Marginal dimensionality

Eq. (27) leads us to the aforementioned singularity concept, the marginal dimensionality, above which the excluded volume effects suddenly vanish For instance, for a branched molecule in the dilution limit (v0 = 1 / 4, q = 0), Eq. (27) gives... 

In light of the concept of the marginal dimensionality, it is expected that this hyperbohc relationship holds more accurately above eight dimensions. 

Second, the failure of the D ring) term should arise from the high dimension expansion based on the premise of Eq. (110). The result of Fig. 24 shows that the expansion works well for d>8. It has been well established that lattice branched clusters have the marginal dimensionality, dc = 8, in the sol phase (bond animals) above which the ideal behavior applies [33]. Now, the critical point shift results from ring formation which is a phenomenon in the sol phase up to the gel point, and the cyclization frequency is influenced by the excluded volume effects. The gel point, therefore, must shift in response to the behavior of the sol clusters. This leads us to a conjecture that a mathematical singularity may arise at dc = 8 on the Dc vs. d curve, in parallel with the phase transition from the excluded volume clusters to the ideal ones. If this is the case, it follows that the high dimension expansion must fail below eight dimensions. To date, there is no experimental evidence that shows the existence of the discontinuity on the Dc vs. d curve, but it is likely that Dc is not a monotonous function of d the result of Fig. 

Als-Nielsen J, Litster J D, Birgeneau R J, Kaplan M and Safinya C R 1980 Lower marginal dimensionality. X-ray scattering from the smectic-A phase of liquid crystals Ordering in Strongly Fluctuating Condensed Matter Systems ed T Riste (New York Plenum)... 

The hexatic, smectic phases have quasi-long-range orientational order, and are discussed in Aharony s lecture at this school, together with a thorough discussion of associated phase transitions. Hexatic and other smectic phases offer a unique possibility for studying the strong fluctuations of low-dimensional systems, even in 3-D samples, and represent condensed matter at the lower marginal dimensionality. 

We must now mention, that traditionally it is the custom, especially in chemo-metrics, for outliers to have a different definition, and even a different interpretation. Suppose that we have a fc-dimensional characteristic vector, i.e., k different molecular descriptors are used. If we imagine a fe-dimensional hyperspace, then the dataset objects will find different places. Some of them will tend to group together, while others will be allocated to more remote regions. One can by convention define a margin beyond which there starts the realm of strong outliers. "Moderate outliers stay near this margin. 

The approaeh taken by Carter (1986, 1997) to determine the reliability when multiple load applieations are experieneed (equation 4.34) is first to present a Safety Margin, SM, a non-dimensional quantity to indieate the separation of the stress and strength distributions as given by ... 

With the application of FIA in the mixture analytical mode for the analysis of environmental samples and after a marginal sample pretreatment by SPE, matrix effects are a high probability. But, as cited previously [27—31], matrix effects were not only observed with FIA but also in LC-MS and MS—MS modes. Advice to overcome these problems by, e.g. an improved sample preparation, dilution of the analyte solution, application of stable isotopic modification of LC conditions [29] or even application of two-dimensional LC separations [27], postcolumn standard addition [29], addition of additives into the mobile phase (e.g. propionic acid, ammonium formate) [34,35] or even matrix compounds [32] were proposed and discussed. 

These minimums reflect the fact of an acquisition of excess electron charge by the fluorine atoms in the crystal and their existence is a consequence of a specific dependence ESP of negative charged atom on the distance (Fig.8). The reliability of existence of two-dimensional minimums in crystals is confirmed by direct calculations of the ESP by the Hartree-Fock method. An analysis shows the more negative charge of isolated ions (deeper negative minimum of potential). At the same time the minimum position shifts to the nucleus. The K-parameter influences the characteristics of the minima only marginally. 

Lastly, we would like to mention here results of the two kinds of large-scale computer simulations of diffusion-controlled bimolecular reactions [33, 48], In the former paper [48] reactions were simulated using random walks on a d-dimensional (1 to 4) hypercubic lattice with the imposed periodic boundary conditions. In the particular case of the A + B - 0 reaction, D = Dq and nA(0) = nB(0), the critical exponents 0.26 0.01 0.50 0.02 and 0.89 0.02 were obtained for d = 1 to 3 respectively. The theoretical value of a = 0.75 expected for d = 3 was not achieved due to cluster size effects. The result for d = 4, a = 1.02 0.02, confirms that this is a marginal dimension. However, in the case of the A + B — B reaction with DB = 0, the asymptotic longtime behaviour, equation (2.1.106), was not achieved at all - even at very long reaction times of 105 Monte Carlo steps, which were sufficient for all other kinds of bimolecular reactions simulated. It was concluded that in practice this theoretically derived asymptotics is hardly accessible. 

Mullins and Sekerka (88, 89) analyzed the stability of a planar solidification interface to small disturbances by a rigorous solution of the equations for species and heat transport in melt and crystal and the constraint of equilibrium thermodynamics at the interface. For two-dimensional solidification samples in a constant-temperature gradient, the results predict the onset of a sinusoidal interfacial instability with a wavelength (X) corresponding to the disturbance that is just marginally stable as either G is decreased... 

Distribution-generated spaces play significant role in the small scale structure. The main idea is as follows. Let S be a set. With each point p of S, associate an -dimensional distribution function 6V whose margins are in P. Then associate a 2n-distribution function /1pq with each pair (p, q) of distinct points such that... 

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