131 research outputs found
Simplicial matrix-tree theorems
We generalize the definition and enumeration of spanning trees from the
setting of graphs to that of arbitrary-dimensional simplicial complexes
, extending an idea due to G. Kalai. We prove a simplicial version of
the Matrix-Tree Theorem that counts simplicial spanning trees, weighted by the
squares of the orders of their top-dimensional integral homology groups, in
terms of the Laplacian matrix of . As in the graphic case, one can
obtain a more finely weighted generating function for simplicial spanning trees
by assigning an indeterminate to each vertex of and replacing the
entries of the Laplacian with Laurent monomials. When is a shifted
complex, we give a combinatorial interpretation of the eigenvalues of its
weighted Laplacian and prove that they determine its set of faces uniquely,
generalizing known results about threshold graphs and unweighted Laplacian
eigenvalues of shifted complexes.Comment: 36 pages, 2 figures. Final version, to appear in Trans. Amer. Math.
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Elements of Petri nets and processes
We present a formalism for Petri nets based on polynomial-style finite-set
configurations and etale maps. The formalism supports both a geometric
semantics in the style of Goltz and Reisig (processes are etale maps from
graphs) and an algebraic semantics in terms of free coloured props: the Segal
space of P-processes is shown to be the free coloured prop-in-groupoids on P.
There is also an unfolding semantics \`a la Winskel, which bypasses the
classical symmetry problems. Since everything is encoded with explicit sets,
Petri nets and their processes have elements. In particular, individual-token
semantics is native, and the benefits of pre-nets in this respect can be
obtained without the need of numberings. (Collective-token semantics emerges
from rather drastic quotient constructions \`a la Best--Devillers, involving
taking of the groupoids of states.)Comment: 44 pages. The math is intended to be in reasonably final form, but
the paper may well contain some misconceptions regarding the place of this
material in the theory of Petri nets. All feedback and help will be greatly
appreciated. v2: fixed a mistake in Section
A Quantitative Study of Pure Parallel Processes
In this paper, we study the interleaving -- or pure merge -- operator that
most often characterizes parallelism in concurrency theory. This operator is a
principal cause of the so-called combinatorial explosion that makes very hard -
at least from the point of view of computational complexity - the analysis of
process behaviours e.g. by model-checking. The originality of our approach is
to study this combinatorial explosion phenomenon on average, relying on
advanced analytic combinatorics techniques. We study various measures that
contribute to a better understanding of the process behaviours represented as
plane rooted trees: the number of runs (corresponding to the width of the
trees), the expected total size of the trees as well as their overall shape.
Two practical outcomes of our quantitative study are also presented: (1) a
linear-time algorithm to compute the probability of a concurrent run prefix,
and (2) an efficient algorithm for uniform random sampling of concurrent runs.
These provide interesting responses to the combinatorial explosion problem
Relation spaces of hyperplane arrangements and modules defined by graphs of fiber zonotopes
We study the exactness of certain combinatorially defined complexes which
generalize the Orlik-Solomon algebra of a geometric lattice. The main results
pertain to complex reflection arrangements and their restrictions. In
particular, we consider the corresponding relation complexes and give a simple
proof of the -formality of these hyperplane arrangements. As an application,
we are able to bound the Castelnouvo-Mumford regularity of certain modules over
polynomial rings associated to Coxeter arrangements (real reflection
arrangements) and their restrictions. The modules in question are defined using
the relation complex of the Coxeter arrangement and fiber polytopes of the dual
Coxeter zonotope. They generalize the algebra of piecewise polynomial functions
on the original arrangement
A Positive Semidefinite Approximation of the Symmetric Traveling Salesman Polytope
For a convex body B in a vector space V, we construct its approximation P_k,
k=1, 2, . . . using an intersection of a cone of positive semidefinite
quadratic forms with an affine subspace. We show that P_k is contained in B for
each k. When B is the Symmetric Traveling Salesman Polytope on n cities T_n, we
show that the scaling of P_k by n/k+ O(1/n) contains T_n for k no more than
n/2. Membership for P_k is computable in time polynomial in n (of degree linear
in k).
We discuss facets of T_n that lie on the boundary of P_k. We introduce a new
measure on each facet defining inequality for T_n in terms of the eigenvalues
of a quadratic form. Using these eigenvalues of facets, we show that the
scaling of P_1 by n^(1/2) has all of the facets of T_n defined by the subtour
elimination constraints either in its interior or lying on its boundary.Comment: 25 page
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