Vector-Valued Function Algebras

In this paper we study the mathematical and mechanical properties of the double- helix structure of DNA in 3dimensional space. Many of the mechanical quantities in the DNA can be described explicitly, which is not the case for the latter models, and the model is computationally efficient in that the super-coiling parameters can be computed from simultaneously solving algebraic equations.

Let L θ be the circular cone in R n which includes a second-order cone as a special case. For any function f from R to R, one can define a corresponding vector-valued function f c (x) on R n by applying f to the spectral values of the spectral decomposition of x ∈ R n with respect to L θ. We show that this vector-valued function inherits from f the properties of continuity, Lipschitz continuity, directional differentiability, Fréchet differentiability, continuous differentiability, as well as semismoothness. These results will play a crucial role in designing solution methods for optimization problem associated with the circular cone.

The concepts of cone extreme points, cone saddle points, and cone saddle values are introduced. The relation of inclusion among the sets minix~x maxy~yf(x, y), maxiy~y minx~x f(x, y), and the set of all weak cone saddle values is investigated in the case where the image space R n o f f is ordered by an acute convex cone.