1,380 research outputs found
An update on the middle levels problem
The middle levels problem is to find a Hamilton cycle in the middle levels,
M_{2k+1}, of the Hasse diagram of B_{2k+1} (the partially ordered set of
subsets of a 2k+1-element set ordered by inclusion). Previously, the best
result was that M_{2k+1} is Hamiltonian for all positive k through k=15. In
this note we announce that M_{33} and M_{35} have Hamilton cycles. The result
was achieved by an algorithmic improvement that made it possible to find a
Hamilton path in a reduced graph of complementary necklace pairs having
129,644,790 vertices, using a 64-bit personal computer.Comment: 11 pages, 5 figure
A constant-time algorithm for middle levels Gray codes
For any integer a middle levels Gray code is a cyclic listing of
all -element and -element subsets of such that
any two consecutive subsets differ in adding or removing a single element. The
question whether such a Gray code exists for any has been the subject
of intensive research during the last 30 years, and has been answered
affirmatively only recently [T. M\"utze. Proof of the middle levels conjecture.
Proc. London Math. Soc., 112(4):677--713, 2016]. In a follow-up paper [T.
M\"utze and J. Nummenpalo. An efficient algorithm for computing a middle levels
Gray code. To appear in ACM Transactions on Algorithms, 2018] this existence
proof was turned into an algorithm that computes each new set in the Gray code
in time on average. In this work we present an algorithm for
computing a middle levels Gray code in optimal time and space: each new set is
generated in time on average, and the required space is
Rearrangements and Tunneling Splittings in Small Water Clusters
Recent far-infrared vibration-rotation tunneling (FIR-VRT) experiments pose
new challenges to theory because the interpretation and prediction of such
spectra requires a detailed understanding of the potential energy surface (PES)
away from minima. In particular we need a global description of the PES in
terms of a complete reaction graph. Hence all the transition states and
associated mechanisms which might give rise to observable tunneling splittings
must be characterized. It may be possible to guess the detailed permutations of
atoms from the transition state alone, but experience suggests this is unwise.
In this contribution a brief overview of the issues involved in treating the
large amplitude motions of such systems will be given, with references to more
detailed discussions and some specific examples. In particular we will consider
the effective molecular symmetry group, the classification of rearrangement
mechanisms, the location of minima and transition states and the calculation of
reaction pathways. The application of these theories to small water clusters
ranging from water dimer to water hexamer will then be considered. More details
can be found in recent reviews.Comment: 15 pages, 5 figures. This paper was prepared in August 1997 for the
proceedings volume of the NATO-ASI meeting on "Recent Theoretical and
Experimental Advances in Hydrogen Bonded Clusters" edited by Sotiris
Xantheas, which has so far not appeare
Bipartite Kneser graphs are Hamiltonian
For integers and the Kneser graph has as vertices all -element subsets of and an edge between any two vertices (=sets) that are disjoint. The bipartite Kneser graph has as vertices all -element and -element subsets of and an edge between any two vertices where one is a subset of the other. It has long been conjectured that all Kneser graphs and bipartite Kneser graphs except the Petersen graph have a Hamilton cycle. The main contribution of this paper is proving this conjecture for bipartite Kneser graphs . We also establish the existence of cycles that visit almost all vertices in Kneser graphs when , generalizing and improving upon previous results on this problem
Precision measurement of light shifts at two off-resonant wavelengths in a single trapped Ba+ ion and determination of atomic dipole matrix elements
We define and measure the ratio (R) of the vector ac-Stark effect (or light
shift) in the 6S_1/2 and 5D_3/2 states of a single trapped barium ion to 0.2%
accuracy at two different off-resonant wavelengths. We earlier found R =
-11.494(13) at 514.531nm and now report the value at 1111.68nm, R = +0.4176(8).
These observations together yield a value of the matrix element,
previously unknown in the literature. Also, comparison of our results with an
ab initio calculation of dynamic polarizability would yield a new test of
atomic theory and improve the understanding of atomic structure needed to
interpret a proposed atomic parity violation experiment.Comment: 12 pages, 11 figures, in submission to PR
Efficient computation of middle levels Gray codes
For any integer a middle levels Gray code is a cyclic listing of
all bitstrings of length that have either or entries equal to
1 such that any two consecutive bitstrings in the list differ in exactly one
bit. The question whether such a Gray code exists for every has been
the subject of intensive research during the last 30 years, and has been
answered affirmatively only recently [T. M\"utze. Proof of the middle levels
conjecture. Proc. London Math. Soc., 112(4):677--713, 2016]. In this work we
provide the first efficient algorithm to compute a middle levels Gray code. For
a given bitstring, our algorithm computes the next bitstrings in the
Gray code in time , which is
on average per bitstring provided that
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