689 research outputs found
Complexity of Some Geometric Problems
We show that recognizing intersection graphs of convex sets has the same complexity as deciding truth in the existential first-order theory of the reals. Comparing this to similar results on the rectilinear crossing number and intersection graphs of line segments, we argue that there is a need to recognize this level of complexity as its own class
Beyond the Existential Theory of the Reals
We show that completeness at higher levels of the theory of the reals is a
robust notion (under changing the signature and bounding the domain of the
quantifiers). This mends recognized gaps in the hierarchy, and leads to
stronger completeness results for various computational problems. We exhibit
several families of complete problems which can be used for future completeness
results in the real hierarchy. As an application we sharpen some results by
B\"{u}rgisser and Cucker on the complexity of properties of semialgebraic sets,
including the Hausdorff distance problem also studied by Jungeblut, Kleist, and
Miltzow
Multi-Sided Boundary Labeling
In the Boundary Labeling problem, we are given a set of points, referred
to as sites, inside an axis-parallel rectangle , and a set of pairwise
disjoint rectangular labels that are attached to from the outside. The task
is to connect the sites to the labels by non-intersecting rectilinear paths,
so-called leaders, with at most one bend.
In this paper, we study the Multi-Sided Boundary Labeling problem, with
labels lying on at least two sides of the enclosing rectangle. We present a
polynomial-time algorithm that computes a crossing-free leader layout if one
exists. So far, such an algorithm has only been known for the cases in which
labels lie on one side or on two opposite sides of (here a crossing-free
solution always exists). The case where labels may lie on adjacent sides is
more difficult. We present efficient algorithms for testing the existence of a
crossing-free leader layout that labels all sites and also for maximizing the
number of labeled sites in a crossing-free leader layout. For two-sided
boundary labeling with adjacent sides, we further show how to minimize the
total leader length in a crossing-free layout
Strong Hanani-Tutte for the Torus
If a graph can be drawn on the torus so that every two independent edges cross an even number of times, then the graph can be embedded on the torus
Strong Hanani-Tutte on the Projective Plane
If a graph can be drawn in the projective plane so that every two non-adjacent edges cross an even number of times, then the graph can be embedded in the projective plane
Design of Low Complexity Model Reference Adaptive Controllers
Flight research experiments have demonstrated that adaptive flight controls can be an effective technology for improving aircraft safety in the event of failures or damage. However, the nonlinear, timevarying nature of adaptive algorithms continues to challenge traditional methods for the verification and validation testing of safety-critical flight control systems. Increasingly complex adaptive control theories and designs are emerging, but only make testing challenges more difficult. A potential first step toward the acceptance of adaptive flight controllers by aircraft manufacturers, operators, and certification authorities is a very simple design that operates as an augmentation to a non-adaptive baseline controller. Three such controllers were developed as part of a National Aeronautics and Space Administration flight research experiment to determine the appropriate level of complexity required to restore acceptable handling qualities to an aircraft that has suffered failures or damage. The controllers consist of the same basic design, but incorporate incrementally-increasing levels of complexity. Derivations of the controllers and their adaptive parameter update laws are presented along with details of the controllers implementations
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