Lifeboat Foundation

face_with_colon_three steps towards infinity getting much closer to the solution with reinmans hypothesis: D.

Just as molecules are composed of atoms, in math, every natural number can be broken down into its prime factors—those that are divisible only by themselves and 1. Mathematicians want to understand how primes are distributed along the number line, in the hope of revealing an organizing principle for the atoms of arithmetic.

“At first sight, they look pretty random,” says James Maynard, a mathematician at the University of Oxford. “But actually, there’s believed to be this hidden structure within the prime numbers.”

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A discrepancy between mathematics and physics has plagued astrophysicists’ understanding of how supermassive black holes merge, but dark matter may have the answer.

A mathematical model suggests there is an unusual region of space where objects can get pulled into the sun’s orbit – meaning we may have to redraw the boundary of the solar system.

By Jonathan O’Callaghan

AI models can easily generate essays and other types of text. However, they’re nowhere near as good at solving math problems, which tend to involve logical reasoning—something that’s beyond the capabilities of most current AI systems.

But that may finally be changing. Google DeepMind says it has trained two specialized AI systems to solve complex math problems involving advanced reasoning. The systems—called AlphaProof and AlphaGeometry 2—worked together to successfully solve four out of six problems from this year’s International Mathematical Olympiad (IMO), a prestigious competition for high school students. They won the equivalent of a silver medal.

Scientists all over the world use modeling approaches to understand complex natural systems such as climate systems or neuronal or biochemical networks. A team of researchers has now developed a new mathematical framework that explains, for the first time, a mechanism behind long transient behaviors in complex systems.

In this thought-provoking exploration, we delve into the profound reflections of Edward Witten, a leading figure in theoretical physics. Join us as we navigate the complexities of dualities, the enigmatic nature of M-theory, and the intriguing concept of emergent space-time. Witten, the only physicist to win the prestigious Fields Medal, offers deep insights into the mathematical and physical mysteries that shape our understanding of reality. From the holographic principle to the elusive (2,0) theory, we uncover how these advanced theories interconnect and challenge our conventional perceptions. This journey is not just a deep dive into high-level physics but a philosophical quest to grasp the nature of existence itself. Read the full interview here: https://www.quantamagazine.org/edward…

#EdwardWitten #TheoreticalPhysics #StringTheory #QuantumFieldTheory #MTheory.

Continue reading “The Mysteries of Physics, Dualities, M theory, and the Emergent Nature of Space-time” »

Did abstract mathematics, such as Pythagoras’s theorem, exist before the big bang?

Simon McLeish Lechlade, Gloucestershire, UK

The notion of the existence of mathematical ideas is a complex one.

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Machine learning influences numerous aspects of modern society, empowers new technologies, from Alphago to ChatGPT, and increasingly materializes in consumer products such as smartphones and self-driving cars. Despite the vital role and broad applications of artificial neural networks, we lack systematic approaches, such as network science, to understand their underlying mechanism. The difficulty is rooted in many possible model configurations, each with different hyper-parameters and weighted architectures determined by noisy data. We bridge the gap by developing a mathematical framework that maps the neural network’s performance to the network characters of the line graph governed by the edge dynamics of stochastic gradient descent differential equations. This framework enables us to derive a neural capacitance metric to universally capture a model’s generalization capability on a downstream task and predict model performance using only early training results. The numerical results on 17 pre-trained ImageNet models across five benchmark datasets and one NAS benchmark indicate that our neural capacitance metric is a powerful indicator for model selection based only on early training results and is more efficient than state-of-the-art methods.

Using AI-generated datasets to train future generations of machine learning models may pollute their output, a concept known as model collapse, according to a new paper published in Nature. The research shows that within a few generations, original content is replaced by unrelated nonsense, demonstrating the importance of using reliable data to train AI models.

Generative AI tools such as large language models (LLMs) have grown in popularity and have been primarily trained using human-generated inputs. However, as these AI models continue to proliferate across the Internet, computer-generated content may be used to train other AI models—or themselves—in a recursive loop.

Ilia Shumailov and colleagues present mathematical models to illustrate how AI models may experience model collapse. The authors demonstrate that an AI may overlook certain outputs (for example, less common lines of text) in training data, causing it to train itself on only a portion of the dataset.