Topological Cone Research Articles

Cones of traces arising from AF $C^{*}$-algebras

We characterize the topological non-cancellative cones that can be expressed as projective limits of finite powers of [0,\infty] . For metrizable cones, these are also the cones of lower semicontinuous extended-valued traces on approximately finite-dimensional (AF) C^{*} -algebras. Our main result may be regarded as a generalization of the fact that any Choquet simplex is a projective limit of finite-dimensional simplices. To obtain our main result, we first establish a duality between certain non-cancellative topological cones and Cuntz semigroups with real multiplication. This duality extends the duality between compact convex sets and complete order unit vector spaces to a non-cancellative setting.

Completing simple valuations in -categories

We prove that Keimel and Lawson's K-completion Kc of the simple valuation monad Vs defines a monad Kc∘Vs on each ▪-category K. We also characterise the Eilenberg-Moore algebras of Kc∘Vs as the weakly locally convex K-cones, and its algebra morphisms as the continuous linear maps.In addition, we explicitly describe the distributive law of Vs over Kc, which allows us to show that the K-completion of any locally convex (resp., weakly locally convex, locally linear) topological cone is a locally convex (resp., weakly locally convex, locally linear) K-cone.We also give an example – the Cantor tree with a top – that shows the dcpo-completion of the simple valuations is not the D-completion of the simple valuations in general, where D is the category of monotone convergence spaces and continuous maps.

CONE PHOTORECEPTOR INTEGRITY ASSESSED WITH ADAPTIVE OPTICS IMAGING AFTER LASER POINTER-INDUCED RETINAL INJURY.

To examine the three-dimensional foveal cone photoreceptor structure in a patient who had suffered laser pointer-induced retinal injury. Patient underwent standard fundus photography and clinical spectral domain optical coherence tomography imaging. High-resolution imaging was performed using an adaptive optics-optical coherence tomography-scanning laser ophthalmoscope. Adaptive optics imaging revealed loss of inner and outer segments of cone photoreceptors whereas the anterior retinal layers appeared healthy. Analysis of cone topology showed an increase in Voronoi domain area and a less regular hexagonal packing structure closer to the lesion site. Exposure to laser pointer radiation, however brief, can result in damage to the retina. Here, repeated imaging nine months later showed a decrease in the size of the lesions (ranging from 3.7 to 23.9%) compared with the first time point. However, the longer-term prognosis is likely permanent scarring.

Monoids, their boundaries, fractals and C*-algebras

AbstractIn this note we establish some connections between the theory of self-similar fractals in the sense of John E. Hutchinson (cf. [3]), and the theory of boundary quotients of C*-algebras associated to monoids. Although we must leave several important questions open, we show that the existence of self-similar ℳ-fractals for a given monoid ℳ, gives rise to examples of C*-algebras (1.9) generalizing the boundary quotients \partial C_\lambda ^*(\mathcal{M}) discussed by X. Li in [4, §7, p. 71]. The starting point for our investigations is the observation that the universal boundary of a finitely 1-generated monoid carries naturally two topologies. The fine topology plays a prominent role in the construction of these boundary quotients, while the cone topology can be used to define canonical measures on the attractor of an ℳ-fractal for a finitely 1-generated monoid ℳ.

Algebras of the Extended Probabilistic Powerdomain Monad

We investigate the Eilenberg-Moore algebras of the extended probabilistic powerdomain monad Vw over the category TOP0 of T0 topological spaces and continuous maps. We prove that every Vw-algebra in our setting is a weakly locally convex sober topological cone, and that a map is the structure map of a Vw-algebra if and only if it is continuous and sends every continuous valuation to its unique barycentre. Conversely, for locally linear sober cones—a strong form of local convexity—, the mere existence of barycentres entails that the barycentre map is the structure map of a Vw-algebra; moreover the algebra morphisms are exactly the linear continuous maps in that case.We also examine the algebras of two related monads, the simple valuation monad Vf and the point-continuous valuation monad Vp. In TOP0 their algebras are fully characterised as weakly locally convex topological cones and weakly locally convex sober topological cones, respectively. In both cases, the algebra morphisms are continuous linear maps between the corresponding algebras.

On semistability of CAT(0) groups

Does every one-ended CAT(0) group have semistable fundamental group at infinity? As we write, this is an open question. Let G be such a group acting geometrically on the proper CAT(0) space X . In this paper we show that in order to establish a positive answer to the question it is only necessary to check that any two geodesic rays in X are properly homotopic. We then show that if the answer to the question is negative, with (G, X) a counter-example, then the boundary of X, \partial X with the cone topology, must have a weak cut point. This is of interest because a theorem of Papasoglu and the second-named author [10] has established that there cannot be an example of (G, X) where \partial X has a cut point. Thus, the search for a negative answer comes down to the difference between cut points and weak cut points. We also show that the Tits ball of radius \frac{\pi}{2} about that weak cut point is a “cut set” in the sense that it separates \partial X . Finally, we observe that if a negative example (G, X) exists then G is rank 1.

On the geometry and fixed point free mappings in hyperbolic spaces

In the paper, we focus on two problems in geodesic hyperbolic spaces. First, we will consider the behavior of Picard iterative sequences of fixed point free nonexpansive mappings in spaces of constant curvature. Second, motivated by Hotchkiss (Proc Am Math Soc 125:1903–1912, 1997) we will show under which additional assumptions the Gromov boundary of a hyperbolic space equipped with the topology defined by the visual metric coincides with the geodesic boundary equipped with the cone topology.

Embedding cones over trees into their symmetric products

Given a metric continuum X and a positive integer n, let Fn(X) be the hyperspace of nonempty sets of X with at most n points and let Cone(X) be the topological cone of X. We say that a continuum X is cone-embeddable in Fn(X) if there is an embedding h from Cone(X) into Fn(X) such that h(x,0)={x} for each x in X. In this paper, we characterize trees X that are cone-embeddable in Fn(X).

Ultrafast spin-polarized electron dynamics in the unoccupied topological surface state of Bi2Se3

The three-dimensional topological insulator Bi2Se3 presents two cone-like dispersive topological surface states centered at the point. One of them is unoccupied in equilibrium conditions and located 1.8 eV above the other one lying close to the Fermi level. In this work we employ time- and angle-resolved photoemission spectroscopy with circularly polarized pump photons to selectively track the spin dynamics of the empty topological states. We observe that spin-polarized electrons flow along the topological cone and recombine towards the unpolarized bulk states on a timescale of few tens of femtoseconds. This provides direct evidence of the capability to trigger a spin current with circularly polarized light.

Dual-Functional Tamm-Dancoff Approximation: A Convenient Density Functional Method that Correctly Describes S1/S0 Conical Intersections.

Time-dependent Kohn-Sham density functional theory has been used successfully to compute vertical excitation energies, especially for large molecular systems. However, the lack of double excitation character in the excited amplitudes produced by linear response in the adiabatic approximation holds it back from broader applications in photochemistry; for example, it shows (3N - 7)-dimensional conical intersection seams (where N is the number of atoms) between ground and excited states, although the correct dimensionality is 3N - 8. In this letter, we present a new, conceptually simple, easy-to-implement, and easy-to-use way to employ time-dependent Kohn-Sham density functional theory that has global accuracy comparable with the conventional single-functional version and that recovers the double cone topology of the potential energy surfaces at S1/S0 conical intersection seams. The new method is called the dual-functional Tamm-Dancoff approximation (DF-TDA).

Locally convex projective limit cones

Abstract Using a family of ordered cones, we define a suitable order to investigate projective limit topologies for cones. Under the strict separation property, the projective limit cone topology is proved to be equivalent to some of the X-topologies embedded on a subcone of the product cone in its topology. Also, we discuss the cones of convex subsets in projective limits; in particular, we show that a cone of convex subsets of the projective limit cone in its topology carries the projective limit of the cones of convex subsets in its components.

Interception in differential pursuit/evasion games

A qualitative criterion for a pursuer to intercept a target in a class of differential games is obtained in terms of future cones: Topological cones that contain all attainable trajectories of target or interceptor originating from an initial position. An interception solution exists after some initial time iff the future cone of the target lies within the future cone of the interceptor. The solution may be regarded as a kind of Nash equilibrium. This result is applied to two examples: 1. The game of Two Cars: The future cone condition is shown to be equivalent to conditions for interception obtained by Cockayne. 2. Satellite warfare: The future cone for a spacecraft or direct-ascent antisatellite weapon (ASAT) maneuvering in a central gravitational field is obtained and is shown to equal that for a spacecraft which maneuvers solely by means of a single velocity change at the cone vertex. The latter result is illustrated with an analysis of the January 2007 interception of the FengYun-1C spacecraft.

Embedding cones over arc-smooth continua into their hyperspace of subcontinua

Given a metric continuum X, let C(X) be the hyperspace of subcontinua of X and Cone(X) the topological cone of X. We say that a continuum X is ordered cone-embeddable in C(X) provided that there is an embedding h from Cone(X) into C(X) such that, for each x in X, h(x,0)={x} and h(x,s) is properly contained in h(x,t) whenever s<t. In this paper, we prove that arc-smooth continua X are ordered cone-embeddable in C(X).

‘Parabolic’ trapped modes and steered Dirac cones in platonic crystals

This paper discusses the properties of flexural waves governed by the biharmonic operator, and propagating in a thin plate pinned at doubly periodic sets of points. The emphases are on the design of dispersion surfaces having the Dirac cone topology, and on the related topic of trapped modes in plates for a finite set (cluster) of pinned points. The Dirac cone topologies we exhibit have at least two cones touching at a point in the reciprocal lattice, augmented by another band passing through the point. We show that these Dirac cones can be steered along symmetry lines in the Brillouin zone by varying the aspect ratio of rectangular lattices of pins, and that, as the cones are moved, the involved band surfaces tilt. We link Dirac points with a parabolic profile in their neighbourhood, and the characteristic of this parabolic profile decides the direction of propagation of the trapped mode in finite clusters.

Compactifications whose cone can be embedded into their hyperspace of subcontinua

Given a metric continuum X, let C(X) be the hyperspace of subcontinua of X and Cone(X) the topological cone of X. We say that a continuum X is cone-embeddable in C(X) provided that there is an embedding h from Cone(X) into C(X) such that h(x,0)={x} for each x in X. In this paper, we present some results concerning compactifications X of rays, union of two rays, and real lines which are cone-embeddable in C(X).

Tuning the pseudo-Zeeman splitting in graphene cones by magnetic field

The effect of a homogeneous magnetic field on the energy spectrum of gapped graphene cones with long range charged Coulomb impurity is investigated within the framework of the perturbation theory. It is found that, besides lifting the degeneracy of angular momentum channels due to topological defects, the size of the corresponding splitting is enhanced by the introduction of a homogeneous magnetic field. This indicates that the level splitting may be controlled by the strength of a magnetic field in topological cones. Therefore, our results show that the magnetic field may be used as a tool to tune the pseudo-Zeeman splitting in graphene cones.

Locally Convex Product and Direct Sum Cones

In Motallebi and Saiflu [3], products and direct sums of cones have been investigated based on convex quasi-uniform structures. In this paper, we define the notion of product and direct sum cone topologies based on a suitable order relation and discuss the duality properties. Using the polars of neighborhoods, we define some of the X-topologies which are isomorphic to each other. In the condition of the strict separation property, these topologies will be isomorphic to a product (or direct sum) cone topology. This leads us to study weak-topologies, topologies of pointwise convergence and the finest X-topologies.

Embedding cones in hyperspaces

Given a metric continuum X, let C(X) be the hyperspace of subcontinua of X and Cone(X) the topological cone of X. We say that a continuum X is cone-embeddable in C(X) provided that there is an embedding h from Cone(X) into C(X) such that h(x,0)={x} for each x in X. In this paper, we present some results concerning continua X that are cone-embeddable in C(X).

Dugundji’s theorem for cone metric spaces

In this work, we will prove the Dugundji extension theorem for the cone metric space. It is heavily reliant on the paracompactness of the cone topology that is proved by Ayse Sönmez in the paper Sönmez (2010) [11].

Topological cones: functional analysis in a T0-setting

Already in his PhD Thesis on compact Abelian semigroups under the direction of Karl Heinrich Hofmann the author was lead to investigate locally compact cones (Keimel in Math. Z. 99:205–428, 1967). This happened in the setting of Hausdorff topologies. The theme of topological cones has been reappearing in the author’s work in a non-Hausdorff setting motivated by the needs of mathematical models for a denotational semantics of languages combining probabilistic and nondeterministic choice. This is in the line of common work with Karl Heinrich Hofmann in Continuous Lattices and Domains (Gierz et al. in Encyclopedia of Mathematics and its Applications, vol. 93, 2003).