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Category: information science – Page 2

Most people’s experiences with polynomial equations don’t extend much further than high school algebra and the quadratic formula. Still, these numeric puzzles remain a foundational component of everything from calculating planetary orbits to computer programming. Although solving lower order polynomials—where the x in an equation is raised up to the fourth power—is often a simple task, things get complicated once you start seeing powers of five or greater. For centuries, mathematicians accepted this as simply an inherent challenge to their work, but not Norman Wildberger. According to his new approach detailed in The American Mathematical Monthly, there’s a much more elegant approach to high order polynomials—all you need to do is get rid of pesky notions like irrational numbers.

Babylonians first conceived of two-degree polynomials around 1800 BCE, but it took until the 16th century for mathematicians to evolve the concept to incorporate three-and four-degree variables using root numbers, also known as radicals. Polynomials remained there for another two centuries, with larger examples stumping experts until in 1832. That year, French mathematician Évariste Galois finally illustrated why this was such a problem—the underlying mathematical symmetry in the established methods for lower-order polynomials simply became too complicated for degree five or higher. For Galois, this meant there just wasn’t a general formula available for them.

Mathematicians have since developed approximate solutions, but they require integrating concepts like irrational numbers into the classical formula.

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For decades, neuroscientists have developed mathematical frameworks to explain how brain activity drives behavior in predictable, repetitive scenarios, such as while playing a game. These algorithms have not only described brain cell activity with remarkable precision but also helped develop artificial intelligence with superhuman achievements in specific tasks, such as playing Atari or Go.

Yet these frameworks fall short of capturing the essence of human and animal behavior: our extraordinary ability to generalize, infer and adapt. Our study, published in Nature late last year, provides insights into how in mice enable this more complex, intelligent behavior.

Unlike machines, humans and animals can flexibly navigate new challenges. Every day, we solve new problems by generalizing from our knowledge or drawing from our experiences. We cook new recipes, meet new people, take a new path—and we can imagine the aftermath of entirely novel choices.

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A UNSW Sydney mathematician has discovered a new method to tackle algebra’s oldest challenge—solving higher polynomial equations.

Polynomials are equations involving a variable raised to powers, such as the degree two polynomial: 1 + 4x – 3x2 = 0.

The equations are fundamental to math as well as science, where they have broad applications, like helping describe the movement of planets or writing computer programs.

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The quantum black hole with (almost) no equations by Professor Gerard ‘t Hooft.

How to reconcile Einstein’s theory of General Relativity with Quantum Mechanics is a notorious problem. Special relativity, on the other hand, was united completely with quantum mechanics when the Standard Model, including Higgs mechanism, was formulated as a relativistic quantum field theory.

Since Stephen Hawking shed new light on quantum mechanical effects in black holes, it was hoped that black holes may be used to obtain a more complete picture of Nature’s laws in that domain, but he arrived at claims that are difficult to use in this respect. Was he right? What happens with information sent into a black hole?

The discussion is not over; in this lecture it is shown that a mild conical singularity at the black hole horizon may be inevitable, while it doubles the temperature of quantum radiation emitted by a black hole, we illustrate the situation with only few equations.

About the Higgs Lecture.

The Faculty of Natural, Mathematical & Engineering Sciences is delighted to present the Annual Higgs Lecture. The inaugural Annual Higgs Lecture was delivered in December 2012 by its name bearer, Professor Peter Higgs, who returned to King’s after graduating in 1950 with a first-class honours degree in Physics, and who famously predicted the Higgs Boson particle.

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A quantum computer can solve optimization problems faster than classical supercomputers, a process known as “quantum advantage” and demonstrated by a USC researcher in a paper recently published in Physical Review Letters.

The study shows how , a specialized form of quantum computing, outperforms the best current classical algorithms when searching for near-optimal solutions to complex problems.

“The way quantum annealing works is by finding low-energy states in , which correspond to optimal or near-optimal solutions to the problems being solved,” said Daniel Lidar, corresponding author of the study and professor of electrical and computer engineering, chemistry, and physics and astronomy at the USC Viterbi School of Engineering and the USC Dornsife College of Letters, Arts and Sciences.

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In recent years, computer scientists have created various highly performing machine learning tools to generate texts, images, videos, songs and other content. Most of these computational models are designed to create content based on text-based instructions provided by users.

Researchers at the Hong Kong University of Science and Technology recently introduced AudioX, a model that can generate high quality audio and music tracks using texts, video footage, images, music and audio recordings as inputs. Their model, introduced in a paper published on the arXiv preprint server, relies on a diffusion transformer, an advanced machine learning algorithm that leverages the so-called transformer architecture to generate content by progressively de-noising the input data it receives.

“Our research stems from a fundamental question in artificial intelligence: how can intelligent systems achieve unified cross-modal understanding and generation?” Wei Xue, the corresponding author of the paper, told Tech Xplore. “Human creation is a seamlessly integrated process, where information from different sensory channels is naturally fused by the brain. Traditional systems have often relied on specialized models, failing to capture and fuse these intrinsic connections between modalities.”

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In a network, pairs of individual elements, or nodes, connect to each other; those connections can represent a sprawling system with myriad individual links. A hypergraph goes deeper: It gives researchers a way to model complex, dynamical systems where interactions among three or more individuals—or even among groups of individuals—may play an important part.

Instead of edges that connect pairs of nodes, it is based on hyperedges that connect groups of nodes. Hypergraphs can represent higher-order interactions that represent collective behaviors like swarming in fish, birds, or bees, or processes in the brain.

Scientists usually use a hypergraph to predict dynamic behaviors. But the opposite problem is interesting, too. What if researchers can observe the dynamics but don’t have access to a reliable model? Yuanzhao Zhang, an SFI Complexity Postdoctoral Fellow, has an answer.

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