Irreducible domain

Figure 2. Oxygen ELF = 0.85 localization domains of oxygen atoms in (a) cristobalite, (b) stishovite and (c) fluorite-type prototype structure. The bond directions which are not indicated on the figure are along the lines joining the oxygen core domain (the small sphere at the center of the pictures) and the disynaptic domains. In cristobalite the disynaptic domains are the two discs on both side of the core. In stishovite the valence domain is not fully resolved in irreducible domains, the disynpatic attractors are located within the three bulges lying in the horizontal plane. In fluorite-type structure, the four disynaptic domains form a tetrahedron.

Another interest of the domain isosurface representation is the clear evidence of multicenter bonds such as in the examples of B4CI4 and diborane (cf. Fig. 7). In boron tetrachloride, the V(B,B,B) trisynaptic basins have their attractors in front of the boron tetrahedron giving rise to the green reducible localization at ELF = 0.85 which splits into four irreducible domains for ELF = 0.875. In diborane, the two V (B,H,B) trisynaptic basins in which a proton is embedded support the protonated double bond picture of Pitzer [61]. 

Detailed description of the domains of convergence of hyper geometric series in terms of amoeba of the discriminant of the polynomial has been given recently in Passare and Tsikh (2004). The discriminant A(a) is an irreducible polynomial with integer coefficients in terms of the coefficients , of polynomial (54) that vanishes if this polynomial has multiple roots. For instance, for cubic polynomial the discriminant is... 

The one-exciton Green functions, the scattering matrix, and the irreducible part of the two-exciton Green function entering these equations are given by a Fourier transformation of Equations (18)—(21) to the frequency domain. 

Theorem. A closed set in k" is irreducible iff its ring of functions is an integral domain. 

Theorem, (a) Spec A is irreducible iff A modulo its nilradical is an integral domain. 

On the other hand, in the condensed phases the concept of supermolecules is not useful because every atom or molecule interacts simultaneously with many neighbors. The many-body nature of the induction process, combined with the statistical mechanics of liquids and a complex local field problem have been serious difficulties for a quantitative description of CILS in dense matter. Furthermore, for an accurate modeling, irreducible (i.e., the pairwise nonadditive) contributions of the intermolecular interaction and induction mechanisms may be significant, which complicate the problem even more. Most treatments of CILS in the dense phases have been undertaken in the time domain, based on correlation functions of the type... 

From the above result, it could be inferred exactly that such irreducible subspaces of the state space establish the proper mathematical domain of the classical physical field quantities. In fact, the demonstration was undertaken by using a relativistic electromagnetic field tensor, F,y, and its antisymmetric property ... 

Going-Down Theorem. Let f X — Y be a finite morphism. Assume that Y is an irreducible normal scheme, i.e., all of its local rings oy are domains, integrally closed in its quotient field. Assume that for all x E X, no non-zero element of g x y is a 0-divisor in ox7. Then for every pair of points x E X,... 

The second group consists of reactions (1,2,3,4,5) and (1,2,3,4,5,6,7). The two sets involve ten species of which five are independent species. The forms of the oscillations in these two sets are identical therefore, the five-step (1,2,3,4,5) is an irreducible set and steps 6 and 7 are (again) not essential steps in the seven NFT steps with the FKN rate coefficients. In the mechanisms of both groups I and II, there are five essential species, HBr02, HOBr, BrO, Br , Ce, which are necessary to have oscillations in the NFT mechanism. Although the above oscillatory domains are near the experimental one, and the periods and shapes of the oscillations are compatible with the experimental results, both group I and II reaction mechanisms with the FKN rate coefficients do not fit experimental results well. 

Step 5. Once the irreducible central domains are constructed, we add the secondary environment to each one of them. We do this by examining the off-diagonal elements of the Fock matrix ( i / l y), where runs over all (central and primary environment) orbitals from a given irreducible central domain [i, ..., / ] and (py runs over all correlated occupied LMOs of the entire system that are not included in [i, ..., ij. The secondary environment of the irreducible central... 

This principle could be taken to mean that every good causal story we can tell has to be couched in physical terms or that all causation is always in virtue of physically definable or reducible properties. Taken this way, however, this principle would just beg the question against nonreductive materialism. Under physical causal closure, it is not that we can never invoke physically irreducible higher-level properties as causes for a physical event. We may very well cite some irreducible mental property as the cause of some physical event. The point is that this mental property, if it is to be causal, must somehow be anchored in the physical world. It must be tied to the physical by some sort of supervenience or realization relation. This close tie to the physical will always make it possible, in principle, for us to tell a causal story in terms of physical properties, even though the best causal story might not always refer only to physical properties. Thus, we will never be forced to go outside the physical domain and refer to physically irreducible properties in order to find a sufficient cause for a physical event. So the principle of physical causal closure requires that if we pick any physical event, it will always be possible, in principle, for us to find a purely physical causal chain for that event. So if we take any physical event P, we should be able to find some property or group of properties P, such that P is physical and P is a sufficient cause for P. ... 

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