225,288 research outputs found
A lower bound on independence in terms of degrees
We prove a new lower bound on the independence number of a simple connected graph
in terms of its degrees
A lower bound on the independence number of a graph in terms of degrees
For a connected and non-complete graph, a new lower bound on its independence number is proved. It is shown that this bound is realizable by the well known efficient algorithm MIN
Degree Sequence Index Strategy
We introduce a procedure, called the Degree Sequence Index Strategy (DSI), by
which to bound graph invariants by certain indices in the ordered degree
sequence. As an illustration of the DSI strategy, we show how it can be used to
give new upper and lower bounds on the -independence and the -domination
numbers. These include, among other things, a double generalization of the
annihilation number, a recently introduced upper bound on the independence
number. Next, we use the DSI strategy in conjunction with planarity, to
generalize some results of Caro and Roddity about independence number in planar
graphs. Lastly, for claw-free and -free graphs, we use DSI to
generalize some results of Faudree, Gould, Jacobson, Lesniak and Lindquester
Relaxed Bell inequalities and Kochen-Specker theorems
The combination of various physically plausible properties, such as no
signaling, determinism, and experimental free will, is known to be incompatible
with quantum correlations. Hence, these properties must be individually or
jointly relaxed in any model of such correlations. The necessary degrees of
relaxation are quantified here, via natural distance and information-theoretic
measures. This allows quantitative comparisons between different models in
terms of the resources, such as the number of bits, of randomness,
communication, and/or correlation, that they require. For example, measurement
dependence is a relatively strong resource for modeling singlet state
correlations, with only 1/15 of one bit of correlation required between
measurement settings and the underlying variable. It is shown how various
'relaxed' Bell inequalities may be obtained, which precisely specify the
complementary degrees of relaxation required to model any given violation of a
standard Bell inequality. The robustness of a class of Kochen-Specker theorems,
to relaxation of measurement independence, is also investigated. It is shown
that a theorem of Mermin remains valid unless measurement independence is
relaxed by 1/3. The Conway-Kochen 'free will' theorem and a result of Hardy are
less robust, failing if measurement independence is relaxed by only 6.5% and
4.5%, respectively. An appendix shows the existence of an outcome independent
model is equivalent to the existence of a deterministic model.Comment: 19 pages (including 3 appendices); v3: minor clarifications, to
appear in PR
Bounds on Portfolio Quality
The signal-noise ratio of a portfolio of p assets, its expected return
divided by its risk, is couched as an estimation problem on the sphere. When
the portfolio is built using noisy data, the expected value of the signal-noise
ratio is bounded from above via a Cramer-Rao bound, for the case of Gaussian
returns. The bound holds for `biased' estimators, thus there appears to be no
bias-variance tradeoff for the problem of maximizing the signal-noise ratio. An
approximate distribution of the signal-noise ratio for the Markowitz portfolio
is given, and shown to be fairly accurate via Monte Carlo simulations, for
Gaussian returns as well as more exotic returns distributions. These findings
imply that if the maximal population signal-noise ratio grows slower than the
universe size to the 1/4 power, there may be no diversification benefit, rather
expected signal-noise ratio can decrease with additional assets. As a practical
matter, this may explain why the Markowitz portfolio is typically applied to
small asset universes. Finally, the theorem is expanded to cover more general
models of returns and trading schemes, including the conditional expectation
case where mean returns are linear in some observable features, subspace
constraints (i.e., dimensionality reduction), and hedging constraints
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