4,079 research outputs found
The Stretch - Length Tradeoff in Geometric Networks: Average Case and Worst Case Study
Consider a network linking the points of a rate- Poisson point process on
the plane. Write \Psi^{\mbox{ave}}(s) for the minimum possible mean length
per unit area of such a network, subject to the constraint that the
route-length between every pair of points is at most times the Euclidean
distance. We give upper and lower bounds on the function
\Psi^{\mbox{ave}}(s), and on the analogous "worst-case" function
\Psi^{\mbox{worst}}(s) where the point configuration is arbitrary subject to
average density one per unit area. Our bounds are numerically crude, but raise
the question of whether there is an exponent such that each function
has as .Comment: 33 page
Composition and Relative Counting
According to the so-called strong variant of Composition as Identity (CAI), the Principle of Indiscernibility of Identicals can be extended to composition, by resorting to broadly Fregean relativizations of cardinality ascriptions. In this paper we analyze various ways in which this relativization could be achieved. According to one broad variety of relativization, cardinality ascriptions are about objects, while concepts occupy an additional argument place. It should be possible to paraphrase the cardinality ascriptions in plural logic and, as a consequence, relative counting requires the relativization either of quantifiers, or of identity, or of the is one of relation. However, some of these relativizations do not deliver the expected results, and others rely on problematic assumptions. In another broad variety of relativization, cardinality ascriptions are about concepts or sets. The most promising development of this approach is prima facie connected with a violation of the so-called Coreferentiality Constraint, according to which an identity statement is true only if its terms have the same referent. Moreover - even provided that the problem with coreferentiality can be fixed - the resulting analysis of cardinality ascriptions meets several difficulties
Contingent composition as identity
When the necessity of identity is combined with composition as identity, the contingency of composition is at risk. In the extant literature, either NI is seen as the basis for a refutation of CAI or CAI is associated with a theory of modality, such that: either NI is renounced ; or CC is renounced. In this paper, we investigate the prospects of a new variety of CAI, which aims to preserve both NI and CC. This new variety of CAI is the quite natural product of the attempt to make sense of CAI on the background of a broadly Kripkean view of modality, such that one and the same entity is allowed to exist at more than one possible world. CCAI introduces a world-relative kind of identity, which is different from standard identity, and claims that composition is this kind of world-relative identity. CCAI manages to preserve NI and CC. We compare CCAI with Gibbardâs and Galloisâ doctrines of contingent identity and we show that CCAI can be sensibly interpreted as a form of Weak CAI, that is of the thesis that composition is not standard identity, yet is significantly similar to it
Mutant knots and intersection graphs
We prove that if a finite order knot invariant does not distinguish mutant
knots, then the corresponding weight system depends on the intersection graph
of a chord diagram rather than on the diagram itself. The converse statement is
easy and well known. We discuss relationship between our results and certain
Lie algebra weight systems.Comment: 13 pages, many figure
- âŠ