135 research outputs found
Condorcet Domains, Median Graphs and the Single Crossing Property
Condorcet domains are sets of linear orders with the property that, whenever
the preferences of all voters belong to this set, the majority relation has no
cycles. We observe that, without loss of generality, such domain can be assumed
to be closed in the sense that it contains the majority relation of every
profile with an odd number of individuals whose preferences belong to this
domain.
We show that every closed Condorcet domain is naturally endowed with the
structure of a median graph and that, conversely, every median graph is
associated with a closed Condorcet domain (which may not be a unique one). The
subclass of those Condorcet domains that correspond to linear graphs (chains)
are exactly the preference domains with the classical single crossing property.
As a corollary, we obtain that the domains with the so-called `representative
voter property' (with the exception of a 4-cycle) are the single crossing
domains.
Maximality of a Condorcet domain imposes additional restrictions on the
underlying median graph. We prove that among all trees only the chains can
induce maximal Condorcet domains, and we characterize the single crossing
domains that in fact do correspond to maximal Condorcet domains.
Finally, using Nehring's and Puppe's (2007) characterization of monotone
Arrowian aggregation, our analysis yields a rich class of strategy-proof social
choice functions on any closed Condorcet domain
On k-Convex Polygons
We introduce a notion of -convexity and explore polygons in the plane that
have this property. Polygons which are \mbox{-convex} can be triangulated
with fast yet simple algorithms. However, recognizing them in general is a
3SUM-hard problem. We give a characterization of \mbox{-convex} polygons, a
particularly interesting class, and show how to recognize them in \mbox{} time. A description of their shape is given as well, which leads to
Erd\H{o}s-Szekeres type results regarding subconfigurations of their vertex
sets. Finally, we introduce the concept of generalized geometric permutations,
and show that their number can be exponential in the number of
\mbox{-convex} objects considered.Comment: 23 pages, 19 figure
Two reasons for the appearance of pushed wavefronts in the Belousov-Zhabotinsky system with spatiotemporal interaction
We prove the existence of the minimal speed of propagation for wavefronts in the Belousov-Zhabotinsky system with a
spatiotemporal interaction defined by the convolution with (possibly,
"fat-tailed") kernel . The model is assumed to be monostable non-degenerate,
i.e. . The slowest wavefront is termed pushed or non-linearly
determined if its velocity . We show that
is close to 2 if i) positive system's parameter is
sufficiently large or ii) if is spatially asymmetric to one side (e.g. to
the left: in such a case, the influence of the right side concentration of the
bromide ion on the dynamics is more significant than the influence of the left
side). Consequently, this reveals two reasons for the appearance of pushed
wavefronts in the Belousov-Zhabotinsky reaction.Comment: 22 pages, submitte
Global attractors for doubly nonlinear evolution equations with non-monotone perturbations
This paper proposes an abstract theory concerned with dynamical systems
generated by doubly nonlinear evolution equations governed by subdifferential
operators with non-monotone perturbations in a reflexive Banach space setting.
In order to construct global attractors, an approach based on the notion of
generalized semiflow is employed instead of the usual semi-group approach,
since solutions of the Cauchy problem for the equation might not be unique.
Moreover, the preceding abstract theory is applied to a generalized Allen-Cahn
equation whose potential is divided into a convex part and a non-convex part as
well as a semilinear parabolic equation with a nonlinear term involving
gradients
On Range Searching with Semialgebraic Sets II
Let be a set of points in . We present a linear-size data
structure for answering range queries on with constant-complexity
semialgebraic sets as ranges, in time close to . It essentially
matches the performance of similar structures for simplex range searching, and,
for , significantly improves earlier solutions by the first two authors
obtained in~1994. This almost settles a long-standing open problem in range
searching.
The data structure is based on the polynomial-partitioning technique of Guth
and Katz [arXiv:1011.4105], which shows that for a parameter , , there exists a -variate polynomial of degree such that
each connected component of contains at most points
of , where is the zero set of . We present an efficient randomized
algorithm for computing such a polynomial partition, which is of independent
interest and is likely to have additional applications
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