The world mourned the seven fallen astronauts from the space shuttle Columbia yesterday, and America's leaders vowed to make sure their dreams of exploring the unknown and expanding human understanding would not die with them.

It was a day to recall the past, as the Columbia's crew was honored in the United States, Israel and India, in public ceremonies, Internet chats and an endless stream of televised reminiscences.

But there was also a grim focus on the present. NASA investigators reported new evidence of an unexpected heat increase on Columbia's left side minutes before its dramatic disintegration 39 miles above the Texas plains Saturday morning, and retired Adm. Harold W. Gehman Jr. was named to lead an independent investigation into the disaster. There were even bold visions for the future, as guests on Sunday talk shows pledged to beef up the U.S. space program and spoke confidently of manned missions to Mars.

"I believe the will of the American people is that we explore space," said Sen. Bill Nelson (D-Fla.), a former shuttle crew member who appeared on several programs. "I hope that can be translated into a vigorous program . . . that will take us to Mars."

Last year, Nelson complained that shuttle safety upgrades were being delayed, and warned that "we are starving the shuttle budget, greatly increasing the chances of catastrophic loss."

NASA's budget has been flat for the last decade, and Bush proposed nearly $800 million in cuts in manned flight programs last year. But yesterday, an administration official said the president today will propose a $470 million increase in NASA's $15 billion overall budget, and that the new investigation will address whether budget problems played a role in the tragedy.

On Tuesday, the president will speak at a memorial service for the Columbia crew in Houston.

"From the president's point of view, the mission of science and the marvels of space exploration will go on," White House press secretary Ari Fleischer said. "He wants to make certain that no stone is unturned, to get to the bottom of this."

The nation yesterday began to pick up the pieces from the tragedy -- emotionally, but also literally. Hundreds of local, state and federal law enforcement officials, as well as military personnel, fanned out across a "debris belt" covering more than 200 square miles of East Texas, where they found chunks of metal, a flight helmet, a 4-foot-wide cylinder, and a mission patch with the names of all seven astronauts. But authorities warned that these early finds in more than 800 locations may represent the shuttle's "low-hanging fruit," and that the area's rugged forest terrain -- known locally as the Big Thicket -- could frustrate more elaborate recovery efforts.

At a late-afternoon briefing in Houston, NASA official Robert Cabana announced that recovery teams have discovered remains from all seven astronauts. "It's still in the process of identification," he said.

But last night, NASA corrected his statement, saying that so far remains from some victims, but not all, had been found. They will be taken to Dover Air Force Base in Delaware, where the bodies of the Challenger astronauts were taken 17 years ago.

Meanwhile, mission director Ronald D. Dittemore said investigators are "gaining ground" as they try to solve the mystery of why the Columbia broke apart about 1,000 miles from the end of a 6-million-mile journey, shortly after reentering the Earth's atmosphere. He revealed an intricate web of new data from the Columbia's final minutes, including sensor recordings of excess heat in the left wheel well and excess drag on the left side. Dittemore said investigators will study whether the problems were caused by a chunk of foam that may have damaged tiles on the left wing during liftoff, but he cautioned that it is much too early to speculate on the cause.

"I've got to be careful not to jump to conclusions," Dittemore said.

The early speculation after the destruction of the Challenger in 1986 focused on possible engine problems, and that turned out to be wrong. NASA Administrator Sean O'Keefe yesterday assigned several internal units to investigate the loss of the Columbia, including a "Mishap Response Team" and a "Contingency Action Team." He also announced the formation of an independent board led by Gehman, who co-chaired the probe of the October 2000 terrorist attack on the USS Cole in Yemen.

O'Keefe yesterday appeared on several TV shows to pledge NASA's commitment to find the problem and fix it. He promised that there would be no rush to judgment, although he repeated his assertion on Saturday that NASA has seen no evidence to suggest terrorism. "We are doing everything we possibly can to find out what caused this accident, for the families of the crew of this courageous group of people who were aboard Columbia," O'Keefe said on NBC's "Meet the Press." "We owe them that."

Those astronauts were remembered yesterday around the world. In Rome, Pope John Paul II promised to remember them in his prayers. In Jerusalem, Prime Minister Ariel Sharon declared Ilan Ramon a national hero, and in Karnal, India, a Hindu shrine was set up at Kalpana Chawla's high school. And in the First United Methodist Church of Titusville, Fla., the Rev. David Waller described the plume of smoke that trailed the doomed shuttle as "a glistening tear across the face of the heavens."

Many spiritual leaders who had planned to speak to their congregations about a possible war in Iraq spoke instead of the Columbia's crew.

At the Washington National Cathedral, the senior canon read a prayer by a volunteer chaplain, the Rev. C. Perrin Radley: "We have taken pride in the achievements of scientists and astronauts of many nations, but have also forgotten the risks with which they reckon."

The president and first lady Laura Bush attended the early service at St. John's Church in Washington, where the rector, Luis Leon, saluted the fallen astronauts, saying the tragedy seemed like "more than we can stomach."

"We grieve because they represented the best in us, because part of us has died," the rector said. "I believe that God's heart is more heartbroken than our own, and I believe they're already resting."

At the same time, the head of Lebanon's Hezbollah guerrilla movement crowed that the demolition of the shuttle showed that the United States was not omnipotent. "What happened yesterday is a message to all humanity, and especially Arab, Muslim and Third World people, a message to those who thought in the past few years that America was a god that couldn't be defeated or defied," Sheik Hassan Nasrallah told thousands of people at a graduation ceremony.

The loss of Columbia will surely increase scrutiny of NASA, but Bush's promise to the nation on Saturday that "our journey into space will go on" was echoed by other politicians -- including Nelson, Senate Majority Leader Bill Frist (R-Tenn.) and the chairmen of the congressional subcommittees overseeing NASA, Rep. Sherwood L. Boehlert (R-N.Y.) and Sen. Sam Brownback (R-Kan.) -- as well as former astronauts, including John Glenn, Sally Ride and Edwin "Buzz" Aldrin. None of them suggested that NASA has ever skimped on safety, and all stressed that manned space flight is a crucial element of the nation's destiny.

"Safety has always been Job One at NASA," Brownback said on ABC's "This Week." "I think the totality of their record bears that out, that they have conducted a very safe program. . . . We have to move forward with this. We need to sow that vision for our people and for the people around the world, so that these lives not be lost in vain."

With Iraq dominating the news, it would have been hard to imagine last week that NBC's "Meet the Press" would feature a lengthy discussion of the possibility of extraterrestrial life. But now that the nation is thinking about space, so are politicians such as Frist, who paid homage to micro-gravity experiments conducted on shuttle missions.

"These heroes were scientists," Frist said on CBS's "Face the Nation." "There is no question in my mind that the world is a better place whether it's CAT scans, MRIs, medicines, pharmaceutical agents, the tennis shoes we wear. The world is a better place because of manned exploration."

Several politicians and former astronauts called for exploration of new frontiers, with some echoing the challenge of President George H.W. Bush in 1989 to put a man on Mars. Aldrin, the second man to step on the moon, said yesterday: "I certainly see that in the next century, people will be on Mars."

NASA has also looked into another mission to the moon, and possible unmanned missions to Pluto or Jupiter's moon, Europa. But its most immediate priority, along with the Columbia probe, is the three astronauts now living on the $95 billion International Space Station. The shuttle program is now on hold, so it can no longer service or resupply the station. NASA officials say that the supplies at the station will last into the summer, and that the astronauts can leave any time on a Russian vehicle kept at the station in case of emergency.

Yesterday, Russia launched a cargo rocket to carry more food and fuel to the station. It went off without a hitch.

Researcher Margot Williams contributed to this report.

Pastor Steve O'Donohoe of Grace Community Church in Houston prays during services for the Columbia crew. Two shuttle astronauts, Rick D. Husband and Michael P. Anderson, were parishioners at the church. A cross marks the site where human remains believed to be from the shuttle were found outside Hemphill, Tex. Robert Cabana, NASA's director of shuttle flight crew operations, announced that searchers had found remains of some of Columbia's crew. Their remains will be flown to Dover Air Force Base in Delaware.Near Hemphill, Tex., Wink Miller looks at a piece of debris believed to be from the space shuttle Columbia he found in the woods.