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Category: mathematics – Page 19

Skoltech researchers have proposed novel mathematical equations that describe the behavior of aggregating particles in fluids. This bears on natural and engineering processes as diverse as rain and snow formation, the emergence of planetary rings, and the flow of fluids and powders in pipes.

Reported in Physical Review Letters, the new equations eliminate the need for juggling two sets of equations that had to be used in conjunction, which led to unacceptable errors for some applications.

Fluid aggregation is involved in many processes. In the atmosphere, agglomerate into rain, and ice microcrystals into snow. In space, particles orbiting come together to form rings like those of Saturn.

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Macquarie University researchers have demonstrated how ordinary supermarket grapes can enhance the performance of quantum sensors, potentially leading to more efficient quantum technologies.

The study, published in Physical Review Applied on 20 December 2024, shows that pairs of grapes can create strong localized magnetic field hotspots of microwaves which are used in quantum sensing applications—a finding that could help develop more compact and cost-effective quantum devices.

“While previous studies looked at the causing the plasma effect, we showed that grape pairs can also enhance magnetic fields, which are crucial for quantum sensing applications,” says lead author Ali Fawaz, a quantum physics Ph.D. candidate at Macquarie University.

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Entanglement is perhaps one of the most confusing aspects of quantum mechanics. On its surface, entanglement allows particles to communicate over vast distances instantly, apparently violating the speed of light. But while entangled particles are connected, they don’t necessarily share information between them.

In quantum mechanics, a particle isn’t really a particle. Instead of being a hard, solid, precise point, a particle is really a cloud of fuzzy probabilities, with those probabilities describing where we might find the particle when we go to actually look for it. But until we actually perform a measurement, we can’t exactly know everything we’d like to know about the particle.

These fuzzy probabilities are known as quantum states. In certain circumstances, we can connect two particles in a quantum way, so that a single mathematical equation describes both sets of probabilities simultaneously. When this happens, we say that the particles are entangled.

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The concept of vectors can be traced back to the 17th century with the development of analytic geometry by René Descartes and Pierre de Fermat. They used coordinates to represent points in a plane, which can be seen as a precursor to vectors. In the early 19th century, mathematicians like Bernard Bolzano and August Ferdinand Möbius began to formalize operations on points, lines, and planes, which further developed the idea of vectors.

Hermann Grassmann is considered one of the key figures in the development of vector spaces. In his 1844 work “Die lineale Ausdehnungslehre” (The Theory of Linear Extension), he introduced concepts that are central to vector spaces, such as linear independence, dimension, and scalar products. However, his work was not widely recognized at the time.

In 1888, Giuseppe Peano gave the first modern axiomatic definition of vector spaces. He called them “linear systems” and provided a set of axioms that precisely defined the properties of vector spaces and linear maps. Hilbert helped to further formalize and abstract the concept of vector spaces, placing it within a broader axiomatic framework for mathematics. He played a key role in the development of functional analysis, which studies infinite-dimensional vector spaces.

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Reliably measuring the polarization state of light is crucial for various technological applications, ranging from optical communication to biomedical imaging. Yet conventional polarimeters are made of bulky components, which makes them difficult to reduce in size and limits their widespread adoption.

Researchers at the Shanghai Institute of Technical Physics (SITP) of the Chinese Academy of Sciences and other institutes recently developed an on-chip full-Stokes polarimeter that could be easier to deploy on a large scale. Their device, presented in a paper in Nature Electronics, is based on optoelectronic eigenvectors, mathematical equations that represent the linear relationship between the incident Stokes vector and a detector’s photocurrent.

“This work was driven by the growing demand for compact, high-performance polarization analysis devices in optoelectronics,” Jing Zhou, corresponding author of the paper, told Phys.org. “Traditional polarimeters, which rely on discrete bulky optical components, present significant challenges to miniaturization and limit their broader applicability. Our main goal is to develop an on-chip solution capable of direct electrical readout to reconstruct full-Stokes polarization states.”

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New research shows that three sites spread along an approximately 620-mile portion of today’s Denali Fault were once a smaller united geologic feature indicative of the final joining of two land masses. That feature was then torn apart by millions of years of tectonic activity.

The work, led by associate professor Sean Regan at the University of Alaska Fairbanks Geophysical Institute and UAF College of Natural Science and Mathematics, is featured on the cover of the December edition of Geology.

Regan is the research paper’s lead author. UAF co-authors include doctoral student McKenzie Miller, recent master’s graduate Sean Marble and research assistant professor Florian Hofmann. Other co-authors are from St. Lawrence University, South Dakota School of Mines and Technology and the University of California, Santa Barbara.

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Mapping the geometry of quantum worlds: measuring the quantum geometric tensor in solids.

Quantum states are like complex shapes in a hidden world, and understanding their geometry is key to unlocking the mysteries of modern physics. One of the most important tools for studying this geometry is the quantum geometric tensor (QGT). This mathematical object reveals how quantum states “curve” and interact, shaping phenomena ranging from exotic materials to groundbreaking technologies.

The QGT has two parts, each with distinct significance:

1. The Berry curvature (the imaginary part): This governs topological phenomena, such as unusual electrical and magnetic behaviors in advanced materials.

2. The quantum metric (the real part): Recently gaining attention, this influences surprising effects like flat-band superfluidity, quantum Landau levels, and even the nonlinear Hall effect.

While the QGT is crucial for understanding these phenomena, measuring it directly has been a challenge, previously limited to simple, artificial systems.

A breakthrough now allows scientists to measure the QGT in real crystalline solids. Using an advanced technique involving polarization-, spin-, and angle-resolved photoemission spectroscopy, researchers have reconstructed the QGT in a material called CoSn, a “kagome metal” with unique quantum properties like topological flat bands. This metal forms patterns resembling a woven basket, hosting quantum effects that were previously only theorized.

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Distant, ancient galaxies are giving scientists more hints that a mysterious force called dark energy may not be what they thought.

Astronomers know that the universe is being pushed apart at an accelerating rate and they have puzzled for decades over what could possibly be speeding everything up. They theorize that a powerful, constant force is at play, one that fits nicely with the main mathematical model that describes how the universe behaves. But they can’t see it and they don’t know where it comes from, so they call it dark energy.

It is so vast it is thought to make up nearly 70% of the universe—while ordinary matter like all the stars and planets and people make up just 5%.

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Neural networks are at the core of artificial intelligence (AI), fueling a variety of applications from spotting objects in photos to translating languages. In this article, we’ll dive into what neural networks are, how they work, and why they’re a big deal in our technology-driven world today.

Index · 1: Understanding the Basics ∘ 1.1: What are Neural Networks? ∘ 1.2: Types of Neural Networks

· 2: The Architecture of Neural Networks ∘ 2.1: The Structure of a Neuron ∘ 2.2: Layers ∘ 2.3: The Role of Layers in Learning.

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