A solution to the perimeter surveillance problem for one intruder and multiple robots based on set-invariance is presented. The robots, constrained move of a polygonal region, intercept intruders as they cross perimeter. proposed closed-form control laws only depend maximum speed their distances endpoints line segments that make sides polygon. presented results allow groups with members differe...

We consider the problem of minimizing the kth Dirichlet eigenvalue of planar domains with fixed perimeter and show that, as k goes to infinity, the optimal domain converges to the ball with the same perimeter. We also consider this problem within restricted classes of domains such as n−polygons and tiling domains, for which we show that the optimal asymptotic domain is that which maximises the ...

It is well known that sets of finite perimeter can be strictly approximated by smooth sets, while, in general, one cannot hope to approximate an open set Ω of finite perimeter in R strictly from within. In this note we show that, nevertheless, the latter type of approximation is possible under the mild hypothesis that the (n−1)-dimensional Hausdorff measure of the topological boundary ∂Ω equals...

In this paper we present a new bidirectional heuristic search algorithm. Our algorithm can be viewed as a perimeter search algorithm, and it uses a new technique for reducing the number of heuristic evaluations. We also prove some general results on the behavior of iterative deepening perimeter search algorithms, and we discuss some new “lazy evaluation” techniques for improving their performan...

For a rank-1 matrix A = a ⊗ b over max algebra, we define the perimeter of A as the number of nonzero entries in both a and b. We characterize the linear operators which preserve the rank and perimeter of rank-1 matrices over max algebra. That is, a linear operator T preserves the rank and perimeter of rank-1 matrices if and only if it has the form T (A) = U ⊗ A ⊗ V , or T (A) = U ⊗ A ⊗ V with ...

Alon, N. and D.J. Kleitman, Partitioning a rectangle into small perimeter rectangles, Discrete Mathematics 103 (1992) 111-119. We show that the way to partition a unit square into kZ + s rectangles, for s = 1 or s = -1, so as to minimize the largest perimeter of the rectangles, is to have k 1 rows of k identical rectangles and one row of k + s identical rectangles, with all rectangles having th...

Joseph Emerson,1 Robert Alicki,2 and Karol Życzkowski1, 3, 4 Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada Institute of Theoretical Physics and Astrophysics, University of Gdańsk, Wita Stwosza 57, PL 80-952 Gdańsk, Poland Institute of Physics, Jagiellonian University, ul. Reymonta 4, 31-059 Cracow, Poland Center for Thoretical Physics, Polish Academy of Sciences, Al Lot...

There exist tilings of the plane with pairwise noncongruent triangles of equal area and bounded perimeter. Analogously, there exist tilings with triangles of equal perimeter, the areas of which are bounded from below by a positive constant. This solves a problem of Nandakumar.

The author presents a simple technique for teaching the Cournot model to first-year students. The approach involves demonstrating to the students that out of all rectangles with a common perimeter, the square has the greatest area. No use is made of derivatives. The same approach can be used to understand some other market forms.

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