The abscissa mapping on the affine variety Mn of monic polynomials of degree n is the mapping that takes a monic polynomial to the maximum of the real parts of its roots. This mapping plays a central role in the stability theory of matrices and dynamical systems. It is well known that the abscissa mapping is continuous on Mn, but not Lipschitz continuous. Furthermore, its natural extension to t...

Lipschitz domains. Our presentations here will almost exclusively be for bounded Lipschitz domains. Roughly speaking, a domain (a connected open set) Ω ⊂ R is called a Lipschitz domain if its boundary ∂Ω can be locally represented by Lipschitz continuous function; namely for any x ∈ ∂Ω, there exists a neighborhood of x, G ⊂ R, such that G ∩ ∂Ω is the graph of a Lipschitz continuous function und...

We give conditions that ensure an operator satisfying a Piestch domination in given setting also satisfies different setting. From this we derive bounded multilinear T T </mml:semant...

We study metric spaces defined via a conformal weight, or more generally measurable Finsler structure, on domain Ω ⊂ ℝ2 that vanishes compact set E and satisfies mild assumptions. Our main question is to determine when such space quasiconformally equivalent planar domain. give characterization in terms of the notion sets are removable for mappings. also quasiconformal mapping can be factored as...

Thurston’s circle packing approximation of the Riemann Mapping (proven to give the Riemann Mapping in the limit by Rodin-Sullivan) is largely based on the theorem that any topological disk with a circle packing metric can be deformed into a circle packing metric in the disk with boundary circles internally tangent to the circle. The main proofs of the uniformization use hyperbolic volumes (Andr...

We quantify the sensitivity of KKT pairs associated with a parameterized family of nonlinear programming problems. Our approach involves proto-derivatives, which are generalized derivatives appropriate even in cases when the KKT pairs are not unique; we investigate what the theory of such derivatives yields in the special case when the KKT pairs are unique (locally). We demonstrate that the gra...

We investigate the value function of an infinite horizon variational problem in infinite-dimensional setting. First, we provide upper estimate its Dini–Hadamard subdifferential terms Clarke Lipschitz continuous integrand and normal cone to graph set-valued mapping describing dynamics. Second, derive a necessary condition for optimality form adjoint inclusion that grasps connection between Euler...