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Kakutani fixed-point theorem

In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e. a point which is mapped to a set containing it. The Kakutani fixed point theorem is a generalization of the Brouwer fixed point theorem. The Brouwer fixed point theorem is a fundamental result in topology which proves the existence of fixed points for continuous functions defined on compact, convex subsets of Euclidean spaces. Kakutani’s theorem extends this to set-valued functions.

The Age of Sustainable Abundance Is Here!

Advancements in AI, robotics, and space exploration are driving us towards a future of sustainable abundance, enabled by innovations such as space-based solar power, humanoid robots, and scalable AI infrastructure. ## ## Questions to inspire discussion.

Terafabs and AI Chips.

🛠️ Q: What are Elon Musk’s plans for terafabs?

A: Musk plans to build terafabs with 10 lines, each producing 100k wafers/month, costing **$10–20 billion/line.

🔋 Q: What challenges do AI chips face for scaling?

A: Scaling AI faces bottlenecks in AI chips and energy, with Musk’s terafabs and solar power as key solutions.

Continue reading “The Age of Sustainable Abundance Is Here!” | >

Antibody therapy foils pancreatic cancer’s sugar-based disguise to reawaken immune system

Pancreatic cancer is notoriously hard to treat and often resists the most advanced immunotherapies. Northwestern Medicine scientists have uncovered a novel explanation for that resistance: Pancreatic tumors use a sugar-based disguise to hide from the immune system. The scientists also created an antibody therapy that blocks the sugar-mediated “don’t-attack” signal.

For the first time, the team identified how this sugar trick works and showed that blocking it with a monoclonal antibody reawakens immune cells to attack cancer cells in preclinical mouse models.

“It took our team about six years to uncover this novel mechanism, develop the right antibodies and test them,” said study senior author Mohamed Abdel-Mohsen, associate professor of medicine in the division of infectious diseases at Northwestern University Feinberg School of Medicine.

How plastics grip metals at the atomic scale: Molecular insights pave way for better transportation materials

What makes some plastics stick to metal without any glue? Osaka Metropolitan University scientists have peered into the invisible adhesive zone that forms between certain plastics and metals—one atom at a time—to uncover how chemistry and molecular structure determine whether such bonds bend or break.

Their insights clarify metal–plastic bonding mechanisms and offer guidelines for designing durable, lightweight, and more sustainable hybrid materials for use in transportation.

Combining the strength of metal with the lightness and flexibility of plastic, polymer–metal hybrid structures are emerging as key elements for building lighter, more fuel-efficient vehicles. The technology relies on bonding metals with plastics directly, without adhesives. The success of these hybrids, however, hinges on how well the two materials stick together.

New AI framework can uncover space physics equations in raw data

Artificial intelligence (AI) systems, particularly artificial neural networks, have proved to be highly promising tools for uncovering patterns in large amounts of data that would otherwise be difficult to detect. Over the past decade, AI tools have been applied in a wide range of settings and fields.

Among its many possible applications, AI systems could be used to discover physical relationships and symbolic expressions (i.e., ) describing these relationships.

To uncover these formulas, physicists currently need to extensively analyze , thus automating this process could be highly advantageous.

Triggering cell death in metastatic melanoma may pave the way for new cancer treatments

Metastatic melanoma cells that have spread to lymph nodes survive by relying on a protein called ferroptosis suppressor protein 1 (FSP1)—a surprising metabolic dependency that could open the door to a new class of cancer treatments, according to a new study led by Harvard T.H. Chan School of Public Health.

The researchers say the study, published in Nature, not only highlights the therapeutic potential of drugs that inhibit FSP1, but also offers new ways to understand cancer and its vulnerabilities.

Ferroptosis is a form of cell death driven by excessive lipid oxidation in cell membranes. When this occurs, the cell’s structural integrity collapses, leading to death. Cancer cells rely heavily on antioxidant proteins like FSP1 to prevent ferroptosis.

PCSK9 inhibitor lowers risk of first heart attack and stroke in high-risk adults

Researchers from Mass General Brigham have unveiled the results of a large clinical trial that found that adding the drug evolocumab to patients’ treatment significantly reduced the risk of major adverse cardiovascular events in those who are at high risk. Results were presented today at the American Heart Association Scientific Sessions and simultaneously published in The New England Journal of Medicine.

“The results of this trial offer hope for preventing a , , or other cardiovascular event in patients who are at high risk,” said corresponding author Erin Bohula, MD, a cardiologist in the Mass General Brigham Heart and Vascular Institute. “Our findings reflect the promise of prevention strategies and reflect our ongoing commitment to conducting rigorous clinical trials to advance patient care with the goal of saving lives and improving quality of life.”

PCSK9 inhibitors, such as evolocumab, are designed to reduce LDL cholesterol, a major risk factor for cardiovascular events. Previous studies have found that PCSK9 inhibitors can prevent subsequent cardiovascular events in patients who have previously had a heart attack or stroke, but the current study—known as The Effect of EVolocumab in PatiEntS at High CArdiovascuLar RIsk WithoUt Prior Myocardial Infarction or Stroke (VESALIUS)-CV trial—is the first to study the drug’s preventive effects in people who have not previously had a heart attack or stroke.

Scientists uncover what delayed Earth’s oxygen boom for a billion years

Billions of years ago, cyanobacteria began releasing oxygen through photosynthesis, but the atmosphere stayed oxygen-poor for ages. Researchers uncovered that trace compounds like nickel and urea may have delayed Earth’s oxygenation for millions of years. Experiments mimicking early Earth revealed how their concentrations controlled cyanobacterial growth, dictating when oxygen began to accumulate. As nickel declined and urea stabilized, photosynthetic life thrived, sparking the Great Oxidation Event. The findings could also guide the search for biosignatures on distant worlds.

The arrival of oxygen in Earth’s atmosphere marked a defining moment in the planet’s history, transforming it into a world capable of supporting complex life. This major shift, known as the Great Oxidation Event (GOE), took place approximately 2.1 to 2.4 billion years ago. However, oxygenic photosynthesis — produced by cyanobacteria — had likely evolved hundreds of millions of years before this event. Despite this early ability to generate oxygen, atmospheric levels remained low for a surprisingly long time. Scientists have long debated the cause of this delay, considering explanations such as volcanic emissions, chemical sinks, and biological interactions. Yet no single factor has fully explained why it took so long for oxygen to build up in Earth’s air.

To tackle this enduring question, researchers focused on an often overlooked element of early Earth chemistry: the role of trace compounds such as nickel and urea in cyanobacterial growth.

DBS Rolls Out GenAI Chatbot, Southeast Asia’s Largest Bank Incorporates AI In Its Workflow

In today’s AI news, Southeast Asia’s largest bank is rolling out an AI chatbot for its corporate clients, giving “round-the-block” access for customer care needs. A pilot version of the generative AI-powered chatbot, named DBS Joy, was rolled out in February. It has since managed over 120,000 unique chats, DBS claimed in a statement. The virtual assistant also cut waiting times, and customer satisfaction scores rose by 23%. The bank’s virtual assistant was first rolled out in 2018, under the same name.

In other advancements, Artificial intelligence giant OpenAI on Monday announced a new initiative that aims to make it easier for service members and veterans to use AI tools when they’re transitioning from military service to the workforce. The ChatGPT-maker announced that service members within 12 months of separation or retirement from, or any veteran within their first year of leaving military service, can access a free year of access to ChatGPT Plus, the company’s subscription-based tool.

Meanwhile, Neurodiverse professionals may see unique benefits from artificial intelligence tools and agents, research suggests. With AI agent creation booming in 2025, people with conditions like ADHD, autism, dyslexia and more report a more level playing field in the workplace thanks to generative AI. A recent study from the UK’s Department for Business and Trade found that neurodiverse workers were 25% more satisfied with AI assistants and were more likely to recommend the tool than neurotypical respondents.

And, San Francisco became a meme — a symbol of American urban decay. Between 2019 and 2024, the city lost 4 per cent of its population, one of the largest declines in America. But thanks to AI — and new “tough on crime” mayor Daniel Lurie — the vibe has, unquestionably, shifted. Since 2020, more than 2,400 AI companies have been founded in San Francisco, a city of just 830,000 people. “Hacker houses”, where bright-eyed coders live communally and build what they hope will be the next Google, have cropped up across the city.

In videos, Six of the most influential minds in artificial intelligence joined FT Live for an exclusive conversation on how their breakthroughs and the current state of AI are shaping our world. Jensen Huang, Yoshua Bengio, Geoffrey Hinton, Fei-Fei Li, Yann LeCun, and Bill Dally spoke with the FT’s AI editor, Madhumita Murgia at the FT Future of AI Summit in London. Together, they reflected on decades of pioneering work — from neural networks to generative AI and discuss the ethical, social, and economic implications of the technology they helped to create.

Then, Ellis Hamburger and Alex Heath chat about the Sources launch party, and living in late-stage extractionism. Then, they are then joined by Runway CEO Cristobal Valenzuela to discuss the hardest part about AI, why golf courses are a bigger problem than data centers, and how world models are changing our world forever.

And, David Sacks, White House AI and Crypto Czar, joins Marc, Ben, and Erik on the a16z podcast to explore what’s really happening inside the Trump administration’s AI and crypto strategy. They expose the regulatory capture playbook being pushed by certain AI companies, explain why open source is America’s secret weapon, and detail the infrastructure crisis that could determine who wins the global AI race.

Continue reading “DBS Rolls Out GenAI Chatbot, Southeast Asia’s Largest Bank Incorporates AI In Its Workflow” | >
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