After their private talk they exchanged gifts, and Mr. Biden gave the pope a presidential challenge coin that featured Delaware, his home state, and Beau’s Army National Guard unit. “I know my son would want me to give this to you,” he said.

As Francis showed Mr. Biden and Jill Biden, the first lady, to the door, Mr. Biden was in no rush to leave.

He unspooled a folksy yarn that referred to both he and the pope ascending to their positions later in life. In a nod to their ages — he is 78 and Francis is 84 — he relayed a story about Satchel Paige, the legendary Black player who pitched the majority of his career in the Negro Leagues, and who was allowed to join the Major Leagues only in his 40s.

“Usually, pitchers lose their arms when they’re 35,” Mr. Biden said to the pope, who seemed a little lost by the baseball reference. “He pitched a win on his 47th birthday.”

As Mr. Biden explained it, reporters asked the pitcher: “‘Satch, no one’s ever pitched a win at age 47. How do you feel about pitching a win on your birthday?’” and the pitcher responded: “‘Boys, that’s not how I look at age. I look at it this way: How old would you be if you didn’t know how old you were?’”

The pope looked at Mr. Biden.

“You’re 65, I’m 60,” the president said. “God love ya.”

Jim Tankersley contributed reporting from Rome, and Ruth Graham from Dallas.

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Tokyo Olympics Highlight the Barriers Japan’s Girls Face in Sports

TOKYO — Kurumi Mochizuki is the kind of skilled soccer player who can roll a ball from between her shoulder blades to the top of her head and onto her right foot, keeping it aloft for more than a dozen kicks. She makes it look so easy.

Yet when she practices with her local club team in southeastern Tokyo, her coaches sometimes advise her to take longer breaks than her teammates, and warn her not to pick up heavy bags of balls when clearing equipment from the field.

All because she is a girl.

Kurumi, 13, is the only girl on her team. She plays with boys because there are no girls’ club teams near her neighborhood and no girls’ team at her middle school. Finding a team in high school will be difficult, too. Only one of the 14 schools in Kurumi’s area offers a girls’ team. Her older brother, who plays soccer at his high school, has had no such trouble — almost all the high schools in the district have boys’ soccer teams.

Tokyo Olympics, which open next month, offer an opportunity to anoint another crop of champions to inspire girls with athletic aspirations. But after the Olympic spotlight dims, those like Kurumi will still face powerful obstacles.

Japan has no law like Title IX, the American statute that requires schools receiving public funding to offer equal opportunities to boys and girls, and there is no public data on how much schools spend on extracurricular sports or how it breaks down on gender lines.

Female athletes who persevere often have to push past stereotypes that they are doing something unladylike, jeopardizing their chances of attracting boys and later becoming wives and mothers. Even their coaches view their participation through this lens, in some cases giving them etiquette lessons to ensure they are ready for domestic life.

2011 Women’s World Cup and claimed the silver medal at the London Olympics in 2012.

She followed her brother into soccer when she was 6. “When I was little, I never thought about it,” she said of being the sole girl on her team. “But once I got a bit older, I was much more aware of it.”

The extracurricular soccer team at her public middle school is technically coed, although not one of the team’s 40 players is a girl. Kurumi decided to stick to the club team she had played with since elementary school rather than try to break into a new group at school.

“There is a difference in strength and aggressiveness between boys and girls,” said Shigeki Komatsu, the middle school’s vice principal, standing on the sidelines as the boys scrimmaged on a gravel pitch, their cleats kicking up puffs of dust.

hopes that the situation would improve for female athletes in Japan.

Before that victory, girls in the United States had flocked to suburban soccer clubs after the U.S. women won the World Cup on American soil in 1999.

Koshien, that is more than 100 years old. Just after New Year’s, huge audiences tune in to watch the Hakone Ekiden, a college-level marathon relay that is restricted to male runners.

There are few vocal advocates for female athletes, and most of their coaches are men who often do not provide support for the physical changes that girls undergo in adolescence.

Hanae Ito, a swimmer who represented Japan at the Beijing Summer Olympics in 2008, said coaches along the way had told her she was “mentally weak” when she gained weight or suffered menstruation-related mood changes as a teenage athlete.

“I thought it was a problem with me or that it was my fault,” she said. “But I think that this all ties back to Japan being a patriarchal society. Even women’s sports is seen from a male gaze.”

The idea that female athletes need to worry about their future prospects with men is deeply rooted.

After Hideko Maehata, an Olympic swimmer, became the first woman to win a gold medal for Japan, The Asahi Shimbun, one of Japan’s largest newspapers, heralded her victory at the 1936 Berlin Summer Games with the headline: “Next Up Is Marriage.”

Such attitudes persist today. Yuki Suzuki, who played in Japan’s Nadeshiko women’s professional soccer league and taught the sport until she gave birth to her son, is frustrated by the rigid gender definitions.

“Girls are often told ‘be feminine, be feminine,’” said Ms. Suzuki, now 34. “I think we have to change the fundamental culture of Japan when it comes to women.”

Even when girls get the chance to play, a bias toward boys emerges in small ways. At the middle school Kurumi attends, the boys’ volleyball and basketball teams get the gym three days a week for practice, while the girls use it the other two days.

Kurumi said she tried not to worry about the unequal treatment. She does not hold it against her coaches, she said, for barring her from carrying heavy equipment during practice.

“I am sure the coaches just care about me,” she said. “But personally, I know I could carry it.”

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Will NFTs Transform Tattoos Into Bankable Art?

Top tattoo artists are highly coveted, their work displayed on some of the world’s most visible real estate: LeBron James’s shoulders, Scarlett Johansson’s back, Post Malone’s face.

But you can’t hang tattoos in a gallery, or auction them at Sotheby’s. They live and (unless previously removed) die with their owner. It also means that the most in-demand tattoo artists are still paid by the hour, just as many were during their apprenticeships decorating the biceps of sailors and bikers.

Artists do not generally get paid by the hour, said Scott Campbell, 44, a Los Angeles tattoo artist who has inked Robert Downey Jr., Jennifer Aniston and Marc Jacobs. “Musicians don’t get paid by how long it takes them to create a song. You’d never go to a gallery and think, ‘How long did it take the artist to paint it? I’ll pay him for his time.’”

Mr. Cartoon) and Brian Woo (Dr. Woo), wants to change this equation.

All Our Best, where tattoo artists can offer their designs as permanent, tradable commodities in the form of NFTs.

To refresh: an NFT, which stands for non-fungible token, is basically a digital stamp of authenticity that can be bought, sold or traded like cryptocurrency on a blockchain. This is a far cry from the tattoo world, where the stars of the field see their earnings capped at around $1,000 an hour for a one- to three-hour session, even when working on Hollywood stars.

In this new marketplace, customers will be buying the exclusive rights to the design of the tattoo, rather than the tattoo itself. “I’m selling you an idea, instead of just hours of my life,” said Mr. Campbell, who has been blurring the line between tattoo and fine arts for years, showing his tattoo-inspired sculptures and paintings at galleries and art fairs. “The NFT is basically a digital baseball card.”

As a perk of ownership, buyers get a guaranteed slot with the tattoo artist — no small thing, since top tattoo artists can be nearly impossible to book for those outside the celebrity orbit.

Mr. Campbell, Mr. Cartoon, Dr. Woo, Grime, Sean from Texas and Tati Compton. Mr. Campbell plans to expand the roster, and eventually open the marketplace for any tattoo artist to sell work.

He is not the only tattoo artist to see opportunity in blockchain. An artist in Portland, Me., named Brad Wooten, for example, is selling photos of digitally designed tattoos as NFTs.

The earning potential is considerable. Prices for the initial round of NFT tattoos on All Our Best will range from $1,000 to $10,000. The blockchain technology also allows artists to make a 10 percent royalty every time a work is resold.

Clients also stand to profit if the work appreciates, unlike the current setup where “the only thing they get out of the deal is an Instagram post and some bragging rights,” Mr. Campbell said. “They actually have something that they can keep and pass onto their kids, that has a life beyond being just that thing on their arm that in 10 years is going to be sunburned and blurry anyway.”

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Oatly, a Maker of Oat Milk, Is About to Have Its IPO

Private equity has a place at the table, and so do Oprah and Jay-Z. Food giants like Nestlé are scrambling to get a foot in the door. There are implications for the climate. There are even geopolitical rumblings.

The unlikely focus of this excitement is Oatly, producer of a milk substitute made from oats that can be poured on cereal or foamed for a cappuccino. Oatly, a Swedish company, will sell shares to the public for the first time this week in an offering that could value it at $10 billion and exemplify the changes in consumer preferences that are reshaping the food business.

It’s no longer enough for food to taste good and be healthy. More people want to make sure that their ketchup, cookies or mac and cheese are not helping to melt the polar ice caps. Food production is a leading contributor to climate change, especially when animals are involved. (Cows belch methane, a potent greenhouse gas.) Milk substitutes made from soybeans, cashews, almonds, hazelnuts, hemp, rice and oats have proliferated in response to soaring demand.

“We have a bold vision for a food system that’s better for people and the planet,” Oatly declared in its prospectus for the offering. The company’s shares are expected to start trading in New York on May 20.

Stephen A. Schwarzman, Blackstone’s chief executive, was a steadfast supporter of former President Donald J. Trump, who has maintained that climate change is a hoax.

Blackstone’s backing also helped lend Oatly credibility on Wall Street. And there was no sign that Blackstone’s involvement slowed Oatly sales, which doubled last year.

Oatly’s image benefited from a roster of celebrity investors, including Oprah Winfrey, Natalie Portman, Jay-Z’s Roc Nation company, and Howard Schultz, the former chief executive of Starbucks. All have some connection to the plant-based or healthy living movement.

Oatly declined to comment, citing regulations that restrict public statements ahead of an initial public offering.

Oat milk is part of a larger trend toward food that mimics animal products. So-called food tech companies like Beyond Meat have raised a little more than $18 billion in venture funding, according to PitchBook, which tracks the industry. Plant-based dairy, which in the United States includes brands like Ripple (made from peas) and Moalla (bananas), raised $640 million last year, more than double the amount raised a year earlier.

In the United States, milk substitutes like oat milk and rice milk make up a $2.5 billion industry that is expected to grow to $3.6 billion by 2025, according to Euromonitor. Globally, the $9.5 billion industry is expected to grow to $11 billion.

Once a niche market, alternate milk has become as American as baseball. A frozen version of Oatly that mimics soft-serve ice cream is being sold this season at Yankee Stadium, Wrigley Field in Chicago and Globe Life Field in Arlington, Texas, where the Rangers play.

China Resources, a state-owned conglomerate with vast holdings in cement, power generation, coal mining, beer, retailing and many other industries. The new financing helped Oatly to expand in Europe and begin exporting to the United States and China, where many people cannot tolerate cow’s milk. China Resources’ involvement undoubtedly helped open doors in the Chinese market. Asia, primarily China, accounted for 18 percent of sales in the first quarter of 2021, and is growing at a rate of 450 percent a year, according to Oatly.

In Europe, there is growing alarm about Chinese investment in strategic industries like autos, batteries and robotics. The European Commission has begun erecting regulatory barriers to companies with financial links to the Chinese government. But so far no one has expressed fear that China will dominate the world’s supply of oat milk.

Just in case, Oatly’s prospectus gives it the option of listing in Hong Kong if the foreign ownership becomes a problem in the United States.

The potential of the market for dairy alternatives is not lost on big food producers. Oatly acknowledged in its offering documents that it faces fierce competition, including from “multinational corporations with substantially greater resources and operations than us.”

That would include British consumer goods maker Unilever, which said last year that it aims to generate revenue of one billion euros, or $1.2 billion, by 2027 from plant-based substitutes for meat and dairy, for example Hellmann’s vegan mayonnaise or Ben & Jerry’s dairy-free ice cream. Unilever has not announced plans for a milk substitute.

dairy alternatives are a poor substitute for cow’s milk because they don’t have nearly as much protein.

Stefan Palzer, the chief technology officer at Nestlé, took issue with those who say a big company can’t move as fast as a bunch of Swedish foodies. A young team at Nestlé developed Wunda in nine months, including three months of market testing in Britain, Mr. Palzer said in an interview.

substitutes for almost any kind of animal product. The next frontier: fish. Nestlé has begun selling a tuna substitute called Vuna and is working on scallops.

“It’s a great opportunity to combine health with sustainability,” Mr. Palzer said of plant-based alternatives to milk and meat. “It’s also a great growth opportunity.”

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Businesses Offer Perks to Vaccinated Customers

At Fort Bragg, soldiers who have gotten their coronavirus vaccines can go to a gym where no masks are required, with no limits on who can work out together. Treadmills are on and zipping, unlike those in 13 other gyms where unvaccinated troops can’t use the machines, everyone must mask up and restrictions remain on how many can bench-press at one time.

Inside Dodgers Stadium in Los Angeles, where lines not long ago snaked for miles with people seeking coronavirus vaccines, a special seating area allows those who are fully inoculated to enjoy games side by side with other fans.

When Bill Duggan reopens Madam’s Organ, his legendary blues bar in Washington, D.C., people will not be allowed in to work, drink or play music unless they can prove they have had their shots. “I have a saxophone player who is among the best in the world. He was in the other day, and I said, ‘Walter, take a good look around because you’re not walking in here again unless you get vaccinated.’”

Evite and Paperless Post are seeing a big increase in hosts requesting that their guests be vaccinated.

actually doughnuts, beers and cheesecake — to prod laggards along. Some have even offered cold hard cash: In Ohio, Gov. Mike DeWine this week went so far as to say that the state would give five vaccinated people $1 million each as part of a weekly lottery program.

On Thursday, federal health officials offered the ultimate incentive for many when they advised that fully vaccinated Americans may stop wearing masks.

Now, private employers, restaurants and entertainment venues are looking for ways to make those who are vaccinated feel like V.I.P.s, both to protect workers and guests, and to possibly entice those not yet on board.

Come summer, the nation may become increasingly bifurcated between those who are permitted to watch sports, take classes, get their hair cut and eat barbecue with others, and those who are left behind the spike protein curtain.

for children ages 12 through 15.

But even without a mandate, a nudge can feel like a shove. The military has been strongly encouraging vaccines among the troops. Acceptance has been low in some branches, like the Marines, with only 40 percent having gotten one or more shots. At Fort Bragg, one of the largest military installations in the country and among the first to offer the vaccine, just under 70 percent have been jabbed.

podcast designed to knock down misinformation — a common misbelief is that the vaccines affect fertility — plays around the base. In addition to their freedom gym, vaccinated soldiers may now eat in groups as they please, while the unvaccinated look on as they grab their grub and go.

With soldiers, experts “talk up to decliners versus talk down,” said Col. Joseph Buccino, a spokesman at Fort Bragg.

promoting inoculations, and stadiums have become a new line of demarcation, where vaccinated sections are highlighted as perks akin to V.I.P. skyboxes.

In Washington, Gov. Jay Inslee recently announced that sporting venues and churches would be able to increase their capacity by adding sections for the vaccinated.

Some businesses — like gyms and restaurants — where the coronavirus was known to spread easily are also embracing a reward system. Even though many gyms have reopened around the country, some still haven’t allowed large classes to resume.

Others are inclined to follow the lead of gyms like solidcore in Washington, D.C., which seeks proof of inoculation to enroll in classes listed as “Vaccine Required: Full Body.” “Our teams are now actively evaluating where else we think there will be client demand and will be potentially introducing it to other markets in the weeks ahead,” said Bryan Myers, chief executive officer of the national fitness studio chain, in an email.

specific invitation designs with the inoculated in mind, vaccinated only please RSVP.

Not everyone endorses this type of exclusion as good public policy. “I worry about the operational feasibility,” said Jennifer Nuzzo, an epidemiologist at the Johns Hopkins Coronavirus Resource Center. “In the U.S., we don’t yet have a standard way to prove vaccination status. I hope we’ll see by fall such low levels of infection in the U.S. that our level of concern about the virus will be very low.”

But few dispute that it is legal. “Having dedicated spaces at events reserved for vaccinated people is both lawful and ethical,” said Lawrence O. Gostin, an expert in health law at Georgetown Law School. “Businesses have a major economic incentive to create safer environments for their customers, who would otherwise be reluctant to attend crowded events. Government recommendations about vaccinated-only sections will encourage businesses and can help us back to more normal.”

so far to impose vaccine mandates for workers, especially in a tight labor market. “Our association came out in favor of masks,” said Emily Williams Knight, president of the Texas Restaurant Association. “We probably will not be taking a position on mandates, which are incredibly divisive.”

But some companies are moving that way. Norwegian Cruise Line is threatening to keep its ships out of Florida ports if the state stands by a law prohibiting businesses from requiring vaccines in exchange for services.

Public health mandates — from smoking bans to seatbelt laws to containing tuberculosis outbreaks by requiring TB patients to take their medicines while observed — have a long history in the United States.

“They fall into a cluster of things in which someone is essentially making the argument that what I do is only my business,” said Dr. Frieden, who is now chief executive of Resolve to Save Lives, a program designed to prevent epidemics and cardiovascular disease. “A lot of times that’s true, unless what you do might kill someone else.”

Dr. Frieden was the main official who pushed for a smoking ban in bars and restaurants in 2003 when he was the New York City health commissioner under former Mayor Michael R. Bloomberg. Other senior aides at the time felt certain the ban would cost Mr. Bloomberg a second term. “When I was fighting for that, a City Council member who was against the ban said of bars, ‘That is my place of entertainment.’ And I said, ‘Well, that’s someone’s place of employment.’ It did have impact.”

Mr. Duggan, the bar owner in Washington, said protecting his workers and patrons are of a piece. “As we hit a plateau with vaccines, I don’t think we can sit and wait for all the nonbelievers,” he said. “If we are going to convince them, it’s going to be through them not being able to do the things that vaccinated people are able to do.”

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Eight people from the New York Yankees organization tested positive, despite having been vaccinated. Here’s what to know.

Gleyber Torres, the Yankees’ two-time All-Star shortstop, became the eighth person involved with the Yankees organization to test positive for the coronavirus this week, the team announced on Thursday. But what has caught the attention of many outside of the baseball world is that Torres, three coaches and four support staff members had all been fully vaccinated.

The Yankees have called all eight instances “breakthrough positives.” According to the Centers for Disease Control and Prevention, a breakthrough case occurs when a fully vaccinated person contracts the virus. It said a small number of such cases would be expected despite the effectiveness of vaccines, because none of the vaccines are capable of preventing illness in 100 percent of cases.

It’s not immediately clear how this happened.

The Yankees had been enjoying relaxed health and safety protocols under rules negotiated by M.L.B. and the players’ union for reaching a threshold at which 85 percent of the team’s players and key personnel were fully vaccinated.

When asked about a possible common thread among the positive cases, Cashman pointed to a long rain delay before a game at Yankee Stadium on Saturday and said that players had a much larger clubhouse space indoors to spread out in compared to what the coaches and the support staff had. A day later, the team flew to Tampa.

Dr. Rochelle P. Walensky, the C.D.C.’s director, said on Thursday that the agency wanted to learn more about the Yankees’ outbreak. She pointed to the fact that six of the Yankees’ seven cases as of Wednesday were asymptomatic, suggesting that proved that the vaccine was indeed effective.

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Looking for Bipartisan Accord? Just Ask About Big Business.

But in recent years, that compact has begun to fracture. Democrats, pushed by progressive activists, have shifted further to the left on a wide range of economic policy issues. Under Mr. Trump, Republicans became more hostile to free trade and immigration. After the Jan. 6 storming of the Capitol, some prominent companies and business groups announced they would cut off donations to Republicans who had joined an effort to challenge in Congress the results of Mr. Trump’s November loss to Mr. Biden, prompting some Republican lawmakers to swear off corporate donations.

Many top executives feel they have little choice. They are being pressured by customers and increasingly by young, progressive employees to speak out publicly on major issues. And in the era of social media, companies can get into just as much trouble by staying silent as by weighing in.

Polling data shows the squeeze. A Gallup poll conducted in January, in the days leading up to and immediately following the Capitol riot, found that just 31 percent of Republicans were satisfied with the “size and influence of major corporations.” That was down from 57 percent a year earlier.

And in a survey conducted last month for The New York Times by the online research platform SurveyMonkey, 81 percent of Republicans who knew enough to form an opinion said it was inappropriate for business leaders to speak out against the Georgia law. And 78 percent of Republicans said large corporations had too much influence over American life in general. (The survey was conducted before two coalitions of business leaders released letters calling for expanded voting rights in Texas.)

Elena Adams, a survey respondent in Northern California, said she began to feel that corporate America was shifting against her a few years ago, when Nike embraced Colin Kaepernick, the former San Francisco 49ers quarterback who drew widespread attention for kneeling during the national anthem to protest police violence.

“Basically I think we’re celebrating people who are not for the United States and pushing the agenda that we should be ashamed if we’re not people of color,” she said. “This whole narrative of the race thing, it’s reverse racism, is what’s happening.”

Ms. Adams, 66, said she had stopped flying Delta and buying Coca-Cola products. Since Major League Baseball relocated the All-Star Game from Atlanta over the Georgia voting law, she has quit following the Oakland Athletics. She has abandoned social media, believing that companies such as Facebook and Twitter are unfair to conservatives, and told the purchasing managers at the emergency response business where she is a partner to avoid buying from companies that espouse liberal positions, although she said it was too difficult to avoid companies like Amazon and Google altogether.

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Shohei Ohtani Is Thriving as a Two-Way Star

“I could never get it out of my own mind what I saw,” Minasian said. “He was as talented as any player, from an age standpoint, as I’ve seen. There are things that he can do on a baseball field that other people can’t.”

Ohtani, whose father played baseball in Japan’s industrial league and whose mother was a standout badminton player, recognized that as a youngster. He hoped to leap straight from high school in Japan to the major leagues in the United States. He told this to clubs before the 2012 Nippon Professional Baseball draft and emphasized it to Hokkaido when it informed him it wanted to make him the No. 1 overall pick. The Los Angeles Dodgers were keenly interested at the time, but Hokkaido lured him with the idea of allowing him to pitch and hit.

Takashi Ofuchi, the Fighters’ amateur scout group leader who evaluated Ohtani for years and helped develop the plan, told Bleacher Report in 2017: “If a person has the possibility to do everything, we need to look at that person and his talent and bring his skills along all at the same time. It’s like Michelangelo and Einstein. They could do art, science, everything.”

Ohtani has been painting with bold, broad brush strokes ever since. And in an age of specialization, the Angels and Ohtani are zigging while everybody else is zagging. Why, indeed, place a fence around creative genius?

Through Wednesday he was ranked second in the majors in home runs (10) and extra-base hits (21) and tied for third in total bases (78). On the mound, he is 1-0 with a 2.10 E.R.A. over five starts with 40 strikeouts and a .126 opponents’ batting average. He even leads the Angels with six steals.

In Seattle a couple of Sundays ago, he was hit by a pitch and immediately stole second and third. “I could not have loved that moment any more,” Maddon said.

In a precautionary move, the Angels moved his start, which had been scheduled for the next day, to midweek, but he still smashed a two-run homer and circled the bases to chants of “M.V.P.! M.V.P.!”

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