Let Pn,k be the number of permutations π on [n] = {1, 2, . . . , n} such that the length of the longest increasing subsequences of π equals k, and let M2n,k be the number of matchings on [2n] with crossing number k. Define Pn(x) = ∑ k Pn,kx k and M2n(x) = ∑ k M2n,kx . We propose some conjectures on the log-concavity and q-log-convexity of the polynomials Pn(x) and M2n(x).

Let pn denote the number of self-avoiding polygons of length n on a regular three-dimensional lattice, and let pn(K) be the number which have knot type K. The probability that a random polygon of length n has knot type K is pn(K)/pn and is known to decay exponentially with length [1, 2]. Little is known rigorously about the asymptotics of pn(K), but there is substantial numerical evidence [3, 4...

Improving the visual appearance of coarse triangle meshes is usually done with graphics hardware with per-pixel shading techniques. Improving the appearance at silhouettes is inherently hard, as shading has only a small influence there and the geometry must be corrected. With the new geometry shader stage released with DirectX 10, the functionality to generate new primitives from an input primi...

Let pn be the number of consecutive positive summands and let qn be the number of consecutive negative summands that appear in the classical Riemann’s rearrangement of the alternating harmonic series to sum a prescribed real number s. Assume that s > log 2 and let x = (1/4)e2s. It is shown that the sequence qn is constant equal to 1, and that the values of pn become stabilized: Eventually pn = ...

The art gallery problem [2] asks for the smallest possible size of a point set S (the guards) to completely guard the interior of a simple polygon P (the art gallery). This thesis treats variations of the original problem that arise when we introduce a coloring of the guards. Rather than asking for the minimum number of guards we ask for the minimum number of colors. In [4] L. H. Erickson and S...

Let M be a compact surface without boundary, and n≥2. We analyse the quotient group Bn(M)/Γ2(Pn(M)) of braid Bn(M) by commutator subgroup Γ2(Pn(M)) pure Pn(M). If is different from 2-sphere S2, we prove that Bn(M)/Γ2(Pn(M))≅Pn(M)/Γ2(Pn(M))⋊φSn, crystallographic if only orientable. orientable, number results regarding structure Bn(M)/Γ2(Pn(M)). characterise finite-order elements this group, dete...

The common envelope interaction is responsible for evolved close binaries. Among them are a minority of central stars of planetary nebula (PN). Recent observational results, however, point to most PN actually being in binary systems. We therefore ask the question if it is feasible that most, or even all Galactic PN derive from a common envelope interaction. Our recent calculation finds that if ...

The l-component (edge) connectivity of a graph G, denoted by cκl(G) (cλl(G)), is the minimum number vertices (edges) whose removal from G results in disconnected with at least l components. pancake Pn popular underlying topology for distributed systems. In paper, we determine cκl(Pn) and cλl(Pn) 3 ≤ 5.