The K(n)K(n)-Euler characteristic of extraspecial pp-groups.

Let p be an odd prime, and let K(n)* denote the nth Morava K-theory at the prime p; we compute the K(n)-Euler characteristic \chi_{n;p}(G) of the classifying space of an extraspecial p-group G. Equivalently, we get the number of conjugacy classes of commuting n-tuples in the group G. We obtain this result by examining the lattice of isotropic subspaces of an even-dimensional Fp-vector space with respect to a non-degenerate alternating form B

Searching for fractal structures in the Universal Steenrod Algebra at odd primes

Unlike the p = 2 case, the universal Steenrod Algebra Q(p) at odd primes does not have a fractal structure that preserves the length of monomials. Nevertheless, when p is odd we detect inside Q(p) two different families of nested subalgebras each isomorphic (as length-graded algebras) to the respective starting element of the sequenc

The multiplicative structure of K(n)*(Ba4)

Let K(n)∗(−) be a Morava K-theory at the prime 2. Invariant theory is used to identify K(n)∗(BA4) as a summand of K(n)∗(BZ/2 × BZ/2). Similarities with H∗(BA4;Z/2) are also discussed

On the Existence of Non-golden Signed Graphs

A signed graph is a pair Γ = (G,σ), where G = (V(G),E(G)) is a graph and σ : E(G)→{+1,−1} is the sign function on the edges of G. For a signed graph we consider the least eigenvector λ(Γ) of the Laplacian matrix defined as L(Γ) = D(G)−A(Γ), where D(G) is the matrix of vertices degrees of G and A(Γ) is the signed adjacency matrix. An unbalanced signed bicyclic graph is said to be golden if it is switching equivalent to a graph Γ satisfying the following property: there exists a cycle C in Γ and a λ(Γ)-eigenvector x such that the unique negative edge pq of Γ belongs to C and detects the minimum of the set Sx(Γ,C) = { |xrxs| | rs ∈ E(C) }. In this paper we show that non-golden bicyclic graphs with frustration index 1 exist for each n ≥ 5

The Iran-United States Claims Tribunal, NAFTA Chapter 11, and the Doctrine of Indirect Expropriation

This essay takes its cue from the recent Interim Award rendered under Chapter 11 of the North American Free Trade Agreement ( NAFTA ) on June 26, 2000 in Pope & Talbot, Inc v Canada. Pope & Talbot implied that the awards of the Iran-United States Claims Tribunal concerning expropriation were not relevant to the interpretation of the expropriation provisions of Chapter 11 of NAFTA. My aim here is to explain why, contrary to this suggestion, the precedents of the Iran-United States Claims Tribunal are legitimate and helpful sources of law in NAFTA Chapter 11 disputes

Ordering signed graphs with large index

The index of a signed graph is the largest eigenvalue of its adjacency matrix. We establish the first few signed graphs ordered decreasingly by the index in classes of connected signed graphs, connected unbalanced signed graphs and complete signed graphs with a fixed number of vertices

A Note on the Algebra of Operations for Hopf Cohomology at Odd Primes

Let $p$ be any prime, and let ${\mathcal B}(p)$ be the algebra of operations on the cohomology ring of any cocommutative $\mathbb{F}_p$-Hopf algebra. In this paper we show that when $p$ is odd (and unlike the $p=2$ case), ${\mathcal B}(p)$ cannot become an object in the Singer category of $\mathbb{F}_p$-algebras with coproducts, if we require that coproducts act on the generators of ${\mathcal B}(p)$ coherently with their nature of cohomology operation

The Fractal Structure of the Universal Steenrod Algebra: An Invariant-theoretic Description

As recently observed by the second author, the mod2 universal Steenrod algebra Q has a fractal structure given by a system of nested subalgebras Qs, for s > N, each isomorphic to Q. In the present paper we provide an alternative presentation of the subalgebras Qs through suitable derivations s, and give an invariant-theoretic description of them

Chasing non-diagonal cycles in a certain system of algebras of operations

The mod 2 universal Steenrod algebra Q is a non-locally finite homogeneous quadratic algebra closely related to the ordinary mod 2 Steenrod algebra and the Lambda algebra. The algebra Q provides an example of a Koszul algebra which is a direct limit of a family of certain non-Koszul algebras Rk's. In this paper we see how far the several Rk's are to be Koszul by chasing in their cohomology non-trivial cocycles of minimal homological degre

A representation of the dual of the Steenrod algebra

In this paper we show how to embed A∗, the dual of the mod 2 Steenrod algebra, into a certain inverse limit of algebras of invariants of the general linear group. The prime 2 is fixed throughout the pape