406 research outputs found
On stability of the Hamiltonian index under contractions and closures
The hamiltonian index of a graph is the smallest integer such that the -th iterated line graph of is hamiltonian. We first show that, with one exceptional case, adding an edge to a graph cannot increase its hamiltonian index. We use this result to prove that neither the contraction of an -contractible subgraph of a graph nor the closure operation performed on (if is claw-free) affects the value of the hamiltonian index of a graph
A consistent treatment of link and writhe for open rods, and their relation to end rotation
We combine and extend the work of Alexander & Antman \cite{alexander.82} and
Fuller \cite{fuller.71,fuller.78} to give a framework within which precise
definitions can be given of topological and geometrical quantities
characterising the contortion of open rods undergoing large deformations under
end loading. We use these definitions to examine the extension of known results
for closed rods to open rods. In particular, we formulate the analogue of the
celebrated formula (link equals twist plus writhe) for open rods and
propose an end rotation, through which the applied end moment does work, in the
form of an integral over the length of the rod. The results serve to promote
the variational analysis of boundary-value problems for rods undergoing large
deformations.Comment: 17 pages, 4 figure
Flat forms, bi-Lipschitz parametrizations, and smoothability of manifolds
We give a sufficient condition for a metric (homology) manifold to be locally
bi-Lipschitz equivalent to an open subset in \rn. The condition is a Sobolev
condition for a measurable coframe of flat 1-forms. In combination with an
earlier work of D. Sullivan, our methods also yield an analytic
characterization for smoothability of a Lipschitz manifold in terms of a
Sobolev regularity for frames in a cotangent structure. In the proofs, we
exploit the duality between flat chains and flat forms, and recently
established differential analysis on metric measure spaces. When specialized to
\rn, our result gives a kind of asymptotic and Lipschitz version of the
measurable Riemann mapping theorem as suggested by Sullivan
Recommended from our members
Market Arbitrage, Social Choice and the Core
This paper establishes a clear connection between equilibrium theory, game theory and social choice theory by showing that, for a well defined social choice problem, a condition which is necessary and sufficient to solve this problem-limited arbitrage-is the same as the condition which is necessary and sufficient to establish the existence of a equilibrium and the core. The connection is strengthened by establishing that a market allocation, which is in the core, can always be realized as a social allocation, i.e. an allocation which is optimal according to an ordering chosen by a social choice rule. Limited arbitrage characterizes those economies without Condorcet triples, and those for which Arrow's paradox can be resolved on choices of large utility values
Exploiting structure to cope with NP-hard graph problems: Polynomial and exponential time exact algorithms
An ideal algorithm for solving a particular problem always finds an optimal solution, finds such a solution for every possible instance, and finds it in polynomial time. When dealing with NP-hard problems, algorithms can only be expected to possess at most two out of these three desirable properties. All algorithms presented in this thesis are exact algorithms, which means that they always find an optimal solution. Demanding the solution to be optimal means that other concessions have to be made when designing an exact algorithm for an NP-hard problem: we either have to impose restrictions on the instances of the problem in order to achieve a polynomial time complexity, or we have to abandon the requirement that the worst-case running time has to be polynomial. In some cases, when the problem under consideration remains NP-hard on restricted input, we are even forced to do both.
Most of the problems studied in this thesis deal with partitioning the vertex set of a given graph. In the other problems the task is to find certain types of paths and cycles in graphs. The problems all have in common that they are NP-hard on general graphs. We present several polynomial time algorithms for solving restrictions of these problems to specific graph classes, in particular graphs without long induced paths, chordal graphs and claw-free graphs. For problems that remain NP-hard even on restricted input we present exact exponential time algorithms. In the design of each of our algorithms, structural graph properties have been heavily exploited. Apart from using existing structural results, we prove new structural properties of certain types of graphs in order to obtain our algorithmic results
Exotica or the failure of the strong cosmic censorship in four dimensions
In this letter a generic counterexample to the strong cosmic censor
conjecture is exhibited. More precisely---taking into account that the
conjecture lacks any precise formulation yet---first we make sense of what one
would mean by a "generic counterexample" by introducing the mathematically
unambigous and logically stronger concept of a "robust counterexample". Then
making use of Penrose' nonlinear graviton construction (i.e., twistor theory)
and a Wick rotation trick we construct a smooth Ricci-flat but not flat
Lorentzian metric on the largest member of the Gompf--Taubes uncountable radial
family of large exotic 's. We observe that this solution of the
Lorentzian vacuum Einstein's equations with vanishing cosmological constant
provides us with a sort of counterexample which is weaker than a "robust
counterexample" but still reasonable to consider as a "generic counterexample".
It is interesting that this kind of counterexample exists only in four
dimensions.Comment: LaTeX, 11 pages, 1 figure, the final published versio
- …