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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). Hence Sahlqvist formulas define only a (decidable) subset of modal formulas with first-order correspondents.
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]
The following table lists several common normal modal systems. The notation refers to the table at Kripke semantics § Common modal axiom schemata.Frame conditions for some of the systems were simplified: the logics are sound and complete with respect to the frame classes given in the table, but they may correspond to a larger class of frames.
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 ⊆ ...
So this Welch bound is met with equality if and only if the set of vectors {} is an equiangular tight frame in . Similarly, the Welch bounds stated in terms of average squared overlap, are saturated for all k ≤ t {\displaystyle k\leq t} if and only if the set of vectors is a t {\displaystyle t} -design in the complex projective space C P n ...
This is a list of formulas encountered in Riemannian geometry. Einstein notation is used throughout this article. This article uses the "analyst's" sign convention for Laplacians, except when noted otherwise.
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The proposition in probability theory known as the law of total expectation, [1] the law of iterated expectations [2] (LIE), Adam's law, [3] the tower rule, [4] and the smoothing theorem, [5] among other names, states that if is a random variable whose expected value is defined, and is any random variable on the same probability space, then