Lifeboat Foundation

Andrew Wiles’ proof of Fermat’s Last Theorem is a famous example. Pierre de Fermat claimed in 1637 – in the margin of a copy of “Arithmetica,” no less – to have solved the Diophantine equation xⁿ + yⁿ = zⁿ, but offered no justification. When Wiles proved it over 300 years later, mathematicians immediately took notice. If Wiles had developed a new idea that could solve Fermat, then what else could that idea do? Number theorists raced to understand Wiles’ methods, generalizing them and finding new consequences.

No single method exists that can solve all Diophantine equations. Instead, mathematicians cultivate various techniques, each suited for certain types of Diophantine problems but not others. So mathematicians classify these problems by their features or complexity, much like biologists might classify species by taxonomy.

Holograms are a staple in science fiction, but creating ones detailed enough to have serious applications in the real world has proved difficult. While scientists have been slowly pushing the field of holographic projection forward, they haven’t been able to overcome a problem called cross-talk. However, in a recent paper published in Nature, they have been able to manipulate the shape of light to overcome this, thus allowing them to produce 3D holograms that are orders of magnitude clearer, larger, and more detailed.

What Are Holograms?

Simple holograms are 2D surfaces that produce the illusion of a 3D object when light is shined through it.

Continue reading “Solving a Math Problem Just Brought Holograms Closer to Reality” »

The multiplication of integers is a problem that has kept mathematicians busy since Antiquity. The “Babylonian” method we learn at school requires us to multiply each digit of the first number by each digit of the second one. But when both numbers have a billion digits each, that means a billion times a billion or 10¹⁸ operations.

At a rate of a billion operations per second, it would take a computer a little over 30 years to finish the job. In 1971, the mathematicians Schönhage and Strassen discovered a quicker way, cutting calculation time down to about 30 seconds on a modern laptop. In their article, they also predicted that another algorithm—yet to be found—could do an even faster job. Joris van der Hoeven, a CNRS researcher from the École Polytechnique Computer Science Laboratory LIX, and David Harvey from the University of New South Wales (Australia) have found that algorithm.

They present their work in a new article that is available to the scientific community through the online HAL archive. But one problem raised by Schönhage et Strassen remains to be solved: proving that no quicker method exists. This poses a new challenge for theoretical computer science.

Continue reading “A faster method for multiplying very big numbers” »

High-performance computing (HPC)—the use of supercomputers and parallel processing techniques to solve large computational problems—is of great use in the scientific community. For example, scientists at the U.S. Department of Energy’s (DOE) Brookhaven National Laboratory rely on HPC to analyze the data they collect at the large-scale experimental facilities on site and to model complex processes that would be too expensive or impossible to demonstrate experimentally.

Modern science applications, such as simulating particle interactions, often require a combination of aggregated computing power, high-speed networks for data transfer, large amounts of memory, and high-capacity storage capabilities. Advances in HPC hardware and software are needed to meet these requirements. Computer and computational scientists and mathematicians in Brookhaven Lab’s Computational Science Initiative (CSI) are collaborating with physicists, biologists, and other domain scientists to understand their data analysis needs and provide solutions to accelerate the scientific discovery process.

Gravitational echoes may be caused by the collision of two black holes, and may indicate that these objects have completely new physical properties. This conclusion was made by RUDN physicists after a series of mathematical calculations. The scientists state that if the existence of the echo phenomenon is confirmed, astrophysicists would have to reconsider their view of compact space objects. The results of the study were published in Physical Review D.

According to the theory of general relativity (GR), any massive object distorts space-time. A similar effect is observed when a heavy metal ball is placed on stretched elastic fabric. The heavier is the ball, the deeper is the depression in the fabric. Similarly, the higher the mass of an object, the more it distorts space-time. Black holes are among the heaviest objects in the universe, and therefore distort space-time the most. When two black holes collide, gravitational waves spread out from the site of collision. They can be compared to rings on the water, or sound waves, but there is one important peculiar feature. Gravitational waves do not propagate spatially—they are themselves the oscillations of space-time.

Gravitational waves from the collision of two black holes decay with time, but on their final stage, they can cause the so-called echo—additional wave scattering. It can be compared to regular acoustic echo. The existence of such gravitational echo has not been confirmed yet, and there are different opinions about its possible source. A RUDN physicist, together with colleagues from the Czech Republic and Russia, assumed that if the existence of gravitational echo is experimentally confirmed, it would be the beginning of the new physics adding to GR.

Continue reading “Gravitational echo phenomenon will become a key to the new physics, physicist says” »

Mathematicians have confirmed that humanity’s collective attention span is getting shorter. And it’s not just social media that’s to blame.

Research reveals ‘social acceleration’ occurring across different domains. Samantha Page reports.

https://iopscience.iop.org/article/10.3847/2041-8213/ab0ec7

For Karen Uhlenbeck, winner of the Abel Prize for math, a whimsical phenomenon offers a window onto higher dimensions.

Credit Credit Kym Cox/Science Source

Most Hackaday readers are no doubt familiar with the Faraday cage, at least in name, and nearly everyone owns one: if you’ve ever stood watching a bag of popcorn slowly revolve inside of a microwave, you’be seen Michael Faraday’s 1836 invention in action. Yet despite being such a well known device, the average hacker still doesn’t have one in their arsenal. But why?

It could be that there’s a certain mystique about Faraday cages, an assumption that their construction requires techniques or materials outside the realm of the home hacker. While it’s true that building a perfect Faraday cage for a given frequency involves math and careful attention to detail, putting together a simple model for general purpose use and experimentation turns out to be quick and easy.

As an exercise in minimalist hacking I recently built a basic Faraday cage out of materials sourced from Home Depot, and thought it would be interesting to not only describe its construction but give some ideas as to how one can put it to practical use in the home lab. While it’s hardly a perfect specimen, it clearly works, and it didn’t take anything that can’t be sourced locally pretty much anywhere in the world.

Continue reading “Building a Hardware Store Faraday Cage” »