We discover a new example of a generic rank 2-distribution on a 5-manifold with a 6-dimensional transitive symmetry algebra, which is not present in Cartan's classical five variables paper. It corresponds to the Monge equation z ′ = y + (y ′′ ) 1/3 with invariant quartic having root type [4], and a 6-dimensional nonsolvable symmetry algebra isomorphic to the semidirect product of sl(2) and the 3-dimensional Heisenberg algebra.
We develop a theory of graph algebras over general fields. This is modelled after the theory developed by Freedman et al. (2007, J. Amer. Math. Soc.20 37–51) for connection matrices, in the study of graph homomorphism functions over real edge weight and positive vertex weight. We introduce connection tensors for graph properties. This notion naturally generalizes the concept of connection matrices. It is shown that counting perfect matchings, and a host of other graph properties naturally defined as Holant problems (edge models), cannot be expressed by graph homomorphism functions with both complex vertex and edge weights (or even from more general fields). Our necessary and sufficient condition in terms of connection tensors is a simple exponential rank bound. It shows that positive semidefiniteness is not needed in the more general setting.
Graph homomorphism has been an important research topic since its introduction [20]. Stated in the language of binary relational structures in that paper [20], Lovász proved a fundamental theorem that, for a graph
H
given by its 0-1 valued adjacency matrix, the graph homomorphism function
G
↦ hom(
G
,
H
) determines the isomorphism type of
H
. In the past 50 years, various extensions have been proved by many researchers [1, 15, 21, 24, 26]. These extend the basic 0-1 case to admit vertex and edge weights; but these extensions all have some restrictions such as all vertex weights must be positive. In this article, we prove a general form of this theorem where
H
can have arbitrary vertex and edge weights. A noteworthy aspect is that we prove this by a surprisingly simple and unified argument. This bypasses various technical obstacles and unifies and extends all previous known versions of this theorem on graphs. The constructive proof of our theorem can be used to make various complexity dichotomy theorems for graph homomorphism
effective
in the following sense: it provides an algorithm that for any
H
either outputs a P-time algorithm solving hom(&sdot,
H
) or a P-time reduction from a canonical #P-hard problem to hom(&sdot,
H
).
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.