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Michael Schumacher skiing in northern Italy in 2005.

BEAVER CREEK, Colo. — The fact that Michael Schumacher was wearing a helmet when he sustained a life-threatening head injury while skiing in France on Sunday probably did not come as a surprise to experts who have charted their increasing presence on slopes and halfpipes in recent years. The fact that the helmet did not prevent Schumacher's injury probably did not surprise them, either.

Schumacher, the most successful Formula One driver in history, sustained a traumatic brain injury when he fell and hit his head on a rock while navigating an off-piste, or ungroomed, area at a resort in Méribel, France. Although he was wearing a helmet, he sustained injuries that have left him fighting for his life in a hospital in Grenoble, France.

Schumacher's injury also focused attention on an unsettling trend. Despite the fact that more skiers and snowboarders in the United States than ever are wearing helmets — 70 percent of all participants, nearly triple the number from 2003 — there has been no reduction in the number of snow-sports-related fatalities or brain injuries in the United States, according to the National Ski Areas Association.

Experts ascribe that seemingly implausible correlation to the inability of helmets to prevent serious head injuries like Schumacher's and to the fact that more skiers and snowboarders are engaging in risky behaviors: skiing faster, jumping higher and riding out of bounds.

"The equipment we have now allows us to do things we really couldn't do before, and people's pushing limits has sort of surpassed people's ability to control themselves," Chris Davenport, a professional big-mountain skier, said.

Dave Byrd, the ski association's director of risk management, attributed the surge in helmet use to grass-roots efforts by resorts, helmet manufacturers and medical professionals to encourage their use. He also cited growing public awareness about brain injuries, a result of persistent news media attention on the issue in sports, particularly in the N.F.L., and several high-profile skiing fatalities, like the ones that killed Natasha Richardson and Sonny Bono. New Jersey is the only state that mandates helmet use, requiring it for children 17 and under.

The increase in helmet use has had positive results. Experts say helmets have reduced the incidence of less serious head injuries, like scalp lacerations, by 30 percent to 50 percent, and Schumacher's doctors say he would not have survived his fall had he not worn a helmet. But growing evidence indicates helmets do not prevent some of the more serious injuries, like the tearing of delicate brain tissue, said Jasper Shealy, a professor emeritus at Rochester Institute of Technology.

Shealy, who has been studying snow-sports-related injuries at Sugarbush resort in Vermont for more than 30 years, said that could be because those injuries typically involve a rotational component that today's helmets cannot mitigate. He said his research had not seen any decline in what he called P.S.H.I.'s, for potentially serious head injury, a classification that includes a diagnosed concussion, skull fracture, closed head injury, traumatic brain injury or death by head injury.

In fact, some studies indicate that the number of snow-sports-related head injuries has increased. A 2012 Western Michigan University School of Medicine study on head injuries among skiers and snowboarders in the United States found that the number of head injuries increased 60 percent in a six-year period, from 9,308 in 2004 to 14,947 in 2010, even as helmet use increased by an almost identical percentage over the same period. A March 2013 study by the University of Washington concluded that the number of snow-sports-related head injuries among youths and adolescents increased 250 percent from 1996 to 2010.

Experts agree that the roots of the trend are complicated and could be related to increased awareness about brain injuries and reporting them. But they also agreed on one element underpinning the trend: an increase in risk-taking behaviors that they said the snow-sports industry had embraced. In recent years, many resorts have built bigger features in their terrain parks and improved access to more extreme terrain. At the same time, advances in equipment have made it easier to ski faster, perform tricks and venture out of bounds.

"There's a push toward faster, higher, pushing the limits being the norm, not the exception," said Nina Winans, a sports medicine physician at Tahoe Forest MultiSpecialty Clinics in Truckee, Calif. "So, all of those factors — terrain parks, jumping cliffs and opening terrain that maybe wasn't open in the past — play into some of these statistics with injuries."

The population most susceptible to this culture is the one that is dying, statistics show. Seventy percent of snow-sports fatalities involve men in their late teenage years to late thirties, according to the ski association, the same population that most often engages in high-risk behaviors like driving fast. Head injuries remain the leading cause of fatalities in skiing and snowboarding, Shealy said, about 30 such deaths in the United States each year.

"The helmet does a very good job at protecting against skull lacerations and skull fractures, but it doesn't seem to have much effect on concussions or T.B.I.'s," Shealy said, referring to traumatic brain injuries. "Our guess is that this is due to the fact that those injuries are occurring at such a high magnitude of energy that they overwhelm what a helmet can do for you."

That could be what happened to Schumacher and to Sarah Burke, a four-time X Games superpipe gold medalist who was fatally injured two years ago while skiing in Park City, Utah.

KABUL, Afghanistan — Just months after American officials ceded control over all detention operations in Afghanistan, President Hamid Karzai's government has quietly planned dozens of prisoner releases that American and Afghan officials said on Tuesday would include committed insurgents who had attacked Americans.

The pending wave of releases has revived one of the most caustic — and seemingly resolved — issues between the allies just as relations have hit a new low over Mr. Karzai's refusal to sign a long-term security agreement with American officials. Officials from both countries who support the security deal say they are worried that the mass release of militants with American blood on their hands could scuttle talks altogether and lead to a complete Western withdrawal from Afghanistan in 2014.

It was out of concern that dangerous militants might be too easily returned to the battlefield that American military commanders initially resisted Mr. Karzai's demands in the spring to hand over all detention operations. As the Americans relented, they said they had received his personal assurances that security would be maintained.

Suddenly, it is a major issue again, though the releases have not been made final.

The Afghan commission charged with reviewing detainee cases at the main military prison near Bagram Air Base, a coalition hub north of Kabul, is planning to release more than 85 prisoners who the coalition and Afghan defense officials say should face trial. The plans were detailed by American and Afghan officials alarmed by the move, and have been confirmed by commission members.

"These are guys that are tied directly to killing and trying to kill our forces and Afghan forces," an American military official said. "This is an issue of deep concern. It is serious."

The commission disagrees. Abdul Shakor Dadras, a member of the three-man panel, said there was no real evidence against any of the men, who are among the 650 prisoners the panel has ordered freed since it was created by Mr. Karzai after the transfer of the prison in March.

In an interview on Tuesday, Mr. Dadras said that Mr. Karzai tentatively approved the releases at a cabinet meeting last week. The president, he said, ordered the commission to give the international military coalition and the main Afghan intelligence agency until Friday to produce evidence against the detainees. If none is produced, the men will be released, Mr. Dadras said.

But American and Afghan security officials, who spoke on the condition of anonymity because of tensions between the countries, said they had already given the commission enough evidence to send all the detainees in question to trial, or at least to hold them pending further investigation. The officials said they have evidence linking the detainees to the killings of 117 American, European and Afghan service members and civilians, and that they are ready to continue investigating in cases where more evidence is need to try the men.

One of the detainees was caught planting a hidden bomb and was linked by biometric evidence to other bomb attacks that killed two Afghan soldiers, the officials said. Another is alleged to have trained and deployed teenage boys for suicide attacks. A third placed a hidden bomb outside a school, killing a student, the officials said.

Both the Afghan defense officials and the Americans argued that under the deal that transferred control of the prison, the commission did not have the authority to order the releases over the objection of the coalition and the intelligence service, which is supposed to make a recommendation on each case.

The commission "has exceeded its mandate and ordered the release of a number of dangerous individuals who are legitimate threats and for whom there is strong evidence supporting prosecution or further investigation," said Col. David Lapan, a spokesman for American forces in Afghanistan.

The judge, William M. Skretny of Federal District Court in Buffalo, called the seven-round limit "an arbitrary restriction" that violated the Second Amendment.

But Judge Skretny said the greater restrictions on assault weapons and high-capacity magazines were constitutional because they served to "further the state's important interest in public safety."

Gov. Andrew M. Cuomo and lawmakers passed the new laws, among the most restrictive in the country, in January in response to the mass shooting at an elementary school in Newtown, Conn., in December.

Gun rights groups have been sharply critical of the measures, holding protests at the Capitol, questioning their legality, and vowing to oppose officeholders like Mr. Cuomo who championed them. The judge's ruling came in response to a lawsuit filed by the New York State Rifle and Pistol Association and other firearms groups.

The seven-round limit on magazines, which Mr. Cuomo, a Democrat, touted when he pushed for the new laws in January, had already run into problems before Tuesday's ruling.

In March, in response to complaints that seven-round magazines were not widely available for sale, Mr. Cuomo said he and legislative leaders would change the restriction so that 10-round magazines could continue to be purchased. But even with that change, gun owners would still be forbidden from loading more than seven rounds into those 10-round magazines.

CHICAGO — A year after this city, the nation's third largest, drew national notice for its staggering number of homicides, killings have slowed here.

Officials in the administration of Mayor Rahm Emanuel, who has faced scrutiny over his handling of the violence, said the improvement was a result of an array of efforts, from new police tactics to after-school and summer programs.

In 2012, Chicago witnessed more than 500 killings, many of them shootings tied to gang rivalries in some of the toughest, struggling neighborhoods. As of Dec. 30, Chicago had reported 413 homicides, a 17 percent drop from the same period a year before and the fewest murders to date since 1965, city officials said.

Shootings were also down by about 24 percent in 2013 from a year earlier, and reports of crimes overall had dropped by about 16 percent. Still, Chicago's death toll remained higher than those in the nation's more populous cities — New York and Los Angeles, both of which reported a decrease in killings as the end of 2013 approached.

"We're pleased, we're not satisfied," Garry F. McCarthy, the superintendent of the Chicago police, said in an interview. "It's progress, not victory. We've put systems in place that are going to keep producing better results moving forward."

As in lots of big American cities, the level of killings in Chicago was far worse a few decades ago; in the 1990s, the yearly death toll here sometimes reached more than 900. But the 506 deaths of 2012 marked an uptick and was a painful reminder of this city's divisions of race and wealth; many of those deaths took place in only about half of the city's police districts, south and west of the prosperous downtown.

By last January, the miserable price of the bloodshed here drew widespread note when a high-school girl who had just attended festivities honoring President Obama's inauguration in Washington was slain as she spent an afternoon with friends in a park not far from the president's home on Chicago's South Side. The girl, Hadiya Pendleton, 15, had no ties to gangs, the authorities said, but had been mistakenly swept into a gang-related retaliation shooting. Michelle Obama attended the girl's funeral here, and President Obama spoke of her during his State of the Union address.

Mr. McCarthy, who was appointed by Mr. Emanuel to run the police department in 2011, made changes aimed at tackling the complications of Chicago's hundreds of increasingly splintered gang factions. He sent more officers — some working overtime, at least initially — into a small number of areas deemed the most volatile. The police created an "audit" of the city's tens of thousands of gang members, aimed at tracking rivalries and where retaliatory shootings might be likeliest.

Critics have questioned whether Chicago, a city facing significant budget pressure thanks to a looming increase in required contributions to workers' pension funds, can ultimately afford to continue to tamp down the violence. The city had spent $96 million this year on overtime for the police by the end of November, city officials said.

But Mr. McCarthy dismissed concerns about the price. "It really worked," he said. "So instead of people saying, 'What are we going to do about the overtime?' they should be saying, 'That was a really great investment in saving lives.' "

About $75 million is budgeted for overtime in 2014, the city said.

"We continue to make significant progress in reducing crime rates to historic levels, and gains are being made in all communities across Chicago," Mayor Emanuel said in a written statement issued by his spokesman. "While this is good news for our children and families, we are not resting. Through strategic policing, smart prevention, sound parenting and stiffer penalties, we will continue working until everyone enjoys the same sense of safety."

And while the year-end homicide statistics marked an improvement citywide, not every neighborhood felt safer — or even noticeably different, residents here said. In September, 13 people, including a 3-year-old boy, were wounded during a shooting involving an assault-style rifle near a pickup basketball game at a South Side park. A gang dispute, the authorities have said, led to the attack.

Outside the New York Public Library, a middle-aged man with a heavy Brooklyn accent walked up to him, hand outstretched.

"Mayor Bloomberg, you did a great job," the man said. "Thank you for being our mayor for the past 12 years."

The mayor broke his stride, turned and smiled.

"I wish you could stay on," the man added.

The mayor's smile widened.

So it went on Tuesday, as the mayor who lives by numbers kept a watchful eye on the clock. At an interfaith breakfast inside the library, he flipped open his black leather iPad and started to read from a speech with a stark heading at the top. "LAST DAY," it read.

One by one, religious leaders took to the microphone, offering him praise from a source that seemed to humble even the billionaire mayor: God.

"This great city of New York has never flourished more than today," declared a rabbi from Staten Island. "We have been blessed by Mayor Michael Bloomberg's presence among us."

"The psalmist declares, 'One who does these things shall never falter,'" the rabbi added.

A sheikh from the Bronx was effusive, praising Allah, then Mr. Bloomberg: "We are here to pray for you. Because under your leadership, Mr. Mayor, the city is safer, cleaner, nicer, healthier and friendlier."

"Mr. Mayor, under your leadership, you did not fail any community in the city."

When it was his turn to talk, Mr. Bloomberg abandoned his usual recitation of legacy-burnishing statistics. Instead, he simply thanked those in the room, reminding them that they, not he, were the on-the-ground soothers and problem-solvers who had made the city work.

"I have always been envious of you," he said. "Because you work at the real level where the real problems are."

"At my level, you talk about them, you look at the grand scheme, you look at averages," Mr. Bloomberg continued. "You have to deal with big numbers. Big numbers are easy to deal with. It's much tougher when you deal one on one, looking at a person right in the eye who has a problem."

It was his final speech as mayor.

A choir of children began to sing "When You Believe," a 1998 ballad made famous by Whitney Houston and Mariah Carey.

Mr. Bloomberg looked on, tapping his foot lightly, but seemed distracted. His head drifted to the crowd, which he took in with a long wistful sweep, his eyes growing glassy.

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