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Category: mathematics

Quantum simulation plays an irreplaceable role in diverse fields, beyond the scope of classical computers. In a recent study, Keren Li and an interdisciplinary research team at the Center for Quantum Computing, Quantum Science and Engineering and the Department of Physics and Astronomy in China, U.S. Germany and Canada. Experimentally simulated spin-network states by simulating quantum spacetime tetrahedra on a four-qubit nuclear magnetic resonance (NMR) quantum simulator. The experimental fidelity was above 95 percent. The research team used the quantum tetrahedra prepared by nuclear magnetic resonance to simulate a two-dimensional (2-D) spinfoam vertex (model) amplitude, and display local dynamics of quantum spacetime. Li et al. measured the geometric properties of the corresponding quantum tetrahedra to simulate their interactions. The experimental work is an initial attempt and a basic module to represent the Feynman diagram vertex in the spinfoam formulation, to study loop quantum gravity (LQG) using quantum information processing. The results are now available on Communication Physics.

Classical computers cannot study large quantum systems despite successful simulations of a variety of physical systems. The systematic constraints of classical computers occurred when the linear growth of quantum system sizes corresponded to the exponential growth of the Hilbert Space, a mathematical foundation of quantum mechanics. Quantum physicists aim to overcome the issue using quantum computers that process information intrinsically or quantum-mechanically to outperform their classical counterparts exponentially. In 1982, Physicist Richard Feynman defined quantum computers as quantum systems that can be controlled to mimic or simulate the behaviour or properties of relatively less accessible quantum systems.

In the present work, Li et al. used nuclear magnetic resonance (NMR) with a high controllable performance on the quantum system to develop simulation methods. The strategy facilitated the presentation of quantum geometries of space and spacetime based on the analogies between nuclear spin states in NMR samples and spin-network states in quantum gravity. Quantum gravity aims to unite the Einstein gravity with quantum mechanics to expand our understanding of gravity to the Planck scale (1.22 x 1019 GeV). At the Planck scale (magnitudes of space, time and energy) Einstein gravity and the continuum of spacetime breakdown can be replaced via quantum spacetime. Research approaches toward understanding quantum spacetimes are presently rooted in spin networks (a graph of lines and nodes to represent the quantum state of space at a certain point in time), which are an important, non-perturbative framework of quantum gravity.

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It is the name of the unknown theory of everything which would combine all five Superstring theories and the Supergravity at 11 dimensions together.

The theory requires mathematical tools which have yet to be invented in order to be fully understood. The theory was proposed by Edward Witten.

The following article is somewhat technical in nature, see M-theory simplified for a less technical article.

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iOS/Android/Desktop: Default calculator apps suck. They work like a traditional handheld calculator, which only displays one value at a time and can only do basic math. If you want to do anything more than calculate a tip, you’re better off with these free and cheap calculator apps.

These apps help you do typical “real life math” or solve basic textbook math problems.

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Cooperation means that one individual pays a cost for another to receive a benefit. Cooperation can be at variance with natural selection: Why should you help a competitor? Yet cooperation is abundant in nature and is an important component of evolutionary innovation. Cooperation can be seen as the master architect of evolution and as the third fundamental principle of evolution beside mutation and selection. I will present mathematical principles of cooperation.

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Circa 2012

Hundreds of the world’s brightest minds — engineers from Google and IBM, hedge funds quants, and Defense Department contractors building artificial intelligence — were gathered in rapt attention inside the auditorium of the San Francisco Masonic Temple atop Nob Hill. It was the first day of the seventh annual Singularity Summit, and Julia Galef, the President of the Center for Applied Rationality, was speaking onstage. On the screen behind her, Galef projected a giant image from the film Blade Runner: the replicant Roy, naked, his face stained with blood, cradling a white dove in his arms.

At this point in the movie, Roy is reaching the end of his short, pre-programmed life, “The poignancy of his death scene comes from the contrast between that bitter truth and the fact that he still feels his life has meaning, and for lack of a better word, he has a soul,” said Galef. “To me this is the situation we as humans have found ourselves in over the last century. Turns out we are survival machines created by ancient replicators, DNA, to produce as many copies of them as possible. This is the bitter pill that science has offered us in response to our questions about where we came from and what it all means.”

The Singularity Summit bills itself as the world’s premier event on robotics, artificial intelligence, and other emerging technologies. The attendees, who shelled out $795 for a two-day pass, are people whose careers depend on data, on empirical proof. Peter Norvig, Google’s Director of Research, discussed advances in probabilistic first-order logic. The Nobel prize-winning economist Daniel Kahneman lectured on the finer points of heuristics and biases in human psychology. The Power Point presentations were full of math equations and complex charts. Yet time and again the conversation drifted towards the existential: the larger, unanswerable questions of life.

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https://youtube.com/watch?v=gHBhMGbJHe8

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.

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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.

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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.”

Yeah, sure! Why not sell me that bridge in Brooklyn while you backtrack?

The new claim makes no sense at all—a retraction of an earlier claim about a discovery by a crack team of research scientists (pun intended). The clues offered in the original claim, which was issued just one day earlier, cast suspicion on the retraction. Something fishy is going on here. Who pressured DuPont into making the retraction—and for what purpose? Something smells rotten in Denmark!

Let’s deconstruct this mess by reviewing the basic facts:

  • Wall Street, financial services firm claims they have solved a de facto contest in math & logic
  • They cracked the code a year ago, yet— incredibly—kept it secret until this week
  • A day later (with no outside review or challenge),* they admit the year-old crack was flawed

Waitacottenpickensec, Mr. DuPont!! The flaw (an ‘equipment issue’) was discovered a year after the equipment was configured and used—but just one day after you finally decided to disclose their past discovery? Poppycock!

I am not given to conspiracy theories (a faked moon landing, suppressing perpetual motion technology, autism & vaccinations, etc)—But I recognize government pressure when I see it! Someone with guns and persuasion convinced DuPont to rescind the claim and offer a silly experimental error.

Consider the fallout, if SHA-256 were to suddenly lose public confidence…

  • A broken SHA-256 would wreak havoc on an entrenched market. SHA-256 is a foundational element in the encryption used by consumers & business
  • But for government, disclosing a crack to a ubiquitous standard that they previously discovered (or designed) would destroy a covert surveillance mechanism—because the market would move quickly to replace the compromised methodology.

I understand why DuPont would boast of an impressive technical feat. Cracking AES, SSL or SHA-256 has become an international contest with bragging rights. But, I cannot imagine a reason to wait one year before disclosing the achievement. This, alone, does not create a conundrum. Perhaps DuPont was truly concerned that it would undermine trust in everyday communications, financial transactions and identity/access verification…

But retracting the claim immediately after disclosing it makes no sense at all. There is only one rational explanation. The original claim undermines the interests of some entity that has the power or influence to demand a retraction. It’s difficult to look at this any other way.

What about the everyday business of TS DuPont?

If the purpose of the original announcement was to generate press for DuPont’s financial services, then they have succeeded. An old axiom says that any press is good press. In this case, I don’t think so! Despite the potential for increased name recognition (Who knew that any DuPont was into brokerage & financial services?) I am not likely to think positively of TS DuPont for my investment needs.

* The cryptographic community could not challenge DuPont’s original claim, because it was not accompanied by any explanation of tools, experimental technique or mathematical methodology. Recognizing that SHA-256 is baked into the global infrastructure banking, of commerce and communications, their opaque announcement was designed to protect the economy. Thank you, Mr. DuPont, for being so noble!

Philip Raymond co-chairs CRYPSA, hosts the Bitcoin Event and is keynote speaker at Cryptocurrency Conferences. He is a top writer at Quora.

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:

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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.

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