HOMOTOPY Research Papers - Academia.edu (original) (raw)

We give algorithms to find shortest paths homotopic to given disjoint paths that wind amongst n point obstacles in the plane. Our deterministic algorithm runs in time O(k log n+n√n), and the randomized version in time O(k log n+n(log... more

We give algorithms to find shortest paths homotopic to given disjoint paths that wind amongst n point obstacles in the plane. Our deterministic algorithm runs in time O(k log n+n√n), and the randomized version in time O(k log n+n(log n)1+ε) where k is the input plus output sizes of the paths.

Abstract. We study Rickard equivalences between p-blocks of twisted group algebras and their local structure, in connection with Dade's conjectures. We prove that an extended form of Broué's conjecture implies Dade's... more

Abstract. We study Rickard equivalences between p-blocks of twisted group algebras and their local structure, in connection with Dade's conjectures. We prove that an extended form of Broué's conjecture implies Dade's Inductive Conjecture in the Abelian defect group case; this is a ...

A fundamental task in geodesy is solving systems of equations. Many geodetic problems are represented as systems of multivariate polynomials. A common problem in solving such systems is improper initial starting values for iterative... more

A fundamental task in geodesy is solving systems of equations. Many geodetic problems are represented as systems of multivariate polynomials. A common problem in solving such systems is improper initial starting values for iterative methods, leading to convergence to solutions with no physical meaning, or to convergence that requires global methods. Though symbolic methods such as Groebner bases or resultants have been shown to be very efficient, i.e., providing solutions for determined systems such as 3-point problem of 3D affine transformation, the symbolic algebra can be very time consuming, even with special Computer Algebra Systems (CAS). This study proposes the Linear Homotopy method that can be implemented easily in high-level computer languages like C++ and Fortran that are faster than CAS by at least two orders of magnitude. Using Mathematica, the power of Homotopy is demonstrated in solving three nonlinear geodetic problems: resection, GPS positioning, and affine transformation. The method enlarging the domain of convergence is found to be efficient, less sensitive to rounding of numbers, and has lower complexity compared to other local methods like Newton–Raphson.

matcont is a matlab continuation package for the interactive numerical study of a range of parameterized nonlinear dynamical systems, in particular ODEs, that allows to compute curves of equilibria, limit points, Hopf points, limit... more

matcont is a matlab continuation package for the interactive numerical study of a range of parameterized nonlinear dynamical systems, in particular ODEs, that allows to compute curves of equilibria, limit points, Hopf points, limit cycles, flip, fold and torus bifurcation points of limit cycles. It is now possible to continue homoclinic-to-hyperbolic-saddle and homoclinic-to-saddle-node orbits in matcont . The implementation is done using the continuation of invariant subspaces, with the Riccati equations included in the defining system. A key feature is the possibility to initiate both types of homoclinic orbits interactively, starting from an equilibrium point and using a homotopy method. All known codimension-two homoclinic bifurcations are tested for during continuation. The test functions for inclination-flip bifurcations are implemented in a new and more efficient way. Heteroclinic orbits can now also be continued and an analogous homotopy method can be used for the initializa...

The synchronization of oscillators is defined step by step from the exemple of an electronic phase-locked loop. Besides, an injection-locked pulsed delay oscillator is built in which the signal is synchronized both in its frequency and... more

The synchronization of oscillators is defined step by step from the exemple of an electronic phase-locked loop. Besides, an injection-locked pulsed delay oscillator is built in which the signal is synchronized both in its frequency and its duration : the local behaviour (the carrier) is thus resonating with the global behaviour (the envelope). A model of the electronic wave function in terms of the fundamental groups of punctured Riemann surfaces is attempted. The structure of synchronization zones goes in favour of that hypothesis.

ABSTRACT A global analysis of Branin's method (originally due to Davidenko) for finding all the real zeros of a vector function is carried out. The analysis is based on a global study of this method perceived of as a dynamical... more

ABSTRACT A global analysis of Branin's method (originally due to Davidenko) for finding all the real zeros of a vector function is carried out. The analysis is based on a global study of this method perceived of as a dynamical system. Since Branin's algorithm is closely related to homotopy methods, this paper sheds some light on the global performance of these methods when employed for locating all the zeros of a vector function. Following the dynamical system approach, the performance of Branin's algorithm is related to the existence of extraneous singularities as well as to the relative spatial distribution of the zeros of the vector function and singular manifolds. Branin's conjectures regarding the types and the role of extraneous singularities are examined and counter-examples are provided to disprove them. We conclude that the performance of Branin's method for locating all the zeros of a vector function is questionable even in the absence of extraneous singularities.

In this dissertation we first study partition posets and their topology. For each composition c we show that the order complex of the poset of pointed set partitions is a wedge of spheres of the same dimension with the multiplicity given... more

In this dissertation we first study partition posets and their topology. For each composition c we show that the order complex of the poset of pointed set partitions is a wedge of spheres of the same dimension with the multiplicity given by the number of permutations with descent composition c. Furthermore, the action of the symmetric group on the top homology is isomorphic to the Specht module of a border strip associated to the composition. We also study the filter of pointed set partitions generated by knapsack integer partitions. In the second half of this dissertation we study descent avoidance in permutations. We extend the notion of consecutive pattern avoidance to considering sums over all permutations where each term is a product of weights depending on each consecutive pattern of a fixed length. We study the problem of finding the asymptotics of these sums. Our technique is to extend the spectral method of Ehrenborg, Kitaev and Perry. When the weight depends on the descent...

In this paper, we propose a novel and coherent framework for fast footstep planning for legged robots on a flat ground with 3-D obstacle avoidance. We use swept volume approximations that are computed offline in order to considerably... more

In this paper, we propose a novel and coherent framework for fast footstep planning for legged robots on a flat ground with 3-D obstacle avoidance. We use swept volume approximations that are computed offline in order to considerably reduce the time spent in collision checking during the online planning phase, in which a rapidly exploring random tree variant is used