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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).
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 ⊆ ...
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]
One can also view the Maurer–Cartan form as being constructed from a Maurer–Cartan frame. Let E i be a basis of sections of TG consisting of left-invariant vector fields, and θ j be the dual basis of sections of T * G such that θ j (E i) = δ i j, the Kronecker delta. Then E i is a Maurer–Cartan frame, and θ i is a Maurer–Cartan coframe.
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.
Let stand for ,, or . The Stiefel manifold () can be thought of as a set of n × k matrices by writing a k-frame as a matrix of k column vectors in . The orthonormality condition is expressed by A*A = where A* denotes the conjugate transpose of A and denotes the k × k identity matrix.
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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 ...