Subdivision

The special boiling point spirits (SBP s) have boiling ranges from 30 to 205°C and are grouped in subdivisions according to Table 6.1. 

Vanadium pentoxide, vanadium(V) oxide, V2O5, is the most important compound in this oxidation state. It is a coloured solid (colour due to charge transfer, p. 60), the colour varying somewhat (red -> brown) with the state of subdivision it is formed when vanadium (or some of its compounds) is completely oxidised, and also by heating ammonium vanadate)V) ... 

Computation of Essential Molecular Dynamics by Subdivision Techniques... 

The basic scheme of this algorithm is similar to cell-to-cell mapping techniques [14] but differs substantially In one important aspect If applied to larger problems, a direct cell-to-cell approach quickly leads to tremendous computational effort. Only a proper exploitation of the multi-level structure of the subdivision algorithm (also for the eigenvalue problem) may allow for application to molecules of real chemical interest. But even this more sophisticated approach suffers from combinatorial explosion already for moderate size molecules. In a next stage of development [19] this restriction will be circumvented using certain hybrid Monte-Carlo methods. 

Our global subdivision approach is not sensitive to such a situation. Over sufficiently long run times of direct simulation both methods eventually yield roughly the same results, see Fig. 6. 

Fig. 6. The density of the invariant measure of the potential Vi for total energy F = 4.5. Results of the subdivision approach (left) and a direct simulation with about 4.5 million steps for stepsize t = 1/30 (right).

Based on observations concerning the dynamical behavior we already conjectured that there exist seven almost invariant sets - a conjecture that we now want to check numerically. We employ the subdivision algorithm for subtrajectories of length mr = 0.1. The final box-collection corresponding to the total energy E = 4.5 after 18 subdivision steps consists of 18963 boxes. 

Fig. 7. Eigenmeasure V2 of the Frobenius-Perron operator to the second largest eigenvalue A2 = 0.9963 for the test system (15) with 7 = 3. iV2 was computed via our new subdivision algorithm (cf. Section 4).

M. Dellnitz and A. Hohmann. A subdivision algorithm for the computation of unstable manifolds and global attractors. Numerische Mathematik 75 (1997) 293-317... 

P. Deuflhard, M. Dellnitz, O. Junge, and Ch. Schiitte. Computation of essential molecular dynamics by subdivision techniques I Basic concept. Preprint SC 96-45, Konrad Zuse Zentrum, Berlin (1996)... 

The system defined by the Liouvillian is called the reference system. Now applying the Trotter factorization to the propagator exp iLs + Fi arising from this subdivision gives the new propagator,[17]... 

To separate the non-bonded forces into near, medium, and far zones, pair distance separations are used for the van der Waals forces, and box separations are used for the electrostatic forces in the Fast Multipole Method,[24] since the box separation is a more convenient breakup in the Fast Multipole Method (FMM). Using these subdivisions of the force, the propagator can be factorized according to the different intrinsic time scales of the various components of the force. This approach can be used for other complex systems involving long range forces. 

The elements of an organic compound are listed in empirical formulas according to the Hill system [8] and the stoichiometry is indicated by index numbers. Hill positioned the carbon and the hydrogen atoms in the first and the second places, with heteroatoms following them in alphabetical order, e.g., C9H11NO2. However, it was recognized that different compounds could have the same empirical formula (see Section 2.8.2, on isomerism). Therefore, fine subdivisions of the empirical... 

The first finite element schemes for differential viscoelastic models that yielded numerically stable results for non-zero Weissenberg numbers appeared less than two decades ago. These schemes were later improved and shown that for some benchmark viscoelastic problems, such as flow through a two-dimensional section with an abrupt contraction (usually a width reduction of four to one), they can generate simulations that were qualitatively comparable with the experimental evidence. A notable example was the coupled scheme developed by Marchal and Crochet (1987) for the solution of Maxwell and Oldroyd constitutive equations. To achieve stability they used element subdivision for the stress approximations and applied inconsistent streamline upwinding to the stress terms in the discretized equations. In another attempt, Luo and Tanner (1989) developed a typical decoupled scheme that started with the solution of the constitutive equation for a fixed-flow field (e.g. obtained by initially assuming non-elastic fluid behaviour). The extra stress found at this step was subsequently inserted into the equation of motion as a pseudo-body force and the flow field was updated. These authors also used inconsistent streamline upwinding to maintain the stability of the scheme. 

In this section the discretization of upper-convected Maxwell and Oldroyd-B models by a modified version of the Luo and Tanner scheme is outlined. This scheme uses the subdivision of elements suggested by Marchal and Crochet (1987) to generate smooth stress fields (Swarbrick and Nassehi, 1992a). 

Depending on the type of elements used appropriate interpolation functions are used to obtain the elemental discretizations of the unknown variables. In the present derivation a mixed formulation consisting of nine-node bi-quadratic shape functions for velocity and the corresponding bi-linear interpolation for the pressure is adopted. To approximate stres.ses a 3 x 3 subdivision of the velocity-pressure element is considered and within these sub-elements the stresses are interpolated using bi-linear shape functions. This arrangement is shown in Edgure 3.1. 

The outlined scheme is shown to yield stable solutions for non-zero Weissenberg number flows in a number of benchmark problems (Swarbric and Nassehi, 1992b). However, the extension of this scheme to more complex problems may involve modifications such as increasing of elemental subdivisions for stress calculations from 3 x 3 to 9 x 9 and/or the discretization of the stress field by biquadratic rather than bi-linear sub-elements. It should also be noted that satisfaction of the BB condition in viscoelastic flow simulations that use mixed formulations is not as clear as the case of purely viscous regimes. 

There is an ever-increasing number of further subdivisions, or what I would call hyphenated branches of chemistry or chemically related sciences. Whether chemical-physics or chemical-biology is more meaningful than physical chemistry or biological chemistry may depend on the point of view one wants to look from. 

A field of such importance and intrinsic difficulty should be made as readily accessible as possible, and the lack of a modern detailed and comprehensive presentation of heterocyclic chemistry is therefore keenly felt. It is the intention of the present senes to fill this gap by expert presentations of the various branches of heterocyclic chemistry. The subdivisions have been designed to cover the field in its entirety by monographs which reflect the importance and the interrelations of the various compounds, and accommodate the specific interests of the authors. 

Capacity, mL Subdivision, mL precision grade standard grade ... 

Imagine subdividing the face of the block into two portions of area A/2. A force only half as large would be required for each face to produce the same net distortion. The same argument could be applied for any degree of subdivision hence it is the quantity F/A which is proportional to AL/Lq. 

U.S. Environmental Protection Agency, subdivision G, ProductPeformance NTIS PB-83-153924, Washington, D.C., Sept. 1982. 

---