764 research outputs found
Greedy energy minimization can count in binary: point charges and the van der Corput sequence
This paper establishes a connection between a problem in Potential Theory and
Mathematical Physics, arranging points so as to minimize an energy functional,
and a problem in Combinatorics and Number Theory, constructing
'well-distributed' sequences of points on . Let be (i) symmetric , (ii) twice differentiable on
, and (iii) such that for all . We study the
greedy dynamical system, where, given an initial set , the point is obtained as We prove that if we start this
construction with the single element , then all arising constructions
are permutations of the van der Corput sequence (counting in binary and
reflected about the comma): \textit{greedy energy minimization recovers the way
we count in binary.} This gives a new construction of the classical van der
Corput sequence. The special case answers a
question of Steinerberger. Interestingly, the point sets we derive are also
known in a different context as Leja sequences on the unit disk. Moreover, we
give a general bound on the discrepancy of any sequence constructed in this way
for functions satisfying an additional assumption.Comment: 18 pages, 7 figures, discrepancy bound adde
Optimal control for diffusions on graphs
Starting from a unit mass on a vertex of a graph, we investigate the minimum
number of "\emph{controlled diffusion}" steps needed to transport a constant
mass outside of the ball of radius . In a step of a controlled diffusion
process we may select any vertex with positive mass and topple its mass equally
to its neighbors. Our initial motivation comes from the maximum overhang
question in one dimension, but the more general case arises from optimal mass
transport problems.
On we show that steps are necessary and
sufficient to transport the mass. We also give sharp bounds on the comb graph
and -ary trees. Furthermore, we consider graphs where simple random walk has
positive speed and entropy and which satisfy Shannon's theorem, and show that
the minimum number of controlled diffusion steps is , where is the Avez asymptotic entropy and is the speed
of random walk. As examples, we give precise results on Galton-Watson trees and
the product of trees .Comment: 32 pages, 2 figure
Polarization and Greedy Energy on the Sphere
We investigate the behavior of a greedy sequence on the sphere
defined so that at each step the point that minimizes the Riesz -energy is
added to the existing set of points. We show that for , the greedy
sequence achieves optimal second-order behavior for the Riesz -energy (up to
constants). In order to obtain this result, we prove that the second-order term
of the maximal polarization with Riesz -kernels is of order in the
same range . Furthermore, using the Stolarsky principle relating the
-discrepancy of a point set with the pairwise sum of distances (Riesz
energy with ), we also obtain a simple upper bound on the -spherical
cap discrepancy of the greedy sequence and give numerical examples that
indicate that the true discrepancy is much lower.Comment: 29 pages, 10 figure
The mobile Boolean model: an overview and further results
This paper offers an overview of the mobile Boolean stochastic geometric
model which is a time-dependent version of the ordinary Boolean model in a
Euclidean space of dimension . The main question asked is that of obtaining
the law of the detection time of a fixed set. We give various ways of thinking
about this which result into some general formulas. The formulas are solvable
in some special cases, such the inertial and Brownian mobile Boolean models. In
the latter case, we obtain some expressions for the distribution of the
detection time of a ball, when the dimension is odd and asymptotics when
is even. Finally, we pose some questions for future research.Comment: 19 page
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