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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]
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.
We know a nice sufficient condition: Henrik Sahlqvist identified a broad class of formulas (now called Sahlqvist formulas) such that a Sahlqvist formula is canonical, the class of frames corresponding to a Sahlqvist formula is first-order definable, there is an algorithm that computes the corresponding frame condition to a given Sahlqvist formula.
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.
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 ...
In statistics, especially in Bayesian statistics, the kernel of a probability density function (pdf) or probability mass function (pmf) is the form of the pdf or pmf in which any factors that are not functions of any of the variables in the domain are omitted. [1] Note that such factors may well be functions of the parameters of the
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JASP (Jeffreys’s Amazing Statistics Program [2]) is a free and open-source program for statistical analysis supported by the University of Amsterdam. It is designed to be easy to use, and familiar to users of SPSS. It offers standard analysis procedures in both their classical and Bayesian form.