Sunisa Lee, the all-around gold medalist in women’s gymnastics at the Tokyo Olympics, woke up one morning last year and was startled by her reflection in the mirror.

Her face looked as if it had been inflated with an air pump. Her leg joints were so swollen that she could hardly bend her knees or ankles. A scale revealed she had gained more than 10 pounds.

Her mind raced: Had she been eating too much? Was it the pollen in the air? Maybe she was allergic to her roommate’s new dog?

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Protests broke out Monday in Caracas, Venezuela’s capital, with hundreds of young people marching through the streets furious over a presidential election in which the incumbent, Nicolás Maduro, declared victory despite widespread accusations of fraud, officially proclaiming the election decided without releasing the full vote counts.

The United States and countries around the world denounced the official results of Sunday’s vote, which did not appear to match statistical estimates based on partial counts and other data that showed the president losing by a wide margin.

By Monday afternoon, the Venezuelan government announced it had kicked out the diplomatic missions of seven Latin American countries that had condemned the official electoral results.

The opposition leader, María Corina Machado, announced on Monday evening that her movement had received paper tallies from 73 percent of the country’s voting stations and refuted the government’s claims. Those tallies showed that Mr. Maduro’s opponent, Edmundo González, had received 3.5 million more votes than the president.

Mr. González called the margin “mathematically irreversible.”

The move by the electoral authority to declare victory but not release detailed voting results, which it had routinely done in past elections, intensified the sense among many Venezuelans and international observers that the election had essentially been stolen.

But Mr. Maduro appeared to dig in, with his government announcing that it was investigating top opposition leaders, accusing them of hacking the electoral computer systems.

Sporadic demonstrations in Caracas on Monday morning grew throughout the day as angry residents headed toward the center of the capital, reaching traditional government strongholds that had not seen political unrest for more than two decades. Large groups of young men walked more than five miles down main roads, tearing down Mr. Maduro’s campaign posters and chanting, “They robbed us!”

Another group of hundreds of people tried to make it to the presidential palace, lighting tires on fire along the way. Pro-government paramilitaries responded by opening fire in the air, and the police used tear gas to disperse the protests.

Protesters in Cumaná, 250 miles east of the capital, tried to reach the country’s election headquarters, but they were pushed back by the National Guard.

The disputed election put renewed attention on the Biden administration’s incentives to Venezuela. United States officials’ negotiations with the authoritarian government and easing of sanctions on the country’s vital oil industry had helped pave the way for Sunday’s voting. For now, the administration said it was not considering revoking any licenses to sell oil.

But the Biden administration also demanded that the Maduro government release vote tabulations, and warned that it risks diplomatic isolation as more countries — including some crucial allies — questioned the lack of transparency of an election that appeared to violate international norms.

The Brazilian government, led by President Luiz Inácio Lula da Silva, distanced itself from Mr. Maduro on Monday, despite years of friendly relations between the two leftist leaders.

In a cautiously worded statement, Mr. Lula’s government praised “the peaceful nature” of the election, but then called for “the impartial verification of results” and the release of detailed results from polling stations.

Colombia, led by Gustavo Petro, a leftist who in the first months of his presidency made drawing closer to Venezuela a priority, also called for detailed tallies to be released and for international observers who monitored the vote to provide their assessment.

“It’s important to clear up any doubts about the results,” Colombia’s foreign minister, Luis Gilberto Murillo, wrote on X.

Brazil’s and Colombia’s responses were noteworthy because they showed that two of Venezuela’s biggest neighbors were seeking answers before they would recognize Mr. Maduro’s claim of re-election.

On Monday night, Brazil, Colombia and Mexico were negotiating a joint statement to call on Venezuela to release voting records from each polling station, hoping that a unified stance from three of the region’s most influential nations would help put pressure on Mr. Maduro, according to two Brazilian diplomatic officials who spoke on the condition of anonymity to describe private talks.

Celso Amorim, Mr. Lula’s chief foreign policy adviser, also met separately with Mr. Maduro and Mr. González in Caracas on Monday. In an hourlong meeting at Venezuela’s presidential palace, Mr. Maduro promised to deliver the full voting results, though he did not provide a clear timeline for doing so, according to one of the Brazilian officials.

The official said Brazil viewed the promise as a small step but was remaining realistic. Mr. Amorim told reporters that he could not trust vote tallies presented by both the Maduro and opposition camps because neither had provided proof.

President Biden and Mr. Lula planned to speak about Venezuela on Tuesday, according to the Brazilian official.

The Venezuelan electoral authority, run by a member of the ruling party, announced early Monday that partial results of Sunday’s election showed that Mr. Maduro had received 51.2 percent of the vote and was the clear winner.

Mr. Maduro, 61, who has been in power since 2013, had faced off against Mr. González, 74, a former diplomat, who the electoral authority claimed had received 44.2 percent of the vote.

Mr. González was essentially a stand-in for Ms. Machado, a popular opposition leader who had been disqualified from running.

Ms. Machado called the official results “impossible.”

“Everybody knows what happened,” she said.

Much of the dispute around Sunday’s election focuses on the transparency of the vote count.

The electoral authority has yet to publish any results on its website, breaking with tradition. Opposition poll witnesses at many voting stations were also prevented by electoral officials and soldiers from receiving a paper tally of results, in breach of the regulations and precedent.

The paper tallies record the votes cast at each voting machine. Without them, it is difficult for the opposition to add up individual tallies to cross-reference — and dispute — the national results.

The opposition also said there were irregularities in the way that results were digitally transmitted from the voting stations to the electoral system.

Ms. Machado said that opposition volunteers scanned and posted the paper tallies online so everyone could see the evidence. By Monday evening, she said they had received 73 percent of the tallies, showing Mr. González had won in a landslide.

The United States has tried to push Mr. Maduro from power for years, and the Trump administration responded to a flawed 2018 presidential vote in which Mr. Maduro claimed victory by imposing a series of tough economic sanctions on Venezuela’s oil industry.

The Biden administration had lifted some of those penalties in exchange for a commitment from the Maduro government to work toward competitive elections.

Mr. Kirby would not discuss whether the United States would respond with further sanctions.

“We’re watching,” he said. “The world’s watching. I won’t get ahead of a decision that hasn’t been made here in terms of consequences.”

The Biden administration faces a difficult choice. Sanctions could deepen Venezuela’s economic woes and increase migration toward the United States ahead of the presidential election in November. But not taking a tough stance could strengthen Mr. Maduro and allow Republicans to attack the U.S. president as weak on autocrats.

Mr. Maduro said the United States should not meddle in other nations’ affairs.

Venezuela’s justice minister, Tarek William Saab, said Monday that the government was looking into acts of vandalism against government installations, and said three opposition leaders, including Ms. Machado, were under investigation for a hack of Venezuela’s electoral system.

Mr. Maduro said the opposition was prepared to use a tired tactic: crying fraud even before the election had taken place.

“I’ve seen this movie a few times,” Mr. Maduro said.

Mr. Maduro did receive support from allied leaders in Cuba, Serbia, Nicaragua, Russia, Bolivia and Honduras, who applauded the results.

Daniel Ortega, who as president of Nicaragua has overseen the end of democracy in his own country, congratulated Mr. Maduro. And Cuba’s leader, Miguel Díaz-Canel Bermúdez, said Mr. Maduro had “defeated the pro-imperialist opposition.”

President Vladimir V. Putin of Russia said he was eager to strengthen ties between the two countries. “Russian-Venezuelan relations have the character of a strategic partnership,” Mr. Putin said in a message to Mr. Maduro, the Kremlin said in a statement.

Iran and China also congratulated Mr. Maduro.

But across Latin America, leaders of Uruguay, Peru, Panama, Costa Rica, Ecuador, Argentina and Guatemala all denounced the results.

“The Maduro regime must understand that the results they publish are difficult to believe,” Chile’s leftist leader, Gabriel Boric, said on X.

By Monday afternoon, Venezuela’s foreign minister, Yván Gil, announced that Venezuela had ousted all diplomats from Argentina, Chile, Costa Rica, Peru, Panama, the Dominican Republic and Uruguay.

Genevieve Glatsky and Julie Turkewitz contributed reporting from Bogotá, Colombia; Anatoly Kurmanaev and Adriana Loureiro Fernandez contributed from Caracas; and Hamed Aleaziz contributed from Washington.

The African National Congress announced on Monday that it had expelled from the party the country’s former president, Jacob Zuma, casting him as a traitor who formed a rival political organization and peddled dangerous rhetoric in an effort to take down the party he once led.

“His platform is dangerous, appeals to extremist instincts in our body politic and riles up a political base that may foment social unrest,” said Fikile Mbalula, the secretary general of the African National Congress, the country’s governing party, during a news conference on Monday.

Mr. Zuma was a celebrated anti-apartheid activist with the African National Congress, or A.N.C., but has been at odds with the party since being forced to resign as president in 2018 amid a series of corruption scandals.

He brought his populist brand of grievance politics to this year’s national election, with his new party, uMkhonto weSizwe, known as M.K., running on a platform of seizing white-owned land to empower the Black majority, and overhauling South Africa’s constitution.

His party’s surprisingly strong performance helped to prevent the A.N.C. from winning an absolute majority for the first time since the end of apartheid 30 years ago. Yet Mr. Zuma still claimed, without providing evidence, that the A.N.C. had manipulated the results.

Mr. Zuma’s new M.K. party is now the leading opposition party, having won 58 out of 400 seats in Parliament. The A.N.C. was forced to partner with other rival parties to form a coalition to govern the country, in what it is calling a government of national unity.

Officials with Mr. Zuma’s party have said that it would never work with the A.N.C. as long as it continues to be led by his successor and nemesis, President Cyril Ramaphosa.

Mr. Zuma joined the A.N.C. as a teenager, when it was a liberation movement resisting the white apartheid government. 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.

The AN.C.’s national disciplinary committee found Mr. Zuma guilty of “prejudicing the integrity or repute of the organization by acting in collaboration with” an opposing political party, Mr. Mbalula said.

He added that Mr. Zuma “has been running on a dangerous platform that casts doubt on our entire constitutional edifice.”

Mr. Zuma will have 21 days to appeal his expulsion. The M.K. party released a statement saying that the former president was denied due process by being sentenced after a virtual hearing in which he was not present. The party called the hearing a “kangaroo court.”

Mr. Mbalula called those criticisms hypocritical, noting that M.K. had dismissed some of its members, including the man who officially founded the party, without holding hearings.

“Who are they to lecture people about a due process?” he said.

Critics say the expulsion should have come long ago. Mr. Zuma became the face of endemic corruption within the A.N.C. after a nine-year tenure riddled with accusations that he sold the interests of the state to benefit himself and his close allies.

The widespread corruption, while many South Africans continue to struggle economically, cost the A.N.C. significant popular support. And many critics said that the party’s failure to deal with Mr. Zuma was a sign that it was not serious about cleaning house.

The A.N.C.’s electoral support plummeted to 40 percent in the election in May, about an 18-percentage point drop from the previous election five years ago. Mr. Mbalula conceded that the rise of Mr. Zuma’s party, which finished third in the election, was a major reason for the A.N.C.’s slide.

The A.N.C. suspended Mr. Zuma in January, about a month after he announced his support for M.K. The A.N.C. had scheduled a disciplinary hearing for early May, less than a month before the election, to determine whether to kick Mr. Zuma out of the party. But the A.N.C. postponed that hearing, citing concerns that protests by M.K. members could lead to unrest.

But some analysts believed that politics also played a factor in the delay.

Mr. Zuma, 82, maintains a fervent following within the A.N.C., especially in his home province, KwaZulu-Natal, home of the Zulu ethnic group. His support comes from those who believe that the government of Mr. Ramaphosa needs to take more aggressive steps to lift Black South Africans out of poverty. Expelling Mr. Zuma right before the election could have deepened his supporters’ belief that he is a political martyr.

Mr. Zuma was jailed three years ago for contempt of court for refusing to testify before a public inquiry into corruption during his time as president — a move that he and his supporters argue was orchestrated by his opponents within the A.N.C. When the nation’s top court ruled that Mr. Zuma could not serve in Parliament because of that jail sentence, it only deepened the grievance among his supporters that he was being unfairly treated.

Even though Mr. Zuma could not hold office, he appeared on campaign posters and his picture was on the ballot for M.K. He also led political rallies that attracted thousands of voters.

As leader of the new opposition, Mr. Zuma has accused Mr. Ramaphosa of corruption and wrongdoing, which the president has denied. 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.

Ongama Mtimka, a political analyst at Nelson Mandela University, in Gqeberha, said of Mr. Ramaphosa, “He got to be perceived as somebody who takes too long to take action against rabble rousers.”

Mr. Zuma’s expulsion could help bolster Mr. Ramaphosa’s ability to appear decisive and in control, he added.

Amelia Nierenberg 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 1980s and ’90s 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 promise was they would swim in the Seine.

So far this year, no Olympian has been able to do so.

After the heavy rains on the weekend, swimming officials at the Paris Games have canceled both familiarization swims for triathletes eager to try out the river before the men’s and women’s individual events set for this week. Each time, poor water quality levels were blamed.

Better news arrived on Monday: Paris 2024 organizers say the hot sun over the past two days most likely means the water will be clean enough on Tuesday morning for competitors to dive in for the first 1.5 kilometers of the men’s race before jumping on their bicycles and then running 10 kilometers.

“We are confident we will be able to hold the competition tomorrow,” Etienne Thobois, the chief executive of the Paris 2024 organizing committee, said at a news conference on Monday. “We have done everything we can to achieve the swimmability of the Seine.”

The city of Paris and its mayor, Anne Hidalgo, in particular, have a lot riding on that promise, and the sight of athletes splashing down the river with the gold dome above Napoleon’s tomb glittering nearby and the Eiffel Tower rising ahead.

The authorities have spent 1.4 billion euros (more than $1.53 billion) on an ambitious, multipronged and labor-intensive plan to clean the Seine — or, more precisely, to prevent filthy water from flowing into it.

They have dug new sewage pipes to homes, connected the city’s refurbished peniches — houseboats — to the sewage system and added special treatments to two upstream sewage plants.

In May, they opened a giant underground water storage tank on the city’s left bank that can hold 13.2 million gallons of rainwater: enough to fill 20 Olympic pools. It is one of five big engineering projects built to hold rainwater during storms — water that was previously dumped into the Seine — so it would not overwhelm the city’s antiquated sewer system.

The result has been a big drop in two indicator bacteria — E. coli and intestinal enterococci — that can cause illness in humans, often meeting the threshold set by the European bathing directive. But that effort was not just to ensure triathletes and open-water swimmers could compete in the river this summer.

The colossal cleanup job was a major part of the Olympics’ intended legacy for the city. The promise is that, if all goes well, next summer, locals will be able to swim in the river — something that just years ago was considered a pipe dream.

Daily tests taken during the first three weeks of July revealed six out of seven samples met those cleanliness standards at the Olympic sites for both triathlon and open-water swimming. (Samples taken at other areas had weaker results.)

As a show of confidence in those results, Ms. Hidalgo plunged into the river before an army of reporters this month.

The problem is rain. When it falls heavily, as it did in Paris during the opening ceremony on Friday and the first official day of Olympic competitions on Saturday, water runs off the city’s concrete roads and sidewalks into the river, and some upstream sewage systems are incapable of dealing with the increased volume.

And for the next day or two after the rain showers, test results in the Seine show a spike in bacteria levels. But as long as the rain subsides, those bacteria are devoured by other organisms in the river, Jean-Marie Mouchel, a hydrologist and professor at Sorbonne University, explained in an interview this spring.

“It’s a weather question,” said Mr. Mouchel, who has studied the Seine for three decades. “If the weather is nice, if the flow of the Seine is calm, if there are no storms, the water quality is good.”

Recent test results posted weekly by the region are so promising that even critics like Surfrider, an environmental conservancy organization, are singing Paris’s praises.

“The city has done a lot of things,” said Marc Valmassoni, Surfrider’s water and health coordinator. “We hope many other cities like Bordeaux, Toulouse, Nantes, will have the same idea and over the next few years make their urban rivers swimmable.”

The decision about whether triathletes dive in the Seine near the Alexandre III bridge on Tuesday will be made around 4 a.m. Paris time, when World Triathlon delegates meet with the local authorities and Paris 2024 officials to examine the most recent results.

“Athletes will find out when they wake up: They will know if they should stay in bed or get up and go for breakfast,” said Benjamin Maze, the high-performance manager for the French Triathlon Federation. But to them, that is routine.

“This is a sport based on adaptation,” Mr. Maze said. “They are used to responding to weather conditions. They are focused on what they can control — their fitness and preparation for the event.”

They also have been well briefed about the currents, which are high, he said. And many, including the American Morgan Pearson and France’s Dorian Coninx, both medal contenders, got to test out the Seine in a race last summer.

Their feedback at that time would make Ms. Hidalgo happy.

“They were amazed by the course,” Mr. Maze said. “They had goose bumps.”

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.”

Venezuelans angered by the outcome took the streets of the capital, Caracas, and elsewhere on Monday afternoon. 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.

On Monday, the Brazilian government distanced itself from Mr. Maduro, calling for more transparency in releasing voting data.

Mr. Lula also 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.

Olha Kharlan went to her knees, as if in disbelief, after a stirring comeback in the women’s saber fencing competition. She kissed the metallic dueling surface. Finally, she jumped into her coach’s arms and then bowed theatrically to the crowd.

She had just won a bronze medal by the thinnest margin, 15-14, on Monday night beneath the vaulted glass dome of the Grand Palais. It was her fifth career Olympic medal and the first of any color for Ukraine at the Paris Games, an emotional moment of celebration and defiance for a nation at war.

“It’s really special, incredible, like infinity special,” Kharlan told reporters, speaking in English, saying her medal was won for her country and its defenders and for the Ukrainian athletes “who couldn’t come here because they were killed by Russia.”

Given the circumstances, it might have been the most meaningful medal of her career. Kharlan’s mere presence confirmed that this niche sport, perhaps more than any other, illustrates the acrimony and caustic feuding that have resulted from Russia’s invasion of Ukraine.

Kharlan, 33, was disqualified from the World Fencing Championships last summer for refusing to shake hands with her Russian opponent. But Thomas Bach, the president of the International Olympic Committee and himself a 1976 Olympic fencing champion, gave Kharlan an exemption to participate in the Paris Games, citing her “unique situation.”

There she was on Monday, competing in the Olympics, while Russia was absent from the biggest international event in fencing, a sport in which it has long been a power athletically and administratively.

With Russia banned from these Games because of its invasion, only 15 of its athletes are competing in Paris, all designated as neutral, without the accompaniment of the country’s flag or national anthem. There are none in fencing, a huge blow to the country’s Olympic prestige given that Russia and the former Soviet Union rank behind only Italy, France and Hungary in fencing’s overall medal count.

Alisher Usmanov, an Uzbek-born Russian magnate, stepped aside days after the war began in February 2022 as president of the International Fencing Federation. This followed economic sanctions levied against him by the European Union, which described Mr. Usmanov as having “particularly close ties” to President Vladimir V. Putin of Russia, and said he “actively supported” Russia’s policies regarding the “destabilization of Ukraine.”

In a statement Monday, Mr. Usmanov said he had only suspended his fencing duties and that he was seeking to lift the E.U. sanctions, which he called “unfair and illegal.”

Stanislav Pozdnyakov, the president of the Russian Olympic Committee and himself a four-time Olympic fencing champion, is barred from the Paris Games, as are other Russian sports officials. Attempts to reach him Monday via phone and text messages went unanswered.

Missing, too, from these Games is his daughter, Sofia Pozdnyakova, a two-time Olympic fencing champion who was unable to qualify as a neutral athlete because she represents the Russian armed forces.

“They have to know the consequences” of the invasion, Kharlan said in an interview.

There was a brief, convivial thaw on Monday. Apparently not wanting to risk another disqualification, or perhaps just greeting a friend in an act of sportsmanship after her decisive victory in the round of 16, Kharlan hugged her vanquished opponent, Anna Bashta, a Russian-born fencer now representing Azerbaijan. Bashta said she and Kharlan had known each other for years, and that she hoped the Ukrainian would win a gold medal later in the day.

But such Ukrainian-Russian relationships are mostly fractured now. At the 1992 Barcelona Olympics, Mr. Pozdnyakov and Vadym Gutzeit, now president of the Ukraine Olympic Committee, won a team gold medal in fencing as part of a collection of former Soviet republics called the Unified Team. But Mr. Gutzeit now refers to Mr. Pozdnyakov, his onetime friend and teammate, as “my enemy.” Last year, Mr. Gutzeit told The Associated Press that, now and forever, “this person does not exist for me.”

The invasion has severed more than friendships. What amounted to a royal fencing marriage also collapsed after only two years, mostly because of the war. Pozdnyakova’s former husband, Konstantin Lokhanov, also a Russian Olympic fencer, moved to the United States in 2022 and denounced the invasion. “I decided I could no longer live in a country that kills innocent Ukrainians,” Lokhanov said in an interview last summer.

Two other prominent Russian fencers also moved to the United States and criticized the war, which resulted in the firing of a top Russian coach and an apparent swipe by Mr. Pozdnyakov at what he considered Western frivolousness. His own daughter’s patriotic upbringing, he said in a Russian sports television interview, spared her “the sad fate of frightened lovers of raspberry frappé and yellow scooters.”

Kharlan, the Ukrainian star, said she had no regrets about refusing to shake hands at the world championships last summer in Milan after defeating her Russian opponent, Anna Smirnova, who protested by sitting in a chair for about 45 minutes in the competition area, known as a strip.

Her refusal to shake hands, Kharlan said, was a message to the world that, given what has happened in Ukraine, “nobody can just close their eyes to that.”

A more volatile confrontation occurred last month at the European fencing championships in Switzerland, when Olena Kryvytska of Ukraine refused to shake hands after defeating a Russian-born fencer, Maia Guchmazova, who was competing for Georgia.

After the Ukrainian walked away, an irate Guchmazova cursed and said, “Why are they allowed to get away with everything?”

Mr. Gutzeit, the president of Ukraine’s Olympic committee, said a day after that incident, in an interview in Kyiv, that Kryvytska’s action was exactly how Ukraine’s athletes should respond at the Olympics toward any Russians who compete in Paris: Don’t speak to them. Don’t shake hands. Don’t pose for a photograph except on the medal podium. Don’t even look at them. They don’t exist.

“While the war is going on, they shouldn’t have a place in international sports,” Mr. Gutzeit said.

Along with tens of thousands of civilian deaths in the war, roughly 500 top-level Ukrainian athletes and coaches have died in the fighting. At last count, 518 stadiums and sports training facilities have been damaged or destroyed. Hundreds of Ukrainian Olympic hopefuls trained outside the country, as did Kharlan, who lives in Italy.

These are Kharlan’s fifth Olympics and, in her words, “the hardest.” She said that she had seen her parents only three times since the war began in February 2022. Her mother and sister have come to Paris to support her, but her father cannot because of a law that prevents most Ukrainian men under 60 from leaving the country.

Her hometown, Mykolaiv, in southern Ukraine, has been besieged by Russian strikes, a water crisis and power shortages. Sometimes, she said, she is afraid to look at her phone because there is a “high chance” it will contain bad news.

“Each of us has been damaged by the war,” she said.

Bach, the I.O.C. president, congratulated Kharlan on Monday night. Another chance to win a medal will come Saturday in the women’s team saber competition. Monday’s bronze, she said, was a “message to all Ukrainians, to all the world, that Ukraine never gives up.”

Valerie Hopkins contributed reporting from Paris.

Huw Edwards, who was until recently one of the BBC’s highest-profile TV anchors, has been charged with three counts of making indecent images of children, London’s Metropolitan Police said on Monday.

Mr. Edwards, 62, presented the BBC’s flagship 10 o’clock evening newscast and led the broadcaster’s coverage of major national events, including the death and funeral of Queen Elizabeth II, and the coronation of King Charles III last year.

He left the broadcaster in April 2024.

On Monday a statement from the Metropolitan Police said that Mr. Edwards would appear at Westminster magistrates’ court on Wednesday charged with sharing illegal images on WhatsApp.

Mr. Edwards, of Southwark, London, “has been charged with three counts of making indecent images of children following a Met Police investigation,” the statement said.

“The offenses, which are alleged to have taken place between December 2020 and April 2022, relate to images shared on a WhatsApp chat,” it said, adding that Mr. Edwards had been arrested on Nov. 8, 2023.

It added: “He was charged on Wednesday, 26 June following authorization from the Crown Prosecution Service.”

Once criminal cases become active, the media in Britain are restricted in what they can report about the background to the case under laws designed to prevent juries from being influenced by material outside of the court hearing. On Monday the police issued a warning about those curbs.

“Media and the public are strongly reminded that this is an active case. Nothing should be published, including on social media, which could prejudice future court proceedings.”

Mr. Edwards was born in Bridgend in Wales in 1961 and worked for the BBC for four decades, ending his career there as one of its best paid performers with a salary of between £475,000 and £479,999 a year, or more than $600,000, according to the broadcaster’s latest annual report.