Increase in Dimensionality

If the modification imparts dimensional stability to the wood, the difference between the water-saturated and oven-dry volume of the wood sample is reduced, resulting in a lower value for S. The increase in dimensional stability can then be evaluated in terms of an anti-swelling efficiency (ASE) ... 

Note that a spring reverb is essentially a one-dimensional structure, plate reverbs are primarily two-dimensional, and concert halls are three-dimensional. Since each of these increases in dimensionality is associated with a significant increase in quality, it is reasonable to conjecture that simulated reverberation in four dimensions or more may be even better. 

Interlocked 2D layers can exhibit different relative orientations of the planes of the independent motifs. When all the entangled sheets are parallel and have a common average plane the resulting array is an n-fold interpenetrated 2D structure, with no increase in dimensionality and a finite number of interweaved motifs (see an example in Fig. 1.3.14). If, otherwise, the independent layers show average planes that are parallel but displaced in a perpendicular direction, they generate a polycatenated 3D architecture. This parallel catenation can occur either because the sheets are undulated simple layers or because they are multiple layers, in both cases exhibiting some thickness (see Fig. 1.3.14, left, and 1.3.15). Many cases are known and have been reviewed previously [3]. 

Polycatenation can also involve different sets of 2D sheets (usually two) that cross at a certain angle, in the mode that has been called inclined [2] (see Fig. 1.3.14, right). The examples of this class are numerous, the majority consisting of two identical sets of 2D parallel layers, of 63 or 44 topology, spanning two different stacking directions. In these species there is an increase in dimensionality (2D —> 3D) and each individual layer is catenated with an infinite number of other inclined layers but, obviously, not with all the frames contained in the 3D array. Moreover,... 

We mentioned above only a very small part of the available ID and 2D NMR techniques. Many other techniques are known not only as ID and 2D but also employing more dimensions. We could for instance consider a 3D experiment H,H-COSY - H,C-COSY in which the peaks are spread inside a cube. Cutting the cube on a slice corresponding to the signal of one carbon atom would produce a COSY spectrum in which would be present only the H-H correlations of the hydrogen atom linked to the carbon atom for which the slice was selected. It is obvious that as we increase in dimensionality the... 

D and 4D experiments will become very popular and the increase in dimensionality will continue. 

These factors serve as an indication that approximations to Rayleigh-Ritz upper bounds equally accurate to that obtained herein for hydrogen should be achievable for three-electron entities (at most an increase in dimensionality of four over that for hydrogen) using an easily manageable number of points. 

The major problem with exact diagonalization methods is the exponential increase in dimensionality of the Hilbert space with the increase in the system size. Thus, the study of larger systems becomes not only CPU intensive but also memory intensive as the number of nonzero elements of the matrix cdso increases with system size. With increasing power of the computers, slightly larger problems have been solved every few years. To illustrate this trend, we consider the case... 

Given the significant increase in dimensionality of the system, we consider only measuring the effect of varying the stepsize on the computed averages. We choose a moderate value for the friction (we fix it at 1/ps in all simulations) which gave... 

Today, most of these problems are being resolved by the use of rigid substrates based on high glass transition temperature resins such as polyimides and modified epoxides. The reduced thermal expansion of these materials at the normal processing or operating temperatures provides for significant increases in dimensional stability compared with GIO or FR4 laminates. 

DT rings, 3-5, were first reported in 1989 (Figure 8.1) [13]. They are considered as a it-extended model of vinylogous TTF 2 [14-17] to reduce the on-site Coulomb repulsion and increase in dimensionality. 

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