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...

We discuss the Schur complement formula for quaternionic matrices, M , and give an efficient method to calculate the matrix inverse. We also introduce the functional D[M ] which extends to quaternionic matrices the non-negative number |det[M ]|. 1. Introduction. Much of the spectral theory of complex matrices does not extend to quaternion matrices without further modifications [1, 2, 3]. In par...

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...

We solve a nonstationary problem of heat conduction for thermosensitive half space in contact with layer under the conditions complex exchange medium constant temperature. To construct solution, we use numerical-analytic approach characterized by application version method successive approximations, linearizing parameters, Laplace integral transformation, and its numerical inversion help Prudni...

We study a new Radon-like transform that averages projected pforms in Rn over affine (n − k)-spaces. We then prove an explicit inversion formula for our transform on the space of rapidly-decaying smooth p-forms. Our transform differs from the one in the work by Gelfand, Graev and Shapiro (1969). Moreover, if it can be extended to a somewhat larger space of p-forms, our inversion formula will al...