3,277 research outputs found
Extremal Problems for Subset Divisors
Let be a set of positive integers. We say that a subset of is
a divisor of , if the sum of the elements in divides the sum of the
elements in . We are interested in the following extremal problem. For each
, what is the maximum number of divisors a set of positive integers can
have? We determine this function exactly for all values of . Moreover, for
each we characterize all sets that achieve the maximum. We also prove
results for the -subset analogue of our problem. For this variant, we
determine the function exactly in the special case that . We also
characterize all sets that achieve this bound when .Comment: 10 pages, 0 figures. This is essentially the journal version of the
paper, which appeared in the Electronic Journal of Combinatoric
Completing Partial Packings of Bipartite Graphs
Given a bipartite graph and an integer , let be the smallest
integer such that, any set of edge disjoint copies of on vertices, can
be extended to an -design on at most vertices. We establish tight
bounds for the growth of as . In particular, we
prove the conjecture of F\"uredi and Lehel \cite{FuLe} that .
This settles a long-standing open problem
Metrics with four conic singularities and spherical quadrilaterals
A spherical quadrilateral is a bordered surface homeomorphic to a closed
disk, with four distinguished boundary points called corners, equipped with a
Riemannian metric of constant curvature 1, except at the corners, and such that
the boundary arcs between the corners are geodesic. We discuss the problem of
classification of these quadrilaterals and perform the classification up to
isometry in the case that two angles at the corners are multiples of pi. The
problem is equivalent to classification of Heun's equations with real
parameters and unitary monodromy.Comment: 68 pges, 25 figure
Rigidity and quasi-rigidity of extremal cycles in Hermitian symmetric spaces
I use local differential geometric techniques to prove that the algebraic
cycles in certain extremal homology classes in Hermitian symmetric spaces are
either rigid (i.e., deformable only by ambient motions) or quasi-rigid (roughly
speaking, foliated by rigid subvarieties in a nontrivial way).
These rigidity results have a number of applications: First, they prove that
many subvarieties in Grassmannians and other Hermitian symmetric spaces cannot
be smoothed (i.e., are not homologous to a smooth subvariety). Second, they
provide characterizations of holomorphic bundles over compact Kahler manifolds
that are generated by their global sections but that have certain polynomials
in their Chern classes vanish (for example, c_2 = 0, c_1c_2 - c_3 = 0, c_3 = 0,
etc.).Comment: 113 pages, 6 figures, latex2e with packages hyperref, amsart,
graphicx. For Version 2: Many typos corrected, important references added
(esp. to Maria Walters' thesis), several proofs or statements improved and/or
correcte
Black Holes and D-branes
D-branes have been used to describe many properties of extremal and near
extremal black holes. These lecture notes provide a short review of these
developments.Comment: 13 pages, 2 figure
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