357 research outputs found
Computational Complexity for Physicists
These lecture notes are an informal introduction to the theory of
computational complexity and its links to quantum computing and statistical
mechanics.Comment: references updated, reprint available from
http://itp.nat.uni-magdeburg.de/~mertens/papers/complexity.shtm
Quivers, YBE and 3-manifolds
We study 4d superconformal indices for a large class of N=1 superconformal
quiver gauge theories realized combinatorially as a bipartite graph or a set of
"zig-zag paths" on a two-dimensional torus T^2. An exchange of loops, which we
call a "double Yang-Baxter move", gives the Seiberg duality of the gauge
theory, and the invariance of the index under the duality is translated into
the Yang-Baxter-type equation of a spin system defined on a "Z-invariant"
lattice on T^2. When we compactify the gauge theory to 3d, Higgs the theory and
then compactify further to 2d, the superconformal index reduces to an integral
of quantum/classical dilogarithm functions. The saddle point of this integral
unexpectedly reproduces the hyperbolic volume of a hyperbolic 3-manifold. The
3-manifold is obtained by gluing hyperbolic ideal polyhedra in H^3, each of
which could be thought of as a 3d lift of the faces of the 2d bipartite
graph.The same quantity is also related with the thermodynamic limit of the BPS
partition function, or equivalently the genus 0 topological string partition
function, on a toric Calabi-Yau manifold dual to quiver gauge theories. We also
comment on brane realization of our theories. This paper is a companion to
another paper summarizing the results.Comment: 61 pages, 16 figures; v2: typos correcte
Steinitz Theorems for Orthogonal Polyhedra
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron
with the topology of a sphere in which three mutually-perpendicular edges meet
at each vertex. By analogy to Steinitz's theorem characterizing the graphs of
convex polyhedra, we find graph-theoretic characterizations of three classes of
simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric
projection in the plane with only one hidden vertex, xyz polyhedra, in which
each axis-parallel line through a vertex contains exactly one other vertex, and
arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz
polyhedra are exactly the bipartite cubic polyhedral graphs, and every
bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of
a corner polyhedron. Based on our characterizations we find efficient
algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure
Visualized Algorithm Engineering on Two Graph Partitioning Problems
Concepts of graph theory are frequently used by computer scientists as abstractions when modeling a problem. Partitioning a graph (or a network) into smaller parts is one of the fundamental algorithmic operations that plays a key role in classifying and clustering. Since the early 1970s, graph partitioning rapidly expanded for applications in wide areas. It applies in both engineering applications, as well as research. Current technology generates massive data (“Big Data”) from business interactions and social exchanges, so high-performance algorithms of partitioning graphs are a critical need.
This dissertation presents engineering models for two graph partitioning problems arising from completely different applications, computer networks and arithmetic. The design, analysis, implementation, optimization, and experimental evaluation of these models employ visualization in all aspects. Visualization indicates the performance of the implementation of each Algorithm Engineering work, and also helps to analyze and explore new algorithms to solve the problems. We term this research method as “Visualized Algorithm Engineering (VAE)” to emphasize the contribution of the visualizations in these works.
The techniques discussed here apply to a broad area of problems: computer networks, social networks, arithmetic, computer graphics and software engineering. Common terminologies accepted across these disciplines have been used in this dissertation to guarantee practitioners from all fields can understand the concepts we introduce
Graph Theory
Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem
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