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Category: mathematics – Page 157

Ever really wanted to know what folks truly are thinking about?

A new experiment advances the idea that brain scans can teach us something about how the human mind works.

By Nathan Collins

Mind reading stands as one of science fiction’s most enduring improbabilities, alongside light-speed space travel and laser guns. But unlike those latter two, mind reading actually has a whiff of reality: In a new demonstration, psychologists have shown they can figure out how far along someone’s brain is in the process of solving a sophisticated math problem—a result that, more than anything else, indicates the promise of new brain-scanning techniques for understanding the human mind.

Continue reading “Can a Brain Scan Tell What You’re Thinking? — Pacific Standard” | >

Her computer, Karin Strauss says, contains her “digital attic”—a place where she stores that published math paper she wrote in high school, and computer science schoolwork from college.

She’d like to preserve the stuff “as long as I live, at least,” says Strauss, 37. But computers must be replaced every few years, and each time she must copy the information over, “which is a little bit of a headache.”

It would be much better, she says, if she could store it in DNA—the stuff our genes are made of.

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The Defense Advanced Research Projects Agency has demonstrated a new mathematical framework that works to help researchers discover patterns in complex scientific and engineering systems. DARPA said Thursday researchers at Stanford University created algorithms designed to explore patterns in data in order to gain insights into network structure and function under the Simplifying Complexity in Scientific Discovery [ ].

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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.

Continue reading “Mathematical framework that prioritizes key patterns in networks aims to accelerate scientific discovery” | >

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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