There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

This paper presents the first randomized approach to kinodynamic planning (also known as trajectory planning or trajectory design). The task is to determine control inputs to drive a robot from an ...

Randomized kinodynamic planning

Real and Complex Analysis

Preface 1. The approximation problem and existence of best approximations 2. The uniqueness of best approximations 3. Approximation operators and some approximating functions 4. Polynomial interpolation 5. Divided differences 6. The uniform convergence of polynomial approximations 7. The theory of minimax approximation 8. The exchange algorithm 9. The convergence of the exchange algorithm 10. Rational approximation by the exchange algorithm 11. Least squares approximation 12. Properties of orthogonal polynomials 13. Approximation of periodic functions 14. The theory of best L1 approximation 15. An example of L1 approximation and the discrete case 16. The order of convergence of polynomial approximations 17. The uniform boundedness theorem 18. Interpolation by piecewise polynomials 19. B-splines 20. Convergence properties of spline approximations 21. Knot positions and the calculation of spline approximations 22. The Peano kernel theorem 23. Natural and perfect splines 24. Optimal interpolation Appendices Index.

Approximation theory and methods, by M. J. D. Powell. Pp 339. £25 (hardcover), £8·50 (paperback). 1981. ISBN 0-521-22472-1/29514-9 (Cambridge University Press)

This paper considers the problem of motion planning for a car-like robot (i.e., a mobile robot with a nonholonomic constraint whose turning radius is lower-bounded). We present a fast and exact planner for our mobile robot model, based upon recursive subdivision of a collision-free path generated by a lower-level geometric planner that ignores the motion constraints. The resultant trajectory is optimized to give a path that is of near-minimal length in its homotopy class. Our claims of high speed are supported by experimental results for implementations that assume a robot moving amid polygonal obstacles. The completeness and the complexity of the algorithm are proven using an appropriate metric in the configuration space R/sup 2//spl times/S/sup 1/ of the robot. This metric is defined by using the length of the shortest paths in the absence of obstacles as the distance between two configurations. We prove that the new induced topology and the classical one are the same. Although we concentrate upon the car-like robot, the generalization of these techniques leads to new theoretical issues involving sub-Riemannian geometry and to practical results for nonholonomic motion planning. >

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A motion planner for nonholonomic mobile robots

We exhibit new links between approximation theory in the complex domain and a family of inverse problems for the 2D Laplacian related to non-destructive testing.

How can the meromorphic approximation help to solve some 2D inverse problems for the Laplacian

We consider approximation of $L^p$ functions by Hardy functions on subsets of the circle. We first derive some properties of traces of Hardy classes on such subsets, and then turn to a generalization of classical extremal problems involving norm constraints on the complementary subset.

https://hal.inria.fr/inria-00074299/document

Hardy approximation to $L^p$ functions on subsets of the circle

We consider approximation of L
 p
 functions by Hardy functions on subsets of the circle for $1 \leq p < \infty$  After some preliminaries on the possibility of such an approximation which are connected to recovery problems of the Carleman type, we prove existence and uniqueness of the solution to a generalized extremal problem involving norm constraints on the complementary subset

Hardy Approximation to L p Functions on Subsets of the Circle with 1< =p< = infinity

We consider approximation ofL\n ∞ functions byH\n ∞ functions on proper substs of the circle. We derive some properties of traces of Hardy classes on such subsets, and then turn to a generalization of classical extremal problems involving norm constraints on the complementary subset.

Hardy Approximation to $L^$ Functions on Subsets of the Circle

We are concerned with non-destructive control issues, namely detection and recovery of cracks in a planar (2D) isotropic conductor from partial boundary measurements of a solution to the Laplace–Neumann problem. We first build an extension of that solution to the whole boundary, using constructive approximation techniques in classes of analytic and meromorphic functions, and then use localization algorithms based on boundary computations of the reciprocity gap.

Juliette Leblond

Papers

How can the meromorphic approximation help to solve some 2D inverse problems for the Laplacian

Hardy approximation to $L^p$ functions on subsets of the circle

Hardy Approximation to L p Functions on Subsets of the Circle with 1< =p< = infinity

Hardy Approximation to $L^$ Functions on Subsets of the Circle

Line segment crack recovery from incomplete boundary data