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Researchers from the University of Warwick, Imperial College London, EPFL (Lausanne) and Sciteb Ltd have found a mathematical means of helping regulators and business manage and police Artificial Intelligence systems’ biases towards making unethical, and potentially very costly and damaging commercial choices—an ethical eye on AI.

In 1911, German mathematician Otto Toeplitz first posed the inscribed square problem, in which he predicted that “any closed curve contains four points that can be connected to form a square,” according to Quanta. For more than a century, it’s remained unsolved.

Updated mathematical techniques that can distinguish between two types of ‘non-Gaussian curve’ could make it easier for researchers to study the nature of quantum entanglement.

Quantum entanglement is perhaps one of the most intriguing phenomena known to physics. It describes how the fates of multiple particles can become entwined, even when separated by vast distances. Importantly, the probability distributions needed to define the quantum states of these particles deviate from the bell-shaped, or ‘Gaussian’ curves which underly many natural processes. Non-Gaussian curves don’t apply to quantum systems alone, however. They can also be composed of mixtures of regular Gaussian curves, producing difficulties for physicists studying quantum entanglement. In new research published in EPJ D, Shao-Hua Xiang and colleagues at Huaihua University in China propose a solution to this problem. They suggest an updated set of equations that allows physicists to easily check whether or not a non-Gaussian state is genuinely quantum.

As physicists make more discoveries about the nature of quantum entanglement, they are rapidly making progress towards advanced applications in the fields of quantum communication and computation. The approach taken in this study could prove to speed up the pace of these advances. Xiang and colleagues acknowledge that while all previous efforts to distinguish between both types of non-Gaussian curve have had some success, their choices of Gaussian curves as a starting point have so far meant that no one approach has yet proven to be completely effective. Based on the argument that there can’t be any truly reliable Gaussian reference for any genuinely quantum non-Gaussian state, the researchers present a new theoretical framework.

Continue reading “Mathematical Breakthrough Makes It Easier to Explore Quantum Entanglement” »

Researchers at Rochester Institute of Technology have developed MathDeck, an online search interface that allows anyone to easily create, edit and lookup sophisticated math formulas on the computer.

Created by an interdisciplinary team of more than a dozen faculty and students, MathDeck aims to make math notation interactive and easily shareable, rather than an obstacle to mathematical study and exploration. The math-aware search interface is free to the public and available to use at mathdeck.cs.rit.edu.

Researchers said the project stems from a growing public interest in being able to do web searches with math keywords and formulas. However, for many people, it can be difficult to accurately express sophisticated math without an understanding of the scientific markup language LaTeX.

When a German retiree proved a famous long-standing mathematical conjecture, the response was underwhelming.

Representation theory was initially dismissed. Today, it’s central to much of mathematics.

Many of them are spinning at nearly the speed of light. When you do the math, there’s no other way it could have been.

When Plato set out to define what made a human a human, he settled on two primary characteristics: We do not have feathers, and we are bipedal (walking upright on two legs). Plato’s characterization may not encompass all of what identifies a human, but his reduction of an object to its fundamental characteristics provides an example of a technique known as principal component analysis.

Now, Caltech researchers have combined tools from machine learning and neuroscience to discover that the brain uses a mathematical system to organize visual objects according to their principal components. The work shows that the brain contains a two-dimensional map of cells representing different objects. The location of each cell in this map is determined by the principal components (or features) of its preferred objects; for example, cells that respond to round, curvy objects like faces and apples are grouped together, while cells that respond to spiky objects like helicopters or chairs form another group.

The research was conducted in the laboratory of Doris Tsao (BS ‘96), professor of biology, director of the Tianqiao and Chrissy Chen Center for Systems Neuroscience and holder of its leadership chair, and Howard Hughes Medical Institute Investigator. A paper describing the study appears in the journal Nature on June 3.

Continue reading “Faces, Bodies, Spiders, and Radios: How the Brain Represents Visual Objects” »

A Greek mathematician has found the answer to a mind boggling maths problem that has remained unanswered for 78 years – until now.

Associate Professor of Mathematics Dimitris Koukoulopoulos together with Oxford University research professor James Maynard, has solved the Duffin and Schaeffer Conjecture.

First expressed in 1941 by mathematicians R J Duffin and A C Schaeffer, the last time a mathematician showed promise in solving the problem was in 1990. But it wouldn’t be until 29 years later that it would be fully proven by Koukoulopoulos and Maynard – two relatively young mathematicians, both aged in their 30’s.

Continue reading “The mystery is over: Greek mathematician successfully solves 78-year-old maths problem” »