Doubly connected

A direct path from R to R2 and the path from R to x on the surface, cot-inuing from the diametrically opposed x to R2. The group 0(3) is therefore doubly connected. 

Thus, we can expect further improvements in the description of multiple bond breaking by the QMMCC method. This statement parallels similar findings by Head-Gordon et al. (27, 28, 128, 129), who considered the quadratic variant of the ECC theory of Arponen and Bishop (114 -123), in which the energy is calculated by imposing the stationary conditions for the asymmetric, doubly connected, energy functional... 

In this section, we examine the possibility of improving the quality of Ti and T2 components in the bond breaking region by solving the ECCSD equations described in ref 124. The ECC formalism introduced in ref 124 is based on the idea of rewriting the electronic Schrddinger equation for the CC wave ftmction 1 0) eq (3), in the following doubly connected form ... 

The group space of 0(3) is doubly connected (i.e., non-simply connected) and can therefore support an Aharonov-Bohm effect (Section V), which is described by a physical inhomogeneous term produced by a rotation in the internal gauge space of 0(3) [24]. The existence of the Aharonov-Bohm effect is therefore clear evidence for an extended electrodynamics such as 0(3) electrodynamics, as argued already. A great deal more evidence is reviewed in this article in favor of 0(3) over U(l). For example, it is shown that the Sagnac effect [25] can be described accurately with 0(3), while U(l) fails completely to describe it. 

Doubly Connected Region. A region of space is called doubly connected when there exists only two irreconcilable paths in this region. In this case, a single irreducible loop can be identified in this region. 

In the case of an infinitely long solenoid, space is halved and dissociated into two distinct regions (1) a simply connected region inside the solenoid and (2) a doubly connected region outside the solenoid. 

The ansatz, upon which these results are based, is that the configuration of the vacuum is described by the doubly connected group 0(3), which supports the Aharonov-Bohm effect in Minkowski spacetime [46]. More generally, the vacuum configuration could be described by an internal gauge space more general than 0(3), such as the Lorentz, Poincare, or Einstein groups. The 0(3)... 

Lecourt, T. Mallet, J.-M. Sinay, P. Efficient synthesis of doubly connected primary face-to-face cyclodextrin homo-dimers. Eur. J. Org. Chem. 2003, 4553-4560. 

Figure 8.12. The two pertinent empty orbitals are of b2g symmetry (cluster and ligand antibonding) and a2g symmetry (cluster antibonding) and cannot mix. Bending the Ge-H bonds to model the doubly connected Ge9 cluster units in the dimer and trimer lowers the symmetry to D2h and these two orbitals now are of big symmetry and can mix. One is stabilized and the other destabilized as shown. The former is now available at low energy to accommodate the extra lone pair. An important consequence of this exocluster orbital mixing is that the oligomers are not viewed as delocalized clusters connected by localized bonds but single delocalized entities.

Figure A.4 The concept of connectedness a) a simply connected domain b) a doubly connected domain and c) a doubly connected domain that has been converted into a simply connected domain by introducing a pair of cuts.

Figure A.6 Demonstration of the integration for a doubly connected domain that has been transformed into a simply connected domain.

GC. Zeolites with chains of doubly-connected 4-membered rings 9.GD. Zeolites with chains of 5-membered rings 9.GE. Zeolites with sheets with 4-4-1-1 structural units 9.GF. Zeolites with Leucite-type frameworks... 

The dimensionless shape factor for the isothermal rectangular annulus is derived from the correlation equation of Schneider [89], who obtained accurate numerical values of the thermal constriction resistance of doubly connected rectangular contact areas by means of the boundary integral equation method ... 

Spreading (constriction) resistance is an important thermal parameter that depends on several factors such as (1) geometry (singly or doubly connected areas, shape, aspect ratio), (2) domain (half-space, flux tube), (3) boundary condition (Dirchlet, Neumann, Robin), and (4) time (steady-state, transient). The results are presented in the form of infinite series and integrals that can be computed quickly and accurately by means of computer algebra systems. Accurate correlation equations are also provided. 

Doubly Connected Isoflux Contact Areas on Half-Space... 

FIGURE 3.15 Doubly connected regular polygonal areas. 

The correlation coefficient a0 represents the dimensionless spreading resistance of the full contact area, in agreement with results presented previously. Since the results for the square and the circle are very close for all values of the parameter e, the correlation equations for the square or the circle may be used for other doubly connected regular polygons such as pentagons, hexagons, etc. 

P. A. A. Laura and G. Sanchez Sarmiento, Analytical Determination of Heat Flow Shape Factors for Composite, Prismatic Bars of Doubly-Connected Cross Section, Nuclear Engineering and Design, Vol. 50, pp. 397-385,1978. 

The preceding boundary conditions are applicable to both singly connected and doubly connected ducts. Detailed descriptions of the various boundary conditions are available in Shah... 

Fully developed laminar flow and heat transfer in several doubly connected ducts are discussed in the following sections. 

Nu local Nusselt number for a doubly connected duct= /(ff1 - 0 )... 

P. N. Shivakumar, Viscous Flow in Pipes Whose Cross Sections are Doubly Connected Regions, Appi. Sci. Res., (27) 355-365,1973. 

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