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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).
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a class C of frames or models, if it is valid in every member of C. We define Thm(C) to be the set of all formulas that are valid in C. Conversely, if X is a set of formulas, let Mod(X) be the class of all frames which validate every formula from X. A modal logic (i.e., a set of formulas) L is sound with respect to a class of frames C, if L ⊆ ...
In statistics, especially in Bayesian statistics, the kernel of a probability density function (pdf) or probability mass function (pmf) is the form of the pdf or pmf in which any factors that are not functions of any of the variables in the domain are omitted. [1] Note that such factors may well be functions of the parameters of the
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:
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In Bayesian statistics, the model is extended by adding a probability distribution over the parameter space . A statistical model can sometimes distinguish two sets of probability distributions. The first set Q = { F θ : θ ∈ Θ } {\displaystyle {\mathcal {Q}}=\{F_{\theta }:\theta \in \Theta \}} is the set of models considered for inference.