623 research outputs found
Interval non-edge-colorable bipartite graphs and multigraphs
An edge-coloring of a graph with colors is called an interval
-coloring if all colors are used, and the colors of edges incident to any
vertex of are distinct and form an interval of integers. In 1991 Erd\H{o}s
constructed a bipartite graph with 27 vertices and maximum degree 13 which has
no interval coloring. Erd\H{o}s's counterexample is the smallest (in a sense of
maximum degree) known bipartite graph which is not interval colorable. On the
other hand, in 1992 Hansen showed that all bipartite graphs with maximum degree
at most 3 have an interval coloring. In this paper we give some methods for
constructing of interval non-edge-colorable bipartite graphs. In particular, by
these methods, we construct three bipartite graphs which have no interval
coloring, contain 20,19,21 vertices and have maximum degree 11,12,13,
respectively. This partially answers a question that arose in [T.R. Jensen, B.
Toft, Graph coloring problems, Wiley Interscience Series in Discrete
Mathematics and Optimization, 1995, p. 204]. We also consider similar problems
for bipartite multigraphs.Comment: 18 pages, 7 figure
Effective Temperature in a Colloidal Glass
We study the Brownian motion of particles trapped by optical tweezers inside
a colloidal glass (Laponite) during the sol-gel transition. We use two methods
based on passive rheology to extract the effective temperature from the
fluctuations of the Brownian particles. All of them give a temperature that,
within experimental errors, is equal to the heat bath temperature. Several
interesting features concerning the statistical properties and the long time
correlations of the particles are observed during the transition.Comment: to be published in Philosophical Magazin
Symmetric photon-photon coupling by atoms with Zeeman-split sublevels
We propose a simple scheme for highly efficient nonlinear interaction between
two weak optical fields. The scheme is based on the attainment of
electromagnetically induced transparency simultaneously for both fields via
transitions between magnetically split F=1 atomic sublevels, in the presence of
two driving fields. Thereby, equal slow group velocities and symmetric
cross-coupling of the weak fields over long distances are achieved. By simply
tuning the fields, this scheme can either yield giant cross-phase modulation or
ultrasensitive two-photon switching.Comment: Modified scheme, 4 pages, 1 figur
Frustrated collisions and unconventional pairing on a quantum superlattice
We solve the problem of scattering and binding of two spin-1/2 fermions on a
one-dimensional superlattice with a period of twice the lattice spacing
analytically. We find the exact bound states and the scattering states,
consisting of a generalized Bethe ansatz augmented with an extra scattering
product due to "asymptotic" degeneracy. If a Bloch band is doubly occupied, the
extra wave can be a bound state in the continuum corresponding to a
single-particle interband transition. In all other cases, it corresponds to a
quasi-momentum changing, frustrated collision.Comment: 4 pages, 2 figure
Estimate of the free energy difference in mechanical systems from work fluctuations: experiments and models
The work fluctuations of an oscillator in contact with a heat reservoir and
driven out of equilibrium by an external force are studied experimentally. The
oscillator dynamics is modeled by a Langevin equation. We find both
experimentally and theoretically that, if the driving force does not change the
equilibrium properties of the thermal fluctuations of this mechanical system,
the free energy difference between two equilibrium states can be
exactly computed using the Jarzynski equality (JE) and the Crooks relation (CR)
\cite{jarzynski1, crooks1, jarzynski2}, independently of the time scale and
amplitude of the driving force. The applicability limits for the JE and CR at
very large driving forces are discussed. Finally, when the work fluctuations
are Gaussian, we propose an alternative empirical method to compute
which can be safely applied, even in cases where the JE and CR might not hold.
The results of this paper are useful to compute in complex systems
such as the biological ones.Comment: submitted to Journal of Statistical Mechanics: Theory and experimen
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