1,205,533 research outputs found
Regular cross sections of Borel flows
Any free Borel flow is shown to admit a cross section with only two possible
distances between adjacent points. Non smooth flows are proved to be Lebesgue
orbit equivalent if and only if they admit the same number of invariant ergodic
probability measures.Comment: Minor improvements in expositio
Hamiltonicity in connected regular graphs
In 1980, Jackson proved that every 2-connected -regular graph with at most
vertices is Hamiltonian. This result has been extended in several papers.
In this note, we determine the minimum number of vertices in a connected
-regular graph that is not Hamiltonian, and we also solve the analogous
problem for Hamiltonian paths. Further, we characterize the smallest connected
-regular graphs without a Hamiltonian cycle.Comment: 5 page
Sphere packings II
An earlier paper describes a program to prove the Kepler conjecture on sphere
packings. This paper carries out the second step of that program. A sphere
packing leads to a decomposition of into polyhedra. The polyhedra are
divided into two classes. The first class of polyhedra, called quasi-regular
tetrahedra, have density at most that of a regular tetrahedron. The polyhedra
in the remaining class have density at most that of a regular octahedron (about
0.7209).Comment: 18 pages. Second of two older papers in the series on the proof of
the Kepler conjecture. See math.MG/9811071. The original abstract is
preserve
On the Structure and Complexity of Rational Sets of Regular Languages
In a recent thread of papers, we have introduced FQL, a precise specification
language for test coverage, and developed the test case generation engine
FShell for ANSI C. In essence, an FQL test specification amounts to a set of
regular languages, each of which has to be matched by at least one test
execution. To describe such sets of regular languages, the FQL semantics uses
an automata-theoretic concept known as rational sets of regular languages
(RSRLs). RSRLs are automata whose alphabet consists of regular expressions.
Thus, the language accepted by the automaton is a set of regular expressions.
In this paper, we study RSRLs from a theoretic point of view. More
specifically, we analyze RSRL closure properties under common set theoretic
operations, and the complexity of membership checking, i.e., whether a regular
language is an element of a RSRL. For all questions we investigate both the
general case and the case of finite sets of regular languages. Although a few
properties are left as open problems, the paper provides a systematic semantic
foundation for the test specification language FQL
- …