New Analysis

We use different analysis techniques for our studying the running time of the several algorithms. Of all we typically use probabilistic analysis to determine the running time of an algorithm in cases in which, due to the presence of an inherent probability distribution, the running time may differ on different inputs of the same size. In some cases, we assume that the inputs conform to a known probability distribution, so that we are averaging the running time over all possible inputs. In other cases, the probability distribution comes not from the inputs but from random choices made during the course of the algorithm. An algorithm whose behavior is determined not only by its input but by the values produced by a random-number generator is a randomized algorithm. We can use randomized algorithms to enforce a probability distribution on the inputs-thereby ensuring that no particular input always causes poor performance-or even to bound the error rate of algorithms that are allowed to produce incorrect results on a limited basis.

Truth about Black holes

Black holes don't emit x-rays themselves, matter falling into black holes heats up and emits x-rays before it is swallowed up. The Hawking radiation is different (but not at all ad hoc, as the article said); it's actually emitted by the black hole itself, and much weaker than the x-rays. Also, black holes are not usually "defined" to be singularities, it’s just that every current theory we have inevitably leads to singularities. String theory is different because the fundamental objects, strings, are extended (they have length), as opposed to points. It's been known for awhile that this would prevent singularities because the strings are spread out. What has come out more recently is the idea that the strings could stretch from the center of the black hole out to the edges, and somehow encode the information there (again, note that the key is that the strings have a length). This way information is not lost, which is VITAL to quantum mechanics.

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