The problem concerns the mathematical properties of solutions to the Navier–Stokes equations, a system of partial differential equations that describe the motion of a fluid in space. Solutions to the Navier–Stokes equations are used in many practical applications. However, theoretical understanding of the solutions to these equations is incomplete. In particular, solutions of the Navier–Stokes equations often include turbulence, which remains one of the greatest unsolved problems in physics, despite its immense importance in science and engineering.
Category: mathematics – Page 4
Reports in Advances of Physical Sciences
In this paper, the authors propose a three-dimensional time model, arguing that nature itself hints at the need for three temporal dimensions. Why three? Because at three different scales—the quantum world of tiny particles, the realm of everyday physical interactions, and the grand sweep of cosmological evolution—we see patterns that suggest distinct kinds of “temporal flow.” These time layers correspond, intriguingly, to the three generations of fundamental particles in the Standard Model: electrons and their heavier cousins, muons and taus. The model doesn’t just assume these generations—it explains why there are exactly three and even predicts their mass differences using mathematics derived from a “temporal metric.”
This paper introduces a theoretical framework based on three-dimensional time, where the three temporal dimensions emerge from fundamental symmetry requirements. The necessity for exactly three temporal dimensions arises from observed quantum-classical-cosmological transitions that manifest at three distinct scales: Planck-scale quantum phenomena, interaction-scale processes, and cosmological evolution. These temporal scales directly generate three particle generations through eigenvalue equations of the temporal metric, naturally explaining both the number of generations and their mass hierarchy. The framework introduces a metric structure with three temporal and three spatial dimensions, preserving causality and unitarity while extending standard quantum mechanics and field theory.
Is Mathematics Mostly Chaos or Mostly Order?
Last winter, at a meeting in the Finnish wilderness high above the Arctic Circle, a group of mathematicians gathered to contemplate the fate of a mathematical universe.
It was minus 20 degrees Celsius, and while some went cross-country skiing, Juan Aguilera, a set theorist at the Vienna University of Technology, preferred to linger in the cafeteria, tearing pieces of pulla pastry and debating the nature of two new notions of infinity. The consequences, Aguilera believed, were grand. “We just don’t know what they are yet,” he said.
Infinity, counterintuitively, comes in many shapes and sizes. This has been known since the 1870s, when the German mathematician Georg Cantor proved that the set of real numbers (all the numbers on the number line) is larger than the set of whole numbers, even though both sets are infinite. (The short version: No matter how you try to match real numbers to whole numbers, you’ll always end up with more real numbers.) The two sets, Cantor argued, represented entirely different flavors of infinity and therefore had profoundly different properties.
Post-Alcubierre Warp-Drives
Researchers are actively exploring and revising the concept of Alcubierre warp drive, as well as alternative approaches, to potentially make superluminal travel feasible with reduced energy requirements and advanced technologies ## ## Questions to inspire discussion.
Practical Warp Drive Concepts.
🚀 Q: What is the Alcubierre warp drive? A: The Alcubierre warp drive (1994) is a superluminal travel concept within general relativity, using a warp bubble that contracts space in front and expands behind the spacecraft.
🌌 Q: How does Jose Natario’s warp drive differ from Alcubierre’s? A: Natario’s warp drive (2001) describes the warp bubble as a soliton and vector field, making it harder to visualize but potentially more mathematically robust.
🔬 Q: What is unique about Chris Van Den Broeck’s warp drive? A: Van Den Broeck’s warp drive (1999) uses a nested warp field, creating a larger interior than exterior, similar to a TARDIS, while remaining a physical solution within general relativity. Energy Requirements and Solutions.
💡 Q: How do Eric Lent’s hyperfast positive energy warp drives work? A: Lent’s warp drives (2020) are solitons capable of superluminal travel using purely positive energy densities, reopening discussions on conventional physics-based superluminal mechanisms.
Japanese Journal of Mathematics
Information geometry has emerged from the study of the invariant structure in families of probability distributions. This invariance uniquely determines a second-order symmetric tensor g and third-order symmetric tensor T in a manifold of probability distributions. A pair of these tensors (g, T) defines a Riemannian metric and a pair of affine connections which together preserve the metric. Information geometry involves studying a Riemannian manifold having a pair of dual affine connections. Such a structure also arises from an asymmetric divergence function and affine differential geometry. A dually flat Riemannian manifold is particularly useful for various applications, because a generalized Pythagorean theorem and projection theorem hold. The Wasserstein distance gives another important geometry on probability distributions, which is non-invariant but responsible for the metric properties of a sample space. I attempt to construct information geometry of the entropy-regularized Wasserstein distance.
Can space and time emerge from simple rules? Wolfram thinks so
Stephen Wolfram joins Brian Greene to explore the computational basis of space, time, general relativity, quantum mechanics, and reality itself.
This program is part of the Big Ideas series, supported by the John Templeton Foundation.
Participant: Stephen Wolfram.
Moderator: Brian Greene.
0:00:00 — Introduction.
01:23 — Unifying Fundamental Science with Advanced Mathematical Software.
13:21 — Is It Possible to Prove a System’s Computational Reducibility?
24:30 — Uncovering Einstein’s Equations Through Software Models.
37:00 — Is connecting space and time a mistake?
49:15 — Generating Quantum Mechanics Through a Mathematical Network.
01:06:40 — Can Graph Theory Create a Black Hole?
01:14:47 — The Computational Limits of Being an Observer.
01:25:54 — The Elusive Nature of Particles in Quantum Field Theory.
01:37:45 — Is Mass a Discoverable Concept Within Graph Space?
01:48:50 — The Mystery of the Number Three: Why Do We Have Three Spatial Dimensions?
01:59:15 — Unraveling the Mystery of Hawking Radiation.
02:10:15 — Could You Ever Imagine a Different Career Path?
02:16:45 — Credits.
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