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Category: mathematics – Page 37

A team of researchers from Peking University and the Eastern Institute of Technology (EIT) in China has developed a new framework to train machine learning models with prior knowledge, such as the laws of physics or mathematical logic, alongside data.

Chinese researchers are on the brink of pioneering a groundbreaking approach to developing ‘AI scientists capable of conducting experiments and solving scientific problems.

Recent advances in deep learning models have revolutionized scientific research, but current models still struggle to simulate real-world physics interactions accurately.

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“Without a fundamental understanding of the world, a model is essentially an animation rather than a simulation,” said Chen Yuntian, study author and a professor at the Eastern Institute of Technology (EIT).

Deep learning models are generally trained using data and not prior knowledge, which can include things such as the laws of physics or mathematical logic, according to the paper.

But the scientists from Peking University and EIT wrote that when training the models, prior knowledge could be used alongside data to make them more accurate, creating “informed machine learning” models capable of incorporating this knowledge into their output.

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A team of Stony Brook University physicists and their collaborators have taken a significant step toward the building of a quantum internet testbed by demonstrating a foundational quantum network measurement that employs room-temperature quantum memories. Their findings are described in a paper published in the Nature journal Quantum Information.

Research with quantum computing and quantum networks is taking place around the world in the hopes of developing a quantum internet, a network of quantum computers, sensors, and communication devices that will create, process, and transmit quantum states and entanglement. It is anticipated to enhance society’s internet system and provide certain services and securities that the current internet does not have.

The field of quantum information combines aspects of physics, mathematics, and classical computing to use quantum mechanics to solve complex problems much faster than classical computing and to transmit information in an unhackable manner. While the vision of a quantum internet system is growing and the field has seen a surge in interest from researchers and the public at large, accompanied by a steep increase in the capital invested, an actual quantum internet prototype has not been built.

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Memory storage technology has come a long way from compact disks. Or has it? In a recent paper, scientists report they were able to fit petabytes of memory onto a compact disk using new laser technologies and advanced material design. Is this the future of data storage? Let’s have a look.

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Draw a line between P and Q. That line will intersect the curve at a third point, R. (Mathematicians have a special trick for dealing with the case where the line doesn’t intersect the curve by adding a “point at infinity.”) The reflection of R across the x-axis is your sum P + Q. Together with this addition operation, all the solutions to the curve form a mathematical object called a group.

Mathematicians use this to define the “rank” of a curve. The rank of a curve relates to the number of rational solutions it has. Rank 0 curves have a finite number of solutions. Curves with higher rank have infinite numbers of solutions whose relationship to one another using the addition operation is described by the rank.

Ranks are not well understood; mathematicians don’t always have a way of computing them and don’t know how big they can get. (The largest exact rank known for a specific curve is 20.) Similar-looking curves can have completely different ranks.

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Correcting 50-year-old errors in the math used to understand how electromagnetic waves scatter electrons trapped in Earth’s magnetic fields will lead to better protection for technology in space.

“The discovery of these errors will help scientists improve their models of artificial radiation belts produced by high-altitude and how an event like that would impact our space technology,” said Greg Cunningham, a space scientist at Los Alamos National Laboratory. “This allows us to make better predictions of what that threat could be and the efficacy of radiation belt remediation strategies.”

Heliophysics models are important tools researchers use to understand phenomena around the Earth, such as how electrons can become trapped in the near-Earth space environment and damage electronics on space assets, or how Earth’s magnetic field shields us from both and particles in solar wind.

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Gödel’s Incompleteness theorems are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system capable of modelling basic arithmetic.

The first incompleteness theorem: No consistent formal system capable of modelling basic arithmetic can be used to prove all truths about arithmetic.

In other words, no matter how complex a system of mathematics is, there will always be some statements about numbers that cannot be proved or disproved within the system.

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