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Category: mathematics – Page 115

For mathematicians and computer scientists, 2020 was full of discipline-spanning discoveries and celebrations of creativity. We’d like to take a moment to recognize some of these achievements.

1. A landmark proof simply titled MIP = RE” establishes that quantum computers calculating with entangled qubits can theoretically verify the answers to an enormous set of problems. Along the way, the five computer scientists who authored the proof also answered two other major questions: Tsirelson’s problem in physics, about models of particle entanglement, and a problem in pure mathematics called the Connes embedding conjecture.

2. In February, graduate student Lisa Piccirillo dusted off some long-known but little-utilized mathematical tools to answer a decades-old question about knots. A particular knot named after the legendary mathematician John Conway had long evaded mathematical classification in terms of a higher-dimensional property known as sliceness. But by developing a version of the knot that yielded to traditional knot analysis, Piccirillo finally determined that the Conway knot is not slice.

3. For decades, mathematicians have used computer programs known as proof assistants to help them write proofs — but the humans have always guided the process, choosing the proof’s overall strategy and approach. That may soon change. Many mathematicians are excited about a proof assistant called Lean, an efficient and addictive proof assistant that could one day help tackle major problems. First, though, mathematicians must digitize thousands of years of mathematical knowledge, much of it unwritten, into a form Lean can process. Researchers have already encoded some of the most complicated mathematical ideas, proving in theory that the software can handle the hard stuff. Now it’s just a question of filling in the rest.

Continue reading “2020’s Biggest Breakthroughs in Math and Computer Science” | >

SpaceX’s fleet of reusable Falcon 9 rockets enabled it to conduct more missions in 2020 than ever before. SpaceX completed a record-breaking launch manifest this year, it conducted 26 rocket launches –the most annual launches it has performed in history. Rocket reusability has played a significant role in increasing launch cadence. Falcon 9 is capable of launching payload to orbit and returning from space to land vertically on landing pads and autonomous droneships at sea. To date, SpaceX has landed 70 orbital-class Falcon 9 boosters and reused 49. This year the company accomplished flying two particular rocket boosters 7 times. Engineers aim to reuse a first-stage booster at least 10 times to reduce the cost of spaceflight. The most reused Falcon 9 rockets that reached 7 reflights this year are two first-stage boosters identified as B1051 and B1049. SpaceX is just three flights away from achieving 10 reflights. SpaceX officials state Falcon 9 [Block 5] is designed to perform up to 100 reflights.

Stephen Marr, a spaceflight photographer who goes by the name @spacecoast_stve on Twitter, shared a photo collage of all the Falcon 9 boosters used in 2020, “SpaceX carried out a record-breaking 26 launches this year, but how many boosters did it take to get it done? The answer is 11. And here they are!” he wrote. SpaceX founder Elon Musk replied to Marr’s tweet –“Falcon was 25% of successful orbital launches in 2020, but maybe a majority of payload to orbit. Anyone done the math?” he said.

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The Big Bang might never have existed as many cosmologists start to question the origin of the Universe. The Big Bang is a point in time defined by a mathematical extrapolation. The Big Bang theory tells us that something has to have changed around 13.7 billion years ago. So, there is no “point” where the Big Bang was, it was always an extended volume of space, according to the Eternal Inflation model. In light of Digital Physics, as an alternative view, it must have been the Digital Big Bang with the lowest possible entropy in the Universe — 1 bit of information — a coordinate in the vast information matrix. If you were to ask what happened before the first observer and the first moments after the Big Bang, the answer might surprise you with its straightforwardness: We extrapolate backwards in time and that virtual model becomes “real” in our minds as if we were witnessing the birth of the Universe.

In his theoretical work, Andrew Strominger of Harvard University speculates that the Alpha Point (the Big Bang) and the Omega Point form the so-called ‘Causal Diamond’ of the conscious observer where the Alpha Point has only 1 bit of entropy as opposed to the maximal entropy of some incredibly gigantic amount of bits at the Omega Point. While suggesting that we are part of the conscious Universe and time is holographic in nature, Strominger places the origin of the Universe in the infinite ultra-intelligent future, the Omega Singularity, rather than the Big Bang.

The Universe is not what textbook physics tells us except that we perceive it in this way — our instruments and measurement devices are simply extensions of our senses, after all. Reality is not what it seems. Deep down it’s pure information — waves of potentiality — and consciousness orchestrating it all. The Big Bang theory, drawing a lot of criticism as of late, uses a starting assumption of the “Universe from nothing,” (a proverbial miracle, a ‘quantum fluctuation’ christened by scientists), or the initial Cosmological Singularity. But aside from this highly improbable happenstance, we can just as well operate from a different set of assumptions and place the initial Cosmological Singularity at the Omega Point — the transcendental attractor, the Source, or the omniversal holographic projector of all possible timelines.

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A team of scientists at Freie Universität Berlin has developed an artificial intelligence (AI) method for calculating the ground state of the Schrödinger equation in quantum chemistry. The goal of quantum chemistry is to predict chemical and physical properties of molecules based solely on the arrangement of their atoms in space, avoiding the need for resource-intensive and time-consuming laboratory experiments. In principle, this can be achieved by solving the Schrödinger equation, but in practice this is extremely difficult.

Up to now, it has been impossible to find an exact solution for arbitrary molecules that can be efficiently computed. But the team at Freie Universität has developed a deep learning method that can achieve an unprecedented combination of accuracy and computational efficiency. AI has transformed many technological and scientific areas, from computer vision to materials science. “We believe that our approach may significantly impact the future of quantum ,” says Professor Frank Noé, who led the team effort. The results were published in the reputed journal Nature Chemistry.

Central to both quantum chemistry and the Schrödinger equation is the —a mathematical object that completely specifies the behavior of the electrons in a molecule. The wave function is a high-dimensional entity, and it is therefore extremely difficult to capture all the nuances that encode how the individual electrons affect each other. Many methods of quantum chemistry in fact give up on expressing the wave function altogether, instead attempting only to determine the energy of a given molecule. This however requires approximations to be made, limiting the prediction quality of such methods.

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In some ways, learning to program a computer is similar to learning a new language. It requires learning new symbols and terms, which must be organized correctly to instruct the computer what to do. The computer code must also be clear enough that other programmers can read and understand it.

In spite of those similarities, MIT neuroscientists have found that reading computer code does not activate the regions of the brain that are involved in language processing.

Instead, it activates a distributed network called the multiple demand network, which is also recruited for complex cognitive tasks such as solving math problems or crossword puzzles.

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This week, I had some amazing discussions with Navajo Nation Math Circle leaders — Dave Auckly and Henry Fowler. The idea of starting a math circle on Navajo land was initially brought up by a wonderful math educator and mathematician raised in Kazakhstan, Tatiana Shubin. Here is a small tribute to their efforts:

Project activities were launched in the Fall of 2012. A team of distinguished mathematicians from all over the US, as well as local teachers and community members, work together to run the outreach. Navajo Nation Math Circles present math in the context of Navajo culture, helping students develop their identity as true Navajo mathematicians. “We want to find kids who would not have discovered their talents without our project, to help them realize that they can change the world,” says Fowler. Having introduced Navajo children to the joy of mathematics, the project also yielded a book, Inspiring Mathematics: Lessons from the Navajo Nation Math Circles, which contain lesson plans, puzzles and activities, and other insights for parents and teachers to embrace.

An extension of Navajo Nation Math Circles is an annual two-week Baa Hózhó summer math camp at Navajo Technical University. “Baa Hózhó” means “balance and harmony,” tying together the ideas of mathematical equilibrium with the way of life embraced by Navajo people. The summer camp is widely popular with parents and children; the older students come back as counselors, making everyone feel like one big family. It is preceded by an annual student-run math festival in local schools across the Navajo Nation, where students share their passion for mathematics with families and friends.

Fowler’s ultimate goal is to create a Mathematical Research institute on Navajo land, where local and international researchers could exchange math ideas and study the best ways of teaching mathematics to Indigenous people, enriching worldwide mathematical sciences. Hopefully, the great strides in the Navajo Nation math education will encourage leading high-tech companies to support the rise of a new generation of diverse, talented and passionate Native American STEM professionals.

Continue reading “How The Navajo Nation Is Transforming Math Education” | >

(Checks math.)

Scientists have new evidence that Earth’s many periodic mass extinctions follow a cycle of about 27 million years, connecting the five major mass extinctions with more minor ones occurring throughout Earth’s life-fostering timespan. The artificial intelligence analysis could also shift how evolutionary scientists think about the aftermath of mass extinctions.

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Neuroscientists find that interpreting code activates a general-purpose brain network, but not language-processing centers.

In some ways, learning to program a computer is similar to learning a new language. It requires learning new symbols and terms, which must be organized correctly to instruct the computer what to do. The computer code must also be clear enough that other programmers can read and understand it.

In spite of those similarities, MIT neuroscientists have found that reading computer code does not activate the regions of the brain that are involved in language processing. Instead, it activates a distributed network called the multiple demand network, which is also recruited for complex cognitive tasks such as solving math problems or crossword puzzles.

Read more | >