Finite Constant Groups

Looking back to the previous section, we find the duals of the finite constant groups are precisely the finite diagonalizable groups the dual algebra of kr is kpT]. In general this would not be one of our Hopf algebras, since it is not commutative. But when T is commutative we can write it as a product of various Z/nZ and compute that the dual of Z/nZ is ft,. 

Let r be a finite constant group scheme over a field k. Show that n-dimensional linear representations of T are given by ordinary homomorphisms of I (fc) into GMfc). 

If T is a finite group, show that the corresponding finite constant group scheme assigns to R the set of locally constant functions Spec R- T. 

Proof. Base-extending to k, we may assume we have a finite constant group scheme, say of order n. When we embed it as an algebraic matrix group, each g in it satisfies the separable equation X" — 1 = 0. If g is also unipotent, g — l. Thus the group is trivial. ... 

The rank of a finite locally free sheaf or a finite flat group scheme over S is a locally constant function on S. 

Extension to arbitrary J follows from the fact that, for any finite point group, there is an integer K such that r(7 - - A ) — r(7) is a constant (reducible) character depending (at most) on the value of J mod K. The constant term is either the regular character of the group, or, for centrosymmetric groups, is alternately the g and u halves of the regular character. 

Quotient formation (in a very special case). Let .X— / be a finite formal /-scheme and a constant group operating on X... 

WASP/TOXIWASP/WASTOX. The Water Quality Analysis Simulation Program (WASP, 3)is a generalized finite-difference code designed to accept user-specified kinetic models as subroutines. It can be applied to one, two, and three-dimensional descriptions of water bodies, and process models can be structured to include linear and non-linear kinetics. Two versions of WASP designed specifically for synthetic organic chemicals exist at this time. TOXIWASP (54) was developed at the Athens Environmental Research Laboratory of U.S. E.P.A. WASTOX (55) was developed at HydroQual, with participation from the group responsible for WASP. Both codes include process models for hydrolysis, biolysis, oxidations, volatilization, and photolysis. Both treat sorption/desorption as local equilibria. These codes allow the user to specify either constant or time-variable transport and reaction processes. 

The above models describe a simplified situation of stationary fixed chain ends. On the other hand, the characteristic rearrangement times of the chain carrying functional groups are smaller than the duration of the chemical reaction. Actually, in the rubbery state the network sites are characterized by a low but finite molecular mobility, i.e. R in Eq. (20) and, hence, the effective bimolecular rate constant is a function of the relaxation time of the network sites. On the other hand, the movement of the free chain end is limited and depends on the crosslinking density 82 84). An approach to the solution of this problem has been outlined elsewhere by use of computer-assisted modelling 851 Analytical estimation of the diffusion factor contribution to the reaction rate constant of the functional groups indicates that K 1/x, where t is the characteristic diffusion time of the terminal functional groups 86. ... 

If a NC of a SRM is considered from a suitably defined FS the coordinate vectors may be expressed as functions of the internal coordinates i, 2 . f- The RNC Xk( ), Zk, Mk is then completely defined by the values of ,..., f and the constant structural parameters. Further classification of SRMs may be based on the 2,3,... rigid parts, whose relative positions are determined by the finite coordinates i, 2,... Such parts are often denoted as frame (F), top (T), invertor (I), etc. Moreover each part may have its own local covering symmetry and the complete NC Xk( ), Zk, Mk may have a proper covering symmetry group eg ( ) for arbitrary values of the internal coordinates. Typical SRMs are listed in Tables 1,2 and 3. 

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