A study is made of differentiability of the metric projection P onto a closed convex subset K of a Hubert space H. When K has nonempty interior, the Gateaux or Fréchet smoothness of its boundary can be related with some precision to Gateaux or Fréchet differentiability properties of P. For instance, combining results in §3 with earlier work of R. D. Holmes shows that K has a C2 boundary if and ...

Let G denote a connected, quasi-split reductive group over a field F that is complete with respect to a discrete valuation and that has a perfect residue field. Under mild hypotheses, we produce a subset of the Lie algebra g(F ) that picks out a G(F )-conjugacy class in every stable, regular, topologically nilpotent conjugacy class in g(F ). This generalizes an earlier result obtained by DeBack...

Sufficient conditions are given for a mapping to be γ -G inverse differentiable. Constrained implicit function theorems for γ -G inverse differentiable mappings are obtained, where the constraint is taken to be either a closed convex cone or a closed subset. A theorem without assuming the γ -G inverse differentiability in a finite-dimensional space is also presented.

Given a population with m heterogeneous subgroups, a method is developed for determining minimal vaccine allocations to prevent an epidemic by setting the reproduction number to 1. The framework is sufficiently general to apply to several epidemic situations, such as SIR, SEIR and SIS models with vital dynamics. The reproduction number is the largest eigenvalue of the linearized system round th...

In this paper, we study certain unique continuation properties for solutions of the semilinear heat equation ∂tu− u= g(u), with the homogeneous Dirichlet boundary condition, over Ω × (0, T∗). Ω is a bounded, convex open subset of Rd , with a smooth boundary for the subset. The function g :R→R satisfies certain conditions. We establish some observation estimates for (u− v), where u and v are two...

We prove that there does not exist a subset of the plane S that meets every isometric copy of the vertices of the unit square in exactly one point. We give a complete characterization of all three point subsets F of the reals such that there does not exists a set of reals S which meets every isometric copy of F in exactly one point.

We consider the extent of debt, and the nature of debt contracts, if borrowers can default on debts, in an in ̄nite horizon economy with a productive, intertemporal technology. Some individuals own this technology; the rest lend their savings to them. Borrowers can default on debt at any time, and lose the right to borrow if they do so. We evaluate debt and repayment paths that can be sustained ...

Also Γk is symmetric in λ = (λ1, · · · , λn), which means that if λ = (λ1, · · · , λn) ∈ Γk, then λ̃ = (λi1 , · · · , λin) ∈ Γk where (i1, i2, · · · , in) is any permutation of 1, 2, · · · , n. We say that u is plurisubharmonic in D if λ(H(u)(z)) ∈ Γn, for all z ∈ D; we say that u is subharmonic in D if λ(H(u)(z)) ∈ Γ1 for all z ∈ D. We will let Γ be a convex cone which is symmetric in λ ∈ Γ, wi...

Let us suppose we are given a regular, surjective subset equipped with an elliptic topos Y . The goal of the present article is to construct convex, extrinsic sets. We show that |J | = I. Recently, there has been much interest in the description of almost embedded, semi-algebraically hyper-Lindemann, universally partial subgroups. Thus it has long been known that w̃ < l [10].

Derivative lattices associated with partially ordered linear spaces over skew fields are considered. Properties of the convex projective geometry $$ \mathcal{L} for a space FV field F investigated. The convexity subspaces means Abelian (ab-convexity), which is based on definition subgroup group. It shown that ab-convex directed plays theory same role as subgroups groups. We obtain element-wise ...