11,865 research outputs found
Proof of the Feldman-Karlin Conjecture on the Maximum Number of Equilibria in an Evolutionary System
Feldman and Karlin conjectured that the number of isolated fixed points for
deterministic models of viability selection and recombination among n possible
haplotypes has an upper bound of 2^n - 1. Here a proof is provided. The upper
bound of 3^{n-1} obtained by Lyubich et al. (2001) using Bezout's Theorem
(1779) is reduced here to 2^n through a change of representation that reduces
the third-order polynomials to second order. A further reduction to 2^n - 1 is
obtained using the homogeneous representation of the system, which yields
always one solution `at infinity'. While the original conjecture was made for
systems of viability selection and recombination, the results here generalize
to viability selection with any arbitrary system of bi-parental transmission,
which includes recombination and mutation as special cases. An example is
constructed of a mutation-selection system that has 2^n - 1 fixed points given
any n, which shows that 2^n - 1 is the sharpest possible upper bound that can
be found for the general space of selection and transmission coefficients.Comment: 9 pages, 1 figure; v.4: final minor revisions, corrections,
additions; v.3: expands theorem to cover all cases, obviating v.2 distinction
of reducible/irreducible; details added to: discussion of Lyubich (1992),
example that attains upper bound, and homotopy continuation method
Resolvent Positive Linear Operators Exhibit the Reduction Phenomenon
The spectral bound, s(a A + b V), of a combination of a resolvent positive
linear operator A and an operator of multiplication V, was shown by Kato to be
convex in b \in R. This is shown here, through an elementary lemma, to imply
that s(a A + b V) is also convex in a > 0, and notably, \partial s(a A + b V) /
\partial a <= s(A) when it exists. Diffusions typically have s(A) <= 0, so that
for diffusions with spatially heterogeneous growth or decay rates, greater
mixing reduces growth. Models of the evolution of dispersal in particular have
found this result when A is a Laplacian or second-order elliptic operator, or a
nonlocal diffusion operator, implying selection for reduced dispersal. These
cases are shown here to be part of a single, broadly general, `reduction'
phenomenon.Comment: 7 pages, 53 citations. v.3: added citations, corrections in
introductory definitions. v.2: Revised abstract, more text, and details in
new proof of Lindqvist's inequalit
Radial Structure of the Internet
The structure of the Internet at the Autonomous System (AS) level has been
studied by both the Physics and Computer Science communities. We extend this
work to include features of the core and the periphery, taking a radial
perspective on AS network structure. New methods for plotting AS data are
described, and they are used to analyze data sets that have been extended to
contain edges missing from earlier collections. In particular, the average
distance from one vertex to the rest of the network is used as the baseline
metric for investigating radial structure. Common vertex-specific quantities
are plotted against this metric to reveal distinctive characteristics of
central and peripheral vertices. Two data sets are analyzed using these
measures as well as two common generative models (Barabasi-Albert and Inet). We
find a clear distinction between the highly connected core and a sparse
periphery. We also find that the periphery has a more complex structure than
that predicted by degree distribution or the two generative models
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