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...

Following the approach of Rota and Taylor [17], we present an innovative theory of Sheffer sequences in which the main properties are encoded by using umbrae. This syntax allows us noteworthy computational simplifications and conceptual clarifications in many results involving Sheffer sequences. To give an indication of the effectiveness of the theory, we describe applications to the well-known...

The first report of the congenital facial diplegia syndrome, or Moebius syndrome, was that of Graefe (1880). Moebius reported cases in 1888 and 1892 and reviewed the prior case reports, thus gaining eponymic distinction. Since that time, some authors have broadened the definition of the Moebius syndrome, as for example, Henderson (1939), who considered some cases of congenital unilateral facial...

Moebius syndrome (MS) is a rare disorder mainly characterized by bi- or unilateral palsy of the facial and abducens nerves. Among the various orofacial characteristics related to this syndrome, the following are pointed out, difficulty with sucking, tongue malformation, mandibular hypoplasia, saliva drooling, arched palate and micrognathia. In spite of the diversity of oral features described f...

Persistent currents in a Moebius ladder are shown to be very sensitive to the effects of intrachain interactions on the hopping of electrons between chains. Their periodicity as a function of flux is doubled for strong enough repulsive interactions because electrons cannot hop coherently between the chains and have to travel along the full edge of the Moebius ladder, thus encircling the flux tw...

Björner and Wachs provided two q-generalizations of Knuth’s hook formula counting linear extensions of forests: one involving the major index statistic, and one involving the inversion number statistic. We prove a multivariate generalization of their inversion number result, motivated by specializations related to the modular invariant theory of finite general linear groups.

By defining a prepotential function for the stationary Schrödinger equation we derive an inversion formula for the space variable x as a function of the wave–function ψ. The resulting equation is a Legendre transform that relates x, the prepotential F , and the probability density. We invert the Schrödinger equation to a third–order differential equation for F and observe that the inversion pro...

In this article we present an improved exact inversion formula for the 3D cone beam transform of vector fields. It is well known that only the solenoidal part of a vector field can be determined by the longitudinal ray transform of a vector field in cone beam geometry. The exact inversion formula, as it was developed in A. Katsevich and T. Schuster, An exact inversion formula for cone beam vect...