6,479 research outputs found
On the van der Waerden numbers w(2;3,t)
We present results and conjectures on the van der Waerden numbers w(2;3,t)
and on the new palindromic van der Waerden numbers pdw(2;3,t). We have computed
the new number w(2;3,19) = 349, and we provide lower bounds for 20 <= t <= 39,
where for t <= 30 we conjecture these lower bounds to be exact. The lower
bounds for 24 <= t <= 30 refute the conjecture that w(2;3,t) <= t^2, and we
present an improved conjecture. We also investigate regularities in the good
partitions (certificates) to better understand the lower bounds.
Motivated by such reglarities, we introduce *palindromic van der Waerden
numbers* pdw(k; t_0,...,t_{k-1}), defined as ordinary van der Waerden numbers
w(k; t_0,...,t_{k-1}), however only allowing palindromic solutions (good
partitions), defined as reading the same from both ends. Different from the
situation for ordinary van der Waerden numbers, these "numbers" need actually
to be pairs of numbers. We compute pdw(2;3,t) for 3 <= t <= 27, and we provide
lower bounds, which we conjecture to be exact, for t <= 35.
All computations are based on SAT solving, and we discuss the various
relations between SAT solving and Ramsey theory. Especially we introduce a
novel (open-source) SAT solver, the tawSolver, which performs best on the SAT
instances studied here, and which is actually the original DLL-solver, but with
an efficient implementation and a modern heuristic typical for look-ahead
solvers (applying the theory developed in the SAT handbook article of the
second author).Comment: Second version 25 pages, updates of numerical data, improved
formulations, and extended discussions on SAT. Third version 42 pages, with
SAT solver data (especially for new SAT solver) and improved representation.
Fourth version 47 pages, with updates and added explanation
Ramsey numbers and adiabatic quantum computing
The graph-theoretic Ramsey numbers are notoriously difficult to calculate. In
fact, for the two-color Ramsey numbers with , only nine are
currently known. We present a quantum algorithm for the computation of the
Ramsey numbers . We show how the computation of can be mapped
to a combinatorial optimization problem whose solution can be found using
adiabatic quantum evolution. We numerically simulate this adiabatic quantum
algorithm and show that it correctly determines the Ramsey numbers R(3,3) and
R(2,s) for . We then discuss the algorithm's experimental
implementation, and close by showing that Ramsey number computation belongs to
the quantum complexity class QMA.Comment: 4 pages, 1 table, no figures, published versio
Almost-rainbow edge-colorings of some small subgraphs
Let be the minimum number of colors necessary to color the edges
of so that every is at least -colored. We improve current bounds
on the {7/4}n-3{5/6}n+1\leq
f(n,4,5)n\not\equiv 1 \pmod 3f(n,4,5)\leq n-1G=K_{n,n}GC_4\subseteq G$ is colored by at least three
colors. This improves the best known upper bound of M. Axenovich, Z. F\"uredi,
and D. Mubayi.Comment: 13 page
Quantum metrology with nonclassical states of atomic ensembles
Quantum technologies exploit entanglement to revolutionize computing,
measurements, and communications. This has stimulated the research in different
areas of physics to engineer and manipulate fragile many-particle entangled
states. Progress has been particularly rapid for atoms. Thanks to the large and
tunable nonlinearities and the well developed techniques for trapping,
controlling and counting, many groundbreaking experiments have demonstrated the
generation of entangled states of trapped ions, cold and ultracold gases of
neutral atoms. Moreover, atoms can couple strongly to external forces and light
fields, which makes them ideal for ultra-precise sensing and time keeping. All
these factors call for generating non-classical atomic states designed for
phase estimation in atomic clocks and atom interferometers, exploiting
many-body entanglement to increase the sensitivity of precision measurements.
The goal of this article is to review and illustrate the theory and the
experiments with atomic ensembles that have demonstrated many-particle
entanglement and quantum-enhanced metrology.Comment: 76 pages, 40 figures, 1 table, 603 references. Some figures bitmapped
at 300 dpi to reduce file siz
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