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Category: mathematics

Mathematical model reveals why cracks sharpen during rapid rubber fracture

A research group from the University of Osaka, Zen University, and the University of Tokyo has mathematically uncovered the mechanism that causes crack tips to sharpen during the rapid fracture of rubber.

The bursting of balloons or tire blowouts is caused by rapid fracture, a phenomenon in which a small crack propagates instantaneously. During this process, the crack tip sharpens, accelerating the fracture. However, the reason behind this sharpening had long remained unexplained. Traditionally, it was believed to result from the material’s complex nonlinear effects.

The research group—comprising Hokuto Nagatakiya, a doctoral student; Shunsuke Kobayashi, assistant professor; and Ryuichi Tarumi, professor at the University of Osaka; along with Naoyuki Sakumichi, associate professor at Zen University and project associate professor at the University of Tokyo—has mathematically solved the problem of crack propagation. They derived equations that describe both the shape of the crack and the overall deformation of the material.

A mathematical ‘Rosetta Stone’ translates and predicts the larger effects of molecular systems

Penn Engineers have developed a mathematical “Rosetta Stone” that translates atomic and molecular movements into predictions of larger-scale effects, like proteins unfolding, crystals forming and ice melting, without the need for costly, time-consuming simulations or experiments. That could make it easier to design smarter medicines, semiconductors and more.

In a recent paper in Journal of the Mechanics and Physics of Solids, the Penn researchers used their framework, stochastic thermodynamics with internal variables (STIV), to solve a 40-year problem in phase-field modeling, a widely used tool for studying the shifting frontier between two states of matter, like the boundary between water and ice or where the folded and unfolded parts of a protein join.

“Phase-field modeling is about predicting what happens at the thin frontier between phases of matter, whether it’s proteins folding, crystals forming or ice melting,” says Prashant Purohit, Professor in Mechanical Engineering and Applied Mechanics (MEAM) and one of the paper’s co-authors. “STIV gives us the mathematical machinery to describe how that frontier evolves directly from first principles, without needing to fit data from experiments.”

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.

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