4,571 research outputs found
On the number of maximal intersecting k-uniform families and further applications of Tuza's set pair method
We study the function which denotes the number of maximal
-uniform intersecting families . Improving a
bound of Balogh at al. on , we determine the order of magnitude of
by proving that for any fixed , holds. Our proof is based on Tuza's set pair
approach.
The main idea is to bound the size of the largest possible point set of a
cross-intersecting system. We also introduce and investigate some related
functions and parameters.Comment: 11 page
Some New Bounds For Cover-Free Families Through Biclique Cover
An cover-free family is a family of subsets of a finite set
such that the intersection of any members of the family contains at least
elements that are not in the union of any other members. The minimum
number of elements for which there exists an with blocks is
denoted by .
In this paper, we show that the value of is equal to the
-biclique covering number of the bipartite graph whose vertices
are all - and -subsets of a -element set, where a -subset is
adjacent to an -subset if their intersection is empty. Next, we introduce
some new bounds for . For instance, we show that for
and
where is a constant satisfies the
well-known bound . Also, we
determine the exact value of for some values of . Finally, we
show that whenever there exists a Hadamard matrix of
order 4d
A manifold of pure Gibbs states of the Ising model on the Lobachevsky plane
In this paper we construct many `new' Gibbs states of the Ising model on the
Lobachevsky plane, the millefeuilles. Unlike the usual states on the integer
lattices, our foliated states have infinitely many interfaces. The interfaces
are rigid and fill the Lobachevsky plane with positive density.Comment: 25 pages, 7 figure
Symmetries of statistics on lattice paths between two boundaries
We prove that on the set of lattice paths with steps N=(0,1) and E=(1,0) that
lie between two fixed boundaries T and B (which are themselves lattice paths),
the statistics `number of E steps shared with B' and `number of E steps shared
with T' have a symmetric joint distribution. To do so, we give an involution
that switches these statistics, preserves additional parameters, and
generalizes to paths that contain steps S=(0,-1) at prescribed x-coordinates.
We also show that a similar equidistribution result for path statistics follows
from the fact that the Tutte polynomial of a matroid is independent of the
order of its ground set. We extend the two theorems to k-tuples of paths
between two boundaries, and we give some applications to Dyck paths,
generalizing a result of Deutsch, to watermelon configurations, to
pattern-avoiding permutations, and to the generalized Tamari lattice. Finally,
we prove a conjecture of Nicol\'as about the distribution of degrees of k
consecutive vertices in k-triangulations of a convex n-gon. To achieve this
goal, we provide a new statistic-preserving bijection between certain k-tuples
of non-crossing paths and k-flagged semistandard Young tableaux, which is based
on local moves reminiscent of jeu de taquin.Comment: Small typos corrected, and journal reference and grant info adde
Special Lagrangian fibrations, wall-crossing, and mirror symmetry
In this survey paper, we briefly review various aspects of the SYZ approach
to mirror symmetry for non-Calabi-Yau varieties, focusing in particular on
Lagrangian fibrations and wall-crossing phenomena in Floer homology. Various
examples are presented, some of them new.Comment: 45 pages; to appear in Surveys in Differential Geometr
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