We consider the three forms of self-duality that can be exhibited by a planar graph G, map self-duality, graph self-duality and matroid self-duality. We show how these concepts are related with each other and with the connectivity of G. We use the geometry of self-dual polyhedra together with the structure of the cycle matroid to construct all self-dual graphs. 1.1. Forms of Self-duality. Given...

Let C be a binary linear code and suppose that its automorphism group contains a non trivial subgroup G. What can we say about C knowing G? In this paper we collect some answers to this question in the cases G ∼= Cp, G ∼= C2p and G ∼= D2p (p an odd prime), with a particular regard to the case in which C is self-dual. Furthermore we generalize some methods used in other papers on this subject. F...

Finally the Lagranage dual function is given by g(~λ, ~ν) = inf~x L(~x,~λ, ~ν) We now make a couple of simple observations. Observation. When L(·, ~λ, ~ν) is unbounded from below then the dual takes the value −∞. Observation. g(~λ, ~ν) is concave1 as it is the infimum of a set of affine2 functions. If x is feasible solution of program (10.2)(10.4), then we have the following L(x,~λ, ~ν) = f0(x)...

Given a preference system (G, ≺) and an integral weight function defined on the edge set of G (not necessarily bipartite), the maximum-weight stable matching problem is to find a stable matching of (G, ≺) with maximum total weight. We study this N P-hard problem using linear programming and polyhedral approaches, and show that the Rothblum system for defining the fractional stable matching poly...

In this paper, we study the geometric RAC simultaneous drawing problem: Given two planar graphs that share a common vertex set but have disjoint edge sets, a geometric RAC simultaneous drawing is a straight-line drawing in which (i) each graph is drawn planar, (ii) there are no edge overlaps, and, (iii) crossings between edges of the two graphs occur at right-angles. We first prove that two pla...

We explore the relationship between the Ising model with inverse temperature β, the q = 2 random-cluster model with edge-parameter p = 1− e−2β , and the random even subgraph with edge-parameter 12p. For a planar graph G, the boundary edges of the + clusters of the Ising model on the planar dual of G forms a random even subgraph of G. A coupling of the random even subgraph of G and the q = 2 ran...

In this note, we prove that H is a g-closed subgroup of the compact group K if and only if every sequentially continuous character of the dual of H is continuous-in other words, if the dual of H is a so-called sk-group. We conclude that every countable subgroup of a compact group is g-closed, and thus give a positive answer to a problem of Dikranjan, Milan and Tonolo. We also show that the rela...

Now choose a Borel subgroup B in G and a maximal torus T in B so that, if B is the opposite Borel subgroup, then B ∩ H and B ∩ H are a Borel subgroups in H and T ∩H is a maximal torus in H . We follow the convention that the roots of B are positive. If λ ∈ X(T ), then indGB(−λ) is the dual Weyl module ∇G(λ ) with highest weight λ = −w0λ and lowest weight −λ. Its dual is the Weyl module ∆G(λ). I...

We give a counterexample to the following conjecture of Kauuman 2]: Conjecture Let K be an amphicheiral alternating knot. Then there exists a reduced alternating knot-diagram D of K, such that G(D) is isomorphic to G (D), where G(D) is a checkerboard-graph of D and G (D) its dual.