The Tangle Hypothesis: A Topological Approach to Link Invariants in Higher Dimensions
This research paper proves the 1-dimensional Tangle Hypothesis, which provides a topological framework for constructing link invariants in any dimension, generalizing the Reshetikhin-Turaev invariants for framed links in 3-dimensional space.
(∞, 2)-Topoi and Descent: A Characterization via Fibrational Descent and its Implications for Synthetic (∞, 1)-Category Theory
This paper introduces the concept of (∞, 2)-topoi, a higher-categorical generalization of topoi, characterized by a "fibrational descent" axiom, and explores its implications for developing a synthetic theory of (∞, 1)-categories.
A Geometric Method for Finding the Roots of Quadratic Equations in the Complex Plane
This paper presents a geometric method for finding the roots of a quadratic equation in one complex variable by constructing a line and a circumference in the complex plane, using the known coefficients of the equation.
SE(3)-invariant Space Diffusion Process Analysis
SE(3)-invariant space diffusion mechanisms are mathematically delineated, leading to projection-free SDE and ODE formulations for efficient 3D coordinate generation.
Understanding the Pseudoinverse of Matrix Product A = CR
Three formulas for the pseudoinverse of a matrix product A = CR are presented, highlighting conditions for correctness and usefulness.
Analyzing Hull Dimensions of Conorm Codes in Algebraic Geometry
Studying the hull dimensions of conorm codes in algebraic geometry reveals insights into code properties and constructions.
Geometric Planted Matchings Analysis Beyond Gaussian Model
Recovering unknown matchings between randomly placed points with perturbations is challenging but achievable.
Exploring Large Language Models for Mathematical Reasoning: Progresses and Challenges
Large Language Models are revolutionizing mathematical reasoning, but face challenges in diverse problem types and dataset evaluations.
Applied Category Theory in the Wolfram Language using Categorica I: Diagrams, Functors, and Fibrations
Categorica introduces a powerful framework for applied category theory, combining algebraic computation with diagrammatic theorem-proving.
Convection-Diffusion Equation: Neural Network Framework
Neural networks can be mathematically modeled using convection-diffusion equations, providing a unified framework for understanding and improving network structures.