Partial Regularity of solutions to the Four-dimensional Navier-Stokes equations at the first blow-up time

The solutions of incompressible Navier-Stokes equations in four spatial dimensions are considered. We prove that the two-dimensional Hausdorff measure of the set of singular points at the first blow-up time is equal to zero.Comment: 19 pages, a comment regarding five or higher dimensional case is added in Remark 1.3. accepted by Comm. Math. Phy

Independent analysis of the orbits of Pioneer 10 and 11

Independently developed orbit determination software is used to analyze the orbits of Pioneer 10 and 11 using Doppler data. The analysis takes into account the gravitational fields of the Sun and planets using the latest JPL ephemerides, accurate station locations, signal propagation delays (e.g., the Shapiro delay, atmospheric effects), the spacecrafts' spin, and maneuvers. New to this analysis is the ability to utilize telemetry data for spin, maneuvers, and other on-board systematic effects. Using data that was analyzed in prior JPL studies, the anomalous acceleration of the two spacecraft is confirmed. We are also able to put limits on any secondary acceleration (i.e., jerk) terms. The tools that were developed will be used in the upcoming analysis of recently recovered Pioneer 10 and 11 Doppler data files.Comment: 22 pages, 5 figures; accepted for publication in IJMP

Conflict-Free Coloring of Planar Graphs

A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well-studied in graph theory. Here we study the natural problem of the conflict-free chromatic number chi_CF(G) (the smallest k for which conflict-free k-colorings exist). We provide results both for closed neighborhoods N[v], for which a vertex v is a member of its neighborhood, and for open neighborhoods N(v), for which vertex v is not a member of its neighborhood. For closed neighborhoods, we prove the conflict-free variant of the famous Hadwiger Conjecture: If an arbitrary graph G does not contain K_{k+1} as a minor, then chi_CF(G) <= k. For planar graphs, we obtain a tight worst-case bound: three colors are sometimes necessary and always sufficient. We also give a complete characterization of the computational complexity of conflict-free coloring. Deciding whether chi_CF(G)<= 1 is NP-complete for planar graphs G, but polynomial for outerplanar graphs. Furthermore, deciding whether chi_CF(G)<= 2 is NP-complete for planar graphs G, but always true for outerplanar graphs. For the bicriteria problem of minimizing the number of colored vertices subject to a given bound k on the number of colors, we give a full algorithmic characterization in terms of complexity and approximation for outerplanar and planar graphs. For open neighborhoods, we show that every planar bipartite graph has a conflict-free coloring with at most four colors; on the other hand, we prove that for k in {1,2,3}, it is NP-complete to decide whether a planar bipartite graph has a conflict-free k-coloring. Moreover, we establish that any general} planar graph has a conflict-free coloring with at most eight colors.Comment: 30 pages, 17 figures; full version (to appear in SIAM Journal on Discrete Mathematics) of extended abstract that appears in Proceeedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2017), pp. 1951-196

Insights into the time course of evidentiality processing in Turkish heritage speakers using a self-paced reading task

Introduction: Studies with heritage language speakers (HLS) have often used offline measurements, investigating the post-interpretive effects which emerge after processing has been completed. Relatively few studies have investigated heritage language processing using time-sensitive methods that allow the collection of evidence regarding real-time language processing rather than post-interpretive judgments. Using a self-paced-reading paradigm, we aimed to expand our understanding of HLS language processing by investigating evidentiality-the linguistic marking of information source, which is grammatically expressed in Turkish, but not in English. Method: Participants were 54 bilingual speakers of Turkish and English: 24 HLS (English onset: 0-5 yrs) and 30 emigrant Turkish speakers (ES) who grew up in Turkey before emigrating to Australia (English onset = 6-17 yrs). Participants read sentences with evidential-marked verb forms that either matched or mismatched to the information source context. Word-by-word reading times and end-of-sentence acceptability judgment speed and accuracy were measured. Results: The results showed that although the HLS' responses were slower and less accurate than the ES in both reading times and end-of-sentence acceptability judgments, they showed similarities in online processing patterns. Both groups were faster at reading the mismatching sentences compared to the matching sentences; however, this pattern emerged during the time course of reading first for the indirect condition for the ES, and only later for the direct condition and for the HLS for both evidential conditions. Only HLS read faster in the target region with the direct evidential that is shown to be acquired earlier in childhood, than they did for the indirect evidential which is mastered later. In contrast, the end-of-sentence judgment data showed that while the ES group responded faster to matching direct sentences than matching indirect, this effect was missing for the HLS. Nevertheless, there were similar patterns for accuracy across evidential conditions: both groups were more accurate with the direct evidential. Discussion: Overall, the use of the self-paced-reading paradigm allowed insights into HLS' evidentiality processing above and beyond their generally slower and less accurate processing compared to the reference group. This study provides further evidence for differences in the patterns observed using online vs. post interpretive measures in HLS, reinforcing the importance of combining these methodologies for further understanding of HLS competence and performance.</p

Insights into the time course of evidentiality processing in Turkish heritage speakers using a self-paced reading task

IntroductionStudies with heritage language speakers (HLS) have often used offline measurements, investigating the post-interpretive effects which emerge after processing has been completed. Relatively few studies have investigated heritage language processing using time-sensitive methods that allow the collection of evidence regarding real-time language processing rather than post-interpretive judgments. Using a self-paced-reading paradigm, we aimed to expand our understanding of HLS language processing by investigating evidentiality-the linguistic marking of information source, which is grammatically expressed in Turkish, but not in English.MethodParticipants were 54 bilingual speakers of Turkish and English: 24 HLS (English onset: 0-5 yrs) and 30 emigrant Turkish speakers (ES) who grew up in Turkey before emigrating to Australia (English onset = 6-17 yrs). Participants read sentences with evidential-marked verb forms that either matched or mismatched to the information source context. Word-by-word reading times and end-of-sentence acceptability judgment speed and accuracy were measured.ResultsThe results showed that although the HLS' responses were slower and less accurate than the ES in both reading times and end-of-sentence acceptability judgments, they showed similarities in online processing patterns. Both groups were faster at reading the mismatching sentences compared to the matching sentences; however, this pattern emerged during the time course of reading first for the indirect condition for the ES, and only later for the direct condition and for the HLS for both evidential conditions. Only HLS read faster in the target region with the direct evidential that is shown to be acquired earlier in childhood, than they did for the indirect evidential which is mastered later. In contrast, the end-of-sentence judgment data showed that while the ES group responded faster to matching direct sentences than matching indirect, this effect was missing for the HLS. Nevertheless, there were similar patterns for accuracy across evidential conditions: both groups were more accurate with the direct evidential.DiscussionOverall, the use of the self-paced-reading paradigm allowed insights into HLS' evidentiality processing above and beyond their generally slower and less accurate processing compared to the reference group. This study provides further evidence for differences in the patterns observed using online vs. post interpretive measures in HLS, reinforcing the importance of combining these methodologies for further understanding of HLS competence and performance

Coordinated Motion Planning: Reconfiguring a Swarm of Labeled Robots with Bounded Stretch

We present a number of breakthroughs for coordinated motion planning, in which the objective is to reconfigure a swarm of labeled convex objects by a combination of parallel, continuous, collision-free translations into a given target arrangement. Problems of this type can be traced back to the classic work of Schwartz and Sharir (1983), who gave a method for deciding the existence of a coordinated motion for a set of disks between obstacles; their approach is polynomial in the complexity of the obstacles, but exponential in the number of disks. Despite a broad range of other non-trivial results for multi-object motion planning, previous work has largely focused on sequential schedules, in which one robot moves at a time, with objectives such as the number of moves; attempts to minimize the overall makespan of a coordinated parallel motion schedule (with many robots moving simultaneously) have defied all attempts at establishing the complexity in the absence of obstacles, as well as the existence of efficient approximation methods. We resolve these open problems by developing a framework that provides constant-factor approximation algorithms for minimizing the execution time of a coordinated, parallel motion plan for a swarm of robots in the absence of obstacles, provided their arrangement entails some amount of separability. In fact, our algorithm achieves constant stretch factor: If all robots want to move at most d units from their respective starting positions, then the total duration of the overall schedule (and hence the distance traveled by each robot) is O(d). Various extensions include unlabeled robots and different classes of robots. We also resolve the complexity of finding a reconfiguration plan with minimal execution time by proving that this is NP-hard, even for a grid arrangement without any stationary obstacles. On the other hand, we show that for densely packed disks that cannot be well separated, a stretch factor Omega(N^{1/4}) may be required. On the positive side, we establish a stretch factor of O(N^{1/2}) even in this case. The intricate difficulties of computing precise optimal solutions are demonstrated by the seemingly simple case of just two disks, which is shown to be excruciatingly difficult to solve to optimality