President Nicolás Maduro was declared the winner in a presidential vote on Sunday that was marred by irregularities. Officials at some polling places refused to release paper tallies of the electronic vote count, and there were widespread reports of fraud and voter intimidation. Here are initial takeaways from Venezuela’s election.

Many fear a return to instability.

The government’s announcement that Mr. Maduro had beaten his opponent, Edmundo González, by seven percentage points instantly created a grim scenario for a country that only recently has started emerging from one of the largest economic collapses in modern history.

The results announced by the government-controlled electoral council varied wildly — by up to 30 percentage points — from most public polls and from the opposition’s sample of results obtained directly from voting centers. And there were many reports of major irregularities and problems at those voting centers.

The opposition leader María Corina Machado, who spearheaded Mr. González’s campaign, on Monday morning called the results “impossible.”

Some opposition supporters could take to the streets to protest the result. That could plunge Venezuela into a new period of political unrest, like those in 2014, 2017 and 2019, when security forces aligned with Mr. Maduro used deadly force to crush demonstrations.

Officials from several countries in the Americas, including the United States, expressed doubts about the announced results, raising the likelihood that a new term for Mr. Maduro would not be widely recognized abroad, either.

The opposition’s monitoring effort was blocked.

After a campaign marked by intensifying efforts by Mr. Maduro’s allies to rein in the opposition — including arrests of opposition campaign workers, intimidation and vote suppression — the opposition bet heavily on an effort to have supporters on hand to get a physical printout of the voting tally from every voting machine after the polls closed.

That access is allowed by Venezuelan election law. But by early Monday morning, Mr. González’s campaign said it had obtained only 40 percent of the tallies. In some places, monitors were barred from entering polling places or they never appeared in the first place. Often, election officials simply refused to hand over the tallies.

That will complicate efforts by the opposition to prove undeniably that the vote had been tampered with.

The results could be disastrous for Venezuela’s economy.

After years of fighting Mr. Maduro and his predecessor, Hugo Chávez, Venezuelan businessmen and foreign investors had largely made peace with his government in recent years. Sanctions imposed by the United States had forced Mr. Maduro to ditch some extreme policies like price and currency controls. The private sector was given an increasingly prominent role, public attacks against business owners had stopped and hyperinflation and rampant crime subsided somewhat.

The increased support from the private sector led to hopes that a credible result would keep the improvements coming and lead to some sort of political settlement. That appears unlikely now, and the dubious election results could test the thaw between Mr. Maduro and business leaders, and could possibly trigger a new wave of international sanctions.

Most critically, the result is unlikely to allow the Biden administration to unwind its sweeping economic sanctions against Venezuela. That would stunt the economic recovery, and is likely to lead to another wave of migration from a nation that has seen the exodus of one in five citizens in the past decade.

A smooth Venezuelan election that would have led to greater economic opening also suited the country’s Latin American neighbors, including Mr. Maduro’s old allies, the leftist governments of Brazil and Colombia.

The region has received the bulk of Venezuelan migration, leading to an anti-immigration political backlash in some places.

President Luiz Inácio Lula da Silva of Brazil took a surprisingly strong stand against Mr. Maduro earlier this week. “When you lose, you leave,” he told reporters. He sent his top foreign policy adviser, Celso Amorim, to Caracas for the election, and Mr. Amorim’s position on the vote could become a bellwether for the region.

Frances Robles and Isayen Herrera contributed reporting from Caracas, Genevieve Glatsky from Bogotá, Colombia, and Edward Wong from Tokyo.

The African National Congress, South Africa’s governing party announced on Monday that it had expelled the country’s former president, Jacob Zuma, officially severing ties with the once-celebrated anti-apartheid fighter after he helped form a rival political party.

Though Mr. Zuma has been a vaunted figure in the A.N.C. for decades, his tenure as president and party leader were marked by a series of corruption scandals that tarnished the party’s image. He was pressured to step down as South Africa’s president in 2018. This year, he used his broad political support to campaign for uMkhonto weSizwe, a rival party known by the initials M.K.

The rival party dealt a blow to the A.N.C. in May when it won the third-highest number of votes in the national election, keeping it from winning an outright majority for the first time since the end of apartheid 30 years ago. Mr. Zuma, 82, is now the face of M.K., which holds 58 out of the 400 seats in Parliament and appears set on challenging his former party’s grip on power.

The secretary-general of the A.N.C., Fikile Mbalula, said on Monday that Mr. Zuma had been expelled because he “actively impugned the integrity of the A.N.C. and campaigned to dislodge the A.N.C. from power while claiming that he had not severed his membership.” Mr. Zuma was given 21 days to appeal the decision.

In January, the A.N.C. suspended Mr. Zuma for what it said were “vitriolic attacks” against the party during his campaigning for M.K. Mr. Zuma maintained that his fight was to save the A.N.C. from its current leadership, which he has accused of being corrupt.

Mr. Zuma himself was not allowed to run for a seat because of a conviction and prison sentence for failing to testify before a corruption inquiry. Still, he appeared on campaign posters and on the ballot for M.K. He also led political rallies that attracted thousands of voters with his charismatic brand of populism.

As leader of the new opposition, Mr. Zuma has accused his successor and former A.N.C. deputy, President Cyril Ramaphosa, of corruption and wrongdoing. Analysts and political rivals believe that Mr. Zuma’s return to politics was driven by a desire to humiliate his former party, and particularly Mr. Ramaphosa, for not supporting him when he was arrested in 2021 on charges of contempt of court.

Mr. Zuma joined the A.N.C. as a liberation movement when he was a teenager. At 21, he was arrested by the apartheid police and spent a decade on Robben Island, serving a prison sentence alongside Nelson Mandela and other political stalwarts.

Yet under the leadership of Mr. Zuma, who was twice elected as A.N.C. president, the party lurched from scandal to scandal, and veteran freedom fighters accused him of attracting new members driven by self-interest. A yearslong corruption inquiry found that he had allowed friends and associates to loot state-owned companies.

Amelia Nierenberg and John Eligon contributed reporting.

Kate Douglass, a statistics graduate student and the second-fastest swimmer in the world this year in two Olympic swimming events, has always been good with numbers. But before she enrolled at the University of Virginia, she never considered that swimming itself was a math problem that she could try to solve.

That changed when she realized the concepts she was studying in the classroom could be used in her sport. These days, Douglass often gets into the pool while wearing a belt that holds an accelerometer, the same device found in smartphones and fitness watches. As she swims, the sensor measures her movement in three spatial directions 512 times per second.

“That’s helped me to figure out areas of my stroke where I can be more efficient,” said Douglass, 22. So far, so good: On Saturday, she began a busy Olympics schedule by winning a silver medal in the 4×100 freestyle relay.

The swimmers at the Paris Olympics all have the same challenge: to swim as fast as they can by moving through the water in a way that maximizes the force propelling them toward the finish line, while minimizing the force that slows them down. Elite swimmers use familiar tricks to reduce the resistance known as drag, like shaving before big meets and wearing swimsuits made from the same material as Formula 1 racing cars.

Though the sport has long relied on a swimmer’s feel in the water or a coach’s eye from the pool deck, Douglass and several of her U.S. Olympic teammates are exploring a new competitive frontier. Under the direction of a Virginia mathematics professor, Dr. Ken Ono, they are measuring and analyzing the forces they create as they swim, to optimize the way they move through the water. Details as seemingly small as Douglass’s head position in her underwater breaststroke pullout, or how her left hand enters the water on her backstroke, have been focal points as she has worked to trim the hundredths of a second that make the difference between medals in the sport.

While Douglass is almost certainly the only swimmer in Paris who has co-written a peer-reviewed research paper about this work, similar ideas are catching on elsewhere. As part of a government-funded research program in France aimed at giving the country an edge in its home Olympics, Léon Marchand was tested last summer to learn his “hydrodynamic profile.” And Kyle Chalmers, the Australian sprinter who is a three-time Olympian, has partnered with a Sydney-based sports technology lab that created a device to measure the force generated by a swimmer’s hands as they stroke through the water.

“It gives us a mental edge knowing that we have access to this information that you can’t see with the naked eye,” said the two-time Olympian Paige Madden, who recalls researchers using plastic wrap to affix a sensor to her back when she was an undergraduate at Virginia.

Dr. Ono’s methods have advanced over time. At a conference in Norway about a decade ago, he met a group of mathematicians from the Norwegian School of Sport Sciences who worked with Olympic cross-country skiers, using accelerometers to analyze their movement patterns. A lightbulb turned on for Dr. Ono, a number theory specialist who is also a triathlete and swim dad.

Dr. Ono, then at Emory University in Atlanta, had a willing test subject in Andrew Wilson, a math student who had walked on to the swim team. They started with accelerometers that were designed to track sharks and learned as they went, developing a protocol to home in on the weaknesses in Wilson’s breaststroke. As Wilson shot up in the sport, becoming a Division III national champion and later winning a medley relay gold at the Tokyo Olympics, other members of the U.S. national team began to learn about his and Dr. Ono’s project.

Dr. Ono has since tested about 100 top American swimmers, but he works most closely with the group at Virginia, where he is a regular presence on the pool deck. He also offers an independent study class for STEM students, who learn to analyze the data collected from swimmers like Douglass and Gretchen Walsh, the world-record holder in the 100 butterfly. Thomas Heilman, who, at 17, is the youngest American male swimmer to qualify for the Olympics since Michael Phelps, said one of the reasons he committed to swim at Virginia was to take part in this work regularly.

The 512 snapshots of data captured per second help researchers to create a digital twin for each swimmer, a numerical representation of how the athlete moves through the water. That data pointed to Douglass’s breaststroke pullout as an area where she was losing time. She looked at video to compare her form with that of Lilly King, a breaststroke specialist, and saw that the forward bend of her head was likely creating extra drag that was slowing her down. Mathematical modeling predicted that with a form adjustment, Douglass, who is now the American-record holder in the 200 breaststroke, could save as much as 0.15 seconds per pullout.

“Swimming is the perfect application of mathematics and physics,” Dr. Ono said. “We were never designed to swim in water. So to swim quickly in water is a really unique and complicated combination of athletic prowess and attention to detail in terms of physics and mechanics. That’s why I like it.”

The hydrodynamic profiling of the French national team is based on similar principles. Dr. Ricardo Peterson Silveira, a scientist from Brazil, came to a university in Rennes, France, three years ago to participate in France’s sports science project. He set up a demonstration in the warm-up pool at last year’s French championships that caught the eye of Bob Bowman, Marchand’s coach, who asked if his swimmer could be tested.

On the final day of the meet, Dr. Silveira tethered Marchand to a motorized device attached to the wall. The first test measured his drag while being pulled through the water in a streamlined position. Marchand registered the lowest value for this attribute, which they call passive drag, of any swimmer Dr. Silveira and the French researchers had tested, an indication that his body was built like a torpedo ready to shoot through the water.

A second test measured his speed while swimming against different resistance levels. Dr. Silveira then calculated how much power Marchand generated while swimming freestyle — the stroke he targeted for improvement in the Olympics — and the percentage that was used to propel him forward. Aquatic creatures like fish are very efficient at swimming, but even the best human swimmers are able to apply only about 60 percent of their effort in the direction of their swim. Based on Marchand’s results, Bowman said they incorporated short bursts of race-speed freestyle against heavy resistance into his training, aiming to boost both his power and his propelling efficiency.

To Bowman and Todd DeSorbo, the coach for Virginia and for the U.S. women for the Paris Games, more information allows them to better help their athletes, though the sport’s embrace of data has happened haltingly. Russell Mark, the former high-performance manager for U.S.A. Swimming, who introduced Dr. Ono to DeSorbo, recalled receiving mixed reactions from coaches 20 years ago when handing out paper printouts with race analytics.

“We are just at the cusp of this data revolution in swimming, and the athletes are hungry for it, the coaches are hungry for it, the sport is hungry for it,” Mark said. “You see it with the excitement over what Ken and Todd are doing and the success that they’re having on a program-wide scale. That’s the dream, and that’s the potential.”

The U.S. swimming team in Paris includes six athletes with ties to the University of Virginia. Madden no longer trains at Virginia, but this spring Dr. Ono and one of his interns visited her in Arizona, where they tried out the force paddles that the Australian sprinter Chalmers has used. The data they captured during their testing showed that Madden’s efficiency plunged when she reacted to a swimmer racing in the lane next to her. She felt as if she were going faster, but she was actually out of sync, like a car whose timing belt is off.

Her takeaway for the Olympic trials: Swim her own race. In the final of the 800 freestyle, the second individual event Madden qualified in, she sensed that Jillian Cox, who ended up finishing third, was creeping up on her. But she reminded herself to stay focused on her stroke form. “I was actually thinking about Dr. Ono during my race,” Madden said.

Given her understanding of statistics, Douglass is careful not to say that any one variable has been the reason for her rise in her sport. Later this week, she will compete in the 200 breaststroke and 200 individual medley. Her preparation for Paris has required rigorous attention to detail in her pool and dry land training, sleep, nutrition, race strategy and more. Using math to become a more efficient swimmer has enhanced that work.

As Douglass wrote in the research paper: “Force applied in any direction other than forward is not helping an athlete achieve their dream of Olympic gold.”

Sinead O’Connor, the Irish singer who shot to fame in the 1990s and was known for her activism, died at age 56 last July of chronic obstructive pulmonary disease and bronchial asthma, according to her death certificate.

In January, a coroner in London said that Ms. O’Connor had died of “natural causes” but did not provide details. The police said at the time of Ms. O’Connor’s death that it was “not being treated as suspicious.”

Ms. O’Connor’s death certificate, which was registered last week, filled in some gaps. The singer died of “exacerbation of chronic obstructive pulmonary disease and bronchial asthma together with low-grade lower respiratory-tract infection,” the report said. It was submitted by John Reynolds, Ms. O’Connor’s first husband.

Ms. O’Connor become a global star in the 1990s with a cover of Prince’s “Nothing Compares 2 U.” The album the song was on won a Grammy Award in 1991 for best alternative music performance.

She also wielded her fame as an activist, speaking out against sexual abuse in the Catholic Church, misogyny, the British subjugation of Ireland and other issues. In her later life, she spoke about her mental struggles and her recovery from child abuse.

Ms. O’Connor’s death shook Ireland, which mourned her as a national treasure even though she had been a controversial figure for her political provocations onstage and off. In 1992, Ms. O’Connor tore up a picture of Pope John Paul II during a “Saturday Night Live” performance to protest sexual abuse of children in the Roman Catholic Church.

In the year since she died, debates have continued over Ms. O’Connor’s legacy and representation.

In March, a risqué performance honoring her life and her first studio album opened in London and drew crowds in New York. And last week, a wax museum in Dublin removed a figure of her after her brother said it was “hideous” and “looked nothing like her.”

“She was something grander than a simple pop star,” Jon Caramanica, a pop music critic for The New York Times, wrote in an appraisal of Ms. O’Connor’s career.

“She became a stand-in for a sociopolitical discomfort that was beginning to take hold in the early 1990s,” he continued, “a rejection of the enthusiastic sheen and power-at-all-costs culture of the 1980s.”

The Olympic flame isn’t a flame.

Well, it’s sort of a flame. But it’s not made of fire. Even if it looks a lot like fire.

Wait. Let me backtrack. Every Olympic host city has a few basic tasks that force it to straddle the line between acknowledging the tradition of the Games while showing that it is keeping up with the times. Essentially, each new host needs to play the hits but still surprise and delight the listeners. That’s how you end up, for example, with an opening ceremony featuring a nearly nude man on a barge, covered in blue paint and glitter.

And that’s how you get an Olympic flame that’s not a flame at all. A flame that is actually “a cloud of mist and beams of light,” according to Paris 2024 organizers. That flame (or is it “flame”?) rests in an enormous cauldron, comprises 40 LED spotlights and 200 misting nozzles and is tethered to what looks like a gigantic hot-air balloon that will rise into the air every night of the Games.

I visited that “flame” on Sunday in the Tuileries Garden, in central Paris, where it exerted a certain planetary gravity on its surroundings. Tourists gathered and held their cameras over their heads. Cyclists hopped off to take photos. Police officers took turns snapping selfies with it.

It flickers like a fire, though as I walked alongside it, I felt a spray of cool mist on my legs, a reminder of the illusion at work. The entire structure — metallic, otherworldly and vaguely futuristic — creates an appealing contrast with the serene setting and the 19th-century sculptures that ring the garden.

Tony Estanguet, the president of Paris 2024, said in a statement the aim of this new flame was to capture the spirit of “daring, creativity, innovation — and sometimes madness! — of France.”

As many as 10,000 visitors a day can request free access to view the “flame” and its orb up close, though preregistration is required online and tickets for certain slots have been hard to come by. At sundown each day, the whole contraption ascends roughly 200 feet into the sky.

Rony Gabali and his son, Nelson, 10, felt compelled to swing by on Sunday after seeing it on television. Gabali thought it would be a “wonderful souvenir” for his son to experience the object up close.

“It’s beautiful,” Nelson said, smiling and trying out some English before adding in French, “It reminds me of a montgolfière.”

That’s the goal. The setup was conceived by the French designer Mathieu Lehanneur as a tribute to the Montgolfier brothers, Joseph-Michel and Jacques-Étienne, who in 1783 invented the first hot-air balloon that carried people. The technology for this version was provided by EDF, a government-owned electric utility company.

The modern homage was another example of how Paris is using the beauty of its city as a stage for the Games. And it may be here to stay.

The Paris mayor, Anne Hidalgo, said on French Bleu radio Monday morning that she hoped the cauldron would become a permanent legacy of the Olympic Games, along with the Olympic rings on the Eiffel Tower and the statues of women that emerged from the Seine during the opening ceremony.

She called the cauldron “extraordinary” and its location above the Tuileries Garden “magnificent.”

Yet the object, alluring as it is, raises another question: What happened to the flame, the actual fire, that had been lit and transported from Greece and ferried around France for weeks?

The press office for the Paris Games wrote in an email that the electric flame should be considered the “true Olympic flame.”

“For the Olympic movement, only the symbol of a flame that does not go out before the end of the Games matters,” it said, adding, “Given the specificity of our cauldron and the technologies involved, we will still keep a lit lantern in the immediate vicinity of the cauldron for the public to admire.”

Sure enough, in one corner of the garden, I saw something curious: a little glass box set atop a white stand, like a museum display. “Lit in Olympia, from the sun’s rays,” a sign affixed to it read. Inside was a flame — a tiny, real flame.

Catherine Porter contributed reporting.