Maximal point spaces of posets with relative lower topology

In domain theory, by a poset model of a T1 topological space X we usually mean a poset P such that the subspace Max(P) of the Scott space of P consisting of all maximal points is homeomorphic to X. The poset models of T1 spaces have been extensively studied by many authors. In this paper we investigate another type of poset models: lower topology models. The lower topology ?(P) on a poset P is one of the fundamental intrinsic topologies on the poset, which is generated by the sets of the form P\?x, x ? P. A lower topology poset model (poset LT-model) of a topological space X is a poset P such that the space Max?(P) of maximal points of P equipped with the relative lower topology is homeomorphic to X. The studies of such new models reveal more links between general T1 spaces and order structures. The main results proved in this paper include (i) a T1 space is compact if and only if it has a bounded complete algebraic dcpo LT-model; (ii) a T1 space is second-countable if and only if it has an ?-algebraic poset LT-model; (iii) every T1 space has an algebraic dcpo LT-model; (iv) the category of all T1 space is equivalent to a category of bounded complete posets. We will also prove some new results on the lower topology of different types of posets.

The Most Likely Object to be Seen Through a Window

We study data structures to answer window queries using stochastic input sequences. The first problem is the most likely maximal point in a query window: Let [Formula: see text] be constants, with [Formula: see text]. Let [Formula: see text] be a set of [Formula: see text] points in [Formula: see text], for some fixed [Formula: see text]. For [Formula: see text], each point in [Formula: see text] is associated with a probability [Formula: see text] of existence. A point [Formula: see text] in [Formula: see text] is on the maximal layer of [Formula: see text] if there is no other point [Formula: see text] in [Formula: see text] such that [Formula: see text]. Consider a random subset of [Formula: see text] obtained by including, for [Formula: see text], each point of [Formula: see text] independently with probability [Formula: see text]. For a query interval [Formula: see text], with [Formula: see text], we report the point in [Formula: see text] that has the highest probability to be on the maximal layer of [Formula: see text] in [Formula: see text] time using [Formula: see text] space. We solve a special problem as follows. A sequence [Formula: see text] of [Formula: see text] points in [Formula: see text] is given ([Formula: see text]), where each point [Formula: see text] has a probability [Formula: see text] of existence associated with it. Given a query interval [Formula: see text] and an integer [Formula: see text] with [Formula: see text], we report the probability of [Formula: see text] to be on the maximal layer of [Formula: see text] in [Formula: see text] time using [Formula: see text] space. The second problem we consider is the most likely common element problem. Let [Formula: see text] be the universe. Let [Formula: see text] be a sequence of random subsets of [Formula: see text] such that for [Formula: see text] and [Formula: see text], element [Formula: see text] is added to [Formula: see text] with probability [Formula: see text] (independently of other choices). Let [Formula: see text] be a fixed real number with [Formula: see text]. For query indices [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], with [Formula: see text] and [Formula: see text], we decide whether there exists an element [Formula: see text] with [Formula: see text] such that [Formula: see text] in [Formula: see text] time using [Formula: see text] space and report these elements in [Formula: see text] time, where [Formula: see text] is the size of the output.

Altered thoracolumbar position during application of craniocaudal spinal mobilisation in clinically sound leisure horses

Manual therapy techniques are commonly used by physiotherapists in the management of back pain to restore a pain-free range of motion and function in humans. However, there is a lack of research to support the proposed kinematic effects of manual therapy in the horse. This study investigated the kinematic effects of craniocaudal spinal mobilisation (CCSM) on the thoracolumbar spine in asymptomatic leisure horses. Markers were fixed to T10, T13, T17, L1, L3, the highest point of the wither and the tuber sacrale on thirteen horses that were positioned squarely. The CCSM technique consisted of two parts: (1) carpal flexion of either forelimb to 90° to maintain the horse in a tripod position, and (2) the application of a cranial to caudal force to the forehand via the ipsilateral point of the shoulder. Movement changes of the thoracolumbar markers from baseline to maximum flexion when the CCSM was applied was recorded as ‘depth’ (mm) relative to a fixed line drawn from the tuber sacrale to the maximal point of the withers. The change in angle (°) of each marker relative to the same markers was also recorded. Data were collected via video and analysed with Dartfish™ software. Increases in maximum thoracolumbar angle (P<0.05) and reductions in thoracolumbar depth (P<0.05) were found with CCSM. These results indicate CCSM induced flexion in the thoracolumbar spine, supporting its potential to improve range of motion and function in horses. Further studies to understand whether the changes observed during CCSM translate to treatment of back pain are warranted.

Well-filtered spaces and their dcpo models

A topological space X is called well-filtered if for any filtered family $\mathcal{F}$ of compact saturated sets and an open set U, ∩ $\mathcal{F}$ ⊆ U implies F ⊆ U for some F ∈ $\mathcal{F}$. Every sober space is well-filtered and the converse is not true. A dcpo (directed complete poset) is called well-filtered if its Scott space is well-filtered. In 1991, Heckmann asked whether every UK-admitting (the same as well-filtered) dcpo is sober. In 2001, Kou constructed a counterexample to give a negative answer. In this paper, for each T1 space X we consider a dcpo D(X) whose maximal point space is homeomorphic to X and prove that X is well-filtered if and only if D(X) is well-filtered. The main result proved here enables us to construct new well-filtered dcpos that are not sober (only one such example is known by now). A space will be called K-closed if the intersection of every filtered family of compact saturated sets is compact. Every well-filtered space is K-closed. Some similar results on K-closed spaces are also proved.