3,293 research outputs found
Finite Volume Spaces and Sparsification
We introduce and study finite -volumes - the high dimensional
generalization of finite metric spaces. Having developed a suitable
combinatorial machinery, we define -volumes and show that they contain
Euclidean volumes and hypertree volumes. We show that they can approximate any
-volume with multiplicative distortion. On the other hand, contrary
to Bourgain's theorem for , there exists a -volume that on vertices
that cannot be approximated by any -volume with distortion smaller than
.
We further address the problem of -dimension reduction in the context
of volumes, and show that this phenomenon does occur, although not to
the same striking degree as it does for Euclidean metrics and volumes. In
particular, we show that any metric on points can be -approximated by a sum of cut metrics, improving
over the best previously known bound of due to Schechtman.
In order to deal with dimension reduction, we extend the techniques and ideas
introduced by Karger and Bencz{\'u}r, and Spielman et al.~in the context of
graph Sparsification, and develop general methods with a wide range of
applications.Comment: previous revision was the wrong file: the new revision: changed
(extended considerably) the treatment of finite volumes (see revised
abstract). Inserted new applications for the sparsification technique
Harmonic functions on multiplicative graphs and interpolation polynomials
We construct examples of nonnegative harmonic functions on certain graded
graphs: the Young lattice and its generalizations. Such functions first emerged
in harmonic analysis on the infinite symmetric group. Our method relies on
multivariate interpolation polynomials associated with Schur's S and P
functions and with Jack symmetric functions. As a by-product, we compute
certain Selberg-type integrals.Comment: AMSTeX, 35 page
The totally nonnegative Grassmannian is a ball
We prove that three spaces of importance in topological combinatorics are
homeomorphic to closed balls: the totally nonnegative Grassmannian, the
compactification of the space of electrical networks, and the cyclically
symmetric amplituhedron.Comment: 19 pages. v2: Exposition improved in many place
On the orthogonal rank of Cayley graphs and impossibility of quantum round elimination
After Bob sends Alice a bit, she responds with a lengthy reply. At the cost
of a factor of two in the total communication, Alice could just as well have
given the two possible replies without listening and have Bob select which
applies to him. Motivated by a conjecture stating that this form of "round
elimination" is impossible in exact quantum communication complexity, we study
the orthogonal rank and a symmetric variant thereof for a certain family of
Cayley graphs. The orthogonal rank of a graph is the smallest number for
which one can label each vertex with a nonzero -dimensional complex vector
such that adjacent vertices receive orthogonal vectors.
We show an exp lower bound on the orthogonal rank of the graph on
in which two strings are adjacent if they have Hamming distance at
least . In combination with previous work, this implies an affirmative
answer to the above conjecture.Comment: 13 page
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