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Category: mathematics – Page 41

Google has released a new Pro model of its latest AI, Gemini, and company sources say it has outperformed GPT-3.5 (the free version of ChatGPT) in widespread testing. According to performance reports, Gemini Ultra exceeds current state-of-the-art results on 30 of the 32 widely-used academic benchmarks used in large language model (LLM) research and development. Google has been accused of lagging behind OpenAI’s ChatGPT, widely regarded as the most popular and powerful in the AI space. Google says Gemini was trained to be multimodal, meaning it can process different types of media such as text, pictures, video, and audio.

Insider also reports that, with a score of 90.0%, Gemini Ultra is the first model to outperform human experts on MMLU (massive multitask language understanding), which uses a combination of 57 subjects such as math, physics, history, law, medicine and ethics for testing both world knowledge and problem-solving abilities.

The Google-based AI comes in three sizes, or stages, for the Gemini platform: Ultra, which is the flagship model, Pro and Nano (designed for mobile devices). According to reports from TechCrunch, the company says it’s making Gemini Pro available to enterprise customers through its Vertex AI program, and for developers in AI Studio, on December 13. Reports indicate that the Pro version can also be accessed via Bard, the company’s chatbot interface.

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Is it possible to invent a computer that computes anything in a flash? Or could some problems stump even the most powerful of computers? How complex is too complex for computation? The question of how hard a problem is to solve lies at the heart of an important field of computer science called computational complexity. Computational complexity theorists want to know which problems are practically solvable using clever algorithms and which problems are truly difficult, maybe even virtually impossible, for computers to crack. This hardness is central to what’s called the P versus NP problem, one of the most difficult and important questions in all of math and science.

This video covers a wide range of topics including: the history of computer science, how transistor-based electronic computers solve problems using Boolean logical operations and algorithms, what is a Turing Machine, the different classes of problems, circuit complexity, and the emerging field of meta-complexity, where researchers study the self-referential nature of complexity questions.

Featuring computer scientist Scott Aaronson (full disclosure, he is also member of the Quanta Magazine Board). Check out his blog: https://scottaaronson.blog/

Read the companion article about meta-complexity at Quanta Magazine: https://www.quantamagazine.org/complexity-theorys-50-year-jo…-20230817/

Continue reading “P vs. NP: The Greatest Unsolved Problem in Computer Science” | >

Self-propelled nanoparticles could potentially advance drug delivery and lab-on-a-chip systems — but they are prone to go rogue with random, directionless movements. Now, an international team of researchers has developed an approach to rein in the synthetic particles.

Led by Igor Aronson, the Dorothy Foehr Huck and J. Lloyd Huck Chair Professor of Biomedical Engineering, Chemistry and Mathematics at Penn State, the team redesigned the nanoparticles into a propeller shape to better control their movements and increase their functionality. They published their results in the journal Small (“Multifunctional Chiral Chemically-Powered Micropropellers for Cargo Transport and Manipulation”).

A propeller-shaped nanoparticle spins counterclockwise, triggered by a chemical reaction with hydrogen peroxide, followed by an upward movement, triggered by a magnetic field. The optimized shape of these particles allows researchers to better control the nanoparticles’ movements and to pick up and move cargo particles. (Video: Active Biomaterials Lab)

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The experiment mirrored the principles of the quantum bomb tester, where a photon’s wave-particle behavior was theorized to detect the presence of a bomb without directly interacting with it.

A new study demonstrated how a droplet’s behavior imitates certain behaviors predicted for quantum particles — particularly photons.

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Scientists have grown a tiny brain-like organoid out of human stem cells, hooked it up to a computer, and demonstrated its potential as a kind of organic machine learning chip, showing it can quickly pick up speech recognition and math predictions.

As incredible as recent advances have been in machine learning, artificial intelligence still lags way behind the human brain in some important ways. For example, the brain happily learns and adapts all day long on an energy budget of about 20 watts, where a comparably powerful artificial neural network needs about 8 million watts to achieve anything remotely comparable.

What’s more, the human brain’s neural plasticity, its ability to grow new nervous tissue and expand existing connective channels, has granted it an ability to learn from noisy, low-quality data streams, with minimal training and energy expenditure. What AI systems accomplish with brute force and massive energy, the brain achieves with an effortless elegance. It’s a credit to the billions of years of high-stakes trial and error that delivered the human brain to the state it’s in today, in which it’s chiefly used to watch vast numbers of other people dancing while we’re on the toilet.

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Conference is exploring burgeoning connections between the two fields.

Traditionally, mathematicians jot down their formulas using paper and pencil, seeking out what they call pure and elegant solutions. In the 1970s, they hesitantly began turning to computers to assist with some of their problems. Decades later, computers are often used to crack the hardest math puzzles. Now, in a similar vein, some mathematicians are turning to machine learning tools to aid in their numerical pursuits.

Embracing Machine Learning in Mathematics.

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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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