38,607 research outputs found
Phase Transition in the Density of States of Quantum Spin Glasses
We prove that the empirical density of states of quantum spin glasses on
arbitrary graphs converges to a normal distribution as long as the maximal
degree is negligible compared with the total number of edges. This extends the
recent results of [6] that were proved for graphs with bounded chromatic number
and with symmetric coupling distribution. Furthermore, we generalise the result
to arbitrary hypergraphs. We test the optimality of our condition on the
maximal degree for -uniform hypergraphs that correspond to -spin glass
Hamiltonians acting on distinguishable spin- particles. At the
critical threshold we find a sharp classical-quantum phase
transition between the normal distribution and the Wigner semicircle law. The
former is characteristic to classical systems with commuting variables, while
the latter is a signature of noncommutative random matrix theory.Comment: 21 pages, 2 figure
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Sex-related differences in chromatic sensitivity
Generally women are believed to be more discriminating than men in the use of colour names and this is often taken to imply superior colour vision. However, if both X-chromosome linked colour deficient males (~8%) and females (<1%) as well as heterozygote female carriers (~15%) are excluded from comparisons, then differences between men and women in red-green colour discrimination have been reported as not being significant (e.g., Pickford, 1944; Hood et al., 2006). We re-examined this question by assessing the performance of 150 males and 150 females on the Colour Assessment and Diagnosis (CAD) test (Rodriguez-Carmona, 2005). This is a sensitive test that yields small colour detection thresholds. The test employs direction-specific, moving, chromatic stimuli embedded in a background of random, dynamic, luminance contrast noise. A four-alternative, forced-choice procedure is employed to measure the subject’s thresholds for detection of colour signals in 16 directions in colour space, while ensuring that the subject cannot make use of any residual luminance contrast signals. In addition, we measured the Rayleigh anomaloscope matches in a subgroup of 111 males and 114 females. All the age-matched males (30.8 ± 9.7) and females (26.7 ± 8.8) had normal colour vision as diagnosed by a battery of conventional colour vision tests. Females with known colour deficient relatives were excluded from the study. Comparisons between the male and female groups revealed no significant differences in anomaloscope midpoints (p=0.709), but a significant difference in matching ranges (p=0.040); females on average tended to have a larger mean range (4.11) than males (3.75). Females also had significantly higher CAD thresholds than males along the red-green (p=0.0004), but not along the yellow-blue discrimination axis. The differences between males and females in red-green discrimination may be related to the heterozygosity in X-linked cone photopigment expression common among females
On topological relaxations of chromatic conjectures
There are several famous unsolved conjectures about the chromatic number that
were relaxed and already proven to hold for the fractional chromatic number. We
discuss similar relaxations for the topological lower bound(s) of the chromatic
number. In particular, we prove that such a relaxed version is true for the
Behzad-Vizing conjecture and also discuss the conjectures of Hedetniemi and of
Hadwiger from this point of view. For the latter, a similar statement was
already proven in an earlier paper of the first author with G. Tardos, our main
concern here is that the so-called odd Hadwiger conjecture looks much more
difficult in this respect. We prove that the statement of the odd Hadwiger
conjecture holds for large enough Kneser graphs and Schrijver graphs of any
fixed chromatic number
Some colouring problems for Paley graphs
The Paley graph Pq, where q≡1(mod4) is a prime power, is the graph with vertices the elements of the finite field Fq and an edge between x and y if and only if x-y is a non-zero square in Fq. This paper gives new results on some colouring problems for Paley graphs and related discussion. © 2005 Elsevier B.V. All rights reserved
Breaking Instance-Independent Symmetries In Exact Graph Coloring
Code optimization and high level synthesis can be posed as constraint
satisfaction and optimization problems, such as graph coloring used in register
allocation. Graph coloring is also used to model more traditional CSPs relevant
to AI, such as planning, time-tabling and scheduling. Provably optimal
solutions may be desirable for commercial and defense applications.
Additionally, for applications such as register allocation and code
optimization, naturally-occurring instances of graph coloring are often small
and can be solved optimally. A recent wave of improvements in algorithms for
Boolean satisfiability (SAT) and 0-1 Integer Linear Programming (ILP) suggests
generic problem-reduction methods, rather than problem-specific heuristics,
because (1) heuristics may be upset by new constraints, (2) heuristics tend to
ignore structure, and (3) many relevant problems are provably inapproximable.
Problem reductions often lead to highly symmetric SAT instances, and
symmetries are known to slow down SAT solvers. In this work, we compare several
avenues for symmetry breaking, in particular when certain kinds of symmetry are
present in all generated instances. Our focus on reducing CSPs to SAT allows us
to leverage recent dramatic improvement in SAT solvers and automatically
benefit from future progress. We can use a variety of black-box SAT solvers
without modifying their source code because our symmetry-breaking techniques
are static, i.e., we detect symmetries and add symmetry breaking predicates
(SBPs) during pre-processing.
An important result of our work is that among the types of
instance-independent SBPs we studied and their combinations, the simplest and
least complete constructions are the most effective. Our experiments also
clearly indicate that instance-independent symmetries should mostly be
processed together with instance-specific symmetries rather than at the
specification level, contrary to what has been suggested in the literature
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