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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]
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,
A space curve; the vectors T, N, B; and the osculating plane spanned by T and N. In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve in three-dimensional Euclidean space, or the geometric properties of the curve itself irrespective of any motion.
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 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).
Unprimed quantities refer to position, velocity and acceleration in one frame F; primed quantities refer to position, velocity and acceleration in another frame F' moving at translational velocity V or angular velocity Ω relative to F. Conversely F moves at velocity (—V or —Ω) relative to F'. The situation is similar for relative ...
With respect to a coordinate frame whose origin coincides with the body's center of mass for τ() and an inertial frame of reference for F(), they can be expressed in matrix form as:
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 ...