Elements, essential invariable

In addition, as far as electron-electron interaction is neglected, the tt electrons are subject to a potential with the full spatial symmetry of the CNT topology. The electronic wavefunctions can then be classified according to their transformation properties under the symmetry operations leaving the CNT invariant. As far as dipole approximation holds, both the linear and nonlinear optical response of a CNT are governed by matrix elements of the dipole operator between two electronic wavefunctions. Being the selection rules of dipole matrix elements essentially determined by the wavefunctions symmetry, it turns out that the optical properties of CNT are essentially rooted in their topology. 

Of the zeolitic materials, AlPO s cut a conspicuous figure because of their structural diversity and the incorporation of other elements into their frameworks. The recently developed VPI-5 (refs. 2, 3) announced the feasibility of synthesis of micoporous structures with windows comprising rings of over 12-T. All AlPO s, SAPO s and MAPO s form a family of microporous structures constructed by or essentially by A1(I) and P(V). Some of them are isostructural with zeolites but a majority have novel structures. The primary building units (PBU) centred by P(V) are invariantly PO4 whereas those centred by A1(I) are AlO in most cases and AIO5 or even AlOs in a few cases. So far all AlPO s, SAPO s and MAPO s have been synthesized exclusively in the presence of amines or... 

Because foodstuffs themselves are difficult to analyse and have a variable composition, testing packaging materials for migration almost invariably uses model foods, known as food simulants , which are designed to mimic the main classes of foodstuffs. There are some minor differences in detail between EU and US-FDA requirements but the essential elements are listed in Table 9.3. 

In the works devoted to MEISs isomerization became a "through" example for explanation of their specific features and efficiency. The example is simple and very obvious, since the isomerization reaction at any mechanism is described by the same material balance because of invariable amounts of substances and elements. This fact essentially facilitates both analytical and graphical interpretations. 

It will be useful to have in mind another way of considering the problem a function on a coset space of G is essentially a function on G invariant under translation by the subgroup. When G is GL and H the upper triangular group, for instance, it is easy to compute that no nonconstant polynomial in the matrix entries is invariant under all translations by elements of H, and thus no affine coset space can exist. (What follows from (16.1) is that there are always semi-invariant functions, ones where each translate of/is a constant multiple of/) Our problem is to prove the existence of a large collection of invariant functions for normal subgroups. 

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