Mathematicians used “magic functions” to prove that two highly symmetric lattices solve a myriad of problems in eight- and 24-dimensional space.
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Category: mathematics – Page 144
Given that everything at its base atom is moving maybe our interpretation of reality may be different than its actuality. From shooting photons bouncing off surfaces the world is a cacophony of all sorts of things happening at once.
A provocative new column in Scientific American floats the idea that what’s fundamentally real in the universe — its actual, base reality — isn’t the quarks, fields, and quantum phenomena that seem to comprise it.
Instead, according to scientist and philosopher Bernardo Kastrup, some are starting to suspect that matter itself is an illusion — and that the only real thing is information.
Information Space
A team of researchers from the University of Central Florida and Michigan Technological University has developed a laser system concept built on the principles of supersymmetry. In their paper published in the journal Science, the group reports that their system is meant to solve the problem of producing more light with a compact laser system. Tsampikos Kottos with Wesleyan University has written a Perspective piece on the work done by the team in the same journal issue.
Kottos points out that there are a lot of physics applications that require use of a compact laser system that also has high-output power requirements. To fulfill this need, many physicists have taken to combining multiple lasers into an array. Unfortunately, this approach suffers from the production of a lesser-quality beam. Kottos notes that one way to overcome this problem is to use selective amplification of a single mode—but doing so has its own drawbacks. In this new effort, the researchers have come up with a different approach—one based on the principles of supersymmetry.
Supersymmetry is a math-based theory that describes the relationship between bosons and fermions—it suggests that for every known elementary particle, there has to be a much heavier “super partner.” To build a new kind of laser system, the researchers used this idea to create a stable array of semiconductor lasers that together offer the power needed for prospective applications. More specifically, they designed a system that emphasizes the fundamental mode by suppressing higher-order modes. They did this by pairing them with low-quality modes—their lossy super-partners. The idea was for the array to support them such that they were phase-matched with the higher order modes.
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In air-breathing vertebrates, the circulatory and pulmonary systems contain separate networks of channels that intertwine but do not intersect with each other. Recreating such structures within cell-compatible materials has been a major challenge; even a single vasculature system can be a burden to create. Grigoryan et al. show that natural and synthetic food dyes can be used as photoabsorbers that enable stereolithographic production of hydrogels containing intricate and functional vascular architectures. Using this approach, they demonstrate functional vascular topologies for studies of fluid mixers, valves, intervascular transport, nutrient delivery, and host engraftment.
Science, this issue p. 458
Solid organs transport fluids through distinct vascular networks that are biophysically and biochemically entangled, creating complex three-dimensional (3D) transport regimes that have remained difficult to produce and study. We establish intravascular and multivascular design freedoms with photopolymerizable hydrogels by using food dye additives as biocompatible yet potent photoabsorbers for projection stereolithography. We demonstrate monolithic transparent hydrogels, produced in minutes, comprising efficient intravascular 3D fluid mixers and functional bicuspid valves. We further elaborate entangled vascular networks from space-filling mathematical topologies and explore the oxygenation and flow of human red blood cells during tidal ventilation and distension of a proximate airway. In addition, we deploy structured biodegradable hydrogel carriers in a rodent model of chronic liver injury to highlight the potential translational utility of this materials innovation.
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With the possibility of millions or an infinite number of problems automating everything will cause all things to be solved digitally into a simple math problem. The problems could essentially be hacked by shores algorithm or maybe a theory of everything like m theory or Stephen Hawking’s theory of everything. Maybe it is just as simple as a basic formula like Einstein created E=mc2. Also like some mathematicians have theorized maybe just one line of code that solves everything.
Automation is a game-changer for modern problem-solving – enabling not only visibility to real-time operations but the ability to effectively project the impact of potential solutions into the future. As problem-solvers become more comfortable using the new tools available to them, companies will be able to effectively isolate (and avoid) the impact of problems to their operations and focus their resources on solving the underlying issues and enabling long-term success. Learn More here.
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We all know that maths is really hard. So hard, in fact, that there’s literally a whole Wikipedia page dedicated to unsolved mathematical problems, despite some of the greatest minds in the world working on them around the clock.
But as Avery Thompson points out at Popular Mechanics, from the outset at least, some of these problems seem surprisingly simple — so simple, in fact, that anyone with some basic maths knowledge can understand them… including us. Unfortunately, it turns out that proving them is a little harder.
Inspired by Thompson’s list, we’ve come up with our own list of deceptively simple maths problems to frustrate (and hopefully inspire) you.
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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.
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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.
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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.
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