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Category: mathematics – Page 3

Statistical mechanics is one of the pillars of modern physics. Ludwig Boltzmann (1844–1906) and Josiah Willard Gibbs (1839–1903) were its primary formulators. They both worked to establish a bridge between macroscopic physics, which is described by thermodynamics, and microscopic physics, which is based on the behavior of atoms and molecules.

The Austrian physicist Boltzmann explained the second law of thermodynamics in statistical terms. He defined the entropy of a system based on the number of possible microstates it could assume.

Unlike Boltzmann, who focused more on the physics of gases and the distribution of particles in equilibrium, the American Gibbs developed a general mathematical formalism that could be extended to more complex systems. Together, their contributions formed the basis of a physics capable of modeling a wide variety of phenomena.

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What If Math uses a relatively new concept to enhance the way math is taught so that kids are given more relevant skills for today’s digital world.

The company says that the way math — and algebra specifically — is taught today is based on a concept developed by Leonardo of Pisa in 1202 as a way to help traders. This, it says, is now redundant thanks to all the digital tools that use spreadsheets to do that part of mathematical working.

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Mathematics is like nothing else. The truths of math seem to be unrelated to anything else—independent of human beings, independent of the universe. The sum of 2 + 3 = 5 cannot not be true; this means that 3 + 2 = 5 would be true even if there were never any human beings, even if there were never a universe! When then, deeply, is mathematics?

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Mark Balaguer is Professor of Philosophy at California State University, Los Angeles. His major book is Platonism and Anti-Platonism in Mathematics.

Continue reading “Mark Balaguer — Is Mathematics Eternal?” | >

Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. The name comes from the Monte Carlo Casino in Monaco, where the primary developer of the method, mathematician Stanisław Ulam, was inspired by his uncle’s gambling habits.

Monte Carlo methods are mainly used in three distinct problem classes: optimization, numerical integration, and generating draws from a probability distribution. They can also be used to model phenomena with significant uncertainty in inputs, such as calculating the risk of a nuclear power plant failure. Monte Carlo methods are often implemented using computer simulations, and they can provide approximate solutions to problems that are otherwise intractable or too complex to analyze mathematically.

Monte Carlo methods are widely used in various fields of science, engineering, and mathematics, such as physics, chemistry, biology, statistics, artificial intelligence, finance, and cryptography. They have also been applied to social sciences, such as sociology, psychology, and political science. Monte Carlo methods have been recognized as one of the most important and influential ideas of the 20th century, and they have enabled many scientific and technological breakthroughs.

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What if black holes weren’t the only things slowly vanishing from existence? Scientists have now shown that all dense cosmic bodies—from neutron stars to white dwarfs—might eventually evaporate via Hawking-like radiation.

Even more shocking, the end of the universe could come far sooner than expected, “only” 1078 years from now, not the impossibly long 101100 years once predicted. In an ambitious blend of astrophysics, quantum theory, and math, this playful yet serious study also computes the eventual fates of the Moon—and even a human.

Black Holes Aren’t Alone

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Plasma—the electrically charged fourth state of matter—is at the heart of many important industrial processes, including those used to make computer chips and coat materials.

Simulating those plasmas can be challenging, however, because millions of math operations must be performed for thousands of points in the simulation, many times per second. Even with the world’s fastest supercomputers, scientists have struggled to create a kinetic simulation—which considers individual particles—that is detailed and fast enough to help them improve those manufacturing processes.

Now, a new method offers improved stability and efficiency for kinetic simulations of what’s known as inductively coupled plasmas. The method was implemented in a developed as part of a private-public partnership between the U.S. Department of Energy’s Princeton Plasma Physics Laboratory (PPPL) and chip equipment maker Applied Materials Inc., which is already using the tool. Researchers from the University of Alberta, PPPL and Los Alamos National Laboratory contributed to the project.

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A broad systematic review has revealed that quantum computing applications in health care remain more theoretical than practical, despite growing excitement in the field.

The comprehensive study published in npj Digital Medicine, which analyzed 4,915 research papers published between 2015 and 2024, found little evidence that quantum machine learning (QML) algorithms currently offer any meaningful advantage over classical computing methods for health care applications.

“Despite in research claiming quantum benefits for health care, our analysis shows no consistent evidence that quantum algorithms outperform classical methods for clinical decision-making or health service delivery,” said Dr. Riddhi Gupta from the School of Mathematics and Physics and the Queensland Digital Health Center (QDHeC) at the University of Queensland.

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Science students and academics wrote papers about the mathematics and physics of Ringworld after it was published. Larry Niven discusses whether this would happen if Ringworld was published today.
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Jakub Pachocki, OpenAI’s chief scientist since 2024, believes artificial intelligence models will soon be capable of producing original research and making measurable economic impacts. In a conversation with Nature, Pachocki outlined how he sees the field evolving — and how OpenAI plans to balance innovation with safety concerns.

Pachocki, who joined OpenAI in 2017 after a career in theoretical computer science and competitive programming, now leads the firm’s development of its most advanced AI systems. These systems are designed to tackle complex tasks across science, mathematics, and engineering, moving far beyond the chatbot functions that made ChatGPT a household name in 2022.

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