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The frame condition was first described by Richard Duffin and Albert Charles Schaeffer in a 1952 article on nonharmonic Fourier series as a way of computing the coefficients in a linear combination of the vectors of a linearly dependent spanning set (in their terminology, a "Hilbert space frame"). [4]
Sahlqvist's definition characterizes a decidable set of modal formulas with first-order correspondents. Since it is undecidable, by Chagrova's theorem, whether an arbitrary modal formula has a first-order correspondent, there are formulas with first-order frame conditions that are not Sahlqvist [Chagrova 1991] (see the examples below).
We know a nice sufficient condition: Henrik Sahlqvist identified a broad class of formulas (now called Sahlqvist formulas) such that a Sahlqvist formula is canonical, the class of frames corresponding to a Sahlqvist formula is first-order definable, there is an algorithm that computes the corresponding frame condition to a given Sahlqvist formula.
A formula holds in a model just in case ,. A formula holds on a frame iff A holds in every model (,). A formula is valid in a class of frames iff A holds on every frame in that class. The class of all Routley–Meyer frames satisfying the above conditions validates that relevance logic B.
Suppose there are m regression equations = +, =, …,. Here i represents the equation number, r = 1, …, R is the individual observation, and we are taking the transpose of the column vector.
These rotational conditions follow from the specific construction of the Eckart frame, see Biedenharn and Louck, loc. cit., page 538. Finally, for a better understanding of the Eckart frame it may be useful to remark that it becomes a principal axes frame in the case that the molecule is a rigid rotor , that is, when all N displacement vectors ...
This condition can often be confirmed by using the fact that integration and differentiation can be swapped when either of the following cases hold: The function f ( x ; θ ) {\\displaystyle f(x;\\theta )} has bounded support in x {\\displaystyle x} , and the bounds do not depend on θ {\\displaystyle \\theta } ;
We can prove that these frames produce the same set of valid sentences as do the frames where all worlds can see all other worlds of W (i.e., where R is a "total" relation). This gives the corresponding modal graph which is total complete (i.e., no more edges (relations) can be added). For example, in any modal logic based on frame conditions: