In a Galton-Watson branching process with o spring distribution (p0; p1; : : :) started with k individuals, the distribution of the total progeny is identical to the distribution of the rst passage time to k for a random walk started at 0 which takes steps of size j with probability pj+1 for j 1. The formula for this distribution is a probabilistic expression of the Lagrange inversion formula f...

We present a new cone beam inversion formula based on a general weighting function n. The formula is theoretically exact and is represented by a 2D integral. If the source trajectory C is complete (and satisfies two other very mild assumptions), then the simplest uniform weight gives a convolution-based FBP algorithm. This choice is not always optimal from the practical point of view. The unifo...

Let (M, deg) be a cancellative monoid M equipped with a discrete degree map deg :M →R≥0, and let PM,deg(t) := ∑ u∈M/∼t deg(u) be its generating series, called the growth function of M . We show the inversion formula PM,deg(t) ·NM,deg(t) = 1 where the second factor of the formula is given by NM,deg(t) := 1 + ∑ T∈Tmcm(M,I0)(−1) #J1+···+#Jn−n+1 ∑ ∆∈|T | t , which we call the skew-growth function o...

ab initio calculations were employed to investigate nitrogen inversion as a configuration change that can supply an infinitely useful switchable control mechanism for some complex systems. in this paper, design of a new artificial molecular nanohinge is discussed based on nitrogen inversion in which reciprocating motion of substituent in effect of inversion phenomenon, led to an open–close moti...

We prove the Plancherel formula for spherical Schwartz functions on a reductive symmetric space. Our starting point is an inversion formula for spherical smooth compactly supported functions. The latter formula was earlier obtained from the most continuous part of the Plancherel formula by means of a residue calculus. In the course of the present paper we also obtain new proofs of the uniform t...

We provide yet another proof of the classical Lagrange-Good multivariable inversion formula using the techniques of quantum field theory.

In this paper, we give a weakly stable algorithm to solve a block Toeplitz system of linear equations. If the \look-ahead" steps taken to compute the parameters of the inversion formula for the block Toeplitz matrix are small compared to the order n of the matrix, the algorithm requires O(n 2) oating point operations. The parameters of the inversion formula are interpreted and computed in a rec...

Let a and b be two polynomials having numerical coeecients. We consider the question: when are a and b relatively prime? Since the coeecients of a and b are approximant, the question is the same as: when are two polynomials relatively prime, even after small perturbations of the coeecients? In this paper we provide a numeric parameter for determining that two polynomials are prime, even under s...

Let F (z) = z−H(z) with order o(H(z)) ≥ 1 be a formal map from C to C and G(z) the formal inverse map of F (z). We first study the deformation Ft(z) = z − tH(z) of F (z) and its formal inverse Gt(z) = z + tNt(z). (Note that Gt=1(z) = G(z) when o(H(z)) ≥ 2.) We show that Nt(z) is the unique power series solution of a Cauchy problem of a PDE, from which we derive a recurrent formula for Gt(z). Se...

Let F (z) = z−H(z) with order o(H(z)) ≥ 1 be a formal map from C to C and G(z) the formal inverse map of F (z). We first study the deformation Ft(z) = z − tH(z) of F (z) and its formal inverse Gt(z) = z + tNt(z). (Note that Gt=1(z) = G(z) when o(H(z)) ≥ 2.) We show that Nt(z) is the unique power series solution of a Cauchy problem of a PDE, from which we derive a recurrent formula for Gt(z). Se...