We prove that computing a minimum-dilation (Euclidean) Hamilton circuit or path on a given set of points in the plane is NP-hard.

The nontrivial transformation of the phase space path integral measure under certain discretized analogues of canonical transformations is computed. This Jacobian is used to derive a quantum analogue of the Hamilton-Jacobi equation for the generating function of a canonical transformation that maps any quantum system to a system with a vanishing Hamiltonian. A formal perturbative solution of th...

TheMonotone Upper Bound Problem (Klee, 1965) asks if the numberM(d, n) of vertices in a monotone path along edges of a d-dimensional polytope with n facets can be as large as conceivably possible: Is M(d, n) = Mubt(d, n), the maximal number of vertices that a d-polytope with n facets can have according to the Upper Bound Theorem? We show that in dimension d = 4, the answer is “yes”, despite the...

The middle levels problem is to find a Hamilton cycle in the middle levels, M2k+1, of the Hasse diagram of B2k+1 (the partially ordered set of subsets of a 2k + 1element set ordered by inclusion). Previously, the best known, from [1], was that M2k+1 is Hamiltonian for all positive k through k = 15. In this note we announce that M33 and M35 have Hamilton cycles. The result was achieved by an alg...

The multi-path algorithm for finding hamilton cycles in a graph G is described in the book by Christofides. It is an intelligent exhaustive search for a hamilton cycle. In this paper we describe how the algorithm can be improved in two ways: (1) by detecting small separating sets M for which G–M has more than |M| components; and (2) by detecting bipartitions (X,Y), where |X|<|Y|.

Tutte has shown that every 4-connected planar graph contains a Hamilton cycle. Grünbaum and Nash-Williams independently conjectured that the same is true for toroidal graphs. In this paper, we prove that every 4-connected toroidal graph contains a Hamilton path. Partially supported by NSF grant DMS-9970514 Partially supported by NSF grants DMS-9970527 and DMS-0245530 Partially supported by RGC ...

The notorious middle two levels problem is to nd a Hamilton cycle in the middle two levels, M 2k+1 , of the Hasse diagram of B 2k+1 (the partially ordered set of subsets of a 2k + 1-element set ordered by inclusion). Previously, the best known result, due to Moews and Reid 11] in 1990, was that M 2k+1 is Hamiltonian for all positive k through k = 11. We show that if a Hamilton path between two ...

After reducing a system of higher-order regular Lagrangian into first-order singular Lagrangian using constrained auxiliary description, the Hamilton-Jacobi function is constructed. Besides, the quantization of the system is investigated using the canonical path integral approximation.