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Many ways to approach the Riemann Hypothesis have been proposed during the past 150 years, but none of them have led to conquering the most famous open problem in mathematics. A new paper in the Proceedings of the National Academy of Sciences (PNAS) suggests that one of these old approaches is more practical than previously realized.

“In a surprisingly short proof, we’ve shown that an old, abandoned approach to the Riemann Hypothesis should not have been forgotten,” says Ken Ono, a number theorist at Emory University and co-author of the paper. “By simply formulating a proper framework for an old approach we’ve proven some new theorems, including a large chunk of a criterion which implies the Riemann Hypothesis. And our general framework also opens approaches to other basic unanswered questions.”

The paper builds on the work of Johan Jensen and George Pólya, two of the most important mathematicians of the 20th century. It reveals a method to calculate the Jensen-Pólya polynomials—a formulation of the Riemann Hypothesis—not one at a time, but all at once.

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A mathematician from the University of Bristol has found a solution to part of a 64-year old mathematical problem – expressing the number 33 as the sum of three cubes.

Since the 1950s, mathematicians have wondered if all whole numbers could be expressed as the sum of three cubes; whether the equation k = x³+ y³+ z³ always has a solution.

Continue reading “Bristol mathematician cracks Diophantine puzzle” »

A prime number theory equation by mathematics professor emeritus Carl Pomerance turned up on The Big Bang Theory, where it was scrawled on a white board in the background of the hit sitcom about a group of friends and roommates who are scientists, many of them physicists at the California Institute of Technology.

In a recent paper, “Proof of the Sheldon Conjecture,” Pomerance, the John G. Kemeny Parents Professor of Mathematics Emeritus, does the math on a claim by fictional quantum physicist Sheldon Cooper that 73 is “the best number” because of several unique properties. Pomerance’s proof shows that 73 is indeed unique.

The Big Bang Theory is known for dressing the set with “Easter eggs” to delight the self-avowed science nerds in the audience. When UCLA physics professor David Saltzberg, technical consultant for The Big Bang Theory, heard about the Sheldon proof, he contacted Pomerance to ask if they could use it in the show, which was broadcast April 18.

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Mathematicians used “magic functions” to prove that two highly symmetric lattices solve a myriad of problems in eight- and 24-dimensional space.

Quanta’s In Theory video series returns with an exploration of a mysterious mathematical pattern found throughout nature.

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.

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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.