where [d.sub.i,j], i = 0, 1, ..., m; j = 0, 1, ..., n are the control vertices that correspond to a topological rectangular array that forms a control mesh, [w.sub.i,j] is the weight associated with the vertices, [N.sub.i,p](u) and [N.sub.j,q](v) are the canonical B-spline basis functions determined by the vector U = [[u.sub.0], [u.sub.1], ..., [u.sub.m-p+1]] of the direction u and the vector V = [[v.sub.0], [v.sub.1], ..., [v.sub.n-q+1]] of the direction v, respectively, according to the de Boor-Cox recursion formula
. The recursion formula
of [N.sub.i,p](u) is defined as
It is well known that Chebyshev polynomials of the first and second kind [T.sub.n](x) and [U.sub.n](x) are defined as follows: [T.sub.0](x) = 1, [T.sub.1](x) = x, and the recursion formula
[T.sub.n+1] (x) = 2x[T.sub.n](x) - [T.sub.n-1] (x) for all integers n [greater than or equal to] 1.
Among the topics are Mirzakhani's recursion formula
on Weil-Patersson volume and applications, cubic differentials in the differential geometry of surfaces, an overview of quasi-conformal mappings on the Heisenberg group, some historical commentaries on TeichmEller's paper Extremale quasikonforme Abbildungen und quandratische Differentiale, and a displacement theorem of quasiconformal mapping.
Trotter & Gleser (1952) (14) calculated regression formula for stature estimation from the average of the bone of both sides from individual and later on in 1958, in another project they considered the bones of both the sides separately and the resulting recursion formula
later being combined.
In Formula 2, [L.sub.m] (t) is the Legendre polynomial, in which [L.sub.0] (t) = 1, [L.sub.1] (t) = t and the others satisfy the Recursion Formula
Derivation of the Recursion Formula
. The following finite difference formula (22) is one format of the Runge-Kutta methods:
and the value recursion formula
of dynamic programming is
A Recursion Formula
for the Probability Distribution of the Sum of k Dice
From the recursion formula
(6) and Lemma 2.1, we know that the following relations between the coefficients hold:
A recursion formula
(for factorizations where the rank of each factor is dictated) is indeed given by Reading in [Rea08], but the proof is very specific to the real case.
However, even here, in the "classical" part of the text, the authors present more modern and advanced approaches and discussions (e.g., they present Panjer's recursion formula
for computing the distribution of certain random sums of random variables, a discussion of "stop-loss" premiums, and certain modern approximations for ruin probabilities and loss probabilities).