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Category: mathematics

Algorithm precisely quantifies flow of information in complex networks

Networks are systems comprised of two or more connected devices, biological organisms or other components, which typically share information with each other. Understanding how information moves between these connected components, also known as nodes, could help to advance research focusing on numerous topics, ranging from artificial intelligence (AI) to neuroscience.

To measure the directional flow of information in systems, scientists typically rely on a mathematical construct known as transfer entropy, which essentially quantifies the rate at which information is transmitted from one node to another. Yet most strategies for calculating transfer entropy developed so far rely on approximations, which significantly limits their accuracy and reliability.

Researchers at AMOLF, a institute in the Netherlands, recently developed a computational algorithm that can precisely quantify transfer entropy in a wide range of complex networks. Their algorithm, introduced in a paper published in Physical Review Letters, opens new exciting possibilities for the study of information transfer in both biological and engineered networks.

These Tiny Robots Can Swarm, Adapt, and Heal Themselves

Scientists designed microrobots that use sound to swarm, adapt, and heal themselves — working together like a living organism. The discovery could transform medicine, environmental cleanup, and robotics.

Nature’s Blueprint for Robot Swarms

Animals such as bats, whales, and insects have long relied on sound to communicate and find their way. Drawing inspiration from this, an international group of scientists has developed a model for tiny robots that use sound waves to move and work together in large, coordinated swarms that behave almost intelligently. According to team leader Igor Aronson, Huck Chair Professor of Biomedical Engineering, Chemistry, and Mathematics at Penn State, these robotic collectives could eventually take on challenging missions like exploring disaster areas, cleaning polluted environments, or performing medical procedures inside the human body.

Generation of harmful slow electrons in water is a race between intermolecular energy decay and proton transfer

When high-energy radiation interacts with water in living organisms, it generates particles and slow-moving electrons that can subsequently damage critical molecules like DNA. Now, Professor Petr Slavíček and his bachelor’s student Jakub Dubský from UCT Prague (University of Chemistry and Technology, Prague) have described in detail one of the key mechanisms for the creation of these slow electrons in water, a process known as Intermolecular Coulombic Decay (ICD). Their powerful mathematical model successfully explains all the data from complex laser experiments conducted at ETH Zurich (Hans-Jakob Woerner team).

The work, which deepens the fundamental understanding of radiation chemistry, has been published in the journal Nature Communications.

A detailed knowledge of the processes in , combined with advances in research technologies using high-energy radiation, is transforming the field of radiation chemistry. In the future, these insights could lead to significant changes in various fields, including medicine, particularly in developing more sensitive and controllable applications for devices based on ionizing radiation.

How do you trust a robot you’ve never met?

Many of the environments where human-facing universal robots can provide benefits — homes, hospitals, schools — are sensitive and personal. A tutoring robot helping your kids with math should have a track record of safe and productive sessions. An elder-care assistant needs a verifiable history of respectful, competent service. A delivery robot approaching your front door should be as predictable and trustworthy as your favorite mail carrier. Without trust, adoption will never take place, or quickly stall.

Trust is built gradually and also reflects common understanding. We design our systems to be explainable: multiple AI modules talk to each other in plain language, and we log their thinking so humans can audit decisions. If a robot makes a mistake — drops the tomato instead of placing it on the counter — you should be able to ask why and get an answer you can understand.

Over time, as more robots connect and share skills, trust will depend on the network too. We learn from peers, and machines will learn from us and from other machines. That’s powerful but just like parents are concerned about what their kids learn on the web, we need good ways to audit and align skill exchange for robots… Governance for human–machine societies isn’t optional; it’s fundamental infrastructure.

Mathematical model could help boost drug efficacy by getting dosing in rhythm with circadian clocks

Researchers at the University of Michigan have developed a mathematical model that reveals how our circadian rhythms can have dramatic impacts on how our bodies interact with medicines.

This could help doctors prescribe medicines to have the best intended effect by syncing the dosing up with the natural clocks of their patients.

“These findings provide a mechanistic basis for chronotherapeutics—optimizing drug efficacy by considering circadian timing,” said the new study’s author Tianyong Yao, an undergraduate researcher in the U-M Department of Mathematics. “This could improve treatment for conditions such as ADHD, depression and fatigue.”

Drip by drip: Research provides first complete mathematical description of stalagmite shapes

Deep inside caves, water dripping from the ceiling creates one of nature’s most iconic formations: stalagmites. These pillars of calcite, ranging from centimeters to many meters in height, rise from the cave floor as drip after drip of mineral-rich water deposits a tiny layer of stone.

When mathematics meets aesthetics: Tessellations as a precise tool for solving complex problems

In a recent study, mathematicians from Freie Universität Berlin have demonstrated that planar tiling, or tessellation, is much more than a way to create a pretty pattern. Consisting of a surface covered by one or more geometric shapes with no gaps and no overlaps, tessellations can also be used as a precise tool for solving complex mathematical problems.

This is one of the key findings of the study, “Beauty in/of Mathematics: Tessellations and Their Formulas,” authored by Heinrich Begehr and Dajiang Wang and recently published in the scientific journal Applicable Analysis. The study combines results from the field of complex analysis, the theory of partial differential equations, and geometric function theory.

A central focus of the study is the “parqueting-reflection principle.” This refers to the use of repeated reflections of geometric shapes across their edges to tile a plane, resulting in highly symmetrical patterns. Aesthetic examples of planar tessellations can be seen in the work of M.C. Escher. Beyond its visual appeal, the principle has applications in mathematical analysis—for example, as a basis for solving classic boundary value problems such as the Dirichlet problem or the Neumann problem.

Michael Freedman | The Poincaré Conjecture and Mathematical Discovery

Millennium Prize Problems Lecture 9/17/2025
Speaker: Michael Freedman, Harvard CMSA and Logical Intelligence.

Title: the poincaré conjecture and mathematical discovery.

Abstract: The AI age requires us to re-examine what mathematics is about. The Seven Millenium Problems provide an ideal lens for doing so. Five of the seven are core mathematical questions, two are meta-mathematical – asking about the scope of mathematics. The Poincare conjecture represents one of the core subjects, manifold topology. I’ll explain what it is about, its broader context, and why people cared so much about finding a solution, which ultimately arrived through the work of R. Hamilton and G. Perelman. Although stated in manifold topology, the proof requires vast developments in the theory of parabolic partial differential equations, some of which I will sketch. Like most powerful techniques, the methods survive their original objectives and are now deployed widely in both three-and four-dimensional manifold topology.

Computer advances and ‘invisibility cloak’ vie for physics Nobel

A math theory powering computer image compression, an “invisibility cloak” or the science behind the James Webb Space Telescope are some achievements that could be honored when the Nobel physics prize is awarded Tuesday.

The award, to be announced at 11:45 am (0945 GMT) in Stockholm, is the second Nobel of the season, after the Medicine Prize was awarded on Monday to a US-Japanese trio for research into the human immune system.

Mary Brunkow and Fred Ramsdell, of the United States, and Japan’s Shimon Sakaguchi were recognized by the Nobel jury for identifying immunological “security guards”

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