We show that for every $\eta \gt 0$ every sufficiently large $n$-vertex oriented graph $D$ of minimum semidegree exceeding $(1+\eta )\frac k2$ contains every balanced antidirected tree with $k$ edges and bounded maximum degree, if $k\ge \eta n$. In particular, this asymptotically confirms a conjecture of the first author for long antidirected paths and dense digraphs.

Further, we show that in the same setting, $D$ contains every $k$-edge antidirected subdivision of a sufficiently small complete graph, if the paths of the subdivision that have length $1$ or $2$ span a forest. As a special case, we can find all antidirected cycles of length at most $k$.

Finally, we address a conjecture of Addario-Berry, Havet, Linhares Sales, Reed, and Thomassé for antidirected trees in digraphs. We show that this conjecture is asymptotically true in $n$-vertex oriented graphs for all balanced antidirected trees of bounded maximum degree and of size linear in $n$.

The present study investigates cross-linguistic differences in the description of motion events using Holistic Spatial Semantics (HSS) as a theoretical framework. In this study, six short video stimuli featuring various motion situations were used to elicit narratives from 35 speakers of French and 29 speakers of Swedish. The proportions of semantic category Path linguistically expressed did not vary significantly between the groups. However, the French narratives had significantly less Manner and Direction expression. Furthermore, they included significantly more unbounded non-translocative events compared to the Swedish narratives. Partly, this was due to the tendency within the Swedish group to construe situations as bounded events that were clearly shown as unbounded in the stimuli. There was also a tendency within the French group to distribute the information about the same situation over two clauses. These findings lend further support to the relevance of distinguishing between the Path and Direction categories, as well as other key features of the HSS framework, such as the distinction between non-verbal motion situations and their linguistic construals.

The understanding and interpretation of simplicty in the context of a navigational path can be ambiguous. Different approaches to path planning are briefly juxtaposed, focusing on their simplicity-based distinctive features. This takes into consideration the response to drift caused by air or water currents and the geometric background. Special attention is paid to the straight-line-based solutions which are preferred by the human convenience and perception, recalling some well-known applications in navigation and including their (dis)advantages. In particular, this refers to the Euclidean simplicity applied at the cost of effectiveness in the standard search and rescue (SAR) patterns in the presence of relatively stronger position-dependent currents.

The aim of this article is to map the relationship between the main words that comprise the Homeric lexicon of roads, journeys, paths and travel. The central task is to explore the relationship between the words hodos and keleuthos; along the way, the article will also address other terms that appear less frequently, such as atarp(it)os and poros. The article first teases out a difference in sense between keleuthos in the singular and in the plural. The discussion of keleuthos provides a key distinction, namely between ‘object-concepts’ and ‘activity-concepts’, that proves valuable in discussing different senses of the word hodos. Rather than differentiating the words keleuthos and hodos as others have suggested, however, this distinction should be used to differentiate domains of meaning within each word. The result will be what might be conceived of as a four-part grid, with the two words hodos and keleuthos split into two distinct parts along the ‘activity-concept’/‘object-concept’ axis. Finally, concepts drawn from discussions of verbal aspect and philosophy of action are deployed heuristically to develop further the analysis of this semantic field.

Vehicle state estimation and path prediction, which usually involve Kalman filter and motion model, are critical tasks for intelligent driving. In vehicle state estimation, the comparative performance assessment, regarding accuracy and efficiency, of the unscented Kalman filter (UKF) and the extended Kalman filter (EKF) is rarely discussed. This paper is devoted to empirically evaluating the performance of UKF and EKF incorporating different motion models and investigating the models’ properties and the affecting factors in path prediction. Extensive real world experiments have been carried out and the results show that EKF and UKF have roughly identical accuracy in state estimation; however, EKF is faster than UKF generally; the fastest filter is about 2⋅6 times faster than the slowest. The path prediction experiments reveal that the velocity estimate and the used motion model affect path prediction; the more realistically the model reflects the vehicle's driving status, the more reliable its predictions.

The packaging of meaning in verbs varies widely across languages since verbs are free to encode different aspects of an event. At the same time, languages tend to display recurrent preferences in lexicalization, e.g. verb-framing vs. satellite-framing in motion. It has been noted, however, that the lexicalization patterns in motion are not carried over to the domain of vision, since gaze trajectory (‘visual path’) is coded outside the main verb even in verb-framed languages. This ‘typological split’ (Matsumoto 2001), however, is not universal. This article contains the first extensive report of verb-framing in the domain of vision based on data from Maniq (Austroasiatic, Thailand). The verbs are investigated using a translation questionnaire and a picture-naming task, which tap into subtle semantic detail. Results suggest the meanings of the verbs are shaped by universal constraints linked to earth-based verticality and bodily mechanics, as well as local factors such as the environment and the cultural scenarios of which looking is a salient part. A broader look across the whole Maniq verb lexicon reveals further cases of verbally encoded spatial notions and demonstrates a pervasive cross-domain systematicity, pointing to the language system itself as an important shaping force in lexicalization.

Let $\Omega $ be a connected open set in the plane and $\gamma : [0,1] \to \overline {\Omega }$ a path such that $\gamma ((0,1)) \subset \Omega $ . We show that the path $\gamma $ can be “pulled tight” to a unique shortest path which is homotopic to $\gamma $ , via a homotopy h with endpoints fixed whose intermediate paths $h_t$ , for $t \in [0,1)$ , satisfy $h_t((0,1)) \subset \Omega $ . We prove this result even in the case when there is no path of finite Euclidean length homotopic to $\gamma $ under such a homotopy. For this purpose, we offer three other natural, equivalent notions of a “shortest” path. This work generalizes previous results for simply connected domains with simple closed curve boundaries.

The Moving Ego and Moving Time metaphors have provided a fertile testing ground for the psychological reality of space–time metaphors. Despite this, little research has targeted the linguistic patterns used in these two mappings. To fill that gap, the current study uses corpus data to examine the use of motion verbs in two typologically different languages, English and Spanish. We first investigated the relative frequency of the two metaphors. Whereas we observed no difference in frequency in the Spanish data, our findings indicated that in English, Moving Time expressions are more prevalent than are Moving Ego expressions. Second, we focused on the patterns of use of the verbs themselves, asking whether well-known typological patterns in the expression of spatial motion would carry over to temporal motion. Specifically, we examined the frequencies of temporal uses of path and manner verbs in English and in Spanish. Contra the patterns observed in space, we observed a preference for path verbs in both languages, with this preference more strongly evident in English than in Spanish. In addition, our findings revealed greater use of motion verbs in temporal expressions in Spanish compared to English. These findings begin to outline constraints on the aspects of spatial conceptualization that are likely to be reused in the conceptualization of time.

The early P-T evolution of a garnet-kyanite gneiss from the Variscan Ulten Zone has been defined by detailed petrographic observations leading to the detection of chlorite-epidote- and staurolite-bearing assemblages enclosed in kyanite porphyroblasts. Calculations of P-T pseudosections in the system NaCaKFeMgAlSiHO allowed us to constrain the evolution of these relics to the earliest metamorphic stages. The overall path shows a P-T increase to a peak of 11–12 kbar and 600–650°C followed by decompressional heating to 720°C and 9–10 kbar and final cooling at 7 kbar, 550–600°C. This clockwise P-T path reflects crustal thickening and subsequent thermal decay related to the continental collision of the Variscan orogeny 330–340 Ma ago. Our study demonstrates that large kyanite porphyroblasts may preserve assemblages related to prograde metamorphic stages. As a result, the detection of mineral inclusions in kyanite can complement many similar studies on mineral suites hosted in garnet and zircon.

The Takab meta-ultramafic rocks of northwestern Iran crop out in association with a variety of metamorphic rocks and they can be divided into three types on the basis of dominant mineral assemblage and degree of serpentinization: metaperidotites, talc-serpentine rocks and serpentinites. The peridotite protoliths were mainly spinel-harzburgites and -dunites. The talc-serpentine rocks formed under low-grade conditions and their dominant mineral assemblage is lizardite/chrysotile + talc + calcite/dolomite. Pseudomorphic textures after olivine and pyroxene are typical. Later, lower amphibolite-facies metamorphism produced antigorite, clinochlore and tremolite. Late alteration, assisted by local shearing, produced more antigorite and late lizardite in the metaperidotites. The mineral chemistry of the serpentine polymorphs is variable. Compositions of coexisting spinel and olivine point to an ophiolitic origin. Thermometric estimates from co-existing relict olivine, pyroxenes and spinel are in the range 1000–1200ºC; clinopyroxene barometry yields a pressure of 24± 2.7 kbar for the mantle conditions. Exhumation and hydration followed, and the stability of lizardite/chrysotile constrains the T and XCO2 to <280ºC and <0.002, respectively. Subsequent prograde regional metamorphism took place in an orogenic setting, reaching peak temperatures in the range 410–540ºC at XCO2 >0.1. A Neoproterozoic–Early Cambrian age for the Takab protoliths is suggested on the basis of correlations with the Central Iran Zone. The strips of ultramafic rock in the Takab complex can be attributed to lithospheric remnants of the Proto-Tethyan ocean which closed during the Pan-African orogeny.

In cognitive linguistics, motion metaphors of time (e.g. Christmas is approaching, We left the crisis behind) have been actively studied during the last decades. In addition to motion verbs, prepositional expressions are an important element in such metaphors. This work combines insights from Cognitive Grammar and Conceptual Metaphor Theory to account for uses of English path prepositions in motion metaphors of time. It is argued that such expressions conceptualize time as a path where a mover is advancing. The nature of the mover varies: it can be an individual entity metaphorically in motion (e.g. We wentTHROUGHa hard winter), an extended period of time (e.g. The period of Daylight Saving Time goes onPASTSeptember), or the temporal profile of a process (e.g. I sleptTHROUGHthe afternoon). The nature of the mover correlates with the grammatical function of the path expression, which alternates between a complement of a motion verb and a free modifier. Accordingly, the time path can relate with figurative (motion-related) or veridical (duration-related) conceptualizations of time. While a spatial path is direction-neutral, a time path can, with few exceptions, only be scrutinized in the earlier $\rightarrow$ later direction.

The vertices of the kth power of a directed path with n vertices are exposed one by one to a selector in some random order. At any time the selector can see the graph induced by the vertices that have already appeared. The selector's aim is to choose online the maximal vertex (i.e. the vertex with no outgoing edges). We give upper and lower bounds for the asymptotic behaviour of pn,kn1/(k+1), where pn,k is the probability of success under the optimal algorithm. In order to derive the upper bound, we consider a model in which the selector obtains some extra information about the edges that have already appeared. We give the exact asymptotics of the probability of success under the optimal algorithm in this case. In order to derive the lower bound, we analyse a site percolation process on a sequence of the kth powers of a directed path with n vertices.

Mapping points on the Riemann sphere to points on the plane of complex numbers by stereographic projection has been shown to offer a number of advantages when applied to problems in navigation traditionally handled using spherical trigonometry. Here it is shown that the same approach can be used for problems involving great circles and/or rhumb lines and it results in simple, compact expressions suitable for efficient computer evaluation. Worked numerical examples are given and the values obtained are compared to standard references.

A direct subspace of a dynamic three-dimensional joint space is found to be useful for robot path planning in workspaces comprised of both static and dynamic objects. Dynamic descriptions permit positioning tables, automated guided vehicles, conveyors and cycling machine tools to be modeled by elements which translate or cycle along rectilinear paths. Graphical path planning procedures use cursor indicators to move the robot configuration point between the desired starting and final configurations while avoiding both the static and dynamic joint space obstacles.