This paper answers a question of Burns, Karrass and Solitar by giving examples of knot and link groups which are not subgroup-separable. For instance, it is shown that the fundamental group of the square knot complement is not subgroup separable. Let L denote the fundamental group of the link consisting of a chain of 4 circles. It is shown that L is not subgroup separable. Furthermore, it is sh...

In this paper, the relationships of induced L-convex spaces with L-hull operators, product spaces, and quotient are discussed. It is shown that structure exactly by convex structure. Moreover, sub-S1, sub- S2, S2 S3 separation axioms introduced in spaces. Some properties them investigated.

A game on a convex geometry is a real-valued function de®ned on the family L of the closed sets of a closure operator which satis®es the ®nite Minkowski±Krein±Milman property. If L is the Boolean algebra 2 N then we obtain a n-person cooperative game. We will introduce convex and quasi-convex games on convex geometries and we will study some properties of the core and the Weber set of these games.

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 ...