227,478 research outputs found
On the ground states of the Bernasconi model
The ground states of the Bernasconi model are binary +1/-1 sequences of
length N with low autocorrelations. We introduce the notion of perfect
sequences, binary sequences with one-valued off-peak correlations of minimum
amount. If they exist, they are ground states. Using results from the
mathematical theory of cyclic difference sets, we specify all values of N for
which perfect sequences do exist and how to construct them. For other values of
N, we investigate almost perfect sequences, i.e. sequences with two-valued
off-peak correlations of minimum amount. Numerical and analytical results
support the conjecture that almost perfect sequences do exist for all values of
N, but that they are not always ground states. We present a construction for
low-energy configurations that works if N is the product of two odd primes.Comment: 12 pages, LaTeX2e; extended content, added references; submitted to
J.Phys.
A Geometric Theory for Hypergraph Matching
We develop a theory for the existence of perfect matchings in hypergraphs
under quite general conditions. Informally speaking, the obstructions to
perfect matchings are geometric, and are of two distinct types: 'space
barriers' from convex geometry, and 'divisibility barriers' from arithmetic
lattice-based constructions. To formulate precise results, we introduce the
setting of simplicial complexes with minimum degree sequences, which is a
generalisation of the usual minimum degree condition. We determine the
essentially best possible minimum degree sequence for finding an almost perfect
matching. Furthermore, our main result establishes the stability property:
under the same degree assumption, if there is no perfect matching then there
must be a space or divisibility barrier. This allows the use of the stability
method in proving exact results. Besides recovering previous results, we apply
our theory to the solution of two open problems on hypergraph packings: the
minimum degree threshold for packing tetrahedra in 3-graphs, and Fischer's
conjecture on a multipartite form of the Hajnal-Szemer\'edi Theorem. Here we
prove the exact result for tetrahedra and the asymptotic result for Fischer's
conjecture; since the exact result for the latter is technical we defer it to a
subsequent paper.Comment: Accepted for publication in Memoirs of the American Mathematical
Society. 101 pages. v2: minor changes including some additional diagrams and
passages of expository tex
Probe light-shift elimination in Generalized Hyper-Ramsey quantum clocks
We present a new interrogation scheme for the next generation of quantum
clocks to suppress frequency-shifts induced by laser probing fields themselves
based on Generalized Hyper-Ramsey resonances. Sequences of composite laser
pulses with specific selection of phases, frequency detunings and durations are
combined to generate a very efficient and robust frequency locking signal with
almost a perfect elimination of the light-shift from off resonant states and to
decouple the unperturbed frequency measurement from the laser's intensity. The
frequency lock point generated from synthesized error signals using either
or laser phase-steps during the intermediate pulse is tightly
protected against large laser pulse area variations and errors in potentially
applied frequency shift compensations. Quantum clocks based on weakly allowed
or completely forbidden optical transitions in atoms, ions, molecules and
nuclei will benefit from these hyper-stable laser frequency stabilization
schemes to reach relative accuracies below the 10 level.Comment: accepted for publication in Phys. Rev.
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