Lifeboat Foundation

RUDN University mathematicians have proven the Hardy-Littlewood-Sobolev (HLS) inequalities for the class of generalized Riesz potentials. These results extend the scope of these potentials in mathematics and physics because the main tools for working with such potentials are based on HLS inequalities. New mathematical tools can greatly simplify calculations in quantum mechanics and other fields of physics. The results of the study are published in the journal Mathematical Notes.

Modern physics describes the world in terms of fields and their potentials—that is, the values of the field at each point. But the physical quantities that we can measure are forces and accelerations, that is, derivatives of the second-order of the potential of the corresponding field. The problem of reconstructing the field configuration with the available values of forces and accelerations observed in experiments is complex and not always analytically solvable. Differentiation operations in multidimensional space—operators are usually used to describe the correlation between the potential of the field and the forces. In particular, electromagnetic and gravitational interactions are described in the language of operators.

Since the potential of the field can be determined up to a constant value, for the convenience of calculations, the initial value of the potential is taken at some point in multidimensional space, or on the border of any spatial area. But in some cases, mathematical models of such fields lead to a singularity, that is, at some points the value of the field becomes infinite, and therefore loses its physical meaning.

AMOLF researchers and their collaborators from the Advanced Science Research Center (ASRC/CUNY) in New York have created a nanostructured surface capable of performing on-the-fly mathematical operations on an input image. This discovery could boost the speed of existing imaging processing techniques and lower energy usage. The work enables ultrafast object detection and augmented reality applications. The researchers publish their results today in the journal Nano Letters.

Image processing is at the core of several rapidly growing technologies, such as augmented reality, autonomous driving and more general object recognition. But how does a computer find and recognize an object? The initial step is to understand where its boundaries are, hence edge detection in an image becomes the starting point for image recognition. Edge detection is typically performed digitally using integrated electronic circuits implying fundamental speed limitations and high energy consumption, or in an analog fashion which requires bulky optics.

In 1884, a schoolmaster and theologian named Edwin Abbott wrote a novella called Flatland, which tells the story of a world populated by sentient two-dimensional shapes. While intended as a satire of rigid Victorian social norms, Flatland has long fascinated mathematicians and physicists and served as the setting for many a thought experiment.

One such thought experiment: How can light be controlled in two dimensions?

When a wave of light is confined on a two-dimensional plane by certain materials, it becomes something known as a polariton—a particle that blurs the distinction between light and matter. Polaritons have exciting implications for the future of optical circuits because, unlike electronic integrated circuits, integrated optics is difficult to miniaturize with commonly used materials. Polaritons allow light to be tightly confined to the nanoscale, even potentially to the thickness of a few atoms.

Too large to be classed as molecules, but too small to be bulk solids, atomic clusters can range in size from a few dozen to several hundred atoms. The structures can be used for a diverse range of applications, which requires a detailed knowledge of their shapes. These are easy to describe using mathematics in some cases; while in others, their morphologies are far more irregular. However, current models typically ignore this level of detail; often defining clusters as simple ball-shaped structures.

In research published in The European Physical Journal B, José M. Cabrera-Trujillo and colleagues at the Autonomous University of San Luis Potosí in Mexico propose a new method of identifying the morphologies of atomic clusters. They have now confirmed that the distinctive geometric shapes of some clusters, as well as the irregularity of amorphous structures, can be fully identified mathematically.

The insights gathered by Cabrera-Trujillo’s team could make it easier for researchers to engineer atomic clusters for specific applications. These could include nanoparticles containing two different metals, which are highly effective in catalysing chemical reactions. Their updated methods provided new ways to determine the structural properties of clusters, the ways in which they convert energy to different forms, and the potential forces between atoms. The technique was also able to distinguish the surrounding environments of atoms in the cores of clusters, and on their surfaces. Ultimately, this allowed the researchers to distinguish between distinctive shapes, including icosahedrons, octahedrons, and simple pancakes. They were also able to identify amorphous shapes, which contain no discernible mathematical order.

This is amazing, it will save many lives!

A 14-year-old Pennsylvania girl has come up with an innovative way to get rid of blind spots before she can even legally get behind the wheel.

Continue reading “Genius, 14, wins $25,000 for car design that would install cameras to make blind spots nonexistent” »

A Portland teen won second place in a national technology contest, taking home $2,500 that he can use to attend science camp next summer.

Rishab Jain, 14, is a freshman at Westview High School. His winning project, which he calls the Pancreas Detective, is an artificial intelligence tool that can help diagnose pancreatic cancer through gene sequencing. The algorithm helps doctors focus on the organ during examinations, which is often obscured because it moves around the abdominal area as patients breathe and other bodily functions shift other organs as well.

Last year, the same project netted $25,000 from 3M when he attended Stoller Middle School. He used that money to fund his nonprofit, Samyak Science Society, which promotes science, technology, engineering and math education for other children, Time Magazine reported.

https://www.youtube.com/watch?v=iNqqOLscOBY

Sean Carroll is a theoretical physicist at Caltech and Santa Fe Institute specializing in quantum mechanics, arrow of time, cosmology, and gravitation. He is the author of several popular books including his latest on quantum mechanics (Something Deeply Hidden) and is a host of a great podcast called Mindscape. This conversation is part of the Artificial Intelligence podcast.

This is the second time Sean has been on the podcast. You can watch the first time here: https://www.youtube.com/watch?v=l-NJrvyRo0c

Continue reading “Sean Carroll: Quantum Mechanics and the Many-Worlds Interpretation” »

Mathematicians have uncovered a big new piece of evidence for one of the most famous unproven ideas in mathematics, known as the twin prime conjecture. But the route they took to finding that evidence probably won’t help prove the twin prime conjecture itself.

The twin prime conjecture is all about how and when prime numbers — numbers that are divisible only by themselves and 1 — appear on the number line. “Twin primes” are primes that are two steps apart from each other on that line: 3 and 5, 5 and 7, 29 and 31, 137 and 139, and so on. The twin prime conjecture states that there are infinitely many twin primes, and that you’ll keep encountering them no matter how far down the number line you go. It also states that there are infinitely many prime pairs with every other possible gap between them (prime pairs that are four steps apart, eight steps apart, 200,000 steps apart, etc.). Mathematicians are pretty sure this is true. It sure seems like it’s true. And if it weren’t true, it would mean that prime numbers aren’t as random as everyone thought, which would mess up lots of ideas about how numbers work in general. But no one’s ever been able to prove it.

Related: Mathematicians Edge Closer to Solving a ‘Million Dollar’ Math Problem.

The three-body problem has vexed mathematicians and physicists for 300 years, but AI can find solutions far faster than any other method anyone has come up with.