We propose a Hamilton-Jacobi equation approach for computing time-optimal trajectories of a vehicle which travels under certain curvature constraints. We derive a class of Hamilton-Jacobi equations which models such motions; it unifies two well-known vehicular models, the Dubins’ and Reeds-Shepp’s cars, and gives further generalizations. We show that the value function of the control problem so...

a Advanced Photon Source Argonne National Laboratory 9700 S. Cass Avenue Argonne, Illinois 60439, USA b James Franck Institute and Department of Physics The University of Chicago 929 E 57 Street Chicago, Illinois 60637, USA c Department of Physics and Astronomy McMaster University 1280 Main St. W, Hamilton Hamilton, Ontario, L8S 4M1, Canada and Canadian Institute for Advanced Research 180 Dunda...

Division of Orthopaedic Surgery, Department of Surgery, McMaster University Medical Centre, Hamilton, Ontario, Canada Michael G. DeGroote School of Medicine, McMaster University, Hamilton, Ontario, Canada Department of Clinical Epidemiology and Biostatistics, McMaster University, Hamilton, Ontario, Canada Department of Orthopaedic Surgery, UPMC Center for Sports Medicine, University of Pittsbur...

Let n ≥ 2 be an integer. The complete graph Kn with a 1-factor F removed has a decomposition into Hamilton cycles if and only if n is even. We show that Kn − F has a decomposition into Hamilton cycles which are symmetric with respect to the 1-factor F if and only if n ≡ 2, 4 mod 8. We also show that the complete bipartite graph Kn,n has a symmetric Hamilton cycle decomposition if and only if n ...

We describe an algorithm which enumerates all Hamilton cycles of a given 3regular n-vertex graph in time O(1.276n), improving on Eppstein’s previous bound. The resulting new upper bound of O(1.276n) for the maximum number of Hamilton cycles in 3-regular n-vertex graphs gets close to the best known lower bound of Ω(1.259n). Our method differs from Eppstein’s in that he considers in each step a n...

Tzitzeica hypersurfaces provided by figuratrices of Hamilton and generalized Hamilton spaces are studied. Mathematics Subject Classification: 53A07, 53C60.

We discuss an extension of the Hamilton–Jacobi theory to nonholonomic mechanics with a particular interest in its application to exactly integrating the equations of motion. We give an intrinsic proof of a nonholonomic analogue of the Hamilton–Jacobi theorem. Our intrinsic proof clarifies the difference from the conventional Hamilton–Jacobi theory for unconstrained systems. The proof also helps...

This thesis contains problems in finding spanning subgraphs in graphs, such as, perfect matchings, tilings and Hamilton cycles. First, we consider the tiling problems in graphs, which are natural generalizations of the matching problems. We give new proofs of the multipartite Hajnal-Szemerédi Theorem for the tripartite and quadripartite cases. Second, we consider Hamilton cycles in hypergraphs....

We develop Hamilton–Jacobi theory for Chaplygin systems, a certain class of nonholonomic mechanical systems with symmetries, using a technique called Hamiltonization, which transforms nonholonomic systems into Hamiltonian systems. We give a geometric account of the Hamiltonization, identify necessary and sufficient conditions for Hamiltonization, and apply the conventional Hamilton–Jacobi theor...

In this paper, we settle Alspach’s problem in the case of Hamilton cycles and 5cycles; that is, we show that for all odd integers n ≥ 5 and all nonnegative integers h and t with hn + 5t = n(n − 1)/2, the complete graph Kn decomposes into h Hamilton cycles and t 5-cycles and for all even integers n ≥ 6 and all nonnegative integers h and t with hn+5t = n(n−2)/2, the complete graph Kn decomposes i...