27,727 research outputs found
On the Levi Graph of Point-Line Configurations
We prove that the well-covered dimension of the Levi graph of a point-line
configuration (v_r, b_k) is equal to 0, whenever r > 2.Comment: 7 pages, 4 figure
Word, graph and manifold embedding from Markov processes
Continuous vector representations of words and objects appear to carry
surprisingly rich semantic content. In this paper, we advance both the
conceptual and theoretical understanding of word embeddings in three ways.
First, we ground embeddings in semantic spaces studied in
cognitive-psychometric literature and introduce new evaluation tasks. Second,
in contrast to prior work, we take metric recovery as the key object of study,
unify existing algorithms as consistent metric recovery methods based on
co-occurrence counts from simple Markov random walks, and propose a new
recovery algorithm. Third, we generalize metric recovery to graphs and
manifolds, relating co-occurence counts on random walks in graphs and random
processes on manifolds to the underlying metric to be recovered, thereby
reconciling manifold estimation and embedding algorithms. We compare embedding
algorithms across a range of tasks, from nonlinear dimensionality reduction to
three semantic language tasks, including analogies, sequence completion, and
classification
Rationality of moduli of elliptic fibrations with fixed monodromy
We prove rationality results for moduli spaces of elliptic K3 surfaces and
elliptic rational surfaces with fixed monodromy groups.Comment: 63 pages, LaTe
Algorithms for Lipschitz Learning on Graphs
We develop fast algorithms for solving regression problems on graphs where
one is given the value of a function at some vertices, and must find its
smoothest possible extension to all vertices. The extension we compute is the
absolutely minimal Lipschitz extension, and is the limit for large of
-Laplacian regularization. We present an algorithm that computes a minimal
Lipschitz extension in expected linear time, and an algorithm that computes an
absolutely minimal Lipschitz extension in expected time .
The latter algorithm has variants that seem to run much faster in practice.
These extensions are particularly amenable to regularization: we can perform
-regularization on the given values in polynomial time and
-regularization on the initial function values and on graph edge weights
in time .Comment: Code used in this work is available at
https://github.com/danspielman/YINSlex 30 page
Generating subgraphs in chordal graphs
A graph is well-covered if all its maximal independent sets are of the
same cardinality. Assume that a weight function is defined on its vertices.
Then is -well-covered if all maximal independent sets are of the same
weight. For every graph , the set of weight functions such that is
-well-covered is a vector space, denoted . Let be a complete
bipartite induced subgraph of on vertex sets of bipartition and
. Then is generating if there exists an independent set such
that and are both maximal independent sets of
. In the restricted case that a generating subgraph is isomorphic to
, the unique edge in is called a relating edge. Generating
subgraphs play an important role in finding . Deciding whether an input
graph is well-covered is co-NP-complete. Hence, finding is
co-NP-hard. Deciding whether an edge is relating is NP-complete. Therefore,
deciding whether a subgraph is generating is NP-complete as well. A graph is
chordal if every induced cycle is a triangle. It is known that finding
can be done polynomially in the restricted case that is chordal. Thus
recognizing well-covered chordal graphs is a polynomial problem. We present a
polynomial algorithm for recognizing relating edges and generating subgraphs in
chordal graphs.Comment: 13 pages, 1 figure. arXiv admin note: text overlap with
arXiv:1401.029
From entangled codipterous coalgebras to coassociative manifolds
We construct from coassociative coalgebras, bialgebras, Hopf algebras, new
objects such as Poisson algebras, Leibniz algebras defined by J-L Loday and M.
Ronco and explore the notion of coassociative manifolds.Comment: 25 pages, 13 figure
Constant mean curvature surfaces in 3-dimensional Thurston geometries
This is a survey on the global theory of constant mean curvature surfaces in
Riemannian homogeneous 3-manifolds. These ambient 3-manifolds include the eight
canonical Thurston 3-dimensional geometries, i.e. R3, H3, S3, H2 \times R, S2
\times R, the Heisenberg space Nil3, the universal cover of PSL2(R) and the Lie
group Sol3. We will focus on the problems of classifying compact CMC surfaces
and entire CMC graphs in these spaces. A collection of important open problems
of the theory is also presented
Some old and new problems in combinatorial geometry I: Around Borsuk's problem
Borsuk asked in 1933 if every set of diameter 1 in can be covered by
sets of smaller diameter. In 1993, a negative solution, based on a
theorem by Frankl and Wilson, was given by Kahn and Kalai. In this paper I will
present questions related to Borsuk's problem.Comment: This is a draft of a chapter for "Surveys in Combinatorics 2015,"
edited by Artur Czumaj, Angelos Georgakopoulos, Daniel Kral, Vadim Lozin, and
Oleg Pikhurko. The final published version shall be available for purchase
from Cambridge University Pres
Graph-directed systems and self-similar measures on limit spaces of self-similar groups
Let be a group and be a contracting homomorphism from a
subgroup of finite index. V.Nekrashevych [25] associated with the pair
the limit dynamical system (\lims,\si) and the limit -space
\limGs together with the covering \cup_{g\in G}\tile\cdot g by the tile
\tile. We develop the theory of self-similar measures on these limit
spaces. It is shown that (\lims,\si,\mu) is conjugated to the one-sided
Bernoulli shift. Using sofic subshifts we prove that the tile \tile has
integer measure and we give an algorithmic way to compute it. In addition we
give an algorithm to find the measure of the intersection of tiles \tile\cap
(\tile\cdot g) for . We present applications to the invariant measures
for the rational functions on the Riemann sphere and to the evaluation of the
Lebesgue measure of integral self-affine tiles.Comment: 25 pages, 2 figure
Lectures on approximate groups and Hilbert's 5th problem
This paper gathers four lectures, based on a mini-course at IMA in 2014,
whose aim was to discuss the structure of approximate subgroups of an arbitrary
group, following the works of Hrushovski and of Green, Tao and the author.
Along the way we discuss the proof of the Gleason-Yamabe theorem on Hilbert's
5th problem about the structure of locally compact groups and explain its
relevance to approximate groups. We also present several applications, in
particular to uniform diameter bounds for finite groups and to the
determination of scaling limits of vertex transitive graphs with large
diameter.Comment: lecture notes to appear in IMA volume of proceeding
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