518 research outputs found
A remarkable sequence of integers
A survey of properties of a sequence of coefficients appearing in the
evaluation of a quartic definite integral is presented. These properties are of
analytical, combinatorial and number-theoretical nature.Comment: 20 pages, 5 figure
Graviton Vertices and the Mapping of Anomalous Correlators to Momentum Space for a General Conformal Field Theory
We investigate the mapping of conformal correlators and of their anomalies
from configuration to momentum space for general dimensions, focusing on the
anomalous correlators , - involving the energy-momentum tensor
with a vector or a scalar operator () - and the 3-graviton vertex
. We compute the , and one-loop vertex functions in
dimensional regularization for free field theories involving conformal scalar,
fermion and vector fields. Since there are only one or two independent tensor
structures solving all the conformal Ward identities for the or
vertex functions respectively, and three independent tensor structures for the
vertex, and the coefficients of these tensors are known for free fields,
it is possible to identify the corresponding tensors in momentum space from the
computation of the correlators for free fields. This works in general
dimensions for and correlators, but only in 4 dimensions for ,
since vector fields are conformal only in . In this way the general
solution of the Ward identities including anomalous ones for these correlators
in (Euclidean) position space, found by Osborn and Petkou is mapped to the
ordinary diagrammatic one in momentum space. We give simplified expressions of
all these correlators in configuration space which are explicitly Fourier
integrable and provide a diagrammatic interpretation of all the contact terms
arising when two or more of the points coincide. We discuss how the anomalies
arise in each approach [...]Comment: 57 pages, 7 figures. Refs adde
Sticky Brownian Rounding and its Applications to Constraint Satisfaction Problems
Semidefinite programming is a powerful tool in the design and analysis of
approximation algorithms for combinatorial optimization problems. In
particular, the random hyperplane rounding method of Goemans and Williamson has
been extensively studied for more than two decades, resulting in various
extensions to the original technique and beautiful algorithms for a wide range
of applications. Despite the fact that this approach yields tight approximation
guarantees for some problems, e.g., Max-Cut, for many others, e.g., Max-SAT and
Max-DiCut, the tight approximation ratio is still unknown. One of the main
reasons for this is the fact that very few techniques for rounding semidefinite
relaxations are known.
In this work, we present a new general and simple method for rounding
semi-definite programs, based on Brownian motion. Our approach is inspired by
recent results in algorithmic discrepancy theory. We develop and present tools
for analyzing our new rounding algorithms, utilizing mathematical machinery
from the theory of Brownian motion, complex analysis, and partial differential
equations. Focusing on constraint satisfaction problems, we apply our method to
several classical problems, including Max-Cut, Max-2SAT, and MaxDiCut, and
derive new algorithms that are competitive with the best known results. To
illustrate the versatility and general applicability of our approach, we give
new approximation algorithms for the Max-Cut problem with side constraints that
crucially utilizes measure concentration results for the Sticky Brownian
Motion, a feature missing from hyperplane rounding and its generalization
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