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Seventeen-year-old Hannah Cairo has overturned the long-standing Mizohata–Takeuchi conjecture, a 40-year-old cornerstone of harmonic analysis. The conjecture, introduced in the 1980s, proposed limits on how energy from certain wave functions could spread. While widely studied, it had resisted proof or disproof, with only partial cases examined.

Cairo broke through by constructing a counterexample: a function whose energy dispersed in a chaotic, fractal-like way that violated the conjecture. Her discovery also challenges the related Stein conjecture, another foundational idea in the field. Mathematicians described the result as “shocking” and unprecedented, with veteran researchers stunned after decades of failed attempts.

Born in the Bahamas and homeschooled, Cairo advanced quickly through Khan Academy and online tutoring, reaching calculus by age 11. She later joined math enrichment programs such as Math Circles of Chicago and UC Berkeley's Berkeley Math Circle, consistently working far above her age level.

In 2023, she studied Fourier restriction theory at UC Berkeley under Associate Professor Ruixiang Zhang, who first introduced her to the problem. Initially a homework assignment, her exploration convinced her the conjecture was false, culminating in her arXiv paper published in February 2025.

Her achievement has earned invitations to present at international conferences and admission to top PhD programs. She will begin doctoral studies at the University of Maryland this fall, marking her as one of the youngest mathematicians to make such a significant impact on the field.

Vietnam is expected to attract 25–30 million foreign visitors by 2026, narrowing the gap with Thailand, thanks to new visa relaxations. On Aug. 8, the government waived visa requirements for 12 more European countries, expanding its unilateral visa waiver list to 24 and total exemptions to 39. Analysts said this reform benefits not only tourism but also Vietnam's overall economy.

Alongside exemptions for scientists, investors, and billionaires, Vietnam is targeting luxury travelers who favor premium, sustainable experiences. The country already offers superyacht tours in Ha Long Bay, helicopter flights, Michelin-starred dining, world-class golf courses, and five-star resorts.

The changes could increase arrivals by 20–25%, benefiting airlines, hospitality, food, and handicrafts. European travelers will especially gain from easier entry, opening opportunities for tailored packages like culinary tours, resorts, or heritage itineraries.

Experts caution, however, that visa reforms alone are not enough. To match Thailand, Malaysia, and Singapore, Vietnam must improve infrastructure, service quality, and global marketing. Competitors already maintain dozens of overseas offices, while Vietnam lags behind. Industry leaders emphasized the need to encourage repeat visits from high-spending tourists to maximize long-term gains.

Pantai Merah (Pink Beach) on Indonesia's Komodo Island has been named the world's most beautiful beach by Forbes, citing a survey from British adventure travel company Explore Worldwide. Its pink hue comes from crushed coral and microscopic Foraminifera, creating rosé-colored sands alongside vibrant reef fish and coral gardens.

The beach, located between Sumbawa and Flores, is renowned for its clear waters and rich marine life, making it a prime snorkeling spot. The best visiting period is May to October for sunny weather, while the rainy season falls in the remaining months. Le Morne in Mauritius ranked second, and Boulders Beach in South Africa placed third.

A Chinese influencer known as “Little Azheng” (“Tail Brother”) said a relative's young son caused over 400,000 yuan (US$56,000) in damage at his home after being refused a Labubu toy decorated with expensive jewelry. The boy threw a tantrum, hurled a remote control, and smashed a mirror-glass ceiling worth 100,000 yuan and an Italian crystal chandelier worth 300,000 yuan.

The influencer said repairs required replacing all ceiling panels. The boy showed no remorse, and his parents—who could only offer 20,000 yuan—asked him not to publicize the event. He declined to sue, believing legal action would be pointless. He criticized the parents for poor boundaries and emotional manipulation. The story drew heated discussion online, with many blaming bad parenting.

Labubu, a popular “ugly-cute” toy by Chinese brand Pop Mart, is sold mainly in blind boxes for $20–30, but rare editions can fetch thousands, with one selling for over 1 million yuan at auction. Pop Mart, founded in 2010, is now valued at $40 billion.

Apple cider vinegar (ACV) may provide modest metabolic benefits, such as lowering cholesterol and fasting glucose, which could indirectly help with non-alcoholic fatty liver disease (NAFLD). NAFLD, often linked to obesity, type 2 diabetes, and metabolic syndrome, can progress from simple fat buildup to severe liver damage. ACV, made by fermenting apple juice into acetic acid, has shown promising effects in animal studies.

Research on rainbow trout and rodents found that low ACV doses (~2%) reduced liver fat, triglycerides, and harmful enzyme markers, while higher doses (~4%) worsened the condition. Rodent trials also showed reduced liver fat, improved lipid profiles, and lower glucose. These benefits are believed to come from acetic acid's role in boosting fat oxidation and reducing metabolic stress.

However, current evidence is largely from animal studies, and human research is limited. Experts stress that ACV should be seen as a supplement rather than a cure, with lifestyle changes like healthy eating, exercise, and medical care being the most effective way to manage fatty liver.