This paper establishes conditions under which one can use integrals of locally defined differential forms to give an asymptotic expansion of the supertrace of the heat operator associated with a perturbed Dirac operator on a complete noncompact even-dimensional manifold with bounded geometry.
One means of avoiding detection by predators is to employ cryptic colouration. In some cases, individuals can rapidly adjust their colouration, enabling them to remain cryptic in a heterogeneous environment or to vary colouration in response to predation threat. This is often achieved physiologically but in some cases animals are able to adjust their external appearance by behavioural means. Hermit crabs could potentially enhance their crypticity by occupying gastropod shells that match the substrate. However, changing shells is risky and the benefits of crypsis must be balanced against the imminent threat of predation. Here we show that naked hermit crabs initially enter the shell that shows the greatest contrast with the substrate, but may subsequently move into a shell that offers a greater degree of background matching. However, in the presence of a predator cue they tend to remain in the shell that offers low crypticity. This indicates that rapid colour change may be effected by behavioural mechanisms but this is restricted by the need to make a behavioural decision on the basis of a range of information sources.
Give the smooth part of a singular curve or normal surface the metric induced from the ambient projective space. On this incomplete manifold the minimal L2 ô-complex of (0, ?)-forms has finite-dimensional cohomology groups. The Euler characteristic of this cohomology equals the Todd genus of any desingularization of the singular variety.
Abstract. On an odd-dimensional manifold with isolated conical singularities, we perturb a Dirac operator by a vector bundle endomorphism whose pointwise norm grows in inverse proportion to the distance from the singular set. We give two proofs of an index formula for the resulting Fredholm operator. We mention an application to the index theory of transversally elliptic operators.
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