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Category: mathematics – Page 49

God’s Number Revealed: 20 Moves Proven Enough to Solve Any Rubik’s Cube Position

Year 2010 😗😁

The world has waited with bated breath for three decades, and now finally a group of academics, engineers, and math geeks has discovered the number that explains life, the universe, and everything. That number is 20, and it’s the maximum number of moves it takes to solve a Rubik’s Cube.

Known as God’s Number, the magic number required about 35 CPU-years and a good deal of man-hours to solve. Why? Because there’s-1 possible positions of the cube, and the computer algorithm that finally cracked God’s Algorithm had to solve them all. (The terms God’s Number/Algorithm are derived from the fact that if God was solving a Cube, he/she/it would do it in the most efficient way possible. The Creator did not endorse this study, and could not be reached for comment.)

A full breakdown of the history of God’s Number as well as a full breakdown of the math is available here, but summarily the team broke the possible positions down into sets, then drastically cut the number of possible positions they had to solve for through symmetry (if you scramble a Cube randomly and then turn it upside down, you haven’t changed the solution).

Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey

Nice figures in this newly published survey on Scaled Optimal Transport with 200+ references.

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Optimal Transport (OT) is a mathematical framework that first emerged in the eighteenth century and has led to a plethora of methods for answering many theoretical and applied questions. The last decade has been a witness to the remarkable contributions of this classical optimization problem to machine learning. This paper is about where and how optimal transport is used in machine learning with a focus on the question of scalable optimal transport. We provide a comprehensive survey of optimal transport while ensuring an accessible presentation as permitted by the nature of the topic and the context. First, we explain the optimal transport background and introduce different flavors (i.e. mathematical formulations), properties, and notable applications.

Freezing Point Phenomena: Unlocking the Strange Secrets of Ice Nucleation

Research unveils a mathematical model for ice nucleation, showing how surface angles affect water’s freezing point, with applications in snowmaking and cloud seeding.

From abstract-looking cloud formations to roars of snow machines on ski slopes, the transformation of liquid water into solid ice touches many facets of life. Water’s freezing point is generally accepted to be 32 degrees Fahrenheit. But that is due to ice nucleation — impurities in everyday water raise its freezing point to this temperature. Now, researchers unveil a theoretical model that shows how specific structural details on surfaces can influence water’s freezing point.

Research Findings and Their Implications.

How Chain-of-Thought Reasoning Helps Neural Networks Compute

“They remove some of the magic,” said Dimitris Papailiopoulos, a machine learning researcher at the University of Wisconsin, Madison. “That’s a good thing.”

Training Transformers

Large language models are built around mathematical structures called artificial neural networks. The many “neurons” inside these networks perform simple mathematical operations on long strings of numbers representing individual words, transmuting each word that passes through the network into another. The details of this mathematical alchemy depend on another set of numbers called the network’s parameters, which quantify the strength of the connections between neurons.

Solving the Hard Problem: A Thermodynamic Theory of Consciousness and Intelligence

This paper introduces a novel theoretical framework for understanding consciousness, proposing a paradigm shift from traditional biological-centric views to a broader, universal perspective grounded in thermodynamics and systems theory. We posit that consciousness is not an exclusive attribute of biological entities but a fundamental feature of all systems exhibiting a particular form of intelligence. This intelligence is defined as the capacity of a system to efficiently utilize energy to reduce internal entropy, thereby fostering increased order and complexity. Supported by a robust mathematical model, the theory suggests that subjective experience, or what is often referred to as qualia, emerges from the intricate interplay of energy, entropy, and information within a system. This redefinition of consciousness and intelligence challenges existing paradigms and extends the potential for understanding and developing Artificial General Intelligence (AGI). The implications of this theory are vast, bridging gaps between cognitive science, artificial intelligence, philosophy, and physics, and providing a new lens through which to view the nature of consciousness itself.

Consciousness, traditionally viewed through the lens of biology and neurology, has long been a subject shrouded in mystery and debate. Philosophers, scientists, and thinkers have pondered over what consciousness is, how it arises, and why it appears to be a unique trait of certain biological organisms. The “hard problem” of consciousness, a term coined by philosopher David Chalmers, encapsulates the difficulty in explaining why and how physical processes in the brain give rise to subjective experiences.

Current research in cognitive science, neuroscience, and artificial intelligence offers various theories of consciousness, ranging from neural correlates of consciousness (NCCs) to quantum theories. However, these theories often face limitations in fully explaining the emergence and universality of consciousness.

Scientists proved the fundamental limits of electromagnetic energy absorption

Until recently, researchers were unsure of the minimum thickness of a transparent substance required to take in a given quantity of light.

Konstantin N. Rozanov of the Institute for Theoretical and Applied Electrodynamics in Russia discovered more than two decades ago the amount of light that a gadget might absorb at various wavelengths if one side of it was coated in metal. This metal establishes a barrier where light is absorbed or bounced back, simplifying the mathematical solution.

US researchers determine the limits of energy absorption in transparent materials

Duke researchers find limits of energy absorption in transparent materials.

Researchers at Duke University in the US have determined the theoretical limits of how much electromagnetic energy a transparent material can absorb. This can help researchers optimize device designs in the future, but it has also ended a 20-year wait for a mathematical solution to the problem.