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Sahlqvist formulas are built up from implications, where the consequent is positive and the antecedent is of a restricted form.. A boxed atom is a propositional atom preceded by a number (possibly 0) of boxes, i.e. a formula of the form (often abbreviated as for <).
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]
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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.
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.
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.
This proposition is (sometimes) known as the law of the unconscious statistician because of a purported tendency to think of the aforementioned law as the very definition of the expected value of a function g(X) and a random variable X, rather than (more formally) as a consequence of the true definition of expected value. [1]
The HJM framework originates from the work of David Heath, Robert A. Jarrow, and Andrew Morton in the late 1980s, especially Bond pricing and the term structure of interest rates: a new methodology (1987) – working paper, Cornell University, and Bond pricing and the term structure of interest rates: a new methodology (1989) – working paper ...