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One more remark on the adjoint polynomial
The adjoint polynomial of G is h(G,x)=∑k=1n(−1)n−kak(G)xk,where ak(G) denotes the number of ways one can cover all vertices of the graph G by exactly k disjoint cliques of G. In this paper we show the adjoint polynomial of a graph G is a simple transformation of the independence polynomial of another graph Ĝ. This enables us to use the rich theory of independence polynomials to study the adjoint polynomials. In particular we give new proofs of several theorems of R. Liu and P. Csikvári. © 2017 Elsevier Lt
The Ising model and Special Geometries
We show that the globally nilpotent G-operators corresponding to the factors
of the linear differential operators annihilating the multifold integrals
of the magnetic susceptibility of the Ising model () are
homomorphic to their adjoint. This property of being self-adjoint up to
operator homomorphisms, is equivalent to the fact that their symmetric square,
or their exterior square, have rational solutions. The differential Galois
groups are in the special orthogonal, or symplectic, groups. This self-adjoint
(up to operator equivalence) property means that the factor operators we
already know to be Derived from Geometry, are special globally nilpotent
operators: they correspond to "Special Geometries".
Beyond the small order factor operators (occurring in the linear differential
operators associated with and ), and, in particular,
those associated with modular forms, we focus on the quite large order-twelve
and order-23 operators. We show that the order-twelve operator has an exterior
square which annihilates a rational solution. Then, its differential Galois
group is in the symplectic group . The order-23 operator
is shown to factorize in an order-two operator and an order-21 operator. The
symmetric square of this order-21 operator has a rational solution. Its
differential Galois group is, thus, in the orthogonal group
.Comment: 33 page
Projective geometry of Wachspress coordinates
We show that there is a unique hypersurface of minimal degree passing through
the non-faces of a polytope which is defined by a simple hyperplane
arrangement. This generalizes the construction of the adjoint curve of a
polygon by Wachspress in 1975. The defining polynomial of our adjoint
hypersurface is the adjoint polynomial introduced by Warren in 1996. This is a
key ingredient for the definition of Wachspress coordinates, which are
barycentric coordinates on an arbitrary convex polytope. The adjoint polynomial
also appears both in algebraic statistics, when studying the moments of uniform
probability distributions on polytopes, and in intersection theory, when
computing Segre classes of monomial schemes. We describe the Wachspress map,
the rational map defined by the Wachspress coordinates, and the Wachspress
variety, the image of this map. The inverse of the Wachspress map is the
projection from the linear span of the image of the adjoint hypersurface. To
relate adjoints of polytopes to classical adjoints of divisors in algebraic
geometry, we study irreducible hypersurfaces that have the same degree and
multiplicity along the non-faces of a polytope as its defining hyperplane
arrangement. We list all finitely many combinatorial types of polytopes in
dimensions two and three for which such irreducible hypersurfaces exist. In the
case of polygons, the general such curves< are elliptic. In the
three-dimensional case, the general such surfaces are either K3 or elliptic
Higher order terms for the quantum evolution of a Wick observable within the Hepp method
The Hepp method is the coherent state approach to the mean field dynamics for
bosons or to the semiclassical propagation. A key point is the asymptotic
evolution of Wick observables under the evolution given by a time-dependent
quadratic Hamiltonian. This article provides a complete expansion with respect
to the small parameter \epsilon > 0 which makes sense within the
infinite-dimensional setting and fits with finite-dimensional formulae
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