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If F is a field, then the only maximal ideal is {0}. In the ring Z of integers, the maximal ideals are the principal ideals generated by a prime number. More generally, all nonzero prime ideals are maximal in a principal ideal domain. The ideal (,) is a maximal ideal in ring [].
In thermodynamics, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature . The change in the Helmholtz energy during a process is equal to the maximum amount of work that the system can perform in a thermodynamic process ...
The maximum work is thus regarded as the diminution of the free, or available, energy of the system (Gibbs free energy G at T = constant, P = constant or Helmholtz free energy F at T = constant, V = constant), whilst the heat given out is usually a measure of the diminution of the total energy of the system (internal energy). Thus, G or F is ...
The generalized force, X, corresponding to the external parameter x is defined such that is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate is given by:
Still more generally, if A is a regular local ring, then the formal power series ring A[[x]] is regular local. If Z is the ring of integers and X is an indeterminate, the ring Z[X] (2, X) (i.e. the ring Z[X] localized in the prime ideal (2, X) ) is an example of a 2-dimensional regular local ring which does not contain a field.
In the case of degenerate energy levels, we can write the partition function in terms of the contribution from energy levels (indexed by j) as follows: =, where g j is the degeneracy factor, or number of quantum states s that have the same energy level defined by E j = E s.
Consider a formal group F(X,Y) with coefficients in the ring of integers in a local field (for example Z p). Taking X and Y to be in the unique maximal ideal gives us a convergent power series and in this case we define F(X,Y) = X + F Y and we have a genuine group law. For example if F(X,Y)=X+Y, then this is the usual addition
The Betz Limit is the maximum possible energy that can be extracted by an infinitely thin rotor from a fluid flowing at a certain speed. [5] In order to calculate the maximum theoretical efficiency of a thin rotor (of, for example, a wind turbine), one imagines it to be replaced by a disc that removes energy from the fluid passing through it ...