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Category: mathematics – Page 10

Quantum Mechanics

Quantum mechanics is, at least at first glance and at least in part, a mathematical machine for predicting the behaviors of microscopic particles — or, at least, of the measuring instruments we use to explore those behaviors — and in that capacity, it is spectacularly successful: in terms of power and precision, head and shoulders above any theory we have ever had. Mathematically, the theory is well understood; we know what its parts are, how they are put together, and why, in the mechanical sense (i.e., in a sense that can be answered by describing the internal grinding of gear against gear), the whole thing performs the way it does, how the information that gets fed in at one end is converted into what comes out the other. The question of what kind of a world it describes, however, is controversial; there is very little agreement, among physicists and among philosophers, about what the world is like according to quantum mechanics. Minimally interpreted, the theory describes a set of facts about the way the microscopic world impinges on the macroscopic one, how it affects our measuring instruments, described in everyday language or the language of classical mechanics. Disagreement centers on the question of what a microscopic world, which affects our apparatuses in the prescribed manner, is, or even could be, like intrinsically; or how those apparatuses could themselves be built out of microscopic parts of the sort the theory describes.[1]

That is what an interpretation of the theory would provide: a proper account of what the world is like according to quantum mechanics, intrinsically and from the bottom up. The problems with giving an interpretation (not just a comforting, homey sort of interpretation, i.e., not just an interpretation according to which the world isn’t too different from the familiar world of common sense, but any interpretation at all) are dealt with in other sections of this encyclopedia. Here, we are concerned only with the mathematical heart of the theory, the theory in its capacity as a mathematical machine, and — whatever is true of the rest of it — this part of the theory makes exquisitely good sense.

Crows can recognize geometric regularity

A trio of animal physiologists at the University of Tübingen, in Germany, has found that at least one species of crow has the ability to recognize geometric regularity. In their study published in the journal Science Advances, Philipp Schmidbauer, Madita Hahn and Andreas Nieder conducted several experiments that involved testing crows on their ability to recognize geometric shapes.

Recognizing regularity in geometric shapes means being able to pick out one that is different from others in a group—picking out a plastic star, for example, when it is placed among several plastic moons. Testing for the ability to recognize geometric regularity has been done with many animals, including chimps and bonobos. Until now, this ability has never been observed in any creature except for humans.

Because of that, the team started with a bit of skepticism when they began testing carrion crows. In their work, the testing was done using computer screens—the birds were asked to peck the outlier in a group; if they chose correctly, they got a food treat. The team chose to test carrion crows because prior experiments have shown them to have exceptional intelligence and mathematical capabilities.

Proving quantum computers have the edge

Quantum computers promise to outperform today’s traditional computers in many areas of science, including chemistry, physics, and cryptography, but proving they will be superior has been challenging. The most well-known problem in which quantum computers are expected to have the edge, a trait physicists call “quantum advantage,” involves factoring large numbers, a hard math problem that lies at the root of securing digital information.

In 1994, Caltech alumnus Peter Shor (BS ‘81), then at Bell Labs, developed a that would easily factor a large number in just seconds, whereas this type of problem could take a classical computer millions of years. Ultimately, when quantum computers are ready and working—a goal that researchers say may still be a decade or more away—these machines will be able to quickly factor large numbers behind cryptography schemes.

But, besides Shor’s algorithm, researchers have had a hard time coming up with problems where quantum computers will have a proven advantage. Now, reporting in a recent Nature Physics study titled “Local minima in ,” a Caltech-led team of researchers has identified a common physics problem that these futuristic machines would excel at solving. The problem has to do with simulating how materials cool down to their lowest-energy states.

The Continuum Hypothesis — The Problem that BROKE Mathematics

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Getting an all-optical AI to handle non-linear math

A standard digital camera used in a car for stuff like emergency braking has a perceptual latency of a hair above 20 milliseconds. That’s just the time needed for a camera to transform the photons hitting its aperture into electrical charges using either CMOS or CCD sensors. It doesn’t count the further milliseconds needed to send that information to an onboard computer or process it there.

A team of MIT researchers figured that if you had a chip that could process photons directly, you could skip the entire digitization step and perform calculations with the photons themselves, which has the potential to be mind-bogglingly faster.

“We’re focused on a very specific metric here, which is latency. We aim for applications where what matters the most is how fast you can produce a solution. That’s why we are interested in systems where we’re able to do all the computations optically,” says Saumil Bandyopadhyay, an MIT researcher. The team implemented a complete deep neural network on a photonic chip, achieving a latency of 410 picoseconds. To put that in perspective, Bandyopadhyay’s chip could process the entire neural net it had onboard around 58 times within a single tick of the 4 GHz clock on a standard CPU.

Instead of sensing photons and processing the results, why not process the photons?

Frustration incorporated: How mismatched geometries can enhance material strength and toughness

Anyone who’s ever tried tiling a floor, a backsplash or even an arts-and-crafts project probably knows the emotional frustration of working with pieces whose shapes don’t perfectly complement each other. It turns out, though, that some creatures may actually rely on similar mismatches to create geometric frustrations that result in complex natural structures with remarkable properties, such as protective shells and sturdy yet flexible bones.

Now, researchers at the University of Michigan have developed mathematical models showing one way that nature achieves this. These models, in turn, could help design advanced materials for medical devices, sustainable construction and more.

“Frustration—using these mismatched building blocks—gives rise to wonderful complexity and that complexity can be useful in providing superior material properties,” said Xiaoming Mao, U-M professor of physics and senior author of the new study.

MIND-STUFF: How William Clifford Connected Geometry, Matter, and Mind — VERSADOCO

What if the key to the universe was discovered over a century ago—and then forgotten?

In the late 19th century, a young math prodigy named William Clifford proposed a radical idea: that reality itself is woven from the same fabric as the mind. Long before Einstein, long before quantum theory, Clifford envisioned a world where matter, consciousness, and geometry are one.

His ideas were largely overlooked, seen as too speculative for the science of his time. Today, they look like the missing blueprint for a true Theory of Everything.

Is Clifford’s path one that science is only now catching up to?

Based on the original research by idb.kniganews “Clifford’s Path”

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Mathematicians Uncover Hidden Patterns Behind $3.5 Billion Cryptocurrency Collapse

A recent study published in ACM Transactions on the Web by researchers at Queen Mary University of London sheds new light on one of the most significant collapses in cryptocurrency history: the crash of the TerraUSD stablecoin and its sister token, LUNA. The research team uncovered evidence of suspicious, large-scale trading activity that may point to a coordinated effort to destabilize the ecosystem, triggering a rapid $3.5 billion loss in market value.

Led by Dr. Richard Clegg, the study uses temporal multilayer graph analysis, an advanced method for tracking dynamic and interconnected systems over time. By applying this technique to transaction data from the Ethereum blockchain, the researchers were able to trace complex relationships between cryptocurrencies and pinpoint how TerraUSD was systematically undermined through a series of calculated trades.

Stablecoins like TerraUSD are designed to maintain a steady value, typically pegged to a fiat currency like the US dollar. However, in May 2022, TerraUSD and its sister currency, LUNA, experienced a catastrophic collapse. Dr. Clegg’s research sheds light on how this happened, uncovering evidence of a coordinated attack by traders who were betting against the system, a practice known as “shorting.”

Timeless Recursion: Nietzsche’s Eternal Return as Block Universe Prescience

Nietzsche’s intuition about time’s nature likely emerged from his engagement with contemporary scientific thought, particularly the work of Johann Friedrich Herbart and Roger Joseph Boscovich, whose atomistic theories influenced Nietzsche’s conception of force and matter (Small, 2001). Additionally, Nietzsche’s reading of Heinrich Czolbe and Otto Caspari exposed him to cyclical cosmological theories that were precursors to modern conceptions of cosmological cycles.

More compelling than these historical influences, however, is the philosophical insight Nietzsche demonstrated in recognizing that a truly eternal cosmos with finite configurations must contain repetition. This insight, while not formulated in the mathematical language of relativity, nevertheless grasped a fundamental consequence of infinite time and finite states — one that would later be encoded in physical theory.

The convergence between Nietzsche’s eternal recurrence and modern physics becomes even more significant when we recognize similar conceptions in numerous cultural and religious traditions. This suggests a perennial human intuition about time’s nature that transcends historical and cultural boundaries.

Scientists Are Mapping the Boundaries of What Is Knowable and Unknowable

“I give you God’s view,” said Toby Cubitt, a physicist turned computer scientist at University College London and part of the vanguard of the current charge into the unknowable, and “you still can’t predict what it’s going to do.”

Eva Miranda, a mathematician at the Polytechnic University of Catalonia (UPC) in Spain, calls undecidability a “next-level chaotic thing.”

Undecidability means that certain questions simply cannot be answered. It’s an unfamiliar message for physicists, but it’s one that mathematicians and computer scientists know well. More than a century ago, they rigorously established that there are mathematical questions that can never be answered, true statements that can never be proved. Now physicists are connecting those unknowable mathematical systems with an increasing number of physical ones and thereby beginning to map out the hard boundary of knowability in their field as well.