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Let $α$ be a real algebraic number of degree $3$ over $ℚ$ whose conjugates are not real. There exists an unit $ζ$ of the ring of integer of $K = ℚ (α)$ for which it is possible to describe the set of all best approximation vectors of $θ = (ζ, ζ^{2})$ .’

Bethe Ansatz and the geography of rigged strings

Tadeusz Lulek (2007)

Banach Center Publications

We demonstrate the way in which composition of two famous combinatorial bijections, of Robinson-Schensted and Kerov-Kirillov-Reshetikhin, applied to the Heisenberg model of magnetic ring with spin 1/2, defines the geography of rigged strings (which label exact eigenfunctions of the Bethe Ansatz) on the classical configuration space (the set of all positions of the system of r reversed spins). We point out that each l-string originates, in the language of this bijection, from an island of l consecutive...

Betti numbers of monomial ideals and shifted skew shapes.

Nagel, Uwe, Reiner, Victor (2009)

The Electronic Journal of Combinatorics [electronic only]

Betti numbers of some circulant graphs

Mohsen Abdi Makvand, Amir Mousivand (2019)

Czechoslovak Mathematical Journal

Let $o (n)$ be the greatest odd integer less than or equal to $n$ . In this paper we provide explicit formulae to compute $ℕ$ -graded Betti numbers of the circulant graphs $C_{2 n} (1, 2, 3, 5, ..., o (n))$ . We do this by showing that this graph is the product (or join) of the cycle $C_{n}$ by itself, and computing Betti numbers of $C_{n} * C_{n}$ . We also discuss whether such a graph (more generally, $G * H$ ) is well-covered, Cohen-Macaulay, sequentially Cohen-Macaulay, Buchsbaum, or $S_{2}$ .

Betweenness spaces and tree algebras

Jánis Círulis (1986)

Časopis pro pěstování matematiky

BG-ranks and 2-cores.

Chen, William Y.C., Ji, Kathy Q., Wilf, Herbert S. (2006)

The Electronic Journal of Combinatorics [electronic only]

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