branched ones. The purpose of the present communication is theoretical investigation of this elfect within the frameworks of fractal analysis. 

Within the framewoiks of fractal analysis fractal (macromolecular coil) branching degree is characterized by spectral (fraction) dimension J, which is object connectivity degree characteristic [24]. For linear polymer J = 1.0, for statistically branched one = 1.33 [24]. For macromolecular coil with arbitrary branching degree the value varies within the limits of 1.0-1.33. Between dimensions D and the following relationship exists, that takes into consideration the excluded volume effects [17] ... 

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