285 research outputs found
Bi-Lipschitz Bijection between the Boolean Cube and the Hamming Ball
We construct a bi-Lipschitz bijection from the Boolean cube to the Hamming
ball of equal volume. More precisely, we show that for all even n there exists
an explicit bijection f from the n-dimensional Boolean cube to the Hamming ball
of equal volume embedded in (n+1)-dimensional Boolean cube, such that for all x
and y it holds that distance(x,y) / 5 <= distance(f(x),f(y)) <= 4 distance(x,y)
where distance(,) denotes the Hamming distance. In particular, this implies
that the Hamming ball is bi-Lipschitz transitive.
This result gives a strong negative answer to an open problem of Lovett and
Viola [CC 2012], who raised the question in the context of sampling
distributions in low-level complexity classes. The conceptual implication is
that the problem of proving lower bounds in the context of sampling
distributions will require some new ideas beyond the sensitivity-based
structural results of Boppana [IPL 97].
We study the mapping f further and show that it (and its inverse) are
computable in DLOGTIME-uniform TC0, but not in AC0. Moreover, we prove that f
is "approximately local" in the sense that all but the last output bit of f are
essentially determined by a single input bit
Liquid interfaces in viscous straining flows: Numerical studies of the selective withdrawal transition
This paper presents a numerical analysis of the transition from selective
withdrawal to viscous entrainment. In our model problem, an interface between
two immiscible layers of equal viscosity is deformed by an axisymmetric
withdrawal flow, which is driven by a point sink located some distance above
the interface in the upper layer. We find that steady-state hump solutions,
corresponding to selective withdrawal of liquid from the upper layer, cease to
exist above a threshold withdrawal flux, and that this transition corresponds
to a saddle-node bifurcation for the hump solutions. Numerical results on the
shape evolution of the steady-state interface are compared against previous
experimental measurements. We find good agreement where the data overlap.
However, the numerical results' larger dynamic range allows us to show that the
large increase in the curvature of the hump tip near transition is not
consistent with an approach towards a power-law cusp shape, an interpretation
previously suggested from inspection of the experimental measurements alone.
Instead the large increase in the curvature at the hump tip reflects a
logarithmic coupling between the overall height of the hump and the curvature
at the tip of the hump.Comment: submitted to JF
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