Mathematics suggested that time travel is physically possible – and Kurt Gödel proved it. Mathematician Karl Sigmund explains how the polymath did it.
By Karl Sigmund
Category: mathematics – Page 39
Researchers propose the next platform for brain-inspired computing
Computers have come so far in terms of their power and potential, rivaling and even eclipsing human brains in their ability to store and crunch data, make predictions and communicate. But there is one domain where human brains continue to dominate: energy efficiency.
“The most efficient computers are still approximately four orders of magnitude — that’s 10,000 times — higher in energy requirements compared to the human brain for specific tasks such as image processing and recognition, although they outperform the brain in tasks like mathematical calculations,” said UC Santa Barbara electrical and computer engineering Professor Kaustav Banerjee, a world expert in the realm of nanoelectronics. “Making computers more energy efficient is crucial because the worldwide energy consumption by on-chip electronics stands at #4 in the global rankings of nation-wise energy consumption, and it is increasing exponentially each year, fueled by applications such as artificial intelligence.” Additionally, he said, the problem of energy inefficient computing is particularly pressing in the context of global warming, “highlighting the urgent need to develop more energy-efficient computing technologies.”
Neuromorphic computing has emerged as a promising way to bridge the energy efficiency gap. By mimicking the structure and operations of the human brain, where processing occurs in parallel across an array of low power-consuming neurons, it may be possible to approach brain-like energy efficiency.
Brain connectivity found to be disrupted in schizophrenia
Schizophrenia, a neurodevelopmental disorder that features psychosis among its symptoms, is thought to arise from disorganization in brain connectivity and functional integration. Now, a recent study in Biological Psychiatry: Cognitive Neuroscience and Neuroimaging, finds differences in functional brain connectivity in people with and without psychosis and schizophrenia that could help researchers understand the neural underpinnings of this disease.
The brain’s cortex is organized in a hierarchical fashion, anchored by the sensorimotor cortex at one end and by multimodal association areas at the other, with the task of integrating incoming sensory information with internal and external sensory signals. The loss of executive control in schizophrenia may stem from disruption of this hierarchical signaling.
Alexander Holmes, a Ph.D. candidate at Monash University who led the study, said, “We used brain imaging and novel mathematical techniques to investigate the hierarchical organization of the brains of individuals with early psychosis and established schizophrenia. This organization is important for brain health, as it regulates how we can effectively respond to and process stimuli from the external world.”
Euler’s Identity: ‘The Most Beautiful Equation’
Euler’s Identity:
The most beautiful equation in mathematics that combines five of the most important constants of nature: 0, 1, π, e and i, with the three fundamental operations: addition, multiplication and exponentiation.
It’s mystical.
Euler’s identity is an equality found in mathematics that has been compared to a Shakespearean sonnet and described as “the most beautiful equation.” It is a special case of a foundational equation in complex arithmetic called Euler’s Formula, which the late great physicist Richard Feynman called in his lectures “our jewel” and “the most remarkable formula in mathematics.”
In an interview with the BBC, Prof David Percy of the Institute of Mathematics and its Applications said Euler’s Identity was “a real classic and you can do no better than that … It is simple to look at and yet incredibly profound, it comprises the five most important mathematical constants.”
Twisting Light Unlocks New Quantum Realms
A research team is studying how light moves through special circuits called optical waveguides, using a concept called topology. They’ve made an important discovery that combines stable light paths with light particle interactions, which could make quantum computers more reliable and lead to new technological advancements.
Scientific innovation often arises as synthesis from seemingly unrelated concepts. For instance, the reciprocity of electricity and magnetism paved the way for Maxwell’s theory of light, which, up until now, is continually being refined and extended with ideas from quantum mechanics.
Similarly, the research group of Professor Alexander Szameit at the Institute of Physics at the University of Rostock explores light evolution in optical waveguide circuits in the presence of topology. This abstract mathematical concept was initially developed to classify solid geometries according to their global properties. Szameit explains: “In topological systems, light only follows the global characteristics of the waveguide system. Local perturbations to the waveguides such as defects, vacancies, and disorder cannot divert its path.”
Mathematicians Accidentally Found a New Way to Represent Pi
Our favorite mathematical constant, pi (π), describing the ratio between a circle’s circumference and its diameter, has taken on new meaning.
The new representation was borne out of the twists and turns of string theory, and two mathematicians’ attempts to better describe particle collisions.
“Our efforts, initially, were never to find a way to look at pi,” says Aninda Sinha of the Indian Institute of Science (IISc) who co-authored the new work with fellow IISc mathematician Arnab Priya Saha.
Significance of Wave Activity for Understanding Titan’s Climate
Lakes and seas of liquid methane exist on Saturn’s largest moon, Titan, due to the moon’s bone-chilling cold temperatures at-290 degrees Fahrenheit (−179 degrees Celsius), whereas it can only exist as a gas on Earth. But do these lakes and seas of liquid methane strewn across Titan’s surface remain static, or do they exhibit wave activity like the lakes and seas of liquid water on Earth? This is what a recent study published in Science Advances hopes to address as a team of researchers have investigated coastal shoreline erosion on Titan’s surface resulting from wave activity. This study holds the potential to help researchers better understand the formation and evolution of planetary surfaces throughout the solar system and how well they relate to Earth.
For the study, the researchers used a combination of shoreline analogs on Earth, orbital images obtained by NASA’s now-retired Cassini spacecraft, coastal evolution models, and several mathematical equations to ascertain the processes responsible for shoreline morphology across Titan’s surface. Through this, the researchers were able to construct coastal erosion models depicting how wave activity could be responsible for changes in shoreline morphology at numerous locations across Titan’s surface.
“We can say, based on our results, that if the coastlines of Titan’s seas have eroded, waves are the most likely culprit,” said Dr. Taylor Perron, who is a Cecil and Ida Green Professor of Earth, Atmospheric and Planetary Sciences at the Massachusetts Institute of Technology and a co-author on the study. “If we could stand at the edge of one of Titan’s seas, we might see waves of liquid methane and ethane lapping on the shore and crashing on the coasts during storms. And they would be capable of eroding the material that the coast is made of.”
Mathematicians find odd shapes that roll like a wheel in any dimension
Not content with shapes in two or three dimensions, mathematicians like to explore objects in any number of spatial dimensions. Now they have discovered shapes of constant width in any dimension, which roll like a wheel despite not being round.
By Alex Wilkins
This New Idea Could Explain Complexity
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The universe creates complexity out of simplicity, but despite many attempts at understanding how, scientists still have not figured it out. We do know that complexity relies on the emergence of new features and laws, but then again we don’t understand emergence either. The first step must be to clearly define what we are talking about and to measure it. A group of scientists now put forward a way to do exactly this. Let’s have a look.
Paper here: https://arxiv.org/abs/2402.
Correction to what I say at 04:07 \.
AI that defeated humans at Go could now help language models master mathematics
👉 Researchers at the Shanghai Artificial Intelligence Laboratory are combining the Monte Carlo Tree Search (MCTS) algorithm with large language models to improve its ability to solve complex mathematical problems.
Integrating the Monte Carlo Tree Search (MCTS) algorithm into large language models could significantly enhance their ability to solve complex mathematical problems. Initial experiments show promising results.
While large language models like GPT-4 have made remarkable progress in language processing, they still struggle with tasks requiring strategic and logical thinking. Particularly in mathematics, the models tend to produce plausible-sounding but factually incorrect answers.
In a new paper, researchers from the Shanghai Artificial Intelligence Laboratory propose combining language models with the Monte Carlo Tree Search (MCTS) algorithm. MCTS is a decision-making tool used in artificial intelligence for scenarios that require strategic planning, such as games and complex problem-solving. One of the most well-known applications is AlphaGo and its successor systems like AlphaZero, which have consistently beaten humans in board games. The combination of language models and MCTS has long been considered promising and is being studied by many labs — likely including OpenAI with Q*.