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Category: mathematics – Page 7

In classical electromagnetism, electric and magnetic fields are the fundamental entities responsible for all physical effects. There is a compact formulation of electromagnetism that expresses the fields in terms of another quantity known as the electromagnetic potential, which can have a value everywhere in space. The fields are easily derived theoretically from the potential, but the potential itself was taken to be purely a mathematical device, with no physical meaning.

In quantum mechanics, shifts in the electromagnetic potential alter the description of a charged particle only by shifting its phase—that is, by advancing or retarding the crests and troughs in its quantum wave function. In general, however, such a phase change does not lead to any difference in the measurable properties of a particle.

But in 1959 Yakir Aharonov and David Bohm of the University of Bristol, UK, devised a thought experiment that linked the potential to a measurable result. In their scenario, a beam of electrons is split, with the two halves made to travel around opposite sides of a cylindrical electromagnet, or solenoid. The magnetic field is concentrated inside the solenoid and can be made arbitrarily weak outside by making the cylinder extremely narrow. So Aharonov and Bohm argued that the two electron paths can travel through an essentially field-free region that surrounds the concentrated field within the electromagnet.

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Georg Ferdinand Ludwig Philipp Cantor (/ ˈ k æ n t ɔːr / KAN-tor; German: [ˈɡeːɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfiːlɪp ˈkantoːɐ̯] ; 3 March [O.S. 19 February] 1845 – 6 January 1918 [ 1 ] ) was a mathematician who played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers. Cantor’s method of proof of this theorem implies the existence of an infinity of infinities. He defined the cardinal and ordinal numbers and their arithmetic. Cantor’s work is of great philosophical interest, a fact he was well aware of. [ 2 ]

Originally, Cantor’s theory of transfinite numbers was regarded as counter-intuitive – even shocking. This caused it to encounter resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré [ 3 ] and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections; see Controversy over Cantor’s theory. Cantor, a devout Lutheran Christian, [ 4 ] believed the theory had been communicated to him by God. [ 5 ] Some Christian theologians (particularly neo-Scholastics) saw Cantor’s work as a challenge to the uniqueness of the absolute infinity in the nature of God [ 6 ] – on one occasion equating the theory of transfinite numbers with pantheism [ 7 ] – a proposition that Cantor vigorously rejected.

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Paper: https://arxiv.org/abs/2502.12110v1

GitHub Page: https://github.com/WujiangXu/AgenticMemory

Current memory systems for large language model (LLM) agents often struggle with rigidity and a lack of dynamic organization. Traditional approaches rely on fixed memory structures—predefined storage points and retrieval patterns that do not easily adapt to new or unexpected information. This rigidity can hinder an agent’s ability to effectively process complex tasks or learn from novel experiences, such as encountering a new mathematical solution. In many cases, the memory operates more as a static archive than as a living network of evolving knowledge. This limitation becomes particularly apparent during multi-step reasoning tasks or long-term interactions, where flexible adaptation is crucial for maintaining consistency and depth in understanding.

Researchers from Rutgers University, Ant Group, and Salesforce Research have introduced A-MEM, an agentic memory system designed to address these limitations. A-MEM is built on principles inspired by the Zettelkasten method—a system known for its effective note-taking and flexible organization. In A-MEM, each interaction is recorded as a detailed note that includes not only the content and timestamp, but also keywords, tags, and contextual descriptions generated by the LLM itself. Unlike traditional systems that impose a rigid schema, A-MEM allows these notes to be dynamically interconnected based on semantic relationships, enabling the memory to adapt and evolve as new information is processed.

At its core, A-MEM employs a series of technical innovations that enhance its flexibility. Each new interaction is transformed into an atomic note, enriched with multiple layers of information—keywords, tags, and context—that help capture the essence of the experience. These notes are then converted into dense vector representations using a text encoder, which enables the system to compare new entries with existing memories based on semantic similarity. When a new note is added, the system retrieves similar historical memories and autonomously establishes links between them. This process, which relies on the LLM’s ability to recognize subtle patterns and shared attributes, goes beyond simple matching to create a more nuanced network of related information.

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Nobel Laureate Andrea Ghez joins Brian Greene to explore her decade’s long pursuit of the supermassive black hole at the center of the Milky Way Galaxy.

This program is part of the Big Ideas series, supported by the John Templeton Foundation.

Participant: Andrea Ghez.
Moderator: Brian Greene.

00:00 Introduction.
00:46 The Discovery of a Supermassive Black Hole.
01:50 Early Influences and Childhood Curiosity.
03:01 The Impact of the Moon Landing.
04:16 From Math to Physics.
05:42 Women in Science.
07:30 Falling in Love with Astronomy and Telescopes.
09:01 Supermassive vs. Stellar Black Holes.
12:05 Overcoming Doubts in Scientific Research.
15:33 The Search for the Black Hole in the Milky Way.
18:00 Developing Techniques for Infrared Astronomy.
21:12 The Long Journey to a Scientific Breakthrough.
24:45 The Moment of Discovery.
30:20 The Future of Black Hole Research.
32:45 Closing Thoughts and Advice for Aspiring Scientists.
35:10 Unlocking the Mysteries of the Universe.
41:30 Q&A

VISIT our Website: http://www.worldsciencefestival.com.
FOLLOW us on Social Media:
Facebook: / worldsciencefestival.
Twitter: / worldscifest.
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TikTok: / worldscifest.
LinkedIn: / world-science-festival.
#worldsciencefestival #briangreene #cosmology #astrophysics

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Is the key to understanding our universe hidden in its mirror image? Are the answers cosmologists seek much simpler than we think? And can we explain the origin of the universe without inflation?

Here today to share his bold new theory is the renowned physicist and cosmologist Neil Turok. Neil, who specializes in mathematical and early-universe physics, is the Higgs Chair of Theoretical Physics at the University of Edinburgh and Director Emeritus of the Perimeter Institute for Theoretical Physics. Recently, he’s been getting a lot of attention for proposing a simpler, more testable cosmological model that replaces inflation with a CPT-symmetric Mirror Universe, explaining dark matter, cosmic flatness, and density variations without adding unnecessary complexity.

Join us as we explore this provocative new theory in depth!

👉 ‘Cosmic Inflation’: did the early cosmos balloon in size? A mirror universe going backwards in time may be a simpler explanation by Neil Turok: https://shorturl.at/jr8kd.

Continue reading “Is Inflation DEAD? Neil Turok’s Startling Mirror Universe Theory” | >

“The Ouroboros Code” explores the intersection of science and spirituality through the lens of digital alchemy and self-simulation. Authored by Antonin Tuynman, the book presents a philosophical framework called “The Transcendental Metaphysics of Pancomputational Panpsychism” exploring how consciousness may be the fundamental ground of existence and the universe a self-modifying code. Tuynman investigates topics like the nature of intelligence, the limits of computation, and the possibility of artificial general intelligence. The book draws on concepts from physics, information theory, mathematics, and various spiritual traditions, aiming to bridge the gap between objective and subjective realities. It builds upon the author’s previous works and incorporates insights from various scientists and thinkers. Ultimately, the book seeks to understand how the universe, through a recursive process, generates and experiences itself. *Available as a Kindle eBook, paperback, and Audible audiobook: https://www.amazon.com/Ouroboros-Code?tag=lifeboatfound-20… #SelfSimulation #Pancomputationalism #DigitalPhysics #ComputationalPhysics

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WCTU CLEVELAND 13 — A new study suggests that faster-than-light travel, once thought to be purely science fiction, may be achievable sooner than expected through the concept of warp drive, challenging Einstein’s Theory of Relativity. This breakthrough builds on the Alcubierre drive model proposed in 1994, which theorizes that a spacecraft could travel faster than light by contracting space-time ahead of it while expanding space-time behind it.

The idea was first introduced by Mexican theoretical physicist Miguel Alcubierre, who suggested that a space-time bubble could allow for faster-than-light travel without violating the laws of physics. However, his model was initially dismissed due to its extreme energy requirements.

Joseph Agnew, a researcher from the University of Alabama, has been re-evaluating the mathematical foundations of Alcubierre’s theory. “If you fulfill all the energy requirements, they can’t prove that it doesn’t work,” Agnew stated in a university press release. His work has rekindled interest in the feasibility of warp drive by focusing on the possibility of warping space-time around a craft.

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Using an approach called DNA origami, scientists at Caltech have developed a technique that could lead to cheaper, reusable biomarker sensors for quickly detecting proteins in bodily fluids, eliminating the need to send samples out to lab centers for testing.

“Our work provides a proof-of-concept showing a path to a single-step method that could be used to identify and measure and proteins,” says Paul Rothemund (BS ‘94), a visiting associate at Caltech in computing and mathematical sciences, and computation and neural systems.

A paper describing the work recently appeared in the journal Proceedings of the National Academy of Sciences. The lead authors of the paper are former Caltech postdoctoral scholar Byoung-jin Jeon and current graduate student Matteo M. Guareschi, who completed the work in Rothemund’s lab.

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In the late 1960s, physicists like Charles Misner proposed that the regions surrounding singularities—points of infinite density at the centers of black holes—might exhibit chaotic behavior, with space and time undergoing erratic contractions and expansions. This concept, termed the “Mixmaster universe,” suggested that an astronaut venturing into such a black hole would experience a tumultuous mixing of their body parts, akin to the action of a kitchen mixer.

S general theory of relativity, which describes the gravitational dynamics of black holes, employs complex mathematical formulations that intertwine multiple equations. Historically, researchers like Misner introduced simplifying assumptions to make these equations more tractable. However, even with these assumptions, the computational tools of the time were insufficient to fully explore the chaotic nature of these regions, leading to a decline in related research. + Recently, advancements in mathematical techniques and computational power have reignited interest in studying the chaotic environments near singularities. Physicists aim to validate the earlier approximations made by Misner and others, ensuring they accurately reflect the predictions of Einsteinian gravity. Moreover, by delving deeper into the extreme conditions near singularities, researchers hope to bridge the gap between general relativity and quantum mechanics, potentially leading to a unified theory of quantum gravity.

Understanding the intricate and chaotic space-time near black hole singularities not only challenges our current physical theories but also promises to shed light on the fundamental nature of space and time themselves.

Physicists hope that understanding the churning region near singularities might help them reconcile gravity and quantum mechanics.

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Google’s second generation of its AI mathematics system combines a language model with a symbolic engine to solve complex geometry problems better than International Mathematical Olympiad (IMO) gold medalists.

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