President George W. Bush signed a bill yesterday requiring that convicted child molesters be listed on a nationwide Internet registry, an idea that got a boost from research by a Long Island-based child advocacy group.  The law is designed to create nationwide sex-offender registration and notification standards.  The Stony Brook group Parents for Megan's Law documented wide disparities in state requirements and presented the data to Justice Department officials to bolster the case for a tougher, national law, said executive director Laura Ahearn.  We found such a lack of uniformity in all 50 states, so that where you live literally determined how well you could protect your children from sex offenders, said Ahearn, who was at the White House bill signing.  As Bush signed the law, he was joined by America's Most Wanted host John Walsh, whose 6-year-old son, Adam, was abducted 25 years ago yesterday and murdered later.  The bill was named for Adam Walsh.  The law is designed to help police find more than 100,000 sex offenders by creating the first national online listing available to the public and searchable by Zip Code.  The legislation also calls for harsh federal punishment for sexually assaulting children, a new federal DNA database of convicted molesters, and tough new penalties for downloading child pornography on the Internet.  Sex offenders also face a felony charge for failing to update their whereabouts.