No way this thread went without reply for 12 hours!
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The classic analogy for a resistor is a pipe through which water (electricity) runs. The width of the pipe is analogous to the resistance of the flow of current—the narrower the pipe, the greater the resistance. Normal resistors have an unchanging pipe size. A memristor, on the other hand, changes with the amount of water that gets pushed through. If you push water through the pipe in one direction, the pipe gets larger (less resistive). If you push the water in the other direction, the pipe gets smaller (more resistive). And the memristor remembers. When the water flow is turned off, the pipe size does not change.
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There has to be some student out there who will have some creative ideas of what to do with something like this.
Insta-on-off memory locations would mean longer battery life for ... nearly everything these days, from phones to laptops to digital media. With memory that keeps its value after power loss, this would be possible. If it is truly as simple as the article makes it sounds, the whole power savings would be much more efficient since it's faster than doing it with something like flash memory.
Eventually someone will apply the mathematical model to some obscure idea and BAM a brand new technology will be born. It's like the guys in that article (and other articles found on this) have a small vision, but there are much broader implications for this. I wonder how HP will release/license this.