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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]
The frame problem is that specifying only which conditions are changed by the actions does not entail that all other conditions are not changed. This problem can be solved by adding the so-called “frame axioms”, which explicitly specify that all conditions not affected by actions are not changed while executing that action.
For this reason, modal logicians sometimes talk about frames, which are the portion of a relational model excluding the valuation function. A relational frame is a pair M = G , R {\displaystyle {\mathfrak {M}}=\langle G,R\rangle } where G {\displaystyle G} is a set of possible worlds, R {\displaystyle R} is a binary relation on G {\displaystyle ...
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 linear isomorphism is determined by its action on an ordered basis or frame. Hence parallel transport can also be characterized as a way of transporting elements of the (tangent) frame bundle GL(M) along a curve. In other words, the affine connection provides a lift of any curve γ in M to a curve γ̃ in GL(M).
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 ...
A square matrix is called a projection matrix if it is equal to its square, i.e. if =. [2]: p. 38 A square matrix is called an orthogonal projection matrix if = = for a real matrix, and respectively = = for a complex matrix, where denotes the transpose of and denotes the adjoint or Hermitian transpose of .
Because this equation holds for all vectors, p, one concludes that every rotation matrix, Q, satisfies the orthogonality condition, Q T Q = I . {\displaystyle Q^{\mathsf {T}}Q=I.} Rotations preserve handedness because they cannot change the ordering of the axes, which implies the special matrix condition,