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Category: mathematics

Nice.

Networks are mathematical representations to explore and understand diverse, complex systems—everything from military logistics and global finance to air traffic, social media, and the biological processes within our bodies. In each of those systems, a hierarchy of recurring, meaningful internal patterns—such as molecules and proteins interacting inside cells, and capacitors and resistors operating within integrated circuits—determines the functions or behaviors of those systems. The larger and more intricate a system is, however, the harder it is for current network modeling techniques to uncover these patterns and represent them in organized, easy-to-understand ways.

Researchers at Stanford University, funded by DARPA’s Simplifying Complexity in Scientific Discovery (SIMPLEX) program, have made progress in overcoming these challenges through a framework they have developed for identifying and clustering what mathematicians call “motifs”: essential but often obscure patterns within systems that are the building blocks of mathematical modeling and that facilitate the computational representation of complex systems.

A research paper describing the team’s achievement was published in Science (“Higher-order organization of complex networks”). At the heart of the team’s success was the creation of algorithms that can automatically explore and prioritize the hidden patterns in data that are fundamental to explaining network structure and function.

a mathematical framework that automatically identifies and prioritizes the patterns that are fundamental to explaining network structure and function

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Thomas Aquinas and other ludicrous pseudo-philosophers (in contradistinction with real philosophers such as Abelard) used to ponder questions about angels, such as whether they can interpenetrate (as bosons do).

Are today’s mathematicians just as ridiculous? The assumption of infinity has been “proven” by the simplest reasoning ever: if n is the largest number, clearly, (n+1) is larger. I have long disagreed with that hare-brained sort of certainty, and it’s not a matter of shooting the breeze. (My point of view has been spreading in recent years!) Just saying something exists, does not make it so (or then one would believe Hitler and Brexiters). If I say:” I am emperor of the galaxy known as the Milky Way!” that has a nice ring to it, but it does not make it so (too bad, that would be fun).

Given n symbols, each labelled by something, can one always find a new something to label (n+1) with? I say: no. Why? Because reality prevents it. Somebody (see below) objected that I confused “map” and “territory”. But I am a differential geometer, and the essential idea there, from the genius B. Riemann, is that maps allow to define “territory”:

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A transistor, conceived of in digital terms, has two states: on and off, which can represent the 1s and 0s of binary arithmetic.

But in terms, the transistor has an infinite number of states, which could, in principle, represent an infinite range of mathematical values. Digital computing, for all its advantages, leaves most of transistors’ informational capacity on the table.

In recent years, analog computers have proven to be much more efficient at simulating biological systems than digital computers. But existing analog computers have to be programmed by hand, a complex process that would be prohibitively time consuming for large-scale simulations.

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In 2009, President Obama pledged to “restore science to its rightful place.” He said, “We will not just meet, but we will exceed the level achieved at the height of the space race, through policies that invest in basic and applied research, create new incentives for private innovation, promote breakthroughs in energy and medicine, and improve education in math and science.”

Today, the White House released an Impact Report listing 100 things that Obama has made happen with the support of many people across research, policy, education, and, yes, maker culture. Here’s the full Impact Report. A few examples from the list:

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When an astronomical observatory detected two black holes colliding in deep space, scientists celebrated confirmation of Einstein’s prediction of gravitational waves. A team of astrophysicists wondered something else: Had the experiment found the “dark matter” that makes up most of the mass of the universe?

The eight scientists from the Johns Hopkins Henry A. Rowland Department of Physics and Astronomy had already started making calculations when the discovery by the Laser Interferometer Gravitational-Wave Observatory (LIGO) was announced in February. Their results, published recently in Physical Review Letters, unfold as a hypothesis suggesting a solution for an abiding mystery in astrophysics.

“We consider the possibility that the black hole binary detected by LIGO may be a signature of dark matter,” wrote the scientists in their summary, referring to the black hole pair as a “binary.” What follows are five pages of annotated mathematical equations showing how the researchers considered the mass of the two objects LIGO detected as a point of departure, suggesting that these objects could be part of the mysterious substance known to make up about 85 percent of the mass of the universe.

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For the first time, scientists have discovered a classic formula for pi in the world of quantum physics. Pi is the ratio between a circle’s circumference and its diameter, and is incredibly important in pure mathematics, but now scientists have also found it “lurking” in the world of physics, when using quantum mechanics to compare the energy levels of a hydrogen atom.

Why is that exciting? Well, it reveals an incredibly special and previously unknown connection between quantum physics and maths.

“I find it fascinating that a purely mathematical formula from the 17th century characterises a physical system that was discovered 300 years later,” said one of the lead researchers, Tamar Friedmann, a mathematician at the University of Rochester in the US. Seriously, wow.

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(Phys.org)—One of the most ambitious endeavors in quantum physics right now is to build a large-scale quantum network that could one day span the entire globe. In a new study, physicists have shown that describing quantum networks in a new way—as mathematical graphs—can help increase the distance that quantum information can be transmitted. Compared to classical networks, quantum networks have potential advantages such as better security and being faster under certain circumstances.

“A worldwide network may appear quite similar to the internet—a huge number of devices connected in a way that allows the exchange of information between any of them,” coauthor Michael Epping, a physicist at the University of Waterloo in Canada, told Phys.org. “But the crucial difference is that the laws of quantum theory will be dominant for the description of that information. For example, the state of the fundamental information carrier can be a superposition of the basis states 0 and 1. By now, several advantages in comparison to classical information are known, such as prime number factorization and secret communication. However, the biggest benefit of quantum networks might well be discovered by future research in the rapidly developing field of theory.”

Quantum networks involve sending entangled particles across long distances, which is challenging because particle loss and decoherence tend to scale exponentially with the distance.

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Wow!

Mobile phone data may reveal an underlying mathematical connection between how we move and how we communicate that could make it easier to predict how diseases—and even ideas—spread through a population, according to an international team of researchers.

“This study really deepens our quantitative understanding of human behavior,” said Dashun Wang, assistant professor of and technology, Penn State. “We would like to think that we control our own behavior and we can do what we want to do. But, what we are starting to see with is that there is a very deep regularity underlying much of what we do.”

In a study, location and communication data collected from three international carriers showed that people move and communicate in predictable patterns, said Wang.

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The writer is referring to D-Wave (not Dwave) in his article.

Dwave Systems and 1QB Information Technologies Inc. (1QBit), a quantum software firm, and financial industry experts today announced the launch of Quantum for Quants (quantumforquants.org), an online community designed specifically for quantitative analysts and other experts focused on complex problems in finance. Launched at the Global Derivatives Trading and Risk Management conference in Budapest, the online community will allow quantitative finance and quantum computing professionals to share ideas and insights regarding quantum technology and to explore its application to the finance industry. Through this community financial industry experts will also be granted access to quantum computing software tools, simulators, and other resources and expertise to explore the best ways to tackle the most difficult computational problems in finance using entirely new techniques.

“Quantum computers enable us to use the laws of physics to solve intractable mathematical problems,” said Marcos López de Prado, Senior Managing Director at Guggenheim Partners and a Research Fellow at Lawrence Berkeley National Laboratory’s Computational Research Division. “This is the beginning of a new era, and it will change the job of the mathematician and computer scientist in the years to come.”

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Math isn’t everyone’s strong suit, especially those who haven’t stretched that part of their brain since college. Thanks to the wonders of image recognition technology, we now have Mathpix, an iOS app that lets you point your phone camera at a problem and calculates solutions in seconds.

The interface looks like any standard camera app: simply drag the on-screen reticle over the equation and the app solves it and provides graph answers where appropriate. More useful is a step-by-step guide offering multiple methods to reach a solution, making this a bona fide educational tool. It uses image recognition to process problems and pings its servers to do the mathematical heavy lifting, so it likely requires an internet connection to work.

Mathpix was envisioned by Stanford PhD student Nico Jimenez, who was advised by Stanford grad Paul Ferrell. The app’s other developers are high schoolers Michael Lee and August Trollback, which is impressive for an app that claims to be the first to visually recognize and solve handwritten math problems.

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