Topological Cone Research Articles

Dirichlet Problem at Infinity for $\mathcal A$ -Harmonic Functions

We study the Dirichlet problem at infinity for \(\mathcal A\)-harmonic functions on a Cartan–Hadamard manifold M and give a sufficient condition for a point at infinity x0∈M(∞) to be \(\mathcal A\)-regular. This condition is local in the sense that it only involves sectional curvatures of M in a set U∩M, where U is an arbitrary neighborhood of x0 in the cone topology. The results apply to the Laplacian and p-Laplacian, 1<p<∞, as special cases.

On the positive tracial cones of simple stably projectionless [formula omitted]-algebras

It is shown that any topological cone, with a base consisting of a metrizable Choquet simplex, arises as the positive tracial cone of a simple stably projectionless C * -algebra inductive limit of certain specified non-unital algebras of matrix-valued functions over the interval [0,1].

Positive solutions for large elliptic systems of interacting species groups by cone index methods

This article considers the existence of positive solutions for various systems of four nonlinear coupled elliptic partial differential equations subjected to zero Dirichlet boundary conditions. In applications, the components can be interpreted as concentrations of four interacting populations or chemicals. The species are divided into two groups which interact in prey-predator or cooperating Volterra–Lotka type of relations. Within each group, the species can interact in other possibilities. Topological cone index method is used for proving the existence of positive coexistence solutions. Sufficient conditions are found in terms of the signs of the principal eigenvalues of various simple related operators.

Some results on Hahn–Banach-type theorems for continuous D-cones

Like the extended non-negative reals R ̄ + equipped with the Scott topology, there are other real topological cones such that the specialisation order yields a directed complete partially ordered set (dcpo). We will call them d-cones. Further examples are the extended probabilistic powerdomain, the set of all lower semicontinuous functions f : X → R ̄ + for any topological space X and arbitrary products of given d-cones. The dual cone C ∗ for a given d-cone C consists of all linear continuous functions Λ : C → R ̄ + . With respect to the pointwise order, addition and scalar multiplication the dual cone becomes also a d-cone. We are interested in obtaining results with our concept of d-cones that are comparable to Hahn–Banach-type theorems in functional analysis. Indeed, we can prove an Extension Theorem and a Separation Theorem for the continuous d-cones. In particular, the second implies that the elements of the dual cone C ∗ separate the points of C. As a consequence of the Extension Theorem, we obtain a Sum Theorem for continuous d-cones. We will give some sufficient conditions when the previous examples of d-cones are continuous and have an additive way-below relation.

Spaces of Valuations

ABSTRACT:Valuations are measurelike functions mapping the open sets of a topological space X into positive real numbers. They can be classified into finite, point continuous, and Scott continuous valuations. We define corresponding spaces of valuations VfX⊂ VpX⊂VX. The main results of the paper are that VpX is the soberification of VfX, and that VpX is the free sober locally convex topological cone over X. From this universal property, the notion of the integral of a real‐valued function over a Scott continuous valuation can be easily derived. The integral is used to characterize the spaces VpX and VX as dual spaces of certain spaces of real‐valued functions on X.

Convergence in the part metric for discrete dynamical systems in ordered topological cones

Convergence in the part metric for discrete dynamical systems in ordered topological cones

Topological cones, operator equations, and the Newton-Kantorovich method

Topological cones, operator equations, and the Newton-Kantorovich method

The cone topology on a manifold without focal points

The cone topology on a manifold without focal points

The Martin boundaries of equivalent sheaves

The Martin compactification of X defined by a Brelot sheaf H 1 satisfying proportionality is shown to be the same as for H 2 if the sheaves agree outside a compact set. Minimal points coincide and hence S 1 + and S 2 + are isomorphic topological cones. Nakai’s result on the extension to X of a function harmonic outside a compact set is extended to Bauer’s theory. The connected components of the Martin boundary Δ correspond to the ends of X which are related to direct decomposition of the cone H + .

On the nature of isolated asymptotic singularities of solutions of a family of quasi-linear elliptic PDE's on a Cartan–Hadamard manifold

Let $M$ be a Cartan-Hadamard manifold with sectional curvature satisfying $-b^2\leq K\leq -a^2<0$, $b\geq a>0.$ Denote by $\partial_{\infty}M$ the asymptotic boundary of $M$ and by $\bar M:= M\cup\partial_\infty M$ the geometric compactification of $M$ with the cone topology. We investigate here the following question: Given a finite number of points $p_{1},...,p_{k}\in\partial_\infty M,$ if $u\in C^{\infty}(M)\cap C^{0}\left( \bar{M}\backslash\left\{ p_{1},...,p_{k}\right\} \right) $ satisfies a PDE $\mathcal Q(u)=0$ in $M$ and if $u|_{\partial_\infty M\setminus\left\{p_{1},...,p_{k}\right\}}$ extends continuously to $p_{i},$ $i=1,...,k,$ can one conclude that $u\in C^{0}\left( \bar{M}\right )?$ When $\dim M=2$, for $\mathcal Q$ belonging to a linearly convex space of quasi-linear elliptic operators $\mathcal{S}$ of the form $$ \mathcal{Q}(u)=\operatorname{div}\left( \frac{\mathcal{A}(|\nabla u|)}{|\nabla u|} \nabla u \right)=0, $$ where $\mathcal{A}$ satisfies some structural conditions, then the answer is yes provided that $\mathcal{A}$ has a certain asymptotic growth. This condition includes, besides the minimal graph PDE, a class of minimal type PDEs. In the hyperbolic space $ \mathbb{H}^n$, $n\ge 2,$ we are able to give a complete answer: we prove that $\mathcal{S}$ splits into two disjoint classes of minimal type and $p-$Laplacian type PDEs, $p>1,$ where the answer is yes and no respectively. These two classes are determined by the asymptotic behaviour of $\mathcal A.$ Regarding the class where the answer is negative, we obtain explicit solutions having an isolated non removable singularity at infinity.