In this paper, we deal with the family I(p, k, r, n) of trinomial arcs defined as the set of roots of the trinomial equation zn = αzk + (1−α), with z = ρ eiθ is a complex number, α is a real number between 0 and 1 and k is an integer such that k = (2p + 1)n/(2r + 1), where n, p and r are three integers satisfying some conditions. These arcs I(p, k, r, n) are continuous arcs inside the unit disk...

Arrangements of (pseudo-)circles have already been studied in connection with algorithms in computational geometry. Thereby information on the numbers v k of intersection points contained in k circles seems to be particularly interesting. On each circle, there is an induced arrangement of arcs. This is why we begin by studying arrangements of arcs, and we arrive at a complete characterization o...

Let X be a complex algebraic variety, and X◦ ⊂ X be the smooth part of X. Consider the scheme L(X) of formal arcs in X. The C-points of L(X) are just maps D = SpecC[[t]] → X (see, for example, [DL] for a definition of L(X) as a scheme). Let L◦(X) be the open subscheme of arcs whose image is not contained in X \X◦. Fix an arc γ : D → X in L◦(X), and let L(X)γ be the formal neighborhood of γ in L...

In this paper, we generalize a result by Ball, Hill, Landjev and Ward on plane arcs to arcs with multiple points in spaces of arbitrary dimension. This result is further applied to the characterization of some non-Griesmer arcs in the 3-dimensional projective geometry over F4.

Let G(V; E) be a graph (either directed or undirected) with a non-negative length`(e) associated with each arc e in E. For two speciied nodes s and t in V , the k most vital arcs (or nodes) are those k arcs (nodes) whose removal maximizes the increase in the length of the shortest path from s to t. We prove that nding the k most vital arcs (or nodes) is NP-hard, even when all arcs have unit len...

First we discuss the description of inverse polynomial images of [−1, 1], which consists of two Jordan arcs, by the endpoints of the arcs only. The polynomial which generates the two Jordan arcs is given explicitly in terms of Jacobi’s theta functions. Then the main emphasis is put on the case where the two arcs are symmetric with respect to the real line. For instance it is demonstrated that t...

We study the strong gravitational lensing properties of galaxy clusters obtained from N-body simulations with standard ΛCDM cosmology. We have used the 32 most massive clusters from a simulation at various redshifts and ray-traced through the clusters to investigate the giant arcs statistics. We have found that ∼ 40 − 50% of the clusters that produce giant arcs give multiple arcs, which agrees ...

By a generalized arc we mean a continuum with exactly two non-separating points; an arc is a metrizable generalized arc. It is well known that any two arcs are homeomorphic (to the real closed unit interval); we show that any two generalized arcs are co-elementarily equivalent, and that co-elementary images of generalized arcs are generalized arcs. We also show that if f : X → Y is a function b...

We introduce an algorithm for computing Voronoi diagrams of points, straight-line segments and circular arcs in the two-dimensional Euclidean plane. Based on a randomized incremental insertion, we achieve a Voronoi algorithm that runs in expected time O(n log n) for a total of n points, segments and arcs, if at most a constant number of segments and arcs is incident upon every point. Our theore...

Quiet time arcs observed from the Greenland all-sky camera network have been ordered in corrected geomagnetic latitude/time mass plots according to the values of the Y and Z components of the IMF. Two different patterns of discrete quiet arcs occur. In one system the arcs are ordered along the statistical auroral oval; in the other, which we have called the polar cap pattern of discrete arcs, t...