279 research outputs found
Minor stars in plane graphs with minimum degree five
The weight of a subgraph in is the sum of the degrees in of
vertices of . The {\em height} of a subgraph in is the maximum
degree of vertices of in . A star in a given graph is minor if its
center has degree at most five in the given graph. Lebesgue (1940) gave an
approximate description of minor -stars in the class of normal plane maps
with minimum degree five. In this paper, we give two descriptions of minor
-stars in plane graphs with minimum degree five. By these descriptions, we
can extend several results and give some new results on the weight and height
for some special plane graphs with minimum degree five.Comment: 11 pages, 3 figure
Tropical Fukaya Algebras
We introduce a tropical version of the Fukaya algebra of a Lagrangian
submanifold and use it to show that tropical Lagrangian tori are weakly
unobstructed. Tropical graphs arise as large-scale behavior of
pseudoholomorphic disks under a multiple cut operation on a symplectic manifold
that produces a collection of cut spaces each containing relative normal
crossing divisors, following works of Ionel and Brett Parker. Given a
Lagrangian submanifold in the complement of the relative divisors in one of the
cut spaces, the structure maps of the broken Fukaya algebra count broken disks
associated to rigid tropical graphs. We introduce a further degeneration of the
matching conditions (similar in spirit to Bourgeois' version of symplectic
field theory) which results in a tropical Fukaya algebra whose structure maps
are sums of products over vertices of tropical graphs. We show the tropical
Fukaya algebra is homotopy equivalent to the original Fukaya algebra. In the
case of toric Lagrangians contained in a toric component of the degeneration,
an invariance argument implies the existence of projective Maurer-Cartan
solutions.Comment: 167 pages, 17 figures. We fixed some issues with framings of broken
maps pointed out to us by Mohammad F. Tehrani, whom we than
On Combinatorics of the Arthur Trace Formula, Convex Polytopes, and Toric Varieties
We explicate the combinatorial/geometric ingredients of Arthur's proof of the
convergence and polynomiality, in a truncation parameter, of his non-invariant
trace formula. Starting with a fan in a real, finite dimensional, vector space
and a collection of functions, one for each cone in the fan, we introduce a
combinatorial truncated function with respect to a polytope normal to the fan
and prove the analogues of Arthur's results on the convergence and
polynomiality of the integral of this truncated function over the vector space.
The convergence statements clarify the important role of certain combinatorial
subsets that appear in Arthur's work and provide a crucial partition that
amounts to a so-called nearest face partition. The polynomiality statements can
be thought of as far extensions of the Ehrhart polynomial. Our proof of
polynomiality relies on the Lawrence-Varchenko conical decomposition and
readily implies an extension of the well-known combinatorial lemma of
Langlands. The Khovanskii-Pukhlikov virtual polytopes are an important
ingredient here. Finally, we give some geometric interpretations of our
combinatorial truncation on toric varieties as a measure and a Lefschetz
number
Cremona maps defined by monomials
Cremona maps defined by monomials of degree 2 are thoroughly analyzed and
classified via integer arithmetic and graph combinatorics. In particular, the
structure of the inverse map to such a monomial Cremona map is made very
explicit as is the degree of its monomial defining coordinates. As a special
case, one proves that any monomial Cremona map of degree 2 has inverse of
degree 2 if and only if it is an involution up to permutation in the source and
in the target. This statement is subsumed in a recent result of L. Pirio and F.
Russo, but the proof is entirely different and holds in all characteristics.
One unveils a close relationship binding together the normality of a monomial
ideal, monomial Cremona maps and Hilbert bases of polyhedral cones. The latter
suggests that facets of monomial Cremona theory may be NP-hard
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