Lifeboat Foundation

New theoretical work establishes an analogy between systems that are dynamically frustrated, such as glasses, and thermodynamic systems whose members have conflicting goals, such as predator–prey ecosystems.

A system is geometrically frustrated when its members cannot find a configuration that simultaneously minimizes all their interaction energies, as is the case for a two-dimensional antiferromagnet on a triangular lattice. A nonreciprocal system is one whose members have conflicting, asymmetric goals, as exemplified by an ecosystem of predators and prey. New work by Ryo Hanai of Kyoto University, Japan, has identified a powerful mathematical analogy between those two types of dynamical systems [1]. Nonreciprocity alters collective behavior, yet its technological potential is largely untapped. The new link to geometrical frustration will open new prospects for applications.

To appreciate Hanai’s feat, consider how different geometric frustration and nonreciprocity appear at first. Frustration defies the approach that physics students are taught in their introductory classes, based on looking at the world through Hamiltonian dynamics. In this approach, energy is to be minimized and states of matter characterized by their degree of order. Some of the most notable accomplishments in statistical physics have entailed describing changes between states—that is, phase transitions. Glasses challenge that framework. These are systems whose interactions are so spatially frustrated that they cannot find an equilibrium spatial order. But they can find an order that’s “frozen” in time. Even at a nonzero temperature, everything is stuck—and not just in one state. Many different configurations coexist whose energies are nearly the same.

Simple rules in simple settings continue to puzzle mathematicians, even as they devise intricate tools to analyze them.

Human coexistence depends on cooperation. Individuals have different motivations and reasons to collaborate, resulting in social dilemmas, such as the well-known prisoner’s dilemma. Scientists from the Chatterjee group at the Institute of Science and Technology Austria (ISTA) now present a new mathematical principle that helps to understand the cooperation of individuals with different characteristics. The results, published in PNAS, can be applied to economics or behavioral studies.

A group of neighbors shares a driveway. Following a heavy snowstorm, the entire driveway is covered in snow, requiring clearance for daily activities. The neighbors have to collaborate. If they all put on their down jackets, grab their snow shovels, and start digging, the road will be free in a very short amount of time. If only one or a few of them take the initiative, the task becomes more time-consuming and labor-intensive. Assuming nobody does it, the driveway will stay covered in snow. How can the neighbors overcome this dilemma and cooperate in their shared interests?

Scientists in the Chatterjee group at the Institute of Science and Technology Austria (ISTA) deal with cooperative questions like that on a regular basis. They use game theory to lay the mathematical foundation for decision-making in such social dilemmas.

In a recent development at Fudan University, a team of applied mathematicians and AI scientists has unveiled a cutting-edge machine learning framework designed to revolutionize the understanding and prediction of Hamiltonian systems. The paper is published in the journal Physical Review Research.

Named the Hamiltonian Neural Koopman Operator (HNKO), this innovative framework integrates principles of mathematical physics to reconstruct and predict Hamiltonian systems of extremely-high dimension using noisy or partially-observed data.

The HNKO framework, equipped with a unitary Koopman structure, has the remarkable ability to discover new conservation laws solely from observational data. This capability addresses a significant challenge in accurately predicting dynamics in the presence of noise perturbations, marking a major breakthrough in the field of Hamiltonian mechanics.

A mathematical historian at Trinity Wester University in Canada, has found use of a decimal point by a Venetian merchant 150 years before its first known use by German mathematician Christopher Clavius. In his paper published in the journal Historia Mathematica, Glen Van Brummelen describes how he found the evidence of decimal use in a volume called “Tabulae,” and its significance to the history of mathematics.

The invention of the decimal point led to the development of the decimal system, and that in turn made it easier for people working in multiple fields to calculate non-whole numbers (fractions) as easily as whole numbers. Prior to this new discovery, the earliest known use of the decimal point was by Christopher Clavius as he was creating astronomical tables—the resulting work was published in 1593.

The new discovery was made in a part of a manuscript written by Giovanni Bianchini in the 1440s—Van Brummelen was discussing a section of trigonometric tables with a colleague when he noticed some of the numbers included a dot in the middle. One example was 10.4, which Bianchini then multiplied by 8 in the same way as is done with modern mathematics. The finding shows that a decimal point to represent non-whole numbers occurred approximately 150 years earlier than previously thought by math historians.

Networks can represent changing systems, like the spread of an epidemic or the growth of groups in a population of people. But the structure of these networks can change, too, as links appear or vanish over time. To better understand these changes, researchers often study a series of static “snapshots” that capture the structure of the network during a short duration.

Network theorists have sought ways to combine these snapshots. In a new paper in Physical Review Letters, a trio of SFI-affiliated researchers describe a novel way to aggregate static snapshots into smaller clusters of networks while still preserving the dynamic nature of the system. Their method, inspired by an idea from quantum mechanics, involves testing successive pairs of network snapshots to find those for which a combination would result in the smallest effect on the dynamics of the system—and then combining them.

Importantly, it can determine how to simplify the history of the network’s structure as much as possible while maintaining accuracy. The math behind the method is fairly simple, says lead author Andrea Allen, now a data scientist at Children’s Hospital of Philadelphia.

Scientists have found that the growth patterns of trees in a forest differ significantly from the way branches expand on an individual tree.

Nature is full of surprising repetitions. In trees, the large branches often look like entire trees, while smaller branches and twigs look like the larger branches they grow from. If seen in isolation, each part of the tree could be mistaken for a miniature version of itself.

It has long been assumed that this property, called fractality, also applies to entire forests but researchers from the University of Bristol have found that this is not the case.

New mathematical models of our Milky Way Galaxy are helping a team of Argentine, Chilean and Spanish astrophysicists trace the origins of our galaxy back through time.

Logical reasoning is still a major challenge for language models. DeepMind has found a way to support reasoning tasks.

A study by Google’s AI division DeepMind shows that the order of the premises in a task has a significant impact on the logical reasoning performance of language models.

They work best when the premises are presented in the same order as they appear in the logical conclusions. According to the researchers, this is also true for mathematical problems. The researchers make the systematically generated tests available in the R-GSM benchmark for further investigation.