Deconstructing functions on quadratic surfaces into multipoles

Deconstructing Functions on Quadratic Surfaces into Multipoles

Any homogeneous polynomial P (x, y, z) of degree d, being restricted to a unit sphere S, admits essentially a unique representation of the form λ + ∑d k=1[ ∏k j=1 Lkj ], where Lkj ’s are linear forms in x, y and z and λ is a real number. The coefficients of these linear forms, viewed as 3D vectors, are called multipole vectors of P . In this paper we consider similar multipole representations o...

On quadratic Lyapunov functions

Exact envelope computation for moving surfaces with quadratic support functions

We consider surfaces whose support function is obtained by restricting a quadratic polynomial to the unit sphere. If such a surface is subject to a rigid body motion, then the Gauss image of the characteristic curves is shown to be a spherical quartic curve, making them accessible to exact geometric computation. In particular we analyze the case of moving surfaces of revolution.

On the quadratic support of strongly convex functions

In this paper, we first introduce the notion of $c$-affine functions for $c> 0$. Then we deal with some properties of strongly convex functions in real inner product spaces by using a quadratic support function at each point which is $c$-affine. Moreover, a Hyers–-Ulam stability result for strongly convex functions is shown.

Quadratic Functions on Torsion Groups

A quadratic function q on an Abelian groupG is a map, with values in an Abelian group, such that the map b : (x, y) 7→ q(x + y) − q(x) − q(y) is Z-bilinear. Such a map q satisfies q(0) = 0. If, in addition, q satisfies the relation q(nx) = nq(x) for all n ∈ Z and x ∈ G, then q is homogeneous. In general, a quadratic function cannot be recovered from the associated bilinear pairing b. Homogeneou...