A general method for rotational averages

The theory of nonlinear spectroscopy on randomly oriented molecules leads to the problem of averaging molecular quantities over random rotation. We solve this problem for arbitrary tensor rank by deriving a closed-form expression for the rotationally invariant tensor of averaged direction cosine products. From it, we obtain some useful new facts about this tensor. Our results serve to speed the inherently lengthy calculations of nonlinear optics.

System of differential equations of aircraft spatial motion dynamics with arbitrary tensor of inertia and center of gravity position

Derivation of analytic expressions making up the basis of a mathematical model of aircraft flight dynamics for the differential equations describing the change in the rate of roll, yaw and pitch, as well as flight velocity components in projections on the body-fixed coordinate axes is presented. The origin of the coordinate system does not in general coincide with the center of mass of the plane, and the axes are not the same as its main central axes of inertia. The differential equations for angular and linear velocities are reduced to the form convenient for the use of numerical methods and computer systems and make it possible to get consistent results of simulating the dynamics of aircraft spatial motion with an arbitrary tensor of inertia and center of gravity position.

Canonical tensor scaling

In this paper we generalize the canonical positive scaling of rows and columns of a matrix to the scaling of selected-rank subtensors of an arbitrary tensor. We expect our results and framework will prove useful for sparse-tensor completion required for generalizations of the recommender system problem beyond a matrix of user-product ratings to multidimensional arrays involving coordinates based both on user attributes (e.g., age, gender, geographical location, etc.) and product/item attributes (e.g., price, size, weight, etc.).

The Formula of Grangeat for Tensor Fields of Arbitrary Order innDimensions

The cone beam transform of a tensor field of orderminn≥2dimensions is considered. We prove that the image of a tensor field under this transform is related to a derivative of then-dimensional Radon transform applied to a projection of the tensor field. Actually the relation we show reduces form=0andn=3to the well-known formula of Grangeat. In that sense, the paper contains a generalization of Grangeat's formula to arbitrary tensor fields in any dimension. We further briefly explain the importance of that formula for the problem of tensor field tomography. Unfortunately, form>0, an inversion method cannot be derived immediately. Thus, we point out the possibility to calculate reconstruction kernels for the cone beam transform using Grangeat's formula.