We consider the polynuclear growth model (PNG) in 1+1 dimension with flat initial condition and no extra constraints. Through the Robinson-Schensted-Knuth (RSK) construction, one obtains the multilayer PNG model, which consists of a stack of non-intersecting lines, the top one being the PNG height. The statistics of the lines is translation invariant and at a fixed position the lines define a p...

Rogue waves in (2+1)-dimensional three-wave resonant interactions are studied by the bilinear KP-reduction method. General rogue arising from a constant background, lump-soliton background and dark-soliton have been derived, their dynamics illustrated. For fundamental line-shaped, multi-rogue exhibit multiple intersecting lines. Higher-order could also be but they parallel distinctive patterns ...

Let L be a line intersecting the sidelines of triangle ABC at X , Y , Z respectively. The lines joining these intercepts to the centroid give rise to six more intercepts on the sidelines which lie on a conic Q(L , G). We show that this conic (i) degenerates in a pair of lines if L is tangent to the Steiner inellipse, (ii) is a parabola if L is tangent to the ellipse containing the trisection po...

This paper presents a powerful tool designed to extract characteristic lines, called 3D thin nets, from 3D volumetric images. 3D thin nets are the lines where the 3D grey level function is locally extremum in a given plane. Recently, we have shown that it is possible to characterize 2D thin nets as the crest lines of the image surface. This paper generalizes this approach to 3D data having thre...

In non-central cameras, the complete geometry of a 3D line is mapped to each corresponding projection, therefore each line can be theoretically recovered from a single view. However, the solution of this problem is ill-conditioned due to the lack of effective baseline between rays. This limitation prevents from a practical implementation of the approach if lines are not close to the visual syst...

The celebrated Erdős-Ko-Rado theorem shows that for n > 2k the largest intersecting k-uniform set family on [n] has size ( n−1 k−1 ) . It is natural to ask how far from intersecting larger set families must be. Katona, Katona and Katona introduced the notion of most probably intersecting families, which maximise the probability of random subfamilies being intersecting. We consider the most prob...

Suppose that 2d − 2 tangent lines to the rational normal curve z 7→ (1 : z : . . . : z) in d-dimensional complex projective space are given. It was known that the number of codimension 2 subspaces intersecting all these lines is always finite; for a generic configuration it is equal to the d-th Catalan number. We prove that for real tangent lines, all these codimension 2 subspaces are also real...

We analyze the problem of recovering the shape of a mirror surface. We generalize the results of [1], where the special case of planar and spherical mirror surfaces was considered, extending that analysis to any smooth surface. A calibrated scene composed of lines passing through a point is assumed. The lines are reflected by the mirror surface onto the image plane of a calibrated camera, where...

R d can be divided into a union of parallel d k ats of the form x g x g xk gk where the gi are constant Let C be a family of parallel d k dimensional convex sets meaning that each is contained in one of the above parallel d k ats We give a parameterization of the set of k ats in R such that the set of k ats which intersect in a point any set c C is convex Parameterizing the lines in R through h...

Let n,r and t be positive integers. A family F of subsets of [n]={1,2, . . . ,n} is called r-wise t-intersecting if |F1∩·· ·∩Fr|≥ t holds for all F1, . . . ,Fr ∈F . An r-wise 1-intersecting family is also called an r-wise intersecting family for short. An r-wise t-intersecting family F is called non-trivial if |⋂F∈F F |<t. Let us define the Brace–Daykin structure as follows. F BD = {F ⊂ [n] : |...