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President Biden was standing in an Upper East Side townhouse owned by the businessman James Murdoch, the rebellious scion of the media empire, surrounded by liberal New York Democrats who had paid handsomely to come hear optimistic talk about the Biden agenda for the next few years.

It was Oct. 6, 2022, but what they heard instead that evening was a disturbing message that — though Mr. Biden didn’t say so — came straight from highly classified intercepted communications he had recently been briefed about, suggesting that President Vladimir V. Putin’s threats to use a nuclear weapon in Ukraine might be turning into an operational plan.

For the “first time since the Cuban Missile Crisis,” he told the group, as they gathered amid Mr. Murdoch’s art collection, “we have a direct threat of the use of a nuclear weapon if in fact things continue down the path they’ve been going.” The gravity of his tone began to sink in: The president was talking about the prospect of the first wartime use of a nuclear weapon since Hiroshima and Nagasaki.

And not at some vague moment in the future. He meant in the next few weeks.

The intercepts revealed that for the first time since the war in Ukraine had broken out, there were frequent conversations within the Russian military about reaching into the nuclear arsenal. Some were just “various forms of chatter,” one official said. But others involved the units that would be responsible for moving or deploying the weapons. The most alarming of the intercepts revealed that one of the most senior Russian military commanders was explicitly discussing the logistics of detonating a weapon on the battlefield.

Fortunately, Mr. Biden was told in his briefings, there was no evidence of weapons being moved. But soon the C.I.A. was warning that, under a singular scenario in which Ukrainian forces decimated Russian defensive lines and looked as if they might try to retake Crimea — a possibility that seemed imaginable that fall — the likelihood of nuclear use might rise to 50 percent or even higher. That “got everyone’s attention fast,” said an official involved in the discussions.

No one knew how to assess the accuracy of that estimate: the factors that play into decisions to use nuclear weapons, or even to threaten their use, were too abstract, too dependent on human emotion and accident, to measure with precision. But it wasn’t the kind of warning any American president could dismiss.

“It’s the nuclear paradox,” Gen. Mark A. Milley, the chairman of the Joint Chiefs of Staff until he retired in September, told me over dinner last summer at his official quarters above the Potomac River, recalling the warnings he had issued in the Situation Room.

He added: “The more successful the Ukrainians are at ousting the Russian invasion, the more likely Putin is to threaten to use a bomb — or reach for it.”

This account of what happened in those October days — as it happened, just before the 60th anniversary of the Cuban Missile Crisis, the closest the United States and the Soviet Union ever came to a nuclear exchange in the Cold War — was reconstructed in interviews I conducted over the past 18 months with administration officials, diplomats, leaders of NATO nations and military officials who recounted the depth of their fear in those weeks.

Though the crisis passed, and Russia now appears to have gained an upper hand on the battlefield as Ukraine runs low on ammunition, almost all of the officials described those weeks as a glimpse of a terrifying new era in which nuclear weapons were back at the center of superpower competition.

While news that Russia was considering using a nuclear weapon became public at the time, the interviews underscored that the worries at the White House and the Pentagon ran far deeper than were acknowledged then, and that extensive efforts were made to prepare for the possibility. When Mr. Biden mused aloud that evening that “I don’t think there’s any such thing as the ability to easily” make use of “a tactical nuclear weapon and not end up with Armageddon,” he was reflecting urgent preparations being made for a U.S. reaction. Other details of extensive White House planning were published in a New York Times opinion piece and by Jim Sciutto of CNN.

Mr. Biden said he thought Mr. Putin was capable of pulling the trigger. “We’ve got a guy I know fairly well,” he said of the Russian leader. “He is not joking when he talks about potential use of tactical nuclear weapons or biological or chemical weapons because his military is, you might say, significantly underperforming.”

Since then, the battlefield advantage has changed dramatically, and October 2022 now looks like the high-water mark of Ukraine’s military performance over the past two years. Yet Mr. Putin has now made a new set of nuclear threats, during his equivalent of the State of the Union address in Moscow in late February. He said that any NATO countries that were helping Ukraine strike Russian territory with cruise missiles, or that might consider sending their own troops into battle, “must, in the end, understand” that “all this truly threatens a conflict with the use of nuclear weapons, and therefore the destruction of civilization.”

“We also have weapons that can strike targets on their territory,” Mr. Putin said. “Do they not understand this?”

Mr. Putin was speaking about Russian medium-range weapons that could strike anywhere in Europe, or his intercontinental ballistic missiles that can reach the United States. But the scare in 2022 involved so-called battlefield nukes: tactical weapons small enough to be loaded into an artillery shell and designed to eviscerate a military unit or a few city blocks.

At least initially, their use would look nothing like an all-out nuclear exchange, the great fear of the Cold War. The effects would be horrific but limited to a relatively small geographic area — perhaps detonated over the Black Sea, or blasted into a Ukrainian military base.

Yet the White House concern ran so deep that task forces met to map out a response. Administration officials said that the United States’ countermove would have to be nonnuclear. But they quickly added that there would have to be some kind of dramatic reaction — perhaps even a conventional attack on the units that had launched the nuclear weapons — or they would risk emboldening not only Mr. Putin but every other authoritarian with a nuclear arsenal, large or small.

Yet as was made clear in Mr. Biden’s “Armageddon speech” — as White House officials came to call it — no one knew what kind of nuclear demonstration Mr. Putin had in mind. Some believed that the public warnings Russia was making that Ukraine was preparing to use a giant “dirty bomb,” a weapon that spews radiological waste, was a pretext for a pre-emptive nuclear strike.

The wargaming at the Pentagon and at think tanks around Washington imagined that Mr. Putin’s use of a tactical weapon — perhaps followed by a threat to detonate more — could come in a variety of circumstances. One simulation envisioned a successful Ukrainian counteroffensive that imperiled Mr. Putin’s hold on Crimea. Another involved a demand from Moscow that the West halt all military support for the Ukrainians: no more tanks, no more missiles, no more ammunition. The aim would be to split NATO; in the tabletop simulation I was permitted to observe, the detonation served that purpose.

To forestall nuclear use, in the days around Mr. Biden’s fund-raiser appearance Secretary of State Antony J. Blinken called his Russian counterpart, as did Defense Secretary Lloyd J. Austin III and the national security adviser, Jake Sullivan. Germany’s chancellor, Olaf Scholz, was going on a planned visit to Beijing; he was prepped to brief Xi Jinping, China’s president, about the intelligence and urge him to make both public and private statements to Russia warning that there was no place in the Ukraine conflict for the use of nuclear weapons. Mr. Xi made the public statement; it is unclear what, if anything, he signaled in private.

Mr. Biden, meanwhile, sent a message to Mr. Putin that they had to set up an urgent meeting of emissaries. Mr. Putin sent Sergei Naryshkin, head of the S.V.R., the Russian foreign intelligence service that had pulled off the Solar Winds attack, an ingenious cyberattack that had struck a wide swath of U.S. government departments and corporate America. Mr. Biden chose William J. Burns, the C.I.A. director and former U.S. ambassador to Russia, who is now his go-to troubleshooter for a variety of the toughest national security problems, most recently getting a temporary cease-fire and the release of hostages held by Hamas.

Mr. Burns told me that the two men saw each other on a mid-November day in 2022. But while Mr. Burns arrived to warn what would befall Russia if it used a nuclear weapon, Mr. Naryshkin apparently thought the C.I.A. director had been sent to negotiate an armistice agreement that would end the war. He told Mr. Burns that any such negotiation had to begin with an understanding that Russia would get to keep any land that was currently under its control.

It took some time for Mr. Burns to disabuse Mr. Naryshkin of the idea that the United States was ready to trade away Ukrainian territory for peace. Finally, they turned to the topic Mr. Burns had traveled around the world to discuss: what the United States and its allies were prepared to do to Russia if Mr. Putin made good on his nuclear threats.

“I made it clear,” Mr. Burns later recalled from his seventh-floor office at the C.I.A., that “there would be clear consequences for Russia.” Just how specific Mr. Burns was about the nature of the American response was left murky by American officials. He wanted to be detailed enough to deter a Russian attack, but avoid telegraphing Mr. Biden’s exact reaction.

“Naryshkin swore that he understood and that Putin did not intend to use a nuclear weapon,” Mr. Burns said.

The Scantron bubbles were gone. So were the page-long passages and the pressure to speed-read them. No. 2 pencils? Optional, and only for taking notes.

On Saturday, students in America took the newest version of the SAT, which was shorter, faster — and most notably, all online.

Some exams were briefly mired by technical glitches, but even so, test takers generally had positive views about the new format. They were especially relieved with the brevity of the exam — which dropped from three hours to a little over two hours — as well as the ability to set their own pace as they worked through the questions.

“It’s here to stay,” said Harvey Joiner, 17, a junior at Maynard H. Jackson High School in Atlanta, referring to the digital format. “Computers are what we’re more comfortable with.”

Given on paper for 98 years, the SAT was updated to reflect the experience of a generation raised in an era of higher anxiety, challenged attention spans and remote learning. The change comes as the College Board, which administers the test, and proponents of standardizing testing say that the exams still have a place in determining college acceptance and aptitude.

Disrupted by the pandemic and rocked by concerns that the tests favor high-income students, the SAT has had a shaky few years, with many colleges removing standardized tests as a requirement for admission. Some selective universities, including Brown, Yale, Dartmouth and M.I.T., have since reinstated the test, but at most schools, it has remained optional.

The current iteration of the test aims to drain some of the intimidation out of the process and evaluate modern students with tools to which they are more accustomed. The test has been trimmed, and students have been given more time for each question. The reading passages are much shorter, and an online graphing calculator is built into the application for the math section, which some see as a way to level the playing field for low-income students.

The tests also are harder to cheat on, with “adaptive” questions that become harder or easier, depending on a student’s performance. Students can bring their own laptops or tablets or use school-issued equipment, but cannot have any other application running in the background, and must take the test at a public test center with a proctor roaming the room.

Many students seemed to welcome this new format on Saturday. Naysa Srivastava, a 17-year-old who took the test in Chicago, found that the brevity of the reading passages and the built-in calculator better reflected her experience as an online learner. “Almost all my classes are digital,” she said.

Elijah McGlory, 18, a senior at Druid Hills High School in Atlanta, said taking the test digitally was “way better” compared with the paper version. “I got more questions done online,” he said.

Sharen Pitts, a retired schoolteacher who has worked for four years as a proctor in and around Chicago, noticed several of her students echoing the sentiment after the test she oversaw on Saturday. But she added that some “preferred paper because digital was harder on the eyes.”

Ms. Pitts said that the main difference she noticed with the new format on Saturday was the shortened test time, which some teachers see as a negative change for students. Critics of the new SAT have said that the shorter exam and reading passages do not help students develop the greater reading stamina they need amid constant distractions from technology.

But the test’s speed was offset by a range of technical issues.

The start of the exam was set back at some test centers, as students had problems connecting to the Wi-Fi. Specifically, test takers at Oak Park River Forest and Georgia State University experienced 30- to 45-minute delays because of connectivity issues.

“It took a little while for everyone to get on the internet,” said Matthew Schmitt, a 16-year-old junior from the Lincoln Park neighborhood of Chicago. “But this is the first time they’re doing the digital SAT.”

On social media, students and parents reported other glitches, including math answers that seemed incorrect and frozen on-screen calculations. In New York, Lida Safa, 15, noticed technical issues such as one student needing a charger at her test center. And she brought her own calculator, just as a backstop in case the online one felt too unfamiliar.

This is not the first time test takers have encountered glitches on digital versions of standardized exams. In recent years, several high school students taking Advanced Placement tests online have had problems with functions like submitting their answers and logging in.

Priscilla Rodriguez, senior vice president of college readiness assessments at the College Board, said that “a vast majority of students” were able to complete the new SAT on Saturday.

“As with paper-and-pencil testing, individual student or test center issues are possible with digital testing,” Ms. Rodriguez said. She added that those who had problems with testing would be able to retake the exam if needed.

And students seemed not too bothered by the snags on Saturday. Naysa, in Chicago, regarded the bugs as an inevitable feature of any new system. And Danny Morrison, 16, who tested in Atlanta, said, “I think as they keep going, they’re going to get more efficient.”

Several also liked a function of the test that puts each student on an automatic timer, rather than leaving stop and start times up to the proctor.

“Before, it was your teacher that had to have all the timing right, and you had to wait for everyone to finish to go on breaks,” said Lora Paliakov, 16, of Atlanta.

Matthew, the 16-year-old in Chicago, noted that “you could work more at your own pace.” This, some found, made the whole testing experience less stressful.

Nerves, however, were another matter. Lida, the 15-year-old in New York who goes to the Razi School, a private Islamic institution, had taken the test on paper in December, and she had a good sense of what to expect. “But I didn’t know about this one,” she said, referring to the new format.

So she leaned on a few home remedies before going into the exam. A light breakfast. A trick she has used to calm her mind — counting her fingers by touching each one to her thumb in order. And a little prayer before opening her MacBook for the test that her math teacher had taught her.

“To be honest? It wasn’t as bad as I thought it would be,” she said. “I feel like I probably did better this time.”

Dana Goldstein contributed reporting.

Nearly two months after undergoing abdominal surgery, Catherine, the princess of Wales, appeared in an official photograph, released on Sunday by Kensington Palace to mark Mother’s Day in Britain.

Catherine, who posed with her three children, George, Charlotte and Louis, has not been seen in public since before she entered the hospital on Jan. 16, provoking a swirl of increasingly far-fetched rumors on social media about her condition.

The image of a smiling Catherine, surrounded by her family, will most likely quiet what had become an extraordinary distraction for the British royal family, which is also dealing with a cancer diagnosis for King Charles III. But it did not completely dispel the questions about the 42-year-old princess.

Kensington Palace, where Catherine and her husband, Prince William, have their offices, did not release any new details on her condition or convalescence. It said the photograph was taken by William this past week in Windsor, where the family lives in Adelaide Cottage, on the grounds of Windsor Castle.

In a post on social media accompanying the photograph, Catherine said, “Thank you for your kind wishes and continued support over the last two months. Wishing everyone a Happy Mother’s Day.”

Last week, a grainy paparazzi shot of Catherine, riding in a car driven by her mother, was posted on the American celebrity gossip site TMZ. British newspapers and broadcasters reported widely on the photograph, but did not publish it, honoring the palace’s appeal that she be allowed to recuperate in private.

Britain’s news media has struggled with how to cover Catherine, torn between a ravenous public appetite for news about a future queen and Britain’s custom of giving even public figures a right to privacy in health matters.

Kensington Palace has released almost no information about Catherine’s surgery or recovery, saying only that she will not go back to official duties until after Easter. She was last seen in public on Christmas Day at Sandringham, the royal family’s country residence, where she attended church services with her family.

Catherine was not seen leaving the London Clinic, in the city’s Marylebone neighborhood, on Jan. 29, where she had spent 13 days after the surgery. There were no pictures of her being visited in the hospital by her husband or children, and there was only a single shot of William driving near the hospital during that period.

Charles, by contrast, was photographed leaving the same hospital after being treated for an enlarged prostate. Doctors discovered he had cancer during that procedure, Buckingham Palace later announced. The disclosure was a break with the family’s tradition of being reticent about making health issues public.

But the palace has also left questions about the king, not specifying what kind of cancer he has or his prognosis. Though he has suspended public appearances to undergo weekly treatment, Charles has continued to work. He was photographed holding his weekly meeting with Prime Minister Rishi Sunak and speaking via video link with Prime Minister Justin Trudeau of Canada.

The health troubles of Charles and Catherine have made this an anxious winter for the royal family. Its ranks have already been slimmed by the death of Queen Elizabeth II in 2022 and her husband, Prince Philip, a year earlier, as well as by the bitter split with Prince Harry and his wife, Meghan, who moved to California in 2020.

Early on Saturday morning, a car crashed into the main gate in front of Buckingham Palace. Neither Charles nor his wife, Queen Camilla, were there at the time, the palace said on Sunday, adding that the police had arrested the driver. London newspapers published images of a kneeling figure, surrounded by police officers who shouted, “Keep your hands on your head.” The police did not cite a motive.

The questions about Catherine reached a crescendo last week with the paparazzi shot, which prompted debate on social media about whether it was actually her. There was also confusion over an announcement by the British Army that Catherine would attend Trooping the Color, a military ceremony marking the birthday of the sovereign, in June.

Kensington Palace raised objections with the Defense Ministry, saying that only it confirms public appearances by the prince and princess of Wales. The army pulled the announcement off its website later that day.

With Catherine’s prolonged absence, the only hint about her condition came from her uncle, Gary Goldsmith, who appeared last week on a British reality TV show, “Celebrity Big Brother.” Mr. Goldsmith, 58, said his niece was getting “the best care in the world,” but did not offer other details about her health.

Conciliatory messages on social media. Open invitations for coffee. Zoom calls inviting attendees to unload.

Even before she announced her campaign for Senate in Arizona, Kari Lake, a Republican and a favorite of former President Donald J. Trump, has been on a mission to make peace. Her failed bid for governor two years ago was defined by her fervent embrace of Mr. Trump’s false claims of a stolen election, and by her relentless attacks on the party’s establishment figures who criticized her for that dishonesty. But now, looking to wrest a seat from Democrats in a key presidential battleground, Ms. Lake is courting former foes and trying to mend fences.

In addition to her public overtures, Ms. Lake has reached out privately to establishment Republicans in the state in recent months — including some she personally offended — seeking their support. The list includes Doug Ducey and Jan Brewer, two of the state’s former governors; Karrin Taylor Robson and former Representative Matt Salmon, two of her 2022 primary rivals; and Meghan McCain, the daughter of the longtime Arizona senator John McCain, according to six people with knowledge of the outreach, some of whom insisted on anonymity to discuss private interactions. In some cases, Ms. Lake has expressed regrets about her past behavior, one of the people said.

“There was some damage done from that primary that clearly bled into the general election,” said Daniel Scarpinato, a Republican consultant in Arizona who worked as a top aide to Mr. Ducey years ago. “I think you clearly see a genuine effort to bring more Republicans into the fold.”

Ms. Lake, a former television anchor and political newcomer in 2022, conducted a scorched-earth campaign to win the G.O.P. primary for governor. She appealed to Mr. Trump’s supporters by championing his baseless theories of election fraud, while lacerating her opponents. She accused Ms. Robson of “trying to buy the election with her 95-yr-old husband’s millions,” and blasted Mr. Ducey as “do-nothing Ducey.”

Perhaps most critically, she angered the family of Mr. McCain, who died in 2018, by declaring that her political rise “drove a stake through the heart of the McCain machine” and by inviting the voters in the state who admired him to “get the hell out.” The divisiveness caused some Republicans to balk at backing Ms. Lake, even if it meant a Democrat would win.

She now says her insults to Mr. McCain were meant “in jest.”

“Things have gotten so much worse under Joe Biden that we’re at a point where we don’t have time to have past arguments getting in the way of us moving forward as a country,” Ms. Lake said in an interview in Phoenix last month. She described herself as someone “who enjoys talking to people and bringing people together.”

If Ms. Lake wins her primary, she can expect a tight race against Representative Ruben Gallego, who is running essentially unopposed for the Democratic nomination. Ms. Lake has a primary opponent, Sheriff Mark Lamb of Pinal County, but she leads him by a wide margin in polls ahead of the contest on July 30. Senator Kyrsten Sinema, a former Democrat who left the party in 2022, is not running for re-election.

Some early signs suggest her effort is bearing fruit, at least on the national level. The National Republican Senatorial Committee, which works to elect Republicans to the Senate, endorsed her, and Senator John Barrasso of Wyoming, the third-ranking member of the Senate Republican leadership, campaigned alongside her in Phoenix last month. She met with Senator Mitch McConnell of Kentucky, the longtime Republican leader, on Capitol Hill on Wednesday, before attending a fund-raiser with about 20 senators in Washington.

Newt Gingrich, the former Republican speaker of the House, said in an interview that he was also endorsing Ms. Lake and that she had “matured” in her approach.

Still, her success appears mixed at home, where some of her overtures have been rebuffed. Ms. Lake sent Ms. McCain, whom she once compared to a “rabid dog,” a public message on X last month inviting her to lunch. Ms. McCain responded: “NO PEACE,” punctuated by a vulgarity.

“These are wounds that are unable to heal for me and my family because people like Kari Lake and Trump continue to debase my family and dad,” Ms. McCain said in an interview. “What she has asked of me is to give her cover for her hideous commentary and her hideous statements about my family, and I would rather die than do that.”

Mr. Ducey is not expected to endorse in the Republican primary, according to a person familiar with his plans. A recent conversation between Ms. Lake and Ms. Robson was productive, according to advisers for the two women, but nothing was decided.

By her own account, Ms. Lake’s efforts to make peace have included meeting with skeptical Republicans at their offices, taking them to coffee or drinks and making calls. In an interview at the Conservative Political Action Conference near Washington last month, Ms. Lake said some of the conversations had been difficult, describing occasionally intense phone and Zoom calls that began with outrage from the other side. She has also held Zoom meetings courting Republican donors and consultants, and attended a lunch with members of the Arizona Chamber of Commerce last month.

“I’m willing to continue to extend olive branches. If somebody does rebuff it and say, ‘No, I’m not interested,’ that’s fine,” she said in the interview in Phoenix. “The olive branch is still out. My door is still open.”

But fresh reservations about Ms. Lake have also cropped up among some grass-roots Republicans, many of whom are vocal supporters of Mr. Trump. Many were dismayed by the publication in January of an audio recording that Ms. Lake secretly captured during a conversation she had last year with Jeff DeWit, the chairman of the Arizona Republican Party, during which Mr. DeWit appeared to offer her a bribe to persuade her not to run for Senate. In the recording, which Mr. DeWit said was “selectively edited,” Ms. Lake can be heard rejecting his request.

Mr. DeWit resigned shortly after the recording surfaced, and Ms. Lake framed the episode as an example of her independence. Some members of the state party reacted with anger, expressing worry about other private conversations Ms. Lake might have recorded. Ms. Lake was greeted with boos at a meeting to elect a new chair.

“Is it really the way we should all be behaving, even as Republicans? To tape-record somebody that trusts you?” asked Jeanne Kentch, the chair of the Mohave County Republican Party. “I love Kari, don’t get me wrong. But I think that’s what people are concerned about.”

Ms. Lake denies that she regularly records private conversations. Still, Mr. Lamb, who trails far behind her in fund-raising, has sensed an opening. Mr. Lamb is the only candidate who “can appeal to all Republicans, conservative independents and disaffected Democrats,” said Ed Morabito, a senior adviser to his campaign.

Despite a newfound desire for party unity, Ms. Lake has not shied away from extreme views, continuing to push baseless theories of voter fraud in news media appearances and telling reporters in Washington last week that “we had a really rigged election in Arizona.” (After losing to Ms. Hobbs, Ms. Lake falsely claimed fraud, filing fruitless lawsuits in attempts to overturn the outcome.) She has also sympathized with and appeared alongside people convicted of crimes related to their participation in the Jan. 6, 2021, attack on the Capitol.

However, she has not made these stances a centerpiece of her Senate campaign — one of a handful of departures from her campaign for governor. On abortion, which she once called the “ultimate sin,” she now opposes a federal ban.

“Kari Lake will say or do anything to gain power,” Hannah Goss, a spokeswoman for Mr. Gallego, said in a statement.

And for some Republicans, the scars that Ms. Lake left may be too deep. Sharon Harper, the chief executive of the real estate firm Plaza Companies who was close friends with Senator McCain, supported Ms. Hobbs in 2022 and has no plans to back Ms. Lake this campaign, either.

“I think people understand who Kari Lake is,” Ms. Harper said. “We’ve seen what she has demonstrated, and I don’t think an opinion changes if someone says, ‘I didn’t really mean what I said.’”

Michael C. Bender and Kayla Guo contributed reporting.

Shortly after Gov. Kathy Hochul announced on Wednesday that hundreds of National Guard soldiers would be deployed to patrol the New York City subway system and check riders’ bags, her office made an adjustment: Soldiers searching bags would not carry long guns.

The change was ordered by Ms. Hochul on Wednesday for implementation on Thursday, according to a spokesman for the governor. Ms. Hochul issued a directive that National Guard members would be prohibited from carrying long guns at bag-check stations, he said. Soldiers not working at the stations would presumably be allowed to carry them.

Donna Lieberman, the executive director of the New York Civil Liberties Union, called the ban on long guns at bag-checking stations a “relief,” but said the Guard’s presence underground remained “an unnecessary overreaction based on fear, not facts.”

“Deploying military personnel to the subways will not make New Yorkers feel safe,” Ms. Lieberman said. “It will, unfortunately, create a perfect storm for tension, escalation and further criminalization of Black and brown New Yorkers.”

Early images of the National Guard’s deployment showed soldiers standing near turnstiles in the subterranean system, wearing camouflage and military gear and holding long guns.

Ms. Hochul, a Democrat, said the move to flood the system with reinforcements — 750 members of the New York National Guard, and an additional 250 personnel from the State Police and the Metropolitan Transportation Authority — would help commuters and visitors feel safe.

Subway safety, a perpetual concern for New Yorkers, has been a challenging issue for public officials, who can be as sensitive to the perception that mass transit is dangerous as they are to an actual rise in crime.

In February, following a 45 percent spike in major crimes in the first month of the year compared with the same period last year, Mayor Eric Adams ordered an additional 1,000 police officers into the subway system. Reported crime rates in the system declined that month, according to city data, and the overall rise in major crimes for the year as of March 3 was 13 percent, Police Department data shows.

Ms. Hochul’s announcement this week drew criticism from public officials and from some members of her own party.

Jumaane N. Williams, the city’s public advocate, warned that the plan would “criminalize the public on public transit.” Emily Gallagher, an assemblywoman and democratic socialist from Brooklyn, said that Ms. Hochul’s move was a “ham-fisted and authoritarian response” that validated “G.O.P. propaganda about urban lawlessness in an election year.”

John Chell, the Police Department’s chief of patrol, cited recent statistics suggesting that transit crime had dropped.

“Our transit system is not a ‘war’ zone!” he wrote on X.

On the front line just outside Bakhmut, Ukraine, a 32-year-old commander of a Ukrainian artillery platoon rocked to and fro in the passenger seat of a beat-up Lada, as another soldier navigated the car through a thick forest, sometimes mowing down young trees. When they reached their destination, a small village less than two miles from Russian lines, all that was left were destroyed houses, their shattered roofs visible in the moonlight.

The commander, a female soldier who uses the call sign Witch, is a former lawyer who, along with two of her brothers and her mother, joined the military the day after Russia invaded in February 2022. Her first experience in combat was in the outskirts of Kyiv that year, and much of what she has learned about weapons systems since has been self-taught and on the fly.

Since early 2023, Witch has been with her platoon in the 241st Brigade in the area around Bakhmut, supervising all of the artillery systems. She is resolute about staying in the military even if the war ends. “People who want to join the armed forces must understand that it’s a way of life,” she said.

As Ukraine struggles against fierce Russian assaults and its losses mount, there has been a surge of women who have enlisted, and they are increasingly volunteering for combat roles. The Ukrainian military has also made a concerted effort to recruit more women to fill its ranks.

About 65,000 women are currently serving in the Ukrainian Armed Forces, about a 30 percent increase since the war began. Roughly 45,000 serve as military personnel, and the rest hold civilian positions, according to the Defense Ministry of defense. Just over 4,000 are in combat positions.

Unlike for Ukrainian men, there is no mandatory draft for women; however, women who have medical or pharmaceutical degrees must register for the service.

These women fill a growing number of positions in the military: combat medics in assault units; senior gunners; snipers; commanders of tank units and artillery batteries; and at least one co-pilot on a medevac team who dreams of becoming Ukraine’s first female helicopter combat pilot. Dozens have been wounded in battle, and some have been killed or captured.

Along the front line, they operate under the same blanket of fear and hardship as male soldiers. In the dank, fortified shelter where Witch and one of her mortar teams spent most of their days, they waited in near darkness in the basement. Turning on lights would mean the crew could not quickly adjust their eyes to the dark if they had to go out and fire.

Farther north, a commander with the call sign Tesla, a former Ukrainian folk singer, sat hunched on a stool in the bare house serving as the field headquarters for the 32nd Mechanized Brigade. Russian forces in the Kupiansk region were sending barrages of artillery raining down on Ukrainian lines.

Tesla was simultaneously sending texts and voice notes to the soldiers in her unit while speaking to the second in command about the battlefield plan. Her oversized pants were rolled up, revealing neon orange socks with cartoon avocados on them.

She was trying to redirect Russian fire on a different battalion to her own soldiers’ position, so that the other unit could evacuate a comrade who had been badly wounded. “Three tourniquets on three different limbs,” came the information on a voice message, she said.

“Send one more,” Tesla ordered over a voice note, giving the command to her soldiers to fire again. “When you finish, inform me.”

Shortly after a Russian assault began in October, overwhelming Ukrainian lines, 24 of her soldiers trained in artillery were ordered to reinforce the infantry troops, which are always situated closest to Russian lines. Tesla spoke to them before they deployed, feeling helpless.

“The worst thing is that I instructed them on completely different things in artillery, and then they get sent to the infantry,” she said. “And, imagine, they’re standing there looking at you as their commander, knowing that they will be sent into the worst possible situation.”

Of the 24 sent forward, 15 were wounded, Tesla said, and one was captured in the fighting. The incident stayed on Tesla’s conscience, but she kept her worries to herself. Her mother still did not know that Tesla was leading an artillery battery, thinking her daughter was working as an instructor at an academy, a safe distance from the front.

Until 2018, women were prohibited from holding combat positions in the Ukrainian military, though a few ignored the rules. Restrictions have been relaxed since Russia’s invasion. The enlistment of thousands more female service members in the military has largely been seen as a welcome step for the country, whose bids to join NATO and the European Union are still under review.

The downside is that the military has not been able to adapt quickly enough to accommodate them. Female soldiers say there is still a dire lack of women’s fatigues and boots, correctly fitted body armor, and feminine hygiene products. That leaves women to acquire many items on their own.

As a result, organizations like Veteranka and Zemliachky have helped fill the gap by raising money to provide items tailored to women.

But the problems go deeper, into issues of gender-based inequality and discrimination.

Many women serving in combat roles said that male soldiers and direct superiors largely did not discriminate by gender — though there are still sexual innuendoes and inappropriate comments.

Rather, it is senior commanders, often holdovers from the Soviet-era, who look down on women in the military, especially those in combat roles. In some cases, women are choosing to join newly formed brigades with younger, more dynamic commanders.

“I didn’t want to join a brigade that was established many years ago because I knew they wouldn’t listen to me as a young officer, and as a woman,” Tesla said.

In one instance, a brigade commander was so incensed by a woman commanding an artillery battery that he belittled her directly. “You will be crawling back to me on your knees begging to leave when you realize the job is too difficult, and I won’t allow you to leave your post,” she recalled him saying, requesting anonymity to speak frankly about a sensitive subject.

Claims of sexual harassment have also surfaced. According to some women, there have been no official channels for reporting harassment other than to battalion commanders, who then have to decide whether to follow up. In some cases, female soldiers said, witnesses may decline to testify for fear of repercussions.

Those impediments, as well as the potential for hurting their military careers, discourage women from reporting harassment, female soldiers said.

Diana Davitian, a spokeswoman for the Defense Ministry, said that on Jan. 1, the military launched a hotline where soldiers can report sexual harassment. The reports would be investigated, she said, and measures would be taken if the accusations were found to be true.

The ministry also said it planned to create a separate unit dedicated to ensuring gender equality and providing educational programs, including one focused on combating sexual violence related to the war.

Back in the basement, Tesla took a call from the command post: It was time to fire. The team scrambled into a partly covered yard a few feet away where a mortar barrel was readied.

A silence descended as Kuzya, 20, a senior gunner with the mortar platoon, looked through the scope and read out the coordinates on her phone. “Fire!” someone called out. Several more rounds were sent off before the team scrambled back into the basement, awaiting a potential return from the Russians.

Only months earlier, Kuzya’s boyfriend was killed in the fighting. She and Witch, who has a 7-year-old son she has barely seen in the past year, seemed to find solace in each other’s company. The two women trained in the same judo club in Kyiv, the capital, and the day after the invasion, they went to the enlistment office together to sign up.

For many women, war and the desire to be in combat feels like something they have prepared for for years. Foxy, 24, a former barista turned gunner and medic, volunteered to make camouflage nets after school throughout her teenage years, before she worked with injured veterans. She joined the military last year after weeks of training.

Her battalion commander gave her two options: “You’re a woman. You can work with documents or cook borscht,” Foxy recalled. “I had no choice but to handle paperwork until I switched battalions.”

She then became part of a mortar team in some of the most intense fights on the front line in Bakhmut, and was treated as an equal by her team. “While I faced some degree of sexism early on,” she said, “I feel like I don’t need to prove anything or convince anyone of what I can do.”

That is a sentiment echoed by Kateryna, 21, a lieutenant and a pilot of an Mi-8 medical evacuation helicopter. Kateryna has yet to fly her first medevac mission, but she hopes to become Ukraine’s first female combat pilot.

Ukrainian society is also gradually overcoming its skepticism of women serving in the military. For now, it is up to the new generation of women and their allies who will also be better placed to address discrimination and sexual harassment.

Evelina Riabenko contributed reporting.

When António Costa, a prime minister well liked by European leaders, handily won his third term as prime minister in 2022, many Portuguese prepared for a lasting, stable government given his Socialist Party’s strong majority in Parliament.

But by late last year, Mr. Costa had resigned, his government embroiled in a corruption investigation involving lithium exploration concessions.

On Sunday, Portugal faces a new election. It has raised the prospect that the Socialist Party could lose power for the first time in more than eight years, as well as the possibility of an unstable minority government.

One of the most significant changes in the election campaign has been the rise of a hard-right populist party.

What set off the vote?

Mr. Costa resigned in November after the opening of a corruption investigation into lithium exploration concessions, hydrogen production and the construction of a data center.

Lithium is key to helping the European Union transition to clean energy, and the bloc mostly relies on imports from China.

In Portugal, the lithium explorations were contentious even before the investigation into the awarding of concessions and had faced public opposition for the environmental damage the mines could cause.

The investigation now threatens to dampen much-needed foreign investment, said Marina Costa Lobo, a political scientist at the University of Lisbon. “And this might be problematic for Portugal,” she added.

Mr. Costa has not been formally charged with any crimes.

In televised remarks in November, he said that “no illicit act weighs on my conscience,” but added that the “dignity” of the prime minister’s office was incompatible with suspicions swirling around it.

Who is running now?

Portugal is electing a new Parliament in the early general election. The Socialist Party is in a tight race with the Democratic Alliance, a coalition of center-right parties led by the Social Democratic Party.

The Socialists are expected to win about 28 percent of the vote, according to an aggregation of polls by Politico, and the Democratic Alliance about 33 percent.

Pedro Nuno Santos, backed by the Socialist Party’s left wing, succeeded Mr. Costa as the head of the party.

Members of the Social Democratic Party have also been affected by corruption inquiries, with the president of a local government recently resigning.

Most surprising has been the rise of Chega, a populist right-wing party. Even though Chega is trailing behind the two mainstream parties, according to an aggregation of polls by Politico, a strong showing in the election could turn Chega into a kingmaker if no others can secure an outright majority.

Luís Montenegro, the Social Democratic Party leader, has ruled out making a coalition with Chega. Chega has made harsher punishment for corruption a pillar of its campaign, with billboards that read: “Portugal needs a cleanup.”

If his party comes first, Mr. Montenegro could form a minority government, a potentially unstable outcome that might not last very long, according to experts.

How popular is the far right?

Unlike other European countries, Portugal for years had not seen hard-right, anti-establishment parties gain traction among voters.

The rise of Chega, which means “enough” in Portuguese, is changing that, and a strong showing in the election could spell the end of the country’s exception.

“For us here in Portugal, it would be a novelty,” said José Santana Pereira, an associate professor of political science at the University Institute of Lisbon.

The party, founded in 2019, received about 1 percent of the vote in the 2019 election. That rose to about 7 percent in 2022, and the party is now predicted to win 16 percent of the vote, according to an aggregation of polls by Politico.

What are the main issues at stake?

Despite the corruption investigation, and Chega’s insistence upon it in its campaign, many voters seemed most preoccupied with issues affecting their everyday lives, Ms. Costa Lobo said.

With persistently low wages not keeping up with inflation, the Portuguese seem mostly preoccupied with the cost-of-living crisis. House prices have doubled in the past eight years, partly because of tourism rentals, making property unaffordable especially to the younger generations.

“Housing has become one of the main issues in the election,” Ms. Costa Lobo said. “Middle-class citizens are no longer able to rent or buy.”

Even with growth faltering in China, Xi Jinping appears imperiously assured that he possesses the right road map to surpass Western rivals.

China’s economy has lurched into a slower gear. Its population is shrinking and aging. Its rival, the United States, has built up a lead in artificial intelligence. Mr. Xi’s pronouncement several years ago that the “East is rising and West is declining” — that his country was on the way up while American power shrank — now seems premature, if not outright hubristic.

The problems have brought growing talk abroad that China could peak before it fully arrives as a superpower. But Mr. Xi seems unbowed in insisting that his policies, featuring extensive party control and state-led industrial investment in new sectors like electric vehicles and semiconductors, can secure China’s rise.

In a mark of that confidence, his government announced last week that China’s economy was likely to grow about 5 percent this year, much the same pace as last year, according to official statistics. And Mr. Xi emphasized his ambitions for a new phase of industrial growth driven by innovation, acting as if the past year or two of setbacks were an aberration.

“Faced with a technological revolution and industrial transformation, we must seize the opportunity,” he told delegates at China’s annual legislative meeting in Beijing, who were shown on television ardently applauding him.

He later told another group at the legislative session that China had to “win the battle for key core technologies,” and he told People’s Liberation Army officers to build up “strategic capabilities in emerging areas,” which, the officers indicated, included artificial intelligence, cyberoperations and space technology.

Mr. Xi’s bullishness may partly be for show: Chinese leaders are, like politicians anywhere, loath to admit mistakes. And some officials have privately conceded that the economic malaise is tamping down China’s ambitions and swagger, for now at least.

Ryan Hass, the director of the John L. Thornton China Center at the Brookings Institution who visited China late last year, said he came away with a sense that “the Chinese are a bit chastened even compared to where they were a year ago. The trajectory of China’s economy overtaking America’s in coming years — that’s been pushed further out on the horizon.”

Even so, Mr. Xi’s determination to stick to his long-term ambitions seems more than a show. “Xi and his team still believe that time and momentum remain on China’s side,” said Mr. Hass, a former director for China at the U.S. National Security Council. “With Xi in power,” he added, it’s hard to envision “any significant re-calibration in the overall trajectory that China’s on.”

Since taking office in 2012, Mr. Xi has tightened the hold of the Communist Party on Chinese society. He has extended state management of the economy, expanded the security apparatus to extinguish potential challenges to party rule, and confronted Washington over technology, Taiwan and other disputes.

To Mr. Xi’s critics, his centralizing, hard-line tendencies are part of China’s problems. He did not cause China’s risky dependence on the property market for growth, and he has worked to end it. But many economists argue he has been too heavy-handed, stifling business and innovation. Critics argue that Mr. Xi has also needlessly antagonized Western governments, prompting them to restrict access to technology and deepen security ties with Washington.

Since last year, the Chinese government moved to ease those strains. It has taken steps aiming to revive confidence among private businesses. Mr. Xi has also sought to dial down tensions with the United States and other countries.

Such moderating gestures point to what Mr. Xi has described as the “tactical flexibility” he expects of Chinese officials in difficult times. But in Mr. Xi’s telling, even as officials make easing steps, they must stick to his long-term objectives. He and his loyal subordinates have been defending his policies in speeches and editorials, suggesting that the doubters are shortsighted. Chinese officials and scholars have also stepped up denunciations of Western analysts who have forecast that China faces an era of decline.

Mr. Xi has stressed that economic and security priorities must work hand in hand even as China grapples with slower growth. Mr. Xi is also betting that investing in manufacturing and technology can deliver new “high quality” growth by expanding industries such as new clean energy and electric vehicles.

The Chinese leadership’s “mantra seems to be that ‘We’re not going to grow as fast as we used to, but we’re going to gain more leverage over trade partners by controlling critical parts of the global economy,’” said Michael Beckley, an associate professor at Tufts University, who has argued that China is a “peaking power,” meaning a country whose economic ascent has slowed but not yet stopped.

Some economists argue that China’s advances in these select industries will not be enough to make up for the drag caused by a fall in consumer confidence, and by developers and local governments straining under debt. China’s broader fortunes will heavily rest on whether Mr. Xi’s wager on technology can pay off.

“They see technology as the solution to every problem they’re facing — economic, environmental, demographic, social,” said Nadège Rolland, a researcher at the National Bureau of Asian Research who studies China’s strategic thinking. “If they cannot make sufficient advances in this domain, it’s going to be very difficult for them.”

Scholars in China and abroad who hope the country might take a more liberal path sometimes look to history for examples of when party leaders made bold changes to defuse domestic and international tensions.

The last time that China was caught in such a painful confluence was after the June 4, 1989, crackdown on pro-democracy protesters. The bloodshed prompted Western countries to impose sanctions on China, which deepened the economic shock. Within several years, however, Deng Xiaoping, then China’s leader, sought to repair relations with Washington and other capitals and unleashed market changes that revived growth and lured back Western investors.

Now, though, China faces much more entrenched antagonism from other major powers, Zhu Feng, a prominent foreign policy scholar at Nanjing University in east China, said in an interview. For example, China’s surging exports of electric cars — which have benefited from extensive government subsidies — could revive trade tensions, as the United States, Japan and Europe fear losing jobs and industrial muscle.

The economic and diplomatic strains are “posing the gravest challenge to China” in decades, Professor Zhu said.

Still, Chinese leaders seem to believe that, whatever their problems, their Western rivals face worsening ones that will ultimately humble and fracture them.

Recent reports from institutes under China’s ruling party, military and state security ministry point to the rancorous polarization in the United States ahead of the next election. Regardless of who wins, Chinese analysts argue, American power is likely to remain troubled by political dysfunction.

Chinese scholars have also focused on fault lines in the Western bloc over Russia’s war in Ukraine. Beijing’s relations with the United States and European governments were badly strained over Mr. Xi’s partnership with President Vladimir V. Putin. But as the war stretches into its third year, the burden of supporting Ukraine is deepening rifts and “fatigue” in the United States and Europe.

“U.S. foreign intervention cannot handle everything it is trying to juggle,” Chen Xiangyang, a researcher at the China Institutes of Contemporary International Relations in Beijing, which is under the state security ministry, wrote last year. “China can exploit the contradictions and leverage them to its own advantage.”

Even in the darkest of times, Sri Lankans held on to their humor.

In 2022, when the island nation’s economy collapsed and the government announced a QR code system to ration gasoline, a meme spread online: “Scanning Fuel QR Code Now Makes You Forget Last Three Months.”

And when public anger forced the strongman president to flee his palace, with protesters venturing inside to fry snacks in his kitchen and jump into his pool, another meme captured the mood upon their departure: “We Are Leaving. The Key Is Under the Flower Pot.”

It is this kind of online expression, which helped fuel the largest citizens’ movement in Sri Lanka in decades, that activists and rights groups fear is now endangered.

They are concerned about a new law, the Online Safety Act, that gives the government wide-ranging powers to deem speech on social media to be “prohibited statements.” Under the law, a committee appointed by the president will rule on what is prohibited, and violations could bring penalties ranging from fines of hundreds of dollars to years in prison.

The public security minister, Tiran Alles, told Parliament that the legislation would protect against online fraud, the spread of false information and the abuse of women and children. But he also made clear its potential political applications, saying it could be used against those who insult members of Parliament on social media.

Sri Lanka is taking a page from other countries in the region that are increasingly policing what people say online, most notoriously Bangladesh, where a 2018 law known as the Digital Security Act has led to the imprisonment of activists and opposition leaders.

The Sri Lankan law “is the newest weapon in the government’s arsenal of tools that could be used to undermine freedom of expression and suppress dissent,” said Thyagi Ruwanpathirana, a regional researcher for South Asia at Amnesty International, adding that the act was “ripe for misuse.”

Ms. Ruwanpathirana said that the Sri Lankan government needed to “demonstrate the political will to uphold” international human rights obligations as the country is set this year to hold its first elections since the 2022 crash.

The main impetus for the new law, analysts say, is the protest movement that toppled the government in 2022.

Political leaders want to make sure there is no repeat, the analysts say, a concern that persists as the movement’s goals remain largely unmet. While the powerful president, Gotabaya Rajapaksa, was forced out of office in 2022, little else changed at the top. The political elite has merely rearranged its seats, and Mr. Rajapaksa’s family-run political party has propped up a new president, Ranil Wickremesinghe, until the election later this year.

Mr. Wickremesinghe, a veteran politician, is trying to put the economy back in order, introducing difficult fiscal changes to improve the government’s balance sheet. But activists and rights groups say he has also gone after civil society leaders who were instrumental in the citizens’ movement.

“We saw many taking to social media to critique, to challenge and to push back on various state initiatives, so social media played a huge role in the people’s mobilization,” said Bhavani Fonseka, a senior researcher at the Center for Policy Alternatives, in the Sri Lankan capital, Colombo. “That gives new incentive for the government to bring in restrictions.”

Nalaka Gunawardena, a Colombo-based analyst, said that the political intentions of the new legislation were made evident by officials’ refusal to adjust it to better balance freedom of expression and the government’s concerns over online abuse.

In rushing through the legislation, Mr. Gunawardena said, the government rejected suggestions from media experts and rights activists who urged an exemption for those engaging in satire and parody.

Historically, satirists have faced trouble, and even exile, in Sri Lanka for targeting the majority Sinhala community or the powerful Buddhist monks. During the decades of the country’s bloody civil war, which ended in 2009, military leaders — particularly Mr. Rajapaksa, who served as defense secretary — were increasingly off limits.

When a coalition government briefly broke the Rajapaksa family’s hold on the country in 2015, political satire began to thrive online — the new president, Maithripala Sirisena, was a favorite of meme makers.

The elevation of the feared Mr. Rajapaksa as president in 2019 initially gave some pause, but as his management of the economy sent the country into a downward spiral, cartoonists and satirists saw little to lose.

The administrator of a popular anonymously run meme page called NewsCurry, which has about 50,000 followers on social media platforms, said that such efforts had brought attention to anti-democratic behavior and lies by politicians, helping to make up for a docile news local media. The new law, said the administrator, who asked not to be named for fear of running afoul of the authorities, should be renamed the Safety for Politicians Act.

Hamza Haniffa, who is part of a group that runs meme pages, said the law had made many of his friends hesitant to continue generating jokes. Posts have become less frequent.

“During the protest movement, we gave our opinions without being afraid,” he said. “But now we are concerned.”