We study the geometry of Gorenstein curve singularities genus two, and their stable limits. These come in two families, corresponding to either Weierstrass or conjugate points on a semistable tail. For every $1\leq m <n$, stability condition - using one markings as reference point, therefore not $\mathfrak S_n$-symmetric defines proper Deligne-Mumford stacks $\overline{\mathcal M}_{2,n}^{(m)}$ ...

We started our research with the intent on answering the following question: can we find a way to calculate all the subgroups of the symmetric group. This is easier said that done, as the number of subgroups for a symmetric group grows quickly with each successive symmetric group. This problem can actually be simplified to finding the subgroup conjugacy classes. So now we have the slightly diff...

Symmetric disk graphs are often used to model wireless communication networks. Given a set S of n points in R (representing n transceivers) and a transmission range assignment r : S → R, the symmetric disk graph of S (denoted SDG(S)) is the undirected graph over S whose set of edges is E = {(u, v) | r(u) ≥ |uv| and r(v) ≥ |uv|}, where |uv| denotes the Euclidean distance between points u and v. ...

—Conjugate symmetry is an entirely new approach to symmetric Boolean functions that can be used to extend existing methods for handling symmetric functions to a much wider class of functions. These are functions that currently appear to have no symmetries of any kind. Such symmetries occur widely in practice. In fact, we show that even the simplest circuits exhibit conjugate symmetry. To demons...

Let G = (V,E) be a graph and let C be a finite set of colours. An edge C-colouring of G is a function λ : E → C. If G is symmetric and λ(x, y) = λ(y, x), for all (x, y) ∈ E, then we say that λ is a symmetric edge C-colouring. Let n be a natural number and let Cn = {f, ci : i < n} be a set of n+1 colours. Using a probabilistic construction and an application of the Local Lemma we prove that ther...

Ž . A Steiner minimum tree SMT in the rectilinear plane is the shortest length tree interconnecting a set of points, called the regular points, possibly using Ž . additional vertices. A k-size Steiner minimum tree kSMT is one that can be split into components where all regular points are leaves and all components have at most k leaves. The k-Steiner ratio in the rectilinear plane, r , is the in...

We give combinatorial proofs of two identities from the representation theory of the partition algebra CAk(n), n ≥ 2k. The first is nk = ∑ λ f mk , where the sum is over partitions λ of n, fλ is the number of standard tableaux of shape λ, and mk is the number of “vacillating tableaux” of shape λ and length 2k. Our proof uses a combination of Robinson-Schensted-Knuth insertion and jeu de taquin....