Regularly log-periodic functions and some applications

We prove a Tauberian theorem for the Laplace--Stieltjes transform and Karamata-type theorems in the framework of regularly log-periodic functions. As an application we determine the exact tail behavior of fixed points of certain type smoothing transforms

Segment length survival functions depending on the incoming direction and support inclination.

<p>The orientation domain has been split in 8 sectors. Segments were partitioned in sectors according to , the incoming direction of the ant. Graphs represent the corresponding survival distributions of the segments' length. The corresponding phase functions are shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0076531#pone-0076531-g007" target="_blank">Fig. 7</a>.</p

Typical example of a segmented trajectory.

<p>A portion of a trajectory is shown in dots (same ant as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0076531#pone-0076531-g001" target="_blank">Fig. 1A</a> for the null inclination). The segments resulting from algorithm 1 are shown as red lines.</p

Mean Square Displacement of ants on the horizontal plane ().

<p>The MSD () is shown as a function of the number of reorientation events along the trajectory. Points and dotted lines report the observed values (mean, 95% CI). The square-curve “ballistic” shape for few events are a trace of the direction persistence of ants, which disappears after some direction changes, yielding then a linear dependence of the MSD to the number of reorientation events, a well-known indication of diffusive behavior. For even larger numbers of reorientation events, the censoring effect of the domain frontier becomes dominant. The red line reports the MSD predicted by simulating the isotropic BW with the parameters estimated from the segmented trajectories for the null inclination.</p

Experiments Timetable.

<p>Experiments Timetable.</p

Effect of the support inclination on A — the average motion speed, B — the average residence time and C — the average trajectory lengths.

<p>These quantities are computed over 69 trajectories for each inclination. The inclination has a major impact on the motion speed, which in turn induces longer residence times. However, since ants move straighter towards the upper or lower edges when the inclination is steeper, their total trajectory length within the disks is lowered.</p

Observed (left) and predicted (right) statistics of the exit heading, for each inclination.

<p>The exit headings were computed for each ant as the direction from the starting point to the point where they exited the disk of radius 0.2(left, N = 69 per value). The arrow indicates the direction of the magnetic North (N). Simulations of trajectories were performed using the sector-based statistics of the phase function and mean free path (N = 100,000 simulations by inclination, see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0076531#s4" target="_blank"><i>Methods</i></a> for the simulation algorithm). The corresponding predicted exit headings distributions are shown on the right. The compatibility of the data with the predictions was tested using the standard Kolmogorov-Smirnov test for the null hypothesis that the two distributions were drawn from the same continuous distribution. This null hypothesis is not rejected for any inclination. The consistency of the data and the predicted distributions indicates that the impact of the inclination on the ant motion is comprehensively captured by the induced change in the BW model features.</p

Phase function depending on the incoming direction and support inclination.

<p>The orientation domain has been split into 8 sectors. Segments were partitioned in sectors according to , the incoming direction of the ant. Graphs represent the corresponding distributions of the next direction deviation . Red arrows are the average in each sector.</p

Effect of the support inclination on A — typical trajectories of ants, B — statistics of headings and C — statistics of positions.

<p>Slopes are indicated by labels , and illustrated by the (arbitrarily) increased length of the vectors on the left, heading uphill. Trajectories are 8.95, 2.28, 1.86, 2.30 and 0.67 meters long respectively. The statistics of headings, shown in B, compiles all ants' headings over time estimated every second. They show that ants are more and more often aligned with the steepest line as the inclination becomes steeper. Over time, this consistently biases the positions of ants towards locations uphill or downhill (up or down on the graphs A). This bias is summarized in C, using as proxies the absolute values of horizontal versus vertical coordinates of ant locations averaged over time for each ant (one dot per ant) for each inclination (red dot: and locations averaged over time and ants, red line: ). The higher values in indicate that the ants are on average further away from the center along the steepest line ( axis) than along the horizontal line ( axis), meaning that ants are more dispersed in the direction. Both types of distributions are significantly different from homogeneity even for the smallest incline .</p

Suppl. Fig. 2. Direction of cell migration after heterotypic and homotypic contact. from Cell segregation and border sharpening by Eph receptor: ephrin-mediated heterotypic repulsion

Eph receptor and ephrin signalling has a major role in cell segregation and border formation, and may act through regulation of cell adhesion, repulsion or tension. To elucidate roles of cell repulsion and adhesion, we combined experiments in cell culture assays with quantitations of cell behaviour which are used in computer simulations. Cells expressing EphB2, or kinase-inactive EphB2 (kiEphB2), segregate and form a sharp border with ephrinB1-expressing cells, and this is disrupted by knockdown of N-cadherin. Measurements of contact inhibition of locomotion reveal that EphB2-, kiEphB2- and ephrinB1-expressing cells have strong heterotypic and weak homotypic repulsion. EphB2 cells have a transient increase in migration after heterotypic activation, which underlies a shift in the EphB2–ephrinB1 border but is not required for segregation or border sharpening. Simulations with the measured values of cell behaviour reveal that heterotypic repulsion can account for cell segregation and border sharpening, and is more efficient than decreased heterotypic adhesion. By suppressing homotypic repulsion, N-cadherin creates a sufficient difference between heterotypic and homotypic repulsion, and enables homotypic cohesion, both of which are required to sharpen borders