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Archive for the ‘mathematics’ category: Page 58

May 6, 2022

A ‘beyond-quantum’ equivalence principle for superposition and entanglement

Posted by in categories: mathematics, particle physics, quantum physics

The physics of the microrealm involves two famous and bizarre concepts: The first is that prior to observation, it is impossible to know with certainty the outcome of a measurement on a particle; rather the particle exists in a “superposition” encompassing multiple mutually exclusive states. So a particle can be in two or more places at the same time, and you can only calculate the probability of finding it in a certain location when you look. The second involves “entanglement,” the spooky link that can unite two objects, no matter how far they are separated. Both superposition and entanglement are described mathematically by quantum theory. But many physicists believe that the ultimate theory of reality may lie beyond quantum theory. Now, a team of physicists and mathematicians has discovered a new connection between these two weird properties that does not assume that quantum theory is correct. Their study appears in Physical Review Letters.

“We were really excited to find this new connection that goes beyond quantum theory because the connection will be valid even for more exotic theories that are yet to be discovered,” says Ludovico Lami, a member of the physics think-tank, the Foundational Questions Institute, FQXi, and a physicist at the University of Ulm, in Germany. “This is also important because it is independent of the mathematical formalism of quantum theory and uses only notions with an immediate operational interpretation,” he adds. Lami co-authored the study with Guillaume Aubrun of Claude Bernard University Lyon 1, in France, Carlos Palazuelos, of the Complutense University of Madrid, in Spain, and Martin Plávala, of Siegen University, in Germany.

While quantum theory has proven to be supremely successful since its development a century ago, physicists have struggled to unify it with gravity to create one overarching “theory of everything.” This suggests that quantum theory may not be the final word on describing reality, inspiring physicists to hunt for a more fundamental framework. But any such ultimate theory must still incorporate superposition, entanglement, and the probabilistic nature of reality, since these features have been confirmed time and again in lab tests. The interpretation of these experiments does not depend on quantum theory being correct, notes Lami.

May 4, 2022

Neurocompositional computing: From the Central Paradox of Cognition to a new generation of AI systems

Posted by in categories: mathematics, robotics/AI

What explains the dramatic progress from 20th-century to 21st-century AI, and how can the remaining limitations of current AI be overcome? The widely accepted narrative attributes this progress to massive increases in the quantity of computational and data resources available to support statistical learning in deep artificial neural networks. We show that an additional crucial factor is the development of a new type of computation. Neurocompositional computing adopts two principles that must be simultaneously respected to enable human-level cognition: the principles of Compositionality and Continuity. These have seemed irreconcilable until the recent mathematical discovery that compositionality can be realized not only through discrete methods of symbolic computing, but also through novel forms of continuous neural computing.

May 4, 2022

Computers could revise past conclusions with AI

Posted by in categories: economics, mathematics, robotics/AI

To better automate reasoning, machines should ideally be able to systematically revise the view they have obtained about the world. Timotheus Kampik’s dissertation work presents mathematical reasoning approaches that strike a balance between retaining consistency with previously drawn conclusions and rejecting them in face of overwhelming new evidence.

When reasoning and when making decisions, humans are continuously revising what their view of the world is, by rejecting what they have previously considered true or desirable, and replacing it with an updated and ideally more useful perspective. Enabling machines to do so in a similar manner, but with logical precision, is a long-running line of artificial intelligence research.

In his dissertation, Timotheus advances this line of research by devising reasoning approaches that balance retaining previously drawn conclusions for the sake of ensuring consistency and revising them to accommodate new compelling evidence. To this end, he applies well-known from to formal argumentation, an approach to logic-based automated reasoning.

May 1, 2022

The basics of decentralized finance

Posted by in categories: blockchains, computing, cryptocurrencies, finance, information science, mathematics

Decentralized finance is built on blockchain technology, an immutable system that organizes data into blocks that are chained together and stored in hundreds of thousands of nodes or computers belonging to other members of the network.

These nodes communicate with one another (peer-to-peer), exchanging information to ensure that they’re all up-to-date and validating transactions, usually through proof-of-work or proof-of-stake. The first term is used when a member of the network is required to solve an arbitrary mathematical puzzle to add a block to the blockchain, while proof-of-stake is when users set aside some cryptocurrency as collateral, giving them a chance to be selected at random as a validator.

To encourage people to help keep the system running, those who are selected to be validators are given cryptocurrency as a reward for verifying transactions. This process is popularly known as mining and has not only helped remove central entities like banks from the equation, but it also has allowed DeFi to open more opportunities. In traditional finance, are only offered to large organizations, for members of the network to make a profit. And by using network validators, DeFi has also been able to cut down the costs that intermediaries charge so that management fees don’t eat away a significant part of investors’ returns.

Apr 26, 2022

The Human Calculator

Posted by in categories: education, mathematics

Thomas Fuller, an African sold into slavery in 1,724 at age 14, was sometimes known as the “Virginia Calculator” for his extraordinary ability to solve complex mathematical problems in his head. He was asked how many seconds there were in a year, he briefly answered 31,536,000 seconds.

He was asked again how many seconds a man who is 70 years old, 17 days and 12 hours lived, he answered in a minute and a half 2,210,500,800. One of the men was doing the problems on paper and informed Fuller that he was wrong because the answer was much smaller. Fuller hastily responded, “Nah, you forgot about leap years. When leap years were added to the account, the sums matched up.”

Fuller was one of the first cases recorded in the literature of the wise man syndrome, when in 1,789, Benjamin Rush, the father of American psychiatry, described his incredible ability to calculate, without having an education and training in mathematics, his ability was used as proof that enslaved African Americans were equal to whites in intelligence, fueling some pro-abolitionist discussions.

Apr 21, 2022

Deep Learning Poised to ‘Blow Up’ Famed Fluid Equations

Posted by in categories: information science, mathematics, robotics/AI

For centuries, mathematicians have tried to prove that Euler’s fluid equations can produce nonsensical answers. A new approach to machine learning has researchers betting that “blowup” is near.

Apr 19, 2022

How Wavelets Allow Researchers to Transform — and Understand — Data

Posted by in category: mathematics

Built upon the ubiquitous Fourier transform, the mathematical tools known as wavelets allow unprecedented analysis and understanding of continuous signals.

Apr 19, 2022

Research team measures the mass of the top quark with unparalleled accuracy

Posted by in categories: mathematics, particle physics, quantum physics

The CMS collaboration at the Large Hadron Collider (LHC) has performed the most accurate ever measurement of the mass of the top quark—the heaviest known elementary particle. The latest CMS result estimates the value of the top-quark mass with an accuracy of about 0.22%. The substantial gain in accuracy comes from new analysis methods and improved procedures to consistently and simultaneously treat different uncertainties in the measurement.

The precise knowledge of the top-quark mass is of paramount importance to understand our world at the smallest scale. Knowing this heaviest as intimately as possible is crucial because it allows testing of the internal consistency of the mathematical description of all elementary particles, called the Standard Model.

For example, if the masses of the W boson and Higgs boson are known accurately, the top-quark mass can be predicted by the Standard Model. Likewise, using the top-quark and Higgs-boson masses, the W-boson mass can be predicted. Interestingly, despite much progress, the theoretical-physics definition of mass, which has to do with the effect of quantum-physics corrections, is still tough to pin down for the top quark.

Apr 19, 2022

Study shows simple, computationally-light model can simulate complex brain cell responses

Posted by in categories: biotech/medical, chemistry, computing, mathematics, neuroscience

The brain is inarguably the single most important organ in the human body. It controls how we move, react, think and feel, and enables us to have complex emotions and memories. The brain is composed of approximately 86 billion neurons that form a complex network. These neurons receive, process, and transfer information using chemical and electrical signals.

Learning how respond to different signals can further the understanding of cognition and development and improve the management of disorders of the brain. But experimentally studying neuronal networks is a complex and occasionally invasive procedure. Mathematical models provide a non-invasive means to accomplish the task of understanding , but most current models are either too computationally intensive, or they cannot adequately simulate the different types of complex neuronal responses. In a recent study, published in Nonlinear Theory and Its Applications, IEICE, a research team led by Prof. Tohru Ikeguchi of Tokyo University of Science, has analyzed some of the complex responses of neurons in a computationally simple neuron model, the Izhikevich neuron model.

“My laboratory is engaged in research on neuroscience and this study analyzes the basic mathematical properties of a neuron model. While we analyzed a single neuron model in this study, this model is often used in computational neuroscience, and not all of its properties have been clarified. Our study fills that gap,” explains Prof. Ikeguchi. The research team also comprised Mr. Yota Tsukamoto and Ph.D. student Ms. Honami Tsushima, also from Tokyo University of Science.

Apr 18, 2022

Fractal Pattern in a Quantum Material Confirmed for the First Time!

Posted by in categories: mathematics, quantum physics

Image by: Arkadiusz Jadczyk.

The word fractal has become increasingly popular, although the concept started more than two centuries ago in the 17th century with prominent and prolific mathematician and philosopher Gottfried Wilhelm Leibnitz is believed to have addressed for the first time the notion of recursive self-similarity, and it wasn’t until 1960 that the concept was formally stabilized both theoretically and practically, through the mathematical development and computerized visualizations by Benoit Mandelbrot, who settled on the name “fractal”.

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