2,013 research outputs found
Combinatorics and Geometry of Transportation Polytopes: An Update
A transportation polytope consists of all multidimensional arrays or tables
of non-negative real numbers that satisfy certain sum conditions on subsets of
the entries. They arise naturally in optimization and statistics, and also have
interest for discrete mathematics because permutation matrices, latin squares,
and magic squares appear naturally as lattice points of these polytopes.
In this paper we survey advances on the understanding of the combinatorics
and geometry of these polyhedra and include some recent unpublished results on
the diameter of graphs of these polytopes. In particular, this is a thirty-year
update on the status of a list of open questions last visited in the 1984 book
by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure
IST Austria Thesis
This thesis considers two examples of reconfiguration problems: flipping edges in edge-labelled triangulations of planar point sets and swapping labelled tokens placed on vertices of a graph. In both cases the studied structures â all the triangulations of a given point set or all token placements on a given graph â can be thought of as vertices of the so-called reconfiguration graph, in which two vertices are adjacent if the corresponding structures differ by a single elementary operation â by a flip of a diagonal in a triangulation or by a swap of tokens on adjacent vertices, respectively. We study the reconfiguration of one instance of a structure into another via (shortest) paths in the reconfiguration graph.
For triangulations of point sets in which each edge has a unique label and a flip transfers the label from the removed edge to the new edge, we prove a polynomial-time testable condition, called the Orbit Theorem, that characterizes when two triangulations of the same point set lie in the same connected component of the reconfiguration graph. The condition was first conjectured by Bose, Lubiw, Pathak and Verdonschot. We additionally provide a polynomial time algorithm that computes a reconfiguring flip sequence, if it exists. Our proof of the Orbit Theorem uses topological properties of a certain high-dimensional cell complex that has the usual reconfiguration graph as its 1-skeleton.
In the context of token swapping on a tree graph, we make partial progress on the problem of finding shortest reconfiguration sequences. We disprove the so-called Happy Leaf Conjecture and demonstrate the importance of swapping tokens that are already placed at the correct vertices. We also prove that a generalization of the problem to weighted coloured token swapping is NP-hard on trees but solvable in polynomial time on paths and stars
Entropy-scaling search of massive biological data
Many datasets exhibit a well-defined structure that can be exploited to
design faster search tools, but it is not always clear when such acceleration
is possible. Here, we introduce a framework for similarity search based on
characterizing a dataset's entropy and fractal dimension. We prove that
searching scales in time with metric entropy (number of covering hyperspheres),
if the fractal dimension of the dataset is low, and scales in space with the
sum of metric entropy and information-theoretic entropy (randomness of the
data). Using these ideas, we present accelerated versions of standard tools,
with no loss in specificity and little loss in sensitivity, for use in three
domains---high-throughput drug screening (Ammolite, 150x speedup), metagenomics
(MICA, 3.5x speedup of DIAMOND [3,700x BLASTX]), and protein structure search
(esFragBag, 10x speedup of FragBag). Our framework can be used to achieve
"compressive omics," and the general theory can be readily applied to data
science problems outside of biology.Comment: Including supplement: 41 pages, 6 figures, 4 tables, 1 bo
Complexity Theory
Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness, and quantum computation. Many of the developments are related to diverse mathematical ïŹelds such as algebraic geometry, combinatorial number theory, probability theory, representation theory, and the theory of error-correcting codes
Arithmetic diagonal cycles on unitary Shimura varieties
We define variants of PEL type of the Shimura varieties that appear in the
context of the Arithmetic Gan-Gross-Prasad conjecture. We formulate for them a
version of the AGGP conjecture. We also construct (global and semi-global)
integral models of these Shimura varieties and formulate for them conjectures
on arithmetic intersection numbers. We prove some of these conjectures in low
dimension.Comment: We removed all mention of an 'exotic level structure". Final version.
To appear in Compositio Mathematic
Deformation classification of real non-singular cubic threefolds with a marked line
We prove that the space of pairs formed by a real non-singular cubic hypersurface with a real line has 18 connected components and give for them several quite explicit interpretations. The first one relates these components to the orbits of the monodromy action on the set of connected components of the Fano surface formed by real lines on . For another interpretation we associate with each of the 18 components a well defined real deformation class of real non-singular plane quintic curves and show that this deformation class together with the real deformation class of characterizes completely the component
Brane Tilings and Their Applications
We review recent developments in the theory of brane tilings and
four-dimensional N=1 supersymmetric quiver gauge theories. This review consists
of two parts. In part I, we describe foundations of brane tilings, emphasizing
the physical interpretation of brane tilings as fivebrane systems. In part II,
we discuss application of brane tilings to AdS/CFT correspondence and
homological mirror symmetry. More topics, such as orientifold of brane tilings,
phenomenological model building, similarities with BPS solitons in
supersymmetric gauge theories, are also briefly discussed.
This paper is a revised version of the author's master's thesis submitted to
Department of Physics, Faculty of Science, the University of Tokyo on January
2008, and is based on his several papers: math.AG/0605780, math.AG/0606548,
hep-th/0702049, math.AG/0703267, arXiv:0801.3528 and some works in progress.Comment: 208 pages, 92 figures, based on master's thesis; v2: minor
corrections, to appear in Fortschr. Phy
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