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How Bees, Beer Cans and Data Solve the Same Packing Problem

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How Bees, Beer Cans and Data Solve the Same Packing Problem

Animation of the same plastic spheres disappearing one at a time.

A holy grail in pure mathematics is sphere packing in higher dimensions. Almost nothing has been rigorously proven about it, except in dimensions 1, 2 and 3.

That’s why it was such a breakthrough when, in 2016, a young Ukrainian mathematician named Maryna Viazovska solved the sphere-packing problem in eight dimensions, and later, with collaborators, in 24 dimensions.

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Fourth measles case confirmed in L.A. County; person visited LAX, restaurants while infectious

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Fourth measles case confirmed in L.A. County; person visited LAX, restaurants while infectious

A fourth measles case has been confirmed in Los Angeles County, prompting renewed calls from health officials for residents to ensure they are protected against the highly contagious virus.

The infected individual flew from Singapore to Los Angeles International Airport on Feb. 9 aboard Singapore Airlines Flight 38. The plane landed at about 7 p.m. following a 14-hour journey, according to the L.A. County Department of Public Health.

Over the following days, the individual visited a few San Gabriel Valley fast food restaurants, potentially exposing others to the measles virus.

“As measles cases increase, it is important that residents take steps to make sure they are fully protected,” L.A. County Health Officer Dr. Muntu Davis said in a statement. “The [measles-mumps-rubella] vaccine is the safest and most reliable way to prevent measles and protect yourself, your family, and your community.”

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The health department did not respond to questions from The Times regarding the sex or age of the infected individual, who was described in a statement as “a resident who recently traveled internationally.”

After arriving at and leaving the Tom Bradley International Terminal, the individual visited restaurants and convenience stores throughout Whittier and Montebello.

The individual ate at a Burger King in Montebello, 1212 West Beverly Blvd., on Feb. 10 between 5:30 and 7 p.m. The next day, the person dined at Taqueria El Atacor, 11156 1/2 Whittier Blvd. in Whittier, between 3 and 5 p.m.

The final two stops in Montebello were at Domino’s Pizza, 803 West Whittier Blvd., between 4 p.m. and 5:15 p.m. on Feb. 12; and 7-Eleven, 1106 West Beverly Blvd., on Feb. 13 between 4:30 p.m. and 5:45 p.m.

Those who were in the Bradley Terminal (Terminal B) on Feb. 9 from 7:30 p.m. to 9:40 p.m., or in the eateries above during the aforementioned time windows, may have been exposed to the measles virus.

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Symptoms typically develop anywhere from seven to 21 days after exposure, according to the health department.

Concerned individuals should confirm whether they have protection against the virus, either through past measles vaccinations or infections.

Those who are not immunized or are unsure of their status should monitor themselves closely for signs of infection. Common symptoms include fever, cough, runny nose or red eyes, as well as a rash.

Those with such symptoms are encouraged to stay home and avoid school, work and any gatherings. They should also call a healthcare provider immediately, but not go into a healthcare facility without informing them.

The monitoring deadline for symptoms ranges from March 2 at LAX until March 6 at 7-Eleven for individuals who visited those spaces around the same time as the infected person.

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Davis said the most effective way to protect against measles is to take the MMR vaccine. Children age 1 year and older are considered fully immunized after receiving two doses.

“Measles spreads easily and can lead to serious complications, including pneumonia, brain swelling, and even death,” Davis said. “We urge everyone to confirm their immunity and get the MMR vaccine if needed, especially before traveling. Taking this simple step helps safeguard your health and strengthens protection for our entire community.”

Previous cases confirmed in L.A. County so far this year also involved individuals who had traveled internationally. One of those individuals ventured to Sherman Oaks on Jan. 24, another to Woodland Hills on Jan. 30. The first case had no identified public exposure locations in L.A. County.

Measles cases have increased in the United States as vaccination rates have fallen in recent years, allowing the highly contagious virus to spread in communities with lower vaccine coverage.

According to the U.S. Centers for Disease Control and Prevention, there were 2,280 confirmed measles cases in the U.S. last year, the highest annual total since 1991. As of Feb. 12, 910 cases had already been confirmed nationwide this year — including 15 in California.

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Shark attacks rose in 2025. Here’s why Californians should take note

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Shark attacks rose in 2025. Here’s why Californians should take note

Shark attacks returned to near-average levels in 2025 after a dip the previous year, according to the latest report from the Florida Museum of Natural History’s International Shark Attack File, published Wednesday.

Researchers recorded 65 unprovoked shark bites worldwide last year, slightly below the 10-year average of 72, but an increase from 2024. Nine of those bites were fatal, higher than the 10-year average of six fatalities.

The United States once again had the highest number of reported incidents, accounting for 38% of global unprovoked bites when assessed on a country by country basis. That said, it’s actually a decline from recent years, including 2024, when more than half of all reported bites worldwide occurred off the U.S. coast.

In 2025, Florida led all states with 11 recorded attacks. California, Hawaii, Texas and North Carolina accounted for the remaining U.S. incidents.

But California stood out in another way: It had the nation’s only unprovoked fatal shark attack in 2025.

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A 55-year-old triathlete was attacked by a white shark after entering the water off the coast of Monterey Bay with members of the open-ocean swimming club she co-founded. It was the sole U.S. fatality among 25 reported shark bites nationwide.

It’s not surprising that the sole U.S. shark-related death occurred in California, said Steve Midway, an associate professor of fisheries at Louisiana State University. “In California, you tend to have year-to-year fewer attacks than other places in the U.S. and in the world,” Midway said. “But you tend to have more serious attacks, a higher proportion of fatal attacks.”

The difference lies in species and geography, Midway said. Along the East Coast, particularly in Florida, many bites involve smaller coastal sharks in shallow water, which are more likely to result in nonfatal injuries. California’s deeper and colder waters are home to larger species, such as the great white shark.

“Great whites just happen to be larger,” Midway said. “You’re less likely to be attacked, but if you are, the outcomes tend to be worse.”

Whether measured over 10, 20 or 30 years, average annual shark bite totals globally are actually very stable.

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“The global patterns change only slightly from one year to the other,” said Gavin Naylor, director of the Florida Program for Shark Research.

Those annual fluctuations are influenced by a combination of shark biology, ocean conditions and the number of people in the water at any given time in any given place, researchers say.

At the same time, global shark populations remain far below historical levels. Naylor categorizes about 30% of shark species as endangered, largely due to overfishing. In some countries, including the United States and Australia, stronger protections have allowed certain shark populations to recover.

Nevertheless, the risk of being bitten by a shark remains extremely low. The report notes that drowning is a far more common cause of death worldwide — and, if it helps you sleep (or swim), the data show that you are much more likely to be killed by lightning than you are by a shark.

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What a Speech Reveals About Trump’s Plans for Nuclear Weapons

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What a Speech Reveals About Trump’s Plans for Nuclear Weapons

Within hours of the expiration last week of the final arms control treaty between Moscow and Washington, the State Department sent its top arms diplomat, Thomas G. DiNanno, to Geneva to lay out Washington’s vision for the future. His public address envisioned a future filled with waves of nuclear arms buildups and test detonations.

The views of President Trump’s administration articulated in Mr. DiNanno’s speech represent a stark break with decades of federal policy. In particular, deep in the speech, he describes a U.S. rationale for going its own way on the global ban on nuclear test detonations, which had been meant to curb arms races that in the Cold War had raised the risk of miscalculation, and war.

This annotation of the text of his remarks aims to offer background information on some of the specialized language of nuclear policymaking that Mr. DiNanno used to make his points, while highlighting places where outside experts may disagree with his and the administration’s claims.

What remains unknown is the extent to which Mr. DiNanno’s presentation represents a fixed policy of unrestrained U.S. arms buildups, or more of an open threat meant to spur negotiations toward new global accords on ways to better manage the nuclear age.

Read the original speech.

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New York Times Analysis

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1

Established in 1979 as Cold War arsenals grew worldwide, the Conference on Disarmament is a United Nations arms reduction forum made up of 65 member states. It has helped the world negotiate and adopt major arms agreements.

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2

In his State Department role, working under Secretary of State Marco Rubio, Mr. DiNanno is Washington’s top diplomat for the negotiation and verification of international arms accords. Past holders of that office include John Bolton during the first term of the George W. Bush administration and Rose Gottemoeller during Barack Obama’s two terms.

3

This appears to be referring to China, which has 600 nuclear weapons today. By 2030, U.S. intelligence estimates say it will have more than 1,000.

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4

Here he means Russia, which is conducting tests to put a nuclear weapon into space as well as to develop an underwater drone meant to cross oceans.

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New York Times Analysis

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5

In this year’s federal budget, the Trump administration is to spend roughly $90 billion on nuclear arms, including basic upgrades of the nation’s arsenal and the replacement of aging missiles, bombers and submarines that can deliver warheads halfway around the globe.

6

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A chief concern of many American policymakers is that Washington will soon face not just a single peer adversary, as in the Cold War, but two superpower rivals, China and Russia.

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New York Times Analysis

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7

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The 1987 Intermediate-Range Nuclear Forces Treaty or I.N.F. banned all weapons capable of traveling between 500 and 5,500 kilometers, or 310 and 3,420 miles, whether armed with nuclear or conventional warheads. The Trump administration is now deploying a number of conventionally armed weapons in that range, including a cruise missile and a hypersonic weapon.

8

The destructive force of the relatively small Russian arms can be just fractions of the Hiroshima bomb’s power, perhaps making their use more likely. The lesser warheads are known as tactical or nonstrategic nuclear arms, and President Vladimir V. Putin of Russia has repeatedly threatened to use them in Ukraine.

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9

Negotiators of arms control treaties have mostly focused on long-range weapons because the delivery vehicles and their deadly warheads are considered planet shakers that could end civilization.

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New York Times Analysis

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10

This underwater Russian craft is meant to cross an ocean, detonate a thermonuclear warhead and raise a radioactive tsunami powerful enough to shatter a coastal city.

11

The nuclear power source of this Russian weapon can in theory keep the cruise missile airborne far longer than other nuclear-armed missiles.

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12

Russia has conducted test launches for placing a nuclear weapon into orbit, which the Biden administration quietly warned Congress about two years ago.

13

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The term refers to the five permanent members of the United Nations Security Council—China, France, Russia, the United Kingdom and the United States.

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New York Times Analysis

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14

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A top concern of American officials is that Beijing and Moscow might form an alliance to coordinate their nuclear forces. Their joint program to develop fuel for atom bombs is seen as an indication of this emerging threat.

15

This Trump administration plan is dated November but was made public in December.

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16

Released last year, this Chinese government document sought to portray Beijing as a leader in reducing the global threat of nuclear weapons.

17

Typically, arms control treaties have not required countries to destroy warheads so their keepers put them into storage for possible reuse. The United States retains something on the order of 20,000 small atom bombs meant to ignite the larger blasts of hydrogen bombs.

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18

An imminent surge centers on the nation’s Ohio-class submarines. The Trump administration has called for the reopening of submarine missile tubes that were closed to comply with the New START limits. That will add as many 56 long-range missiles to the fleet. Because each missile can hold multiple arms, the additional force adds up to hundreds more warheads.

19

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This refers to weapons meant for use on a battlefield or within a particular geographic region rather than for aiming at distant targets. It is often seen as synonymous with intermediate-range weapons.

20

Here, the talk turns to the explosive testing of nuclear weapons for safety, reliability and devising new types of arms. The United States last conducted such a test in 1992 and afterwards adopted a policy of using such nonexplosive means like supercomputer simulations to evaluate its arsenal. In 1996, the world’s nuclear powers signed a global ban on explosive testing. A number of nations, including the United States and China, never ratified the treaty, and it never officially went into force.

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21

In new detail, the talk addresses what Mr. Trump meant last fall when he declared that he had instructed the Pentagon “to start testing our Nuclear Weapons on an equal basis” in response to the technical advances of unnamed foreign states.

22

Outside experts say the central issue is not whether China and Russia are cheating on the global test ban treaty but whether they are adhering to the U.S. definition. From the treaty’s start in 1996, Washington interpreted “zero” explosive force as the compliance standard but the treaty itself gives no definition for what constitutes a nuclear explosion. Over decades, that ambiguity led to technical disputes that helped block the treaty’s ratification.

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23

By definition, all nuclear explosions are supercritical, which means they split atoms in chain reactions that become self-sustaining in sufficient amounts of nuclear fuel. The reports Mr. DiNanno refers to told of intelligence data suggesting that Russia was conducting a lesser class of supercritical tests that were too small to be detected easily. Russian scientists have openly discussed such small experiments, which are seen as useful for assessing weapon safety but not for developing new types of weapons.

24

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This sounds alarming but experts note that the text provides no evidence and goes on to speak of preparations, not detonations, except in one specific case.

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New York Times Analysis

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25

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The talk gave no clear indication of how the claims about Russian and Chinese nuclear testing might influence U.S. arms policy. But it repeated Mr. Trump’s call for testing “on an equal basis,” suggesting the United States might be headed in that direction, too.

26

The talk, however, ended on an upbeat but ambiguous note, giving no indication of what Mr. DiNanno meant by “responsible.” Even so, the remark came in the context of bilateral and multilateral actions to reduce the number of nuclear arms in the world, suggesting that perhaps the administration’s aim is to build up political leverage and spur new negotiations with Russia, China or both on testing restraints.

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