Computer-aided calculations have played a crucial part in producing the proofs of several high-profile results. And more recently, some mathematicians have made progress towards AI that doesn’t just perform repetitive calculations, but develops its own proofs. Another growing area has been software that can go over a mathematical proof written by humans and check that it is correct.
Algorithm named after mathematician Srinivasa Ramanujan suggests interesting formulae, some of which are difficult to prove true.
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What is this mysterious quantum tunneling effect, where does it come from? And why is it one of the most important phenomena in physics?
Quantum mechanics shows that quantum objects have a wave-particle duality. What we think of as an electron particle actually behaves like a wave, a probability wave. This means that its position is not a precise location in space. It is defined by a wave function that can only tell us the probability of finding it a particular location when measured. The wave function of a particle exists in all of space, in the entire universe up to infinity. So there is always a non-zero probability of finding the electron anywhere, including outside a barrier.
We can attribute this behavior to the Heisenberg uncertainty principle. It states that the uncertainty in a particle’s position times the uncertainty in its momentum has to be greater than a finite number. Practically this means we cannot know with 100% certainty what the position of that electron is. And the wave function of the electron, which gives us the probability of finding it at any location can be found using the Schrodinger equation.
This equation was developed by Erwin Schrödinger in 1926, and it is the equation that describes the wave nature of matter. The Greek letter psi in the equation is the wave function. The wave function depends on both time and position. It can be both positive or negative, but the square is always positive. The square of the wave function as a function of position is the probability of finding the particle at that position. The Schrödinger equation is a statement of conservation of energy. It says that kinetic energy plus potential energy equals the total energy—But instead of just energies, we have energy operators acting on the wave function of the particle.
For all its comparisons to the human brain, AI still isn’t much like us. Maybe that’s alright. In the animal kingdom, brains come in all shapes and sizes. So, in a new machine learning approach, engineers did away with the human brain and all its beautiful complexity—turning instead to the brain of a lowly worm for inspiration.
Turns out, simplicity has its benefits. The resulting neural network is efficient, transparent, and here’s the kicker: It’s a lifelong learner.
Whereas most machine learning algorithms can’t hone their skills beyond an initial training period, the researchers say the new approach, called a liquid neural network, has a kind of built-in “neuroplasticity.” That is, as it goes about its work—say, in the future, maybe driving a car or directing a robot—it can learn from experience and adjust its connections on the fly.
In a world that’s noisy and chaotic, such adaptability is essential. ## **Worm-Brained Driver**
The algorithm’s architecture was inspired by the mere 302 neurons making up the nervous system of *C. elegans*, a tiny nematode (or worm).
In work published last year, the group, which includes researchers from MIT and Austria’s Institute of Science and Technology, said that despite its simplicity, *C. elegans* is capable of surprisingly interesting and varied behavior. So, they developed equations to mathematically model the worm’s neurons and then built them into a neural network.
Their worm-brain algorithm was much simpler than other cutting-edge machine learning algorithms, and yet it was still able to accomplish similar tasks, like keeping a car in its lane.
But the human eye can only see so much, even with the help of a microscope; despite embryologists’ efforts to select the “best” embryo, success rates are still relatively low. “Many decisions are based on gut feeling or personal experience,” said Embryonics founder and CEO Yael Gold-Zamir. “Even if you go to the same IVF center, two experts can give you different opinions on the same embryo.”
This is where Embryonics’ technology comes in. They used 8,789 time-lapse videos of developing embryos to train an algorithm that predicts the likelihood of successful embryo implantation. A little less than half of the embryos from the dataset were graded by embryologists, and implantation data was integrated when it was available (as a binary “successful” or “failed” metric).
The algorithm uses geometric deep learning, a technique that takes a traditional convolutional neural network—which filters input data to create maps of its features, and is most commonly used for image recognition—and applies it to more complex data like 3D objects and graphs. Within days after fertilization, the embryo is still at the blastocyst stage, essentially a microscopic clump of just 200–300 cells; the algorithm uses this deep learning technique to spot and identify patterns in embryo development that human embryologists either wouldn’t see at all, or would require massive collation of data to validate.
The human eye can only see so much, and despite embryologists’ efforts to select the “best” embryo, IVF success rates are still relatively low.
Fast-forwarding quantum calculations skips past the time limits imposed by decoherence, which plagues today’s machines.
A new algorithm that fast forwards simulations could bring greater use ability to current and near-term quantum computers, opening the way for applications to run past strict time limits that hamper many quantum calculations.
“Quantum computers have a limited time to perform calculations before their useful quantum nature, which we call coherence, breaks down,” said Andrew Sornborger of the Computer, Computational, and Statistical Sciences division at Los Alamos National Laboratory, and senior author on a paper announcing the research. “With a new algorithm we have developed and tested, we will be able to fast forward quantum simulations to solve problems that were previously out of reach.”
Very interesting.
Albert Einstein’s theory of general relativity profoundly changed our thinking about fundamental concepts in physics, such as space and time. But it also left us with some deep mysteries. One was black holes, which were only unequivocally detected over the past few years. Another was “wormholes” – bridges connecting different points in spacetime, in theory providing shortcuts for space travellers.
Wormholes are still in the realm of the imagination. But some scientists think we will soon be able to find them, too. Over the past few months, several new studies have suggested intriguing ways forward.
Black holes and wormholes are special types of solutions to Einstein’s equations, arising when the structure of spacetime is strongly bent by gravity. For example, when matter is extremely dense, the fabric of spacetime can become so curved that not even light can escape. This is a black hole.