## Archive for the ‘mathematics’ category: Page 77

In music, “portamento” is a term that’s been used for hundreds of years, referring to the effect of gliding a note at one pitch into a note of a lower or higher pitch. But only instruments that can continuously vary in pitch—such as the human voice, string instruments, and trombones—can pull off the effect.

Now an MIT student has invented a novel algorithm that produces a portamento effect between any two audio signals in real-time. In experiments, the algorithm seamlessly merged various audio clips, such as a piano note gliding into a human voice, and one song blending into another. His paper describing the algorithm won the “best student paper” award at the recent International Conference on Digital Audio Effects.

The algorithm relies on “optimal transport,” a geometry-based framework that determines the most efficient ways to move objects—or data points—between multiple origin and destination configurations. Formulated in the 1700s, the framework has been applied to supply chains, fluid dynamics, image alignment, 3D modeling, , and more.

Linear algebra is a field of mathematics that has been thoroughly investigated for many centuries, providing invaluable tools used not only in mathematics, but also across physics and engineering as well as many other fields. For years physicists have used important theorems in linear algebra to quickly calculate solutions to the most complicated problems.

This August, three theoretical physicists—Peter Denton, a scientist at Brookhaven National Laboratory and a scholar at Fermilab’s Neutrino Physics Center; Stephen Parke, at Fermilab; and Xining Zhang, a University of Chicago graduate student working under Parke—turned the tables and, in the context of particle physics, discovered a fundamental in .

The identity relates eigenvectors and eigenvalues in a direct way that hadn’t been previously recognized. Eigenvectors and eigenvalues are two important ways of reducing the properties of a matrix to their most basic components and have applications in many math, physics and real-world contexts, such as in analyzing vibrating systems and facial recognition programs. The eigenvectors identify the directions in which a transformation occurs, and the eigenvalues specify the amount of stretching or compressing that occurs.

SHA-256 is a one way hashing algorithm. Cracking it would have tectonic implications for consumers, business and all aspects of government including the military.

It’s not the purpose of this post to explain encryption, AES or SHA-256, but here is a brief description of SHA-256. Normally, I place reference links in-line or at the end of a post. But let’s get this out of the way up front:

One day after Treadwell Stanton DuPont claimed that a secret project cracked SHA-256 more than one year ago, they back-tracked. Rescinding the original claim, they announced that an equipment flaw caused them to incorrectly conclude that they had algorithmically cracked SHA-256.

All sectors can still sleep quietly tonight,” said CEO Mike Wallace. “Preliminary results in this cryptanalytic research led us to believe we were successful, but this flaw finally proved otherwise.

Generally girls lose interest in STEM careers as they get older. But, according to a new study, small changes at school and at home can have a profound impact on how girls perceive STEM careers, how confident they feel in class and how likely they are to pursue STEM academically and into their careers.

The study, “Closing the STEM Gap,” published today by Microsoft, surveyed more than 6,000 girls and young women on their interests and perceptions of science, technology, engineering and math. It found that girls tended to lose interest in STEM as they headed toward adulthood. And, by the time they’d finished high school, their interest had dropped substantially. For example, the report found that interest in computer science among females dropped 27 percentage points between middle school and college. According to the report: “In middle school … 31 percent of girls believe that jobs requiring coding and programming are ‘not for them.’ In high school, that percentage jumps up to 40. By the time they’re in college, 58 percent of girls count themselves out of these jobs.”

But, the study found, countermeasures both large and small can have a profound effect, including:

The Electric Coin Company (ECC) says it discovered a new way to scale blockchains with “recursive proof composition,” a proof to verify the entirety of a blockchain in one function. For the ECC and zcash, the new project, Halo, may hold the key to privacy at scale.

A privacy coin based on zero-knowledge proofs, referred to as zk-SNARKs, zcash’s current underlying protocol relies on “trusted setups.” These mathematical parameters were used twice in zcash’s short history: upon its launch in 2016 and first large protocol change, Sapling, in 2018.

Zcash masks transations through zk-SNARKs but the creation of initial parameters remains an issue. By not destroying a transaction’s mathematical foundation – the trusted setup – the holder can produce forged zcash.

Breakthroughs in physics sometimes require an assist from the field of mathematics—and vice versa.

Using a global network of computers, mathematicians have finally solved a decades-old math conundrum — and possibly the meaning of life.

Women innovators across the United States have been selected as AAAS IF/THEN® Ambassadors by the American Association for the Advancement of Science and Lyda Hill Philanthropies to share their stories and serve as high-profile role models for middle-school girls.

IF/THEN®, a national initiative of Lyda Hill Philanthropies, seeks to further women in science, technology, engineering and math by empowering current innovators and inspiring the next generation of pioneers.

An exotic physical phenomenon, involving optical waves, synthetic magnetic fields, and time reversal, has been directly observed for the first time, following decades of attempts. The new finding could lead to realizations of what are known as topological phases, and eventually to advances toward fault-tolerant quantum computers, the researchers say.

The new finding involves the non-Abelian Aharonov-Bohm Effect and is published in the journal Science by MIT graduate student Yi Yang, MIT visiting scholar Chao Peng (a professor at Peking University), MIT graduate student Di Zhu, Professor Hrvoje Buljan at University of Zagreb in Croatia, Francis Wright Davis Professor of Physics John Joannopoulos at MIT, Professor Bo Zhen at the University of Pennsylvania, and MIT professor of physics Marin Soljačić.

The finding relates to gauge fields, which describe transformations that particles undergo. Gauge fields fall into two classes, known as Abelian and non-Abelian. The Aharonov-Bohm Effect, named after the theorists who predicted it in 1959, confirmed that gauge fields — beyond being a pure mathematical aid — have physical consequences.

Quantum many-body systems (QMBs), which are physical systems made up of multiple interacting particles, are among the most challenging structures to reproduce in numerical simulations. In the past, researchers have attempted to simulate these systems using a variety of techniques, including Monte Carlo simulations and even exact diagonalizations.

Methods involving networks (TNs), mathematical concepts that can be applied in a variety of scientific fields, have also shown some potential for the simulation of QMBs. However, so far, these techniques have only been successfully applied to small systems or those with a simple geometry.

In a recent study, researchers at the University of Central Florida were able to simulate QMBs on Amazon Web Services using a TN-based method. Their paper, pre-published on arXiv, highlights some of the potential advantages and implications of using for research purposes.

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