Induction of autophagy by spermidine promotes longevity.

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Approximating minimum-area rectangular and convex containers for packing convex polygons

We investigate the problem of finding a minimum-area container for the disjoint packing of a set of convex polygons by translations. In particular, we consider axis-parallel rectangles or arbitrary convex sets as containers. For both optimization problems which are NP-hard we develop efficient constant factor approximation algorithms

Approximating minimum-area rectangular and convex containers for packing convex polygons

We investigate the problem of finding a minimum-area container for the disjoint packing of a set of convex polygons by translations. In particular, we consider axis-parallel rectangles or arbitrary convex sets as containers. For both optimization problems which are NP-hard we develop efficient constant factor approximation algorithms

Covering many points with a small-area box

\u3cp\u3eLet P be a set of n points in the plane. We show how to find, for a given integer k >0, the smallest-area axis-parallel rectangle that covers k points of P in O(nk\u3csup\u3e2\u3c/sup\u3elogn+ n log\u3csup\u3e2\u3c/sup\u3e n) time. We also consider the problem of, given a value α > 0, covering as many points of P as possible with an axis-parallel rectangle of area at most α. For this problem we give a probabilistic (1-ε)-approximation that works in near-linear time: In O((n/ε\u3csup\u3e4\u3c/sup\u3e) log\u3csup\u3e3\u3c/sup\u3e n log(1/ε)) time we find an axis-parallel rectangle of area at most α that, with high probability, covers at least (1-ε)κ* points, where κ* is the maximum possible number of points that could be covered.\u3c/p\u3

Covering many points with a small-area box

Let P be a set of n points in the plane. We show how to find, for a given integer k>0 , the smallest-area axis-parallel rectangle that covers k points of P in O(nk 2 logn+nlog 2 n) time. We also consider the problem of, given a value α>0 , covering as many points of P as possible with an axis-parallel rectangle of area at most α . For this problem we give a randomized (1−ε) -approximation that works in near-linear time: in O((n/ε 4 )log 3 nlog(1/ε)) time we find an axis-parallel rectangle of area at most α that covers at least (1−ε)κ ∗ points, where κ ∗ is the maximum possible number of points that could be covered

In Situ Localization of Small RNAs in Plants

Small RNAs have vital roles in numerous aspects of plant biology. Deciphering their precise contributions requires knowledge of a small RNA's spatiotemporal pattern of accumulation. The in situ hybridization protocol described here takes advantage of locked nucleic acid (LNA) oligonucleotide probes to visualize small RNA expression at the cellular level with high sensitivity and specificity. The procedure is optimized for paraffin-embedded plant tissue sections, is applicable to a wide range of plants and tissues, and can be completed within 2-6 days