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Category: mathematics

Open-source framework lets drones dodge obstacles in milliseconds while minimizing travel time

In the aftermath of a devastating earthquake, unpiloted aerial vehicles (UAVs) could fly through a collapsed building to map the scene, giving rescuers information they need to quickly reach survivors. But this remains an extremely challenging problem for an autonomous robot, which would need to swiftly adjust its trajectory to avoid sudden obstacles while staying on course.

Researchers from MIT and the University of Pennsylvania developed a new trajectory-planning system that tackles both challenges at once. Their technique enables a UAV to react to obstacles in milliseconds while staying on a smooth flight path that minimizes travel time.

Their system uses a new mathematical formulation that ensures the robot travels safely to its destination along a feasible path, and that is less computationally intensive than other techniques. In this way, it generates smoother trajectories faster than state-of-the-art methods.

This Common Houseplant Is Secretly Using Advanced Geometry

Scientists have discovered that the Chinese money plant hides a remarkable geometric system inside its leaves, revealing that nature may solve complex problems using mathematical rules similar to those found in computer science and city planning.

People often see meaningful shapes and patterns in random things. Maybe you have looked at clouds and spotted a sailboat, a seahorse, or even your great-aunt Rosemary. Scientists call this tendency “apophenia,” the human habit of finding patterns that are not really there. But in some cases, nature truly does follow hidden mathematical rules. Cold Spring Harbor Laboratory Associate Professor Saket Navlakha studies these kinds of patterns and recently uncovered one inside a familiar houseplant.

Hidden geometry in chinese money plants.

Learning physics can derail some students: New research shows the best way to keep them on track

For many undergraduate students, exploring the complexities of physics for the first time, from wading through advanced mathematics, to absorbing information in a large lecture format, can be a daunting endeavor—one that dissuades many students from continuing their studies.

Educators have known for some time that students tend to learn these subjects better in hands-on, or “active learning,” environments—but some are more effective than others.

Roadmap charts three paths to room-temperature quantum materials for cooler computing

Imagine a laptop that never gets hot, a phone that holds its charge for days, or a computer memory chip designed to permanently retain data, even when the power goes out. This is the possibility sitting inside a remarkable family of materials that a team of researchers from the University of Ottawa and the Massachusetts Institute of Technology (MIT) has spent years trying to understand, and they just published a comprehensive roadmap of the field to date in the journal Newton.

Magnetic topological materials sit at the crossroads of magnetism and topology in modern physics. Topology is the mathematical study of shapes that cannot be continuously deformed into one another. In these materials, that idea protects the flow of electrons in a way that normal materials simply cannot.

“Magnetic topological materials offer a unique platform where magnetism and quantum physics work together in ways we are only beginning to fully understand,” explains Hang Chi, Canada Research Chair in Quantum Electronic Devices and Circuits and Assistant Professor at uOttawa’s Department of Physics. “This review brings together the field’s most significant advances and gives researchers a shared foundation to build on.”

Generalization Dynamics of LM Pre-training

An AI has a limited amount of “capacity” (brainpower). Early in training, it develops quick, shallow circuits to memorize data because that’s the easiest way to get the right answer. Later, it develops complex circuits for actual reasoning. Because space is limited, these two internal systems are constantly competing for control. Whichever type of data the AI happens to be reading in a specific moment determines which circuit wins the battle.

People typically assume that LMs stably mature from pattern-matching parrots to generalizable intelligence during pre-training. We build a toy eval suite and show this mental model is wrong: throughout pre-training, LMs frequently and suddenly hop between parrot-like and intelligence-like modes, i.e. distinct algorithms implemented by distinct circuits. We call this mode-hopping. Across our suite, LMs can suddenly latch onto memorized or in-context patterns instead of in-context learning, use System 1 instead of System 2 thinking, pick up what sounds true instead of what is true, fail at multi-hop persona QA, out-of-context reasoning, and emergent misalignment — then just as suddenly revert and generalize. Mode-hopping is not explained by standard optimization dynamics: it is locally stable and can not be fixed by checkpoint averaging. We instead think of it as a capacity allocation problem: in a capacity-bounded model, generalizable circuits must compete with the shallow ones learned early in training, and the data in each pre-training window decides which circuits win. Our suite provides a cheap set of pre-training monitors and a new lens on generalization. Building upon our insights, we demonstrate three applications: (i) select intermediate pre-training checkpoints that strongly generalize reasoning and alignment, better than the final pre-or mid-training checkpoints, (ii) select pre-training data that controls and stabilizes generalization dynamics, and (iii) test prior generalization predictors, falsifying the monolithic belief that “simpler solutions generalize better”

Building general AI without generalization is doable but meh. We want an intelligence that learns deep, transferable structure, not a parrot that matches shallow patterns. Real generalization would unblock many today’s key open problems: data-efficient (online) learning, shortcut learning, transfer capabilities from verifiable domains (math, coding) to broader non-verifiable yet economically valuable domains, and maintain a coherent character that truly aligns with human values.

The distinction between parrots and intelligence is computational. Parrots repeat in-context patterns; intelligence infers in-context functions. Parrots encode a persona as bags of disconnected facts and traits; intelligence learns a shared persona representation that connects all. Parrots memorize reasoning steps; intelligence forms general reasoning circuits for entity tracking, backtracking, or even for highly abstract concepts like truth.

Liad Mudrik: Using Prediction Maps to Guide Theory Testing and the Search for the NCC

This talk is part of the “New Ideas in NCC Research” workshop of the Bamberg Mathematical Consciousness Science Initiative (BAMΞ). For more talks and details, see https://www.uni-bamberg.de/en/bamxi/r…

Abstract: In recent years, the search for the neural correlates of consciousness (NCC) has been complemented, and influenced, by the ongoing efforts to test neuroscientific theories of consciousness. A key insight from these efforts, though, is that many theories remain underdeveloped and not fully specified, making it harder to establish stringent tests for their predictions. In this talk, I will present a novel methodological approach that represents scientific theories as networks of beliefs structured in a core-periphery manner. These Prediction Maps visualize theoretical claims and empirical predictions, and illustrate their inferential relations. This framework further facilitates systematic theory testing by allowing researchers to evaluate the evidential weight of different components of a theory, and to identify which experimental results would constitute the most informative tests. To do so, we apply graph-theoretic and network analysis metrics, quantifying the centrality of specific predictions. I argue that this approach can advance efforts to arbitrate between theories of consciousness and to identify their most promising candidate mechanisms as NCCs.

String theory is uniquely derived from basic assumptions about the universe, physicists show

If you could take an apple and break it into smaller and smaller parts, you would find molecules, then atoms, followed by subatomic particles like protons and the quarks and gluons that make them up. You might think you hit the bottom, but, according to string theorists, if you keep going to even smaller scales—about a billion billion times smaller than a proton—you will find more: tiny vibrating strings.

Developed in the 1960s, string theory proposes that everything in the universe is made from invisible strings. The theory arose as a possible solution to the problem of “quantum gravity,” the quest to align quantum mechanics, which describes our world at the smallest scales, with the general theory of relativity, which explains how our universe works on the largest scales (and includes gravity). Researchers have tried to reconcile the two theories—asking, for example, how gravity behaves in the quantum realm—but their equations go berserk, or in mathematical terms, go to infinity.

String theory is a mathematical solution that tames the unruly infinities. It purports that all particles, including the graviton—the hypothetical particle believed to convey the force of gravity—are generated by very small vibrating strings. The math behind string theory requires the strings to vibrate in at least 10 dimensions, rather than the four we live in (three for space and one for time), which is one of the reasons some scientists are not convinced that string theory is correct. But perhaps the biggest challenge for the theory is the ultrahigh energies required for testing it: Such an experiment would require a particle collider the size of a galaxy.

String Theory Emerges from “Almost Nothing”

Developed in the 1960s, string theory proposes that everything in the universe is made from invisible strings. The theory arose as a possible solution to the problem of “quantum gravity,” the quest to align quantum mechanics, which describes our world at the smallest scales, with the general theory of relativity, which explains how our universe works on the largest scales (and includes gravity). Researchers have tried to reconcile the two theories—asking, for example, how gravity behaves in the quantum realm—but their equations go berserk, or in mathematical terms, go to infinity.

String theory is a mathematical solution that tames the unruly infinities. It purports that all particles, including the graviton—the hypothetical particle believed to convey the force of gravity—are generated by very small vibrating strings. The math behind string theory requires the strings to vibrate in at least 10 dimensions, rather than the four we live in (three for space and one for time), which is one of the reasons some scientists are not convinced that string theory is correct. But perhaps the biggest challenge for the theory is the ultrahigh energies required for testing it: Such an experiment would require a particle collider the size of a galaxy.

What is a physicist to do? One way they can probe the theory is to turn to a “bootstrap” approach, in which researchers start with certain assumptions they believe to be true about the universe, and then see what laws emerge out of those assumptions. In a new paper titled “Strings from Almost Nothing,” accepted for publication in Physical Review Letters, Caltech researchers, and their colleagues at New York University and Institut de Fisica d’Altes Energies in Barcelona, have done just that. From a couple of basic assumptions about how particles should scatter off one another at very high energies, they derived the elements of string theory.

Amino Acid Patterns Help Scientists Distinguish Alien Life

Dr. Fabian Klenner: “We’re showing that life does not only produce molecules. Life also produces an organizational principle that we can see by applying statistics.” [ https://www.labroots.com/trending/space/30534/amino-acid-pat…ien-life-2](https://www.labroots.com/trending/space/30534/amino-acid-pat…ien-life-2)

What methods can scientists use to correctly identify biosignatures, aka signs of life beyond Earth? This is what a recent study Nature Astronomy hopes to address as a team of researchers from the University of California, Riverside (UC Riverside) and Israel investigated a new pattern-based method for identifying biosignatures. This study has the potential to help scientists develop new methods for finding life beyond Earth, which could narrow the scope for both how and where to find life.

For the study, the researchers used mathematics to suggest that instead of looking for specific molecules when searching for biosignatures scientists should instead look for organizational patterns. The primary motivation behind the study was to challenge longstanding methods regarding how to search for biosignatures, which have traditionally been focused on finding individual and specific molecules. In the end, the researchers found that the amino acids in biological (biotic) samples exhibited a much larger range of diversity compared to non-biological (abiotic) samples. These could shape a new generation of astrobiology, which is the study of searching for life beyond Earth.

“We’re showing that life does not only produce molecules,” said Dr. Fabian Klenner, who is an assistant professor of planetary sciences at UC Riverside and a co-author on the study. “Life also produces an organizational principle that we can see by applying statistics.”

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