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Category: mathematics – Page 51

Lean Co-pilot for LLM-human collaboration to write formal mathematical proofs that are 100% accurate.

Top right: LeanDojo extracts proofs in Lean into datasets for training machine learning models. It also enables the trained model to prove theorems by interacting with Lean’s proof environment.

Top left: The proof tree of a Lean theorem ∀n∈N, gcd n n = n, where gcd is the greatest common divisor. When proving the theorem, we start from the original theorem as the initial state (the root) and repeatedly apply tactics (the edges) to decompose states into simpler sub-states, until all states are solved (the leaf nodes). Tactics may rely on premises such as mod_self and gcd_zero_left defined in a large math library. E.g., mod_self is an existing theorem ∀n∈N, n % n = 0 used in the proof to simplify the goal.

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A quantum property dubbed “magic” could be the key to explaining how space and time emerged, a new mathematical analysis by three RIKEN physicists suggests. The research is published in the journal Physical Review D.

It’s hard to conceive of anything more basic than the fabric of spacetime that underpins the universe, but have been questioning this assumption. “Physicists have long been fascinated about the possibility that space and time are not fundamental, but rather are derived from something deeper,” says Kanato Goto of the RIKEN Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS).

This notion received a boost in the 1990s, when theoretical physicist Juan Maldacena related the gravitational theory that governs spacetime to a theory involving . In particular, he imagined a hypothetical space—which can be pictured as being enclosed in something like an infinite soup can, or “bulk”—holding objects like that are acted on by gravity. Maldacena also imagined particles moving on the surface of the can, controlled by . He realized that mathematically a used to describe the particles on the boundary is equivalent to a gravitational theory describing the black holes and spacetime inside the bulk.

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A RIKEN physicist and two colleagues have found that a wormhole—a bridge connecting distant regions of the Universe—helps to shed light on the mystery of what happens to information about matter consumed by black holes.

Einstein’s theory of predicts that nothing that falls into a black hole can escape its clutches. But in the 1970s, Stephen Hawking calculated that black holes should emit radiation when , the theory governing the microscopic realm, is considered. “This is called black hole evaporation because the black hole shrinks, just like an evaporating water droplet,” explains Kanato Goto of the RIKEN Interdisciplinary Theoretical and Mathematical Sciences.

This, however, led to a paradox. Eventually, the black hole will evaporate entirely—and so too will any information about its swallowed contents. But this contradicts a fundamental dictum of quantum physics: that information cannot vanish from the Universe. “This suggests that general relativity and quantum mechanics as they currently stand are inconsistent with each other,” says Goto. “We have to find a unified framework for quantum gravity.”

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With a processor that has fewer qubits, IBM has improved error correction, paving the way for the use of these processors in real life.

IBM has unveiled its much-awaited 1,000+ qubit quantum processor Condor, alongside a utility-scale processor dubbed IBM Quantum Heron at its Quantum Summit in New York. The latter is the first in the series of utility-scale quantum processors that IBM took four years to build, the company said in a press release.

Quantum computers, considered the next frontier of computing, have locked companies big and small in a race to build the platform that everybody would want to use to solve complex problems in medicine, physics, mathematics, and many more.

Even the fastest supercomputers of today are years behind the potential of quantum computers, whose capabilities keep improving with the addition of quantum bits or qubits in the processor. So, a 1,000+ qubit processor is a big deal, and even though a startup may have beaten IBM to this milestone, the latter’s announcement is still significant for what else IBM brings to the table.

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Order vs Disorder, Jordan Peterson’s Yin Yang analogy, & Stephen Wolfram’s 4 classes of cellular automata are explored. The edge of chaos is the phase transition zone between order and disorder which is found across a broad range of complex systems. We discuss Norman Packard, Christopher Langton, John Beggs, Stuart Kauffman, Mihaly Csikszentmihalyi, and M. Mitchell Waldrop. Wolfram’s Rule 110 and John Conway’s Game of Life, both Turing complete, make appearances.

0:00 Intro.
0:59 Lambda & Wolfram’s 4 Classes.
3:32 Criticality, Avalanches, & John Beggs.
4:44 Homework? More like FUNwork!
5:08 Flow by Mihaly Csikszentmihalyi.
5:35 Jordan Peterson (Yin-Yang)
9:39 M. Mitchell Waldrop’s Complexity.

Play with cellular automata here:
https://math.hws.edu/eck/js/edge-of-chaos/CA.html.
David J. Eck, Hobart and William Smith Colleges flow state by MIHALY CSIKSZENTMIHALYI ► The Secret to Happiness & Psychology of Optimal Experience https://www.youtube.com/watch?v=9e31Tdvz_FU&list=PLyQeeNuuRL…sWGsWQOG6H

🚾 Works Cited.
Jeremy Avnet (brainsik); Senior Thesis, Mathematics; University California, Santa Cruz; 6th June 2000 https://theory.org/complexity/cdpt/html/node5.html#foot561

Jfromm (2009) http://wiki.cas-group.net/index.php?title=File: Edge_of_Chaos.png.

Hill, Sean. (2017). Toward Conceptualizing Race and Racial Identity Development Within an Attractor Landscape. SAGE Open. 7. 10.1177÷2158244017719310.

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Has OpenAI invented an AI technology with the potential to “threaten humanity”? From some of the recent headlines, you might be inclined to think so.

Reuters and The Information first reported last week that several OpenAI staff members had, in a letter to the AI startup’s board of directors, flagged the “prowess” and “potential danger” of an internal research project known as “Q*.” This AI project, according to the reporting, could solve certain math problems — albeit only at grade-school level — but had in the researchers’ opinion a chance of building toward an elusive technical breakthrough.

There’s now debate as to whether OpenAI’s board ever received such a letter — The Verge cites a source suggesting that it didn’t. But the framing of Q* aside, Q* in actuality might not be as monumental — or threatening — as it sounds. It might not even be new.

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Carlos Bravo-Prieto1,2,3, Ryan LaRose4, M. Cerezo1,5, Yigit Subasi6, Lukasz Cincio1, and Patrick J. Coles1

1Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87,545, USA. 2 Barcelona Supercomputing Center, Barcelona, Spain. 3 Institut de Ciències del Cosmos, Universitat de Barcelona, Barcelona, Spain. 4 Department of Computational Mathematics, Science, and Engineering & Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48,823, USA. 5 Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM, USA 6 Computer, Computational and Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos, NM 87,545, USA

Get full text pdfRead on arXiv Vanity.

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