21 research outputs found
Graphs, Friends and Acquaintances
As is well known, a graph is a mathematical object modeling the
existence of a certain relation between pairs of elements of a given set.
Therefore, it is not surprising that many of the first results concerning
graphs made reference to relationships between people or groups of
people. In this article, we comment on four results of this kind, which
are related to various general theories on graphs and their applications:
the Handshake lemma (related to graph colorings and Boolean
algebra), a lemma on known and unknown people at a cocktail party
(to Ramsey theory), a theorem on friends in common (to distanceregularity
and coding theory), and Hall’s Marriage theorem (to the
theory of networks). These four areas of graph theory, often with
problems which are easy to state but difficult to solve, are extensively
developed and currently give rise to much research work. As examples
of representative problems and results of these areas, which are
discussed in this paper, we may cite the following: the Four Colors
Theorem (4CTC), the Ramsey numbers, problems of the existence of
distance-regular graphs and completely regular codes, and finally the
study of topological proprieties of interconnection networks.Preprin
On the classical Ramsey Number R(3,3,3,3)
The classical Ramsey Number R(3, 3, 3, 3), which is the smallest positive integer n such that any edge coloring with four colors of the complete graph on n vertices must contain at least one monochromatic triangle, is discussed. Basic facts and graph theoretic definitions are given. Papers concerning triangle-free colorings are presented in a historical overview. Mathematical theory underlying the main result of the thesis, which is Richard Kramers unpublished result, i?(3,3,3,3) \u3c 62, is given. The algorithms for the com putational verification of this result are presented along with a discussion of the software tools that were utilized to obtain it
THE ELECTRONIC JOURNAL OF COMBINATORICS (2014), DS1.14 References
and Computing 11. The results of 143 references depend on computer algorithms. The references are ordered alphabetically by the last name of the first author, and where multiple papers have the same first author they are ordered by the last name of the second author, etc. We preferred that all work by the same author be in consecutive positions. Unfortunately, this causes that some of the abbreviations are not in alphabetical order. For example, [BaRT] is earlier on the list than [BaLS]. We also wish to explain a possible confusion with respect to the order of parts and spelling of Chinese names. We put them without any abbreviations, often with the last name written first as is customary in original. Sometimes this is different from the citations in other sources. One can obtain all variations of writing any specific name by consulting the authors database of Mathematical Reviews a
Revealing the Landscape of Globally Color-Dual Multi-loop Integrands
We report on progress in understanding how to construct color-dual multi-loop
amplitudes. First we identify a cubic theory, semi-abelian Yang-Mills, that
unifies many of the color-dual theories studied in the literature, and provides
a prescriptive approach for constructing -dimensional color-dual numerators
through one-loop directly from Feynman rules. By a simple weight counting
argument, this approach does not further generalize to two-loops. As a first
step in understanding the two-loop challenge, we use a -dimensional
color-dual bootstrap to successfully construct globally color-dual local
two-loop four-point nonlinear sigma model (NLSM) numerators. The double-copy of
these NLSM numerators with themselves, pure Yang-Mills, and
super-Yang-Mills correctly reproduce the known unitarity constructed integrands
of special Galileons, Born-Infeld theory, and Dirac-Born-Infeld-Volkov-Akulov
theory, respectively. Applying our bootstrap to two-loop four-point pure
Yang-Mills, we exhaustively search the space of local numerators and find that
it fails to satisfy global color-kinematics duality, completing a search
previously initiated in the literature. We pinpoint the failure to the bowtie
unitarity cut, and discuss a path forward towards non-local construction of
color-dual integrands at generic loop order.Comment: 42 pages, 4 figures, ancillary fil
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
Introduction to Vassiliev Knot Invariants
This book is a detailed introduction to the theory of finite type (Vassiliev)
knot invariants, with a stress on its combinatorial aspects. It is intended to
serve both as a textbook for readers with no or little background in this area,
and as a guide to some of the more advanced material. Our aim is to lead the
reader to understanding by means of pictures and calculations, and for this
reason we often prefer to convey the idea of the proof on an instructive
example rather than give a complete argument. While we have made an effort to
make the text reasonably self-contained, an advanced reader is sometimes
referred to the original papers for the technical details of the proofs.
Version 3: some typos and inaccuracies are corrected.Comment: 512 pages, thousands picture
Exact Quantum Algorithms for Quantum Phase Recognition: Renormalization Group and Error Correction
We explore the relationship between renormalization group (RG) flow and error
correction by constructing quantum algorithms that exactly recognize 1D
symmetry-protected topological (SPT) phases protected by finite internal
Abelian symmetries. For each SPT phase, our algorithm runs a quantum circuit
which emulates RG flow: an arbitrary input ground state wavefunction in the
phase is mapped to a unique minimally-entangled reference state, thereby
allowing for efficient phase identification. This construction is enabled by
viewing a generic input state in the phase as a collection of coherent `errors'
applied to the reference state, and engineering a quantum circuit to
efficiently detect and correct such errors. Importantly, the error correction
threshold is proven to coincide exactly with the phase boundary. We discuss the
implications of our results in the context of condensed matter physics, machine
learning, and near-term quantum algorithms.Comment: 10 pages + appendices v2: extended discussion on RG convergence;
added ref