Lifeboat Foundation

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.

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 polarization 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.”

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.

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:

Continue reading “Physicists measure quantum geometry for first time” »

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%.

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.

Data is the new oil, as they say, and perhaps that makes Harvard University the new Exxon. The school announced Thursday the launch of a dataset containing nearly one million public domain books that can be used for training AI models. Under the newly formed Institutional Data Initiative, the project has received funding from both Microsoft and OpenAI, and contains books scanned by Google Books that are old enough that their copyright protection has expired.

Wired in a piece on the new project says the dataset includes a wide variety of books with “classics from Shakespeare, Charles Dickens, and Dante included alongside obscure Czech math textbooks and Welsh pocket dictionaries.” As a general rule, copyright protections last for the lifetime of the author plus an additional 70 years.

Foundational language models, like ChatGPT, that behave like a verisimilitude of a real human require an immense amount of high-quality text for their training—generally the more information they ingest, the better the models perform at imitating humans and serving up knowledge. But that thirst for data has caused problems as the likes of OpenAI have hit walls on how much new information they can find—without stealing it, at least.

The possibilities might be endless—literally.

📝 — Bertran, et al.

Full text is available 👇

Nonalcoholic fatty liver disease (NAFLD) is the most prevalent chronic hepatic disease; nevertheless, no definitive diagnostic method exists yet, apart from invasive liver biopsy, and nor is there a specific approved treatment. Runt-related transcription factor 1 (RUNX1) plays a major role in angiogenesis and inflammation; however, its link with NAFLD is unclear as controversial results have been reported. Thus, the objective of this work was to determine the proteins involved in the molecular mechanisms between RUNX1 and NAFLD, by means of systems biology. First, a mathematical model that simulates NAFLD pathophysiology was generated by analyzing Anaxomics databases and reviewing available scientific literature.